i PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS Except where reference is made to the work of others, the work described in this dissertation my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. ________________________ Roland Horv?th Certificate of approval: ________________________ ________________________ Subhash C. Sinha George T. Flowers, Chair Professor Professor Mechanical Engineering Mechanical Engineering ________________________ ________________________ John E. Cochran, Jr. Dan B. Marghitu Professor Professor Aerospace Engineering Mechanical Engineering ________________________ Stephen L. McFarland Acting Dean Graduate School ii PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS Roland Horv?th A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 11, 2006 iii PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS Roland Horv?th Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon request of individuals or institutions and at their expense. The author reserves all publication rights. ______________________________ Signature of Author ______________________________ Date of Graduation iv VITA Roland Horv?th, son of J?nos Horv?th and Erzs?bet Fazekas, was born on August 16, 1972, in Kaposv?r, Hungary. He entered the Budapest University of Technology and Economics (former Technical University of Budapest), September 1994, where he received Master of Science degree in Mechanical Engineering. He joined the doctoral program in Mechanical Engineering at Auburn University, Auburn, Alabama, in the fall of 2000. v DISSERTATION ABSTRACT PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS Roland Horv?th Doctor of Philosophy, May 11, 2006 (M.E., the Budapest University of Technology and Economics, 2000) 152 Typed Pages Directed by George T. Flowers Automatic, passive self-balancing systems are important tools for reducing the effects of synchronous vibration in a variety of rotating machinery. Such systems are ideally capable of precise balancing, subject to certain dynamic restrictions. There are a number of designs that are used, but the most common type is the ball balancer system that employs balls that move inside a cylindrical race or channel. However, such systems may be subject to a variety of effects that arise due to rolling resistance. An alternative approach uses pendulums rather than balls to provide the balancing. In the present work, a passive pendulum balancer system is investigated from several aspects. A mathematical model has been developed to discover the stability characteristic of the pendulum balancer. Because of the obvious potential for practical application pendulum balancers this system was investigated from engineering point of view. These investigations tried to cover all the possible differences that could arise when the mathematical model would be materialized as a real passive balancing device. The application of non identical vi pendulums was studied in detail and its advantages and disadvantages are discussed. The influences of rolling resistance and shaft misalignment on the functional capability of pendulum self-balancing systems are specifically examined. The study of a passive pendulum balancer with non-isotropic suspension is also presented. The existence of two natural frequencies results in two distinct areas of stability. These stable areas are determined by Floquet analysis and verified by numerical simulations and experimental measurements. vii ACKNOWLEDGMENTS The author would like to express his appreciation and thanks to his advisor Professor George T. Flowers, Department of Mechanical Engineering for his guidance and support toward the completion of this dissertation. The author also wishes to acknowledge the following committee members: Dr. Subhash C. Sinha, Professor, Dr. Dan Marghitu, Associate Professor, Department of Mechanical Engineering; and Dr. John Cochran, Jr., Department Head and Professor of Department of Aerospace Engineering. The author is thankful for the invaluable initial guidance to his former professor Dr. G?bor St?p?n, DSc, Department of Applied Mechanics Budapest University of Technology and Economics. As well, the author would like to thank his friend, Dr. Tam?s Insperger, Associate Professor, Department of Applied Mechanics Budapest University of Technology and Economics, for his great advice and ideas. Finally, the author would like to thank J?nos Torma and B?la Pallos, his former teachers for supporting and believing in him. viii Journal used: Journal of Sound and Vibration Computer software used: Microsoft Office Word 2003 ix TABLE OF CONTENTS LIST OF TABLES............................................................................................................ xii LIST OF FIGURES .........................................................................................................xiii 1 INTRODUCTION ...................................................................................................... 1 1.1 Background......................................................................................................... 1 1.2 Motivation for research....................................................................................... 4 1.3 Organization of Dissertation............................................................................... 5 2 BASIC TWO DEGREES OF FREEDOM (DOF) MATHEMATICAL MODEL ..... 6 3 PASSIVE PENDULUM BALANCER WITH ISOTROPIC SUSPENSION .......... 12 3.1 Balancing boundaries (relative balancing areas) of a two-pendulum balancer with non-identical pendulums....................................................................................... 14 Visualization of balancing capability: ?CG circles?............................................. 17 Singular Points and Stability Characteristics........................................................ 20 Stability Analysis.................................................................................................. 29 3.2 EXPERIMENTAL INVESTIGATION............................................................ 41 3.2.1 Pendulum Balancer Experimental Facility ............................................... 41 3.2.2 Experimental validation for non-identical pendulums.............................. 45 Subcritical operation ............................................................................................. 45 x Supercritical operation.......................................................................................... 47 3.2.3 Numerical and Experimental validation ................................................... 50 Special case: Identical pendulums ........................................................................ 52 Numerical and Experimental validation ............................................................... 54 3.3 Influence of Pendulum Shaft Misalignment ..................................................... 58 3.3.1 Experimental investigation of pendulum shaft misalignment .................. 62 Pendulum Balancer Experimental Facility ........................................................... 62 Implementation of shaft misalignment ................................................................. 62 Description of experimental procedure................................................................. 63 3.3.2 Experimental comparison of the sensitivity and consistency of ball and pendulum balancers .................................................................................................. 67 Ball Balancer Experimental Facility..................................................................... 67 Ball Balancer Experimental Results ..................................................................... 68 Pendulum Balancer Experimental Results............................................................ 74 4 PASSIVE PENDULUM BALANCER WITH NON-ISOTROPIC SUSPENSION 78 4.1 Analytical investigation .................................................................................... 78 4.1.1 Stability of homogeneous linear system with time periodic coefficients . 84 4.1.2 Floquet analysis: piecewise approximation .............................................. 86 4.1.3 Floquet analysis with numerical integration: single pass scheme ............ 88 4.1.4 The result of Floquet analysis................................................................... 90 Floquet characteristic multipliers for the Type I singular point ........................... 93 xi 4.1.5 Stability changes for different damping coefficients................................ 96 4.2 Experimental facility with non-isotropic suspension........................................ 99 4.2.1 Description of experimental facility ......................................................... 99 4.2.2 Description of experimental procedure................................................... 105 4.3 Numerical and experimental investigation ..................................................... 106 4.4 Summary of validation.................................................................................... 112 5 CONCLUSIONS AND FUTURE WORK ............................................................. 113 BIBLIOGRAPHY........................................................................................................... 115 APPENDIX A PARAMETERS OF EXPERIMENTAL FASCILITY .......................... 119 APPENDIX B MATLAB SOURCE CODES ................................................................ 121 5.1 MatLab source code: Model40.m ................................................................... 122 5.2 MatLab source code: ODE01.m ..................................................................... 131 5.3 MatLab source code: wRunUp.m ................................................................... 132 xii LIST OF TABLES Table 3.1 Stability table of Type I singular point ............................................................ 32 Table 3.2 Stability table of Type II singular point........................................................... 34 Table 3.3 Stability table of Type III singular point ......................................................... 35 Table 3.4 Stable configuration for two operational and three constructional cases ........ 36 xiii LIST OF FIGURES Figure 2.1 Two DOF model with radial mass imbalance: M P ........................................... 6 Figure 2.2 Magnitude and phase shift of the frequency response...................................... 9 Figure 2.3 Configurations of the frequency response and the three forces of the pendulum as a function of phase shift............................................................................... 11 Figure 3.1 4 DOF model of rotor with two-non-identical pendulum balancer in a rotating coordinate system.............................................................................................................. 13 Figure 3.2 Balancing boundaries and relative balancing areas of two-pendulum set...... 16 Figure 3.3 Balancing boundaries of two-pendulum set for a series of mass imbalance S P ........................................................................................................................................... 17 Figure 3.4 CG circles relative to a certain radial mass imbalance S P .............................. 18 Figure 3.5 Illustrations of the three types of singular point............................................. 26 Figure 3.6 Singular point loss as a function of relative damping and operational speed for properly oversized pendulums ........................................................................................ 38 Figure 3.7 Singular point loss as a function of relative damping and operational speed for improperly oversized pendulums.................................................................................... 39 Figure 3.8 The side view of pendulum balancer experimental facility............................ 43 Figure 3.9 Top view of pendulum balancer experimental facility................................... 44 Figure 3.10 Side view of pendulum assembly................................................................. 44 Figure 3.11 Analytical and experimental results for the amplitude of vibration (subcritical operation) ....................................................................................................... 46 xiv Figure 3.12 Analytical and experimental results for the position of pendulums (subcritical operation) ....................................................................................................... 47 Figure 3.13 Analytical and experimental results for the amplitude of vibration (supercritical operation).................................................................................................... 49 Figure 3.14 Analytical and experimental results for the position of pendulums (supercritical operation).................................................................................................... 49 Figure 3.15 Results of numerical simulation and experimental measurement of two-non- identical pendulum............................................................................................................ 51 Figure 3.16 Analytical and experimental results for the amplitude of vibration (supercritical operation).................................................................................................... 53 Figure 3.17 Analytical and experimental results for the position of pendulums (supercritical operation).................................................................................................... 54 Figure 3.18 Results of numerical simulation and experimental measurement of two- identical pendulum............................................................................................................ 55 Figure 3.19 Possible design solutions for pendulum balancing systems......................... 57 Figure 3.20 Mathematical model of rotor system with non-centered pendulums ........... 58 Figure 3.21 Simulation results showing the non-dimensionalized rotor vibration level for the system with pendulum shaft misalignment................................................................. 60 Figure 3.22 Simulation results showing the absolute and relative positions of the pendulums for the system with pendulum shaft misalignment......................................... 61 Figure 3.23 Experimental results showing the non-dimensionalized rotor vibration level for the system with pendulum shaft misalignment ........................................................... 64 Figure 3.24 Experimental results showing the absolute and relative positions of the pendulums for the system with pendulum shaft misalignment......................................... 65 Figure 3.25 Side view of ball balancer experimental facility .......................................... 69 Figure 3.26 Top view of ball balancer experimental facility........................................... 70 xv Figure 3.27 The final positions of balancing balls and vibration level for different startups .............................................................................................................................. 71 Figure 3.28 Deformation of contact surfaces and force distribution of the balancing ball and channel ....................................................................................................................... 73 Figure 3.29 The final positions of balancing pendulums and the level of vibration for different startups ............................................................................................................... 75 Figure 3.30 The final positions of balancing pendulums and the level of vibration on the zoomed plot for different startups..................................................................................... 76 Figure 3.31 Force distribution of pendulum and ball balancer........................................ 77 Figure 4.1 Coarse Floquet stability map of Type I singular point................................... 91 Figure 4.2 Fine Floquet stability map of Type I singular point with the points used for numerical and experimental validation............................................................................. 92 Figure 4.3 The unit cylinder and the biggest Floquet characteristic multiplier as a function of relative running speed. ................................................................................... 94 Figure 4.4 Three different views of unit cylinder and the Floquet characteristic multipliers of Type I singular point .................................................................................. 95 Figure 4.5 Floquet stability map of Type I singular point for different relative damping coefficients of rotor suspension ........................................................................................ 97 Figure 4.6 Floquet stability map of Type I singular point for different damping coefficients of pendulums................................................................................................. 98 Figure 4.7 Side view of pendulum balancer experimental facility with non-isotropic suspension....................................................................................................................... 102 Figure 4.8 Top view of rotor assembly.......................................................................... 103 Figure 4.9 Side view of rotor assembly ......................................................................... 104 Figure 4.10 Numerical and experimental validation of point A .................................... 107 Figure 4.11 Numerical and experimental validation of point B .................................... 108 xvi Figure 4.12 Numerical and experimental validation of point C .................................... 109 Figure 4.13 Numerical and experimental validation of point D .................................... 110 Figure 4.14 Numerical and experimental validation of point E..................................... 111 xvii NOMENCLATURE t = Time [s] T = Time period [s] ? = Phase angle [deg] ? = Angular displacement of disk [deg] ?(t) = Angular velocity of the disk [rad/s] ? n = Natural angular velocity [rad/s] f ni = i th Natural frequency [Hz] ? 1,2 = Linear degrees of freedom [m] ? 3,4 = Angular degrees of freedom [deg] Y 1,2E = Linear coordinates of singular points, (linearized system) [m] Y 3,4E = Angular coordinates of singular points (linearized system) [deg] M D = Mass of disk [kg] M P = Mass imbalance [kg] M BA = Mass of pendulum A [kg] M BB = Mass of pendulum B [kg] M S = Mass of the entire rotor system [kg] I D = Mass moment of inertia of disk [kgm 2 ] P 3 = Radial perturbation variable [m] R D = Radius of disk [m] xviii R BA = Length of pendulum A [m] R BB = Length of pendulum B [m] e = Linear distance of CG of imbalanced rotor [m] k 1,2 = Linear spring stiffness [N/m] c 1,2 = Linear damping coefficients [Ns/m] c 3,4 = Angular damping coefficients [Nms/rad] ? = Relative damping coefficient [-] ? C = Non-dimensionalized shaft misalignment [-] R C = Shift of suspension of pendulums [m] e = Linear distance of CG of imbalanced rotor [m] ? = Non-dimensionalized amplitude of vibration [-] A PR = Amplitude of vibration with released pendulums [m] A PL = Amplitude of vibration with locked pendulums [m] N = Amplification factor [-] A = Amplitude of vibration [m] A Stat = Static amplitude of vibration [m] [I] = Identity matrix [M] = Inertia matrix [D] = Damping matrix [G] = Gyroscopic matrix [K] = Elastic matrix [N] = Non-conservative force matrix [A] = Coefficient matrix xix [?] = Fundamental matrix [C] = Floquet Principal Matrix ? = Characteristic multiplier ? = Relative running speed [-] K = Relative spring stiffness [-] S A,B = First order moment of inertia of pendulums [kgm] S P = First order moment of inertia of mass imbalance [kgm] T = Sum of all kinetic energies [J] V = Sum of all potential energies [J] D = Sum of dissipation energies [J] Q 1,2 = Generalized forces [N] Q 3,4 = Generalized torques [Nm] F CF = Centrifugal force [N] F C = Constraining force [N] F F = Friction force [N] F R = Force of rolling resistance [N] F B = Balancing force [N] 1 1 INTRODUCTION 1.1 Background Rotating machinery is commonly used in civil, military and industrial applications including vehicle wheels, machining tools, industrial rotating machinery aircraft gas turbine engines and helicopter blades. One of the primary sources of vibration is mass imbalance, which occurs when the principal axis of inertia of the rotor is not coincident with its rotational axis. There are two common balancing methods which are used to align the principal inertia axis and rotational axis. One method is off-line balancing in which the rotating machine is stopped for the adjustment of mass distribution. The second method is on-line balancing in which the mass distribution rearrangement happens continuously during rotation. Such automatic balancing can be either active or passive. Active balancing systems use sensors to measure the unbalance level and actuators to shift the mass distribution. Passive balancing systems perform a similar task but without sensors, control laws and external power supplies. The simplicity, reliability and relatively low cost of passive balancing systems make them a very attractive solution and thus, have been the subject of significant past research. The first documented appearance of an auto balancing device was by A. Fesca who patented improved centrifugal machine equipped with three ring balancer in 1872 [1] The second documented automatic balancer is also a ring balancer patented by G. W. Ledyard in 1896. Ledyard used a series of rings around the outer diameter of his centrifugal machine [2]. In the same year a new type of rotor balancing system was registered by 2 United States Patent Office. M. Leblanc patented his automatic balancer for rotating bodies. In his design, the balancer consists of a simple cylindrical chamber field with a heavy liquid [3]. Thearle [4] in 1932 published a detailed experimental study. Probably this design was the first to use self-aligning balls to achieve passive balancing. The first documentation of pendulum type balancer is also a patented invention by K. Clark 1946 [5]. Clark used four non-centrally attached pendulums to reduce the level of centrifugal machines. Thearle [6] investigated in detail the Leblanc balancer and summarized the requirements of ideal balancers. In the same journal, Thearle [7] compared several different types of automatic dynamic balancers, such as a ring, pendulum and ball balancers. This is the first appearance of an automatic pendulum-balancer in the literature. In his paper Thearle concluded that by placing the pivot of pendulums at the center of rotation the pendulum balancer become equivalent to the ring balancer, his analysis was heuristic and did not include a rigorous and detailed analytical study. In addition, he concluded that ball balancers were a superior system. Since the 1950?s, the majority of researchers have concentrated their efforts on investigations of ball-balancing systems. Sharp [8] provided a stability analysis of the balanced condition for a two-ball balancer on a planar rotor, and the presented the results of a parametric study of that system. Conclusions were drawn regarding the satisfactory operation of such a balancer. B?vic and H?gfors [10] in 1986 by using the method of multiple scales showed that an automatic ball-balancer reduces vibration for planar and non-planar 6 DOF rotor systems. Their research is a detailed analytical study but it was not verified by any experimental measurements. 3 Wettergren [11] investigated a ball-balancing system with one and two balls in a cylindrical groove. He also examined the effect of oil viscosity and found a relationship between the viscosity and the stability properties of the balancing system. However, his research was analytical in nature and not verified experimentally. Huang and his colleagues [12] in 2002 presented a combined analytical and experimental study of the loss of balancing capability for a ball-type balancing system due to runway eccentricity and rolling resistance. Other investigators [13] have considered the effects of dry friction on ball balancer systems. This is a specific concern if such units are operated in an un- lubricated condition, which is desirable for some applications (such as optical drives) where lubricants can cause contamination and damage. It was demonstrated that, ?even for very low friction coefficients? the balancing behavior can deteriorate considerably. K. Green and his colleagues demonstrated a nonlinear bifurcation analysis of an automatic ball balancer [14]. Applying only analytical methods they discovered large regions in the parameter space where the ball balancer shows instability. They also investigated the effect of perturbations and transient dynamics. The only substantial study in the current literature on pendulum-based passive balancing systems was conducted by Kubo and his colleagues, who presented their research on an automatic balancer using pendulums [9]. This paper was concerned with theoretical and experimental investigations on the dynamic behavior and stability of an automatic balancer using centrifugal pendulums. Although they noted the most important requirement that has to be satisfied for proper balancing such as the pivot of the pendulums have to be placed to the center of rotor, the experimental facility was equipped with non-centrally attached pendulums. Additionally this paper has a 4 fundamental flaw. In their study they model the suspension of disk as single DOF. This is not an adequate model for a rotating disk especially when a centrifugal force has a balancing effect. 1.2 Motivation for research Providing reliable, on line balancing for wide range of applications is a great challenge. The complexity of active on line balancing reduces its reliability. This complexity does not suit zero tolerance applications and it requires tremendous research hours to develop a somewhat reliable and robust active balancing system, which is expensive production cost and this way rejects any application on mass production level. Application of automatic balancers on a mass production level causes a focus of efforts on implementation of the simplest passive balancing system: the ball balancer. The extreme simplicity of ball balancer systems has generated a huge amount of research effort but the widespread utilization of ball balancer systems has not taken place. The effect of rolling resistance greatly reduces number of potential applications of automatic ball balancers. 5 The desire to design and build a simple, reliable passive balancing device has not been fulfilled. This research purposefully targeted this technological deficiency. The discovery of such a system would initiate numerous potential applications: - Providing artificial gravity in space - Centrifuges for Gravitational biology research - Helicopter blade balancing - Centrifugal casting - Handheld power tools 1.3 Organization of Dissertation This research effort is a study of a passive pendulum balancer. The research work was performed in the Vibration Analysis Laboratory of the Department of Mechanical Engineering at Auburn University. Specifically, the work includes: ? Development of mathematical model ? Development of numerical procedures ? Development and testing of experimental facilities 6 2 BASIC TWO DEGREES OF FREEDOM (DOF) MATHEMATICAL MODEL In order to provide an appropriate background for the current work, this discussion begins by considering the behavior of a 2 DOF thin disk model rotating in a horizontal plane with a radial mass imbalance represented by a point mass M P that is shifted from the center of disk by P 3 in the 'I null direction, as shown in Figure 2.1. ?1 ?2 ?(t) P3 MP k1 k2 c1 c2 ?(t) I' I J 0 0 0I 0J J' Figure 2.1 Two DOF model with radial mass imbalance: M P The disk model has two degrees of freedom, ? 1 and ? 2 , which are mutually orthogonal linear displacements in the same horizontal plane. The model is symmetric, having the same spring stiffness k 1,2 and damping coefficient c 1,2 in both directions, which yields coincident natural angular velocity ? n . 7 The differential equations of motion are: 2 1113 cos( ) P M ckPM t? ??+?+?= nullnull null (2-1) 2 2223 sin( ) P M ckPM t? ??+?+?= nullnull null (2-2) D P M MM= + (2-3) 12 cc c= = (2-4) 12 kk k= = (2-5) The solution of radially unbalanced system is: () 2 1 22 2 cos( ) e tt kc MM ? ?? ? ? ?= ? ???? ?+ ???? ???? (2-6) () 2 2 22 2 sin( ) e tt kc MM ? ? ? ? ? ?= ? ???? ?+ ???? ???? (2-7) 3 : P D P PM Where e M M = + (2-8) The phase angle of frequency response is: 1 2 () tan c k M M ? ?? ? ? ? ? ? ? ? ?= ?? ? ? ? ?? ? ? ?? ? ? (2-9) 8 The magnitude of frequency response is: 2 22 2 2 4 2 () 2 M N ck kM M ? ? ??? = +? + (2-10) || () Stat A N A ? = (2-11) 22 12 ||A = ?+? (2-12) 3 P Stat M A P M = (2-13) The rotating position vector, directed from the center of rotation to the center of disk, has a lag angle (phase angle ?) relative to the forcing vector. The forms of the phase angle and magnitude relations are shown in Equations (2-9) and (2-10), respectively, with plots for several different relative damping coefficients shown in Figure 2.2. As is well known, the response amplitude and phase are sensitive to damping level. 9 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 ? / ? n N [-] Magnitude of the frequency response ? = 0.05 ? = 0.2 ? = 0.5 ? = 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0 50 100 150 ? / ? n ? [d eg ] Phase angel of the frequency response ? = 0.05 ? = 0.2 ? = 0.5 ? = 1 Figure 2.2 Magnitude and phase shift of the frequency response A series of thought experiments using this 2 DOF model provides substantial insight into self-balancing using pendulum balancers. Figure 2.3 is a series of snapshots of the basic system for three operating speeds. The center of the centrifugal force field is the origin of the first two DOF, ? 1 and ? 2. An imaginary (almost massless) pendulum is also shown as a part of this discussion to illustrate the force distribution along the path where a real pendulum would move. 10 For each case, the combined effects of centrifugal force and the constraining force acting on the system will cause the position of the imaginary pendulum to change until these two forces become parallel. For subcritical operating speeds, the steady-state phase delay of the displacement related to the exciting force is less than 90?. In this case, the stable pendulum position will be between 0? and 90? as measured from the horizontal axis (as shown in Figure 2.3.a) which exacerbates the mass imbalance of the system. For the resonance case, the phase delay is 90?, and the equilibrium position of the pendulum is 90? behind the position of the imbalance mass, as shown in Figure 2.3.b. Again, the effect of the pendulum is to shift the CG further from the origin though not quite as far as in the subcritical case. Figure 2.3.c shows the system operating at a supercritical running speed. The phase delay is between 90? and 180?. The resultant effect of the centrifugal and the constraining force acting on the pendulum will drive it toward a position on the opposite side of the disk from the mass imbalance, partially compensating the unbalancing effect. 11 Mp Mp Mp Pendulum ? ? ? a. b. c. ?1?1 ?1 Figure 2.3 Configurations of the frequency response and the three forces of the pendulum as a function of phase shift 12 3 PASSIVE PENDULUM BALANCER WITH ISOTROPIC SUSPENSION In this chapter, the passive pendulum balancer is investigated with isotropic suspension equipped with a pendulum balancing system consisting of two non-identical pendulums. From a practical point-of-view, basic differences in the pendulums (mass or length) may be incidental due to manufacturing variations or accidental damage. On the other hand, differences may be an intentionally designed into the system if they are determined to be advantageous to the operation and/or performance of the balancing system. In the following paragraphs, a model of rotor with a two balancing system with non-identical pendulums is developed and analyzed. The basic elements of the mechanical model are shown in Figure 3.1. The first two degrees of freedom (? 1 , ? 2 ) are linear displacements that describe the position of the rotor center. The third and the fourth are angular displacements that describe the positions of pendulum A and pendulum B, respectively. 13 I' ?1 0J J' ?(t) ?(t) I0 Mp P 3 ?2 I' J' B A ?3 ?4 Figure 3.1 4 DOF model of rotor with two-non-identical pendulum balancer in a rotating coordinate system 14 3.1 Balancing boundaries (relative balancing areas) of a two-pendulum balancer with non-identical pendulums First, consider the balancing capability of a pendulum balancing system with non- identical pendulums. Because the pendulums are not identical, the ability of the system to counter a given imbalance level depends very strongly on the relationship between the parameters of the individual pendulums and the initial imbalance level. A useful parametric description for pendulums is the first mass moment of inertia with unit: [kg- m]. It should be noted that two non-identical pendulums may have the same first moment of inertia. For each of the two pendulums, A and B, and for the mass imbalance, the following relations define the first mass moments of inertia. ABABA SMR= (3-1) BBBB SMR= (3-2) 3PP SMP= (3-3) The balancing boundaries of the pendulum system for a variety of S A and S B combinations and for a specified mass imbalance, S P , are shown in Figure 3.2. The shaded area ABCD with the dark line border shows the parametric values which can compensate for the given level of imbalance. If the parametric values are in this shaded area (as described by Equation (3-4)) then the pendulums are able to counterbalance the mass imbalance. For values outside this area, the pendulums are unable to completely compensate for the mass imbalance because they are either undersized or improperly oversized. 15 || AB P AB P SS S and SS S+? ?? (3-4) For parametric combinations either above line AD or below line BC in Figure 3.2, the pendulums are improperly oversized relative to the mass imbalance, S P . The magnitude of the difference between the first mass moments of the pendulums is greater than the mass imbalance, as described by Equation (3-5). If this is the case, there are no pendulum locations where the net unbalancing forces resulting from the two pendulums and the rotor mass imbalance can completely cancel and the rotor will remain unbalanced. The final level of net imbalance may be higher or lower than the initial level without a balancing system depending upon the relative differences between the two pendulums. || AB P SS S? > (3-5) If the parametric combination is located in the OAB triangle (Equation (3-6)) then the pendulums can only partially counterbalance the mass imbalance. In this case, the pendulum set is undersized relative to the mass imbalance. AB P SS S+ < (3-6) A series of similar relative balancing areas for a variety of mass imbalances are illustrated in Figure 3.3. If the two pendulums have the same first mass moment (S A = S B ), then the parametric configuration will be located on a line that starts from the origin with 16 a 45? slope, as shown in figure 6 with a dashed line. For this case, the pendulums are either undersized or properly oversized relative to any mass imbalance S P . 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 S A [kgm] S B [ kgm ] Pr o p e r l y O v e r s i z e d I mp r o p er l y O ve r s i ze d Im p r o p e r ly O v e r s i z ed U. S. O A B C D S P = 0.010 [kgm] Figure 3.2 Balancing boundaries and relative balancing areas of two-pendulum set 17 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 S A [kgm] S B [ kgm ] S P = 0.010 [kgm] S P = 0.015 [kgm] S P = 0.020 [kgm] S P = 0.025 [kgm] S P = 0.030 [kgm] S P = 0.035 [kgm] Figure 3.3 Balancing boundaries of two-pendulum set for a series of mass imbalance S P Visualization of balancing capability: ?CG circles? The balancing capability of a pendulum balancing system can also be characterized by the resulting steady-state location of the center of gravity of the entire rotor system (rotor, pendulums and mass imbalance). Figure 3.4 shows a relative balancing boundary plot which uses a series of circular symbols with a darkened area, which are referred to herein as CG circles where the pendulums are able to relocate the CG of the entire system. If these dark CG circles include the center of the rotor, then balancing can be archived for proper operational conditions. Again, for the case of same first mass moment, the CG circles are located on the 45? dashed line. Leaving this line the CG circles have an internal ?blind spot?. Having this 18 blind spot not necessarily means the pendulums cannot do the balancing. If the chosen pendulum combination stays inside the ABCD area the pendulums would balance regardless of the existence of blind spot. Under the BC or above the AD line the blind spot is over the center of the rotor this means balancing cannot be archived because the pendulums bother each other they are insufficiently oversized. In order to achieve balancing the point of chosen pendulums combination has to stay in the ABCD area. This condition is specified by Equations (3-4). SA SB A B C D Figure 3.4 CG circles relative to a certain radial mass imbalance S P O 19 In general, all of the relative-balancing-area plots shown above (Figure 3.2 - Figure 3.4) contain three distinct areas: Rectangular region ABCD If the parametric combination is located in this area, the pendulum system is capable of counterbalancing the entire mass imbalance. The system is properly oversized relative to the mass imbalance. Triangular region OAB If the parametric combination is located in this area, the pendulum system is not capable of counterbalancing the entire mass imbalance. They are undersized relative to the mass imbalance. Regions above the AD line and below the BC line If the parametric configuration of the pendulum system is located in either of these regions, the pendulums are not capable of counterbalancing the entire mass imbalance. They are improperly oversized relative to the mass imbalance and may partially reduce the level of vibration if: || AB P SS S? < (3-7) Otherwise, such an improperly oversized system will exacerbate the synchronous vibration resulting in a higher amplitude of vibration than if the balancer was not present. 20 Singular Points and Stability Characteristics The above discussion has examined the fundamental capability of a balancing system with two non-identical pendulums. The actual dynamic behavior of such a system depends on its equilibrium points and their associated stability characteristics. So, the next step in this investigation considers the occurrence of singular points and the stability of motion in the vicinity of such points. The governing equations of motion for a symmetric rotor suspension (c = c 1 = c 2 and k = k 1 = k 2 ) in a disk-fixed coordinate system can be represented as: 1, 2, 3, 4 i iiii dT T V D Qi dt ?????? ?++= = ?? ?? ?? ?? ?? ?? nullnull (3-8) Where: T = Sum of all kinetic energies V = Sum of all potential energies D = Rayleigh dissipation function i ? = Generalized coordinates i Q = Generalized forces 21 () ()() () () ()( )() ()( () () 2 1323 1424 22 2 31 12 21 2 3 3 21 1 4 4 3 21 21 2313 1 1 2cos 2sin 2 2cos 2sin 1 2 2 sin cos cos sin ABA BBB P D BA BB P D D BA A P B A BA BB D PA B TSR SR SP M M M M MR MS M SSR M SS S ? ? ? =+??+??+ ? ? ++??+??+ ? ++?++++?+?+ + ? ? +??+???+?+ +? ??+ ?+ ??+ ??+ +?+?? ??? nullnullnull nullnullnullnull nullnull null () () () () () ) ()() () () 4 244 122 3 2412 4 12 1312 22 21 22 34233244 14 sin cos cos sin 11 11 cos cos 22 si BBAA BDBB AP DBABBP DBABBP ABA BBB A B B SM SRM SM MM M M MM M M SR SR S S S ?+ +? ??? ??+? ?+ +? ??? ?+ ?? ??? ?? ??? ?+ ++++?++++?+ + ?+ ?+?? ?+?? ?+ ??? nullnull null nullnull nullnull nullnull nullnull null null nullnull nullnull nullnull () () 4133 nsin A S???? ? nullnull (3-9) () 22 12 1 2 Vk=?+? (3-10) ()( )()() 22 222 2 2 12 1221 12 3344 111 Dc c c c c??= ? +? ? ?? ??? + ? +? + ? + ? nullnull null null null null (3-11) 0 i Q = (3-12) 22 The resulting equations of motion are: () ( ) ( ) ( ) () () () () () ()()() 22 33 33 3 33 44 44 44 4 22 11 2 3 1 2 12 1 2 1 2 cos sin cos cos 2cos cos sin cos 2 20 A B P D BA BB DBAB S S MPMM ckM ?? ??? ? ???+?+?+?? ???+?+?+?? ????+?++++?+ +???+?? + + ?+?= nullnullnull null nullnullnull nullnull null nullnull nullnull (3-13) ( ) ( ) ( ) ( )( ) () () () () ()()() 22 33 33 3 33 44 44 44 4 2 22 1 2 2 21 2 1 2 2 sin cos sin sin 2 sin sin cos sin 2 20 A B P D BA BB DBAB S S MMM ckM ?? ?? ? ?????+?+?? ? ? ?+ ? ?? ? ?+ ? ? ??????+++?+ +?+?+?+ + + ???= nullnullnull null nullnullnull nullnull null nullnull nullnull (3-14) ( ) ( )( ( ) () () ()) 31 3 3 2 3 2 22 31 32 3133 sin cos 2 sin 2cos cos sin 0 ABA SR c ? ?? ? ???+?+??+ ??+ +??? ??+ ??+?= nullnull nullnull nullnull null nullnull (3-15) ( ) ( ) ( )( () () ()) 41 4 4 2 4 2 22 41 42 4144 sin cos 2 sin 2cos cos sin 0 BBB SR c ? ?? ? ? ? ?+ ?+ ? ?+ ? ?+ +??? ??+ ??+?= nullnull nullnull nullnull null nullnull (3-16) 23 The singular points are found by setting the time derivative terms in Equations (3-13) - (3-16) to zero. This results in the following set of algebraic equations. ( ) ( )( ) 2 13421 cos cos 0 ES A E B E E E YMSYSY cYkY???+ + ?+= (3-17) ( ) ( )( ) 2 23412 sin sin 0 ES A E B E E E Y M S Y S Y cY kY?+ + ++= (3-18) ( ) ( )( ) 2 132 3 sin cos 0 AE E E E SY Y Y Y ?? = (3-19) ( ) ( )( ) 2 142 4 sin cos 0 BE E E E SY Y Y Y ?? = (3-20) These equations are nonlinear and are not, in general, easy to solve in a closed form. However, the possible solutions are of three distinct types. The different types of singular points and the associated stability characteristics are detailed below. 24 Type I: The Type I singular points represent a configuration where the rotor base motions (Y 1E and Y 2E ) are zero, as illustrated in Figure 3.5.a. The pendulums settle into the positions where they counter the mass imbalance, M P . The resulting closed form solutions for Y 3E and Y 4E , in terms of a four quadrant inverse tangent function, are shown in Equations (3-21) and (3-22). Of course, this configuration only occurs when the pendulums are physically capable of balancing the system, which means the parametric configuration is in the ABCD region. When the parametric configuration is outside of that region, the terms under the square roots in Equations (3-21) and (3-22) takes on a negative value and there is no real solution. From the perspective of balancing effectiveness, this singular point is the most important of the three types. If this singular point is stable, the center of rotor is not vibrating because the relative locations of the pendulums and mass imbalance result in a balanced system. 222 34 arctan , 22 ABP E PA PA SSSAB Y SS SS ?? ?+? =?? ?? ?? (3-21) 222 44 arctan , ABP E PB PB SSSAB Y SS SS ?? ??? = ?? ?? (3-22) ( )( ) ()() : A BPABP ABPABP WhereASSSSSS B SSSSSS =+? ++ =?? ?+ 25 Type II: The Type II singular points (as shown in (3-24) and illustrated in Figure 3.5.b) represent a configuration where Y 4E = Y 3E + 180?. If this type of singular point is stable, the rotor exhibits a steady state synchronous whirl, because Y 1E and Y 2E are not zero in the rotor fixed coordinate system. Please note that, although closed form symbolic solutions were developed for both the Type II and the Type III singular points, they are not shown here because of the large size of those expressions. 26 Type III: The Type III singular points (as shown in Equation (3-25) and illustrated in Figure 3.5.c). In the algebraic equation system Y 4E = Y 3E . The Type III singular point also includes a steady state synchronous whirling of the rotor, but for this case the pendulums are overlapping one other. [ ] 1234 :00 EEEE Type I Y Y Y Y== (3-23) 0 1E 2E 3E 4E 3E :00 18Type II Y Y Y Y Y???? =+ ?? (3-24) [ ] 1E 2E 3E 4E 3E :00Type III Y Y Y Y Y?? ? (3-25) a. ? ? b. I' Type I J' Type III I' Type II J' Mp Mp Mp ? J' I' c. A B B AB A Y3E Y4E Y3E Y1E Y2E Y4E Y1E Y2E Y3E Y4E Figure 3.5 Illustrations of the three types of singular point 27 In order to assess the stability characteristics of each type of singular point, the differential equations of motions were linearized in the neighborhood of the singular points. The resulting equations in vector-matrix form are: [] [ ] [ ] M0YDGYKNY+ +++= nullnullnull nullnull null (3-26) The inertia matrix is: [] ( ) ( ) () () () () () () 34 33 34 0sinsin 0cosco sin cos 0 sin cos 0 SAEBE SAEBE AEA E ABA BEB E BB MSYSY MSYSY M SYS Y SR SYS Y SR ???? ?? = ? ? ?? (3-27) : S D BA BB P Where M M M M M= +++ The damping and the gyroscopic matrix is: [ ] () () () () () () () () 34 333 44 4 2 2 cos 2 cos 22sinin 2cos 2sin 0 2cos 2sin 0 SAEBE SA AE AE BE BE DG c M SY SY M c SY SY SY SY c SY SY c ?? ? ?? ?? += ?? ??? ?? ?? ? ? ?? (3-28) 28 The elastic and the non-conservative force matrix is: [] ( )() () () () () () () 222 34 22 22 343 44 4 sin sin cos cos sin cos 0 sin cos 0 SAEBE SA AE A E A BE B E B Mk c SY SY cMk KN SY S Y SKN SY S Y SKN ??? ? ?? ???+ ? ?? ?+? ? += ? ? ?? (3-29) ( ) ( ) () () 33 2 3 1 3 44 2 4 1 4 :sincos sin cos EEE E EEE E Where KN Y Y Y Y KN Y Y Y Y =+ The Cauchy transformation of equation (3-26) results in a system of first order linear differential equation system represented by 8x8 coefficient matrix [A]. Where: X= Y Y ? ? ? ? ? ? ? ? null null null null (3-30) X=[A]X null null null (3-31) [] [] [][ ] [][ ] 11 0 [] I A M KN M DG ?? ?? ?? = ?+?+ ?? (3-32) 29 Stability Analysis For the purposes of this stability investigation, all three types of singular points were examined separately for subcritical and supercritical operation. Because of the symmetric rotor suspension (k = k 1 = k 2 ), the undamped first and the second natural frequencies are the same, as indicated by the following expression: n D BA BB P k M MMM ? = +++ (3-33) The sub and supercritical stability investigation also had to be further divided for the three relative balancing areas. This means for one certain singular point there is 2x3 cases to investigate. Each case was also further examined separately for each of the three relative balancing areas. So, six cases are considered for each type of singular point. The specific cases that were investigated - subcritical or supercritical operation and each of the three relative balancing areas are shown in a series of Tables (1-3) in the following pages. The specific parametric combinations were chosen to encompass all three relative balancing areas, with the specific location of each test case marked by a dot. The real part of the calculated eigenvalues as a function of pendulum damping, c 34 , is also shown on the right side of each table. 30 Stability investigation of Type I singular point From a balancing perspective, this is the most important singular point. If it is stable for a given operating condition, the synchronous vibration caused by mass imbalance will be eliminated. The stability characteristics of this singular point are only investigated inside the ABCD relative balancing area. The Type I singular point does not exist outside of that region. The analysis results are shown in Table 3.1. They demonstrate that the Type I singular point is stable for supercritical operation and unstable for subcritical operation. In addition, it is important to note that the stable position of the pendulums could be mirrored about axis X. This operation has no effect on the location of the common CG of the pendulums. Basically, Type I singular point consists of two sets that produce identical balancing results. Stability investigation of Type II singular point For the Type II singular points, the steady-state locations of the pendulums are directly opposite to one another. Solving the associated equations resulted in two sets of explicit symbolic solutions. Table 3.2 summarizes the stability analysis results for each of the six cases that were considered (as described above). This singular point is only stable for supercritical operation and when the parametric configuration is improperly oversized relative to the mass imbalance. For the other five cases, this singular point is unstable. 31 Stability investigation of Type III singular point For the Type III singular points, the steady-state locations of the pendulums are directly on top of each other. Table 3.3 summarizes the stability analysis results for each of the six cases that were considered, in a fashion similar to that done for the other two types of singular points. This singular point is stable for subcritical operation regardless of the parametric configuration with regard to the relative balancing areas. In addition, it is also stable for supercritical operation when the parametric configuration is undersized relative to the mass imbalance. For the other two cases, this singular point is unstable. 32 Type I x y -5 -4 -3 -2 -1 0 1 0 1 2 3 4 5 x 10 -4 Real(?) c 34 S A [kgm] S B [ kgm] UNSTABLE S A [kgm] S B [ kgm] Subcritical operation S A [kgm] S B [kg m ] Singular point does not exist x y -20 -15 -10 -5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 -3 Real(?) c 34 S A [kgm] S B [ kgm] STABLE S A [kgm] S B [ kgm] Supercritical operation S A [kgm] S B [kg m ] Singular point does not exist Table 3.1 Stability table of Type I singular point 33 Further investigation has demonstrated, that for certain cases, the stable singular point is a function of the damping coefficient of the rotor suspension. Table 3.4 summarizes the stable singular points for all of the six cases and their dependence on rotor suspension damping. 34 Type II Set I Set II x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 S A [kgm] S B [k g m ] UNSTABLE UNSTABLE x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 S A [kgm] S B [kgm ] UNSTABLE UNSTABLE x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.5 1 1.5 2 x 10 -3 Real(?) c 34 Sub c ritical op eratio n S A [kgm] S B [k gm ] UNSTABLE UNSTABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 S A [kgm] S B [k g m ] UNSTABLE UNSTABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 S A [kgm] S B [kgm ] UNSTABLE UNSTABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 Superc ritical operation S A [kgm] S B [k gm ] UNSTABLE STABLE Table 3.2 Stability table of Type II singular point 35 Type III Set I Set II x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 S A [kgm] S B [k g m ] UNSTABLE STABLE x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 S A [kgm] S B [kgm ] UNSTABLE STABLE x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 x y -8 -6 -4 -2 0 0 0.2 0.4 0.6 0.8 1 x 10 -3 Real(?) c 34 Sub c ritical op eratio n S A [kgm] S B [k gm ] UNSTABLE STABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 S A [kgm] S B [k g m ] UNSTABLE UNSTABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 S A [kgm] S B [kgm ] UNSTABLE STABLE x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 x y -40 -30 -20 -10 0 10 0 1 2 3 4 5 6 x 10 -3 Real(?) c 34 Superc ritical operation S A [kgm] S B [k gm ] UNSTABLE UNSTABLE Table 3.3 Stability table of Type III singular point 36 S A [kgm] S B [k g m ] Type III: (function of rotor damping) x y S A [kgm] S B [k g m ] Type III: x y Subcritical operation S A [kgm] S B [k gm ] Type III: (function of rotor damping) x y S A [kgm] S B [k g m ] Type I: x y S A [kgm] S B [k g m ] Type III: x y Supercritical operation S A [kgm] S B [k gm ] Type II: (function of rotor damping) x y Table 3.4 Stable configuration for two operational and three constructional cases 37 A close examination of the characteristics of the various singular points shows that for a certain level of rotor suspension damping there is a rotor speed domain where certain singular points become complex. Such a result has no physical meaning and the singular points do not exist for those cases. Figure 3.6 shows a highlighted area in which the singular point is lost for properly oversized pendulums. For supercritical operation, the stable singular point is Type I and it is independent of damping. This configuration is represented by the horizontal upper boundary line of this area. For subcritical operation, the stable singular point is Type III and it is a function of damping. The Type III singular point does not exist when the system is operating slightly below the critical speed and the damping is relatively high. 38 Figure 3.6 Singular point loss as a function of relative damping and operational speed for properly oversized pendulums Figure 3.7 shows the dark area where the singular point is lost for improperly oversized pendulums. For supercritical operation, the stable singular point is Type II and for subcritical operation the stable singular point is type III. Both are a function of damping. The damping dependence results in a singular point loss near the critical speed for cases with a relatively high rotor damping. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 25 50 75 100 c 12 [Ns/m] ? ? / ? n Singular point does not exist Together: Set II Separated x y x y Singular Point Type I Singular Point Type III 39 Figure 3.7 Singular point loss as a function of relative damping and operational speed for improperly oversized pendulums 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 0 25 50 75 100 c 12 [Ns/m] ? ? / ? n Singular point does not exist Together: Set II Opposite: Set II x y x y Singular Point Type II Singular Point Type III 40 Table 4 shows that the stable singular points for relatively undersized pendulum combinations have no damping dependence. This means there is no singular point loss for any operational speed or rotor damping. In summary, for each of the six cases, there is only one stable singular point. For subcritical operation, only the Type II singular point is stable regardless of location within the relative balancing areas. When the pendulums are oversized, either properly or improperly, relative to the mass imbalance the singular point is a function of damping when the system is operating subcritically. When the pendulums are undersized and the rotor system is operating subcritically, there is no damping dependence. For supercritical operation, if the pendulums are undersized relative to the mass imbalance the stable singular point is damping independent. When the pendulums are oversized, the stable singular point is Type III and is independent of damping. Improperly oversized pendulums for supercritical operation have a stable Type II singular point which is damping dependent. 41 3.2 EXPERIMENTAL INVESTIGATION 3.2.1 Pendulum Balancer Experimental Facility In order to gain detailed insight into the dynamic characteristics and performance of a pendulum balancing system, an experimental facility was designed, fabricated and used for a series of tests. The design of this test rig is such that the weights of the pendulums can be changed while maintaining the same lengths. Figure 3.8 shows a side view of the entire experimental system. The disk rotation is in the horizontal plane. It is supported by a base plate (5) which is mounted on the shaft of a DC motor (11). The DC motor is supported by a column attached to the ground with a flexible joint. This vertical column is supported from in the horizontal (x and y) directions by springs attached at their opposite ends to fixed supports. This arrangement allows the disk to move in nearly horizontal plane with minimal friction. Figure 3.9 shows a top-view and Figure 3.10 a side view of the pendulum assembly. In these figures, the shaft of the pendulums (1) and the pendulums themselves (2) and (3) can be seen. Each pendulum is attached to an aluminum disk and supported at the center by two small ball bearings. The pendulums can be locked at the 0 and 180? positions by plastic pins (6) to the frame of the pendulum assembly (4) when the system is stopped. The motion of the pendulums is regulated by a damping system, which consists of two magnet magazine rims (9) which can accommodate up to 12 small rear earth magnets (10). One of the damping magazines is placed beneath the bottom pendulum and the other is placed above the top pendulum. Between the pendulums there is a steel magnetic field guide (12). The pendulums, the damping rims and the magnetic field guide comprise a sandwich structure. This arrangement generates eddy currents inside the aluminum disk 42 of each pendulum, producing a velocity proportional damping force that can be controlled by the number and the polarity of the magnets (10). The pendulum assembly (4) is fixed to the base plate (5) by two bolts from underneath. The gross imbalance of the system center of mass is set by adjusting the brass weights (7, 8), M P , and its radial position, P 3 . Weights (7) are basically identical brass nuts on threaded radial rods. When these weights (7) are twisted to the base (showed on Figure 3.9) and the auxiliary mass imbalance (8) is removed from the system the rotor is balanced because the main brass weights (7) are counterbalancing each other. By changing the location of one of these brass weights the desired mass imbalance can be added to the rotor system. 43 Figure 3.8 The side view of pendulum balancer experimental facility 44 Figure 3.9 Top view of pendulum balancer experimental facility Figure 3.10 Side view of pendulum assembly 45 Using this pendulum balancing facility, a series of experiments were conducted to validate the analytical results obtained in the previous section. The details and results of these experiments are presented in the following sections. 3.2.2 Experimental validation for non-identical pendulums For the first series of experimental measurements, the facility was equipped with two non-identical pendulums. The parameters of the pendulums were not changed. But, the magnitude of the rotor imbalance was adjusted by changing the radial position of the brass weight. In this way, the three relative balancing areas were investigated for both subcritical and supercritical operation. Subcritical operation As shown in Table 4, for subcritical operation only the Type III singular point is stable regardless of balancing area. For this configuration, the center of rotor will have a steady-state whirling motion even for zero mass imbalance. The analysis results are shown with a continuous line on Figure 3.11, along with the results form the experimental measurements. The horizontal axis is the first moment of the mass imbalance, S P , and the vertical axis is the vibration amplitude. In similar fashion, Figure 3.12 also shows the analytical and experimental results but the vertical axis represents the steady-state position of the pendulums. The ?circle? symbols represent pendulum ?A? which has the higher first mass moment, S A , compared to pendulum ?B? which is represented by the ?x? symbols. The experiments show that the pendulums stayed together regardless of the location of the system within the relative 46 balancing areas, which tended to verify quite well the previously described analysis results for subcritical operation. 0 1 2 3 4 5 6 7 8 9 x 10 -3 0.8 1 1.2 1.4 1.6 1.8 2 Sp[kgm] |Vi b .Am p | [ m m ] Analysis Experiment Figure 3.11 Analytical and experimental results for the amplitude of vibration (subcritical operation) 47 0 1 2 3 4 5 6 7 8 9 x 10 -3 260 270 280 290 300 310 320 330 340 350 360 Sp[kgm] ? 3 , ? 4 [d e g ] Analysis: Pend. A & B Experiment: Pend. A Experiment: Pend. B Figure 3.12 Analytical and experimental results for the position of pendulums (subcritical operation) Supercritical operation In contrast to the situation just described for subcritical operation, it is important to distinguish the three relative balancing areas for supercritical operation. As summarized in Table 3.4, for supercritical operation each relative balancing area has a different type of stable singular point. Figure 3.13 and Figure 3.14 show the three relative balancing areas and the transition regions using five small relative balancing area plots. For both the analytical results and for the experimental results described, the pendulum parametric combination is held constant and the mass imbalance was varied. This is why the point of pendulum combination shown is at the same location on all of the relative balancing plots, while the ABCD open area changes. The first configuration that will be considered is for low mass imbalance, which results in the pendulums being improperly oversized relative to the mass imbalance. This 48 situation is represented by the leftmost section of Figure 3.13 and Figure 3.14. The pendulums do not have the same first mass moment, and cannot properly counterbalance each other. This results in a stable Type II singular point, with the rotor engaging in a steady-state whirling motion. As the mass imbalance is increased, the rotor vibration amplitude decreases until the configuration transitions to a properly oversized condition in which the steady-state rotor vibration ceases. The middle section of Figure 3.13 and Figure 3.14 shows the results for pendulum combinations that are properly oversized. The relative balancing area plot shows the point of pendulum combination on the S A S B plane is inside the ABCD rectangular area. For this configuration, the Type I singular point is stable. As the mass imbalance is increased further, the pendulums do not have sufficient balancing capability to properly compensate and the pendulum combination is undersized relative to the mass imbalance. The pendulums come together opposite to the mass imbalance, resulting in a stable Type III singular point. The pendulum system is able to counterbalance only part of the mass imbalance. In this case, if the mass imbalance is increased further, the rotor vibration amplitude increases also. 49 Figure 3.13 Analytical and experimental results for the amplitude of vibration (supercritical operation) Figure 3.14 Analytical and experimental results for the position of pendulums (supercritical operation) 0 1 2 3 4 5 6 7 8 9 x 10 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sp[kgm] |V i b .A m p .| [ m m ] Analysis Experiment 0 1 2 3 4 5 6 7 8 9 x 10 -3 -100 -50 0 50 100 150 200 Sp[kgm] ? 3 , ? 4 [d e g ] Analysis: Pend. A Analysis: Pend. B Experiment: Pend. A Experiment: Pend. B 50 3.2.3 Numerical and Experimental validation In this chapter the result of numerical simulation and experimental measurements are discussed and compared. After several repeated measurement the system parameters were identified. These parameters were used as input parameters of simulations. Both in the numerical simulation and the experimental measurement the after the startup a proper amount of time was spent to let the transients die out. During this time the pendulums were locked at their base position 0? and 180?. From a balancing point of view, this initial position of pendulum would be neutral if the pendulums had the same first order moment of inertial. This part of research investigated non identical pendulums with different first order moment of inertia. In the numerical simulation and in the experiment initially the pendulum, with higher first moment of inertia was locked at 0? location. This initial configuration added extra imbalance to the system and resulted higher amplitude of vibration. The results of numerical simulation and the experimental measurement the logged coordinates ? 1..4 showed in time from prior pendulum release 3 seconds. The comparison of numerical simulation and experiment had resulted in quite similar system response for pendulum release proving the success of the model development and system parameter identification. 51 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 Time[s] ? 1 [mm ] Experiment 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 Time[s] ? 2 [m m ] 0 1 2 3 4 5 6 7 8 9 10 0 60 120 180 240 Time[s] ? 3 , ? 4 [d e g ] Pendulum A Pendulum B 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 Time[s] ? 1 [mm ] Numerical Simulation 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 Time[s] ? 2 [m m ] 0 1 2 3 4 5 6 7 8 9 10 0 60 120 180 240 Time[s] ? 3 , ? 4 [d e g ] Pendulum A Pendulum B Figure 3.15 Results of numerical simulation and experimental measurement of two-non- identical pendulum 52 Special case: Identical pendulums Now, consider the special case in which the pendulums are identical. For such a configuration, the pendulums can counterbalance each other even if there is no mass imbalance in the rotor system. So, the improperly oversized condition cannot exist and there are only two relative balancing areas ? (1) properly oversized, and (2) undersized relative to the mass imbalance. Figure 3.16 and Figure 3.17 show these two domains separated by a vertical dash line. On the leftmost side, the pendulums are properly oversized. For this configuration, the Type I singular point is stable and the center of the rotor is not vibrating. On the rightmost side, the rotor mass imbalance is beyond the balancing capability of the two pendulums and the system is undersized. The Type III singular point is stable and the center of the rotor moves on a steady-state circular path. The pendulums overlap each other in an angular location opposite to that of the mass imbalance and partially counterbalance the system. For the case of identical pendulums, the ?wishbone? shaped section of the curve in Figure 3.17 is symmetric, as compared to the non-symmetric appearance of the similar region for the non-identical pendulum system, as shown in Figure 3.14. An interesting phenomenon can be observed in the responses illustrated by Figure 3.16 and Figure 3.17. The first pendulum, indicated by an ?x?, generally settles in the location described by the upper curve and the pendulum, indicated by an ?o?, generally settles in the location represented by the lower curve of Figure 3.17 as the mass imbalance is increased. However, the two pendulums occasionally switch positions, with the ?x? pendulum associated with the lower curve and the ?o? pendulum associated with the upper curve. Inspection of Figure 3.16 shows this switching behavior results in an 53 abrupt change in the magnitude of the rotor vibration. This is in spite of the fact that the pendulums are theoretically identical and should be interchangeable. However, in practice the pendulums are not identical and have some small differences due to the fabrication process. These differences produce the observed sensitivity to the switching behavior described above, which is itself a result of initial conditions for the experimental system, which are random for every startup of the pendulums. From a balancing point of view, these switched positions are not equivalent because the residual mass imbalance is different for the two cases. However, as the mass imbalance of the system is increased to a sufficiently high value (around S P = 0.005[kg-m] for the experimental system) this sensitivity to initial conditions disappears and the settled positions of the pendulums are consistent and appear independent of the initial conditions. Figure 3.16 Analytical and experimental results for the amplitude of vibration (supercritical operation) 0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.1 0.2 0.3 0.4 0.5 Sp[kgm] | V i b .A mp | [ m m] Analysis Experiment 54 Figure 3.17 Analytical and experimental results for the position of pendulums (supercritical operation) Numerical and Experimental validation In this chapter, the result of numerical simulation and experimental measurements are discussed and compared in similar fashion as it was showed for non-identical pendulums. Because of the application of identical pendulums, the initially locked pendulums had no effect on the level of mass imbalance. They simple counterbalanced each other. The results of numerical simulation and experimental measurement are summarized in Figure 3.18. 0 0.002 0.004 0.006 0.008 0.01 0.012 100 150 200 250 300 Sp[kgm] ? 3 , ? 4 [d e g ] Analysis: Pend. A Analysis: Pend. B Experiment: Pend. A Experiment: Pend. B 55 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 Time[s] ? 1 [m m ] Experiment 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 Time[s] ? 2 [mm ] 0 1 2 3 4 5 6 7 8 9 10 0 60 120 180 240 Time[s] ? 3 , ? 4 [d e g ] Pendulum A Pendulum B 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 Time[s] ? 1 [m m ] Numerical Simulation 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 Time[s] ? 2 [mm ] 0 1 2 3 4 5 6 7 8 9 10 0 60 120 180 240 Time[s] ? 3 , ? 4 [d e g ] Pendulum A Pendulum B Figure 3.18 Results of numerical simulation and experimental measurement of two- identical pendulum 56 It is important to notice the learned property of identical pendulums is also valid for pendulum combinations where the pendulums are not physically identical but they have a same first order moment of inertia. The value of this discovery is important from engineering point of view. Two pendulums can be manufactured to be quite identical, but the CG of these two pendulums never will move on the same plane as it is shown by Figure 3.19.a. These pendulums will produce a dynamic imbalance on the rotor as they are rotating around the shaft. Figure 3.19.b. shows another (better) possible design solution based upon the above results. In this case, each of the pendulums has a different length but they have the same first order moment of inertia around the axis of the shaft. In this design, the CG of each individual pendulum moves in the same plane and will not dynamically imbalance the system. This also provides a better force distribution on the bearing system. The experimental setup had similar design to the engineering design showed by Figure 3.19.a. 57 a. b. Figure 3.19 Possible design solutions for pendulum balancing systems 58 3.3 Influence of Pendulum Shaft Misalignment It was demonstrated analytically above that a pendulum self-balancing system is ideally capable of exact radial balancing. However, imperfections in the fabrication and assembly of such a system will compromise some of the modeling assumptions that provided this result. One major imperfection is shaft misalignment, which can easily occur due to improper design, fabrication and assembly. In the following sections, the effect of misalignment between the center of rotation of the pendulums and rotor shaft is examined in detail. Figure 3.20 illustrates the basic configuration for a system with non-centered pendulums. The position of the center of the shaft is described in a disk-fixed rotating coordinate system. The shaft of the pendulums is shifted from the center of the rotor by an amount, R C , in the ? 1 direction. Mp 0J ?4 ?1 ?2 ?(t) ?3 J' B A ?(t) I0 I' R c P 3 J' I' e Figure 3.20 Mathematical model of rotor system with non-centered pendulums 59 Using this model, a series of simulation studies were performed for a variety of R C . In order to better generalize the results, the non-dimensional parameter, ? C and ? are used. 3 ,: CP C D P RPM Where e eM ? == + (3-34) || || PR PL A A ? = (3-35) R C is the offset of the pendulum center of rotation and e is the imbalance eccentricity of imbalanced rotor system. Where A PR is the amplitude of the steady state vibration with the pendulums free to rotate and A PL is the amplitude of vibration with the pendulums locked at the 0? and 180? positions, respectively. Figure 3.21 and Figure 3.22 summarize the results of these simulations. The angular velocity of the rotor system is higher than the first critical speed of the system, which is a basic requirement for this type of passive balancing system to work properly. In the numerical simulations, the operational speed was set at almost 20 times higher than the first critical speed and the damping ratio for the rotor suspension was small, which resulted in the frequency transfer function having a magnitude close to one. 60 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ? c ? Numerical simulation Figure 3.21 Simulation results showing the non-dimensionalized rotor vibration level for the system with pendulum shaft misalignment The upper part of Figure 3.22 shows the angular positions of the pendulums (? 3 and ? 4 ) as a function of ? C . The lower part of this figure shows the closest relative angular distance between the two pendulums. Examination of this figure shows that when ? C is zero, the pendulums are close to the 180? position and almost overlapping one other, which indicates that the radial imbalance of the system is only slightly less than the balancing limit of the pendulums. At ? C = 1.0, the two pendulums are the farthest apart at 180 o . For higher values of ? C , they move closer together and for ? C = 2.0, the pendulums converge to overlapping angular positions and stay in that configuration for further increases of ? C . At this ? C value, there is a break-point in the vibration level of the center 61 of the rotor (as shown in Figure 3.21). The rate of increase for the vibration amplitude changes abruptly and almost settles into a plateau. 0 0.5 1 1.5 2 100 200 300 400 ? c ? 3 [d e g ], ? 4 [d e g ] Numerical simulation ? 3 ? 4 0 0.5 1 1.5 2 0 50 100 150 200 ? c ? 4 - ? 3 [d e g ] Figure 3.22 Simulation results showing the absolute and relative positions of the pendulums for the system with pendulum shaft misalignment 62 3.3.1 Experimental investigation of pendulum shaft misalignment Pendulum Balancer Experimental Facility To gain detailed insight into the dynamic characteristics and performance of a pendulum balancing system with misalignment shaft the same experimental facility was used that was described in chapter 3.2.1. The design of this test rig is such that the pendulum rotation center could be offset in the ? 1 direction from the center of the rotor by an amount R C . Implementation of shaft misalignment The pendulum assembly (4) is fixed to the base plate (5) by two bolts from underneath. When these bolts are loosened, the pendulum assembly can be shifted in the ? 1 direction by an amount, R C , while still maintaining the shaft of the pendulums parallel to the shaft to the DC motor (11). However, shifting the pendulum assembly and the structural parts surrounding the pendulums, itself also produces an additional mass imbalance, which is undesirable for the purposes of the present study. This imbalance can be offset by adjusting the brass weights (7) in the radial direction. The gross imbalance of the system center of mass is set by adjusting the brass weight (8), M P , and its radial position, P 3 . 63 Description of experimental procedure Using the pendulum balancing facility described previously, a series of experiments was conducted to validate the simulation results obtained in the previous section and to gain further insight into the influence of specific imperfections and non-idealities on the dynamic performance of such passive balancing system. The steps in the testing process are described below: 1. Pendulums are locked at 0 o and 180? respectively with plastic pins 2. The pendulum assembly is shifted by an amount R C relative to the base plate 3. The rotor was spun-up to the test operating speed and balanced. 4. A specific gross mass imbalance was set and the pendulums were released 5. The rotor was spun-up to the test operating speed and three sets of data were measured and recorded: - The magnitude of vibration [mm], measured by a laser displacement system - The final angular position of each pendulum ? 3 and ? 4 [deg] Figure 3.23 and Figure 3.24 summarize the results for a typical set of experiments, which are quite similar to the predicted behavior from the simulation results (Figure 3.21 and Figure 3.22). Inspection of these figures shows that increasing the shift of the pendulums, ? C , produces a proportional increase in rotor vibration level until the pendulums are overlapping one other. At ? C = 2.0, there is again a break-point, after which further shifting of the pendulum axis has only a slight effect on the rotor vibration, in a fashion similar to that observed in the numerical simulations. 64 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 ? c ? Measurement Figure 3.23 Experimental results showing the non-dimensionalized rotor vibration level for the system with pendulum shaft misalignment 65 0 0.5 1 1.5 2 2.5 100 200 300 400 ? c ? 3 [d e g ], ? 4 [d e g ] Measurement ? 3 ? 4 0 0.5 1 1.5 2 2.5 0 50 100 150 200 ? c ? 4 - ? 3 [d e g ] Figure 3.24 Experimental results showing the absolute and relative positions of the pendulums for the system with pendulum shaft misalignment Top part of Figure 3.24 shows the absolute position of the pendulums and the bottom part of this figure shows the relative position to each other. Again, obvious similarity can be found between these experimental results and the numerical simulation results shown in Figure 3.24. In particular, the lower graphs (showing the relative position of pendulums) are almost identical. The upper graphs are also similar. However, for values of ? C near 1.0, the experimental and simulation results differed for the absolute position of the pendulums. This area is marked by dashed lines on the upper part of Figure 3.24. Since the relative angular velocity (?/? n ) is nearly 20 and the damping ratio 66 is quite low, the transfer function is close to 1. Thus, the center of rotation is very close to the CG of the imbalanced rotor system, which is also the location of the pendulum when ? C is near 1.0. This means the center of pendulums is near the source of the centrifugal force field. So at this location, the relative position of the pendulums will be 180? but the absolute position of the pendulum configuration is indeterminate from the perspective of balancing the system. When the suspension point of the pendulums is moved slightly from the CG of the unbalanced rotor, the relative distance of the pendulums slightly decreases and this neutral state will change. Thus, small effects that are not included in the simulation model but that are present in the experimental facility, such as rolling resistance and friction, will tend to produce a different absolute position of the pendulums from that predicted by the simulations. This explains the differences between the numerical simulations and the experimental measurements in the region where ? C is near 1.0. 67 3.3.2 Experimental comparison of the sensitivity and consistency of ball and pendulum balancers This investigation begins by considering the effects of rolling resistance. Previous investigators have noted that rolling-resistance and dry friction are significant problems that tend to degrade the consistent performance of ball balancer systems [12], [13]. However, previous work has not considered the influence of such effects on pendulum balancers. A basic question is: Are pendulum balancers less susceptible to such effects (rolling resistance and dry friction) than ball balancer systems? Accordingly, in the following sections, a comparison is made between the performance of these two types of systems with the goal of providing some insight into this question. Two comparable test rigs are developed, one with a ball-balancer system and the other with a pendulum balancer system, and experimental results from each are compared. The observed results are discussed and some insights into the expected relative performance of such systems are presented. Ball Balancer Experimental Facility Photographs of the ball balancer test facility are shown in Figure 3.25 (side view) and Figure 3.26 (top view). The same central suspension was used as for the pendulum balancer described earlier. This suspension allows the center of the disk to move in a nearly horizontal plane without significant friction. Distinct from the pendulum balancer system, the moving elements are two steel balls (2, 3) guided by a cylindrical channel (4) (also a secondary channel (5) can be seen for later investigations) machined in an aluminum disk (1). The channels are covered by a 68 plexi-glass cover (7) for safety reasons and to support the dial plate. The rotating disk and the plexi-glass cover together are well balanced. The imbalance is generated by component (6) which is a brass block whose position in the radial direction can be adjusted. Ball Balancer Experimental Results Using this facility, a series of experiments was performed to evaluate the performance and consistency of this system. The experimental test procedure consisted of the following steps. First, the balance state of the rotor was set using the following procedure. The steel balls were locked at ?30? (as shown in Figure 3.26) and the system was balanced by adjusting the radial position of the brass block (6). Accordingly, the magnitude of the imbalance and the desired settling position of the balancing balls (when released) are precisely known. Next, the balancing balls were released and repeated startups were performed without changing any of the other physical properties of the system. For each startup, the system was started from rest and was driven until the rotor disk had reached 1500 rpm. The amplitude of the steady-state vibration and the position of the balancing balls were then recorded. 69 Figure 3.25 Side view of ball balancer experimental facility 70 Figure 3.26 Top view of ball balancer experimental facility Figure 3.27 serves to graphically summarize the results of these experiments. The horizontal axis shows the positions of the balancing balls. Each pair is shown by a different symbol and is connected by a dashed horizontal line. The vertical axis shows the measured amplitude of the center of disk in the ? 1 direction. The smallest measured vibration level was recorded when the balancing balls were locked at ?30?. The highest vibration level was recorded when the rotating disk was not equipped with the balancing balls. This level of vibration (0.80475 [mm]) would be the amplitude of the steady state motion of the rotating system without any passive balancing mechanism. For all of the other test cases, the balancing balls tended to reduce the overall vibration but they settled 71 to different locations each time with a relatively large scattering. The above experiment was repeated with a variety of different sizes and numbers of balls, with similar results. -60 -40 -20 0 20 40 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Vibration amplitude without balancing balls: 0.80475[mm] -30 30 Position of balancing balls [deg] R e la tiv e a m p litu d e o f v i b r a tio n [- ] Figure 3.27 The final positions of balancing balls and vibration level for different startups These results clearly indicated that there are some physical effects that prevent the balancing balls from settling into the proper positions and achieving more consistent results. This effect can be traced to the deformation of the contact point of the ball and channel surface due to the high normal forces generated by the centripetal acceleration and the resulting dry friction. For the specific experimental setup used for the present tests, the mass of each 15[mm] diameter chrome steel ball was 13.8[gram] moving in 114[mm] radius and the material of the channel was alloy 6061 aluminum. This material 72 and geometrical combination resulted in an approximately 16 [?m] deformations at the ball contact points when the disk was rotating at 1500 rpm. Figure 3.28 illustrates a graphical explanation of rolling resistance. The deformation of both channel and ball forms a pocket at contact point A and the idealized point-to- point contact becomes a surface as a result of high centrifugal force F CF . This elastically formed pocket changes the direction of constraining force, F CS , that was directed toward the center of rotation in the idealized model. So, the constraining force has a component that tends to oppose the balancing force, F B . In the ball balancing system that was investigated, the centrifugal force, F C , is much greater than the gravitational force, mg. Because of this high force ratio, the ball contact motion will primarily be rolling at point A and slipping at point B. Relative motion at the contacting surfaces at point B will result in a friction force, F F , which also tends to oppose the balancing force, F B. The action of these combined forces tends to stop the motion of the balancing balls before they reach the complete balancing position, resulting in a nonzero balance state for this system. This remaining mass imbalance produces synchronous vibration, which is thus the indirect result of rolling resistance. 73 A B mg FC FC FCS FB FF C A C FCB FCS S L O P E Figure 3.28 Deformation of contact surfaces and force distribution of the balancing ball and channel 74 Pendulum Balancer Experimental Results A similar series of experiments were performed for the pendulum balancing system. The rotor was repeatedly started without changing any parameters of the system. The rotor vibration level and the position of the pendulums were captured when the system reached the final speed and exhibited steady-state behavior. The results of these experiments are summarized in Figure 3.29 and Figure 3.30, in a fashion similar to that which was used for the foregoing ball balancer experiments. In Figure 3.29, the amplitude of the steady-state periodic motion is also shown to illustrate the relationship of the different counterbalancing levels. When the pendulums are locked at the 0 o and 180? positions respectively, the overall vibration level of the system is 1.75[mm]. Figure 3.30 shows a closer (zoomed) view of the settled pendulum positions for better insight. A comparison of the results for the ball balancer system and the pendulum balancer system shows that the pendulum balancing system has a much greater consistency. 75 -60 -40 -20 0 20 40 60 0 0.2 0.4 0.6 0.8 1 Vibration amplitude without balancing pendulums: 1.75[mm] -30 30 Position of balancing pendulums [deg] R e l a t i v e a m pl i t ude of v i br a t i on [ - ] Figure 3.29 The final positions of balancing pendulums and the level of vibration for different startups 76 -32 -31 -30 -29 -28 0.0105 0.011 0.0115 0.012 0.0125 Position of balancing pendulums [deg] R e la tiv e a m p litu d e o f v i b r a tio n [ - ] 27 28 29 30 31 0.0105 0.011 0.0115 0.012 0.0125 Figure 3.30 The final positions of balancing pendulums and the level of vibration on the zoomed plot for different startups Figure 3.31 illustrates the force distribution of a pendulum balancer system and of a ball balancer system. F RR is the rolling resistance generated by the deformation of the contact surface, friction and other less significant forces. F CF is the centrifugal force. F C is the constraint force which guides the moving masses, pendulums or balls, in a circular path. F B is the balancing force. Inspection of these diagrams show that the pendulum balancing system has a better overall force distribution in the sense that (in general) a larger percentage of the forces are directed toward achieving relocation of the balancing mass. This serves to dramatically improve the overall sensitivity of the pendulum balancer system with respect to a similar ball balancer system. Thus, pendulum balancer 77 systems tend to be inherently less susceptible to inconsistencies and errors as a result of rolling resistance. ?1 MP ? FCF FB FC FRR FCF ?1 FB FRR MP FC ? b.a. Figure 3.31 Force distribution of pendulum and ball balancer 78 4 PASSIVE PENDULUM BALANCER WITH NON-ISOTROPIC SUSPENSION 4.1 Analytical investigation The previous studies have targeted isotopic suspension of rotor system where the spring stiffness and the damping were identical. Identical suspension results two identical natural frequencies. It was demonstrated that the Type I singular point of an isotropic pendulum system showed stability when the system was operating supercritically. This part of the research has investigated a rotor system equipped with a pendulum balancer and supported from ? 1 and ? 2 directions by non-identical springs and dampers. The application of two different supports results in two distinct natural frequencies. The analytical investigation starts with the development of the equations of motion. 1, 2, 3, 4 i i iii dT T V D Qi dt ?????? ?++= = ?? ?? ?? ?? ?? ?? nullnull (4-1) Where: T = Sum of all kinetic energies V = Sum of all potential energies D = Rayleigh dissipation function i ? = Generalized coordinates i Q = Generalized forces 79 () ()() () () ()( )() ()( () () 2 1323 1424 22 2 31 12 21 2 3 3 21 1 4 4 3 21 21 2313 1 1 2cos 2sin 2 2cos 2sin 1 2 2 sin cos cos sin ABA BBB P D BA BB P D D BA A P B A BA BB D PA B TSR SR SP M M M M MR MS M SSR M SS S ? ? ? =+??+??+ ? ? ++??+??+ ? ++?++++?+?+ + ? ? +??+???+?+ +? ??+ ?+ ??+ ??+ +?+?? ??? nullnullnull nullnullnullnull nullnull null () () () () () ) ()() () () 4 244 122 3 2412 4 12 1312 22 21 22 34233244 14 sin cos cos sin 11 11 cos cos 22 si BBAA BDBB AP DBABBP DBABBP ABA BBB A B B SM SRM SM MM M M MM M M SR SR S S S ?+ +? ??? ??+? ?+ +? ??? ?+ ?? ??? ?? ??? ?+ ++++?++++?+ + ?+ ?+?? ?+?? ?+ ??? nullnull null nullnull nullnull nullnull nullnull null null nullnull nullnull nullnull () () 4133 nsin A S???? ? nullnull (4-2) () () () () 22 11 221 2 cos sin sin cos 22 Vk t t k t t?? ??=???+?+? (4-3) () () ( ) () ()()() ( () ()()() ) () ()() () ()() 22 2 11 2 11 2 2 1 2221 2 2 12213 2 2 22214 1 cos sin 2 cos sin cos sin cos sin cos sin 11 sin cos Dctct ct t t ct t t ct t c ct t c ?? ?? ? ? ? ?? ? ? ? ?? ? ? ?? ? ? =? + + +? ? ? ? ? +? + + ? ?+ ?+? + + ? ?? ?+? + ? + + ? ?+ ?+? + ? null nullnull null nullnull nullnull (4-4) 0 i Q = (4-5) 80 The most general form of the equations of motion are: () ( ) ( ) ( ) () () () () () () ()() () () ()() () () 22 33 33 3 33 44 44 44 4 22 11 2 3 1 122 221 11 2cos sin cos cos 2cos cos sin cos 2 cos cos sin sin sin cos cos cos A B P D BA BB S S MPMM ct t t ct t t kt t ?? ??? ????? ????? ?? ???+?+?+?? ???+?+?+?? ????+?++++?+ +????+?+ + ? ?+ ?+? + +? nullnullnull null nullnullnull nullnull null nullnull null null () () () () () 2 21 2 21 sin sin sin cos 20 DBAB t kt t t MM M ? ?? ? ?? ??+ +?? ?++ ?+?= null (4-6) () ( ) ( ) ( ) () () () () () () ()() () () ()() () () 22 33 33 3 33 44 44 44 4 2 22 1 2 121 222 11 2 sin cos sin sin 2 sin sin cos sin 2 sin sin cos cos cos cos sin sin cos A B P D BA BB S S MMM ct t t ct t t kt t ?? ?? ????? ????? ?? ? ???+?+?+?? ? ? ?+ ? ?? ? ?+ ? ? ????+?+++?+ +????+?+ + ? ?+ ?+? + +?? nullnullnull null nullnullnull nullnull null nullnull null null () () () () () 2 21 2 12 cos cos cos 20 DBAB t kt t t MM M ?? ? ?? ?+ +?+ +++ ???= null (4-7) ( ) ( )( () () () ()) 31 3 3 2 32 31 22 32 31 33 sin cos 2sin 2cos cos sin 0 ABA SR c ?? ???+?+??+ ??+ ??? ???+??+?= nullnull nullnull nullnull nullnull null (4-8) ( ) ( )( () () () ()) 41 4 4 2 42 41 22 42 41 44 sin cos 2sin 2cos cos sin 0 BBB SR c ?? ???+?+??+ +??+??? ???+??+?= nullnull nullnull nullnull nullnull null (4-9) 81 In order to verify that the Type I singular point exists for the non-isotropic system the, time derivative terms are set to zero in Equations (4-6)-(4-9). This results in the following set of algebraic equations. ( ) ( )( ) () () () () () () () () () () () () 2 134 21 2 12 1 11 2 21 2 cos cos cos sin sin cos sin cos cos sin cos sin cos sin 0 ES A E B E EE EE EE YMSYSY ctY tY t ctY tY t ktY tY t ktY tY t ? ?? ? ? ?? ? ? ??? ?+ + + +? ? ?+ + +? ++ = (4-10) ( ) ( )( ) () () () () () () () () () () () () 2 134 11 2 21 2 12 1 22 2 sin sin sin cos sin cos sin cos cos sin cos sin cos sin 0 ES A E B E EE EE EE YMSYSY ctY tY t ctY tY t ktY tY t ktY tY t ? ?? ? ? ?? ? ? ??? ?+ + + ++ + +? ++ = (4-11) ( ) ( )( ) 2 132 3 sin cos 0 AE E E E SY Y Y Y ?? = (4-12) ( ) ( )( ) 2 142 4 sin cos 0 BE E E E SY Y Y Y ?? = (4-13) It is easy to prove that the Type I singular point, found earlier for the isotropic system, is also a singular point for the non-isotropic system, especially when the pendulums have the same first moment of inertia S A = S B . Further investigation showed Type II and Type III singular points also satisfy Equations (4-10) - (4-13) for an arbitrary chosen non-identical pendulum combination. 82 In order to assess the stability characteristics of the Type I singular point, the differential equations of motions were linearized around this point. The resulting equations, in a matrix vector form, are: [] () () M0YDtGYKtNY++++=???? ???? nullnullnull nullnull null (4-14) The inertia matrix is: [] ( ) ( ) () () () () () () 34 33 34 0sinsin 0cosco sin cos 0 sin cos 0 SAEBE SAEBE AEA E ABA BEB E BB MSYSY MSYSY M SYS Y SR SYS Y SR ?? ?? = ? ? ?? (4-15) : S D BA BB P Where M M M M M= +++ The time periodic damping and the gyroscopic matrix is: ( ) () () () () () () () () 22134 21 1 3 4 33 44 4 2()2cos 2cos 2 ( ) 2 sin 2 sin 2cos 2sin 0 2cos 2sin 0 SAEBE SA AE AE BE BE Dt G Mcc SY SY Mcc SY SY SY SY c SY SY c ????? ?? ? ?? +=?? ?? ?+?? ? ?? +? ? ? ? ? (4-16) ( ) ( ) () () () () 22 11 2 22 21 2 :sincos cos sin sin cos Where t t ctct ct t ??? ?? ? ? ?? = =+ =+ 83 The time periodic elastic and the non-conservative force matrix is: ( ) () () () () () () () () () () () () 2 22 121 112 3 4 222 212 212 3 4 22 34 44 sin sin cos cos sin cos 0 sin cos 0 AE BE AE BE AE A E A BE B E B Kt N Mcc kk SYSY kk M cc S Y S Y SY S Y SKN SY S Y SKN ?? ? ?? ? ? ? ?? ? ? ? ? ? ? ?? +=?? ?? ? ??++? ??? ? ? ?? ? ++? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (4-17) ( ) ( ) () () () () () () () () () () () () 22 11 2 22 21 2 22 11 2 22 21 2 33 2 3 1 3 44 2 4 1 4 :sincos cos sin sin cos cos sin sin cos sin cos sin cos EEE E EEE E Where t t ktkt ktk t ctct ct t KN Y Y Y Y KN Y Y Y Y ??? ?? ? ?? ? ?? ? ?? ? = =+ =+ =+ =+ =+ 84 The Cauchy transformation of Equation (4-14) results in a system of first order differential equations with an 8x8 coefficient matrix ( )At? ? ? ? . () =tX AX? ? ? ? null null null (4-18) Where: = Y X Y ? ? ? ? ? ? ? ? null null null null (4-19) The coefficient matrix of the linearized differential equations with time periodic coefficients is: () [] [] [] () [] () 11 0 [] I At M Kt N M Dt G ?? ?? ?? = ?+?+? ??? ? ??? ?? (4-20) 4.1.1 Stability of homogeneous linear system with time periodic coefficients This section describes the mathematical background that was applied for the stability analysis. The general time periodic linear system is defined by equation (4-18) where the coefficient matrix [ ] ()A t is periodic with period 0>T , that is [ ] [ ] () ()At T At+= . Of interest is the stability of the equilibrium point 0X ? nullnull . To solve the problem, we apply the Floquet theorem [28[29]. 85 The fundamental matrix of system of the linearized (4-18) is [ ] ()t? , if the [ ] [][] () () () dt At t dt ? =? (4-21) matrix differential equation is satisfied. The following statements can be proved: - All solutions of equation (4-18) can be written in the form ( )tc?? ? ? ? null , where c null is a constant vector. - There exists a fundamental matrix ( ) 0 t?? ? ? ? , that all solutions of (4-18) come up in the form () 00 tX??? ?? null , where ( ) 0 0XX= nullnull is the initial condition, that is () [ ] 0 tI?=?? ?? , where [ ] I is the identity matrix. - All fundamental matrix can be written in the form ( )tC ? ? ??? ?? ? ? null , where C ?? ?? null is a constant matrix. - For any fundamental matrix ( )t?? ? ? ? , ( )tT?+? ? ? ? is also a fundamental matrix. - There exists constant matrix C ? ? ? ? null for which ( ) ( )tT t C ? ? ?+ =????? ???? ? ? null , where C ?? ?? null is called the principal matrix of (4-18), () ( ) 1 CttT ? ??=? ?+ ? ?? ? ? ?? ? ?? null . - The principal matrix belonging to the fundamental matrix () 0 t??? ?? assumes the form [] () ( ) () ( ) () 11 00 0 0 0 00Ct tT T ?? =? ? + =? ? + =???? ???? ??? ??? ???? ??? . - All principal matrices are similar to each other, consequently the eigenvalues of the principal matrix - called the characteristic multipliers (notation: n ??? ?,, 21 ) - are invariant, and determined by the system. 86 - System (4-18) is asymptotically stable if and only if 1< i ? , ni ?,2,1= . - System (4-18) is stable in the Liapunov sense if and only if 1? i ? , ni ?,2,1= , and if 1= i ? , than i ? is simple in the minimal polynomial of the system. In general the principal matrix can not be determined in an analytic way, but there are several methods to approximate it [16]. 4.1.2 Floquet analysis: piecewise approximation If the coefficient matrix [ ] ()At is piecewise constant, then - by the coupling of solutions - the complete solution at time Tt = is obtained in the form: [ ]() [ ]( ) [ ]( ) 11 110 () exp exp exp nn n n X TtAtA tAX ?? = nullnull ? (4-22) () [ ] [] [] 11 2112 12 1 12 if 0 if if nnn Att At A tttttttT ? ? ?? ?