Mitigation of the Effects of High Levels of High-Frequency
Noise on MEMS Gyroscopes
by
Pregassen Soobramaney
A dissertation submitted to the Graduate Faculty of Engineering
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 03, 2013
Keywords: MEMS gyroscopes, noise mitigation,
microfibrous media, modeling
Copyright 2013 by Pregassen Soobramaney
Approved by
George Flowers, Co-chair, Professor of Mechanical Engineeing
Subhash Sinha,Co-chair, Professor of Mechanical Engineering
Malcolm Crocker, Professor Emeritus of Mechanical Engineering
Robert Dean, Associate Professor of Electrical and Computer Engineering
ii
Abstract
MEMS gyroscopes are being used in a variety of applications such as camcorders,
vehicle stability control and game controllers. Sometimes they are used in harsh environments
such as high levels of high-frequency noise. If the frequency of the noise coincides with the
natural frequency of the gyroscope, the output of the latter is corrupted. Experiments have been
performed to demonstrate the effects of noise on MEMS gyroscopes. The objective of this
dissertation is to suggest ways to mitigate those effects.
In the first part of this research work, a mathematical model has been developed to
include the effects of an external noise signal on a MEMS gyroscope. The model is simulated to
assist with understanding its dynamics and it is found that the effects of the external noise signal
are superimposed on the normal gyroscope?s output. This finding leads to the solution of using a
non-driven gyroscope to measure the superimposed effects. Thus, a differential-measurement
system consisting of two gyroscopes is implemented, and simulation of the mathematical model
is performed to successfully demonstrate the mitigation of the effects of noise. Experiments were
performed on five gyroscopes to confirm the simulation results, and the superposition of the
effects of noise has been confirmed implying that the differential measurement is a viable
solution.
The second part of this research provides a study of passive approaches to attenuate the
effects of noise. In this regard, four types of nickel microfibrous material were made using three
diameters of nickel fibers and a wet-lay papermaking process. The Delany-Bazley analytical
iii
acoustical model was used to determine the optimum acoustical properties of the material. The
properties were then used to calculate the absorption coefficients of the microfibrous media.
Damping characterization of the media was performed using the vibration transmissibility
concept. Enclosures have been designed and made from the materials to attenuate the effects of
noise on MEMS gyroscopes. Acoustical tests performed in a reverberation room show up to 90%
reduction in the amplitude of the effects of noise.
In conclusion, two different approaches have been suggested to mitigate the undesirable
effects of noise on MEMS gyroscopes. In the first approach an active design is proposed by
utilizing a pair of gyroscopes whose outputs can be manipulated to yield the desired uncorrupted
results. In the second approach, a passive design is proposed using nickel microfibrous material
as an acoustical enclosure. Considerable reductions in the effects of noise have been achieved,
showing that the nickel microfibrous material can be used to construct an acoustical enclosure.
iv
Acknowledgments
I thank my advisors, Dr. George T. Flowers and Dr. Subhash C. Sinha, for their guidance,
support, encouragement and patience in completing this work. I wish to acknowledge Dr.
Malcolm Crocker, Dr. Robert N. Dean, and Dr. John L. Evans for serving on my committee. I
am thankful to Haoyue Yang, James Jantz, Fuxi Zhang, Colin Stevens, Wesley Smith, and
Chong Li, for their friendship and assistance.
I thank my beautiful wife, Selvina, for her love, patience, motivation and unfailing
support. I am grateful to my parents, Devarajen and Kisnamah, for giving me a great education
and inculcating good values in me.
I thank God for keeping me on the right path, showing me the way and giving me the
strength to complete this work.
v
Table of Contents
Abstract ......................................................................................................................................... ii
Acknowledgements ...................................................................................................................... iv
List of Tables ................................................................................................................................ x
List of Illustrations ..................................................................................................................... xiii
List of Abbreviations .................................................................................................................. xx
Chapter 1 Literature Review ...................................................................................................... 1
1.1 Introduction .............................................................................................................. 1
1.2 Gyroscopes ............................................................................................................... 2
1.3 MEMS Gyroscopes .................................................................................................. 2
1.4 Dynamics of a MEMS Gyroscope ........................................................................... 3
1.5 Design Challenges for MEMS Gyroscopes ............................................................. 5
1.6 Effects of Noise on MEMS Gyroscopes .................................................................. 6
1.7 Previous Mitigation Efforts...................................................................................... 7
1.8 Acoustical Materials ................................................................................................ 7
1.9 Sound Absorption Coefficient (?)............................................................................ 9
1.10 Acoustic Models ...................................................................................................... 9
1.10.1 Delany-Bazley Model .............................................................................. 9
1.10.2 Dunn-Davern Model .............................................................................. 10
vi
1.10.3 Voronina Model ..................................................................................... 10
1.11 Vibration Isolation and Damping........................................................................... 11
1.12 Elasto-Damping Materials ..................................................................................... 13
1.12.1 Elastomers (Rubbers) ............................................................................ 14
1.12.2 Wire-mesh Materials ............................................................................. 14
1.12.3 Felt ........................................................................................................ 14
1.13 Microfibrous Metallic Cloth .................................................................................. 15
1.14 Dissertation Organization ...................................................................................... 16
Chapter 2 Mathematical Model for Noise Simulation ............................................................. 18
2.1 Dynamics of a Vibratory Gyroscope ...................................................................... 18
2.1.1 Equations of Motion ............................................................................... 18
2.2 Gyroscope Model for Noise Simulation ................................................................. 22
2.2.1 Equations of Motion ............................................................................... 26
2.2.2 Model Simulation ................................................................................... 28
Chapter 3 Mitigation of the Effects of Noise ........................................................................... 32
3.1 The Mitigation Procedure ....................................................................................... 32
3.2 Quantifying the Effects of an External Noise Signal .............................................. 33
3.2.1 Pure Tone Case ....................................................................................... 34
3.2.2 Random Noise Case ................................................................................ 35
3.3 Conclusions ............................................................................................................. 38
Chapter 4 Experimental Verification of Gyroscope Model and Mitigation of the Effects
of Noise .................................................................................................................. 41
4.1 Gyroscope Assembly .............................................................................................. 41
4.1.1 ADXRS652 Gyroscope ........................................................................... 41
vii
4.1.2 PCB Design ............................................................................................. 42
4.1.3 Surface Mounting of Capacitors and Gyroscopes .................................. 44
4.2 Experimental Verification ....................................................................................... 46
4.2.1 Equipment Setup ..................................................................................... 47
4.2.2 Testing..................................................................................................... 50
4.2.3 Statistical Analysis of Results ................................................................. 52
4.3 Conclusions ............................................................................................................. 55
Chapter 5 Microfibrous Material Fabrication .......................................................................... 56
5.1 Introduction ............................................................................................................. 56
5.2 Fabrication Procedure ............................................................................................. 56
5.3 Optical Microscopy of Microfibrous Sheets .......................................................... 59
Chapter 6 Determination of Acoustical Properties .................................................................. 64
6.1 Material Porosity ..................................................................................................... 64
6.2 Delany-Bazley Model ............................................................................................. 64
6.3 Flow Resistance Measurement ............................................................................... 65
6.4 Absorption Coefficient............................................................................................ 71
Chapter 7 Damping Characterization ....................................................................................... 74
7.1 Test Design ............................................................................................................. 74
7.2 Test Fixture ............................................................................................................. 74
7.3 Equipment Setup ..................................................................................................... 76
7.4 Experimental Procedure .......................................................................................... 76
7.5 Stiffness and Damping Ratio .................................................................................. 79
7.6 Analysis of Results ................................................................................................. 85
viii
Chapter 8 Mitigation of the Effects Noise Using Nickel Microfibrous Materials ................... 87
8.1 Enclosure Design and Fabrication .......................................................................... 87
8.2 Experimental Validation of Enclosures .................................................................. 89
8.3 Conclusions ............................................................................................................. 90
Chapter 9 Conclusions and Scope for Further Work ............................................................... 95
References ................................................................................................................................. 99
Appendix A ............................................................................................................................. 104
A.1 Maximum Displacement Transmissibility ........................................................... 104
A.2 Damping Ratio as a Function of Maximum Amplitude ....................................... 105
Appendix B ............................................................................................................................. 106
B.1 Solving Equations of Motion ............................................................................... 106
B.1.1 Drive Motion ........................................................................................ 106
B.1.2 Sense Motion ........................................................................................ 107
B.2 Natural Frequency Computation .......................................................................... 109
B.2.1 Basic Vibratory Gyroscope .................................................................. 109
B.2.2 Gyroscope for Noise Simulation .......................................................... 110
B.3 Force Approximation from Sound Pressure Level ............................................... 112
Appendix C ............................................................................................................................. 113
C.1 Matlab Codes for Solving the Basic Vibratory Gyroscope Model ...................... 113
C.2 Matlab Codes for Solving the Four-degree-of-freedom Gyroscope Model ......... 114
Appendix D ............................................................................................................................. 117
D.1 Mechanical Drawings for Fixture Parts ............................................................... 118
D.2 Plots of Vibration Tests........................................................................................ 122
ix
D.3 Experimental Data for the Natural Frequencies of 4 Microns Media .................. 142
D.4 Experimental Data for the Maximum Amplitude of the Transfer Functions
of 4 Microns Media ............................................................................................. 144
D.5 Experimental Data for the Natural Frequencies of 4/8 Microns Media ............... 146
D.6 Experimental Data for the Maximum Amplitude of the Transfer Functions
of 4/8 Microns Media.......................................................................................... 148
D.7 Experimental Data for the Natural Frequencies of 8 Microns Media .................. 150
D.8 Experimental Data for the Maximum Amplitude of the Transfer Functions
of 8 Microns Media ............................................................................................. 152
D.9 Experimental Data for the Natural Frequencies of 12 Microns Media ................ 154
D.10 Experimental Data for the Maximum Amplitude of the Transfer Functions
of 12 Microns Media ........................................................................................... 156
Appendix E ............................................................................................................................. 158
E.1 Matlab Codes for Stiffness and Damping Calculations ....................................... 158
Appendix F .............................................................................................................................. 168
F.1 Acoustical Test Results for Gyroscopes G3-G7 ................................................... 168
x
List of Tables
Table 2.1 Displacements and velocities of the proof mass and the gyroscope frame ........... 26
Table 4.1 Natural frequencies and sound pressure levels of the gyroscopes ........................ 52
Table 4.2 Statistical analysis of results .................................................................................. 55
Table 6.1 Media thickness measurements and porosity calculations ................................... 66
Table 6.2 Pressure drop measurements for 4 microns media ................................................ 69
Table 6.3 Pressure drop measurements for 4/8 microns media ............................................. 69
Table 6.4 Pressure drop measurements for 8 microns media ................................................ 70
Table 6.5 Pressure drop measurements for 12 microns media .............................................. 70
Table 8.1 Summary of experimental results .......................................................................... 96
Table D.1a Natural frequencies of 1 layer of 4 microns media ............................................. 142
Table D.1b Natural frequencies of 1 layer of 4 microns media ............................................. 142
Table D.2 Natural frequencies of 2 layers of 4 microns media ........................................... 142
Table D.3 Natural frequencies of 3 layers of 4 microns media ........................................... 143
Table D.4 Natural frequencies of 4 layers of 4 microns media ........................................... 143
Table D.5 Natural frequencies of 5 layers of 4 microns media ........................................... 143
Table D.6a Maximum amplitude of the transfer functions of 1 layer of 4 microns media .... 144
Table D.6b Maximum amplitude of the transfer functions of 1 layer of 4 microns media .... 144
Table D.7 Maximum amplitude of the transfer functions of 2 layers of 4 microns media ... 144
Table D.8 Maximum amplitude of the transfer functions of 3 layers of 4 microns media ... 145
xi
Table D.9 Maximum amplitude of the transfer functions of 4 layers of 4 microns media .. 145
Table D.10 Maximum amplitude of the transfer functions of 5 layers of 4 microns media .. 145
Table D.11a Natural frequencies of 1 layer of 4/8 microns media ......................................... 146
Table D.11b Natural frequencies of 1 layer of 4/8 microns media ......................................... 146
Table D.12 Natural frequencies of 2 layers of 4/8 microns media ....................................... 146
Table D.13 Natural frequencies of 3 layers of 4/8 microns media ....................................... 147
Table D.14 Natural frequencies of 4 layers of 4/8 microns media ....................................... 147
Table D.15 Natural frequencies of 5 layers of 4/8 microns media ....................................... 147
Table D.16a Maximum amplitude of the transfer functions of 1 layer of 4/8 microns
media .................................................................................................................. 148
Table D.16b Maximum amplitude of the transfer functions of 1 layer of 4/8 microns
media .................................................................................................................. 148
Table D.17 Maximum amplitude of the transfer functions of 2 layers of 4/8 microns
media .................................................................................................................. 148
Table D.18 Maximum amplitude of the transfer functions of 3 layers of 4/8 microns
media .................................................................................................................. 149
Table D.19 Maximum amplitude of the transfer functions of 4 layers of 4/8 microns
media .................................................................................................................. 149
Table D.20 Maximum amplitude of the transfer functions of 5 layers of 4/8 microns
media .................................................................................................................. 149
Table D.21a Natural frequencies of 1 layer of 8 microns media ............................................ 150
Table D.21b Natural frequencies of 1 layer of 8 microns media ............................................ 150
Table D.22 Natural frequencies of 2 layers of 8 microns media .......................................... 150
Table D.23 Natural frequencies of 3 layers of 8 microns media .......................................... 151
Table D.24 Natural frequencies of 4 layers of 8 microns media .......................................... 151
Table D.25 Natural frequencies of 5 layers of 8 microns media .......................................... 151
xii
Table D.26a Maximum amplitude of the transfer functions of 1 layer of 8 microns media ... 152
Table D.26b Maximum amplitude of the transfer functions of 1 layer of 8 microns media ... 152
Table D.27 Maximum amplitude of the transfer functions of 2 layers of 8 microns media .. 152
Table D.28 Maximum amplitude of the transfer functions of 3 layers of 8 microns media .. 153
Table D.29 Maximum amplitude of the transfer functions of 4 layers of 8 microns media .. 153
Table D.30 Maximum amplitude of the transfer functions of 5 layers of 8 microns media .. 153
Table D.31a Natural frequencies of 1 layer of 12 microns media .......................................... 154
Table D.31b Natural frequencies of 1 layer of 12 microns media .......................................... 154
Table D.32 Natural frequencies of 2 layers of 12 microns media ........................................ 154
Table D.33 Natural frequencies of 3 layers of 12 microns media ........................................ 155
Table D.34 Natural frequencies of 4 layers of 12 microns media ........................................ 155
Table D.35 Natural frequencies of 5 layers of 12 microns media ........................................ 155
Table D.36a Maximum amplitude of the transfer functions of 1 layer of 12 microns
media .................................................................................................................. 156
Table D.36b Maximum amplitude of the transfer functions of 1 layer of 12 microns
media .................................................................................................................. 156
Table D.37 Maximum amplitude of the transfer functions of 2 layers of 12 microns
media .................................................................................................................. 156
Table D.38 Maximum amplitude of the transfer functions of 3 layers of 12 microns
media .................................................................................................................. 157
Table D.39 Maximum amplitude of the transfer functions of 4 layers of 12 microns
media .................................................................................................................. 157
Table D.40 Maximum amplitude of the transfer functions of 5 layers of 12 microns
media .................................................................................................................. 157
xiii
List of Illustrations
Figure 1.1 A conventional gyroscope ..................................................................................... 3
Figure 1.2 Schematic of a vibratory gyroscope ...................................................................... 4
Figure 1.3 Vibration isolation system ................................................................................... 11
Figure 2.1 Schematic of a MEMS Gyroscope ....................................................................... 19
Figure 2.2 Counterclockwise rotation .................................................................................... 23
Figure 2.3 Clockwise rotation ................................................................................................ 23
Figure 2.4 Relation between sense response and rotation rate .............................................. 24
Figure 2.5 Effects of drive force frequency on sense response ............................................. 24
Figure 2.6 Gyroscope model with external noise .................................................................. 25
Figure 2.7 Effects of noise frequency on sense response ...................................................... 30
Figure 2.8 Effects of noise amplitude on sense response ...................................................... 31
Figure 2.9 Effects of noise on sense response at different rotation rates ............................... 31
Figure 3.1 Gyroscope model without drive force FD ............................................................. 33
Figure 3.2 Difference between Gyro 1 outputs compared to the output of Gyro 2 ............... 34
Figure 3.3 Simulink model of gyroscope ............................................................................... 36
Figure 3.4 Unfiltered output of gyroscope in presence of a random noise signal ................. 37
Figure 3.5 Filtered output of gyroscope in presence of a random noise signal ..................... 37
Figure 3.6 Simulink model of three subsystems .................................................................... 39
Figure 3.7 Difference between the outputs of Gyro 1 subsystems ........................................ 40
xiv
Figure 3.8 Output of Gyro 2 .................................................................................................. 40
Figure 4.1 ADXRS652 Functional Block Diagram ............................................................... 42
Figure 4.2 FreePcb interface showing the designed PCB ...................................................... 43
Figure 4.3 Gerber files: (a) Top copper, (b) Top mask, and (c) Top silk .............................. 44
Figure 4.4 Picture of boards coming out of the oven with mounted capacitors .................... 45
Figure 4.5 (a) Alignment using flip-chip bonder, and (b) X-ray of alignment ...................... 45
Figure 4.6 Testing of gyroscope after assembly .................................................................... 46
Figure 4.7 Schematic of rate table and acoustical test setup.................................................. 48
Figure 4.8 Picture of rate table and speakers ......................................................................... 49
Figure 4.9 Picture of control and data acquisition systems ................................................... 49
Figure 4.10 Finding the natural frequency of gyroscope G1 ................................................... 51
Figure 4.11 Amplitude of pure tone at the natural frequency of Gyroscope G1 ..................... 51
Figure 4.12 Experimental results for gyroscope G1 ................................................................ 52
Figure 4.13 Experimental results for gyroscope G2 ................................................................ 53
Figure 4.14 Experimental results for gyroscope G3 ................................................................ 53
Figure 4.15 Experimental results for gyroscope G4 ................................................................ 54
Figure 4.16 Experimental results for gyroscope G5 ................................................................ 54
Figure 5.1 (a) Blender and speed controller, (b) HEC sieved into water, and (c) Turbid
mixture ................................................................................................................ 57
Figure 5.2 (a) Turbid mixture turns clear, (b) Cellulose mixture, and (c) Aqueous
suspension of nickel fibers .................................................................................. 58
Figure 5.3 (a) Paddle agitating the mixture and (b) Perform after water is drained .............. 59
Figure 5.4 Optical microscopic images of 4 microns diameter material at
magnification factors 100 X, 300 X, 500 X and 1000 X ................................... 60
xv
Figure 5.5 Optical microscopic images of 8 microns diameter material at
magnification factors 100 X, 300 X, 500 X and 1000 X .................................... 61
Figure 5.6 Optical microscopic images of 12 microns diameter material at
magnification factors 100 X, 300 X, 500 X and 1000 X .................................... 62
Figure 5.7 Optical microscopic images of 4 and 8 microns diameters material (mixed
in a ratio of 1:1) at magnification factors 100 X, 300 X, 500 X and 1000 X .... 63
Figure 6.1 Equipment for measuring media thickness........................................................... 65
Figure 6.2 Schematic of equipment setup for airflow resistivity measurement .................... 67
Figure 6.3 Plot of pressure drop as a function of flow velocity ............................................. 68
Figure 6.4 Absorption coefficients of 4 microns media ........................................................ 72
Figure 6.5 Absorption coefficients of 4/8 microns media ..................................................... 72
Figure 6.6 Absorption coefficients of 8 microns media ........................................................ 73
Figure 6.7 Absorption coefficients of 12 microns media ...................................................... 73
Figure 7.1 Vibration transmitted through base motion .......................................................... 75
Figure 7.2 Photograph of test fixture ..................................................................................... 75
Figure 7.3 Schematic of vibration test equipment setup ........................................................ 77
Figure 7.4 Photograph of test setup ....................................................................................... 78
Figure 7.5 Picture of analyzer screen showing a Bode plot ................................................... 79
Figure 7.6 Transfer function at different vibration amplitudes .............................................. 80
Figure 7.7 Stiffness of 4 microns media ................................................................................ 81
Figure 7.8 Damping ratio of 4 microns media ....................................................................... 82
Figure 7.9 Stiffness of 4/8 microns media ............................................................................. 82
Figure 7.10 Damping ratio of 4/8 microns media .................................................................... 83
Figure 7.11 Stiffness of 8 microns media ................................................................................ 83
Figure 7.12 Damping ratio of 8 microns media ....................................................................... 84
xvi
Figure 7.13 Stiffness of 12 microns media .............................................................................. 84
Figure 7.14 Damping ratio of 12 microns media ..................................................................... 85
Figure 8.1 (a) Stacked sheets, and (b) Wire mesh frame ....................................................... 88
Figure 8.2 Photograph of furnace .......................................................................................... 88
Figure 8.3 Photograph of enclosures showing top and bottom surfaces ............................... 89
Figure 8.4 Photograph of mass sitting on top of enclosure.................................................... 90
Figure 8.5 Experimental results of the 4 microns enclosure on gyroscope G1 ..................... 91
Figure 8.6 Experimental results of the 4/8 microns enclosure on gyroscope G1 .................. 92
Figure 8.7 Experimental results of the 8 microns enclosure on gyroscope G1 ..................... 92
Figure 8.8 Experimental results of the 12 microns enclosure on gyroscope G1 ................... 93
Figure 8.9 Experimental results of the 4 microns enclosure on gyroscope G2 ..................... 93
Figure 8.10 Experimental results of the 4/8 microns enclosure on gyroscope G2 .................. 94
Figure 8.11 Experimental results of the 8 microns enclosure on gyroscope G2 ..................... 94
Figure 8.12 Experimental results of the 12 microns enclosure on gyroscope G2 ................... 95
Figure D.1 Mechanical drawing of top fixture ..................................................................... 118
Figure D.2 Mechanical drawing of bottom fixture ............................................................... 119
Figure D.3 Mechanical drawing of sliding pin .................................................................... 120
Figure D.4 Mechanical drawing of lower screw ................................................................... 121
Figure D.5 Transfer functions for 1 layer (sample 08) of 4 microns media ......................... 122
Figure D.6 Transfer functions for 1 layer (sample 09) of 4 microns media ......................... 122
Figure D.7 Transfer functions for 2 layers (samples 03/04) of 4 microns media ................. 123
Figure D.8 Transfer functions for 2 layers (samples 07/08) of 4 microns media ................. 123
Figure D.9 Transfer functions for 3 layers (samples 10/02/04) of 4 microns media ............ 124
xvii
Figure D.10 Transfer functions for 3 layers (samples 14/05/09) of 4 microns media ............ 124
Figure D.11 Transfer functions for 4 layers (samples 01/02/04/06) of 4 microns media ....... 125
Figure D.12 Transfer functions for 4 layers (samples 07/02/08/14) of 4 microns media ....... 125
Figure D.13 Transfer functions for 5 layers (samples 08/06/07/09/10) of 4 microns media ..... 126
Figure D.14 Transfer functions for 5 layers (samples 14/09/11/13/15) of 4 microns media ..... 126
Figure D.15 Transfer functions for 1 layer (sample 03) of 4/8 microns media ...................... 127
Figure D.16 Transfer functions for 1 layer (sample 11) of 4/8 microns media ...................... 127
Figure D.17 Transfer functions for 2 layers (samples 01/02) of 4/8 microns media .............. 128
Figure D.18 Transfer functions for 2 layers (samples 10/11) of 4/8 microns media .............. 128
Figure D.19 Transfer functions for 3 layers (samples 09/10/11) of 4/8 microns media ......... 129
Figure D.20 Transfer functions for 3 layers (samples 07/03/11) of 4/8 microns media ......... 129
Figure D.21 Transfer functions for 4 layers (samples 02/13/09/06) of 4/8 microns media ...... 130
Figure D.22 Transfer functions for 4 layers (samples 14/08/03/10) of 4/8 microns media ...... 130
Figure D.23 Transfer functions for 5 layers (samples 01/02/03/05/06) of 4/8 microns
media ................................................................................................................. 131
Figure D.24 Transfer functions for 5 layers (samples 07/08/09/10/11) of 4/8 microns
media ................................................................................................................. 131
Figure D.25 Transfer functions for 1 layer (sample 10) of 8 microns media ......................... 132
Figure D.26 Transfer functions for 1 layer (sample 15) of 8 microns media ......................... 132
Figure D.27 Transfer functions for 2 layers (samples 04/05) of 8 microns media ................. 133
Figure D.28 Transfer functions for 2 layers (samples 14/15) of 8 microns media ................. 133
Figure D.29 Transfer functions for 3 layers (samples 01/02/04) of 8 microns media ............ 134
Figure D.30 Transfer functions for 3 layers (samples 06/07/08) of 8 microns media ............ 134
Figure D.31 Transfer functions for 4 layers (samples 12/13/14/15) of 8 microns media ....... 135
xviii
Figure D.32 Transfer functions for 4 layers (samples 09/11/13/15) of 8 microns media ....... 135
Figure D.33 Transfer functions for 5 layers (samples 02/04/06/08/10) of 8 microns media ..... 136
Figure D.34 Transfer functions for 5 layers (samples 12/14/01/05/07) of 8 microns media ..... 136
Figure D.35 Transfer functions for 1 layer (sample 07) of 12 microns media ....................... 137
Figure D.36 Transfer functions for 1 layer (sample 13) of 12 microns media ....................... 137
Figure D.37 Transfer functions for 2 layers (samples 02/04) of 12 microns media ............... 138
Figure D.38 Transfer functions for 2 layers (samples 05/06) of 12 microns media ............... 138
Figure D.39 Transfer functions for 3 layers (samples 01/03/05) of 12 microns media .......... 139
Figure D.40 Transfer functions for 3 layers (samples 06/08/09) of 12 microns media .......... 139
Figure D.41 Transfer functions for 4 layers (samples 07/05/06/08) of 12 microns media ....... 140
Figure D.42 Transfer functions for 4 layers (samples 10/11/12/13) of 12 microns media ....... 140
Figure D.43 Transfer functions for 5 layers (samples 06/08/09/10/11) of 12 microns
media ................................................................................................................. 141
Figure D.44 Transfer functions for 5 layers (samples 13/12/04/05/03) of 12 microns
media ................................................................................................................. 141
Figure F.1 Experimental results of the 4 microns enclosure on gyroscope G3 ................... 168
Figure F.2 Experimental results of the 4/8 microns enclosure on gyroscope G3 ................ 169
Figure F.3 Experimental results of the 8 microns enclosure on gyroscope G3 ................... 169
Figure F.4 Experimental results of the 12 microns enclosure on gyroscope G3 ................. 170
Figure F.5 Experimental results of the 4 microns enclosure on gyroscope G4 ................... 170
Figure F.6 Experimental results of the 4/8 microns enclosure on gyroscope G4 ................ 171
Figure F.7 Experimental results of the 8 microns enclosure on gyroscope G4 ................... 171
Figure F.8 Experimental results of the 12 microns enclosure on gyroscope G4 ................. 172
Figure F.9 Experimental results of the 4 microns enclosure on gyroscope G5 ................... 172
xix
Figure F.10 Experimental results of the 4/8 microns enclosure on gyroscope G5 ................ 173
Figure F.11 Experimental results of the 8 microns enclosure on gyroscope G5 ................... 173
Figure F.12 Experimental results of the 12 microns enclosure on gyroscope G5 ................. 174
Figure F.13 Experimental results of the 4 microns enclosure on gyroscope G6 ................... 174
Figure F.14 Experimental results of the 4/8 microns enclosure on gyroscope G6 ................ 175
Figure F.15 Experimental results of the 8 microns enclosure on gyroscope G6 ................... 175
Figure F.16 Experimental results of the 12 microns enclosure on gyroscope G6 ................. 176
Figure F.17 Experimental results of the 4 microns enclosure on gyroscope G7 ................... 176
Figure F.18 Experimental results of the 4/8 microns enclosure on gyroscope G7 ................ 177
Figure F.19 Experimental results of the 8 microns enclosure on gyroscope G7 ................... 177
Figure F.20 Experimental results of the 12 microns enclosure on gyroscope G7 ................. 178
xx
List of Abbreviations
HEC hydroxethycellulose
MEMS Microelectromechanical systems
PCB printed circuit board
SOI silicon on insulator
SPL sound pressure level
1
CHAPTER 1
LITERATURE REVIEW
1.1 Introduction
Microelectromechanical Systems (MEMS) technology is relatively new and
combines mechanical and electrical systems at the micro scale. MEMS devices are being
widely used because of their small size, low cost, and low power consumption. One of
the most interesting MEMS devices is the MEMS gyroscope which is replacing its
traditional counterpart and is being used in applications such as image stabilization in
cameras and camcorders, vehicle stability control, game controllers, and motion control
of robots. MEMS gyroscopes are sometimes used in harsh environments such as high
levels of high-frequency noise. If the frequency of the noise coincides with the natural
frequency of the gyroscope, the proof mass is overexcited giving rise to a corrupted
gyroscope output. Experiments have been performed to demonstrate the effects of noise
on MEMS gyroscopes. Progress has also been made to mitigate the effects of the high
levels of high-frequency noise, but there is still room for improvement.
The objective of this dissertation is to mitigate the effects of noise on MEMS
gyroscopes. To meet this objective, two approaches are considered. In the first approach
a mathematical model is developed to include the effects of external noise on a MEMS
gyroscope and then the model is used to provide a solution to mitigate the effects of noise
on the gyroscope. In the second approach the mitigation of the effects of noise is
2
attempted using nickel microfibrous material as an acoustical material. Nickel
microfibrous materials have a porosity of above 90% making them suitable for noise
attenuation. They also have interesting properties such as flame and temperature
resistance. In addition they can be used in chemically-harsh and high-humidity
environments.
1.2 Gyroscopes
A gyroscope is a device that measures the rotation rate of an object. The name
gyroscope was coined by a French physicist, Leon Foucault, who combined the Greek
words ? gyros? and ?skopeein? meaning rotation and to see respectively. The analysis of
gyroscopic motion is one of the most fascinating problems in dynamics. As illustrated in
Fig. 1.1, a conventional gyroscope consists of a flywheel which is attached to a pair of
gimbal rings within the gyroscope frame. In this configuration the spinning axis of the
flywheel is free to take any orientation. The flywheel has a relatively large mass and is
made to spin at high speed giving it a high angular momentum. The latter counteracts
externally applied torques so that the orientation of the spin axis is constant.
Conventional gyroscopes are precise, robust, and are widely used in navigation. But those
gyroscopes have some disadvantages, in particular, the rotation bearings wear out, they
are relatively bulky, and they are expensive [1-3].
1.3 MEMS Gyroscopes
MEMS gyroscopes are micromachined sensors that are used to measure angular
rates. Microfabrication techniques such as lithography, deposition and etching are used
to mass-produce MEMS devices on silicon-on-insulator (SOI) wafers. Most MEMS gyro-
3
Figure 1.1 A conventional gyroscope.
scopes consist of a vibrating mass which upon rotation experiences a Coriolis force,
which is proportional to the rotation rate. They do not have rotating parts and therefore
they do not require bearings, and are not affected by friction and wear. Therefore MEMS
gyroscopes are cheap, small, and light, have low power requirements and are
maintenance free. Thus, they are finding many new applications in the automotive
industry as well as consumer products such as mobile phones, games and cameras [1, 3].
1.4 Dynamics of a MEMS Gyroscope
The dynamics of the MEMS gyroscope depicted in Fig. 1.2 are governed by Eqns.
(1.1) and (1.2). The proof mass m is forced into oscillation in the drive direction x by a
sinusoidal force of amplitude F and rotates about the out-of-plane axis at a rotation rate
. A Coriolis force is therefore coupled in the sense direction, y, causing the proof mass to
oscillate in that direction at the drive force frequency ?. Also, the amplitude of the sense
4
Figure 1.2 Schematic of a vibratory gyroscope.
response is proportional to the Coriolis force and therefore to the angular rate. The proof
mass is held by a suspension system with stiffness kx and damping cx in the drive
direction, and stiffness ky and damping cy in the sense direction [1, 4, 5].
The dynamics can be represented by [1]
? ? , (1.1)
and
? ? ? . (1.2)
For maximum sense response, the proof mass is made to resonate in the sense
direction, and the drive force frequency is made equal to the natural frequency of the
drive mode. Therefore for high sensitivity, the natural frequencies in both directions are
designed and tuned to match [1, 6]. This applies only for a constant angular rate signal,
5
and the detection bandwidth is zero. In practice, the sense mode frequency is separated
from the drive force frequency (by approx. 100 Hz) to achieve larger bandwidth and to
make the gyroscope sensitive to angular acceleration. However, this causes the gyroscope
to have lower sensitivity [47].
1.5 Design Challenges for MEMS Gyroscopes
The main challenge in designing MEMS gyroscopes is that the output amplitude
is very small relative to the drive force because of the small magnitude of the Coriolis
force. To overcome this issue, gyroscopes are designed with high quality factor (Q) to
mechanically amplify the sense amplitude. To achieve a high value of Q, high vacuum is
required during packaging of a gyroscope so as to minimize damping, and also the sense
mode resonance frequency is tuned to the drive force frequency by using electrostatic
forces. A gyroscope with a high value of Q has a small bandwidth and is sensitive to
external vibrations having frequencies at or near to the natural frequency of the
gyroscope. To mitigate the effects of the vibration forces, gyroscopes are designed with
high natural frequencies (10 kHz - 20 kHz). But they are still susceptible to high levels of
high-frequency noise [3, 7-9].
Quadrature errors in the design of gyroscopes are due to manufacturing
imperfections and consist of stiffness and damping cross-coupling between the sense and
drive modes. Because of the great difference in magnitude between drive response and
sense response amplitudes, a small quadrature coupling from the drive motion into the
sense motion causes the Coriolis force to be corrupted. Quadrature errors are reduced by
more precise and careful microfabrication, and are nullified by the control system of the
gyroscope [1, 5, 9].
6
1.6 Effects of Noise on MEMS Gyroscopes
MEMS gyroscopes are sometimes used in harsh environments such as high levels
of high-frequency noise with sound pressure levels up to 120 dB and frequencies up to
and above 20 kHz. Examples of such environments are supersonic aerospace vehicles,
missiles, rockets and machinery using high pressure nozzles [10, 11].
Dean et al. demonstrated that high levels of noise corrupt the output of MEMS
gyroscopes. In the study, noise was generated by a supersonic wind tunnel and sound
pressure levels up to 110 dB were generated within a frequency range of 0-24 kHz. Four
commercially available gyroscopes were tested near the supersonic nozzle and in the
presence of the noise, degradation of the output of the gyroscopes was observed [11].
Dean et al. conducted another set of experiments in a reverberation room where a speaker
system was used to generate noise levels up to 130 dB within a frequency range of 0-20
kHz. Twenty four similar gyroscopes were tested using a single tone frequency sweep at
the maximum output of the speakers. It was observed that in the frequency region near to
and at the natural frequency of the gyroscopes, there was a large change in the output of
the gyroscopes. It was concluded that the effects of the noise were frequency dependent
as for frequencies of the noise away from the natural frequency, the gyroscopes were
insensitive to the noise level [12]. Yunker et al. investigated the underwater acoustic
effects on a MEMS gyroscope. The gyroscope was waterproofed and a hydrophone was
used to generate a single tone sweep from 14 kHz to 20 kHz with a sound pressure level
of around 140 dB. The sound pressure level was measured using a second hydrophone in
a sensor configuration. Similar observations were made as when the medium was air. It
7
was found that the gyroscope output was corrupted near to and at its natural frequency
[13].
1.7 Previous Mitigation Efforts
Castro et al. attempted the mitigation of the effects of noise using acoustical
foam. The foam was designed to surround the gyroscope on all sides and an aluminum
plate was placed on its top. Another aluminum plate was used at the bottom of the
gyroscope for added mass. For testing purposes the noise was generated using a
supersonic wind tunnel and considerable reductions in the effects of the noise on the
MEMS gyroscopes were observed [14]. Roth performed another set of tests for the
mitigation of noise in a reverberation room using several types of acoustical foams.
Different attenuation levels were achieved by the use of the different foams but the
effects of the noise were not completely mitigated [15]. Recently, Yunker et al. have used
acoustical metamaterials for noise attenuation. Micro-Helmholtz resonators were
designed and manufactured to have a frequency of 14.5 kHz. Microfabrication techniques
were used to manufacture the sides of a cube from a 100 mm diameter silicon wafer, with
each side consisting of many such resonators. Testing of the cube was done in an
acoustical environment and an attenuation of the noise level of 18 dB at 14.5 kHz was
achieved [16].
1.8 Acoustical Materials
Acoustical materials have the ability to absorb a considerable fraction of the total
acoustic energy striking their surfaces. They are widely used in noise control and are
normally placed near the noise source, near the receiver, or in the noise path. Examples of
their applications are: in ships and aircraft construction, in machine enclosures, inside
8
earmuffs, and as the finished room surfaces in hospitals, schools and offices. Because of
their wide range of application, acoustical materials are required to have other properties
apart from having high sound absorptivity. They should be able to fit in the environment
in which they are used and so their appearance, color, shapes, and light reflectivity are
important. Also they need to be reliable, maintainable, flame resistant and durable [17,
18].
For sound absorption to take place, the exposed surface of the material should
have some passages or openings for the sound waves to go through and the internal
structure of the material should be able to transform the acoustic energy into heat energy.
Having high porosity, which is the ratio of the void volume to the total volume, porous
and fibrous materials are therefore good acoustical materials. In porous materials, the air
molecules within the pores vibrates in the presence of noise and in doing so energy is lost
between the molecules and the pore walls. In fibrous materials acoustic energy is
converted into heat from the scattering of sound waves, the vibration of individual fibers
and the rubbing of the fibers against each other [17, 18].
In acoustical applications, where high temperature resistance, flame resistance,
high durability and weathering resistance are required, the use of metal foams has been
investigated. Metal foams are made mostly from aluminum which is light, has good
thermal conductivity and is relatively cheap. Other metals used are steel, titanium and
copper. The structure of metal foams is predominantly closed cells and this has made the
acoustical absorption characteristics of the material to be poor. To improve acoustical
characteristics, rolling of the foams is done so as to break open the closed cells but they
are still not as good as other commonly used acoustical materials [19, 17]. Alternatives to
9
metal foams, for high temperature applications, are aerogels which can sustain
temperatures up to 1200 oC. Aerogels are nanoporous making them good acoustic
absorbers. They are also claimed to be the best thermal insulators. The drawback of these
materials is that they are very costly and are only used for some high-tech applications
[20, 17].
1.9 Sound Absorption Coefficient (?)
The absorption coefficient of a material is defined as the ratio of the absorbed
sound intensity to the total incident sound intensity. It is dependent on the frequency of
the sound and the angle of incidence of the sound waves. Typical values of ? vary from 1
to 100 percent. It is normally calculated using the Sabine empirical formula from the
measured reverberation time (time for the sound pressure level to drop by 60 dB) with
and without the presence of the material in a reverberation room [21, 22]. The absorption
coefficient can also be measured using an impedance tube. The material under test is
mounted at the end of the tube against a flat rigid surface and a speaker is used to
generate sound waves at the other end of the tube. A two-microphone system is used for
the acoustical measurements from which the absorption coefficient is derived [23].
1.10 Acoustic Models
1.10.1 Delany-Bazley Model [24, 25]
Delany and Bazley developed a model to determine the acoustical properties, the
characteristic impedance Zc and the propagation coefficient ?, of a range of fibrous
materials having a porosity factor near unity. Extensive experiments were carried out to
measure these properties using the impedance-tube method with plane-wave conditions.
10
Using the dimensional variable frequency/flow-resistance (f/?), the measured data were
normalized and empirical expressions (Eqns. (1.3) and (1.4)) were formulated in terms of
power-law relationships [24, 25].
{[ ( ) ] [ ( ) ]} , (1.3)
{[ ( ) ] [ ( ) ]}, (1.4)
where ?0 is the density of air, c0 is the speed of sound in air, f is frequency of sound and ?
is the airflow resistivity per unit thickness.
This model can be used with confidence when the ratio of the frequency to the air
flow resistivity is between 0.01 and 1.0 m3/kg. Extrapolation outside this range is not
recommended as other power-law relationships may be required. Miki improved the
Delany-Bazley model to predict the acoustical properties at low frequencies when f/? is
less than 0.01 m3/kg [24, 21].
1.10.2 Dunn-Davern Model [26]
Using a similar approach to Delany and Bazley, Dunn and Davern developed a
model for the acoustical properties of foam materials. They also developed a
methodology to calculate the overall acoustical impedance of multi-layered acoustical
materials, using the single layer properties [26].
1.10.3 Voronina Model [27]
The relationship between the acoustical properties and the physical structural
parameters of acoustical materials was investigated by Voronina. Experimental
measurements of the characteristic impedance and the propagation constant were carried
out for several materials with different fiber diameters and values of porosity. Empirical
11
formulae were developed as a function of the material properties. Unlike the Delany-
Bazley model, the Voronina model does not consider the ratio of frequency to air flow
resistivity as the latter is already a function of the fiber size and porosity. Voronina
compared her model with experimental values and showed that her model is accurate for
materials made of relatively thick fibers [27].
1.11 Vibration Isolation and Damping
Vibration isolation can be either passive or active depending on whether external
power is required or not for the isolator to work. A passive vibration isolation system
consists of three main parts; the payload which requires isolation, the vibration isolators
and the supporting surface or foundation. This is illustrated in Fig. 1.3 where the
supporting surface, the base, is subjected to a motion y(t), the payload, represented by a
rigid mass m, and the isolator, a linear spring/damper pair [28, 29]. The equation of
motion for this system is
? ( ? ?) ( ) . (1.5)
Figure 1.3 Vibration isolation system.
12
Eqn. (1.5) can be rewritten as
? ( ? ?) ( ) , (1.6)
where ? is the damping ratio, and ?n is the angular natural frequency.
The displacement transmissibility Td is the ratio of the displacement amplitude of
object (X) to that of the base motion (Y), and is given by
| | [ ( ) ( ) ( ) ]
?
, (1.7)
where r is the frequency ratio [29].
The maximum displacement transmissibility occurs at a frequency ratio [29, 30]
?(? ) , (1.8)
where ?m is the frequency at which there is maximum transmissibility.
Also at maximum transmissibility, the amplitude ratio (see Appendix A) is
| |
[ ? ]
?
, (1.9)
which is a function of the damping ratio only.
The system showed in Fig. 1.3 is for the case when vibration is transmitted from
the foundation to the equipment. Another common type of vibration isolation is when the
equipment itself generates the vibration and the isolation prevents the generated vibration
to be transmitted to the foundation. Examples of machine generated vibrations are
unbalanced forces in rotating machinery, impact forces in forging or stamping and
vibration shakers [28, 29].
Vibration isolators are chosen so that the natural frequency of the system is much
lower than the operating frequency of the equipment, therefore reducing the magnitude of
13
the force transmitted. For frequency ratios greater than ? , a smaller damping ratio leads
to a smaller magnitude of transmitted vibration. Therefore, the damping ratio should be
as small as possible. But some damping is required in the system because the machine
has to go through resonance during start-up and stopping. In the case when the system is
exposed several excitation frequencies and harmonics, it is not easy to isolate all those
frequencies and it is better to use a high damping ratio [28, 29].
Damping is the dissipation of energy by the conversion of mechanical (vibration)
energy into heat energy. Damping can be modeled as viscous damping, Coulomb
damping (dry friction) or hysteretic damping. Viscous damping occurs when a system
vibrates in a viscous medium such as oil or air and the resistance of the medium to the
vibrating body causes the dissipation of energy. Viscous damping is proportional to the
velocity of the vibrating body. Coulomb or sliding-friction damping happens due to the
relative motion of surfaces. It is of constant amplitude and opposite to the direction of
motion of the vibrating body. Hysteretic or material damping occurs within a material as
it is deformed and the internal planes slide or slip losing energy by friction. The stress-
strain diagram of material damping forms a hysteresis loop and the area of the loop is
indicative of the amount of energy lost. [29, 31]
1.12 Elasto-Damping Materials
An elasto-damping material (EDM) is one that has a relatively small value of
elastic modulus and therefore has relatively large deformations when subjected to small
stresses. Because of this property, an EDM dissipates a lot of energy which is of hysteric
nature when subjected to vibration. The use of an EDM as a vibration isolator has the
14
advantage of providing good damping not requiring additional dampers. Types of EDMs
are elastomers and meshed fibrous materials [28].
1.12.1 Elastomers (Rubbers)
Elastomers have a small value of elastic modulus and some of them can be
stretched to ten times their original length without failure. Because they can experience
large deformations, rubbers dissipate a lot of energy. The properties of an elastomer can
be tailored by blending a base elastomer with some additives allowing elastomers to be
made in many shapes [28].
1.12.2 Wire-mesh Materials
Wire-mesh materials consist of relatively thick wires (0.1-0.6 mm diameter)
which are interwoven, then rolled into small balls followed by compression to the desired
shape. The amount of compression determines the stiffness and damping properties of the
final product. Under a dynamic load, the wires slip against each other dissipating energy.
The energy dissipation and the material stiffness are vibration-amplitude dependent. An
increase in vibration amplitude leads to greater energy dissipation and a reduction in
stiffness [28].
1.12.3 Felt
?Felt is a fabric produced by meshing natural or synthetic fibers by a combination
of mechanical movements, chemical actions, and application of moisture and heat, but
without using looms or knitting operation? [28]. The internal structure of felts is very
similar to wire-mesh materials and therefore they exhibit similar dynamic characteristics.
15
1.13 Microfibrous Metallic Cloth
Microfibrous metallic cloth is made by intermingling and fusing micro metal
fibers together. The preparation consists of making an aqueous suspension of the metal
fibers and cellulose, which acts as a binding agent both in the liquid and dried state.
Using a wet-lay papermaking technique the mixture is then cast into a preform sheet
followed by drying to remove moisture. The dried sheet is sintered in a continuous
hydrogen furnace at 1000 oC, which removes the cellulose and causes the metallic fibers
to sinter-bond at their junctures. A felt-like cloth with a high void volume of more than
90% is obtained. The manufacturing process of microfibrous cloth can be tailored to
produce varying surface areas, void volumes and pore sizes [32-34]. Damping is achieved
in this type of material because of the sliding contact between the fibers as they are
compressed causing dissipation of energy. Microfibrous materials are notably insensitive
to temperature variations and have demonstrated resistance to harsh environments,
including extreme temperature and humidity [35].
The high-speed papermaking process used in the manufacture of microfibrous
materials is a low cost one compared to other traditional ways of making metal fibrous
materials. This has made the microfibrous materials more affordable for use as filters.
Also the wet-lay process permits the use of different fiber size in a single sheet. This
gives rise to asymmetric porosity leading to better filtration as entrapment of finer
particles is achieved [37].
The use of microfibrous cloth has been experimentally investigated for vibration
isolation and noise attenuation. Layers of nickel microfibrous cloth were sandwiched
between a fixture and a printed circuit board which was subjected to vibration. Effective
16
vibration damping was achieved. A similar set up was used for the acoustic tests except
that a gyroscope was mounted on the printed circuit board. The effects of the noise on the
gyroscope were considerably reduced [36, 38, 39]. Sintered fibrous metals have also been
utilized in muffler designs of internal combustion engines and successful noise
attenuation was achieved which was comparable to commercially available perforated
duct silencers [40].
1.14 Dissertation Organization
In Chapter 2 the dynamics of a vibratory gyroscope are analyzed and the
equations of motion have been derived. The model is then modified to include the effects
of noise on the gyroscope and corresponding governing equations are developed.
In Chapter 3 the mitigation of the effects of noise on a MEMS gyroscope was
investigated using a differential measurement system. Two types of noise signals are
considered: pure tones and random noise.
Chapter 4 describes the mounting of gyroscopes on printed circuit boards. The
gyroscopes are used to experimentally verify the model and the mitigation presented in
Chapters 2 and 3, respectively.
Chapter 5 describes the fabrication of nickel microfibrous sheets. A wet-lay
papermaking process is used to make a preform of nickel and cellulose fibers. The
preform is then dried and sintered to give a highly porous microfibrous media
In Chapter 6 the airflow resistivity of nickel microfibrous materials (media) was
experimentally determined. The Delany-Bazley model was then used to obtain the
acoustical properties of the media using the airflow resistivity.
17
In Chapter 7 the damping characterization of nickel microfibrous materials was
performed using the displacement transmissibility concept. A fixture was designed for
attachment to the shaker head and to hold the material during testing.
In Chapter 8 nickel microfibrous enclosures were designed and fabricated by
sintering together layers of the microfibrous material. Experiments were then done in a
reverberation room to validate the effectiveness of the enclosures for noise attenuation.
18
CHAPTER 2
MATHEMATICAL MODEL FOR NOISE SIMULATION
In this chapter, the dynamics of a vibratory MEMS gyroscope are analyzed, and
the equations of motion are derived. The model is then modified to include external noise
affecting a gyroscope, and the corresponding governing equations are developed.
2.1 Dynamics of a Vibratory Gyroscope
A vibratory gyroscope (illustrated in Fig. 2.1) consists of a proof mass which is
attached to the gyroscope frame by a suspension system with stiffness kx in the drive
direction x and ky in the sense direction y. During operation, the proof mass is made to
oscillate in the drive direction by a sinusoidal electrostatic drive force (FD) generated by a
pair of comb drive actuators, one on each side of the proof mass. When the gyroscope
rotates about the out-of-plane axis at a rate ? rad/s, a sinusoidal orthogonal force, the
Coriolis force, is induced in the sense direction. The sense motion is measured using
capacitive sensors and is proportional to the rotation rate. The suspension system is such
that the proof mass does not rotate with respect to the frame.
2.1.1 Equations of Motion
Lagrange?s equations (Eqn. (2.1)) are used to derive the equations of motion for
the vibratory gyroscope.
(
? )
? , , (2.1)
19
Figure 2.1 Schematic of a MEMS Gyroscope
where T is the kinetic energy, V is the potential energy, R is the Rayleigh dissipation
function, are the generalized forces and are the generalized coordinates.
Kinetic energy
The total kinetic energy is given by
( ? ?) , (2.2)
where m is the mass, ? is the velocity and is the second moment of inertia of the proof
mass.
The velocity of the proof mass is
? ? ?? ? ?? ? ( ?? ??)
20
( ? ) ?? ( ? ) ?? (2.3)
Substituting the velocity in Eqn. (2.2) gives
? ? ( ) ? ? . (2.4)
Potential Energy
The potential energy V of the gyroscope is given by
( ) ( ) (2.5)
Rayleigh Dissipation Function
The Rayleigh dissipation function for the vibratory gyroscope is
( ?) ( ?) . (2.6)
Virtual work
The virtual work and the generalized forces are related by
? . (2.7)
Equations of motion
Applying the Lagrange equations for the generalized coordinate gives
(
?)
( ? ) ? ?, (2.8)
?, (2.9)
, (2.10)
? ?, (2.11)
and
. (2.12)
Therefore, the equation of motion in the x-direction is
21
? ? ? . (2.13)
Similarly the equation of motion for the sense motion was derived:
? ? ? (2.14)
Dividing Eqns. (2.13) and (2.14) by m and rearranging give
? ? ( ) ? , (2.15)
and
? ? ( ) ? (2.16)
The natural frequencies of the gyroscope are much greater than the rotation rate,
and therefore, the and terms are negligible. Also the amplitude of the sense
motion is small compared to that of the amplitude of the drive motion causing the
Coriolis component ? to be negligible. To maximize sensitivity and response gain, the
natural frequencies of the drive and sense modes are made equal, and resonance is used in
both modes (by making drive force frequency equal to the natural frequency). Similar
results have been obtained in previous studies where Newtonian Mechanics were used to
derive the equations of motion [1].
The equations of motion can therefore be simplified to:
? ? , (2.17)
and
? ? ? . (2.18)
The solutions (Appendix B) to the above Eqns. (2.17) and (2.18) are
, (2.19)
and
22
( )
. (2.20)
Eqn. (2.20) shows that the amplitude of the sense response is directly proportional
to the rotation rate ?. The phase angle between the two displacements is used to
determine the direction of rotation. For a positive rotation rate (counterclockwise
rotation), the displacements are out of phase, and for a clockwise rotation, the
displacements are in phase. The mathematical model, represented by Eqns. (2.13) and
(2.14), was integrated using the Matlab solver ODE45 (Appendix C), and comparison
plots for counterclockwise and clockwise rotation were produced as shown in Figs. 2.2
and 2.3 respectively. Both of the natural frequencies of the gyroscope were designed to
be 14 kHz (Appendix B), and the rotation rate was set at 1 rad/s.
The Matlab model was also used to show that the amplitude of the sense motion
was directly proportional to the angular rate (Fig. 2.4). The effect of the frequency of the
drive force on the sense response amplitude was then investigated (Fig. 2.5), and as
expected, the displacement was found to be maximum when the drive force frequency
was equal to the natural frequency of the system.
2.2 Gyroscope Model for Noise Simulation
To simulate the effects of noise on a gyroscope, the basic model of Fig. 2.1 was
modified as shown in Fig. 2.6. The new model consists of two masses, the proof mass
(mp) and the mass of the gyroscope frame and packaging (mf), each with two degrees of
freedom so that the model is a four-degree-of-freedom system. Because the electrostatic
drive force is generated by two pairs of comb drive actuators with half of each
interdigitating comb attached to the proof mass and the other half attached to the frame,
both masses are affected by the generated drive force but in opposite directions. The
23
Figure 2.2 Counterclockwise rotation.
Figure 2.3 Clockwise rotation
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
x 1 0
-3
-3
-2
-1
0
1
2
3
x 1 0
-5 D r i v e M o d e R e s p o n s e
T i m e , s
D
i
sp
l
a
ce
m
e
n
t
(
x)
,
m
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
x 1 0
-3
-1
- 0 . 5
0
0 . 5
1
x 1 0
-6 S e n s e M o d e R e s p o n s e
T i m e , s
D
i
sp
l
a
ce
m
e
n
t
(
y)
,
m
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
x 1 0
-3
-3
-2
-1
0
1
2
3
x 1 0
-5 D r i v e M o d e R e s p o n s e
T i m e , s
D
i
sp
l
a
ce
m
e
n
t
(
x)
,
m
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
x 1 0
-3
-1
- 0 . 5
0
0 . 5
1
x 1 0
-6 S e n s e M o d e R e s p o n s e
T i m e , s
D
i
sp
l
a
ce
m
e
n
t
(
y)
,
m
24
Figure 2.4 Relation between sense response and rotation rate.
Figure 2.5 Effects of drive force frequency on sense response.
-4 -3 -2 -1 0 1 2 3 4
- 2 . 5
-2
- 1 . 5
-1
- 0 . 5
0
0 . 5
1
1 . 5
2
2 . 5
x 1 0
-6 S e n s e R e s p o n s e a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , r a d / s
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
1 3 . 7 1 3 . 8 1 3 . 9 14 1 4 . 1 1 4 . 2 1 4 . 3
0
1
2
3
4
5
6
7
8
x 1 0
-7 S e n s e R e s p o n s e a g a i n s t D r i v e F o r c e F r e q u e n c y
D r i v e F o r ce F r e q u e n cy , kH z
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
25
Figure 2.6 Gyroscope model with external noise.
damping and stiffness between the proof mass and the frame are the same as in the basic
model. The suspension system is such that the proof mass does not rotate with respect to
the frame.
The packaging of the gyroscope is fixed onto a printed circuit board (PCB) which
rotates at a constant angular rate ?. The stiffness and damping between the packaging
and the PCB are kz and cz, respectively. For modeling purposes, the packaging is
subjected to high levels of noise, FN, as shown in Fig. 2.6.
26
2.2.1 Equations of motion
The displacements and velocities of the proof mass and the gyroscope frame with
reference to the inertial frame are summarized in Table 2.1.
Mass Displacement
in x direction
Displacement
in y direction
Velocity in
x direction
Velocity in
y direction
Gyro frame ? ?
Proof mass ? ?
Table 2.1 Displacements and velocities of the proof mass and the gyroscope frame.
Lagrange?s equations were used to derive the equations of motion.
Kinetic energy
The total kinetic energy T is given by
, (2.21)
where are the kinetic energy of the frame and the proof mass respectively.
( ? ? ) , (2.22)
and
( ? ? ) , (2.23)
where are the second moments of inertia of the frame and the proof mass
respectively.
The velocity of the frame is
? ? ?? ? ?? ? ( ?? ??)
= ( ? ) ?? ( ? ) ?? . (2.24)
27
The velocity of the proof mass is
? ? ?? ? ?? ? ( ?? ??)
= ( ? ) ?? ( ? ) ?? . (2 .25)
Substituting Eqn. (2.24) in Eqn. (2.22) gives
? ? ( ) ? ? . (2.26)
Substituting Eqn. (2.25) in Eqn. (2.23) gives
? ? ( ) ? ? . (2.27)
Therefore the total kinetic energy is
? ? ( ) ? ?
?
?
(
) ? ?
. (2.28)
Potential Energy
The total potential energy for the gyroscope is
( ) ( ) ( ) ( ) (2.29)
Rayleigh Dissipation Function
The total Rayleigh dissipation function for the gyroscope is
? + ( ? ? ) ? + ( ? ? ) . (2.30)
Virtual work
To obtain the generalized forces, the virtual work was determined.
( ) . (2.31)
The virtual work and generalized forces are related by
? . (2.32)
28
Equations of motion
For the generalized coordinate ,
(
? )
( ? ) ? ? , (2.33)
? , (2.34)
( ) , (2.35)
? ( ? ? ) ? , (2.36)
and
(2.37)
Therefore, the equation of motion for the generalized coordinate is
? ( ? ? ) ? ( ) ?
(2.38)
Similarly the equations of motion for the other three generalized
coordinates , and are obtained as:
? ( ? ? ) ( ) ? , (2.39)
? ( ? ? ) ? ( ) ? , (2.40)
and
? ( ? ? ) ( ) ? . (2.41)
2.2.2 Model Simulation
The four equations were integrated using Matlab solver ODE45 (Appendix C). To
simulate the model, the following parameters were considered:
1. Stiffness, 77.38 N/m.
29
2. Damping, 5 x 10-7 Ns/m.
3. Stiffness, 200 N/m.
4. Damping, 5 x 10-3 Ns/m.
5. Proof mass, mp = 1 x 10-8 kg.
6. Mass of frame and packaging, mf = 3.8 x 10-4 kg.
7. Drive force amplitude, FD = 1 x 10-6 N.
8. Noise amplitude, FN= 1.26 x 10-3 N.
The computed natural frequencies using the above parameters (Appendix B) were
14,001 Hz, 14,000 Hz, 115.3 Hz and 115.62 Hz.
The first simulation was carried out to show the effects of the frequency of the
external noise on the gyroscope output. The rotation rate was set at 1 rad/s, the frequency
of the drive force was fixed at the natural frequency of the proof mass, 14, 000 Hz, and
the frequency of the noise varied from 13,000 Hz to 15,000 Hz. As expected, it was
observed that the output was maximum when the frequency of the noise was equal to the
natural frequency of the gyroscope (Fig. 2.7). Also, when the frequency of the noise did
not coincide with the natural frequency, no effects on the output of the gyroscope were
observed.
The effects of the amplitude of the noise on the gyroscope?s output were then
investigated. To maximize the effects of the noise, its frequency was kept constant at the
natural frequency of the gyroscope. The rotation rate was also kept constant at 1 rad/s and
the noise level amplitude was varied from 0 to 126 dB. For simulation purposes, the noise
level was approximated to a force as shown in Appendix B. The simulation results (Fig.
2.8) showed a linear increase in sense response with an increase in noise amplitude.
The model was then run at different rotation rates in the presence of external noise
having fixed amplitude and frequency. The rotation rate was varied from -3 to +3 rad/s in
30
steps of 1 rad/s. The simulation results (dashed line) were plotted in Fig. 2.9. It was
observed that at zero rotation rate the gyroscope had a positive amplitude but a zero
output amplitude at a rotation rate of ? 0.92 rad/s. The sense response amplitude also
increased with increasing rotation rates. To explain these observations, the model was run
at the same rotation rates but without external noise, that is FN = 0, and the results were
compared in Fig. 2.9. The comparison showed that the presence of the noise caused an
upward shift in the sense response amplitude. Also the shift was greater when the
rotation was negative. This was due to the Coriolis component (Refer to Eqn. (2.41).)
which caused an increase in amplitude when the rotation was negative and a decrease in
amplitude when the rotation rate was positive. It was therefore concluded that the effects
of the noise were superimposed on the normal gyroscope?s output.
Figure 2.7 Effects of noise frequency on sense response.
1 3 . 8 1 3 . 8 5 1 3 . 9 1 3 . 9 5 14 1 4 . 0 5 1 4 . 1 1 4 . 1 5 1 4 . 2
0 . 7
0 . 8
0 . 9
1
1 . 1
1 . 2
1 . 3
1 . 4
1 . 5
x 1 0
-6 S e n s e R e s p o n s e a g a i n s t N o i s e F r e q u e n c y
F r e q u e n cy o f N o i se , kH z
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
31
Figure 2.8 Effects of noise amplitude on sense response.
Figure 2.9 Effects of noise on sense response at different rotation rates.
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4
x 1 0
-5
0 . 7
0 . 8
0 . 9
1
1 . 1
1 . 2
1 . 3
1 . 4
x 1 0
-6 S e n s e R e s p o n s e a g a i n s t N o i s e A m p l i t u d e
N o i se A m p l i t u d e , N
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
-4 -3 -2 -1 0 1 2 3 4
-3
-2
-1
0
1
2
3
x 1 0
-6 S e n s e R e s p o n s e a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , r a d / s
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
N o N o i s e
N o i s e
32
CHAPTER 3
MITIGATION OF THE EFFECTS OF NOISE
In this chapter the effects of noise on a MEMS gyroscope are quantified and a
differential-measurement system is used to mitigate the effects of the noise. Two types of
noise signals are considered: a pure tone and random noise.
3.1 The Mitigation Procedure
The simulation of the MEMS gyroscope model (section 2.2.2) showed that in the
presence of an external noise signal, the output of the gyroscope was augmented by
almost a constant value. It was concluded that the effects of the noise were superimposed
on the normal gyroscope?s output.
In order to measure the superimposed effects of noise, a non-driven gyroscope
was used as shown in Fig. 3.1. Being affected by the external noise signal only, its output
is assumed to be equal to the superimposed effects of noise that needed to be determined.
Therefore, if the superimposed effects of noise can be quantified, a differential-
measurement system can be proposed to consist of a pair of similar gyroscopes to
mitigate the effects of the noise. In this system, the first gyroscope (Gyro 1) will be a
normal gyroscope affected by noise, and the second gyroscope (Gyro 2) will measure the
effects of noise only (no drive force). Subtracting the output of Gyro 2 from the output of
Gyro 1 will give an uncorrupted output.
33
Figure 3.1 Gyroscope model without drive force FD.
3.2 Quantifying the Effects of an External Noise Signal
The approach used to show that the superposition of the effects of an external
noise signal on the normal output of a gyroscope could be quantified was to simulate
Gyro 1 in the presence of the noise signal and in the absence of the noise signal. Then the
difference between the two measured outputs of Gyro 1 can be compared to the output of
Gyro 2 in the presence of the same noise signal.
No drive force
34
3.2.1 Pure Tone Case
A pure tone was first considered as the external noise signal. Since the
simulations of section 2.2.2 used a pure tone of constant amplitude as the noise signal, the
results were used in this section and are reproduced in Fig. 3.2 as ?Gyro 1 + noise? and
?Gyro 1 without noise?. The difference between the outputs is represented by a nearly
horizontal line showing that the effects of the pure tone caused the amplitude of the
gyroscope to be augmented by almost a constant amplitude. The model of section 2.2.2
was then simulated as Gyro 2 by making FD equal to zero. The same noise signal was
used, and the results are plotted in Fig. 3.2. It is observed that the output of Gyro 2 is
exactly the same as the difference between the outputs of Gyro 1, implying that the
effects of the noise signal have been quantified.
Figure 3.2 Difference between Gyro 1 outputs compared to the output of Gyro 2.
-4 -3 -2 -1 0 1 2 3 4
-3
-2
-1
0
1
2
3
x 1 0
-6 S e n s e R e s p o n s e a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , r a d / s
S
e
n
se
R
e
sp
o
n
se
A
m
p
l
i
t
u
d
e
,
m
G y r o 1 + n o i s e ( 1 )
G y r o 1 w i t h o u t n o i s e ( 2 )
( 1 ) - ( 2 )
G y r o 2 + n o i s e
35
3.2.2 Random Noise Case
Since real life situations consist mostly of random noise, another set of
simulations was performed using a random noise signal. Since the latter could be
generated in ?Simulink?, a graphical modeling tool in Matlab, the governing equations of
the gyroscope were modeled using the software shown in Fig. 3.3. Basic modeling
blocks such as addition, integration and gain were used along with the two specific
blocks, band-limited white noise block and peak-notch filter. The band-limited white
noise block generates normally distributed random numbers at a specific sample rate
which is related to the correlation time of the noise. Based on the bandwidth of the
system, the correlation time tc is given by
, (3.1)
where fmax is the bandwidth of the system in rad/sec [44].
Since the natural frequency of the gyroscope?s model was designed to be 14 kHz,
the bandwidth was set at 20 kHz, and therefore tc was set at 5 x 10-7 s. Also because the
output component at the natural frequency of the gyroscope was needed, a peak filter was
used for filtering out all the remaining frequency components affecting the gyroscope?s
output and was implemented in the Simulink model by the peak-notch filter block.
Using the same parameters as in section 2.2.2, the Simulink model was simulated,
and the results are plotted in Figs. 3.11 and 3.12 for the unfiltered and filtered gyroscope
outputs, respectively. As expected the unfiltered signal is noisy showing the random
nature of the noise signal. The filtered signal on the other hand is smooth showing that
successful filtration was achieved.
36
Figure 3.3 Simulink model of gyroscope.
Z e r o - O r d e r
H o l d
Yp
Yf
Xp
Xf
S co p e 6
P r o d u ct 7
P r o d u ct 6
P r o d u ct 5
P r o d u ct 4
P r o d u ct 3
P r o d u ct 2
P r o d u ct 1
P r o d u ct
P e a k - N o t ch
P e a k - N o t ch F i l t e r
u
2
O m e g a
S q u a r e
1
O m e g a
- K -
K z1 / m f
- K -
K z / m f
- K -
K y / m p
- K -
K y / m f
- K -
K x / m p
- K -
K x / m f
1
s
I n t e g r a t o r 7
1
s
I n t e g r a t o r 6
1
s
I n t e g r a t o r 5
1
s
I n t e g r a t o r 4
1
s
I n t e g r a t o r 3
1
s
I n t e g r a t o r 2
1
s
I n t e g r a t o r 1
1
s
I n t e g r a t o r
-1
G5
2
G4
2
G3
-1
G2
-2
G1
-2
G
D r i ve F o r ce
- K -
C z1 / m f
- K -
C z / m f
- K -
C y / m p
- K -
C y / m f
- K -
C x / m p
- K -
C x / m f
B a n d - L i m i t e d
W h i t e N o i s e
- K -
1 / m p
- K -
1 / m f 2
- K -
1 / m f 1
- K -
1 / m f
37
Figure 3.4 Unfiltered output of gyroscope in presence of a random noise signal.
Figure 3.5 Filtered output of gyroscope in presence of a random noise signal.
0 1 2 3 4 5 6 7 8 9 10
-8
-6
-4
-2
0
2
4
6
8
x 1 0
-6 U n f i l t e r e d G y r o s c o p e O u t p u t
T i m e , s
S
e
n
s
e
R
e
s
p
o
n
s
e
A
m
p
li
t
u
d
e
,
m
0 1 2 3 4 5 6 7 8 9 10
-8
-6
-4
-2
0
2
4
6
8
x 1 0
-7 F i l t e r e d G y r o s c o p e O u t p u t
T i m e , s
S
e
n
s
e
R
e
s
p
o
n
s
e
A
m
p
li
t
u
d
e
,
m
38
For the pure tone case, Gyro 1 and Gyro 2 were run independently, since the noise
signal could be reproduced. But in the case of a random noise signal, for the gyroscopes
to be affected by the same noise signal, it should be applied to both gyroscopes
simultaneously. Therefore for quantifying the effects of noise, a new Simulink model was
made (Fig. 3.6) consisting of three subsystems: Gyro 1 affected by a random noise signal,
Gyro 1 not affected by the noise signal and Gyro 2 (no drive force) affected by the same
noise signal as Gyro 1. Each of the subsystems had the same internal structure as that of
the gyroscope of Fig. 3.3. The model was also designed to take the difference between
the outputs of Gyro 1 subsystems (to determine the superimposed effects of noise on the
gyroscope output), and compare it to the output of Gyro 2.
The model was simulated, and the results, the difference between the outputs of
Gyro 1 subsystems and the output of Gyro 2, are shown in Figs. 3.7 and 3.8, respectively.
The data of both figures are exactly the same from which it was concluded that Gyro 2
was able to measure the superimposed effects of noise due to a random noise signal on
Gyro 1.
3.3 Conclusions
The model simulations described in this Chapter have shown that Gyro 2 is able
to quantify the superimposed effects of a pure tone and random noise signals on Gyro 1.
Therefore a differential-measurement system consisting of two gyroscopes, Gyro 1 and
Gyro 2, can be used to provide an uncorrupted output of a gyroscope in the presence of
noise signal by subtracting the output of Gyro 2 from the output of Gyro 1.
39
Figure 3.6 Simulink model of three subsystems.
Z e r o - O r d e r
H o l d 3
Z e r o - O r d e r
H o l d 2
Z e r o - O r d e r
H o l d 1
Y p 3Y p 2Y p 1
Y f 3Y f 2Y f 1
X p 3X p 2X p 1
X f 3X f 2X f 1
U n f i l t Y p 3
U n f i l t Y p 2
U n f i l t Y p 1
S u b t r a ct
P e a k - N o t ch
P e a k - N o t ch F i l t e r 3
P e a k - N o t ch
P e a k - N o t ch F i l t e r 2
P e a k - N o t ch
P e a k - N o t ch F i l t e r 1
1
O m e g a
N o i s e
I n1
I n2
I n4
Out 1
Out 2
Out 3
Out 4
G yr o 2
I n1
I n2
I n4
Out 1
Out 2
Out 3
Out 4
G yr o 1 _ N o N o i s e
I n1
I n2
I n4
Out 1
Out 2
Out 3
Out 4
G yr o 1 + N o i s e
F i l t Y p 3F i l t Y p 2F i l t Y p 1
D r i ve F o r ce
0
C o n s t a n t 2
0
C o n s t a n t
B a n d - L i m i t e d
W h i t e N o i s e
40
Figure 3.7 Difference between the outputs of Gyro 1 subsystems.
Figure 3.8 Output of Gyro 2.
0 2 4 6 8 10 12 14
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 1 0
-9
D i f f e r e n c e b e t w e e n ( G y r o 1 + N o i s e )
a n d ( G y r o 1 W i t h o u t N o i s e )
T i m e , s
S
e
n
s
e
R
e
s
p
o
n
s
e
A
m
p
li
t
u
d
e
,
m
0 2 4 6 8 10 12 14
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 1 0
-9
G y r o 2 O u t p u t
T i m e , s
S
e
n
s
e
R
e
s
p
o
n
s
e
A
m
p
li
t
u
d
e
,
m
41
CHAPTER 4
EXPERIMENTAL VERIFICATION OF GYROSCOPE
MODEL AND MITIGATION OF THE EFFECTS OF NOISE
In this chapter the assembly of gyroscopes on printed circuit boards is described.
The gyroscopes are used to experimentally verify the model and the mitigation solutions
presented in Chapters 2 and 3, respectively.
4.1 Gyroscope Assembly
The MEMS gyroscope selected for the experimental verification was an
automotive grade one from Analog Devices, model ADXRS652. For it to be functional it
had to be mounted on a printed circuit board (PCB) so that the unit could be supplied
with power and the output signal could be read.
4.1.1 ADXRS652 Gyroscope
The ADXRS652 gyroscope comes as ball grid array (BGA) chip-scale package
with dimensions 7 mm x 7 mm x 3 mm and produces an output voltage which is
proportional to the angular rate about the Z-axis, which is perpendicular to the top surface
of the package. The sensor as well as all the required electronics are integrated on a
single chip. Seven external capacitors are required for the operation of the gyroscope as
illustrated in Fig. 4.1. The typical natural frequency and measurement range of the
gyroscope are 14.5 kHz and ? 300 o/sec respectively [41].
42
Figure 4.1 ADXRS652 Functional Block Diagram [41].
Being a vibratory gyroscope, the proof mass is driven into resonance
electrostatically yielding the velocity required to produce a Coriolis force, which is
sensed by a capacitive pickoff structure, during rotation. The voltage required to drive the
proof mass is 18 to 20 V. Since 5 V is normally available in most applications, a charge
pump is built in the chip which requires two external capacitors for operation. The
gyroscope is designed to reject external vibration. Experiments performed by the
manufacturer using a sinusoidal vibration (with amplitude 100 m.s-2) from 100 Hz to
3000 Hz showed little effect on the gyroscope output [41]. The frequency range used in
the experiments was much below the natural frequency of the sensor.
4.1.2 PCB Design
A PCB was designed to mount the gyroscope and the capacitors using Surface
Mount Technology (SMT). FreePcb which is a free software for designing circuit boards
was used. Fig. 4.2 shows the software interface along with the design which consists of
the gyroscope land patterns at the center, all the wiring land patterns on the right, and
43
those for the capacitors around the board at a distance of 13 mm from the gyroscope. The
space between the gyroscope and the capacitors was provided so as to fit in an acoustical
enclosure made of nickel microfibrous material (the second approach of this dissertation
to mitigate the effects of noise on a MEMS gyroscope).
Figure 4.2 FreePcb interface showing the designed PCB.
The landing pattern geometry is a very important factor in SMT. The proper
geometry promotes self-alignment of components during the reflow-solder process and
provides good soldering of the component. Therefore specific design guidelines were
used to determine the landing pattern size and spacing [42]. For the ceramic capacitors
which had dimensions 1.6 mm x 3.2 mm, the landing pattern consisted of two pads 1.8
mm x 1.27 mm with 2 mm separation between the pads. The landing pattern of the
gyroscope was governed by its BGA arrangement. Design rules were also followed for
44
the proper conductor routing [43]. Two layers of copper were needed for the conductor
network and the layers were connected by through-hole vias.
After designing the PCB, the software (FreePcb) was used to generate Computer
Aided Manufacturing (CAM) files that are needed for the manufacturing of the PCBs.
Six files were generated; top copper, top mask, top silk, bottom copper, bottom mask and
drill file. Those files are also known as ?Gerber? files and can only be viewed using a
?Gerber? viewer. Fig. 4.3 illustrates some of the ?Gerber? files. The fabrication of the
board was contracted out to a commercial PCB manufacturer, ?Advance Circuits?.
Figure 4.3 Gerber files: (a) Top copper, (b) Top mask, and (c) Top silk.
4.1.3 Surface Mounting of Capacitors and Gyroscopes
Ten gyroscopes were mounted on boards. The first step was to populate the
boards with capacitors. This was done by first applying a small quantity of solder paste
on each capacitor landing pad by using a pneumatic syringe and then carefully placing
the capacitors on their respective pads. A tray was used to hold all the ten boards which
were passed through a reflow oven in one go as shown in Fig. 4.4.
Then the gyroscopes were mounted one at a time on the boards. Flux was applied
to the array of landing pads so as to facilitate the placement of the gyroscope on the board
(a) (b) (c)
45
Figure 4.4 Picture of boards coming out of the oven with mounted capacitors.
and for good soldering. No additional solder material was applied onto those pads. A flip
chip bonder with a split alignment system was used to carefully align the gyroscopes to
the landing pads. Fig. 4.5(a) illustrates the alignment. Once a good position was obtained
the gyroscope was carefully lowered into place. Before the reflow process, an x-ray (Fig.
4.5(b)) was done to confirm the positioning of the gyroscope.
Figure 4.5 (a) Alignment using flip-chip bonder, and (b) X-ray of alignment.
(a) (b)
46
The PCB was then sent through the reflow oven to solder the gyroscope onto the
board. Another x-ray of the mounted gyroscope was taken to check the correctness of the
soldering process. During the reflow process, based on the design of the PCB, self-
alignment of the gyroscope should have taken place but the second x-ray revealed that it
did not happen. Further investigations revealed that the mask layer overlapped slightly on
the pads preventing the perfect alignment. Since no contacts were made between adjacent
balls and all balls were soldered to their respective pads, the alignment obtained was
considered as satisfactory. Wires were then manually soldered to the boards using a
heating iron and the boards were tested using a power supply, Agilent E3631A, and an
oscilloscope, Agilent DSO1004A. The testing is illustrated in Fig. 4.6 where 5 V is
supplied to the gyroscope and the oscilloscope displays the output signal.
Figure 4.6 Testing of gyroscope after assembly.
4.2 Experimental Verification
Five gyroscopes, labeled G1-G5, were used for verifying the superposition of the
effects of noise and therefore the mitigation of these effects.
47
4.2.1 Equipment Setup
The setup used for the experimental tests on the gyroscopes consisted of four
main parts: the sound generation system, the rate table system, the acquisition of the
gyroscopes? output and the measurement of the sound pressure level. The setup is
illustrated in Fig. 4.7.
A software package, NCH Tone Generator, was used to generate sound source
types such as a pure tone, a tone sweep and random noise. A converter (model TX-AFC 1
M) was used to change the connector type from the computer output connector to the
amplifier (model Crown XTi 1000) connector. The sound was amplified and fed to the
two speakers, model VHF100.
The top surface of the rate table, model Aerotech ADR 160-MA-RTAS-HM, was
used to mount the gyroscopes. A controller, model Aerotech A3200, was used to control
the rate table and a computer software was used to communicate with the controller. The
latter precisely controlled the rotation rate and direction of the rate table.
A power supply was used to provide power to the gyroscopes. A total of five
gyroscopes could be tested at a time and a data acquisition box, model TEXAS BNC
2110, was used to collect the gyroscopes? output. A Labview program was used to read
and display all measured data.
A quarter inch free-field microphone, model Bruel and Kjaer (B&K) Type 4939,
was placed near the gyroscopes to measure the sound pressure level. A B & K system, the
PULSE LabShop, was used to acquire and display the microphone?s output.
The rate table as well as the speakers were in a reverberation room while the data
acquisition systems and computers were outside the room (Figs. 4.8 and 4.9).
48
Figure 4.7 Schematic of rate table and acoustical test setup.
Computer
Amplifier
VHF100
VHF100
Speakers Converter
Rate table Controller
A3200
Rate table
Power
supply
Gyros
5 V
DAQ
Connectors
5 V Gyro Output
Gyro
Output
PULSE LABSHOP
Computer
D
A
Q
Microphone
49
Figure 4.8 Picture of rate table and speakers.
Figure 4.9 Picture of control and data acquisition systems.
50
4.2.2 Testing
The testing procedure consisted of running the gyroscopes at different rotation
rates in the presence and the absence of a noise field. Then the superimposed effects of
noise on the gyroscope?s output was determined by taking the difference of the measured
outputs in the presence and the absence of the noise field. Since the maximum effects of
the noise field is experienced when the noise frequency is equal to the natural frequency
of the gyroscope, a pure tone at the natural frequency of the gyroscope was used as the
noise field.
The first step was to find the natural frequency of the gyroscope which was
determined by sweeping a pure tone from 13000 to 16000 Hz. The output of gyroscope
G1 is shown in Fig. 4.10, where the big change in the output slightly above 15 kHz
indicates the natural frequency. The sweep was repeated few more times with a smaller
bandwidth to narrow down the exact natural frequency, which was 15031 Hz.
Gyroscope G1 was then run at different rotation rates (without any noise field)
from -30 deg/s to +30 deg/s in steps of 10 deg/s and the gyroscope output amplitude
recorded. A pure tone at 15301 Hz was generated at the maximum power of the amplifier
which gave a recorded sound pressure level of 115 dB (Fig. 4.11). The gyroscope was
run in the presence of the noise field at the same rotation rates and the maximum output
amplitude recorded. The plot of the gyroscope?s outputs is shown in Fig. 4.12 and the
difference between the outputs is shown as the near horizontal line.
The same procedures were repeated for gyroscopes G2-G5. Table 4.1 summarizes
the natural frequencies and the recorded sound pressure levels for the gyroscopes and
Figs. 4.13-4.16 illustrate the results.
51
Figure 4.10 Finding the natural frequency of gyroscope G1.
Figure 4.11 Amplitude of pure tone at the natural frequency of Gyroscope G1.
1 . 3 1 . 3 5 1 . 4 1 . 4 5 1 . 5 1 . 5 5 1 . 6
x 1 0
4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
G yr o sco p e O u t p u t a g a i n st F r e q u e n cy o f P u r e T o n e
F r e q u e n c y , H z
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
52
Figure 4.12 Experimental results for gyroscope G1.
Gyroscope Natural Frequency, Hz Sound Pressure Level, dB
G1 15031 115
G2 15114 117
G3 14492 116
G4 15105 118
G5 15227 115
Table 4.1 Natural frequencies and sound pressure levels of the gyroscopes.
4.2.3 Statistical Analysis of Results
A statistical analysis was also performed to show the accuracy of the experimental
results and is summarized in Table 4.2. It was concluded that the measurements were
accurate as the standard deviation was within 5 % of the mean for 4 out the 5 gyroscopes.
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
O u t p u t o f G y r o s c o p e G 1 a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , d e g / s
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
G y r o G 1 + N o i s e ( 1 )
G y r o G 1 W i t h o u t N o i s e ( 2 )
D i f f e r e n c e ( 2 ) - ( 1 )
53
Figure 4.13 Experimental results for gyroscope G2.
Figure 4.14 Experimental results for gyroscope G3.
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
O u t p u t o f G y r o s c o p e G 2 a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , d e g / s
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
G y r o G 2 + N o i s e ( 1 )
G y r o G 2 W i t h o u t N o i s e ( 2 )
D i f f e r e n c e ( 2 ) - ( 1 )
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
O u t p u t o f G y r o s c o p e G 3 a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , d e g / s
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
G y r o G 3 + N o i s e ( 1 )
G y r o G 3 W i t h o u t N o i s e ( 2 )
D i f f e r e n c e ( 2 ) - ( 1 )
54
Figure 4.15 Experimental results for gyroscope G4.
Figure 4.16 Experimental results for gyroscope G5.
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
O u t p u t o f G y r o s c o p e G 4 a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , d e g / s
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
G y r o G 4 + N o i s e ( 1 )
G y r o G 4 W i t h o u t N o i s e ( 2 )
D i f f e r e n c e ( 2 ) - ( 1 )
- 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 4
0 . 6
0 . 8
1
O u t p u t o f G y r o s c o p e G 5 a g a i n s t R o t a t i o n R a t e
R o t a t i o n R a t e , d e g / s
G
y
r
o
s
c
o
p
e
O
u
t
p
u
t
,
V
G y r o G 5 + N o i s e ( 1 )
G y r o G 5 W i t h o u t N o i s e ( 2 )
D i f f e r e n c e ( 2 ) - ( 1 )
55
Gyroscope
Amplitude of Effects of noise
Mean, V Standard deviation Standard deviation as a percentage of the mean
G1 0.1156 0.0058 5.01 %
G2 0.3894 0.0105 2.70 %
G3 0.2363 0.0052 2.21 %
G4 0.3480 0.0357 10.25 %
G5 0.2229 0.0045 2.01 %
Table 4.2 Statistical analysis of results.
4.3 Conclusions
The results of the experiments for each of the five gyroscopes show that the
difference between the outputs in the presence and absence of the external sound field has
little variance. This shows that the effects of the noise are superimposed on the normal
output of the gyroscope and therefore confirming the simulation results of Chapter 3.
Also, because the superposition of the effects of noise has been proven experimentally, a
MEMS gyroscope not driven into oscillation will be affected by the noise field only and
will measure the amplitude of the effects of noise. Therefore mitigation of the effects of
noise can be achieved using the proposed differential measurement of two similar
gyroscopes discussed in Chapter 3. Also, it is observed that for gyroscopes G2 and G4,
the amplitude of the effects of noise on the gyroscope output at zero rotation rate is
slightly smaller than when the gyroscope is rotating.
56
CHAPTER 5
MICROFIBROUS MATERIAL FABRICATION
This chapter describes the fabrication of nickel microfibrous sheets. A wet-lay
papermaking process was used to make a preform of nickel and cellulose fibers. The
preform was then dried and sintered to give a highly porous microfibrous media.
5.1 Introduction
Nickel microfibrous sheets were made using nickel fibers and cellulose, which
acts as binding agent both in the liquid and dried states. The diameters of fibers used
were: 4, 8 and 12 microns. Forty square sheets of 20 cm were made; ten for each fiber
diameter, and ten for a mixture of 8 and 12 microns fibers in a ratio of 1:1 by mass.
5.2 Fabrication Procedure
A domestic blender was used to agitate 800 ml of water at low speed to remove
air from it. The blender was controlled by a variable speed controller and the agitation
was done at low speed to prevent turbulence, which would add more air to the water.
When air bubbles were no longer perceived in the water, 7.5 g of hydroxyethyl cellulose
(HEC) were added as a dispersion agent to breakdown cellulose into fibers. A fine sieve
was used to ensure that the added HEC was in the powdered state. 3-5 ml of a 10%
sodium hydroxide (NaOH) solution, made from 100 ml of water and 0.4 g of NaOH,
were then progressively added to the blender until a weakly alkaline solution mixture was
obtained (determined by a litmus test). The NaOH was a catalyst for breaking down
57
cellulose, and made the solution more viscous and turbid, requiring an increase in the
mixing speed. The above steps are illustrated in Fig. 5.1.
Figure 5.1 (a) Blender and speed controller, (b) HEC sieved into water, and (c) Turbid
mixture.
The turbid mixture was agitated vigorously, causing the viscosity to increase
continuously, until it turned into a clear solution. At this point 1.58 g of damped cellulose
cut into small strips were added to the blender. The speed of the latter had to be further
increased so as to break down the cellulose into fibers. When a homogeneous cellulose
mixture was obtained, 7 g of nickel fibers, broken down into small lumps, were added.
An aqueous suspension of nickel fibers was eventually obtained. Furthermore, it was
ensured that fiber lumps did not cling to the blades of the blender. These steps are
illustrated in Fig. 5.2.
A conventional papermaker was used to make a preform sheet from the aqueous
suspension of nickel fibers. The paper maker was lined at the bottom with a very fine
screen to filter out the nickel and cellulose fibers from the mixture. A water test was done
to make sure that the papermaker was leak proof, and that there was no air trapped within
(a) (b) (c)
58
Figure 5.2 (a) Turbid mixture turns clear, (b) Cellulose mixture, and (c) Aqueous
suspension of nickel fibers.
its piping system. For this purpose 2 cm of water was left at the bottom of the papermaker
and the prepared mixture was added to it. Additional water was used to ensure that all the
fibers were removed from the blender. A paddle was used to agitate the mixture and to
press the fibers together (Fig. 5.3(a)), before the system valve was opened to release all
the fluid leaving a square preform of fibers behind (Fig. 5.3(b)). A rolling pin was used to
press the excess of water out of the sheet, and the latter was then dried overnight in an
oven at 70 oC so as to remove all moisture from it. The drying process was done to
prevent the tearing of the sheet during the sintering process. The forty sheets, that were
made, were then sintered in a continuous hydrogen furnace at 1000 oC at a speed of 10
cm/min for a total sintering time of 30 minutes. The sintering is done in the middle of the
furnace, allowing time for the sheets to warm after they are introduced and to cool down
after the sintering. Therefore it took a total of one hour from the time the sheets were
inserted on the conveyor belt of the furnace to the time they came out of it. Also the
furnace required 6 hours to warm up and 6 hours to cool down.
(a) (b) (c)
59
Figure 5.3 (a) Paddle agitating the mixture and (b) Perform after water is drained.
5.3 Optical Microscopy of Microfibrous Sheets
The different types of microfibrous sheets produced were observed under an
optical microscope study their fibrous structure. For each type, four magnification factors
were used; 100X, 300X, 500X and 1000X. The optical microscopic pictures are shown in
Figs. 5.4-5.7. At 100X, a general view of the fiber network is obtained where the
randomness in the layout of the fibers can be observed. The 4 microns material has a
compact structure and with an increase in fiber size, it is observed that the microfibrous
structure is looser as the spaces between the fibers become larger. Another way to
visualize the space between the fibers is to look at a light source through the material.
The 4 microns material is so compact that it is opaque, but the 12 microns material allows
a lot of light to pass through so that the spaces between the fibers are seen as minute
pores. At 500 X and 1000X, the sintering (the fusion) of the fibers ends can be seen. It is
also clear from Fig. 5.7 that the sheet consists of two fiber diameters.
(a) (b)
60
Figure 5.4 Optical microscopic images of 4 microns diameter material at magnification
factors 100 X, 300 X, 500 X and 1000 X.
500 X 1000 X
100 X 300 X
61
Figure 5.5 Optical microscopic images of 8 microns diameter material at magnification
factors 100 X, 300 X, 500 X and 1000 X.
500 X 1000 X
100 X 300 X
62
Figure 5.6 Optical microscopic images of 12 microns diameter material at magnification
factors 100 X, 300 X, 500 X and 1000 X.
500 X 1000 X
100 X 300 X
63
Figure 5.7 Optical microscopic images of 4 and 8 microns diameters material (mixed in
a ratio of 1:1) at magnification factors 100 X, 300 X, 500 X and 1000 X.
500 X 1000 X
100 X 300 X
64
CHAPTER 6
DETERMINATION OF ACOUSTICAL PROPERTIES
In this chapter the airflow resistivity of nickel microfibrous materials (media) is
experimentally determined. The Delany-Bazley model is then used to obtain the
acoustical properties of the media using the airflow resistivity.
6.1 Material Porosity
The porosity of the materials was determined using the bulk volume of a sheet
and the mass of nickel fibers used for making the sheet. The bulk volume was calculated
from the surface area and the thickness of the sheet. The latter was obtained by taking the
average of nine measurements at different places on the sheet, as the thickness of the
sheet was not uniform. The equipment used for thickness measurement is shown in Fig.
6.1. Since a known mass of nickel (7g) was used in a sheet, its volume was computed,
and therefore, the porosity was calculated as the percentage of the void volume to the
bulk volume. The calculations for the four media types were summarized in Table 6.1.
6.2 Delany-Bazley Model
Delany and Bazley developed a model to predict the acoustical properties; the
characteristic impedance Zc and the propagation constant ?, for fibrous materials having
porosity factors near unity [24, 25]. Since the calculated porosity of the media in this
study was at least 98%, the Delany-Bazley model, Eqns. (6.1) and (6.2), were expected to
be accurate.
65
Figure 6.1 Equipment for measuring media thickness.
{[ ( ) ] [ ( ) ]} , (6.1)
{[ ( ) ] [ ( ) ]} , (6.2)
where ?0 is the density of air, c0 is the speed of sound in air, f is the frequency of sound
and ? is the airflow resistivity.
Therefore, to obtain the acoustical properties, the airflow resistivity of the media
was required and was sufficient.
6.3 Flow Resistance Measurement
The flow resistance for each of the four media types was determined
experimentally. For each type, 5 samples of 50 mm diameter were cut from a sheet using
a circular leather punch. The experimental set up is illustrated in Fig. 6.2.
66
Fiber
Size
(?m)
Sheet
No.
Thickness Measurement (mm) Sheet dim. (mm) Bulk
Volume
(mm3)
Vol. of
Nickel
Fibers
(mm3)
Porosity
(%) 1 2 3 4 5 6 7 8 9 Ave. Length Width
4 1 1.85 1.87 1.89 1.80 1.66 1.75 2.05 1.92 1.90 1.85 180 178 59416.40 786.3 98.68
4 2 2.18 1.91 1.83 2.13 1.91 1.92 1.94 1.99 1.92 1.97 171 175 58952.25 786.3 98.67
4 3 2.25 2.01 2.12 2.10 1.93 1.90 2.15 1.90 2.12 2.05 178 176 64326.83 786.3 98.78
4 4 2.09 1.93 1.95 2.01 1.83 1.86 2.05 1.94 1.98 1.96 178 176 61402.88 787.3 98.72
4/8 1 1.44 1.46 1.52 1.40 1.37 1.54 1.71 1.48 1.43 1.48 170 175 44129.17 786.3 98.22
4/8 2 1.74 1.72 1.82 1.70 1.65 1.62 1.99 1.89 1.80 1.77 170 173 52055.70 786.3 98.49
4/8 3 1.77 1.72 1.79 1.49 1.56 1.56 1.78 1.53 1.52 1.64 175 175 50088.89 786.3 98.43
4/8 4 1.72 1.53 1.69 1.46 1.47 1.60 1.57 1.53 1.52 1.57 170 175 46575.28 786.3 98.31
4/8 5 1.55 1.70 1.87 1.59 1.53 1.50 1.63 1.57 1.65 1.62 165 174 46542.10 786.3 98.31
8 1 1.40 1.37 1.32 1.37 1.32 1.30 1.26 1.25 1.47 1.34 170 170 38726.00 786.3 97.97
8 2 1.49 1.42 1.53 1.35 1.29 1.35 1.64 1.62 1.46 1.46 165 170 40984.17 786.3 98.08
8 3 1.72 1.62 1.86 1.70 1.57 1.71 1.87 1.78 1.75 1.73 177 175 53548.03 786.3 98.53
8 4 1.82 1.53 1.96 1.66 1.54 1.87 1.82 1.61 1.65 1.72 180 180 55656.00 786.3 98.59
8 5 1.66 1.65 1.62 1.47 1.45 1.44 1.42 1.26 1.59 1.51 170 166 42518.13 786.3 98.15
12 1 1.31 1.24 1.42 1.42 1.42 1.28 1.39 1.31 1.41 1.36 170 172 39636.44 786.3 98.02
12 2 1.52 1.44 1.58 1.29 1.45 1.43 1.42 1.24 1.35 1.41 172 171 41568.96 786.3 98.11
12 3 1.42 1.35 1.21 1.39 1.44 1.30 1.41 1.32 1.29 1.35 175 175 41275.69 786.3 98.10
12 4 1.21 1.28 1.39 1.28 1.30 1.30 1.19 1.22 1.27 1.27 175 175 38927.78 786.3 97.98
12 5 1.07 1.22 1.24 1.15 1.12 1.17 1.18 1.33 1.27 1.19 172 171 35131.00 786.3 97.76
Table 6.1 Media thickness measurements and porosity calculations.
67
Figure 6.2 Schematic of equipment setup for airflow resistivity measurement.
One sample was clamped between the two sections of a 25.4 mm diameter pipe
system, where the contact surfaces of the flanges had a layer of foam to prevent air
leakage when the system was pressurized. Airflow through the pipe was precisely
controlled by a mass airflow meter, ALICAT MC-5SLPM-D/5M. According to ASTM
C522 [45], which is the standard for airflow resistivity measurement of porous acoustical
materials, the airflow velocity should be between 0.5 to 50 mm/s, and the pressure drop
across the media should be between 0.1 to 250 Pa. If higher airflow velocity is used, the
flow characteristic will shift from laminar to turbulent. For a desired airflow velocity, the
corresponding volumetric airflow rate was obtained by multiplying the velocity by the
effective area of the media, and therefore, the airflow meter was programmed accordingly
Compressed
Air
Airflow
Meter
Differential Pressure
Transmitter
Piping
System
Flange
Surface
68
in standard liters per minute. The two sections of central piping system were connected to
a low differential pressure transmitter, OMEGA PX154-010DI, which measured the
pressure drop across the media in inches of water (which was then converted to Pascals).
The results of the experiments were recorded in Tables 6.2-6.5.
Using the experimental results, a graph of pressure drop against the airflow
velocity was plotted (Fig. 6.3) for each media type. It is observed that there is a linear
relation between pressure drop and velocity. Also for bigger fiber diameters, the pressure
drop is smaller as the resistance to the airflow is less. The gradient of the plotted lines of
Fig. 6.3 gives the airflow resistivity of the material, which is ratio of the pressure drop
per unit thickness to the airflow velocity. The airflow resistivity values obtained are
138 844, 109 499, 54 814 and 32 732 Pa.s/m2 for fiber diameters 4, 4/8, 8 and 12
microns, respectively.
Figure 6.3 Plot of pressure drop as a function of flow velocity.
0 10 20 30 40 50 60
0
1000
2000
3000
4000
5000
6000
7000
F l o w V e l o ci t y, m m / s
P
r
e
ssu
r
e
D
r
o
p
,
P
a
/
m
P r e ssu r e D r o p a g a i n st F l o w V e l o ci t y
4 m i c r o n s
4 / 8 m i c r o n s
8 m i c r o n s
1 2 m i c r o n s
69
Fiber
dia.
(?m)
Material
Thickness
(mm)
Flow
Rate
(L/min)
Face
Vel.
(mm/s)
Pressure drop ( inch of water) Mean
Press.
drop ( Pa)
Press. drop per
unit thickness
(Pa/m) Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean
4 1.85 0.304 10.0 0.010 0.010 0.007 0.009 0.011 0.009 2.35 1270.73
4 1.85 0.608 20.0 0.020 0.021 0.017 0.019 0.022 0.020 4.95 2676.64
4 1.85 0.912 30.0 0.031 0.030 0.026 0.030 0.033 0.030 7.50 4055.51
4 1.85 1.216 40.0 0.042 0.040 0.036 0.040 0.043 0.040 10.05 5434.39
4 1.85 1.52 50.0 0.053 0.051 0.044 0.050 0.054 0.050 12.60 6813.26
Table 6.2 Pressure drop measurements for 4 microns media.
Fiber
dia.
(?m)
Material
Thickness
(mm)
Flow
Rate
(L/min)
Face
Vel.
(mm/s)
Pressure drop ( inch of water) Mean
Press.
drop ( Pa)
Press. drop per
unit thickness
(Pa/m) Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean
4/8 1.48 0.304 10.0 0.004 0.004 0.006 0.006 0.005 0.005 1.25 844.90
4/8 1.48 0.608 20.0 0.010 0.010 0.013 0.013 0.012 0.012 2.90 1960.16
4/8 1.48 0.912 30.0 0.015 0.016 0.019 0.018 0.018 0.017 4.30 2906.45
4/8 1.48 1.216 40.0 0.022 0.023 0.028 0.026 0.025 0.025 6.20 4190.70
4/8 1.48 1.52 50.0 0.028 0.029 0.034 0.033 0.030 0.031 7.70 5204.58
Table 6.3 Pressure drop measurements for 4/8 microns media.
70
Fiber
dia.
(?m)
Material
Thickness
(mm)
Flow
Rate
(L/min)
Face
Vel.
(mm/s)
Pressure drop ( inch of water) Mean
Press.
drop ( Pa)
Press. drop per
unit thickness
(Pa/m) Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean
8 1.46 0.304 10.0 0.004 0.002 0.002 0.002 0.002 0.002 0.60 411.11
8 1.46 0.608 20.0 0.007 0.005 0.005 0.005 0.005 0.005 1.35 924.99
8 1.46 0.912 30.0 0.011 0.009 0.008 0.008 0.008 0.009 2.20 1507.39
8 1.46 1.216 40.0 0.015 0.012 0.010 0.011 0.011 0.012 2.95 2021.28
8 1.46 1.52 50.0 0.018 0.016 0.014 0.014 0.014 0.015 3.80 2603.68
Table 6.4 Pressure drop measurements for 8 microns media.
Fiber
dia.
(?m)
Material
Thickness
(mm)
Flow
Rate
(L/min)
Face
Vel.
(mm/s)
Pressure drop ( inch of water) Mean
Press.
drop ( Pa)
Press. drop per
unit thickness
(Pa/m) Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean
12 1.36 0.304 10.0 0.002 0.003 0.002 0.001 0.002 0.002 0.50 367.78
12 1.36 0.608 20.0 0.005 0.005 0.003 0.003 0.004 0.004 1.00 735.56
12 1.36 0.912 30.0 0.007 0.006 0.005 0.005 0.005 0.006 1.40 1029.78
12 1.36 1.216 40.0 0.009 0.008 0.007 0.006 0.007 0.007 1.85 1360.78
12 1.36 1.52 50.0 0.011 0.010 0.009 0.008 0.008 0.009 2.30 1691.79
Table 6.5 Pressure drop measurements for 12 microns media.
71
6.4 Absorption Coefficient
At a particular sound frequency, the characteristic impedance Zc and the
propagation constant ? were computed by using the measured air flow resistivity in Eqns.
(6.1) and (6.2). The impedance Zl of a rigidly-backed layer of thickness l was then
calculated using [24]
. (6.3)
The normal-incidence energy absorption coefficient ?n was obtained by
|
| . (6.4)
?n was computed as a function of frequency for material thicknesses 0.5, 1, 2 and
5 cm. As a requirement of the Delany-Bazley model the ratio of frequency to the airflow
resistivity was kept between 0.01 and 1 m3/kg. The graphs for each media type were
plotted in Figs. 6.4 ? 6.7. The graphs show that the absorption coefficient increases with
frequency and material thickness which are typical of acoustical materials. With an
increase in fiber size, an increase in the absorption coefficient especially in the low
frequency range is observed (for thicknesses 2 and 5 cm). Also, it is observed that for a
thin layer of the media (0.5 cm), the absorption coefficient decreased with an increase in
fiber diameter. For a material thickness of 1 cm and for frequencies above 6 kHz, all the
media types have good absorption coefficients between 0.9 and 1.0.
72
Figure 6.4 Absorption coefficients of 4 microns media.
Figure 6.5 Absorption coefficients of 4/8 microns media.
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0 . 2
0 . 4
0 . 6
0 . 8
1
4 M i cr o n s M e d i a
F r e q u e n cy, H z
A
b
so
r
p
t
io
n
C
o
e
f
f
ici
e
n
t
0 . 5 c m
1 c m
2 c m
5 c m
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0 . 2
0 . 4
0 . 6
0 . 8
1
4 / 8 M i cr o n s M e d i a
F r e q u e n cy, H z
A
b
so
r
p
t
io
n
C
o
e
f
f
ici
e
n
t
0 . 5 c m
1 c m
2 c m
5 c m
73
Figure 6.6 Absorption coefficients of 8 microns media.
Figure 6.7 Absorption coefficients of 12 microns media.
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0 . 2
0 . 4
0 . 6
0 . 8
1
8 M i cr o n s M e d i a
F r e q u e n cy, H z
A
b
so
r
p
t
io
n
C
o
e
f
f
ici
e
n
t
0 . 5 c m
1 c m
2 c m
5 c m
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 2 M i cr o n s M e d i a
F r e q u e n cy, H z
A
b
so
r
p
t
io
n
C
o
e
f
f
ici
e
n
t
0 . 5 c m
1 c m
2 c m
5 c m
74
CHAPTER 7
DAMPING CHARACTERIZATION
In this chapter, the damping characterization of nickel microfibrous materials was
performed using the displacement transmissibility concept. A fixture was designed for
attachment to the shaker head and to hold the material during testing.
7.1 Test Design
In a displacement transmissibility plot, the natural frequency is related to the
frequency ratio at the maximum transmissibility while the damping ratio is related to the
amplitude of the maximum transmissibility. Therefore displacement transmissibility was
used to characterize the damping in the microfibrous media. The experiment was
designed to emulate the simple vibration isolation system illustrated in Fig. 7.1. The
material under test, of stiffness k and damping c, is sandwiched between a known mass
and vibrating base. The displacement of the mass is measured and compared to the
measured displacement of the base so as to obtain the displacement transmissibility.
7.2 Test Fixture
A test fixture was required to hold the material under test, for positioning the
mass on top of material, and for attachment to the shaker head. Therefore the fixture (Fig.
7.2) was designed with three main parts; an aluminum bracket, a sliding mass and a
Teflon top. A cap screw was used to attach the aluminum bracket to the shaker head.
Also the flat interior surface of the bracket had two dowels press fitted near the corners to
75
Figure 7.1 Vibration transmitted through base motion.
hold the test material, preventing any side movement of the latter during vibration. The
sliding mass was designed with a flat bottom surface to sit squarely on top of the test
material. The mass could be altered by using thick washers, and the aluminum slider part
was highly polished to reduce friction during motion within the Teflon top part. The latter
was made of Teflon for minimum friction and was attached to the aluminum brackets by
four cap screws. Mechanical drawings of the fixture are shown in Appendix D.
Figure 7.2 Photograph of test fixture.
76
7.3 Equipment Setup
A schematic of the equipment setup is shown in Fig. 7.3. The test fixture is
attached to the electromagnetic shaker, model LDS 408, which vibrates vertically based
on the input from the amplifier, model LDS PA500L. The input to the amplifier is the
source signal generated by the analyzer, model HP 35665A. Two laser vibrometers are
used; one for measuring the input (the displacement of the shaker head), and one for
measuring the output (the displacement of the mass). Each of the vibrometers consists of
a sensor head, model Polytec OFV 353, and a controller, model Polytec OFV 2610. The
controller provides power to the sensor head and decodes the measured signal from the
sensor head. The sensor head emits a laser beam, which is reflected from a small
reflective sticker placed on the surface being measured. The input and output
vibrometers? signals are respectively fed to channels 1 and 2 of the analyzer, and the
transfer function of the output/input is displayed on the analyzer. To improve signal to
noise ratio, the laser is focused at the center of the reflective material for maximum
reflection. The test set up is illustrated in Fig. 7.4.
7.4 Experimental Procedure
For each of the four media types, a series of experiments was conducted using
different vibration amplitudes and different number of layers of the material. The
generated source signal was a random noise and its amplitude and bandwidth were
specified in the analyzer. The vibration amplitude was varied by increasing the source
amplitude from 20 mVrms to 60 mVrms in steps of 10 mVrms while keeping the gain of the
amplifier constant. An accelerometer was used to measure the acceleration corresponding
to the source amplitudes and the respective values were 0.694 m/s2, 1.103 m/s2, 1.468 m/s2,
77
Figure 7.3 Schematic of vibration test equipment setup.
Source
SHAKER
LDS V408
ANALYZER
AMPLIFIER
Output Vibrometer
Input
Controller
Shaker head
Output
Controller
Output Laser Input Laser
Input Vibrometer
Output
Signal
Input
Signal
Input
Signal
78
Figure 7.4 Photograph of test setup.
1.831 m/s2 and 2.202 m/s2. The number of layers was varied from 1 to 5.
The 20 cm x 20 cm microfibrous sheets were cut according to the dimensions and
shape of the fixture. As such, 15 samples were obtained from one sheet and were labeled
1-15. The samples were first tested individually at the different vibration amplitudes and
their transfer functions saved on the analyzer. Averaging of the transfer function was
done for noise rejection. A typical transfer function is illustrated in Fig. 7.5, where a
phase shift of 90o is clearly observed at the natural frequency. An example of the transfer
functions at the different vibration amplitudes is illustrated in Fig. 7.6 for sample number
9 made of 4 micron diameter fibers. It is observed that the amplitude of the transfer
function increases, and the natural frequency decreases, with increasing vibration
79
amplitude. For each test performed, the frequency and amplitude of the maximum of the
transfer function were recorded and the average computed for the 15 samples. The
number of layers was then incremented and the same process repeated until all the tests
were completed. The transfer functions for the different material types and the different
number of layers used are shown in Appendix D. A series of tables were then made
(Appendix D) for recording the frequency and amplitude of the transfer functions.
Figure 7.5 Picture of analyzer screen showing a Bode plot.
7.5 Stiffness and Damping Ratio
The average values of the frequency and amplitude of the transfer functions were
used to determine the stiffness and damping ratio of the materials. The relation between
the frequency ratio at maximum amplitude rm and the damping ratio ? is given by Eqn.
(7.1), and the relation between the maximum amplitude Am and the damping ratio is given
by Eqn. (7.2). The equations were derived from the displacement transmissibility as
shown in Appendix A.
80
Figure 7.6 Transfer function at different vibration amplitudes.
?? . (7.1)
[ ? ]
?
. (7.2)
Using the maximum amplitude, iteration of Eqn. (7.2) was done to obtain the
damping ratio. Subsequently the frequency ratio at resonance was obtained using Eqn.
(7.1). The frequency ratio at resonance is also given by
, (7.3)
where ?m is the frequency of maximum amplitude, obtained from the transfer function, in
rad/s and ?n is the natural frequency.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n c y , H z
T
r
a
n
s
f
e
r
F
u
n
c
t
io
n
4 M i c r o n s - 1 S h e e t - T e s t N o . 9 ( 4 D 0 9 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
81
Therefore from Eqn. (7.3), the natural frequency of the system was determined.
Using the mass of the slider m and the natural frequency, the stiffness k was determined
using Eqn. (7.4).
. (7.4)
For all the tests performed, the damping ratio and stiffness were computed and
plotted in Figs. 7.7 ? 7.14 as a function of the number of layers and vibration amplitude.
Figure 7.7 Stiffness of 4 microns media
1 2 3 4 5
0
0 . 5
1
1 . 5
2
2 . 5
3
x 1 0
4
N u m b e r o f L a y e r s
S
t
if
f
n
e
s
s
,
N
/
m
4 M i c r o n s - S t i f f n e s s a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
82
Figure 7.8 Damping ratio of 4 microns media.
Figure 7.9 Stiffness of 4/8 microns media
1 2 3 4 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
N u m b e r o f L a y e r s
D
a
m
p
in
g
R
a
t
io
4 M i c r o n s - D a m p i n g R a t i o a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
1 2 3 4 5
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
x 1 0
4
N u m b e r o f L a y e r s
S
t
if
f
n
e
s
s
,
N
/
m
4 / 8 M i c r o n s - S t i f f n e s s a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
83
Figure 7.10 Damping ratio of 4/8 microns media.
Figure 7.11 Stiffness of 8 microns media.
1 2 3 4 5
0 . 1
0 . 1 1
0 . 1 2
0 . 1 3
0 . 1 4
0 . 1 5
0 . 1 6
0 . 1 7
0 . 1 8
0 . 1 9
N u m b e r o f L a y e r s
D
a
m
p
in
g
R
a
t
io
4 / 8 M i c r o n s - D a m p i n g R a t i o a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
1 2 3 4 5
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
x 1 0
4
N u m b e r o f L a y e r s
S
t
if
f
n
e
s
s
,
N
/
m
8 M i c r o n s - S t i f f n e s s a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
84
Figure 7.12 Damping ratio of 8 microns media.
Figure 7.13 Stiffness of 12 microns media.
1 2 3 4 5
0 . 1
0 . 1 1
0 . 1 2
0 . 1 3
0 . 1 4
0 . 1 5
0 . 1 6
0 . 1 7
0 . 1 8
0 . 1 9
0 . 2
N u m b e r o f L a y e r s
D
a
m
p
in
g
R
a
t
io
8 M i c r o n s - D a m p i n g R a t i o a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
1 2 3 4 5
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
x 1 0
4
N u m b e r o f L a y e r s
S
t
if
f
n
e
s
s
,
N
/
m
1 2 M i c r o n s - S t i f f n e s s a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
85
Figure 7.14 Damping ratio of 12 microns media.
7.6 Analysis of Results
With an increase in the number of layers, a decrease in the stiffness was observed.
This is similar to adding springs in series causing the overall stiffness to decrease. The
stiffness also decreased with an increase in vibration amplitude. This shows that the
stiffness of the materials is nonlinear and that they behave a soft springs.
The damping ratio was found to be almost constant with an increase in the
number of layers. The damping ratio is a property of the material, and therefore is not
affected by the number of layers. The magnitude of the damping ratio is found to
decrease with an increase in the vibration amplitude. This can be explained by the fact
that damping in the system is proportional to the material damping and friction between
1 2 3 4 5
0 . 1
0 . 1 2
0 . 1 4
0 . 1 6
0 . 1 8
0 . 2
0 . 2 2
N u m b e r o f L a y e r s
D
a
m
p
in
g
R
a
t
io
1 2 M i c r o n s - D a m p i n g R a t i o a g a i n s t N u m b e r o f L a y e r s
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
86
the slider and top frame. An increase in the vibration amplitude causes the friction to be
less, reducing the total damping in the system.
87
CHAPTER 8
MITIGATION OF THE EFFECTS OF NOISE USING
NICKEL MICROFIBROUS MATERIALS
In this chapter nickel microfibrous enclosures are designed and fabricated by
sintering together layers of the microfibrous material. Experiments are then done in a
reverberation room to validate the effectiveness of the enclosures for noise attenuation.
8.1 Enclosure Design and Fabrication
Based on the results obtained using the Delany-Bazley model (section 6.4), the
absorption coefficient of a 10 mm thick microfibrous media at a sound frequency of 15
kHz was between 0.9 and 1.0. Therefore to obtain good noise attenuation, the enclosures
were designed so that the gyroscope could be surrounded by 12 mm of the media on all
sides. Since the media was fabricated in sheets, the enclosure was obtained by cutting the
sheets into the desired dimension and stacking them as illustrated in Fig. 8.1(a). A wire
mesh was then bent and shaped so as to hold all the stacked sheets (Fig. 8.1(b)), which
were then sintered to give the final enclosure.
The sintering was done in a small continuous hydrogen furnace at 950 oC for 40
minutes. The wire mesh was placed at the bottom of a sintering tube which was then
positioned inside the furnace (Fig. 8.2). Alumina fibers which can sustain a temperature
of 1700 oC were placed on top of the furnace to prevent heat loss. The four enclosures
made (one for each media type) are illustrated in Fig. 8.3.
88
Figure 8.1 (a) Stacked sheets, and (b) Wire mesh frame.
Figure 8.2 Photograph of furnace.
(b) (a)
Sintering tube
89
Figure 8.3 Photograph of enclosures showing top and bottom surfaces.
8.2 Experimental Validation of Enclosures
The equipment setup for the acoustical tests of section 4.2 was used for validating
the effectiveness of the microfibrous enclosures. Seven gyroscopes, labeled G1-G7, were
tested. For each gyroscope, a reference signal (without noise) was recorded. A pure tone
was then generated at the natural frequency of the gyroscope, and at the maximum power
of the amplifier. The effects of the noise on the gyroscope?s output were recorded. Then
one enclosure was placed on the gyroscope to attenuate the effects of the noise, but no
reduction in the effects of noise was achieved. Investigations showed that the bottom
surface of the enclosure was not perfectly flat and was not making full contact with the
PCB, allowing noise to leak through. To solve this issue, a mass was placed on top of the
enclosure as depicted in Fig. 8.4. With the mass on, considerable attenuation in the
effects of noise was observed. The test was then repeated using the other enclosures and
the other gyroscopes. Another problem that was encountered during testing was that the
enclosure, being made of metal, was short-circuiting the surface mounted capacitors
required for the gyroscope operation. Therefore liquid tape was used to electrically
isolate the capacitors.
90
Figure 8.4 Photograph of mass sitting on top of enclosure.
The results of the acoustical tests performed on gyroscopes G1 and G2 are shown
in Figs. 8.5 - 8.12, where each figure illustrates the gyroscope?s outputs, namely, the
reference signal, the output in the presence of noise, and the output with the enclosure on.
Considerable reduction in the effects of noise on the gyroscopes is observed. The results
for gyroscopes G3-G7 are found in Appendix F. Table 8.1 is a summary of all test results,
in which the amplitudes of the effects of noise are compared to the attenuated amplitudes,
and the attenuation as a percentage of the amplitude of the effects of noise computed.
8.3 Conclusions
The experimental results show that the enclosures have been effective in
attenuating the effects of noise on the MEMS gyroscopes. Though the effects are not
completely mitigated, up to 90% reduction in the amplitude of the effects of noise has
been observed. On the average, 65% reduction in the effects of noise can be easily
obtained. No major difference in attenuation was observed between the different
91
enclosures. This is due to the fact that they all have similar absorption coefficient at
around 15 kHz.
Figure 8.5 Experimental results of the 4 microns enclosure on gyroscope G1.
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
4 M i cr o n s E n cl o su r e o n G yr o sco p e G 1
N o N o i s e
N o i s e
E n c l o s u r e
92
Figure 8.6 Experimental results of the 4/8 microns enclosure on gyroscope G1.
Figure 8.7 Experimental results of the 8 microns enclosure on gyroscope G1.
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 1
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
8 M i cr o n s E n cl o su r e o n G yr o sco p e G 1
N o N o i s e
N o i s e
E n c l o s u r e
93
Figure 8.8 Experimental results of the 12 microns enclosure on gyroscope G1.
Figure 8.9 Experimental results of the 4 microns enclosure on gyroscope G2.
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 1
N o N o i s e
N o i s e
E n c l o s u r e
3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 7 . 5 8
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
4 M i cr o n s E n cl o su r e o n G yr o sco p e G 2
N o N o i s e
N o i s e
E n c l o s u r e
94
Figure 8.10 Experimental results of the 4/8 microns enclosure on gyroscope G2.
Figure 8.11 Experimental results of the 8 microns enclosure on gyroscope G2.
3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 7 . 5 8
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 2
N o N o i s e
N o i s e
E n c l o s u r e
3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 7 . 5 8
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
8 M i cr o n s E n cl o su r e o n G yr o sco p e G 2
N o N o i s e
N o i s e
E n c l o s u r e
95
Figure 8.12 Experimental results of the 12 microns enclosure on gyroscope G2.
3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 7 . 5 8
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
O
u
t
p
u
t
A
m
p
li
t
u
d
e
,
V
1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 2
N o N o i s e
N o i s e
E n c l o s u r e
96
Gyro
Natural
Frequency
(Hz)
SPL
(dB)
Amplitude
of Noise
Effects
(mV)
Amplitude of Noise Effects with
Enclosure (mV)
Percentage Reduction in the Amplitude
of Noise Effects due to Enclosure (%)
4
microns
4/8
microns
8
microns
12
microns
4
microns
4/8
microns
8
microns
12
microns
G1 15031 119 128.5 20.5 22.0 24.0 18.0 84.0 82.9 81.3 86.0
G2 15114 118 207.5 42.5 32.0 60.0 48.0 79.5 84.6 71.1 76.9
G3 14492 115 93.5 43.0 24.0 53.0 42.0 54.0 74.3 43.3 55.1
G4 15105 118 136.0 32.5 13.0 25.5 30.0 76.1 90.4 81.3 77.9
G5 15227 119 201.5 118.5 103.5 150.5 105.0 41.2 48.6 25.3 47.9
G6 14883 123 123.0 51.0 34.0 34.5 46.0 58.5 72.4 72.0 62.6
G7 15066 119 137.0 62.5 72.0 73.0 73.0 54.4 47.5 46.7 46.7
Table 8.1 Summary of experimental results.
95
CHAPTER 9
CONCLUSIONS AND SCOPE FOR FURTHER WORK
In this dissertation, ways to mitigate the effects of noise on MEMS gyroscopes
have been suggested using two approaches. In the first approach, a mathematical model
has been developed that includes an external noise signal affecting the gyroscopes. The
simulations of the model show that the magnitude of the effects of noise on the
gyroscope?s output depends on the frequency contents of the noise signal. As expected,
the maximum effect occurs when the frequency is equal to the natural frequency of the
gyroscope. Also, when the frequency of noise is not near the natural frequency, no effect
on the output of the gyroscope is observed. This concurs with the results of previous
experimental tests performed on MEMS gyroscopes.
The simulations at different rotation rates in the presence of external noise show
that at a zero rotation rate the gyroscope has a positive output amplitude but a zero output
amplitude at a rotation rate of ? 0.92 rad/s. Further, an increase in rotation rate causes the
sense response to increase. To explain these observations, the model was run at the same
rotation rates but without the external noise. A comparison of the two cases shows that
the presence of the noise causes an increase in the sense response amplitude. Also, the
shift is greater when the rotation is negative. This effect is due to the Coriolis component
of acceleration, which causes an increase in the amplitude when the rotation is negative
(clockwise rotation) and a decrease in the amplitude when the rotation rate is positive
96
(counterclockwise). It is therefore concluded that the effects of the noise is superimposed
on the normal output of the gyroscope.
To mitigate the superimposed effects of noise, a differential- measurement system
consisting of two gyroscopes is suggested. The first gyroscope (Gyro 1) is a regular
MEMS gyroscope, while the second gyroscope (Gyro 2) is a non-driven gyroscope and
thus affected by the noise field only. In this configuration, the Gyro 2 is able to quantify
the superimposed effects and subtracting its output from the output of Gyro 1 yields an
uncorrupted output. Simulations were performed for two types of noise signals: a pure
tone and band-limited white noise. In both cases, Gyro 2 was able to quantify the effects
of noise and the differential-measurement system mitigated these effects on a MEMS
gyroscope.
Experiments were performed on five gyroscopes to verify the simulations. The
experimental results show that the difference between the outputs of the gyroscopes in
the presence and absence of the external noise is almost constant. Since the amplitude of
the supplied noise was made constant, it is concluded that the effects of noise were
superimposed on the normal output of the gyroscope as indicated by the simulations.
Further, because the superposition of the effects of noise was experimentally shown, a
MEMS gyroscope not driven into oscillation is affected by the noise field only, and
quantifies the amplitude of the effects of noise. Therefore, the experimental results
confirm that mitigation of effects of noise can be achieved using the proposed differential
measurement.
The second approach to mitigate the effects of noise was to use a nickel
microfibrous material for noise attenuation. Four types of the material were made using
97
three diameters of fibers and a wet-lay papermaking process. The sheets had a porosity of
at least 98%. The Delany-Bazley analytical model was used to determine the optimum
acoustical properties of the media and the absorption coefficients were then computed.
The results show that the four types of media have an absorption coefficient between 0.9
and 1.0 in the region of interest, which is 15 kHz.
The damping characterization of the materials was carried out using the
displacement transmissibility concept. Different vibration amplitudes were used to test
five stacked layers of the material. With an increase in the number of layers, a decrease in
the stiffness was observed. This is similar to adding springs in series causing the overall
stiffness to decrease. The stiffness also decreased with an increase in vibration amplitude.
This shows that the stiffness of the materials is nonlinear and they behave as a soft
spring. Being a property of the material, the damping ratio is not affected by the number
of layers. The magnitude of the damping ratio is found to decrease with an increase in the
vibration amplitude. This can be explained by the fact that damping in the system is
proportional to the material damping and friction between the slider and the top frame of
the fixture. An increase in the vibration amplitude causes the friction to decrease,
reducing the total of the damping in the system.
Microfibrous enclosures were designed so that the gyroscope could be surrounded
by 12 mm of the material on all sides. Four enclosures, one for each material type, were
made by sintering together layers of the material. The experimental results show that the
enclosures have been effective in attenuating the effects of noise on the MEMS
gyroscopes. Though the effects are not completely mitigated, up to 90% reduction in the
amplitude of the effects of noise has been observed. On average, 65% reduction in the
98
amplitude of the effects of noise can be easily obtained. No major difference in
attenuation was observed between the different enclosures. This is due to the fact that
they all have similar absorption coefficients at around 15 kHz.
In summary this research describes two different approaches that have been
studied to mitigate the undesirable effects of noise fields on MEMS gyroscopes. In the
first approach an active design is proposed by utilizing a pair of gyroscopes whose
outputs can be manipulated to yield the desired uncorrupted results. In the second
approach, a passive design is proposed using nickel microfibrous material as an
acoustical enclosure. Considerable reductions in the effects of noise have been achieved,
showing that the nickel microfibrous material can be used to construct an acoustical
enclosure.
As further work to this dissertation, a gyroscope can be designed and fabricated to
carry out the differential measurement. It is suggested to design Gyro 1 and Gyro 2 to
have the same natural frequency and to fabricate them side by side on the same silicon-
on-insulator wafer.
99
REFERENCES
1. Acar, C., Shkel, A., ?MEMS Vibratory Gyroscopes Structural Approaches to
Improve Robustness,? Springer, 2009.
2. Meriam, J.L., Kraige, L.G., ?Engineering Mechanics Dynamics,? John Wiley and
Sons, 2002.
3. Beeby, S., Ensell, G., Kraft, M., White, N., ?MEMS Mechanical Sensors,? Artech
House Inc., 2004.
4. Mochida, Y., Tamura, M., Ohwada, K., ?A micromachined vibrating rate
gyroscope with independent beams for the drive and detection modes,? Elsevier,
Sensors and Acuators 80, 170-178, 2000.
5. Rashed, R., Momeni, H., ?System Modeling of MEMS Gyroscopes,?
Mediterranean Conference on Control and Automation, 2007.
6. Acar, C., Shkel, A., ?MEMS Gyroscopes with Structurally Decoupled 2-DOF
Drive and Sense Mode Ocillators,? Nanotech, Vol. 1, 2003.
7. Yunker, W.N., ?Sound Attenuation Using MEMS Fabricated Acoustic
Metamaterials,? MS Thesis, Auburn University, 2012.
8. Brown, T. G.,? Harsh Military Environments and Microelectromechanical
(MEMS) Devices,? Proceedings of IEEE Sensors, 2, pp. 753-760, 2003.
100
9. Weinberg, M. S., and Kourepenis, A., ? Error Sources in In-plane Silicon Tuning-
fork MEMS Gyroscopes,? Microelectromechanical Systems, 15(3), pp. 479-491,
2006.
10. Dean, R., Flowers, G., Hodel, S., MacAllister, K., Horvath, R., Matras, A.,
Robertson, G., and Glover, R., ? Vibration Isolation of MEMS Sensors for
Aerospace Applications,? Proceedings of the IMAPS International Conference and
Exhibition on Advanced Packaging and Systems, Reno, NV, pp. 166-170, 2002.
11. Dean, R., Flowers, G., Ahmed, A., Hodel, A., Roth, G., Castro, S., Zhou, R., Rifki,
R., Moreira, A., Grantham, B., Bittle, D., Brunsch, J., "On the degradation of
MEMS gyroscope performance in the presence of high power acoustic noise," Proc.
of ISIE, 1435-1440, 2007.
12. Dean, R., Castro, S., Flowers, G.T., Roth, G., Ahmed, A., Hodel, A.S., Grantham,
B.E., Bittle, D. A., Brunsch, J.P., ?A Characterization of the Performance of a
MEMS Gyroscope in Acoustically Harsh Environments,? Industrial Electronic,
IEEE Transactions on, Vol 58, Issue 7, 2591-2596, 2011.
13. Yunker, W. N., Soobramaney, P., Black, M., Dean, R. N., Flowers, G. T., Ahmed,
A., ? The underwater effects of high power, high frequency acoustic noise on
MEMS gyroscopes,? Proc. of ASME IDETC2011-47180, 2011.
14. Castro, S., Roth, G., Dean, R., Flowers, G.T., Grantham, B., ?Influence of acoustic
noise on the dynamic performance of MEMS gyroscopes,? ASME IMECE2007-
42108, 2007.
15. Roth, G., ?Simulation of the Effects of Acoustic Noise on MEMS Gyroscope,? MS
Thesis, Auburn University, 2009.
101
16. Yunker, W.N., Stevens, C.B., George, T. F., Dean R.N, ?Sound attenuation using
microelectromechanical systems fabricated acoustic metamaterials,? Journal of
Applied Physics 113, 024906, 2013.
17. Arenas, J.P., Crocker M.J., ?Recent Trends in Porous Sound-Absorbing Materials,?
Sound and Vibration, 44(7), 12-18, 2010.
18. Harris, C.M., ?Handbook of noise control,? McGraw-Hill, Inc, 1957.
19. Ashby, M.F., Lu, T., ?Metal foams: A survey, Science in China,? Series B, Vol. 46,
No. 6, 521-532, 2003.
20. Schmidt, M., Schwertfeger, F., ?Applications for silica aerogel products,? Journal
of Non-Crystalline Solids, 225, 364-368, 1998.
21. Crocker M.J., ?Handbook of acoustics,? John Wiley & Sons, Inc., 1998.
22. Wilson, C.E., ?Noise control,? Krieger Publishing Company, 2006.
23. Vissamraju, K., ?Measurement of Absorption Coefficient of Road Surfaces Using
Impedance Tube Method,? MS Thesis, Auburn University, 2005.
24. Delany, M.E., Bazley, E.N., ?Acoustical properties of fibrous absorbent materials,?
Applied Acoustics, 3, 105-116, 1970.
25. Komatsu, T., ?Improvement of the Delany-Bazely and Miki models for fibrous
sound-absorbing materials,? Acoustical Science and Technology, 29(2), 121-129,
2008.
26. Dunn, I.P., Davern W.A., ?Calculation of Acoustic Impedence of Multi-layer
Absorbers,? Applied Acoustics 19, 321-334, 1986.
27. Voronina, N., ?Acoustic Properties of Fibrous Materials,? Applied Acoustics 42,
165-174, 1994.
102
28. Rivin, E.I., ?Passive Vibration Isolation,? ASME Press, New York, 2003.
29. Rao, S.S., ?Mechanical Vibrations,? Pearson Prentice Hall, New Jersey, 228-243,
2004.
30. Crede, C.E., ? Vibration and Shock Isolation,? John Wiley and Sons Inc., New
York, 176-186, 1951.
31. Inman, D., ?Encyclopedia of Vibration,? Volume 1, Academic Press, 2002.
32. Meffert, M. W., ?Preparation & characterization of sintered metal microfiber based
composite materials for heterogeneous catalyst application,? PhD dissertation,
Auburn University, 60-64, (1998).
33. Tatarchuk, B. J., Rose, M. F., Krishnagopalan, A., ?Mixed fiber composite
structures,? US Patent 5,102,745, April 7, 1992.
34. Tatarchuk, B. J., Rose, M. F., Krishnagopalan, A., Zabasajja, J. N., Kohler, D.
?Preparation of mixed fiber composite structures,? US Patent 5,304,330, April 19,
1994.
35. Burch, N. H., Black, M. N., Dean, R. N., Flowers, G. T., ?Microfibrous metallic
cloth for damping enhancement in printed circuit boards,? Proc. SPIE 7643, 2010.
36. Zhu, W. H., Flanzer, M. E., Tatarchuk, B.J., ?Nickel?zinc accordion-fold batteries
with microfibrous electrodes using a papermaking process?, Journal of Power
Sources, 112, 353-366, 2002.
37. Harris, D. K., Cahela, D. R., Tatarchuk, B.J, ?Wet layup and sintering of metal-
containing microfibrous composites for chemical processing
opportunities,? Composites Part A - Applied Science And Manufacturing, 32(8),
1117-1126, 2001.
103
38. Dean, R., Burch, N., Black, M., Flowers, G., ?Microfibrous metallic cloth for
acoustic isolation of a MEMS gyroscope,? Proc. SPIE 7979, 2011.
39. Hyatt, N., Black, M., Dean, R., Flowers, G., Grantham, B., Garner, R., ?Damping
enhancement in printed circuit boards with potting materials or microfibrous
metallic cloth,? Proc. of IDETC/VIB-87846, 2009.
40. Storm, M. C., ?Prediction of sintered fibrous metal liner influence on muffler sound
attenuation performance and noise emission for single-cylinder motorcycle engine
exhaust,? Proc. of NCAD2008-73022, 2008.
41. Datasheet, ?ADXRS652 ? 250o/sec Yaw Rate Gyro,? Analog Devices, Inc., Rev A.
42. Solberg, V., ?Design Guidelines for Surface Mount and Fine-pitch Technology,?
McGraw-Hill, 1996.
43. Lindsey, D., ?Digital Printed Circuit Design and Drafting,? Bishop Graphics, Inc.,
1986.
44. http://www.mathworks.com/help/simulink/slref/bandlimitedwhitenoise.html
45. http://enterprise.astm.org/filtrexx40.cgi?+REDLINE_PAGES/C522.htm
46. Hatch, M.R., ?Vibration simulation using MATLAB and ANSYS,? Chapman &
Hall/CRC, 2001.
47. Bao, M.H., ? Handbook of Sensors and Actuators,? Elsevier, 2000.
104
APPENDIX A
A.1 Maximum Displacement Transmissibility
For a vibration system with base motion, the displacement transmissibility which
is the ratio of the output displacement X to the input displacement Y is given by
| | [ ( ) ]
?
, (A.1)
where r is the frequency ratio and z is the damping ratio.
The frequency ratio rm at the maximum displacement transmissibility is obtained
by solving
[|
|] . (A.2)
Differentiating the displacement transmissibility with respect to r:
{[
( ) ]
?
}
[
( ) ]
?
{[(
) ]( ) ( )[ ( ) ]}
[( ) ] . (A. 3)
Simplifying Eqn. (A.3) and equating to zero give
. (A.4)
or
. (A.5)
105
Solving Eqn. (A.5) gives
? ( ) , (A.6)
or
?? . (A.7)
A.2 Damping Ratio as a Function of Maximum Amplitude
The maximum transmissibility amplitude Am occurs at the frequency ratio rm
given by Eqn. (A.7). Substituting rm in Eqn. (A.1) gives
[ (
)
] . (A.8)
Substituting Eqn. (A.7) in Eqn. (A.8) gives
?
? ( ? )
( ? ) . (A.9)
Therefore
[ ? ]
?
. (A.10)
106
APPENDIX B
B.1 Solving Equations of Motion
B.1.1 Drive Motion
The equation of motion for the drive direction is
? ? . (B.1)
Let
( ) . (B.2)
? ( ) , (B.3)
and
? ( ) . (B.4)
Substituting Eqns. (B.2) ? (B.4) in Eqn. (B.1) gives
. (B.5)
Separating the coefficients of and gives
, (B.6)
and
. (B.7)
Simplifying Eqns. (B.6) and (B.7) gives
[( ) ] , (B.8)
107
and
[( ) ] . (B.9)
Solving Eqns. (B.8) and (B.9) gives
[( ) ( ) ] , (B.10)
and
( ( ) ) . (B.11)
At drive force frequency w = wn
, (B.12)
and
. (B.13)
Therefore
? . (B.14)
B.1.2 Sense Motion
The equation of motion for the sense direction is
? ? ? . (B.15)
Substituting Eqn. (B.14) in Eqn. (B.15) gives
? ? (B.16)
Let
( ) . (B.17)
Therefore
? ( ) , (B.18)
and
108
? ( ) (B.19)
Substituting Eqns. (B.17) - (B.19) in Eqn. (B.16) gives
( ).
(B.20)
Separating the coefficients of and gives
,
(B.21)
and
. (B.22)
Simplifying Eqns. (B.21) and (B.22) gives
[( ) ] , (B.23)
and
[ ( ) ] . (B.24)
Solving Eqns. (B.23) and (B.24) gives
(
)
, (B.25)
and
[( ) ( ) ] . (B.26)
At w=wn
, (B.27)
and
( ) . (B.28)
109
. (B.29)
B.2 Natural Frequency Computation
B.2.1 Basic Vibratory Gyroscope
The equations of motion for the gyroscope of section 2.1.1 are reproduced below:
? ? ( ) ? , (B.30)
and
? ? ( ) ? . (B.31)
The equations of motion were converted to state space form
? , (B.32)
where
[
] [
?
?
] , (B.34)
[
]
, (B.35)
and
[
( )
( ) ]
. (B.36)
The eigenvalues of matrix A were numerically computed in Matlab using the
following parameters: m=1.0 x 10-8 kg, kx =ky = 77.38 N/m, cx = cy = 5 x 10-7 N.s/m and ?
= 1 rad/s. The computed eigenvalues were -25 ? 87967i and -25 ? 87965i. The damped
110
natural frequencies were obtained from the imaginary part of the eigenvalues [46] and
were equal to 14000 Hz.
B.2.2 Gyroscope for Noise Simulation
The equations of motion of the gyroscope model of section 2.2.1 are reproduced
below:
? ( ? ? ) ? ( ) ?
, (B.37)
? ( ? ? ) ( ) ? , (B.38)
? ( ? ? ) ? ( ) ? , (B.39)
and
? ( ? ? ) ( ) ? . (B.40)
The equations of motion were converted to state space form
? (B.41)
where
[
]
[
?
?
?
? ]
, (B.42)
111
[
]
(B.43)
and
[
]
. (B.44)
The eigenvalues of matrix A were numerically computed in Matlab using the
following parameters: mf =3.8 x 10-4 kg, mp =1.0 x 10-8 kg, kx =ky = 77.38 N/m, cx = cy = 5
x 10-7 N.s/m, kz = 200 N/m, cz = 5 x 10-3 N.s/m, and ? = 1 rad/s.
The computed eigenvalues were: -25.001 ? 87968i, -25.001 ? 87966i, -6.5697 ?
724.44i, and -6.5878 ? 726.44i. The damped natural frequencies were obtained from the
imaginary part of the eigenvalues and were equal to 14001 Hz, 14000 Hz, 115.3 Hz and
115.62 respectively.
?
112
B.3 Force Approximation from Sound Pressure Level
The sound pressure level LP is given by
(
), (B.45)
where is the mean square sound pressure and is the hearing threshold
pressure [22].
For a sound pressure level of 126 dB
(
) . (B.46)
Therefore
? (B.47)
? ( ) .
The peak pressure for a pure tone is given by [22]
? . (B.48)
Therefore
? Pa
The force due to the sound pressure, F, was approximated by
, (B.49)
where A is the surface area of one side of a gyroscope.
Therefore the approximated force was obtained:
( ) ( )
= 1.26 x 10-3 N.
113
APPENDIX C
C.1 Matlab Codes for Solving the Basic Vibratory Gyroscope Model
% Basic Vibratory Gyroscope
% 2 DOF
clc;
clear all;
close all;
format compact;
format short g;
diary off;
global B fd w mp kx ky cx cy;
% mass 'mp' of proof mass
mp = 1e-8;
%'fd' frequency of driving force
fd=14003;
% 'kx' stiffness in x direction
kx=77.38;
% 'ky' stiffness in y direction
ky=77.38;
% 'cx' damping in y direction
cx=5e-7;
% 'cy' damping in y direction
cy=5e-7;
% 'B' input amplitude of drive force signal
B=1e-6;
%'w' angular velocity of gyroscopic frame
w=1;
init = [0,0,0,0];
[t,z] = ode45(@newgyro2dof, (0:0.000001:.6),init);
z1 = z(:,1);
z2 = z(:,2);
z3 = z(:,3);
z4 = z(:,4);
114
Q=[z1(3.90023e5:3.91023e5) z2(3.90023e5:3.91023e5)];
j=(1:1001)*0.000001;
figure(1)
subplot (2,1,1)
plot(j,Q(:,1),'LineWidth',1.5);
grid on
title('Drive Mode Response','FontSize',14);
xlabel ('Time, s','FontSize',12);
ylabel ('Displacement (x), m','FontSize',12);
axis([0,1e-3,-3e-5,3e-5])
subplot (2,1,2)
plot(j,Q(:,2),'LineWidth',1.5);
grid on
title('Sense Mode Response','FontSize',14);
xlabel ('Time, s','FontSize',12);
ylabel ('Displacement (y), m','FontSize',12);
axis([0,1e-3,-1.2e-6,1.2e-6])
% Basic Vibratory Gyroscope function
% 2 DOF
function zprime = newgyro2dof(t,z)
global B fd w mp kx ky cx cy;
zprime = zeros(4,1);
Fd=B*sin(2*pi*fd*t);
zprime(1)= z(3);
zprime(2)= z(4);
zprime(3)= -1/mp*(cx*z(3)+ kx*z(1))+ 2*w*z(4)+ w^2*z(1)+ Fd/mp;
zprime(4)= -1/mp*(cy*z(4)+ ky*z(2))- 2*w*z(3)+ w^2*z(2);
return;
C.2 Matlab Codes for Solving the Four-degree-of-freedom Gyroscope Model
% Simulation of the effects of noise on a gyrosocpe
% 4 DOF
clc;
clear all;
close all;
format compact;
115
format short g;
diary off;
global B C fd w mp mf fn kx ky kz cx cy cz;
% mass 'mp' of proof mass
mp = 1e-8;
% mass 'mf' of frame
mf = 3.8e-4;
%'fn' frequency of noise
fn=14003;
%'fd' frequency of driving force
fd=14003;
% 'kx' stiffness in x direction
kx=77.38;
% 'ky' stiffness in y direction
ky=77.38;
% 'kz' stiffness between gyro and pcb
kz=200;
% 'cx' damping in y direction
cx=5e-7;
% 'cy' damping in y direction
cy=5e-7;
% 'cz' damping between gyro and pcb
cz=5e-3;
% 'B' input amplitude of drive force signal
B=1e-6;
% 'C' input amplitude of disturbance force
C=1.26e-3;
%'w' angular velocity of gyroscopic frame
w=1;
init = [0,0,0,0,0,0,0,0];
[t,z] = ode45(@newgyro4dof, (0:0.000001:1.2),init);
z1 = z(:,1);
z2 = z(:,2);
z3 = z(:,3);
z4 = z(:,4);
z5 = z(:,5);
z6 = z(:,6);
z7 = z(:,7);
z8 = z(:,8);
z9=B*sin(2*pi*fd*t);
figure(1)
plot(t,z1,'b');
grid on
xlabel ('Time');
ylabel ('Displacement(Xf)');
figure(2)
116
plot(t,z2);
grid on
xlabel ('Time');
ylabel ('Displacement(xp)');
figure(3)
plot(t,z3);
grid on
xlabel ('Time');
ylabel ('Displacement(Yf)');
figure(4)
plot(t,z4);
grid on
xlabel ('Time');
ylabel ('Displacement(Yp)');
figure(5)
Youtput = z4-z3;
plot (t,Youtput);
grid on
xlabel ('Time');
ylabel ('Displacement(Youtpout)');
% Function for simulating noise on gyroscope
% 4 DOF
function zprime = newgyro4dof(t,z)
global B C fd w mp mf fn kx ky kz cx cy cz;
zprime = zeros(8,1);
Fd=B*sin(2*pi*fd*t);
FN=C*sin(2*pi*fn*t);
zprime(1)= z(5);
zprime(2)= z(6);
zprime(3)= z(7);
zprime(4)= z(8);
zprime(5)= -1/mf*(cx*(z(5)-z(6))+ cz*z(5)+ kx*(z(1)-z(2))+ kz*z(1))+
2*w*z(7)+ w^2*z(1)+ FN/mf - Fd/mf;
zprime(6)= -1/mp*(cx*(z(6)-z(5))+ kx*(z(2)-z(1)))+ 2*w*z(8)+ w^2*z(2)+
Fd/mp;
zprime(7)= -1/mf*(cy*(z(7)-z(8))+ cz*z(7)+ky*(z(3)-z(4))+ kz*z(3))-
2*w*z(5)+ w^2*z(3)+ FN/mf;
zprime(8)= -1/mp*(cy*(z(8)-z(7))+ ky*(z(4)-z(3)))- 2*w*z(6)+ w^2*z(4);
return;
117
APPENDIX D
The engineering drawings for the design and manufacture of the text fixtures are
shown in the first section of this appendix. The second section contains a series of plots
for the vibration experiments performed to characterize the nickel microfibrous materials.
Each plot contains five transfer functions based on the source amplitudes used during the
tests. The total number of plots for all the tests performed is 170 but only 40 plots are
shown for brevity. As such there are two plots for each number of layers (1-5) for each of
the four different material types used.
From each plot the natural frequencies and the maximum amplitudes of the
transfer functions were recorded in tabular form. The third section of this appendix
contains the tables for all the experiments performed (that is the 170 plots).
118
D.1 Mechanical Drawings for Fixture Parts
Figure D.1 Mechanical drawing of top fixture.
119
Figure D.2 Mechanical drawing of bottom fixture.
120
Figure D.3 Mechanical drawing of sliding pin.
121
Figure D.4 Mechanical drawing of lower screw.
122
D.2 Plots of Vibration Tests
Figure D.5 Transfer functions for 1 layer (sample 08) of 4 microns media.
Figure D.6 Transfer functions for 1 layer (sample 09) of 4 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 1 L a y e r - T e s t N o . 8 ( 4 D 0 8 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 1 L a y e r - T e s t N o . 9 ( 4 D 0 9 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
123
Figure D.7 Transfer functions for 2 layers (samples 03/04) of 4 microns media.
Figure D.8 Transfer functions for 2 layers (samples 07/08) of 4 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 2 L a y e r s - T e s t N o . 2 ( 4 D 0 3 0 4 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 2 L a y e r s - T e s t N o . 4 ( 4 D 0 7 0 8 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
124
Figure D.9 Transfer functions for 3 layers (samples 10/02/04) of 4 microns media.
Figure D.10 Transfer functions for 3 layers (samples 14/05/09) of 4 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 3 L a y e r s - T e s t N o . 5 ( 4 D 1 0 0 2 0 4 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 3 L a y e r s - T e s t N o . 7 ( 4 D 1 4 0 5 0 9 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
125
Figure D.11 Transfer functions for 4 layers (samples 01/02/04/06) of 4 microns media.
Figure D.12 Transfer functions for 4 layers (samples 07/02/08/14) of 4 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 4 L a y e r s - T e s t N o . 1 ( 4 D 0 1 0 2 0 4 0 6 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 4 L a y e r s - T e s t N o . 4 ( 4 D 0 7 0 2 0 8 1 4 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
126
Figure D.13 Transfer functions for 5 layers (samples 08/06/07/09/10) of 4 microns media.
Figure D.14 Transfer functions for 5 layers (samples 14/09/11/13/15) of 4 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 5 L a y e r s - T e s t N o . 4 ( 4 D 0 8 0 6 0 7 0 9 1 0 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 M i c r o n s - 5 L a y e r s - T e s t N o . 7 ( 4 D 1 4 0 9 1 1 1 3 1 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
127
Figure D.15 Transfer functions for 1 layer (sample 03) of 4/8 microns media.
Figure D.16 Transfer functions for 1 layer (sample 11) of 4/8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 1 L a y e r - T e s t N o . 3 ( 4 8 A 0 3 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 1 L a y e r - T e s t N o . 1 0 ( 4 8 A 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
128
Figure D.17 Transfer functions for 2 layers (samples 01/02) of 4/8 microns media.
Figure D.18 Transfer functions for 2 layers (samples 10/11) of 4/8 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 2 L a y e r s - T e s t N o . 1 ( 4 8 A 0 1 0 2 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 2 L a y e r s - T e s t N o . 5 ( 4 8 A 1 0 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
129
Figure D.19 Transfer functions for 3 layers (samples 09/10/11) of 4/8 microns media.
Figure D.20 Transfer functions for 3 layers (samples 07/03/11) of 4/8 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 3 L a y e r s - T e s t N o . 3 ( 4 8 A 0 9 1 0 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 3 L a y e r s - T e s t N o . 7 ( 4 8 A 0 7 0 3 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
130
Figure D.21 Transfer functions for 4 layers (samples 02/13/09/06) of 4/8 microns media.
Figure D.22 Transfer functions for 4 layers (samples 14/08/03/10) of 4/8 microns media.
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 4 L a y e r s - T e s t N o . 6 ( 4 8 A 0 2 1 3 0 9 0 6 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
1
2
3
4
5
6
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 4 L a y e r s - T e s t N o . 7 ( 4 8 A 1 4 0 8 0 3 1 0 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
131
Figure D.23 Transfer functions for 5 layers (samples 01/02/03/05/06) of 4/8 microns media.
Figure D.24 Transfer functions for 5 layers (samples 07/08/09/10/11) of 4/8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 5 L a y e r s - T e s t N o . 1 ( 4 8 A 0 1 0 2 0 3 0 5 0 6 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
4 / 8 M i c r o n s - 5 L a y e r s - T e s t N o . 2 ( 4 8 A 0 7 0 8 0 9 1 0 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
132
Figure D.25 Transfer functions for 1 layer (sample 10) of 8 microns media.
Figure D.26 Transfer functions for 1 layer (sample 15) of 8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 1 L a y e r - T e s t N o . 9 ( 8 A 1 0 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 1 L a y e r - T e s t N o . 1 4 ( 8 A 1 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
133
Figure D.27 Transfer functions for 2 layers (samples 04/05) of 8 microns media.
Figure D.28 Transfer functions for 2 layers (samples 14/15) of 8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 2 L a y e r s - T e s t N o . 2 ( 8 A 0 4 0 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 2 L a y e r s - T e s t N o . 7 ( 8 A 1 4 1 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
134
Figure D.29 Transfer functions for 3 layers (samples 01/02/04) of 8 microns media.
Figure D.30 Transfer functions for 3 layers (samples 06/07/08) of 8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 3 L a y e r s - T e s t N o . 1 ( 8 A 0 1 0 2 0 4 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 3 L a y e r s - T e s t N o . 7 ( 8 A 0 6 0 7 0 8 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
135
Figure D.31 Transfer functions for 4 layers (samples 12/13/14/15) of 8 microns media.
Figure D.32 Transfer functions for 4 layers (samples 09/11/13/15) of 8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 4 L a y e r s - T e s t N o . 3 ( 8 A 1 2 1 3 1 4 1 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 4 L a y e r s - T e s t N o . 6 ( 8 A 0 9 1 1 1 3 1 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
136
Figure D.33 Transfer functions for 5 layers (samples 02/04/06/08/10) of 8 microns media.
Figure D.34 Transfer functions for 5 layers (samples 12/14/01/05/07) of 8 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 5 L a y e r s - T e s t N o . 4 ( 8 A 0 2 0 4 0 6 0 8 1 0 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
8 M i c r o n s - 5 L a y e r s - T e s t N o . 5 ( 8 A 1 2 1 4 0 1 0 5 0 7 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
137
Figure D.35 Transfer functions for 1 layer (sample 07) of 12 microns media.
Figure D.36 Transfer functions for 1 layer (sample 13) of 12 microns media.
0 50 100 150 200 250
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 1 L a y e r - T e s t N o . 7 ( 1 2 A 0 7 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 1 L a y e r - T e s t N o . 1 3 ( 1 2 A 1 3 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
138
Figure D.37 Transfer functions for 2 layers (samples 02/04) of 12 microns media.
Figure D.38 Transfer functions for 2 layers (samples 05/06) of 12 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 2 L a y e r s - T e s t N o . 1 ( 1 2 A 0 2 0 4 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 2 L a y e r s - T e s t N o . 2 ( 1 2 A 0 5 0 6 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
139
Figure D.39 Transfer functions for 3 layers (samples 01/03/05) of 12 microns media.
Figure D.40 Transfer functions for 3 layers (samples 06/08/09) of 12 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 3 L a y e r s - T e s t N o . 1 ( 1 2 A 0 1 0 3 0 5 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 3 L a y e r s - T e s t N o . 2 ( 1 2 A 0 6 0 8 0 9 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
140
Figure D.41 Transfer functions for 4 layers (samples 07/05/06/08) of 12 microns media.
Figure D.42 Transfer functions for 4 layers (samples 10/11/12/13) of 12 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 4 L a y e r s - T e s t N o . 4 ( 1 2 A 0 7 0 5 0 6 0 8 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 4 L a y e r s - T e s t N o . 5 ( 1 2 A 1 0 1 1 1 2 1 3 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
141
Figure D.43 Transfer functions for 5 layers (samples 06/08/09/10/11) of 12 microns media.
Figure D.44 Transfer functions for 5 layers (samples 13/12/04/05/03) of 12 microns media.
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 5 L a y e r s - T e s t N o . 5 ( 1 2 A 0 6 0 8 0 9 1 0 1 1 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
0 50 100 150 200 250
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
F r e q u e n cy H z
T
r
a
n
sf
e
r
F
u
n
ct
i
o
n
1 2 M i c r o n s - 5 L a y e r s - T e s t N o . 6 ( 1 2 A 1 3 1 2 0 4 0 5 0 3 )
0 . 6 9 4 m / s
2
1 . 1 0 3 m / s
2
1 . 4 6 8 m / s
2
1 . 8 3 1 m / s
2
2 . 2 0 2 m / s
2
142
D.3 Experimental Data for the Natural Frequencies of 4 Microns Media
Microfibrous Sample ? 4 Microns - 1 Layer - (Natural Frequency / Hz)
4D01 4D02 4D03 4D04 4D05 4D06 4D07 4D08
Input
Vibration
Amplitude
(m/s2)
0.694 119 135.5 120.5 126.5 133.5 153 133.5 138.5
1.103 115.5 124.5 117.5 119 128 141 129 128.5
1.468 114 122.5 113 113.5 125.5 137 126.5 125
1.831 113.5 118.5 108.5 107 124.5 134.5 122.5 121.5
2.202 111 117.5 105.5 104 123 132 118.5 117
Table D.1a Natural frequencies of 1 layer of 4 microns media.
Microfibrous Sample ? 4 Microns - 1 Layer - (Natural Frequency / Hz)
4D09 4D10 4D11 4D12 4D13 4D14 4D15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 144 123.5 128.5 134 134 131.5 108 130.90
1.103 141 118 124.5 127 126 123 104.5 124.47
1.468 138.5 114 123 120.5 122 122 101 121.20
1.831 132 108.5 120 117.5 120.5 118.5 96.5 117.60
2.202 128 107 117 114 117.5 116.5 96 114.97
Table D.1b Natural frequencies of 1 layer of 4 microns media.
Microfibrous Sample ? 4 Microns - 2 Layers - (Natural Frequency / Hz)
4D0102 4D0304 4D0506 4D0708 4D0910 4D1112 4D1314 AV
Input
Vibration
Amplitude
(m/s2)
0.694 80.5 84 95 93.5 103 87.5 86.5 90.00
1.103 78.5 80.5 91.5 88 98.5 81.5 84 86.07
1.468 78 77 90.5 82.5 94 79.5 82.5 83.43
1.831 76 74.5 89 80.5 92 77 81.5 81.50
2.202 75.5 73.5 87 78 89 76.5 80 79.93
Table D.2 Natural frequencies of 2 layers of 4 microns media.
143
Microfibrous Sample ? 4 Microns - 3 Layers - (Natural Frequency / Hz)
4D020103 4D040507 4D060911 4D081315 4D100204 4D120810 4D140509 AV
Input
Vibration
Amplitude
(m/s2)
0.694 70.5 74 84 71 70.5 72 72 73.43
1.103 68 73 79 68 68.5 69.5 67.5 70.50
1.468 66 68.5 75.5 63.5 66 65.5 67 67.43
1.831 63.5 64.5 73.5 63 63.5 63.5 65.5 65.29
2.202 63 64 70.5 60.5 62.5 62 65 63.93
Table D.3 Natural frequencies of 3 layers of 4 microns media.
Microfibrous Sample ? 4 Microns - 4 Layers - (Natural Frequency / Hz)
4D01020406 4D03081012 4D05141501 4D07020814 4D09041015 4D11061203 4D13050709 AV
Input
Vibration
Amplitude
(m/s2)
0.694 65.5 64.5 66 64 68 59.5 68 65.07
1.103 64.5 60.5 64.5 61.5 65.5 56.5 66.5 62.79
1.468 64 57.5 62 61 63.5 55 65 61.14
1.831 62.5 56.5 61 57 63 54.5 63 59.64
2.202 60.5 56 58.5 56 61.5 53 61.5 58.14
Table D.4 Natural frequencies of 4 layers of 4 microns media.
Microfibrous Sample ? 4 Microns - 5 Layers - (Natural Frequency / Hz)
4D0203050709 4D0411131501 4D0602030405 4D0806070910 4D1008111213 4D1214150107 4D1409111315 AV
Input
Vibration
Amplitude
(m/s2)
0.694 59.5 58 51 63.5 54.5 58 58 57.50
1.103 58 57 50.5 61.5 53.5 56 57.5 56.29
1.468 57 55 48.5 60.5 53 54.5 57 55.07
1.831 56.5 53 47.5 59.5 50.5 53 55.5 53.64
2.202 55.5 53 46.5 58 50 52 54.5 52.79
Table D.5 Natural frequencies of 5 layers of 4 microns media.
144
D.4 Experimental Data for the Maximum Amplitude of the Transfer Functions of 4 Microns Media
Microfibrous Sample ? 4 Microns - 1 Layer - (Amplitude)
4D01 4D02 4D03 4D04 4D05 4D06 4D07 4D08
Input
Vibration
Amplitude
(m/s2)
0.694 3.0906 3.7144 1.9611 1.6786 2.3325 2.2602 2.1566 1.8801
1.103 4.3385 4.2733 2.4696 2.6624 2.8484 2.5716 2.3079 2.4526
1.468 6.1616 4.6092 3.5353 3.5892 3.8969 3.3723 3.0277 3.4060
1.831 7.4348 4.8140 4.4414 4.2806 4.3210 3.7910 3.6251 4.0203
2.202 7.6475 5.4012 4.7919 5.0536 4.8450 4.3823 4.3592 4.5349
Table D. 6a Maximum amplitude of the transfer functions of 1 layer of 4 microns media.
Microfibrous Sample ? 4 Microns - 1 Layer - (Amplitude)
4D09 4D10 4D11 4D12 4D13 4D14 4D15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.2526 1.9761 3.0345 1.9479 1.9378 2.1205 2.5095 2.3235
1.103 2.5783 3.1093 4.3659 2.5410 2.5373 2.8789 2.9664 2.9934
1.468 3.4371 4.6848 4.4764 3.3434 3.2398 3.4154 4.1609 3.8904
1.831 4.5057 5.1291 5.3373 4.3205 4.1470 3.8866 4.6509 4.5804
2.202 4.9089 5.5017 5.8304 5.0538 4.4711 4.5283 4.9279 5.0825
Table D.6b Maximum amplitude of the transfer functions of 1 layer of 4 microns media.
Microfibrous Sample ? 4 Microns - 2 Layers - (Amplitude)
4D0102 4D0304 4D0506 4D0708 4D0910 4D1112 4D1314 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.0815 2.5941 2.8776 2.2791 2.7100 2.7707 3.4040 2.6739
1.103 3.0537 3.6641 4.2323 2.7880 3.1256 3.8479 4.7485 3.6372
1.468 4.0552 4.7214 4.9805 3.6915 3.5776 4.3763 5.0449 4.3496
1.831 4.3658 5.1260 5.4391 4.4754 4.2920 4.9676 5.4514 4.8739
2.202 5.1141 5.4461 5.7392 4.5309 4.3824 4.7233 5.5318 5.0668
Table D.7 Maximum amplitude of the transfer functions of 2 layers of 4 microns media.
145
Microfibrous Sample ? 4 Microns - 3 Layers - (Amplitude)
4D020103 4D040507 4D060911 4D081315 4D100204 4D120810 4D140509 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.7648 2.1660 3.2985 4.3174 2.8953 1.6257 2.7888 2.8366
1.103 3.7801 3.1807 3.9217 5.0933 3.7945 2.5380 3.6814 3.7128
1.468 4.4609 3.6866 4.6750 5.2577 4.2903 3.0294 4.3715 4.2531
1.831 4.8002 4.2225 4.7711 5.4083 4.6736 3.5481 4.5896 4.5733
2.202 5.2976 4.5409 4.9636 5.6972 4.8398 3.9298 4.8574 4.8752
Table D.8 Maximum amplitude of the transfer functions of 3 layers of 4 microns media.
Microfibrous Sample ? 4 Microns - 4 Layers - (Amplitude)
4D01020406 4D03081012 4D05141501 4D07020814 4D09041015 4D11061203 4D13050709 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.9355 1.7069 4.5592 2.0036 3.4572 3.3105 1.8302 2.8290
1.103 4.0726 3.0210 4.8421 3.2552 4.4678 4.0032 2.8356 3.7854
1.468 4.6526 3.5525 5.3350 3.8075 4.9897 4.4782 3.0747 4.2700
1.831 5.0513 3.9273 5.4117 4.5031 5.2796 4.7799 3.2839 4.6053
2.202 5.4084 4.1802 5.6864 4.9223 5.9215 4.8345 3.8058 4.9656
Table D.9 Maximum amplitude of the transfer functions of 4 layers of 4 microns media.
Microfibrous Sample ? 4 Microns - 5 Layers - (Amplitude)
4D0203050709 4D0411131501 4D0602030405 4D0806070910 4D1008111213 4D1214150107 4D1409111315 AV
Input
Vibration
Amplitude
(m/s2)
0.694 1.9444 3.7801 1.7548 3.1909 2.4954 3.6432 2.6633 2.7817
1.103 2.1140 4.7523 3.0103 4.0512 3.1741 4.4452 3.8501 3.6282
1.468 3.1812 5.2947 3.8618 4.2500 3.6832 4.8266 4.4288 4.2180
1.831 3.7390 5.3362 4.2704 4.6847 4.0178 5.0223 4.5247 4.5136
2.202 4.4816 5.4410 4.6572 5.2049 4.3602 5.0625 4.6173 4.8321
Table D.10 Maximum amplitude of the transfer functions of 5 layers of 4 microns media.
146
D.5 Experimental Data for the Natural Frequencies of 4/8 Microns Media
Microfibrous Sample ? 4/8 Microns - 1 Layer - (Natural Frequency / Hz)
48A01 48A02 48A03 48A05 48A06 48A07 48A08
Input
Vibration
Amplitude
(m/s2)
0.694 144.5 135 141.5 149 142.5 150.5 141.5
1.103 140 131.5 134 142 138.5 145 138.5
1.468 132 128.5 126.5 133 130.5 143 130
1.831 129.5 121 123.5 130.5 126.5 137.5 128.5
2.202 121 116 117.5 126.5 123.5 131 122
Table D.11a Natural frequencies of 1 layer of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 1 Layer - (Natural Frequency / Hz)
48A09 48A10 48A11 48A12 48A13 48A14 48A15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 158 135.5 146.5 155 157.5 140 164 147.21
1.103 146 128.5 142 144.5 145.5 131.5 149.5 139.79
1.468 141 124.5 139 141.5 136.5 127 147.5 134.32
1.831 137.5 119 129.5 131 130.5 124 139 129.11
2.202 134 112 123.5 127 126 119.5 134 123.82
Table D.11b Natural frequencies of 1 layer of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 2 Layers - (Natural Frequency / Hz)
48A0102 48A0305 48A0607 48A0809 48A1011 48A1213 48A1415 AV
Input
Vibration
Amplitude
(m/s2)
0.694 107.5 102 107 113 108 115 104 108.07
1.103 102 95 101 108 103 103.5 97.5 101.43
1.468 98 91.5 98 106 97.5 100 92.5 97.64
1.831 94 91 95.5 97 94.5 97.5 91.5 94.43
2.202 90.5 89 90.5 95 92.5 95 91 91.93
Table D.12 Natural frequencies of 2 layers of 4/8 microns media.
147
Microfibrous Sample ? 4/8 Microns - 3 Layers - (Natural Frequency / Hz)
48A010203 48A050607 48A091011 48A131415 48A150612 48A020913 8A070311 AV
Input
Vibration
Amplitude
(m/s2)
0.694 84 88 87.5 88 85.5 89.5 90.5 87.57
1.103 80 82 86.5 86 83 83 85 83.64
1.468 78 78.5 84 82.5 81 80 82.5 80.93
1.831 74.5 75.5 82 78.5 78 78.5 81.5 78.36
2.202 73 74 79.5 73.5 74 76 76 75.14
Table D.13 Natural frequencies of 3 layers of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 4 Layers - (Natural Frequency / Hz)
48A03050607 48A08091011 48A12131415 48A01020305 48A06071215 48A02130906 48A14080310 AV
Input
Vibration
Amplitude
(m/s2)
0.694 72 78.5 78 77.5 79 71.5 79 76.50
1.103 68.5 73.5 74.5 74 77 67.5 76 73.00
1.468 64.5 72 69 72.5 74 65.5 74.5 70.29
1.831 63 71 66 71.5 72 62.5 73.5 68.50
2.202 63 68.5 65 69 70 61.5 68 66.43
Table D.14 Natural frequencies of 4 layers of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 5 Layers - (Natural Frequency / Hz)
48A0102030506 48A0708091011 48A1213141501 48A1103060912 48A1405071013 48A0502080115 48A0308091013 AV
Input
Vibration
Amplitude
(m/s2)
0.694 71 73 69 75 67.5 70.5 71 71.00
1.103 69.5 71 67.5 73.5 64 67.5 68 68.71
1.468 66 66.5 64.5 65.5 64 65.5 66 65.43
1.831 63 65 62.5 66.5 60 62 62 63.00
2.202 61 61.5 61.5 65 58 58.5 61 60.93
Table D.15 Natural frequencies of 5 layers of 4/8 microns media.
148
D.6 Experimental Data for the Maximum Amplitude of the Transfer Functions of 4/8 Microns Media
Microfibrous Sample ? 4/8 Microns - 1 Layer - (Amplitude)
48A01 48A02 48A03 48A05 48A06 48A07 48A08
Input
Vibration
Amplitude
(m/s2)
0.694 1.6613 3.4781 2.2435 3.1152 2.8738 3.5628 3.5206
1.103 2.3955 4.2372 2.6155 3.9464 4.1916 4.1830 3.9786
1.468 3.1147 4.8284 3.5998 4.4392 4.6276 4.7799 4.4578
1.831 3.7156 5.2507 4.4419 4.7320 5.7542 4.7245 4.8737
2.202 3.8593 5.4351 4.7403 5.0436 5.6432 5.2181 4.8286
Table D.16a Maximum amplitude of the transfer functions of 1 layer of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 1 Layer - (Amplitude)
48A09 48A10 48A11 48A12 48A13 48A14 48A15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.6779 2.9903 2.3377 2.6119 3.4014 2.8155 3.3337 2.9017
1.103 4.3481 3.1136 2.5181 3.5739 3.8092 3.4930 3.4610 3.5618
1.468 4.9753 3.8357 3.2300 4.1377 4.3383 4.0583 3.9381 4.1686
1.831 4.9391 3.9191 3.6537 4.3239 4.4676 4.1574 4.4773 4.5308
2.202 4.9882 3.9600 3.9758 4.4971 4.6768 4.4570 4.5066 4.7021
Table D.16b Maximum amplitude of the transfer functions of 1 layer of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 2 Layers - (Amplitude)
48A0102 48A0305 48A0607 48A0809 48A1011 48A1213 48A1415 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.2453 3.8604 3.6616 3.0982 2.4096 3.6172 3.2725 3.3093
1.103 4.1549 4.6097 4.1393 3.2795 3.2959 4.3511 3.9587 3.9699
1.468 4.6644 4.5466 4.6298 4.0313 4.1377 4.9050 4.4207 4.4765
1.831 5.0011 5.1756 5.1710 4.4038 4.5209 5.3878 4.6146 4.8964
2.202 5.1344 5.2836 5.2302 4.7311 4.8224 5.4290 4.5187 5.0213
Table D.17 Maximum amplitude of the transfer functions of 2 layers of 4/8 microns media.
149
Microfibrous Sample ? 4/8 Microns - 3 Layers - (Amplitude)
48A010203 48A050607 48A091011 48A131415 48A150612 48A020913 48A070311 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.6602 2.9076 4.1094 4.7700 2.2000 4.2550 2.7300 3.3760
1.103 3.4137 3.8229 4.7599 5.1100 3.3035 5.1145 3.6005 4.1607
1.468 3.9472 4.6222 5.2012 5.1700 4.0169 5.2669 4.3534 4.6540
1.831 4.6195 4.7330 5.5483 5.4500 4.3129 5.4008 4.6354 4.9571
2.202 4.7154 5.3048 5.3627 5.4600 4.6653 5.0933 4.8769 5.0683
Table D.18 Maximum amplitude of the transfer functions of 3 layers of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 4 Layers - (Amplitude)
48A03050607 48A08091011 48A12131415 48A01020305 48A06071215 48A02130906 48A14080310 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.5834 2.3592 4.2788 3.5343 3.4349 3.3051 4.1830 3.5255
1.103 4.0868 2.9278 4.5903 4.1240 3.8919 3.9258 4.5892 4.0194
1.468 4.4825 3.0668 5.1826 4.2942 3.8511 4.6616 4.8064 4.3350
1.831 4.6117 3.6929 5.4189 4.7259 4.0847 4.9581 5.3214 4.6877
2.202 4.7357 3.8145 5.3496 5.1054 4.4250 5.0020 5.1387 4.7958
Table D.19 Maximum amplitude of the transfer functions of 4 layers of 4/8 microns media.
Microfibrous Sample ? 4/8 Microns - 5 Layers - (Amplitude)
48A0102030506 48A0708091011 48A1213141501 48A1103060912 48A1405071013 48A0502080115 48A0308091013 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.2265 2.4380 3.8941 3.6943 4.0077 3.3721 3.2358 3.4098
1.103 3.7736 3.3857 4.2278 4.0336 4.5383 4.4692 3.8920 4.0457
1.468 4.2505 3.9116 4.5580 4.3469 4.9806 4.8435 4.4542 4.4779
1.831 4.7230 4.7868 4.8022 4.5705 5.1187 4.9807 5.2069 4.8841
2.202 4.8974 4.9661 4.7773 4.6399 5.5292 5.1828 5.4256 5.0598
Table D.20 Maximum amplitude of the transfer functions of 5 layers of 4/8 microns media.
150
D.7 Experimental Data for the Natural Frequencies of 8 Microns Media
Microfibrous Sample ? 8 Microns - 1 Layer - (Natural Frequency / Hz)
8A01 8A02 8A04 8A05 8A06 8A07 8A08
Input
Vibration
Amplitude
(m/s2)
0.694 160.5 155 163 171.5 143.5 142.5 150.5
1.103 140.5 126.5 153 165.5 135.5 132 141
1.468 133.5 120 146.5 151 124.5 127.5 129.5
1.831 128.5 116 137.5 146 116 122 126
2.202 124 114.5 129 137.5 113 118 122
Table D.21a Natural frequencies of 1 layer of 8 microns media.
Microfibrous Sample ? 8 Microns - 1 Layer - (Natural Frequency / Hz)
8A09 8A10 8A11 8A12 8A13 8A14 8A15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 170 157 155.5 169.5 169 165 167.5 160.00
1.103 154 142.5 143 152 149 153.5 145.5 145.25
1.468 141 134.5 133 141.5 145 147.5 136 136.50
1.831 137.5 132 131 135.5 136.5 138 134 131.18
2.202 127.5 124 124.5 132 129.5 134.5 127 125.50
Table D.21b Natural frequencies of 1 layer of 8 microns media.
Microfibrous Sample ? 8 Microns - 2 Layers - (Natural Frequency / Hz)
8A0102 8A0405 8A0607 8A0809 8A1011 8A1213 8A1415 AV
Input
Vibration
Amplitude
(m/s2)
0.694 103.5 119 126.5 110 110 116 106 113.00
1.103 99 112.5 116 103 101.5 108.5 96.5 105.29
1.468 93.5 109 109 96 99 103.5 92 100.29
1.831 89 105.5 106 94.5 93 97.5 89 96.36
2.202 88 102 101 89 89 93 87 92.71
Table D.22 Natural frequencies of 2 layers of 8 microns media.
151
Microfibrous Sample ? 8 Microns - 3 Layers - (Natural Frequency / Hz)
8A010204 8A050607 8A080910 8A111213 8A141501 8A020405 8A060708 AV
Input
Vibration
Amplitude
(m/s2)
0.694 91.5 96.5 88 94.5 89.5 88.5 90 91.21
1.103 90.5 90.5 81 91 83 85.5 87.5 87.00
1.468 84 86 79.5 86.5 80 83 83.5 83.21
1.831 80 84.5 76 83 77.5 78.5 78.5 79.71
2.202 76 80.5 72.5 78 75 76 76 76.29
Table D.23 Natural frequencies of 3 layers of 8 microns media.
Microfibrous Sample ? 8 Microns - 4 Layers - (Natural Frequency / Hz)
8A04050607 8A08091011 8A12131415 8A01020405 8A06081012 8A09111315 8A14020406 AV
Input
Vibration
Amplitude
(m/s2)
0.694 83.5 87 75.5 76 77.5 78.5 81.5 79.93
1.103 78.5 76.5 73 72.5 75.5 74 77 75.29
1.468 73.5 71.5 70 71 63.5 69 71.5 70.00
1.831 71 69.5 66 67.5 64 67 68 67.57
2.202 68.5 68 63 66 62.5 64.5 67 65.64
Table D.24 Natural frequencies of 4 layers of 8 microns media.
Microfibrous Sample ? 8 Microns - 5 Layers - (Natural Frequency / Hz)
8A0102040506 8A0708091011 8A1213141501 8A0204060810 8A1214010507 8A0507111315 8A0204051011 AV
Input
Vibration
Amplitude
(m/s2)
0.694 73.5 73 74.5 74.5 69.5 71.5 71 72.50
1.103 70.5 66.5 71.5 70 66 65 66.5 68.00
1.468 67 64.5 65.5 67.5 62.5 63 65 65.00
1.831 65 62 64 65 60.5 61.5 62.5 62.93
2.202 63.5 61 62.5 61.5 57.5 59.5 59.5 60.71
Table D.25 Natural frequencies of 5 layers of 8 microns media.
152
D.8 Experimental Data for the Maximum Amplitude of the Transfer Functions of 8 Microns Media
Microfibrous Sample ? 8 Microns - 1 Layer - (Amplitude)
8A01 8A02 8A04 8A05 8A06 8A07 8A08
Input
Vibration
Amplitude
(m/s2)
0.694 1.8168 1.6219 3.3471 3.2278 3.9138 4.2153 3.1339
1.103 2.2542 2.4243 4.0314 3.3063 4.1937 4.4444 4.0060
1.468 3.0421 2.9171 3.9951 3.5789 4.3535 4.7416 4.2988
1.831 3.3491 3.2706 4.1762 3.8211 4.6407 4.6500 4.6000
2.202 3.8528 3.4241 4.2373 3.9089 5.2687 4.6285 4.7512
Table D.26a Maximum amplitude of the transfer functions of 1 layer of 8 microns media.
Microfibrous Sample ? 8 Microns - 1 Layer - (Amplitude)
8A09 8A10 8A11 8A12 8A13 8A14 8A15 AV
Input
Vibration
Amplitude
(m/s2)
0.694 1.9406 2.4912 3.5257 3.0915 2.2652 2.0777 2.0587 2.7662
1.103 3.1367 3.5144 4.0523 3.5985 3.3695 3.1814 2.9999 3.4652
1.468 3.4426 4.0848 4.3206 4.0393 3.9416 3.9266 3.3456 3.8592
1.831 3.6016 4.1717 4.6399 4.0919 4.2404 4.1419 3.3844 4.0557
2.202 3.5798 4.6697 4.6325 4.1879 4.1721 4.4500 3.6842 4.2463
Table D.26b Maximum amplitude of the transfer functions of 1 layer of 8 microns media.
Microfibrous Sample ? 8 Microns - 2 Layers - (Amplitude)
8A0102 8A0405 8A0607 8A0809 8A1011 8A1213 8A1415 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.9575 2.7311 1.8776 3.7443 3.8738 3.9258 2.9248 3.1478
1.103 3.7938 3.4278 2.4706 4.1830 4.6410 5.0249 3.3829 3.8463
1.468 4.4058 3.9415 3.0344 4.3811 4.5737 4.8113 3.7915 4.1342
1.831 4.8427 3.9822 3.5844 4.4137 5.0025 5.0946 4.0975 4.4311
2.202 4.5757 4.5109 3.7535 4.5991 4.7505 5.0733 4.1536 4.4881
Table D.27 Maximum amplitude of the transfer functions of 2 layers of 8 microns media.
153
Microfibrous Sample ? 8 Microns - 3 Layers - (Amplitude)
8A010204 8A050607 8A080910 8A111213 8A141501 8A020405 8A060708 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.1912 2.9269 4.6532 3.7737 3.5800 3.6534 3.3942 3.4532
1.103 2.9501 3.7109 4.8094 3.9606 3.9300 4.3992 3.9002 3.9515
1.468 3.6906 3.9511 4.8496 4.4242 4.1800 4.1128 4.3319 4.2200
1.831 4.2793 4.5487 5.0698 4.8015 4.3400 4.3191 4.7141 4.5818
2.202 4.5501 4.6063 5.1036 5.1478 4.3500 4.6244 4.8655 4.7497
Table D.28 Maximum amplitude of the transfer functions of 3 layers of 8 microns media.
Microfibrous Sample ? 8 Microns - 4 Layers - (Amplitude)
8A04050607 8A08091011 8A12131415 8A01020405 8A06081012 8A09111315 8A14020406 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.7871 3.1168 3.0890 4.1101 3.0679 2.9321 3.6528 3.3937
1.103 4.3980 3.4022 3.6572 4.9104 3.6384 3.5695 4.2914 3.9810
1.468 4.8257 3.7233 3.9397 4.5674 3.8267 4.1864 4.6423 4.2445
1.831 4.6936 3.9159 4.1460 4.7589 4.2299 4.3069 4.9550 4.4295
2.202 4.5746 4.1216 4.1911 5.1976 4.5468 4.5207 4.2780 4.4901
Table D.29 Maximum amplitude of the transfer functions of 4 layers of 8 microns media.
Microfibrous Sample ? 8 Microns - 5 Layers - (Amplitude)
8A0102040506 8A0708091011 8A1213141501 8A0204060810 8A1214010507 8A0507111315 8A0204051011 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.8538 4.0165 2.4793 2.5761 3.7959 2.3961 3.4571 3.2250
1.103 3.9913 4.3262 3.2929 3.1001 4.1278 2.8647 3.8997 3.6575
1.468 4.1220 4.4464 3.8178 3.5607 4.7041 3.3761 4.0842 4.0159
1.831 4.2514 4.6672 4.0882 3.9097 4.7613 2.9225 4.5009 4.1573
2.202 4.2210 4.5710 4.4387 3.9983 4.5597 3.5791 4.3146 4.2403
Table D.30 Maximum amplitude of the transfer functions of 5 layers of 8 microns media.
154
D.9 Experimental Data for the Natural Frequencies of 12 Microns Media
Microfibrous Sample ? 12 Microns-1 Layer- (Natural Frequency / Hz)
12A01 12A02 12A03 12A04 12A05 12A06 12A07
Input
Vibration
Amplitude
(m/s2)
0.694 178.5 159.5 174 163.5 162.5 159 157
1.103 158.5 153 150.5 154 144 148.5 144.5
1.468 155 142.5 140.5 144 136.5 139.5 141
1.831 146.5 136.5 136 135.5 131.5 137 135
2.202 140.5 131 131.5 133 126.5 133.5 132
Table D.31a Natural frequencies of 1 layer of 12 microns media.
Microfibrous Sample ? 12 Microns - 1 Layer- (Natural Frequency / Hz)
12A08 12A09 12A10 12A11 12A12 12A13 AV
Input
Vibration
Amplitude
(m/s2)
0.694 160 175.5 160.5 162 158.5 156 163.58
1.103 144.5 163 146.5 144 148.5 144.5 149.54
1.468 137.5 147.5 138.5 132 139 136 140.73
1.831 133.5 140.5 134 125.5 129 132.5 134.85
2.202 126.5 130.5 126 124 124 126 129.62
Table D.31b Natural frequencies of 1 layer of 12 microns media.
Microfibrous Sample ? 12 Microns - 2 Layers- (Natural Frequency / Hz)
12A0204 12A0506 12A0809 12A1011 12A1213 12A0507 12A0103 AV
Input
Vibration
Amplitude
(m/s2)
0.694 121 117.5 118 119.5 125 125 125.5 121.64
1.103 109.5 108 105.5 114.5 117 113 112.5 111.43
1.468 107.5 104 101 106 113 108.5 109 107.00
1.831 104 96 97.5 103 107.5 103 105 102.29
2.202 101 94.5 93.5 98 105 101 101.5 99.21
Table D.32 Natural frequencies of 2 layers of 12 microns media.
155
Microfibrous Sample ? 12 Microns - 3 Layers- (Natural Frequency / Hz)
12A010305 12A060809 12A101112 12A130204 12A050607 12A091011 12A121302 AV
Input
Vibration
Amplitude
(m/s2)
0.694 98.5 100.5 111 96.5 102 103 98 101.36
1.103 93.5 97 103.5 82 96.5 98.5 91 94.57
1.468 88 90.5 98 79 82 94.5 86 88.29
1.831 85.5 86 92.5 76.5 79.5 90.5 81.5 84.57
2.202 81.5 81.5 91 71 76 87.5 80 81.21
Table D.33 Natural frequencies of 3 layers of 12 microns media.
Microfibrous Sample ? 12 Microns - 4 Layers- (Natural Frequency / Hz)
12A02040506 12A08091011 12A12130103 12A07050608 12A10111213 12A05060712 12A01020308 AV
Input
Vibration
Amplitude
(m/s2)
0.694 82.5 96 81.5 83.5 90 87.5 86 86.71
1.103 74.5 87.5 78 78 84.5 84.5 80 81.00
1.468 68.5 85.5 76 75.5 81 82 75 77.64
1.831 65 80.5 71.5 73.5 77.5 76 74 74.00
2.202 63.5 79.5 70 71 76 74.5 67.5 71.71
Table D.34 Natural frequencies of 4 layers of 12 microns media.
Microfibrous Sample ? 12 Microns - 5 Layers - (Natural Frequency / Hz)
12A0204050608 12A0910111213 12A0103071011 12A0506081213 12A0608091011 12A1312040503 12A0203010704 AV
Input
Vibration
Amplitude
(m/s2)
0.694 74.5 81 82 76 86.5 75 82.5 79.64
1.103 68.5 76 75.5 71.5 83 71 75 74.36
1.468 67 71.5 72.5 67.5 80 67.5 73 71.29
1.831 65.5 69 71 65.5 74 64.5 72.5 68.86
2.202 62 67.5 70 62 71 61.5 69 66.14
Table D.35 Natural frequencies of 5 layers of 12 microns media.
156
D.10 Experimental Data for the Maximum Amplitude of the Transfer Functions of 12 Microns Media
Microfibrous Sample ? 12 Microns - 1 Layer - (Amplitude)
12A01 12A02 12A03 12A04 12A05 12A06 12A07
Input
Vibration
Amplitude
(m/s2)
0.694 2.0495 3.5800 1.9042 3.9558 1.9443 3.9160 2.0292
1.103 2.3504 4.1300 2.4677 3.9896 2.8385 4.6428 2.4926
1.468 2.7667 4.6900 3.1181 4.3923 3.4916 5.4440 2.9100
1.831 3.4957 4.7800 3.5546 4.5018 3.9971 5.3383 3.1437
2.202 3.9384 5.0300 3.9243 4.4972 4.1019 5.3424 3.3277
Table D.36a Maximum amplitude of the transfer functions of 1 layer of 12 microns media.
Microfibrous Sample ? 12 Microns - 1 Layer - (Amplitude)
12A08 12A09 12A10 12A11 12A12 12A13 AV
Input
Vibration
Amplitude
(m/s2)
0.694 3.2300 1.6850 2.8016 2.7820 2.0847 2.1445 2.6236
1.103 3.9290 2.0005 3.3023 3.8014 2.3553 2.8409 3.1647
1.468 4.1230 2.7859 3.9767 4.2652 3.4505 3.2726 3.7451
1.831 4.8763 3.4645 4.2305 4.7040 3.7589 3.5710 4.1090
2.202 4.4379 3.8150 4.2957 4.5952 4.5733 3.9318 4.2931
Table D.36b Maximum amplitude of the transfer functions of 1 layer of 12 microns media.
Microfibrous Sample ? 12 Microns - 2 Layers - (Amplitude)
12A0204 12A0506 12A0809 12A1011 12A1213 12A0507 12A0103 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.4696 2.0014 2.0588 1.7499 3.5661 3.5371 2.8589 2.6060
1.103 3.4051 2.7616 2.5380 2.8302 4.2177 2.7608 3.6513 3.1664
1.468 3.8047 3.4303 3.3585 3.4927 4.4662 3.5798 3.8722 3.7149
1.831 4.5932 3.8478 3.6706 3.7436 5.0211 4.3420 4.1913 4.2014
2.202 3.9369 4.0533 4.1108 4.1881 4.8551 4.3703 4.5110 4.2894
Table D.37 Maximum amplitude of the transfer functions of 2 layers of 12 microns media.
157
Microfibrous Sample ? 12 Microns - 3 Layers - (Amplitude)
12A010305 12A060809 12A101112 12A130204 12A050607 12A091011 12A121302 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.1974 1.6436 3.9253 1.7999 2.1174 3.1394 3.4919 2.6164
1.103 2.9582 2.5480 4.4592 2.9087 2.9242 3.9868 4.6102 3.4850
1.468 3.7821 3.1592 4.6922 3.3311 2.9575 4.2257 4.3424 3.7843
1.831 4.1892 3.6381 4.9063 3.8118 3.5322 4.6353 4.8278 4.2201
2.202 4.3654 3.6886 4.9060 3.8524 3.6567 4.5360 5.2855 4.3272
Table D.38 Maximum amplitude of the transfer functions of 3 layers of 12 microns media.
Microfibrous Sample ? 12 Microns - 4 Layers - (Amplitude)
12A02040506 12A08091011 12A12130103 12A07050608 12A10111213 12A05060712 12A01020308 AV
Input
Vibration
Amplitude
(m/s2)
0.694 2.7904 2.0298 3.4514 2.5435 1.8958 2.4594 3.1469 2.6167
1.103 3.7587 3.8872 4.0485 3.2821 2.8058 3.2303 3.5271 3.5057
1.468 3.8798 3.5610 4.2641 3.7016 3.2963 3.8428 4.5457 3.8702
1.831 4.4711 3.9846 4.5062 3.9430 3.4764 4.3841 4.4863 4.1788
2.202 4.2047 4.0148 4.7552 4.1076 3.5986 4.5082 4.6791 4.2669
Table D.39 Maximum amplitude of the transfer functions of 4 layers of 12 microns media.
Microfibrous Sample ? 12 Microns - 5 Layers - (Amplitude)
12A0204050608 12A0910111213 12A0103071011 12A0506081213 12A0608091011 12A1312040503 12A0203010704 AV
Input
Vibration
Amplitude
(m/s2)
0.694 1.9417 3.1633 2.1100 2.1272 2.1405 3.8677 3.1737 2.6463
1.103 2.6882 4.1413 3.1470 3.1239 3.1925 4.1280 4.3927 3.5448
1.468 3.4718 4.5964 3.4724 3.6476 3.3152 4.4652 4.5538 3.9318
1.831 3.5811 4.9393 4.0801 3.9558 3.6838 4.7228 4.6759 4.2341
2.202 3.9811 4.8485 4.0821 3.8749 3.9487 4.7198 5.0201 4.3536
Table D.40 Maximum amplitude of the transfer functions of 5 layers of 12 microns media.
158
APPENDIX E
E.1 Matlab Codes for Stiffness and Damping Calculations
%% Computation of stiffness and Damping
%4 MICRONS
% Enter mass 'm' kg
m = 35.34e-3;
% Enter Frequencies of Maximum Transmissibility
% Example Wx1: Maximum frequency for 1 layer and with increasing
amplitude
% Fibre size x microns
W41 = [130.90 124.47 121.20 117.60 114.97]*2*pi;
W42 = [90.00 86.07 83.43 81.50 79.93]*2*pi;
W43 = [73.43 70.50 67.43 65.29 63.93]*2*pi;
W44 = [65.07 62.79 61.14 59.64 58.14]*2*pi;
W45 = [57.50 56.29 55.07 53.64 52.79]*2*pi;
% Enter Maximum Amplitudes of Transmissibility
% Example, Ax1 = Maximum Amplitudes for 1 layer and with increasing
% amplitude. Fibre size x microns
A41 = [2.3235 2.9934 3.8904 4.5804 5.0825];
A42 = [2.6739 3.6372 4.3496 4.8739 5.0668];
A43 = [2.8366 3.7128 4.2531 4.5733 4.8752];
A44 = [2.8290 3.7854 4.2700 4.6053 4.9656];
A45 = [2.7817 3.6282 4.2180 4.5136 4.8321];
% Calculating Damping Ratio Zeta
A4 = [A41 A42 A43 A44 A45];
Z4 = zeros;
for r=1:25
A = A4(r);
n=0;
x=7;
while (x-A) >= 0.0001
n = n+1;
z = 0.1 + (n*0.0001);
x = sqrt ((8*z^4)/(8*z^4 - 4*z^2 + sqrt(1+8*z^2)-1));
end
Z4(r) = z;
end
159
Z41 = [Z4(1) Z4(2) Z4(3) Z4(4) Z4(5)];
Z42 = [Z4(6) Z4(7) Z4(8) Z4(9) Z4(10)];
Z43 = [Z4(11) Z4(12) Z4(13) Z4(14) Z4(15)];
Z44 = [Z4(16) Z4(17) Z4(18) Z4(19) Z4(20)];
Z45 = [Z4(21) Z4(22) Z4(23) Z4(24) Z4(25)];
%Calculating Natural frequency Wn
Wn41 = W41./(sqrt((sqrt(1 + 8*Z41.^2)-1)/(4*Z41.^2)));
Wn42 = W42./(sqrt((sqrt(1 + 8*Z42.^2)-1)/(4*Z42.^2)));
Wn43 = W43./(sqrt((sqrt(1 + 8*Z43.^2)-1)/(4*Z43.^2)));
Wn44 = W44./(sqrt((sqrt(1 + 8*Z44.^2)-1)/(4*Z44.^2)));
Wn45 = W45./(sqrt((sqrt(1 + 8*Z45.^2)-1)/(4*Z45.^2)));
%Calculating Stiffness
K41 = (Wn41.^2)*m;
K42 = (Wn42.^2)*m;
K43 = (Wn43.^2)*m;
K44 = (Wn44.^2)*m;
K45 = (Wn45.^2)*m;
%Plotting figures
N = [1 2 3 4 5];
%Rearranging Stiffness with respect to amplitude for plotting
K420 = [K41(1) K42(1) K43(1) K44(1) K45(1)];
K430 = [K41(2) K42(2) K43(2) K44(2) K45(2)];
K440 = [K41(3) K42(3) K43(3) K44(3) K45(3)];
K450 = [K41(4) K42(4) K43(4) K44(4) K45(4)];
K460 = [K41(5) K42(5) K43(5) K44(5) K45(5)];
%Rearranging Damping with respect to amplitude for plotting
Z420 = [Z41(1) Z42(1) Z43(1) Z44(1) Z45(1)];
Z430 = [Z41(2) Z42(2) Z43(2) Z44(2) Z45(2)];
Z440 = [Z41(3) Z42(3) Z43(3) Z44(3) Z45(3)];
Z450 = [Z41(4) Z42(4) Z43(4) Z44(4) Z45(4)];
Z460 = [Z41(5) Z42(5) Z43(5) Z44(5) Z45(5)];
figure(1)
plot (N,K420,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Stiffness K, N/m','FontSize',12);
title('4 Microns - Stiffness against Number of Layers','FontSize',14)
hold on
plot (N,K430,'--^R','LineWidth',2.5);
hold on
plot (N,K440,':sG','LineWidth',2.5);
hold on
plot (N,K450,'-.dB','LineWidth',2.5);
hold on
160
plot (N,K460,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
figure(2)
plot (N,Z420,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Damping Ratio','FontSize',12);
title('4 Microns - Damping Ratio against Number of
Layers','FontSize',14)
hold on
plot (N,Z430,'--^R','LineWidth',2.5);
hold on
plot (N,Z440,':sG','LineWidth',2.5);
hold on
plot (N,Z450,'-.dB','LineWidth',2.5);
hold on
plot (N,Z460,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
Z4 = [Z41;Z42;Z43;Z44;Z45]
%% Computation of stiffness and Damping
%48 MICRONS
% Enter mass 'm' kg
m = 35.34e-3;
% Enter Frequencies of Maximum Transmissibility
% Example Wx1: Maximum frequency for 1 layer and with increasing
amplitude
% Fibre size x microns
W481 = [147.21 139.79 134.32 129.11 123.82]*2*pi;
W482 = [108.07 101.43 97.64 94.43 91.93]*2*pi;
W483 = [87.57 83.64 80.93 78.36 75.14]*2*pi;
W484 = [76.50 73.00 70.29 68.50 66.43]*2*pi;
W485 = [71.00 68.71 65.43 63.00 60.93]*2*pi;
% Enter Maximum Amplitudes of Transmissibility
% Example, Ax1 = Maximum Amplitudes for 1 layer and with increasing
% amplitude. Fibre size x microns
A481 = [2.9017 3.5618 4.1686 4.5308 4.7021];
A482 = [3.3093 3.9699 4.4765 4.8964 5.0213];
A483 = [3.3760 4.1607 4.6540 4.9571 5.0683];
A484 = [3.5255 4.0194 4.3350 4.6877 4.7958];
A485 = [3.4098 4.0457 4.4779 4.8841 5.0598];
161
% Calculating Damping Ratio Zeta
A48 = [A481 A482 A483 A484 A485];
Z48 = zeros;
for r=1:25
A = A48(r);
n=0;
x=7;
while (x-A) >= 0.0001
n = n+1;
z = 0.1 + (n*0.0001);
x = sqrt ((8*z^4)/(8*z^4 - 4*z^2 + sqrt(1+8*z^2)-1));
end
Z48(r) = z;
end
Z481 = [Z48(1) Z48(2) Z48(3) Z48(4) Z48(5)];
Z482 = [Z48(6) Z48(7) Z48(8) Z48(9) Z48(10)];
Z483 = [Z48(11) Z48(12) Z48(13) Z48(14) Z48(15)];
Z484 = [Z48(16) Z48(17) Z48(18) Z48(19) Z48(20)];
Z485 = [Z48(21) Z48(22) Z48(23) Z48(24) Z48(25)];
%Calculating Natural frequency Wn
Wn481 = W481./(sqrt((sqrt(1 + 8*Z481.^2)-1)/(4*Z481.^2)));
Wn482 = W482./(sqrt((sqrt(1 + 8*Z482.^2)-1)/(4*Z482.^2)));
Wn483 = W483./(sqrt((sqrt(1 + 8*Z483.^2)-1)/(4*Z483.^2)));
Wn484 = W484./(sqrt((sqrt(1 + 8*Z484.^2)-1)/(4*Z484.^2)));
Wn485 = W485./(sqrt((sqrt(1 + 8*Z485.^2)-1)/(4*Z485.^2)));
%Calculating Stiffness
K481 = (Wn481.^2)*m;
K482 = (Wn482.^2)*m;
K483 = (Wn483.^2)*m;
K484 = (Wn484.^2)*m;
K485 = (Wn485.^2)*m;
%Plotting figures
N = [1 2 3 4 5];
%Rearranging Stiffness with respect to amplitude for plotting
K4820 = [K481(1) K482(1) K483(1) K484(1) K485(1)];
K4830 = [K481(2) K482(2) K483(2) K484(2) K485(2)];
K4840 = [K481(3) K482(3) K483(3) K484(3) K485(3)];
K4850 = [K481(4) K482(4) K483(4) K484(4) K485(4)];
K4860 = [K481(5) K482(5) K483(5) K484(5) K485(5)];
%Rearranging Damping with respect to amplitude for plotting
Z4820 = [Z481(1) Z482(1) Z483(1) Z484(1) Z485(1)];
Z4830 = [Z481(2) Z482(2) Z483(2) Z484(2) Z485(2)];
Z4840 = [Z481(3) Z482(3) Z483(3) Z484(3) Z485(3)];
Z4850 = [Z481(4) Z482(4) Z483(4) Z484(4) Z485(4)];
162
Z4860 = [Z481(5) Z482(5) Z483(5) Z484(5) Z485(5)];
figure(3)
plot (N,K4820,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Stiffness K, N/m','FontSize',12);
title('4/8 Microns - Stiffness against Number of Layers','FontSize',14)
hold on
plot (N,K4830,'--^R','LineWidth',2.5);
hold on
plot (N,K4840,':sG','LineWidth',2.5);
hold on
plot (N,K4850,'-.dB','LineWidth',2.5);
hold on
plot (N,K4860,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
figure(4)
plot (N,Z4820,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Damping Ratio','FontSize',12);
title('4/8 Microns - Damping Ratio against Number of
Layers','FontSize',14)
hold on
plot (N,Z4830,'--^R','LineWidth',2.5);
hold on
plot (N,Z4840,':sG','LineWidth',2.5);
hold on
plot (N,Z4850,'-.dB','LineWidth',2.5);
hold on
plot (N,Z4860,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
Z48 = [Z481;Z482;Z483;Z484;Z485]
%% Computation of stiffness and Damping
%8 MICRONS
% Enter mass 'm' kg
m = 35.34e-3;
163
% Enter Frequencies of Maximum Transmissibility
% Example Wx1: Maximum frequency for 1 layer and with increasing
amplitude
% Fibre size x microns
W81 = [160.00 145.25 136.50 131.18 125.50]*2*pi;
W82 = [113.00 105.29 100.29 96.36 92.71]*2*pi;
W83 = [91.21 87.00 83.21 79.71 76.29]*2*pi;
W84 = [79.93 75.29 70.00 67.57 65.64]*2*pi;
W85 = [72.50 68.00 65.00 62.93 60.71]*2*pi;
% Enter Maximum Amplitudes of Transmissibility
% Example, Ax1 = Maximum Amplitudes for 1 layer and with increasing
% amplitude. Fibre size x microns
A81 = [2.7662 3.4652 3.8592 4.0557 4.2463];
A82 = [3.1478 3.8463 4.1342 4.4311 4.4881];
A83 = [3.4532 3.9515 4.2200 4.5818 4.7497];
A84 = [3.3937 3.9810 4.2445 4.4295 4.4901];
A85 = [3.2250 3.6575 4.0159 4.1573 4.2403];
% Calculating Damping Ratio Zeta
A8 = [A81 A82 A83 A84 A85];
Z8 = zeros;
for r=1:25
A = A8(r);
n=0;
x=7;
while (x-A) >= 0.0001
n = n+1;
z = 0.1 + (n*0.0001);
x = sqrt ((8*z^4)/(8*z^4 - 4*z^2 + sqrt(1+8*z^2)-1));
end
Z8(r) = z;
end
Z81 = [Z8(1) Z8(2) Z8(3) Z8(4) Z8(5)];
Z82 = [Z8(6) Z8(7) Z8(8) Z8(9) Z8(10)];
Z83 = [Z8(11) Z8(12) Z8(13) Z8(14) Z8(15)];
Z84 = [Z8(16) Z8(17) Z8(18) Z8(19) Z8(20)];
Z85 = [Z8(21) Z8(22) Z8(23) Z8(24) Z8(25)];
%Calculating Natural frequency Wn
Wn81 = W81./(sqrt((sqrt(1 + 8*Z81.^2)-1)/(4*Z81.^2)));
Wn82 = W82./(sqrt((sqrt(1 + 8*Z82.^2)-1)/(4*Z82.^2)));
Wn83 = W83./(sqrt((sqrt(1 + 8*Z83.^2)-1)/(4*Z83.^2)));
Wn84 = W84./(sqrt((sqrt(1 + 8*Z84.^2)-1)/(4*Z84.^2)));
Wn85 = W85./(sqrt((sqrt(1 + 8*Z85.^2)-1)/(4*Z85.^2)));
%Calculating Stiffness
K81 = (Wn81.^2)*m;
K82 = (Wn82.^2)*m;
K83 = (Wn83.^2)*m;
K84 = (Wn84.^2)*m;
164
K85 = (Wn85.^2)*m;
%Plotting figures
N = [1 2 3 4 5];
%Rearranging Stiffness with respect to amplitude for plotting
K820 = [K81(1) K82(1) K83(1) K84(1) K85(1)];
K830 = [K81(2) K82(2) K83(2) K84(2) K85(2)];
K840 = [K81(3) K82(3) K83(3) K84(3) K85(3)];
K850 = [K81(4) K82(4) K83(4) K84(4) K85(4)];
K860 = [K81(5) K82(5) K83(5) K84(5) K85(5)];
%Rearranging Damping with respect to amplitude for plotting
Z820 = [Z81(1) Z82(1) Z83(1) Z84(1) Z85(1)];
Z830 = [Z81(2) Z82(2) Z83(2) Z84(2) Z85(2)];
Z840 = [Z81(3) Z82(3) Z83(3) Z84(3) Z85(3)];
Z850 = [Z81(4) Z82(4) Z83(4) Z84(4) Z85(4)];
Z860 = [Z81(5) Z82(5) Z83(5) Z84(5) Z85(5)];
figure(5)
plot (N,K820,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Stiffness K, N/m','FontSize',12);
title('8 Microns - Stiffness against Number of Layers','FontSize',14)
hold on
plot (N,K830,'--^R','LineWidth',2.5);
hold on
plot (N,K840,':sG','LineWidth',2.5);
hold on
plot (N,K850,'-.dB','LineWidth',2.5);
hold on
plot (N,K860,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
figure(6)
plot (N,Z820,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Damping Ratio','FontSize',12);
title('8 Microns - Damping Ratio against Number of
Layers','FontSize',14)
hold on
plot (N,Z830,'--^R','LineWidth',2.5);
hold on
plot (N,Z840,':sG','LineWidth',2.5);
hold on
plot (N,Z850,'-.dB','LineWidth',2.5);
hold on
165
plot (N,Z860,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
Z8 = [Z81;Z82;Z83;Z84;Z85]
%% Computation of stiffness and Damping
%12 MICRONS
% Enter mass 'm' kg
m = 35.34e-3;
% Enter Frequencies of Maximum Transmissibility
% Example Wx1: Maximum frequency for 1 layer and with increasing
amplitude
% Fibre size x microns
W121 = [163.54 149.54 140.73 134.85 129.62]*2*pi;
W122 = [121.64 111.43 107.00 102.29 99.21]*2*pi;
W123 = [101.36 94.57 88.29 84.57 81.21]*2*pi;
W124 = [86.71 81.00 77.64 74.00 71.71]*2*pi;
W125 = [79.64 74.36 71.29 68.86 66.14]*2*pi;
% Enter Maximum Amplitudes of Transmissibility
% Example, Ax1 = Maximum Amplitudes for 1 layer and with increasing
% amplitude. Fibre size x microns
A121 = [2.6236 3.1647 3.7451 4.1090 4.2931];
A122 = [2.6060 3.1664 3.7149 4.2014 4.2894];
A123 = [2.6164 3.4850 3.7843 4.2201 4.3272];
A124 = [2.6167 3.5057 3.8702 4.1788 4.2669];
A125 = [2.6463 3.5448 3.9318 4.2341 4.3536];
% Calculating Damping Ratio Zeta
A12 = [A121 A122 A123 A124 A125];
Z12 = zeros;
for r=1:25
A = A12(r);
n=0;
x=7;
while (x-A) >= 0.0001
n = n+1;
z = 0.1 + (n*0.0001);
x = sqrt ((8*z^4)/(8*z^4 - 4*z^2 + sqrt(1+8*z^2)-1));
end
Z12(r) = z;
end
Z121 = [Z12(1) Z12(2) Z12(3) Z12(4) Z12(5)];
166
Z122 = [Z12(6) Z12(7) Z12(8) Z12(9) Z12(10)];
Z123 = [Z12(11) Z12(12) Z12(13) Z12(14) Z12(15)];
Z124 = [Z12(16) Z12(17) Z12(18) Z12(19) Z12(20)];
Z125 = [Z12(21) Z12(22) Z12(23) Z12(24) Z12(25)];
%Calculating Natural frequency Wn
Wn121 = W121./(sqrt((sqrt(1 + 8*Z121.^2)-1)/(4*Z121.^2)));
Wn122 = W122./(sqrt((sqrt(1 + 8*Z122.^2)-1)/(4*Z122.^2)));
Wn123 = W123./(sqrt((sqrt(1 + 8*Z123.^2)-1)/(4*Z123.^2)));
Wn124 = W124./(sqrt((sqrt(1 + 8*Z124.^2)-1)/(4*Z124.^2)));
Wn125 = W125./(sqrt((sqrt(1 + 8*Z125.^2)-1)/(4*Z125.^2)));
%Calculating Stiffness
K121 = (Wn121.^2)*m;
K122 = (Wn122.^2)*m;
K123 = (Wn123.^2)*m;
K124 = (Wn124.^2)*m;
K125 = (Wn125.^2)*m;
%Plotting figures
N = [1 2 3 4 5];
%Rearranging Stiffness with respect to amplitude for plotting
K1220 = [K121(1) K122(1) K123(1) K124(1) K125(1)];
K1230 = [K121(2) K122(2) K123(2) K124(2) K125(2)];
K1240 = [K121(3) K122(3) K123(3) K124(3) K125(3)];
K1250 = [K121(4) K122(4) K123(4) K124(4) K125(4)];
K1260 = [K121(5) K122(5) K123(5) K124(5) K125(5)];
%Rearranging Damping with respect to amplitude for plotting
Z1220 = [Z121(1) Z122(1) Z123(1) Z124(1) Z125(1)];
Z1230 = [Z121(2) Z122(2) Z123(2) Z124(2) Z125(2)];
Z1240 = [Z121(3) Z122(3) Z123(3) Z124(3) Z125(3)];
Z1250 = [Z121(4) Z122(4) Z123(4) Z124(4) Z125(4)];
Z1260 = [Z121(5) Z122(5) Z123(5) Z124(5) Z125(5)];
figure(7)
plot (N,K1220,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Stiffness K, N/m','FontSize',12);
title('12 Microns - Stiffness against Number of Layers','FontSize',14)
hold on
plot (N,K1230,'--^R','LineWidth',2.5);
hold on
plot (N,K1240,':sG','LineWidth',2.5);
hold on
plot (N,K1250,'-.dB','LineWidth',2.5);
hold on
plot (N,K1260,'-vC','LineWidth',2.5);
hold on
167
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
figure(8)
plot (N,Z1220,'-oK','LineWidth',2.5);
%pbaspect([1.6;1;1])
grid on
set(gca,'XTick',[0 1 2 3 4 5]);
xlabel ('Number of layers','FontSize',12);
ylabel ('Damping Ratio','FontSize',12);
title('12 Microns - Damping Ratio against Number of
Layers','FontSize',14)
hold on
plot (N,Z1230,'--^R','LineWidth',2.5);
hold on
plot (N,Z1240,':sG','LineWidth',2.5);
hold on
plot (N,Z1250,'-.dB','LineWidth',2.5);
hold on
plot (N,Z1260,'-vC','LineWidth',2.5);
hold on
h = legend('0.694 m/s^2','1.103 m/s^2','1.468 m/s^2','1.831
m/s^2','2.202 m/s^2');
Z12 = [Z121;Z122;Z123;Z124;Z125]
168
APPENDIX F
F.1 Acoustical Test Results for Gyroscopes G3-G7
Figure F.1 Experimental results of the 4 microns enclosure on gyroscope G3.
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
T i m e , s
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4 M i cr o n s E n cl o su r e o n G yr o sco p e G 3
N o N o i s e
N o i s e
E n c l o s u r e
169
Figure F.2 Experimental results of the 4/8 microns enclosure on gyroscope G3.
Figure F.3 Experimental results of the 8 microns enclosure on gyroscope G3.
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
T i m e , s
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4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 3
N o N o i s e
N o i s e
E n c l o s u r e
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
T i m e , s
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8 M i cr o n s E n cl o su r e o n G yr o sco p e G 3
N o N o i s e
N o i s e
E n c l o s u r e
170
Figure F.4 Experimental results of the 12 microns enclosure on gyroscope G3.
Figure F.5 Experimental results of the 4 microns enclosure on gyroscope G4.
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
T i m e , s
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1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 3
N o N o i s e
N o i s e
E n c l o s u r e
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
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4 M i cr o n s E n cl o su r e o n G yr o sco p e G 4
N o N o i s e
N o i s e
E n c l o s u r e
171
Figure F.6 Experimental results of the 4/8 microns enclosure on gyroscope G4.
Figure F.7 Experimental results of the 8 microns enclosure on gyroscope G4.
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
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4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 4
N o N o i s e
N o i s e
E n c l o s u r e
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
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8 M i cr o n s E n cl o su r e o n G yr o sco p e G 4
N o N o i s e
N o i s e
E n c l o s u r e
172
Figure F.8 Experimental results of the 12 microns enclosure on gyroscope G4.
Figure F.9 Experimental results of the 4 microns enclosure on gyroscope G5.
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
2 . 8
T i m e , s
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1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 4
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6
2 . 2 5
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
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4 M i cr o n s E n cl o su r e o n G yr o sco p e G 5
N o N o i s e
N o i s e
E n c l o s u r e
173
Figure F.10 Experimental results of the 4/8 microns enclosure on gyroscope G5.
Figure F.11 Experimental results of the 8 microns enclosure on gyroscope G5.
0 1 2 3 4 5 6
2 . 2 5
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
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4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 5
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6
2 . 2 5
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
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N o N o i s e
N o i s e
E n c l o s u r e
174
Figure F.12 Experimental results of the 12 microns enclosure on gyroscope G5.
Figure F.13 Experimental results of the 4 microns enclosure on gyroscope G6.
0 1 2 3 4 5 6
2 . 2 5
2 . 3
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
2 . 7 5
T i m e , s
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1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 5
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6 7 8
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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4 M i cr o n s E n cl o su r e o n G yr o sco p e G 6
N o N o i s e
N o i s e
E n c l o s u r e
175
Figure F.14 Experimental results of the 4/8 microns enclosure on gyroscope G6.
Figure F.15 Experimental results of the 8 microns enclosure on gyroscope G6.
0 1 2 3 4 5 6 7 8
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 6
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6 7 8
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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8 M i cr o n s E n cl o su r e o n G yr o sco p e G 6
N o N o i s e
N o i s e
E n c l o s u r e
176
Figure F.16 Experimental results of the 12 microns enclosure on gyroscope G6.
Figure F.17 Experimental results of the 4 microns enclosure on gyroscope G7.
0 1 2 3 4 5 6 7 8
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 6
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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4 M i cr o n s E n cl o su r e o n G yr o sco p e G 7
N o N o i s e
N o i s e
E n c l o s u r e
177
Figure F.18 Experimental results of the 4/8 microns enclosure on gyroscope G7.
Figure F.19 Experimental results of the 8 microns enclosure on gyroscope G7.
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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4 / 8 M i cr o n s E n cl o su r e o n G yr o sco p e G 7
N o N o i s e
N o i s e
E n c l o s u r e
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
. 5
. 6
2 . 6 5
2 . 7
T i m e , s
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8 M i cr o n s E n cl o su r e o n G yr o sco p e G 7
N o N o i s e
N o i s e
E n c l o s u r e
178
Figure F.20 Experimental results of the 12 microns enclosure on gyroscope G7.
0 1 2 3 4 5 6 7 8
2 . 3 5
2 . 4
2 . 4 5
2 . 5
2 . 5 5
2 . 6
2 . 6 5
2 . 7
T i m e , s
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1 2 M i cr o n s E n cl o su r e o n G yr o sco p e G 7
N o N o i s e
N o i s e
E n c l o s u r e