Thresholds of Axial Vibration Induced Fretting Corrosion in Electrical 
Connectors 
 
by 
 
Haoyue Yang 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
May 5, 2013 
 
 
 
 
Keywords: electrical contacts, fretting corrosion, modeling 
 
Copyright 2013 by Haoyue Yang 
 
 
Approved by 
 
George T. Flowers, Chair, Professor of Mechanical Engineering 
Subhash C. Sinha, Professor of Mechanical Engineering 
Robert L. Jackson, Associate Professor of Mechanical Engineering 
 
ii 
 
 
 
Abstract 
 
      Vibration induced fretting corrosion has been considered an important cause 
of failure in electrical contacts. In previous research, the basic mechanism that 
controls the inchoation and development of such fretting degradation has been well 
understood. However the vibration exciting the motion of connector systems is 
limited in the direction that is perpendicular to the connector and wire lead with it. In 
the present research, random noise and single frequency vibration tests in the axial 
direction of the connector system are used to induce the fretting corrosion in the 
connector contacts. Vibration characteristics and vibration threshold amplitudes of 
the connector systems with various wire lead lengths are studies by the vibration 
tests. 
      The experimental results exhibit the transverse vibration of the connector 
pairs and wire leads with them induced by the axial vibration excitation. Several 
vibration modes were found in the vibration tests. The axial vibration with 
micrometer-scale amplitude is able to trigger the fretting degradation of this kind of 
connector systems. The threshold behavior at various frequencies for the onset of 
fretting corrosion is displayed by the experimental results. Typically the threshold 
amplitudes at natural frequencies are apparently lower than the ones at other 
frequencies. With the increase of lengths of wire leads of the connector systems, 
natural frequency of the major mode decreases, and the threshold amplitudes of the 
 
iii 
 
mode vary as a function with a non-monotonic curve. This threshold behavior is 
explained experimentally and by latter modeling results. 
      A simple mathematical model is developed to relate the threshold amplitude 
to the dynamic properties of connectors and wiring configuration. This 
spring-mass-damper vibration model is built according to the vibration 
characteristics and fretting mechanism of the connector systems that are validated by 
vibration tests. The results of this model are well correlated with the experimental 
results. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
iv 
 
 
 
Acknowledgements 
 
      Sincerely, I hope to express my deep appreciation to my Advisor George T. 
Flowers for his guidance in the completion of my research work in Auburn 
University. His wise instruction and patient support provides me with constant help 
to deal with every academic problem. 
      I also would like to thank my committee members, Dr. Subhash C. Sinha and 
Dr. Robert L. Jackson for their academic instruction and assistance. 
      Special thanks would be given to all my friends, especially Mr. Pregassen 
Subramany, Mr. Yang Xu, Mr. Rujian Fu, which without their constant support, this 
?journey? would have been very difficult to accomplish.   
      Finally, I would like to express my greatest gratitude and gratefulness to my 
parents, Donghai Yang and Jie Wen, whose love, care and perpetual support have 
been invaluable for the completion of this degree (Master of Science). 
 
 
 
 
 
 
 
 
 
 
 
v 
 
 
 
 
Table of Contents 
 
Abstract ........................................................................................................................ ii 
Acknowledgements ..................................................................................................... iv 
List of Tables ............................................................................................................ viii 
List of Figures ............................................................................................................. ix 
Nomenclature ............................................................................................................ xiii 
Chapter 1     Introduction and Literature Review ................................................... 1 
 1.1 Fretting Corrosion ................................................................................................ 1 
 1.2 Mechanism of Fretting Corrosion ........................................................................ 3 
 1.3 Literature Review of Fretting Corrosion in Electrical Contacts .......................... 5 
 1.4 Overview of Current Work  .............................................................................. 19 
Chapter 2     Experimental Study of Threshold of Fretting Corrosion in  
              Electrical Connectors under Axial Vibration ................................. 21 
 2.1 Experimental Configuration............................................................................... 21 
      2.1.1 Experimental Equipment ...................................................................... 21 
        2.1.1.1 Keithley Model 2010 Multimeter .................................................. 22 
        2.1.1.2 Polytec Laser Vibrometer System ................................................. 22 
        2.1.1.3 HP 35665A Dynamic Signal Analyzer .......................................... 24 
        2.1.1.4 Vibration System ........................................................................... 25 
      2.1.2 Experimental Sample ............................................................................ 27 
 
vi 
 
 2.2 A Study on Vibration Threshold Amplitudes at Various Frequencies .............. 29 
      2.2.1 Experimental Setup ............................................................................... 29 
        2.2.1.1 Experimental Samples ................................................................... 29 
        2.2.1.2 Experimental Setup of Transfer Function Measurement ............... 31 
        2.2.1.3 Experimental Setup of Fretting Threshold Measurement .............. 38 
      2.2.2 Experimental Results ............................................................................ 40 
        2.2.2.1 Experimental Results of Transfer Function Measurement ............ 40 
        2.2.2.2 Experimental Results of Fretting Threshold Measurement ........... 42 
      2.2.3 Discussion ............................................................................................. 43 
  2.3 A Study on Vibration Thresholds of Electrical Connectors with Various 
       Wire Lengths ................................................................................................ 44 
      2.3.1 Experimental Setup ............................................................................... 45 
        2.3.1.1 Experimental Samples ................................................................... 45 
        2.3.1.2 Experimental Setup for Vibration and Threshold Tests ................ 46 
      2.3.2 Experimental Results ............................................................................ 50 
      2.3.3 Discussion ............................................................................................. 61 
Chapter 3     Mathematical Modeling and Analysis of a Blade/Receptacle                
              Connector Pair of the Prediction of Axial Vibration- Induced                
              Fretting Corrosion .......................................................................... 63 
 3.1 Model Development........................................................................................... 63 
 3.2 Measurement of Nominal Static Friction Force................................................. 71 
 3.3 Modeling Results ............................................................................................... 74 
Chapter 4     Conclusion and Future Work ............................................................ 87 
 
vii 
 
References .................................................................................................................. 89 
Appendix A   MATLAB Code Example of Mathematical Model .......................... 96 
Appendix B   Testing Fixture Dimensions ............................................................ 102 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
viii 
 
 
 
 
 
List of Tables 
 
Table 2-1: Displacement decoder specification of the laser vibrometer system ....... 23 
Table 2-2: Experimental Threshold Amplitudes of Electrical Connectors with 
          Various Wire Lengths ............................................................................. 61 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ix 
 
 
 
 
List of Figures 
 
Figure 1-1: Idealized Model of Fretting Action (Feng et al. [4])................................. 3 
Figure1-2: Mechanics of Wear- Oxidation Theory (Fontana [2]) ............................... 4 
Figure1-3: Mechanics of Oxidation- Wear Theory (Fontana [2]) ............................... 4 
Figure 1-4: Mechanism of Connector Degradation due to Fretting (Bryant [5]) ........ 5 
Figure 1-5: Typical Fretting Curve in Sine Vibration Test (Xie [31]) ...................... 12 
Figure 1-6: Experimental Setup Used by Flowers et al. [36] .................................... 14 
Figure 1-7: Experimental Setup Used by Xie et al. [37]............................................ 14 
Figure 1-8: Electrical Resistance Transients under Excitation Levels (Xie [31]) ..... 15 
Figure 1-9: Corrosion Spots on Connector Pins Caused by Fretting 
          (Flowers et al, [34])................................................................................ 16 
Figure 2-1 Resistance Measurement (Ketithley Instruments Inc. [43]) ..................... 22 
Figure 2-2: Photograph of the Polytec laser Vibrometer System .............................. 23 
Figure 2-3: Photograph of HP 35665A Dynamic Signal Analyzer ........................... 24 
Figure 2-4: PA 500L Amplifier ................................................................................. 26 
Figure 2-5: V408 Shaker and Fixture Table .............................................................. 26 
Figure 2-6: Typical Vibration System ....................................................................... 27 
Figure 2-7: Photograph of the Blade and Receptacle Parts ....................................... 28 
Figure 2-8: Photograph of the Mated Blade/Receptacle Pair .................................... 28 
Figure 2-9: Photograph of the Receptacle with a Single Wire Lead ......................... 28 
 
x 
 
Figure 2-10: Receptacle with the Strengthened Wire Leads Bundled by Thermal 
           Shrink Tubing ...................................................................................... 30 
Figure 2-11: Photograph of Blade Attached with an Annulus and Washer ............... 31 
Figure 2-12: Experimental Setup of the Transfer Function Measurement ................ 34 
Figure 2-13: Photograph of Fixture for Experiments ................................................ 35 
Figure 2-14: Reflector Positions for Transfer Function Test ..................................... 36 
Figure 2-15: Displacement Measurement System Laser Beams Directions and 
           Positions ............................................................................................... 37 
Figure 2-16: Photograph of the Fretting Threshold Amplitudes Experimental 
           Setup .................................................................................................... 38 
Figure 2-17: Complete Experimental Setup for Threshold Measurement ................. 39 
Figure 2-18: Transfer Function Plot with Input A (Sample with Wire Lead 4 cm in  
           Length) ................................................................................................. 41 
Figure 2-19: Transfer Function Plot with Input B- Output A (Sample with Wire 
           Lead 4 cm in Length) ........................................................................... 42 
Figure 2-20: Averaged Thresholds at Various Frequencies (Sample with Wire  
           Lead 4 cm in Length) ........................................................................... 43 
Figure 2-21: Connector Samples with Strengthened Wire Leads .............................. 46 
Figure 2-22: Reflector Positions on the Blade and Receptacle.................................. 47 
Figure 2-23: Measuring Positions of Laser Beams on the Wire Lead ....................... 48 
Figure 2-24: Fixture Setup for Connectors with Various Wire Lengths ................... 49 
Figure 2-25: Transfer Function of Connector with Wire Lead 2cm in Length ......... 51 
Figure 2-26: Transfer Function of Connector with Wire Lead 3cm in Length ......... 52 
Figure 2-27: Transfer Function of Connector with Wire Lead 5cm in Length ......... 53 
Figure 2-28: Transfer Function of Connector with Wire Lead 6cm in Length ......... 54 
Figure 2-29: Transfer Function of Connector with Wire Lead 8cm in Length ......... 55 
 
xi 
 
Figure 2-30: Natural Frequencies of Connectors with Various Wire Lengths 
           (Mode A- A) ........................................................................................ 56 
Figure 2-31: Transfer Functions with Outputs on Blade and Receptacle at Different  
           Height (Wire Length = 2cm) ............................................................... 57 
Figure 2-32: Estimation of Vibration of Connector Pair for Connectors with 
           Short Wires .......................................................................................... 58 
Figure 2-33: Vibration Postures of Connectors with Various Wire Lengths ............. 59 
Figure 2-34: Vibration Threshold Amplitudes of Electrical Connectors with Various  
           Wire Lengths ........................................................................................ 60 
Figure 3-1: Schematic of Model ................................................................................ 64 
Figure 3-2: Coordinates Used for Calculation ........................................................... 66 
Figure 3-3: Free Body Diagram of Transfer Function Fitting ................................... 67 
Figure 3-4: Free Body Diagram for Threshold Calculation....................................... 70 
Figure 3-5: Photograph of Small Scale Tensile Machine .......................................... 72 
Figure 3-6: Setup of Connector Pair for Tensile Test ................................................ 73 
Figure 3-7: Tensile Force Change of Friction Force Measurement ........................... 74 
Figure 3-8: Transient of   ......................................................................................... 75 
Figure 3-9: Transient of    ......................................................................................... 76 
Figure 3-10: Transient of    ....................................................................................... 76 
Figure 3-11: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 2cm) ........................................................................... 77 
Figure 3-12: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 3cm) ........................................................................... 77 
Figure 3-13: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 4cm) ........................................................................... 78 
Figure 3-14: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 5cm) ........................................................................... 78 
 
xii 
 
Figure 3-15: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 6cm) ........................................................................... 79 
Figure 3-16: Comparison of Experimental and Modeling Transfer Functions 
           (Wire Length = 8cm) ........................................................................... 79 
Figure 3-17: Transient of Friction Force at Contacting Interface .............................. 80 
Figure 3-18: Maximal Absolute Value of Friction Force at Various Frequencies  
            under Vibration with Threshold Amplitude (Wire Length = 2cm).... 81 
Figure 3-19: Maximal Absolute Value of Friction Force at Various Frequencies  
            under Vibration with Threshold Amplitude (Wire Length = 3cm).... 81 
Figure 3-20: Maximal Absolute Value of Friction Force at Various Frequencies               
            under Vibration with Threshold Amplitude (Wire Length = 4cm).... 82 
Figure 3-21: Maximal Absolute Value of Friction Force at Various Frequencies              
            under Vibration with Threshold Amplitude (Wire Length = 5cm).... 82 
Figure 3-22: Maximal Absolute Value of Friction Force at Various Frequencies              
            under Vibration with Threshold Amplitude (Wire Length = 6cm).... 83 
Figure 3-23: Maximal Absolute Value of Friction Force at Various Frequencies              
            under Vibration with Threshold Amplitude (Wire Length = 8cm).... 83 
Figure 3-24: Comparison of Experimental and Modeling Thresholds at Mode A- A              
            for Connectors with Various Wire Lengths ....................................... 84 
Figure 3-25: Total Stiffness of Connectors with Various Wire Lengths ................... 85 
Figure 3-26: Total Stiffness of Connectors with Various Wire Lengths ................... 86 
 
 
 
 
 
 
 
 
 
xiii 
 
 
 
 
 
Nomenclature 
 
    input vibration amplitude (?m) (page 69), a point in modeling 
    acceleration of shaker head  (    ) 
     acceleration of center of mass of receptacle (    ) 
AO  simplified connector 
AG?  simplified blade 
     horizontal damping (     ) 
     vertical damping (     ) 
     horizontal force of spring and damper  (N) 
     vertical force of spring and damper  (N) 
Ff   friction force (N) 
     normal force at contacting interface (N) 
     x axis component of support force of shaker head  
     y axis component of support force of shaker head  
G   center of mass of connector 
G'   center of mass of receptacle 
     gravity (    ) 
I     moment of inertia of connector about its root 
k    stiffness (N/m) 
     ngular stiffness (N/rad) 
 
xiv 
 
     horizontal stiffness (N/m) 
     vertical stiffness (N/m) 
     length of AO (m) 
     length of OG (m) 
      length of OG? (m) 
M    magnitude of Transfer Function (non-dimensional) 
      moment of angular spring (   ) 
     moment about point O (   ) 
m    mass (kg) (Page 63), mass of connector  (kg) 
m?    mass of receptacle (kg) 
OG   simplified receptacle 
      initial position of point A in x axis (m) 
      position of point A in x axis (m) 
      position of point G in x axis (m) 
      position of point G? in x direction (m) 
      initial position of point A in x axis (m) 
      position of point A in y axis (m) 
      position of point G in y axis (m) 
      position of point G? in y direction (m) 
      position of point O in y direction (m) 
      relative motion (mm) 
Greek Letters: 
 
xv 
 
      angular acceleration of AO about point O (        
      initial angle between AO and y axis (rad) 
?     initial angle between OG and y axis (rad) 
?     intial angle between OG? and y axis (rad) 
      angle of vibration (rad) 
      phase angle (rad) 
      angluar velocity of natural frequency         
      angular velocity         
Superscript: 
         index in the x direction 
         index in the y direction 
        index in the x? direction 
         index in the y? direction 
 
 
 
 
 
 
 
 
 
 
 
1 
 
 
 
CHAPTER 1   INTRODUCTION AND LITERATURE REVIEW 
      Fretting corrosion is widely considered a serious cause of failures of 
electrical contact. In recent decades, many researchers have worked and contributed 
to understanding causes, mechanisms and precaution of the fretting degradation. As 
the fundamental factor of such corrosion, the chemical reaction between contact 
surface material and the corrosive environments around it, like oxygen and chlorine, 
has been deeply researched and well understood. From the mechanical engineering 
prospective, vibration, thermal and tribology research has been focused on the small 
amplitude relative motion of contact surfaces, the other vital cause of fretting 
corrosion. 
 
1.1 Fretting Corrosion 
      Fretting Corrosion is a phenomenon that happens due to mechanical stresses 
and chemical reactions; primarily because of chemical reactions (Waterhouse [1]). 
Fretting corrosion occurs at the interface of contacting metal surfaces that are subject 
to relative slip. The slipping is usually oscillatory motion, with small amplitudes, 
which can be resulted from mechanical vibration, changing temperatures, differential 
thermal expansion of contacting metals, load relaxation and junction heating as the 
power is switched on and off. When slipping happens at the interface, oxidation 
usually takes place as well. 
 
2 
 
      The after effect of the fretting corrosion is characterized by discoloration of 
the metal surfaces by the formation of pits and grooves along with various corrosion 
products surrounding them (Fontana [2]). ?Reddish brown debris? is found if the 
materials of one or both of the contacting surfaces contain iron (Williams [3]). With 
the rapid chemical conversion from metal to oxide, fretting corrosion is detrimental to 
dimensional accuracy and satisfactory tolerance. The corrosion products could also 
cause seizing or clogging for moving parts.  
      In multiple fields, the fretting corrosion has been causing serious problems. It 
usually leads to failures of suspension springs, bolt and rivet heads, king pins in auto 
steering mechanisms, jewel bearings, variable- pitch propellers, shrink fits, contacts of 
electrical relays, connecting rods, and other parts of vibrating machinery. Multiple 
bridges have collapsed caused by fretting at essential parts of the bridge design. As 
one reason of the substitute of artificial hips every nine years, fretting corrosion 
loosens femoral stems. One of the earliest documented examples of this corrosion was 
about automobiles shipped by railroad from Detroit to the West Coast. Because ball- 
bearing races of wheels were severely pitted by fretting, the automobiles were not 
operable on arrival. Based on its significance and influences, fretting corrosion is one 
of the topics under research at the Corrosion Technology Laboratory at the NASA 
Kennedy Space Center.  
      Multiple environmental factors have been researched by laboratory 
experiments. Moist air causes less corrosion damages compared to dry air. Nitrogen 
decreases the rate of progression of fretting corrosion. Parameters, including 
 
3 
 
amplitudes of slippage, the number of cycle slippage and normal load that keeps 
contacting surfaces compressed, are directly proportional to the rate of occurrence of 
fretting corrosion (Waterhouse [1]). 
 
1.2 Mechanism of Fretting Corrosion 
      From the prospective of tribology, any flat surface is full of micro-scale 
asperities. Contact of two surfaces compressed together only occurs at the asperities. 
During relative motion of the surfaces, asperities rub clean tracks on the opposite 
surface. The track may oxidize superficially. By the repeated slip, the oxides are 
wiped off by the asperities. This slip activates a repeated conversion from metal to 
oxides, which is wiped off in turn, forming another fresh metal track. As shown in 
Figure 1-1, the oxide particles moving relatively to the metal surface fret the 
contacting surface after a run-in period [4]. Thus electrical resistance between the 
surfaces increases from a low value and remains high.  
 
Figure 1-1: Idealized Model of Fretting Action (Feng et al. [4]) 
      Oxidation usually takes place at interfaces of metallic surfaces that relatively 
 
4 
 
slip. Two mechanisms called wear- oxidation and oxidation- wear theories are used to 
explain how oxidation occurs. As shown in Figure 1-2 which illustrates the wear- 
oxidation theory, the oxidation of particles is caused by the asperities that are broken 
by frictional forces. The oxidation- wear theory, shown in Figure 1-3, is the opposite 
of wear-oxidation theory. It is believed that the wear- oxidation theory takes place first 
with the alternate occurrence of these theories after that (Fontana [2]).  
 
Figure 1-2: Mechanics of Wear- Oxidation Theory (Fontana [2]) 
 
Figure 1-3: Mechanics of Oxidation- Wear Theory (Fontana [2]) 
      As shown in Figure 1-4, Bryant introduced a mechanism of fretting corrosion 
on electrical connectors [5]. During the process of fretting, the initial rubbing of 
 
5 
 
asperities is ignored. Fretting motions expose the asperity contacts to corrosive attack 
and films are formed. The reversal fretting motions shears off corrosive layers and/or 
drag layers into contact. Further motion exposes contact to repeated corrosive attack. 
After a while, fretting motions induce plastic deformation of asperities, breaking the 
corrosive layers between the contacting asperities and mix this corroded material with 
the asperity metal. The asperities become contaminated with corrosion product and 
electrical resistance increases. Finally, after this series of events, the connector fails. 
 
Figure 1-4: Mechanism of Connector Degradation due to Fretting (Bryant [5]) 
 
1.3 Literature Review of Fretting Corrosion in Electrical Contacts 
      Fretting corrosion is a significant problem affecting performance of various 
electrical and electronic equipments that contains expensive components. The 
detrimental influence of fretting to the electrical connections was not widely 
recognized until some failures of electrical connectors and Bock and Whitley?s 
 
6 
 
subsequent work [6] in 1974. In their work, commonly used non- noble contact 
materials experienced large increases of electrical resistance during fretting actions, 
which occurs on electrical contacts as a result of vibration, mechanical motion or 
differential thermal expansion. Lubricants were proved to be able to exclude oxygen 
and other oxidizing gases from the contacts, thus preventing oxidation of the wear. 
After these findings, a lot of research was focused on the influences of different 
material properties and coatings of the contact surfaces. Some research was focused 
on the thermal, vibrational, tribologcical factors affecting the fretting corrosion [7-9]. 
Furthermore, mathematical models and FEA models have been widely built to predict 
the onset and progression of the fretting corrosion [10-14]. 
      In 1984, a comprehensive review of fretting corrosion is offered [15]. In the 
study, the mechanism of phenomenon of this corrosion was discussed, such as transfer, 
wear, oxidation, and frictional forces that influence the fretting; the influence of the 
cycle rate and force on electrical resistance change of contact surfaces; an analysis of 
fretting behaviors of different materials; and a survey of lubricants that control 
fretting based on contact materials. The determination factors of connector designs, 
the expected connector lifetime and the reliability requirement were investigated. 
Shock and vibration tests of hardware were recommended to be conducted for new 
material incorporation and contact design of electrical connectors. By fretting 
corrosion, the contact resistance ordinarily becomes more unstable with longer time, 
and resistance transients are considerably higher than static contact resistance. 
Lubricants that can reduce the oxidation rate by retarding the wear rate of thin noble 
 
7 
 
metals on film- forming substrates are able to stabilize the contact resistance. By the 
category of fretting process based on regimes [16], stick regime happens for motions 
of approximate 1 ?m according to the contacting material, geometry and other factors. 
In this regime, relative motion between contact surfaces is accommodated by elastic 
deformation of the parts near contacting regions. Not until the parts are separated, the 
asperities are joined and there is no surface damage. Mixed stick- slip regime occurs 
with central stick areas surrounded by ring slip regions at which crack formation, 
fretting fatigue and wear debris may take place. In gross slip regime with 10 to 100 
?m movement of contact surfaces, all asperity contacts are broken and asperities rub 
others of the opposing surface during each cycle. Fretting corrosion occurs in 
reciprocating sliding regime with large displacements of surface movement more than 
100 to 200 ?m. In research of Hannel et al. [17], the relative motion is divided into 
partial slip and gross slip conditions. The stabilized partial slip sliding condition 
allows direct metal/metal interactions operating where inner stick domain maintains 
low and stable contact resistance. Under the stabilized gross slip condition, wear leads 
to debris that oxidizes, and generates an insulating oxide layer. Stable metal/metal 
interactions are no longer possible and the electrical contact resistance becomes high 
and unstable, controlled by random oxide debris movement through the interface. 
During gross slip process, the coating is necessary to display noble properties to 
prevent the formation of oxide debris and a high wear resistance. 
      As one of the most critical fields affected by fretting corrosion, vehicle 
electronics are in the environment combing mechanical vibration, harmful 
 
8 
 
temperature and humidity. In 1987, Lee and Mamrick [18] reported a study about the 
fretting corrosion of tin- plated copper alloy used for stepping motor/precision stage 
assembly. The objective of this study is to explain the physical phenomenon of 
electrical conduction by tin- plated contacts subjected to fretting corrosion and the 
effect of electric load on the process of fretting corrosion. The rise of contact 
resistance of tin- plated copper alloy was studied under minute cyclical motion. By 
the tests over the range of circuit voltage and current, electrical load did not affect the 
electrical conduction of slightly corroded contacts. With the heat from further 
corrosion, the temperature rose more and resulted in the melting, sublimation and 
decomposition of oxides and finally vaporizing tin. The complexity of fretting 
corrosion in vehicle environment was reported by Swingler et al. in 2000 [19]. In their 
study, the temperature of automotive connectors was being monitored during real 
operation to evaluate stress levels. According to real environmental data gained from 
tests, an empirical model of temperature behaviors of various electrical connectors 
was presented. For improving this study, some micro-sensors were embedded in a 
connector housing, which provided real information about phenomena taking place at 
the connector interface for the first time. 
      In recent years, a lot of research has been focused on the performance of 
different materials as coatings of electrical connectors to retard the fretting corrosion. 
Song et al. indicated that there is a strong correlation between the rate of wear and 
lifetime of electrical contacts [20]. In the research of Bouzera et al. [21], the 
amplitude of transition where fretting begins was measured by a method of 
 
9 
 
incrementing amplitude at the micro- displacement scale test. Different types of 
materials are considered in this research. It was found that hard materials like nickel 
needs higher motion amplitude than softer material like tin to enable fretting 
corrosion whereas coating material in connector needs higher amplitude than pure 
metal without coating. The lifetime of electrical contact of noble materials is 
unlimited. For non-noble material, the lifetime is limited, and the contact resistance 
sharply increased at amplitude as low as a few micrometers. Fouvery et al. explained 
the reason and mechanism that noble coatings like gold and silver last longer than 
non- noble material like tin [22]. It?s concluded that electrical contact endurance 
under established gross slip conditions is controlled by the kinetics of oxide debris 
layer formation and by the evolution of its composition of the debris. For non-noble 
coatings, like tin, the electrical conductivity decays as soon as the debris layer fully 
separates the interface, resulted from the high current resistivity of tin oxides. And 
for noble materials, electrical failure is controlled by the progressive elimination of 
the noble element from the oxide debris layer. When the content of noble atoms in 
the lay has been low, the conductivity of the interface decreases and the electrical 
contact fails. BNi- Au contact pair was proved to be able to replace Au-Au contact 
pair only under limited conditions of short fretting lifetime with several thousand 
cycles [23]. And it?s better for BNi to be used as probe than as coupon, where 
contact pair displays rather low and stable fretting resistance within during 
thousands of cycles, longer than BNi(coupon)- Au(probe). Park et al. [24-27] proved 
that higher current load was useful to break oxide film and establish a good electrical 
 
10 
 
contact. Tin plated copper was suggested not to be used in over 85?  as the tin 
would soften and cause an increase in the thickness of oxidation film. Neol et al. did 
the first study that aimed at relating the electro-deposition process with the 
microstructure of the layers [28], characterizing the properties of different layers for 
a given application. Bright nickel was compared to matte nickel in a fretting test that 
simulated the micro displacements resulted from vibrations. A drastic increase of the 
electrical resistance was observed with bright nickel. This behavior is linked to the 
microstructure of the coating that depends on the mechanisms of texture 
development in nickel electro-deposits.  
      For predicting contact resistance behavior of electrical connectors under 
fretting corrosion, mathematical models are necessary but difficult to be developed. In 
the early 1990s, two previously developed models, which dealt with contact resistance 
and oxide built- up under fretting corrosion, were combined into an analytical model 
by Malucci [29-30]. This model is able to predict the average influences of contact 
forces and fretting amplitudes on contact degradation. By modeling results, the results 
of fretting tests on tin plated contacts were explained. By this study, the fretting 
degradation is proved to be increased by increasing fretting amplitude or decreasing 
contact force. This phenomenon is attributed to either more asperity deformation by 
larger contact force or an increased number of asperity deformations per cycle under 
longer fretting amplitudes, which are able to increase the buildup of oxides. In 
addition, the relationship between the temperature swing and acceleration factor was 
studied by the analysis of the data of thermal shock experiments. By these results and 
 
11 
 
analysis, a basis of modeling of fretting corrosion based on fretting cycles and 
temperature swing is provided. To sum up, according to the necessary experimental 
parameters, the field degradation is able to be estimated by this model which 
simulates the acceleration factor of fretting corrosion. 
      A mathematical model, combing the tribology properties of contact surfaces 
and the chemical reaction of such corrosion, was also put forward by Bryant [5] in 
1994. This comprehensive model is able to predict the contact resistance during 
specific fretting cycles and the lifetime of the ultimate failure of contacts. In this 
model, contact wipe, fretting vibration amplitude and frequency, contaminant 
chemistry, material properties, plating thickness, asperity deformations, normal load, 
electrical load, surface topography were taken into consideration. The fretting 
mechanism proposed by him is the basis of the model. Both the filling of surface 
valleys with debris produced by fretting and contamination of surface asperities by 
the corrosion products are considered to be the causes of fretting corrosion. Based on 
the mechanism, the model calculated the amount of corrosive products generated on 
the exposed surfaces during cycles of fretting. Assumptions which correlate mixing 
to plastic flow and modern composite theory are used to estimate the conductivity in 
the contaminated asperity. The asperity resistance that produced by the integration 
over the asperity volume and the total resistance were predicted by Greenwood?s 
theory.  
      Bryant?s theory and model is able to explain the general trend of the 
progression of electrical resistance, as shown in Figure 1-5 [31]. This figure reveals 
 
12 
 
transients of the resistance of electrical connectors under the vibration test with 
single frequency and single amplitude. During cycles of relative motion of 
contacting surfaces, the electrical resistance increased slowly first, and then rose 
tremendously after a critical fretting cycle. Based on Bryant?s model and mechanism 
of fretting corrosion, the phenomenon can be generally divided into three stages. At 
the first stage, the contact micro structures- asperities and valleys - only have the 
elastic deformation. Secondly, the fretting motion begins and wear debris is 
generated. The escape of debris from the metal to metal contacts was facilitated by 
fretting actions. So the low contact resistance is maintained. At the third stage, 
insulating layers containing oxides and wear debris accumulate. Metallic contacts are 
lost and contact resistance fluctuates. Eventually, contact resistance becomes high 
and the connector fails.  
 
Figure1-5: Typical Fretting Curve in Sine Vibration Test (Xie [31]) 
 
      In recent years, many new applications of the studies of fretting corrosion in 
 
13 
 
electrical contacts are extended by computer simulations. This technology has been 
being used to assist and guide the experiments and analyses of many companies and 
researchers. In 1996, Villeneuve et al. [32] at Ford Motor Company used finite 
element method to model the terminal crimping process of vehicle connectors. In his 
model, the terminal grip cross section, the punch tooling and wire strands were 
involved. For simulating the real application, the grip was forced into the punch as it 
sits on an anvil. In terms of results, the friction between the grip surface and the 
punch surface was found to be crucial to a ?good? crimp. In 2005, Monnier et al. [33] 
applied FEM for simulating the performance of a sphere-plane electrical contact as a 
high current flowed through it. In this simulation, the mechanical, electrical and 
thermal coupling was involved. The validity of the model was confirmed by the high 
consistency of the simulation and experimental results. The contact terminal voltage, 
the contact resistance of the system and the solid temperature, which are not able to 
be measured in experiments, were provided by the simulation. 
      Compared to the research on multiple fields, relatively few researchers are 
focused on the vibration threshold in electrical connectors. From 2002?  Flowers et 
al. [34], [35], [36] researched on the relationship between vibration excitation levels 
and fretting rates under single frequency vibration tests. In their works, the flexible 
cables connected with metal connectors was taken into account in vibration tests, as 
shown in Figure 1-6, and figure 1-7.  
 
14 
 
 
Figure 1-6: Experimental Setup Used by Flowers et al. [36] 
 
Figure 1-7: Experimental Setup Used by Xie et al. [37] 
      By the effect of the cables, relative motions between contact surfaces of 
connector pair are avoided under the vibration with small amplitudes. The threshold 
levels of excitation for fretting corrosion, above which the fretting corrosion occurs, 
were revealed by the results of the study. With the increase of excitation levels above 
the threshold, fretting rates rise monotonically. This phenomenon is illustrated by 
Figure 1-8. With the vibration excitation, smaller than 5 Grms, the electrical 
 
15 
 
resistance did not rise. With excitation levels that are larger than 6 Grms, the rate of 
the increase of the resistance rises monotonically. The threshold g-level should be 
between 5 to 6 Grms, and the accuracy of the measurement depends on the period of 
the test run- in. The researchers also found that the threshold g-level varies as a 
function of the dynamic behavior of the connector, the tie-off configurations and the 
mass and stiffness properties.  
 
Figure 1-8: Electrical Resistance Transients under Excitation Levels (Xie [31]) 
      In this study, the direction of exciting vibration, which is perpendicular to 
cable and connector pair, results in the rocking motion between contact pair. The 
primary mode of connector interface motion (rocking- type motion), the relative 
motion at the contacting location, served as an indicator of the fretting rate. The 
rocking motion leads one side of connector pin to be exposed to air, and to isolate 
another side from the air. As shown in Figure 1-9, the fretting corrosion mostly 
happened on the side of the pin that contacts air. It also indicated that the moment 
applied as the result of the given excitation level and frequency was reasonably 
 
16 
 
estimated by a transfer matrix model. An empirical fit of the data was in a good 
correlation with the model as the damping was considered. The analysis 
demonstrated that the bending moment caused by the excitation levels and tie- off 
configurations at the contact interface was significant to analyze and predict the 
threshold level. The researchers concluded that the dynamic behavior of the 
mechanical system under different excitation levels and tie- off configuration 
influences the performance of the connector system under vibration stresses. 
 
(a) 
 
(b) 
Figure 1-9: Corrosion Spots on Connector Pins Caused by Fretting (Flowers et al. [34]) 
 
17 
 
      Chen et al. [38-39] continued these research topics in FEA field. They have 
made two-dimensional and three dimensional FEA models to predict the vibration 
characteristics and thresholds of rocking- motion style vibration of connector systems. 
With the threshold levels measured experimentally, approximate amplitudes of 
relative motion at contact interface were obtained by the FEM model. In addition, the 
calculation cost of the two- dimensional model, with the comparative calculation 
accuracy, has been proven to be much better than the three- dimensional model. 
Ibrahim et al. [11] studied the influence of thermal factors on the fretting in electrical 
connectors. The results exhibited the non effect of temperature on the threshold and 
early stage performance. Gao et al. [40] researched on fretting corrosion on electrical 
connectors with contaminants between contact interfaces. It was proved that the 
failure of fretting degradation occurs much more easily than with the clean surfaces. 
      Another approach to measure the relative motion inducing fretting corrosion 
was reported by Lam et al. in 2005 [41]. In the study, a novel thick film sensor 
measuring the displacement at the connector interface was used to test for 
environmental effects on fretting corrosion in electrical contacts. This sensor was 
assembled into a connector sample to replace the male component. While the relative 
motion at the interface happened, the relative displacement of the contact point 
resulted in a corresponding electrical resistance change, which was measured by the 
connection of male and female parts. After the validation by a series of experiments, 
the sensors were subsequently used in a field test to research on the relationships 
between the fretting influences and temperature, humidity and differential pressure. 
 
18 
 
The researchers found that the differential pressure dominates the relative movement 
behavior at the interface of connector samples in condition that the influences of 
temperature and humidity on measured relative motion are negligible.  
      Fu et al. investigated the fretting mechanisms of silver-plated high power 
connectors induced by vibrations [42]. His results indicated that the variation of 
electrical contact resistance (ECR) of connectors subjected to vibration is primarily 
due to periodic changes of contact area caused by relative motion between the contact 
pairs, rather than other fretting corrosions. This finding was reinforced by observing a 
good correlation between relative motion and the variance of ECR under vibration. 
When a vibration stopped, the ECR decreased to a value that was slightly higher than 
the original value. A surface analysis showed no apparent degradation until the 
coating was worn away. An FEA model to predict the instant electrical resistance 
change during vibration was developed in his research. 
      In 2008, Angadi et al. [7] proposed a multi-physics coupled simulation with 
the mechanical, thermal and electrical interactions predicting contact forces, 
electrical contact resistance (ECR) and thermal contact resistance (TCR) in the rough 
surface contact regions. This model was developed in terms of multi- scale 
sinusoidal rough surface (MSRS) contact model using a MATLAB code that 
communicates with ANSYS. In the bulk connector material, it provided predictions 
of the stresses, strains, Joule heat effects, current flow, electric potential and the 
temperature distributions between the spring and pin parts of the connector. Finally 
the results illustrated the significant proportional rise in voltage drop and 
 
19 
 
temperature across the bulk regions of connector parts. 
 
1.4 Overview of Current Work 
      This thesis discusses an experimental study and a mathematical model which 
are focused on axial vibration thresholds of fretting corrosion of electrical 
connectors. 
      The thesis has been divided into four chapters. Chapter one is the literature 
review and introduction. It demonstrated the concept and mechanism of fretting 
corrosion. The research on fretting corrosion in electrical connectors in recent years 
has been also introduced in this chapter. 
      In chapter two, two experiments that utilized the same apparatus are 
described. In the first experiment, the transfer function test and the threshold test 
were conducted on one type of electrical connectors. The experimental results are 
analyzed to study the relationship between the single frequency thresholds for 
fretting corrosion and the excitation frequency. In the second experiment, transfer 
function and threshold tests were conducted on sample connectors with various wire 
lead lengths. Vibration characteristics of the connector systems and influence of the 
tie-off configuration to thresholds of fretting corrosion are analyzed based on the 
testing results. 
      Chapter three describes a discrete mass-spring-damper vibration model. This 
model was programmed and run in MATLAB. In this model, the beginning of the 
relative motion at the interface of connector pair was assumed as the indicator of 
 
20 
 
fretting corrosion. A couple of experiments were conducted to measure several input 
dynamic properties of connectors, including stiffness and static friction force at the 
contact interface. 
      Chapter four provides a summary and conclusion of this work. Suggested 
future work is also discussed. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
21 
 
 
 
CHAPTER 2   EXPERIMENTAL STUDY OF THRESHOLD OF 
FRETTING CORROSION IN ELECTRICAL CONNECTORS UNDER 
AXIAL VIBRATION 
2.1 Experimental Configuration 
      In this chapter, two experiments are performed to study the vibration and 
threshold behaviors and how tie-off configurations affect threshold of fretting 
corrosion in electrical connectors under axial vibration. For studying vibration 
characteristics and threshold behavior of the connector samples, a vibration system 
and electrical resistance measuring equipment are used for both experiments.  
 
2.1.1 Experimental Equipment 
      The experimental equipment used in this study consist of a Keithley Model 
2010 multimeter that is applied for the tests of the values of connector resistance; 
Polytec laser vibrometers systems measuring the displacements of the selected 
locations on connector samples; an HP 35665A dynamic signal analyzer, which 
generates different types of excitation signals (such as sine wave, random noise, etc) 
for the shaker and analyzes the frequency response; a vibration system that consists of 
a Dactron vibration control system, a PA500L Amplifier and a V408 Shaker. The 
details of these major components are mentioned below. 
 
 
22 
 
2.1.1.1 Keithley Model 2010 Multimeter 
      The Model 2010 is a     -digit high-performance digital multimeter. It has 
0.0018% 90-day basic DC voltage accuracy and 0.0032% 90-day basic resistance 
accuracy [43]. During tests, the multimeter was working in a four-wire mode with a 
resistance measurement range of 1u? to 120M?. Furthermore, the dry circuit method 
is indicated in Figure 2-1. It is an efficient type of low resistance measurement 
method which is made on contact devices such as switches and relay contacts. The 
purpose of this method is to determine whether fretting corrosion increases the 
resistance of the contacts. If the voltage across the contacts during the test is too high, 
the oxidation film might be punctured and render the test meaningless. Therefore dry 
circuit testing limits the voltage across the DUT to 20mV or less. 
 
Figure 2-1: Resistance Measurement (Keithley Instruments Inc. [43]) 
 
2.1.1.2 Polytec Laser Vibrometer System 
      As shown in Figure 2-2, the Polytec laser vibrometer system is used in the 
 
23 
 
experiment for the non-contact displacement measurements [44]. The system consists 
of two Polytec OFV 353 sensor heads (lasers), and two Manfrotto tripods to hold the 
lasers and two Polytec OFV 2610 controllers. The system provides the methods for a 
displacement measurement based on the fringe counter principle. An analog voltage 
signal is available at its output that is proportional to the vibration amplitude of the 
measurement object. This signal can be visualized with an oscilloscope or can be 
entered into a data acquisition and processing system, such as the HP 35665A 
dynamic signal analyzer. Table 2-1 shows its displacement decoder specifications. 
 
Figure 2-2: Photograph of the Polytec Laser Vibrometer System 
Table 2-1: Displacement Decoder Specification of the Laser Vibrometer System 
Measure
ment Range 
Full 
Scale 
Output(Peak 
to Peak) 
Resol
ution 
Max 
Vibration 
Frequency 
Ma
x. 
Velocity 
Max. 
Acceleration 
?m/V Mm ?m kHz m/s G 
20 0.32 0.08 20 1.6 20,000 
80 1.3 0.32 20 1.6 20,000 
320 5.2 1.3 20 1.6 20,000 
1280 20.5 5 20 1.6 20,000 
5120 82 20 20 1.6 20,000 
 
 
24 
 
2.1.1.3 HP 35665A Dynamic Signal Analyzer 
      As shown in Figure 2-3, HP 35665A dynamic signal analyzer is a two-channel 
fast Fourier transform (FFT) spectrum/network analyzer with a frequency range that 
extends from 0.19531 Hz to 102.4 kHz in the single channel mode and from 0.097656 
to 51.2 kHz in the two channel mode [45]. The analyzer has a built-in signal source 
providing random noise, burst random noise, periodic chirp, pink noise, and fixed sine. 
Measurements can be saved to an internal 3.5-inch flexible disk drive, or an external 
HP SS-80 disk drive, or can be directly printed out. The main characters of this 
analyzer applied in this study are listed as below: 
? Input Noise Level: < -140 db 
? Full Span FFT Noise Floor: < -76 db (-85 db typical) 
? FFT Cross-Channel Gain Accuracy: ? 0.04 db (0.46%) 
? FFT Cross-Channel Phase Accuracy: ? 0.5 degree 
? Minimum Frequency Resolution: 122 ?Hz (Two Channel Mode) 
 
Figure 2-3: Photograph of HP 35665A Dynamic Signal Analyzer 
 
25 
 
 
2.1.1.4 Vibration System 
      The vibration system utilized in this research comprises a PA500L Amplifier 
and a V408 Shaker, as shown in Figure 2-4 and Figure 2-5 respectively. The shaker 
provides the vertical vibration and a set of fixtures are designed and attached on the 
shaker for the experimental setup. 
      The configuration of the vibration control system is shown in Figure 2-6. The 
vibration system generates time-domain signals containing the required frequency 
domain characters or profile and sends them to the amplifier. Then the powerful 
amplified signal drives the shaker to vibrate with a larger current and voltage. The 
accelerometer installed on the payload senses the vibration on the shaker and converts 
it into an electrical signal that is then sent back to the vibration control system. The 
time-domain signal is processed by the vibration control system and is transferred into 
the frequency domain by FFT. And then the frequency-domain characteristics of this 
signal are compared with the original setting by this control system. If differences 
exist, the output signal will be revised by the vibration control system accordingly. 
This process repeats until the frequency-domain characteristics of the signal on the 
shaker match the required characteristics specified by the original setting. 
 
26 
 
 
Figure 2-4: PA 500L Amplifier 
 
 
Figure 2-5: V408 Shaker and Fixture Table 
 
27 
 
 
Figure 2-6: Typical Vibration System 
 
2.1.2 Experimental Samples 
      As shown in Figure 2-7, a type of single blade/receptacle connector pair is 
applied in this research. The sample fundamentally comprises two components: the 
blade part on the left and the receptacle part on the right. Inside the receptacle, there is 
a spring part connected to the top of the inner layer of the receptacle. A gap exists 
between the end of this spring and the inner side of the bottom of part. Some degree 
of normal force should be produced between the blade and receptacle to keep the 
stable contact between the two parts, thus the gap is designed shorter than thickness of 
blade. As shown in Figure 2-8, when the blade is inserted into the receptacle, the 
spring bends and the contact force leads to friction force forming the contact pair.  
 
28 
 
 
Figure 2-7: Photograph of the Blade and Receptacle Pair 
 
Figure 2-8: Photograph of the Mated Blade/Receptacle Pair 
 
Figure 2-9: Photograph of the Receptacle with a Single Wire Lead 
      As shown in Figure 2-9, a single wire lead is connected with the receptacle. 
Inside the rubber cover, a copper wiring bundle is connected with the metal part of 
 
29 
 
receptacle to transmit electronic power. For a high quality of the measurement, the 
samples are processed at several positions. The processes on connector samples and 
detailed experimental setup for two experiments are demonstrated below.  
 
2.2 A Study on Vibration Threshold Amplitudes at Various Frequencies  
      In the author?s work, single frequency vibration tests are used to measure the 
thresholds of fretting corrosion of electrical connectors. During each test, the 
connector system is forced to vibrate at the frequency of that of the vibration 
excitation. At various frequencies, different threshold amplitudes are possibly 
necessary to trigger fretting corrosion of the electrical connectors. Vibration transfer 
functions are measured as the significant vibration characteristics of the connector 
system. Comparison between the thresholds and transfer functions is conducted to 
study the relation between the threshold behavior and frequency response. 
 
2.2.1 Experimental Setup 
2.2.1.1 Experimental Samples 
      In the initial trials of vibration tests, increase of electrical resistance induced 
by fretting corrosion hardly happened. Based on the observation and analysis of 
processes and results of the experiment, it is attributed to the plastic deformation of 
wire leads connected with receptacles, which is resulted from the axial vibration and 
causes the blade to get stuck in receptacle. Thus it is necessary to increase the 
stiffness of the single wire lead to avoid the plastic deformation. As shown in Figure 
 
30 
 
2-10, two more wire leads are bundled with the original one and they are tied up by 
thermal shrink tubing. In this experiment, the connector samples with 4 cm cable are 
used for transfer function and threshold tests. The lengths of additional wire leads 
and the thermal shrink tubing are 4 cm.  
 
(a) 
 
(b) 
 
(c) 
Figure 2-10: Receptacle with the Strengthened Wire Lead Bundled by Thermal 
             Shrink Tubing  
 
31 
 
For high- quality laser signal used to measure the vibration behaviors of the 
connector system, reflectors made by laser sensitive material are necessary to be 
attached on the measurement positions. In this study, reflectors are fixed on the blade 
and receptacle to obtain the relative motion between blade and receptacle. To 
measure the axial vibration amplitudes of the blade, an annulus and a washer with 
light weights, which provide with the space for attachment of reflectors, are fixed on 
it by super glue, as shown in Figure 2-11.  
 
Figure 2-11: Photograph of Blade Attached with an Annulus and Washer 
 
2.2.1.2 Experimental Setup of Transfer Function Measurement 
      It?s commonly agreed that relative motion between contacting interfaces is 
one of the fundamental causes of fretting corrosion. For the case of electrical 
 
32 
 
connectors, the fretting degradation resulted from micro- scale relative motions 
between the blade and receptacle leads to the rise of the electrical resistance. In 
previous study, Flowers et al. [12-16] have described the measurement methods and 
testing procedures for determining the transfer functions and fretting threshold. The 
transfer function for each sample is determined by comparing the output vibration 
response at the connector interface to the input excitation at the vibratory shaker 
head or the connector part fixed on the fixture on the shaker head. The ratio of the 
steady-state output amplitude to the input amplitude at a certain excitation frequency 
is the magnitude of the transfer function at that frequency. The phase difference 
between the steady-state output and input responses for a given frequency is the 
phase of the transfer function at that frequency. 
      From a dynamic modeling perspective, a transfer function is a functional 
relationship, in the frequency domain, between the magnitude (M) and phase (?) of an 
input motion and the corresponding output motion. For example, a transfer function 
magnitude of 1 and zero phase shift indicates that there is no relative motion between 
the respective input and output measurement points. Similarly, a transfer function 
with a magnitude of 2 and a 45?phase indicates that the output motion is twice as 
large as that of the input and that it lags the input response by 45?. For most dynamic 
systems, there is a peak magnitude for input excitations at one or more of the resonant 
frequencies. 
      Typically, the phase shift is small for low frequency excitations and increases 
dramatically as the driving frequency approaches and exceeds the primary resonance 
 
33 
 
value. Another way of thinking of the transfer function is in terms of a complex 
number where the real part is M cos (?) and the imaginary part is M sin (?). Therefore, 
the transfer function can be expressed by the following equation: 
                                            ?   
Based on the above transfer function, the relative motion between the two-half 
interfaces is defined by: 
                              
where    is the relative motion, and M cos (?) and M sin (?) are the real part and 
imaginary part of the transfer functions, respectively. 
      As shown in Figure 2-12, the vibration behavior of the connector system is 
measured by the basic setup of the transfer function measurement for the 
blade/receptacle connector pair. By emitting and receiving laser rays, laser heads and 
the controllers process and send the data of dynamic displacements to the analyzer. 
The analyzer processes the data to show them on a screen. When testing by a single 
laser head, vibration displacement of one measurement position is tested. By the 
cooperation of two laser heads, the dynamic displacements of two positions are tested, 
and the transfer function of them are calculated by the analyzer. Though the laser 
heads are fixed by tripods horizontally, the laser rays can be reflected into different 
directions with the mirrors attached in front of them.  
 
 
34 
 
 
Figure 2-12: Experimental Setup of the Transfer Function Measurement 
      As shown in Figure 2-13, a set of fixture is fabricated to fix the electrical 
connectors for the axial vibration tests. The connector pair is assembled vertically. As 
source of the vibration excitation, a shaker head is set to vibrate vertically. A blade 
that is mated with the receptacle is fixed on a fixture on the shaker head. Without the 
connection of wire leads with the fixture, the metal part of the blade is clamped on the 
fixture. The end of the wire lead connected with the receptacle is fixed to a movable 
fixture. The movable fixture can be fixed at different heights in a groove of a stand 
which is stationary on a testing table. As the height of the movable fixture changes, 
different wire lengths can influence the vibration characteristics of the connector 
system.  
 
35 
 
 
Figure 2-13: Photograph of Fixture for Experiments 
      The fretting corrosion of electrical connectors is considered to be triggered by 
micro-scale relative motions between the contact pair, the contact interfaces of blade 
and receptacle. Thus, the output and input of vibration positions for the transfer 
function tests are focused on the blade and receptacle. The axial amplitude of the 
blade is measured by the attached reflector as the vibration input. As the results of the 
initial trials of vibration tests, the transverse vibration of connector systems is 
observed. The reflectors are taped on the blade and receptacle to measure the 
transverse vibration as another input and outputs. The positions of reflectors are 
 
36 
 
shown in Figure 2-14. 
 
Figure 2-14: Reflector Positions for Transfer Function Test 
      After the process of wires by the thermal shrink tubing, the shape of cross 
section of flexible wire turns to be similar to rectangular, which may bring about 
various vibration modes in different vibration directions. Therefore the output 
reflectors are attached on two sides of receptacle to test the vibration behaviors of 
the connector system. As shown in Figure 2-14, the reflectors taped on the blade and 
receptacle respectively are set to measure the vibration in different directions as 
Input A and B and Output A and B. The directions of the laser beams emitted from 
the laser heads are shown in Figure 2-15 (a) and (b). 
 
37 
 
 
(a) 
 
(b) 
Figure 2-15: Displacement Measurement System Laser Beam Directions and 
             Positions  
 
38 
 
2.2.1.3 Experimental Setup of Fretting Threshold Measurement 
      The setup of the fretting threshold level measurement is shown in Figure 2-16. 
For measuring the electrical resistance, the most significant indicator of fretting 
corrosion in electrical connectors, Keithly Model 2010 Multimeter connects the wires 
of connector systems. Free wire leads, which are not under vibration that are shown in 
Figure 2-13, are used to connect with the measurement equipment.  
 
 
Figure 2-16: Photograph of the Fretting Threshold Amplitudes Experimental Setup 
 
39 
 
      The electrical resistance is measured during vibration by the same 
experimental setup of vibration test. Sinusoidal signals are generated from the 
vibration control system, and the real- time resistance change is then observed and 
recorded by a computer system with a Labview program. The complete experimental 
setup of the threshold test is shown in Figure 2-17. 
 
Figure 2-17: Complete Experimental Setup for Threshold Measurement 
      In this threshold test, single frequency vibration is used to excite electrical 
connectors with wire leads 4cm in length at various frequencies. The transients of 
electrical resistances are inspected under incremental excitation amplitudes. The 
lowest vibration amplitude that induces the fretting corrosion is recorded as the 
threshold. At each frequency, repeated measurements are conducted to obtain an 
average result. The range of frequency is limited between 100 HZ to 550 HZ 
 
40 
 
according to the power limitation of the vibration system. Generally there is just one 
frequency measured about per 50 HZ. But if the thresholds at two adjacent 
frequencies are close, one more frequency between them is chosen to be measured 
again. In each test, vibration amplitude increments from zero by amplified power of 
signal of the analyzer. With the each vibration amplitude, the test is run for at least 
five minutes for observing the transients of electrical resistance. If no obvious 
increase of the resistance is found, the vibration amplitude is increased in a small 
scale. By the initial trials, the fretting corrosion can be triggered by micro-meter 
scale amplitudes. Thus, the increase scale should be lower than 0.1?m, which is 
limited by the accuracy of the vibration system.  
      By initial trials, it is found that apparent corrosion spots appear on the contact 
surfaces of the connector pair if the fretting corrosion happens no matter how long 
the vibration takes. A fresh connector sample is used for each threshold test. If failed 
samples are used for the test again, the vibration amplitude that is much lower than 
the threshold can lead to the obvious fluctuation and increase of electrical resistance.  
 
2.2.2 Experimental Results 
2.2.2.1 Experimental Results of Transfer Function Measurement 
      Figure 2-18 shows two transfer functions measured by Input A-Output A, 
and Input A-Output B shown in Figure 2-14. Two natural frequencies of the 
connector system are found around 310 HZ and 450 HZ. In Figure 2-18 (a), two 
 
41 
 
peaks of magnitude exist and there are two 180-degree phase shift appear at the 
corresponding frequencies in Figure 2-18 (b). Figure 2-19 shows another transfer 
function measured by Input B-Output A, which indicates the natural frequency of 
relative rocking motion between receptacle and blade. 
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-18: Transfer Function Plot with Input A (Sample with Wire Lead 4 cm 
             in Length) 
 
42 
 
 
 
Figure 2-19: Transfer Function Plot with Input B- Output A (Sample with Wire lead  
            4cm in Length) 
 
2.2.2.2 Experimental Results of Fretting Threshold Measurement 
      As shown in Figure 2-20, threshold amplitudes of axial vibration on electrical 
connectors are different at various frequencies. The threshold decreases 
tremendously from 100 HZ to 300HZ, where the amplitude is lowest. From 300 HZ 
to about 350 HZ, the threshold increases, and the second valley appears at about 450 
HZ. Then threshold amplitude increases largely at frequencies higher than about 470 
HZ. The two lowest threshold amplitudes appear around two natural frequencies as 
shown in Figure 2-18. As the frequency ?gets away? from the natural frequencies, the 
threshold amplitude increases largely. The threshold amplitude at the natural 
frequency of the mode measured by Input A- Output A (Mode A-A) is the lowest. 
 
43 
 
For this type of connector sample, the mode of relative rocking motion between 
blade and receptacle does not influence the threshold behavior obviously.  
 
Figure 2-20: Averaged Thresholds at Various Frequencies (Sample with Wire 
              Lead 4cm in Length) 
 
2.2.3 Discussion 
      Based on the results of the transfer function test, three natural frequencies are 
found. The reason that the two modes (as shown in Figure 2-18) exist should be 
linked to the cross section shape of the strengthened cable. After the strengthening 
process by thermal shrink tubing and supplementary cables, the cross-section of the 
cable turns to be close to rectangular from circle. The structural change induces the 
 
44 
 
variation of the moments of inertia of the cable, stiffness and damping characteristics 
of the transverse vibration in two directions. By the influence of these factors, the 
natural frequency of the second mode is higher than that of the first mode. 
      As shown in Figure 2-20, the threshold amplitudes gets lower as the vibration 
frequency is closer to the natural frequencies as shown in Figure 2-18. At the natural 
frequency of Mode A-A, the threshold amplitude is lowest. In other words, the 
relative motion of the contact pair is easiest to occur at this frequency. In contrast, 
the relative rocking motion between blade and receptacle did not contribute major 
effect to the threshold behavior. 
      A large number of electrical connector samples are involved to obtain the 
average results in the transfer function and threshold tests. The natural frequencies 
and threshold amplitudes of the repeated trials fluctuate in a small range. The 
experimental results shown above are all averaged values. 
 
2.3 A Study on Vibration Thresholds of Electrical Connectors with Various 
    Wire Lengths  
      In last experiment, random noise and single frequency vibration tests were 
used to measure transfer functions and thresholds of fretting corrosion of electrical 
connectors with wire leads 4cm in lengths. By testing the vibration in two directions, 
the natural frequencies of three vibration modes of the connector systems were found. 
Compared to the thresholds at other frequencies, the threshold amplitude at Mode 
A-A is lowest.  
 
45 
 
      With the increase of the lengths of wire leads of electrical connectors, the 
natural frequencies of the Mode A-A possibly change and threshold amplitudes at 
this natural frequencies are also possibly different. Thus, in this part, transfer 
functions and thresholds of electrical connectors with multiple lengths of wire leads 
are tested. The influence of wire lengths on vibration behaviors and threshold 
amplitudes of this type of electrical connector is studied. 
 
2.3.1 Experimental Setup 
      In this experiment, transfer function measurement and threshold test are 
conducted by random noise and single frequency vibration. The vibration system and 
threshold measurement equipment used in last test are also applied for the tests in 
this experiment. The operation of these equipment is almost as the same as that of 
the last research.  
 
2.3.1.1 Experimental Samples 
      Since the axial vibration induces plastic deformation on the wire leads, which 
causes the blade to get stuck in receptacle, it is necessary to strengthen wire leads 
connected with electrical connectors. By the same method mentioned in last 
experiment, the originally single wire lead connected with the receptacle is bundled 
with two more wire leads by thermal shrink tubing. The electrical connector samples 
involved in the test are with multiple lengths of wire leads. The lengths of 
supplementary wires and shrink tubing are equal to the lengths of the original cables. 
The processed samples with various lengths of wire leads are shown in Figure 2-21. 
 
46 
 
In the tests, lengths of wire leads connected with the receptacles are chosen from 2 
cm to 18 cm. 
 
Figure 2-21: Connector Samples with Strengthened Wire Leads 
      Through the same process described in last experiment, an annulus and a 
washer are attached on the blade. The reflector, used for reflecting the laser ray back 
to laser head, is taped on the annulus. The weights and shapes of them are small 
enough to have nearly no influence on the vibration behavior of the blade. 
 
2.3.1.2 Experimental Setup for Vibration and Threshold Tests  
      Aiming to obtain the better testing accuracy by the non-contact displacement 
 
47 
 
measurements, the reflectors, made by laser- sensitive material, are attached on the 
measured objects to reflect the laser rays. Based on the experimental results of last 
experiment, the threshold amplitudes of single frequency vibration at different 
frequencies are different. For this type of electrical connector systems with wire 
leads 4cm in length, threshold at the natural frequency of Mode A-A is lower than 
the threshold amplitudes at other frequencies. Thus natural frequencies of Mode A-A 
of connector systems with multiple lengths of wires are tested in this part.  
 
Figure 2-22: Reflector Positions on the Blade and Receptacle 
      As shown in Figure 2-22, the reflector measuring vibration input is taped on 
annulus that is attached on the blade. Because of the axial excitation of the shaker 
head, the input reflector measures the axial vibration of the blade (Input A). The 
output reflector A is taped on one side of receptacle to measure the transverse 
 
48 
 
response of the connector system. Based on experimental results of last experiment, 
the Mode A-A is tested in this transfer function test. For a comprehensive 
understanding of the vibration behavior, output reflector B (as shown in Figure 2-14) 
is attached to inspect the relative rocking motion of the connector pair. For the 
connector samples with different lengths of wire leads, the structures of connector 
pairs are the same. Blades are fixed on the fixture at the same position. The output 
reflectors are taped at the same positions of receptacles. 
 
Figure 2-23: Measuring Positions of Laser Beams on the Wire Lead 
  
 
49 
 
 
(a) Wire Length = 6cm 
 
(b) Wire Length = 18cm 
Figure 2-24: Fixture Setup for Connectors with Various Wire Lengths  
      In addition of the transfer function measurement, the detailed vibration 
 
50 
 
behavior of the connector system is studied by measurement of the transverse 
vibration of the whole sample body. As shown in Figure 2-23, reflectors are attached 
on receptacles and wire leads to measure the vibration posture of the Mode A-A.  
      The electrical connectors with wire lengths of 2cm, 3cm, 4 cm, 5 cm, 6 cm, 8 
cm, 10 cm, 12 cm, 14 cm, 16 cm and 18 cm are tested. The fixture shown in Figure 
2-13 is able to fix the connector samples with different wire lengths by the height 
changes of the movable fixture. The fixture setup for different samples is shown in 
Figure 2-24. 
      In the threshold test, the threshold amplitude at the natural frequency of the 
Mode A-A of the connector system is measured. The transfer function of each 
connector sample is not the same. Thus, the threshold test of the connector sample is 
conducted after the transfer function measurement of it. In the threshold test, the 
vibration amplitude increases until the electrical resistance rises continuously. The 
resistance increase under one certain vibration amplitude is observed for at least five 
minutes. Natural frequencies and threshold amplitudes are averaged after all testing 
trials. 
 
2.3.2 Experimental Results 
      The transfer function measurement of connectors with various lengths of 
wire leads is conducted. Figure 25- 29 show transfer functions of the connector 
samples with wires 2cm, 3cm, 5cm, 6cm, 8cm in lengths. The transfer function of 
one electrical connector sample with 2 cm wire, which is close to the averaged result, 
 
51 
 
is shown in Figure 2-25. In Figure 2-25 (a), an obvious peak appears around 565 HZ. 
In Figure 2-24 (b), a typical 180 degrees phase shift is around 565 HZ, too.  
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-25: Transfer Function of Connector with Wire Lead 2cm in Length 
 
52 
 
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-26: Transfer Function of Connector with Wire Lead 3cm in Length 
 
53 
 
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-27: Transfer Function of Connector with Wire Lead 5cm in Length 
 
54 
 
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-28: Transfer Function of Connector with Wire Lead 6cm in Length 
 
55 
 
 
(a) Magnitude Plot 
 
(b) Phase Plot 
Figure 2-29: Transfer Function of Connector with Wire Lead 8cm in Length 
 
56 
 
      The averaged natural frequencies tested of electrical connectors with multiple 
wire lengths are shown in Figure 2-30. With the longer wires, the natural frequencies 
of the connector systems decrease. For the connector sample with wire lead 18cm in 
length, the natural frequency is around 100 HZ. By Input B- Output A, natural 
frequencies of relative rocking motion between blade and receptacle of connectors 
with various wire lengths are measured. It is found that the natural frequency of this 
mode is not affected by the tie- off lengths, which is constantly around 120HZ, as 
shown in Figure 2-19.                                                                                                      
 
Figure 2-30: Natural Frequencies of Connectors with Various Wire Lengths  
               (Mode A- A) 
      The transfer functions with outputs at various positions on the blades and 
receptacles are illustrated in Figure 2-31 (wire length = 2cm). The output positions 
 
57 
 
are at the height of 1mm, 3mm, 11mm and 14 mm above the root of blade, the part 
clamped by the fixture on shaker head. For a single sample, the natural frequencies 
at different output positions are almost the same, except for the ones at the root of 
the blade, which is close to the fixture on shaker head. In addition, it?s indicated that 
the change of magnitude is close to a linear increase with the heightened position of 
output on connector. However this phenomenon only exist for the connector with 
wire length shorter than 10 cm, because the relative rocking motion is activated 
while the natural frequency of Mode A-A moves closer to 120HZ. 
 
Figure 2-31: Transfer Functions with Outputs on Blade and Receptacle at Different 
            Heights (Wire Length = 2cm)  
      Based on the vibration information above, the estimation of the vibration 
posture of blade and receptacle at Mode A- A is showed in Figure 3-32. Only the 
 
58 
 
root of the blade bends during vibrating. For connectors with short wires, relative 
rocking motion between receptacle and blade can be ignored according to the 
linearly increasing vibration magnitude which is illustrated in Figure 2-31. Only the 
tangentially translational relative motion between connector pair is considered. For 
connectors with long wires, translational and rocking motions at the contacting 
interface are both possibly activated by the single frequency vibration at Mode A- A. 
The vibration information above and this estimation provide significant foundation 
of the mathematical model simulating the vibration and fretting behavior of the 
connector system. 
  
(a) Initial Posture without Deformation    (b) Deformation of Blade 
Figure 2-32: Estimation of Vibration of Connector Pair for Connectors with 
               Short Wires  
 
59 
 
      
(a) Wire Length = 2cm         (b) Wire Length = 4cm 
      
(c) Wire Length = 6cm           (d) Wire Length = 8cm 
Figure 2-33: Vibration Postures of Connectors with Various Wire Lengths 
 
60 
 
      Figure 2-33 demonstrates the transverse vibration postures of the connector 
systems including connector pairs and wire leads, at the natural frequencies of Mode 
A- A. The top and end of the connector systems are fixed at fixtures, which do not 
allow any transverse vibration amplitudes. The vibration patterns of the wire leads 
convert continuously as the wire length increases, which becomes obvious while the 
length is 8cm.  
      The experimental results: thresholds with the error bands of electrical 
connector samples with different lengths of wires are shown in Figure 2-34. As 
shown in the figure, the threshold amplitude of sample with 4 cm wire is lowest. As the 
lengths of wire leads increase from 4 to 18cm, threshold increases monotonously. The 
results are also illustrated in Table 2-2. 
 
Figure 2-34: Vibration Threshold Amplitudes of Electrical Connectors with 
               Various Wire Lengths  
 
 
61 
 
Table 2-2: Experimental Threshold Amplitudes of Electrical Connectors with 
Various Wire Lengths 
Wire length(cm) 2 3 4 5 6 8 10 12 14 16 18 
Averaged 
Threshold (?m) 
19.5 15.5 12.4 16.7 21.3 30.1 39.7 45.2 47.3 54.6 56.5 
Largest 
Threshold (?m) 
21.1 17.9 13.8 19.2 23.2 32.9 44.0 48.4 51.1 59.2 58.0 
Smallest 
Threshold (?m) 
16.3 14.4 9.0 13.4 18.2 27.4 35.4 42.0 44.5 50.2 54.8 
 
2.3.3 Discussion 
      In this experiment, the transfer function and threshold tests are conducted by 
the vibration system and electrical resistance measurement equipment used for last 
tests. Natural frequencies of Mode A- A and relative rocking motion of connector 
pairs for sample with various wire lengths are conducted. The threshold tests are 
conducted at natural frequencies of Mode A- A. The vibration amplitude of the 
fixture holding the blade that triggers the fretting corrosion is the threshold. For the 
convenience of threshold tests, wires with connectors are strengthened by 
supplementary wires and thermal shrink tubing. For samples with different lengths 
of wire leads, the blade and receptacle parts have no differences. The laser reflectors 
are located at the same positions on connectors with different cable lengths.  
      As shown in Figure 2-30, the averaged natural frequency of the Mode A-A of 
the electrical connector decreases with the increase of the length of cable. This 
decreasing trend can be attributed to the declined stiffness and larger mass of the connector 
system with the lengthened wires based on the equation for calculating natural frequency   
 
62 
 
       
where   , k and m are respectively natural frequency, stiffness and mass of a vibration 
system,. In addition, it is found that the natural frequency of relative rocking-style motion of 
connector pair is not influenced by different tie- off lengths, which is constantly around 
120HZ for different samples.    
      As shown in Figure 2-33, threshold amplitude varies as a function of the wire 
lengths. This non-monotonic trend of threshold amplitude behavior can be explained 
by the vibration postures for connectors with various wires, as shown in Figure 2-32. 
As the vibration mode evolves with the increased wire length, fretting corrosion may 
occur with lower excitation amplitude. The decreased stiffness of the system with the 
lengthened wire results in a larger vibration amplitude necessary to induce the 
relative motion at the interfaces of connectors. While wire lengths are 2- 4cm, the 
influence of mode conversion is major to lead to a decreasing trend of threshold. For 
longer wires, the declined stiffness affect the threshold greatly to increase the 
threshold as the wire length is increased. It is estimated that for connectors with 
wires shorter than 8cm in lengths, of which fretting corrosion is caused by 
translational relative motion between blade and receptacle. In the conditions that 
wire is longer than 8cm, relatively translational and relative rocking motion at the 
contacting interface both contribute to the degradation. 
 
 
 
 
63 
 
 
 
CHAPTER 3   MATHEMATICAL MODELING AND ANALYSIS OF A 
BLADE/RECEPTACLE CONNECTOR PAIR OF THE PREDICTION OF 
AXIAL VIBRATION- INDUCED FRETTING CORROSION 
      In this chapter, a mathematical model is developed to predict the threshold 
behavior of certain type of electrical connectors. Based on the experimental study of 
the vibration behavior, this model simplifies the dimensions and vibration modes of 
the connector pair to focus on the threshold behavior of the fretting corrosion caused 
by Mode A- A. Some supplementary experiments are conducted to provide 
necessary parameters for the modeling.  
 
3.1 Model Development 
      A spring-mass-damper model is developed to explain and predict the 
threshold behavior of the connector system. In previous work, the relative rocking 
motion at contacting interface and fretting behavior caused by it was simulated by 
FEA modeling method. In current study, modeling objective is fretting degradation 
caused by translational relative motion at the interface of connector pairs with short 
wires. The schematic for the model is shown in Figure 3-1.   
 
64 
 
? + ?
? + ?
? + ?
?
G ?
G
O K 0
C 2K 2
K 1
C 1
A
 
Figure 3-1: Schematic of Model 
      The connector pair is simplified into two rigid parts including bar OG and 
bar AG?. Each of them has a friction pad. The two bars are connected by this pair of 
frictional pads modeling the contacting interface between blade and receptacle where 
relatively rocking motion is not allowed. As the same as the experimental setup, Bar 
OG is fixed on the shaker head at point O, and bar AG? is connected with the wire at 
point A. Point G and G? are respectively the centers of masses of the connector pair 
and receptacle. According to the experimental results, the blade bends at the root of 
it during vibrating at natural frequency of Mode A- A. Thus while there is no relative 
slip at the contacting interface, bars AG? and OG rotate about Joint O without 
 
65 
 
deformation. For the design of sample connectors, the blade and wire of the sample 
connector are axially eccentric as shown in Figure 2-31a. While connector is 
vertically fixed as the initial posture, bars AG? and OG are positioned obliquely with 
an angle ? between AO and vertical axis; the angle between GO and vertical axis is ?; 
the angle between G?O and vertical axis is ?; the frictional pad interface is vertical. 
The angle of vibration is ?. The angle between AO and interface of frictional pad is 
constantly ?. Spring    is located at point O to model the moment produced by the 
deformation of blade root under vibration. The stiffness of    is tested by 
experiments. The wire is simplified to two springs   ,    and two dampers   ,    
which represent the effect of vertical and transverse deformation of the wire. The 
mass of wire is ignored in this model, of which effect is converted into that of 
springs and dampers. 
      There are two steps for running this model, including transfer function fitting 
and threshold calculation. In the first step, springs   ,    and dampers   ,   , 
parameters that are difficult to measure experimentally, are determined by fitting 
transfer functions of the modeling mechanical system with those tested 
experimentally. In transfer function test with small vibration amplitudes, transients 
of friction forces at contacting interface are constantly lower than static friction force. 
Without any relative motion between blade and receptacle, AO is considered to 
vibrate as a rigid bar. In the second step, the friction force at the contacting interface 
is considered as the indicator of the occurrence of fretting corrosion. Relative motion 
occurs at the contacting interface and leads to fretting corrosion only if the friction 
 
66 
 
force rises over the nominal value of the static friction force with the increase of 
vibration amplitude. The determined parameters and transient dynamic data obtained 
from step one are plugged into the equation for calculating transient of the friction 
force during vibration. The vibration input amplitude that caused the friction force to 
be equal to the nominal static friction force is recognized as the threshold. 
      As shown in Figure 3-2, three Cartesian coordinates, including x-y-z, n-t and 
x?-y? coordinates, are set for the ease of calculation. The n axis is along with vector 
        ; y? axis is parallel to the interface of frictional pad. The x-y-z coordinate is fixed 
with the origin at the initial position of point O before vibration. 
G ?
A
O
t
n
x ?
y
x
y ?
z
 
Figure 3-2: Coordinates Used for Calculation 
 
67 
 
     
A
O
G
F 2  
( K 2 , C 2 )
F 1  ( K 1 , C 1 )
m g
? O
F o y
F o x
M 0
?
 
Figure 3-3: Free Body Diagram for Transfer Function Fitting 
      Free body diagram about ?bar? AO for transfer function fitting is shown in 
Figure 3-3.     ,     are x and y axes components of the force that the shaker head 
acts on ?bar? AO at point O;    is the moment produced by   ;    and     are 
forces produced respectively by spring   , damper   , and spring   , damper   ; 
  is angular acceleration of AO about point O;    is acceleration of point O. The 
resultant moment of ?bar? AO about point O is  
                                                          
where    is moment of inertia of ?bar? AO about point O, m is mass of ?bar? AO. In 
the initial condition without vibration, a tension in cable in vertical direction keeps 
the connector system straight. In non-vibration condition, the above equation is 
simplified: 
                          
 
68 
 
Matching like components in the preceding conditions leads to 
      ,               , 
from which we find that  
                              
                        
                             
 
               
                                   
where   ,    are the position of point G;   ,    are the position of point A;    
is the position of point O on y axis;   ,    are initial position of point A before 
vibration. The equation of motion for transfer function fitting is  
     
 
                                           
 
  
                                                        
The position, velocity, and acceleration items in above equations are 
             
                ;                     
              ;                     
                               
                                 
                                         
                                  
 
69 
 
          ;            
where A is input vibration amplitude;   is angular velocity of single frequency 
vibration;   is length of OA;    is length of OG. For this model, ? is assumed a 
small angle. By assuming    ? and    ? to be 
       and         
in equations expressing the position, velocity, and acceleration items, these equations 
are linearized: 
                   
                     ) 
                   
                        
                                     
                        
                            
                                         
      For each calculation trial, the equations are run by a sine vibration input with 
single frequency. The frequency response of mechanical system is modeled by inputs 
over a frequency range of interest.  
 
70 
 
G ?
A
F 1  ( K 1 , C 1 )
m ?g
F
f
 
F
n
? G ?
 
Figure 3-4: Free Body Diagram for Threshold Calculation 
      Figure 3-4 shows free body diagram of bar AG? for threshold calculation. Ff is 
the friction force tangent to the contacting interface;    is a force normal to the 
interface acted from bar OG, m? is mass of bar AG?. The resultant force on bar AG? is  
                               
Matching like components in the preceding conditions leads to 
                         
where 
                          ,                       
from which we find that 
                                                   
                             
       
 
                                 
      For cancelling out the unknown item   , the equation multiplies     . Then 
we find the equation for threshold calculation as shown below. 
 
71 
 
                         
                            
 
     
                             
                             
      For the configuration of the sample connectors, the dimensional parameters 
are                ,        ,         , ?           , ?  
         , ?           ,          ,           ,   ?         . 
Based on the free body diagrams and equations of motion, the nominal static friction 
force at the contacting interface between blade and receptacle is an essential 
parameter. The experimental measurement of it is described in the latter part. 
 
3.2 Measurement of Nominal Static Friction Force 
      An MT-200 tension/torsion test system from Wisdom Technology, Inc., 
shown in Figure 3-5, is applied for sample testing in this research. The system 
provides an axial displacement resolution of 0.1 microns and a rotation resolution of 
0.001?. Tests can be performed in tension, shear, torsion, bending, and combinations 
of these loadings on small specimens such as thin films, solder joints, gold wire, 
fibers, etc. Cyclic (fatigue) testing can also be performed at frequencies of up to 5 Hz. 
In addition, a universal 6-axis load cell was utilized to simultaneously monitor three 
forces and three moments/torques during sample mounting and testing. In this study, 
the tensile testing function is applied to measure the nominal static friction force 
between the contact pair of electrical connector. 
 
72 
 
  
 
Figure 3-5: Photograph of Small Scale Tensile Machine 
       The basic setup of the tensile test for measuring the static friction force is 
shown in Figure 3-6. With two ends of the connector pairs clamped at two aligned 
fixtures, the tensile machine drove the two ends apart with 0.3 mm/s in speed. 
During this process, the transient of the force separating the connector pair is 
recorded. The nominal static friction force is evaluated and averaged by 20 samples 
and repeated testing trials. In order to decrease the moment at the interface that is 
potentially resulted from unaligned setup and influence the friction force value, wire 
leads with connectors were removed and flexible fishing wires were attached to align 
the axial direction of the connector pair with the driving direction of the tensile 
machine. 
 
73 
 
 
Figure 3-6: Setup of Connector Pair for Tensile Test 
      Figure 3-7 shows the result of fiction force at the contacting interface of 
connector pair in the tensile test. As the connector pair is stretched apart, the tensile 
force needed for keeping the uniformly slow movement is recorded. At the 0.0 mm 
horizontal displacement point, the machine did not start to move conversely. In the 
first period of tensile motion, the fish lines with connectors were tightened and the 
relative slip of contact pair did not occur, thus the tensile force increases 
monotonically from 0.0 mm to nearly 2.0 mm distance. After that, the fishing lines 
are already tightened and the blade and receptacle moved apart. The small scale 
variance of the force is mainly due to stick-slip phenomenon [46], the spontaneous 
jerking motion that can occur while two objects are sliding over each other. This 
phenomenon is typically caused by the reality that static friction coefficient is higher 
 
74 
 
than kinetic friction coefficient at one interface. In this case, as the tensile force 
became larger than static friction force, relative slip of connector pair occurred and 
the reduction of the friction to the kinetic friction leaded to a dropped tensile value. 
In the last period, the tensile force decreases simultaneously and becomes zero as the 
contact is separated. Based on the mechanism of stick-slip, the tensile force values 
that indicate the static friction coefficient are focused on the peak magnitudes during 
jerking motions. Through dozens of samples, the averaged static friction force 
between blade and receptacle is 2.19N.  
 
Figure 3-7: Tensile Force Change of Friction Force Measurement 
 
3.3 Modeling Results 
      As the output of the transfer function fitting, the dynamic responses of 
mechanical system like   ,   ,  , and determined stiffness and damping parameters 
 
75 
 
are plugged into the equation for threshold prediction. As shown in Figures 3-8 to 
Figure 3-10, with a sine vibration input,   ,   ,   are all at periodic functions, with 
stable magnitudes after a short time fluctuation. The results for transfer function 
fitting are shown in latter figures. 
 
Figure 3-8: Transient of ? 
 
76 
 
 
Figure 3-9: Transient of ?  
 
Figure 3-10: Transient of ?  
 
77 
 
 
Figure 3-11: Comparison of Experimental and Modeling Transfer Functions  
               (Wire Length = 2cm) 
 
Figure 3-12: Comparison of Experimental and Modeling Transfer Functions 
               (Wire Length = 3cm) 
 
78 
 
 
Figure 3-13: Comparison of Experimental and Modeling Transfer Functions 
               (Wire Length = 4cm)  
 
Figure 3-14: Comparison of Experimental and Modeling Transfer Functions  
               (Wire Length = 5cm) 
 
79 
 
 
Figure 3-15: Comparison of Experimental and Modeling Transfer Functions 
               (Wire Length = 6cm)  
 
Figure 3-16: Comparison of Experimental and Modeling Transfer Functions 
               (Wire Length = 8cm)  
 
80 
 
      The output of the threshold prediction equation is the transient of friction 
force at the contacting interface of the connector pair. For each trial of the 
calculation, the output friction force is in the condition of single frequency and 
single amplitude vibration input. As shown in Figure 3-17, the transient of friction 
force with a specific amplitude and frequency input is a periodic function with a 
fixed magnitude. For this result, the fretting corrosion happens while the maximal 
absolute value of the transient friction force is over than the nominal static friction 
force. By the iteration of frequencies and increased input amplitudes in the range of 
interest, the friction force at various frequencies with the modeling threshold 
amplitudes are shown in figure 3-18 to figure 3-23. 
 
Figure 3-17: Transient of Friction Force at Contacting Interface 
 
81 
 
 
Figure 3-18: Maximal Absolute Value of Friction Force at Various Frequencies 
             under Vibration with Threshold Amplitude (Wire Length = 2cm)  
 
Figure 3-19: Maximal Absolute Value of Friction Force at Various Frequencies 
           under Vibration with Threshold Amplitude (Wire Length = 3cm)    
 
82 
 
 
Figure 3-20: Maximal Absolute Value of Friction Force at Various Frequencies  
           under Vibration with Threshold Amplitude (Wire Length = 4cm) 
 
Figure 3-21: Maximal Absolute Value of Friction Force at Various Frequencies 
           under Vibration with Threshold Amplitude (Wire Length = 5cm)  
 
83 
 
 
Figure 3-22: Maximal Absolute Value of Friction Force at Various Frequencies 
           under Vibration with Threshold Amplitude (Wire Length = 6cm)  
 
Figure 3-23: Maximal Absolute Value of Friction Force at Various Frequencies 
           under Vibration with Threshold Amplitude (Wire Length = 8cm)  
      Based on the results as shown in figures above, the peaks of modeling 
friction forces are located at the natural frequencies of Mode A- A, which obeys the 
 
84 
 
threshold behavior measured experimentally, as shown in Figure 2-20. The 
comparison of experimental and modeling threshold amplitudes at Mode A-A for 
connectors with various wire lengths. A high correlation of the experimental and 
modeling results are shown. However this well correlation only exists for the 
connectors with short wires, and as the wire length is increased, the modeling 
threshold is larger and larger than the experimental thresholds. This phenomenon is 
in accord with the estimation of the styles of relative motions that induce fretting 
corrosion for connector with different wire lengths. Under the single frequency 
vibration at the frequency of Mode A- A, relative rocking motion happens in the 
connectors with long wires.  
 
Figure 3-24: Comparison of Experimental and Modeling Thresholds at 
        Mode A- A for Connectors with Various Wire Lengths  
 
85 
 
      For the models of connectors with various wire lengths, the stiffness and 
damping parameters are different due to the variance of wires. The total stiffness and 
damping data for connector with various wire lengths are shown in figures 3-25 and 
3-26. For a vibration model, higher stiffness and lower damping tend the system to 
vibrate more strongly under a given input amplitude and frequency. In the case of 
the mechanical system of electrical connectors, a smaller input amplitude can cause 
the relative motion between blade and receptacle with the higher stiffness and lower 
damping.  As shown in these figures, with the increase of wire lengths, stiffness 
and damping both decrease. The combination of the two factors can be an 
explanation of the non-monotonic trend of threshold amplitudes, as shown in Figure 
2-33.   
 
Figure 3-25: Total Stiffness of Connectors with Various Wire Lengths 
 
86 
 
 
Figure 3-26: Total Stiffness of Connectors with Various Wire Lengths 
 
 
 
 
 
 
 
 
 
 
 
 
 
87 
 
 
 
CHAPTER 4   CONCLUSION AND FUTURE WORK 
      Two experiments and one mathematical model have been presented in this 
thesis to research the single frequency vibration threshold amplitudes that can trigger 
the fretting corrosion in electrical connector systems. In the experimental parts, 
frequency response and threshold behavior of the connector system was studied. 
Some processes are operated on connector samples for convenience of tests.  The 
first experiment was aimed to find the frequency where threshold is lowest, and the 
second experiment was focused on the lowest thresholds for connector systems with 
various wire lengths. A spring-mass-damper discrete vibration model was applied to 
estimate the threshold behavior of fretting corrosion of the sample connectors. After 
synthesizing the experimental and modeling results, the following conclusions can 
be obtained for the present research: 
? The electrical connectors with various wire lengths vibrate transversely 
under the axial vibration excitation. 
? The electrical connector systems vibrate at multiple modes. 
? Micro-meter scale amplitudes of the axial vibration trigger the fretting 
corrosion of this type of electrical connectors.  
? At natural frequencies, threshold amplitudes are relatively low. And the 
lowest threshold appears at Mode A-A of this type of connector samples. 
? With the increase of the wire length, the natural frequency of Mode A- A of 
 
88 
 
connector system decreases. 
? With the increase of the wire length, the threshold amplitudes vary as a 
function with a non- monotonic trend. This phenomenon can be explained 
by the change of vibration modes based on experimental results, and 
decreased stiffness and damping according to the modeling results. 
? The results of the mathematical model are well correlated with the 
experimental results. 
 
      Progress has been made in the area of fretting corrosion, but more can be 
done. The following are some suggestions for future research in this field: 
? Research on threshold behavior of modes, other than Mode A-A. 
? Create 2-D and 3-D FEA models to validate the vibration and threshold 
behaviors of connector systems with various wire lengths 
? Design an experiment combining the axial and transverse excitation, and 
FEA models to validate the experimental results. 
 
 
 
 
 
 
 
 
89 
 
 
 
REFERENCES 
 
[1]   R. B. Waterhouse, ?Fretting Corrosion?, Oxford Pergamon Press, USA.1972 
[2]   M. G. Fontana, Corrosion Engineering, McGraw-Hill Book, New. York, 1986 
[3]   J. Williams, Engineering Tribologhy, New York: Cambridge University Press, 
2005 
[4]   I-Ming Feng and H. Uhlig, Journal of Applied Mechanics, 21, 395 (1954) 
[5]   M. D. Bryant, ?Resistance Buildup in Electrical Connectors Due to Fretting 
Corrosion of Rough Surfaces?, IEEE Transactions on Components, Packaging 
And Manufacturing Technology, Part A, vol. 17, 1994, pp. 86-95 
[6]   E.M. Bock and J.H. Whitley, ?Fretting Corrosion in Electric Contacts?, 
Proceedings of Holm Conference on Electrical Contacts, 1974, pp. 128-138. 
[7]   S. V. Angadi, W. E. Wilson, R. L. Jackson, G. T. Flowers, B.I. Rickett ?A 
Multi-physics Finite Element Model of an Electrical Connector Considering 
Rough Surface Contact?, Proceedings of Holm Conference on Electrical 
Contacts ,vol. 54 ,2008, pp. 168-177. 
[8]   S. Wada, K. Sawa, ?Degradation Phenomena of Electrical Contacts Using 
Hammering Oscillating Mechanism and Micro-sliding Mechanism- Contact 
Resistance and Its Model?, Proceedings of Holm Conference on Electrical 
Contacts, vol. 57 ,2011, pp. 11-14. 
 
90 
 
[9]   P. Jedrzejczyk, S. Fouvry, P. Chalandon, ?A Fast Methodology to Quantify 
Electrical-contact Behaviour under Fretting Loading Conditions?, Wear, vol. 
267, 2009, pp. 1731-1740 
[10]  L. Tristani, E.M. Zindine, L. Boyer, G. Klimek, ?Mechanical Modeling of 
Fretting Cycles in Electrical Contacts?, Wear, vol. 249, 2001, pp. 12?19 
[11]  R. D. Ibrahim, C. Chen, and G. T. Flowers, ?Modeling and Analysis of a 
Blade/Receptacle Pair for the Prediction of Thermal Cycling and Temperature 
Dependent Vibration Driven Fretting Corrosion?, Proceedings of Holm 
Conference on Electrical Contacts, vol.56, 2010, pp. 4-7 
[12]  K. Lee, D. Jeong, J. Kim, ?Simulational Study of Electrical Contact 
Degradation under Fretting Corrosion?, Tribology International, vol. 44, 2011, 
pp. 1651?1658 
[13]  S. Hsu, K. Liao ?Wear Analysis and Verification of Metallic Terminals for 
Electronic Connectors?, Engineering Failure Analysis, vol. 25, 2012, pp. 71?80 
[14]  K. Mashimo, Y. Ishimaru, ?Computational Modeling and Analysis of a Contact 
Pair for the Prediction of Fretting Dependent Electrical Contact Resistance?, 
Proceedings of Holm Conference on Electrical Contacts, vol. 57, 2011, pp. 
11-14  
[15]  M. Antler, ?Survey of Contact Fretting in Electrical Contacts?, Proceedings of           
the Twenty Ninth IEEE Holm Conference on Electrical Contacts, 1984, pp 
3-22. 
[16]  Ed P. Slade, ?Electrical Contacts: Principles and Applications?, Chapter 6, 
 
91 
 
1990 
[17]  S. Hannel S. Fouvry. Ph. Kaaspa, L. Vincent ?The Fretting Sliding Transition 
as a Criterion for Electrical Contact Performance?, Wear, vol. 249, 2001, 
761-770 
[18]  A. Lee and M.S. Mamrick ?Fretting Corrosion of Tin-Plated Copper Alloy?, 
IEEE Trans. CHMT- 10, 1987, p63 
[19]  J. Swingler, J.W. Bride, and C. Maul, ?The Degradation of Road Tested 
Automotive Connectors?, IEEE Transactions on Components and Packaging 
Technology (CPMT), 2000, 23(1), p. 157- 164 
[20]  J. Song, C. Koch, and L. Wang, ?Correlation betweenWear Resistance and 
Lifetime of Electrical Contacts?, Hindawi Publishing Corporation Advances in 
Tribology, vol. 2012, pp. 9. 
[21]  A. Bouzera, E. Carvou, L. Tristani, E.M. Zindine, N. Ben Jemaa and R. Elabdi  
?Minimum Fretting Amplitude in Medium Force for Connector Coated Material 
and Pure Metals?, Proceedings of Holm Conference on Electrical Contacts, 
VOL. 56 ,2010, pp. 4-7 
[22]  S. Fouvry, P. Jedrzejczyk, P. Chalandon, ?Introduction of an Exponential 
Formulation to Quantify the Electrical Endurance of Micro-contacts Enduring 
Fretting Wear: Application to Sn, Ag and Au Coatings?, Wear, vol. 271, 2011, 
pp. 1524? 1534 
[23]  X. Lin, L. Xu, Y. Shao, G. Luo, H. Zhang ?Research on Fretting Resistance 
and Fretting Wear Property of Ni-Au Contact Pair?, Proceedings of Holm 
 
92 
 
Conference on Electrical Contacts, vol. 57, 2011, pp. 11-14 
[24]  T.S.N. S. Narayanan, Y. Park, K. Lee, ?Fretting Corrosion of Lubricated Tin 
Plated Copper Alloy Contacts: Effect of Temperature?, Tribology International, 
vol. 41, 2008, pp. 87?102 
[25]  Y. Park, T.S.N. S. Narayanan, K. Lee, ?Fretting Corrosion of Tin-plated 
Contacts?, Tribology International, vol. 41, 2008, pp. 616?628 
[26]  Y. Park, T.S.N. S. Narayanan, K. Lee, ?Effect of Temperature on the Fretting 
Corrosion of Tin Plated Copper Alloy Contacts?, Wear, vol. 262, 2007, pp. 
320?330 
[27]  K. Y. Lee, D. K. Jeong, H. G. Joo and Y. W. Park, ?Simulation for Fretting 
Corrosion of Tin-coated Copper Contacts?, Materials and Corrosion, vol. 62 
2011, pp. 352-356 
[28]  S. No?l, D. Alamarguy, S. Correia, P. Laurat, ?Fretting Behavior of Nickel 
Coatings for Electrical Contact Applications?, Proceedings of Holm Conference 
on Electrical Contacts, vol. 57, 2011, pp. 11-14 
[29]  R. D. Malucci, ?Multispot Model of Contacts Based on Surface Features?, 
Electric Contacts- 1990, IEEE- HOLM, Montreal, 1990, p.625 
[30]   R.D. Malucci, ?Impact of Fretting Parameters on Contact Degradation?, 
Proceedings of the 42nd IEEE Holm Conference on Electrical Contacts, 1996, 
pp. 395- 403 
[31]   F. Xie, Thesis ?A Study of Electrical Connector Performance in Fretting?, 
Auburn University, 2003 
[32]  G Villeneuve, D. Kulkami, P. Batnagel and D. Berry, ?Dynamic Finite Element 
 
93 
 
Analysis Simulation of the Terminal Crimping Process?, Electrical Contacts, 
1996, Proceedings of the Forty- Second IEEE Holm Conference Joint with the 
18th International Conference on Electrical Contacts, Sept. 1996, pp. 156-172 
[33]  A. Monnier, B. Froidurot, C. Jarrige, R. Meyer and P. Teste, ?A Mechanical 
Electrical, Thermal Coupled- field Simulation of a Sphere- plane Electrical 
Contact?, Electrical Contacts, 2005. Proceedings of the Fifthy- First IEEE 
Holm Conference on Sept. 26- 28, 2005, pp: 224-231 
[34]   G. T. Flowers, F. Xie, and M. J. Bozack, Robert D. Malucci ?Vibration 
Thresholds for Fretting Corrosion in Electrical Connectors? IEEE 
Transactions on Components and Packaging Technologies, vol.27, 2004, 
pp.65-71. 
[35]  G. T. Flowers, F. Xie, M. J. Bozack, R. Horvath, R. D. Malucci, B. I. Rickett 
?Modeling Early Stage Fretting of Electrical Connectors Subjected to Random 
Vibration?, Proceedings of Holm Conference on Electrical Contacts, vol. 49, 
2003, pp. 45-50. 
[36]  G. T. Flowers, F. Xie, M. J. Bozack, X. Hai, B. I. Rickett, and R. D. Malucci,? 
A Study of the Physical Characteristics of Vibration-Induced Fretting 
Corrosion? . Proceedings of Holm Conference on Electrical Contacts, vol. 50, 
2004, pp 31-319. 
[37]  F. Xie, G. T. Flowers, C. Chen, M. Bozack, J. Suhling, B. I. Rickett, R. D. 
Malucci, and C. Manlapaz, ?Analysis and Prediction of Vibration-Induced 
Fretting Motion in a Blade/Receptacle Connector Pair?, Proceedings of Holm 
 
94 
 
Conference on Electrical Contacts, vol. 53, 2007, pp. 222-228 
[38]  C. Chen, G. T. Flowers, M, J. Bozack, J. Suhling, B.I. Rickett; B.J. Malucci, C. 
Manlapaz, ?Modeling and Analysis of a Blade/Receptacle Pair for the 
Prediction of Vibration-Induced Fretting Degradation? Proceedings of Holm 
Conference on Electrical Contacts, vol. 54, 2008, pp: 276-283 
[39]  C. Chen, G. T. Flowers, M. J. Bozack; J. Suhling, ?Modeling and Analysis of a 
Connector System for the Prediction of Vibration-induced Fretting 
Degradation?, Proceedings of Holm Conference on Electrical Contacts, vol. 55,  
2009, pp. 131-137 
[40]  J. Gao, C. Chen, G.T. Flowers, R.L. Jackson and M.J. Bozack ?The Influence 
of Particulate Contaminants on Vibration- Induced Fretting Degradation in 
Electrical Connectors?, Proceedings of Holm Conference on Electrical Contacts, 
vol. 56, 2010, pp. 1-5  
[41] L. Lam, J. W. McBride, C. Maul and J. K. Atkinson, ?Displacement 
Measurement at the Connector Contact Interface Employing a Novel Thick 
Film Sensor?, Electrical Contacts, 2005. Proceedings of the Fifty- First IEEE 
Holm Conference on 26-28, 2005 pp: 89-96 
[42]  R. Fu, S. Choe, R. L. Jackson, G. T. Flowers. ?Experimental Study of the 
Vibration-induced Fretting of Silver-plated High Power Automotive 
Connectors?, Proceedings of Holm Conference on Electrical Contacts, vol. 56, 
2010, pp. 4-7 
[43]  Model 2010 Multimeter User?s Manual, Keithley Instruments, Inc., USA, 1999 
 
95 
 
[44]  Laser Vibrometer User Manual, Controller OFV-2610/2620, Sensor Head 
OFV-353, POLYTEC Co., Germany 
[45]  HP 35665A Dynamic Signal Analyzer Installation and Verification Guide, 
Hewlett-Packard Company, USA, 1991 
[46]   S. N. Patek (2001). "Spiny lobsters stick and slip to make sound". Nature 411 (6834): 
153?154 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
96 
 
 
 
APPENDIX A   MATLAB CODE OF EXAMPLE OF MATHEMATICAL 
MODEL 
 
%%%% Main File 
 
global g I m1 m2 L Lg Lgg beta gama delta k0 w a k1 k2 c1 c2 
 
g= 9.81; % gravity 9.81 m/s^2 
 
I=264e-9; % moment of inertia is 246e-9 m^2*kg 
 
m1=1.1e-3; % total mass is 1.1e-3 kg 
 
m2=0.45e-3; % receptacle mass is 0.45e-3 kg 
 
L=36.84e-3; % length of connector is 36.84e-3 m 
 
Lg= 15.77e-3; % length of (G) mass center of connector is 15.77e-3 m 
 
Lgg= 24.11e-3; % length of (G') mass center of receptacle is 24.11e-3 m 
 
beta=0.0283; % initial angle of bar is 0.0283 rad 
 
gama= 0.0205; % initial angle of OG to Y axis is 0.0205 rad 
 
delta= 0.0208; % initial angle of OG' to Y axis is 0.0208 rad 
 
k1=-1403; % stiffness 1 
 
c1=0.019; % damping 1 
 
k2=0.083e6; % stiffness  2 
 
c2=0.0014; % damping 2 
 
I0= 36.9229e-9; % moment of inertia of blade 
 
f0=1275;   
 
97 
 
w0=f0*2*pi; % connector beam natural frequeny 
 
k0=I0*w0^2; % stiffness of connector root 
 
a=5e-6; % threshold amplitude e-6m=um. 
 
n=0; % number of magnitude 
 
spm= [390:5:400]; % frequency spectrum 
 
for f=spm 
     
    n=n+1; % number of magnitude 
    f  
     
    w=f*2*pi; 
   
     
    options = odeset('RelTol',1e-6,'AbsTol',[1e-6 1e-6 1e-6 1e-6 1e-6 1e-6]); 
    [S,Y] = ode45(@sub,[0 0.3],[0 0 0 0 a/2*w 0],options); 
     
    ly=length(Y(:,1)); % length of numerical result 
      
    for i=1:1:ly  
         input(n,i)=Y(i,4)+cos(Y(i,1))*10.54e-3-10.54e-3; 
         
    end 
     
         
     
magnitude(n)=range(sin(Y(ly-500:ly,1)+beta))*15e-3/range(input(n,ly-1e3:ly)); 
 
    i=0; % number of friction force 
     
    for i=1:1:length(Y(:,1))   
               
 
        forc(n,i)= (-(-L*sin(beta + Y(i,1))-(-L* sin(beta)))*k1 - (-L*cos(beta +   
Y(i,1))*Y(i,2))*c1)*sin(Y(i,1)) - (-m2*g - (Y(i,4)+ L*cos(beta 
+ Y(i,1)) - L*cos(beta))*k2- (Y(i,5)- L*sin(beta + 
Y(i,1))*Y(i,2))*c2)*cos(Y(i,1))+ m2*( ( Y(i,6) - 
Y(i,2)^2*(Lgg*cos(Y(i,1) + delta)) + Y(i,3)*(-Lgg*sin(Y(i,1) 
+ delta)) )*cos(Y(i,1)) - ( -Y(i,2)^2*(-Lgg*sin(Y(i,1) + delta)) 
- Y(i,3)*(Lgg*cos(Y(i,1) + delta)) )*sin(Y(i,1)) ); 
 
98 
 
 
    end 
     
    force(n)=max(abs(forc(n,:))); 
  
end 
 
% experimental result comparison 
load('/home/me_h3/hzy0011/Desktop/connector_testing/GOODSHAPE/goodshapeto
o/54.TXT'); 
 
for i=1:401 
    mg(i)=sqrt(X54(i,2)^2+X54(i,3)^2); 
end 
 
 
figure(1) 
 
subplot(2,1,1) 
plot(spm,magnitude,X54(1:401,1),mg(1:401)) 
title('magnitude') 
 
subplot(2,1,2) 
plot(spm,force) 
title('force') 
 
 
 
 
 
%%%% Function File 
 
function dy=sub(t,y) 
 
dy=zeros(6,1); 
 
global  I  m1  m2  L  Lg  Lgg  beta  gama  delta  k0  w  a  g  k1  
k2  c1  c2 
 
%  y(1) is angle theta 
dy(1)=y(2); 
 
%  y(2) is anglular velocity theta' 
dy(2)=y(3); 
 
99 
 
 
%  y(3) is anglular acceleration theta'' 
 
% Xg= -Lg*sin(gama + theta); 
% position of G at X axis  
 
% Xg'= -Lg*cos(gama + theta)*theta'; 
% velocity of G at X axis 
 
% XA= -L*sin(beta + theta); 
% position of A at X axis 
 
% XA0= -L* sin(beta); 
% intial XA 
 
% XA'= -L*cos(beta + theta)*theta'; 
% velocity of XA 
 
% XA''= L*theta'*theta'*sin(beta + theta) - L*theta''*cos(beta + theta); 
% acceleration of XA 
 
% YA= s0+ L*cos(beta + theta); 
% position of A at Y axis 
 
% YA0= L*cos(beta); 
% initial position of YA 
 
% YA'= v0- L*sin(beta + theta)*theta' 
% velocity of YA 
 
% YA''= a0 - L*theta'*theta'*cos(beta + theta) - L*theta''*sin(beta + theta) 
% acceleration of YA 
 
% Y0= s0; 
% position of shaker head at Y axis 
 
 
% dy(3)= ( -m1*Xg'*a0 - m1*Xg*v0 - k0*theta' + XA'*(YA-Y0)*k1 + 
(XA-XA0)*(YA'-Y0')*k1 + XA''*(YA-Y0)*c1 + XA'*(YA'-Y0')*c1 - 
YA'*XA*k2- (YA-YA0)*XA'*k2 - YA''*XA*c2 - YA'*XA'*c2 - 
m1*g*Xg' )/I 
 
% dy(3)=( -m1*(-Lg*cos(gama + theta)*theta')*a0 - m1*(-Lg*sin(gama + theta))*v0 
- k0*theta' + (-L*cos(beta + theta)*theta')*(L*cos(beta + theta))*k1 + 
 
100 
 
(-L*sin(beta + theta)+L* sin(beta))*(- L*sin(beta + theta)*theta')*k1 + 
(L*theta'*theta'*sin(beta + theta) - L*theta''*cos(beta + theta))*(L*cos(beta 
+ theta))*c1 + (-L*cos(beta + theta)*theta')*(-L*sin(beta + 
theta)*theta')*c1 - (v0- L*sin(beta +theta)*theta')*(-L*sin(beta + theta))*k2 
- (s0+ L*cos(beta + theta) - L*cos(beta))*(-L*cos(beta + theta)*theta')*k2 - 
(a0 -% L*theta'*theta'*cos(beta + theta) - L*theta''*sin(beta + 
theta))*(-L*sin(beta + theta))*c2 - (v0- L*sin(beta + 
theta)*theta')*(-L*cos(beta + theta)*theta')*c2 - m1*g*(-Lg*cos(gama + 
theta)*theta') )/I; 
 
 
% dy(3)= ( -m1*(-Lg*cos(gama + y(1))*y(2))*y(6) - m1*(-Lg*sin(gama + 
y(1)))*y(7) - k0*y(2) + (-L*cos(beta + y(1))*y(2))*(L*cos(beta + 
y(1)))*k1 + (-L*sin(beta + y(1))+L* sin(beta))*(- L*sin(beta + 
y(1))*y(2))*k1 + (L*y(2)*y(2)*sin(beta + y(1)) - L*y(3)*cos(beta + 
y(1)))*(L*cos(beta + y(1)))*c1+ (-L*cos(beta + y(1))*y(2))*(-L*sin(beta 
+ y(1))*y(2))*c1 - (y(5)- L*sin(beta + y(1))*y(2))*(-L*sin(beta + 
y(1)))*k2 - (y(4)+ L*cos(beta + y(1)) - L*cos(beta))*(-L*cos(beta + 
y(1))*y(2))*k2- (y(6) - L*y(2)*y(2)*cos(beta + y(1)) - L*y(3)*sin(beta + 
y(1)))*(-L*sin(beta + y(1)))*c2 - (y(5)- L*sin(beta + 
y(1))*y(2))*(-L*cos(beta + y(1))*y(2))*c2 - m1*g*(-Lg*cos(gama + 
y(1))*y(2)) )/I; 
 
dy(3)= ( -m1*(-Lg*cos(gama + y(1))*y(2))*(-a/2*w*w*sin(w*t)) - 
m1*(-Lg*sin(gama + y(1)))*(-a/2*w*w*w*cos(w*t)) - k0*y(2) + 
(-L*cos(beta + y(1))*y(2))*(L*cos(beta + y(1)))*k1 + (-L*sin(beta + 
y(1))+L* sin(beta))*(- L*sin(beta + y(1))*y(2))*k1 + (L*y(2)*y(2)*sin(beta 
+ y(1)) - L*y(3)*cos(beta + y(1)))*(L*cos(beta + y(1)))*c1 + (-L*cos(beta + 
y(1))*y(2))*(-L*sin(beta + y(1))*y(2))*c1 - (y(5)- L*sin(beta + 
y(1))*y(2))*(-L*sin(beta + y(1)))*k2 - (y(4)+ L*cos(beta + y(1)) - 
L*cos(beta))*(-L*cos(beta + y(1))*y(2))*k2 - (y(6) - L*y(2)*y(2)*cos(beta + 
y(1)) - L*y(3)*sin(beta + y(1)))*(-L*sin(beta + y(1)))*c2 - (y(5)- L*sin(beta 
+ y(1))*y(2))*(-L*cos(beta + y(1))*y(2))*c2 - m1*g*(-Lg*cos(gama + 
y(1))*y(2)) )/I; 
    
     
dy(4)=y(5);    %  s0         y(4) = a/2*sin(w*t) 
% y(4) is displacement of shaker head  
 
dy(5)=y(6);  %  v0  y(5) = a/2*w*cos(w*t)   
% y(5) is velocity of shaker head 
 
dy(6)=-a/2*w*w*w*cos(w*t);  % a0   y(6) = -a/2*w*w*sin(w*t)                                           
% y(6) is acceleration of shaker head 
 
101 
 
  
% dy(7)=a/2*w*w*w*w*sin(w*t); % a0' y(7)= -a/2*w*w*w*cos(w*t) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
102 
 
 
 
APPENDIX B   TESTING FIXTURE DIMENSIONS  
 
      The following pages include the dimensions of fixture used for vibration and 
threshold experiments. They are intended for use in machining a fixture which allows 
axial vibration of electrical connectors with different lengths of cables. The unit of the 
dimensions is millimeter. All of the materials used in this fixture are aluminum. The 
assembly and operation is described in Chapter 2.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
103 
 
 
 
 
 
 
 
104 
 
 
 
 
 
 
 
 
 
 
 
105 
 
 
 
 
 
 
 
 
 
 
 
106