THE EFFECTS OF VIBRATION ON AN INERTIAL MEASUREMENT UNIT Thesis For The Degree Of M. S. MICHIGAN STATE UNIVERSITY LOUIS RALPH PAPALE 1966 THESIS LIBRARY Michigan State University THE EFFECTS OF VIBRATION ON AN INERTIAL MEASUREMENT UNIT Louis Ralph Papale AN ABSTRACT Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1966 ABSTRACT THE EFFECTS OF VIBRATION ON AN INERTIAL MEASUREMENT UNIT By Louis Ralph Papale In this thesis, the errors produced by both sinusoidal and random vibration acting on an inertial measurement unit are considered. The sources of vibration in missiles are presented and the vibration characteristics are established. A brief description of an inertial measurement unit and its major components, which consist of the gimbal system, the accelerometers and the gyros, are presented. A state model of the gimbal system is deve10ped and an example of the state model to the design of an inertial measurement unit is pre- sented. The particular design considerations associated with vibra- tion are indicated. Analyses of the accelerometers and the gyros are performed. The error equations for the vibropendulous error, the nonlinearity error and the scale factor error of a pendulous, pulse-rebalance accelerometer are derived. The error equation is developed for the anisoelastic effects of a gyro under sinusoidal vibration. The system error equations are derived for both the sin- usoidal and random vibration effects on an inertial measurement unit. The significance of the vibration effects are illustrated and means for minimizing the effects are indicated. THE EFFECTS OF VIBRATION ON AN INERTIAL MEASUREMENT UNIT Louis Ralph Papale A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1966 "The aim of science is to seek the simplest explanation of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quests. The guiding model in the life of every natural philosopher should be 'Seek simplicity and distrust it'.” by Alfred North Whitehead "Concepts of Nature" ACKNOWLEDGEMENTS I wish to extend my graditude to several persons who so willingly provided assistance during the preparation of this thesis. Special thanks go to my advisor, Dr. H. Hedges of the MSU Electrical Engi- neering Department, for reviewing the material and offering many helpful suggestions. My graditude goes to Dr. R. Dubes, also of the MSU Electrical Engineering Department, who reviewed the portion on random vibration and provided many improvements. I would also like to thank Mrs. Ann La Comte for typing the manuscript, and LSI Publications Department for its publication. Finally, a "thank you" to my wife Jo for her admirable patience throughout the pre- paration of this thesis. ii TABLE OF CONTENTS LIST OF ILLUSTRATIONS . 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 INTRODUCTION . SOURCES OF VIBRATION DESCRIPTION OF AN INERTIAL MEASUREMENT UNIT STATE MODEL OF A STABLE PLATFORM . AN APPLICATION OF THE PLATFORM STATE MODEL . 5.1 Terminal Equation Parameters 5.2 Solution of the State Model 5.3 Design Considerations EFFECT OF SINUSOIDAL VIBRATION ON PLATFORM INSTRUMENTS . 6.1 Accelerometers 6.1.1 Vibropendulous Errors 6.1.2 Nonlinearities . 6.1.3 Scale Factor Error 6.2 Gyros . 6.2.1 Mass Unbalance Drift . 6.2.2 Anisoelastic Drift . SYSTEM ERROR DUE TO SINUSOIDAL VIBRATION . EFFECT OF RANDOM VIBRATION ON PLATFORM INSTRUMENTS . 8.1 Accelerometers 8.1.1 Vibropendulous Error . 8.1.2 Nonlinearities . iii Page 14 18 25 25 31 36 39 42 49 51 53 54 56 59 68 68 68 78 9.0 10.0 TABLE OF CONTENTS (cont) 8.1.3 Scale Factor Error . 8.2 Gyros . 8.2.1 Mass Unbalance Drift . 8.2.2 Anisoelastic Drift . SYSTEM ERROR DUE TO RANDOM VIBRATION . CONCLUSIONS References iv Page 81 86 86 88 91 99 100 Figure 10 11 12 13 14 15 16 17 18 LIST OF ILLUSTRATIONS High Speed Rocket Sled . Mean Power Level Measured on a Rocket Sled . Power Spectral Density Measured on a Rocket Sled 8 Seconds After First Motion Power Spectral Density Measured on a Rocket Sled 14 Seconds After First Motion Noise Levels for Boost Flight Generalized System Block Diagram . Two Types of IMU . IMU Gimbal System A Typical IMU Axial Deflection vs. Axial Load for a Single . Bearing Radial Deflection vs. Radial Load for a Single Bearing Platform - Theoretical Vibratory Response Pendulous Force - Balance Accelerometer with . Analog Output Pendulous Force - Balance Accelerometer with . Pulse Output System Error Model . Contribution to System Error by Platform . Instruments Due to Sinusoidal Vibration Effect of Accelerometer Scale Factor Error . on Signal Distribution Contribution to System Error by Platform . Instruments Due to Random Vibration Page 12 16 17 20 21 29 3O 37 40 4s 60 66 82 98 1.0 INTRODUCTION The major part of the missile target error produced by an inertial guidance system is, in general, caused by instrument im— perfections. However, a significant consideration, from the stand- point of both the error contribution and the design, is the effect of vibration on the instruments. This thesis considers the effects of vibration on the instruments as utilized in an inertial measure- ment unit from the standpoint of accuracy and design considerations. The errors in an inertial guidance system can be allocated to four major groups: 1. Instrument imperfections. 2. Initial alignment. 3. Simplification of guidance equations. 4. Vibration. The first two are generally included in the error analyses considered for each system. The third is evaluated when establish- ing the guidance equations and should represent a small error relative to the first two error sources. The fourth error source may be a significant amount depending on the characteristics of the instruments and the vibration environment. It is imperative in the design of an inertial measurement unit to establish the errors aris- ing from vibration. Such a determination provides the means of selection of the type of instruments to be used, such as floated or non-floated gyros; it is important from the standpoint of reli- ability as well as for establishing the need for vibration isolation. In the deve10pment of inertial measurement units, the usual shock and vibration tests are performed in the laboratory to establish the functional capability of the system. A more elabo- rate test prior to actual flight test is to subject the IMU to a high speed sled test environment. This allows the unit to be tested in an environment much like that of a missile. The material herein, although quite general with respect to inertial measurement units, presents an example of the development of a specific type of an IMU. This development program was carried to a point beyond the normal laboratory testing phase and included a high Speed sled test program at Holloman Air Force Base, New Mexico. Examples of actual data obtained at the track are included. Specifically, the material herein presents a unified treat- ment of the effects of vibration on an IMU from the standpoint of both sinusoidal and random vibration. The particular points of accomplishment are as follows: 1. A state model of the IMU gimbal system is derived. 2. An example of the application of the state model to a Specific IMU is presented, and the design considerations for future engineering efforts are outlined. 3. The error equations for the vibropendulous error of a pulse-rebalance accelerometer are derived. 4. The error equations for the nonlinearity errors for an accelerometer are deveIOped. 5. The error equations for the scale factor error of a pulse-rebalance accelerometer are derived. 6. The error equation for the anisoelastic effects of a gyro under sinusoidal vibration is developed. 7. The system error equations for both the sinusoidal and random vibration effects on an IMU are derived. In summary, the contribution of vibration to the IMU system error can be significant, and can range to as high as 25% of the allowable miss distance, depending on the level of vibration. The particular features to be considered in minimizing the vibration effects are pointed out. The arrangement of the material is based upon the logical flow of the vibration from the missile structure to the platform instruments. Consequently, the first consideration is the missile environment and the characteristics of vibration. This, then, describes the input signal at the case of the IMU. Following this discussion is a brief description of the IMU and the major items considered in the study which were the gimbal system, the acceler- ometers, and the gyros. Continuing with the signal flow, the next presentation is the development of the state model of the IMU gimbal system. It is the gimbal system that shapes the environment seen by the platform instruments and is illustrated by an example. The specific error equations for the platform instruments (acceler- ometers and gyros) due to sinusoidal vibration are developed, and the system error equation due to sinusoidal vibration is formulated. Similarly, the error equations for the platform instruments due to random vibration are derived, and finally, the system error equation for random vibration is developed. 2.0 SOURCES OF VIBRATIONS Prior to any considerations of the effects of vibrations, it is helpful to review the sources of vibrations in missiles. This section presents a discussion of the three main sources of vibra- tions in missiles and points out the various characteristics of the vibration which most likely represent the environment of the IMU. Vibrations in missiles are generated by three main sources: 1. The missile power plant. 2. Aerodynamic effects, such as boundry layer and turbulence. 3. Internal operating components. In addition to these sources, a high speed sled used in testing inertial guidance systems has one additional vibration source. This is the effect of the slipper and slipper suspension on the track (see Figure 1). Examples of the vibration presented on a rocket sled are shown in Figures 2, 3, and 4. The missile power plant may consist of a single rocket or a cluster of rockets, depending on the vehicle. All rocket thrust chambers vibrate with variations depending upon the design. The physical mechanism of all these vibrations is not clearly estab- lished but basic types of vibrations have been observed in various rocket thrust chamber assemblies. The first is believed to be a mum! _/ comma! IE5! ,, smlm COIPARIICII swim SUSPENSION SLIPPERS FIGURE 1 HIGH SPEED ROCKET SLED omgm meuom < 20 omz3mm4 mwzom z g aghggg 20:.02 hmmHm $3.12 mazoumm w Guam meuom < 20 amm3mHHmqu fiEHUmmm mmzom m mmzomm The? .p :5.» a» Sp fife s ,,x.. .3113 .HIFflwMHFVFPHWTIEFIMHflAHFU.31.: r.» A . . _ . a . _ p . . Cipw. mm. R: 0! 60d _ Prim “ 8n n 0%» _ 00W , 68% OWN. a 55.. 53:20]. oyliillill -, oi oI , ? all IOIIII > I l ‘ 1 .. I}, «I‘ll OIIC'I. I c v-uFI,’ .IIO-u': Dis-I.- l c I. 16. ~_! ZOHHOZ mm%u< mmzouum vH Guam meuom < 20 omm3mP~mzmo 4<¢H0mgm muzom v mmDUHm +4? :AFPPRHPVSL F9519..." Hufire+p PHF 1+ P: 1.1.1511---) II o , on x 6.1 oou _ and 08 65:3: _ A, w _ a: lud o Iw a! 0.!!! . u 3 93 . n U! O.a I 1 IN. Rioolcsu call. 0"... "ogl - U a. I.‘ 10 chamber pressure and feed system oscillation with relatively low frequencies in the range of 0.1 cps to 15 cps. The second type of vibration is that caused by the excitation of the natural fre- quency of the metal parts, such as the chamber, pipelines, and structural parts. This frequency is usually below 100 cps. The third type is a high pitch, high energy vibration and is associated with the combustion. Overall, the vibration Spectrum is of a random nature with the power levels concentrated at the points indicated. In considering the aerodynamic effects, it can be shown7* that the vibrations due to the boundary layer are caused by surface pressure fluctuations created by turbulence within the boundary layer. In addition, atmospheric turbulence may cause surface pres- sure fluctuations. Further considerations of aerodynamic effects must take into account the flight conditions (dynamic pressure). These are the vibration conditions that exist in the subsonic state. However, in the transonic region, a condition known as transonic buffeting takes place. This is created by a highly turbulent con- dition with a build-up of dynamic pressure. One kind of buffet is identified by a white distribution of power in the power spectrum. The transonic region is of major concern because of possible structural and equipment damage . However, the duration of flight through this region is relatively short for missile flights. * Superscripts refer to the references. 11 Another kind of buffeting - depending upon the re-entrant angles - is that which takes place due to the detachment of the air flow on bulbous configurations. The pressure fluctuations act over a greater area than the first type of buffet and the frequency dis- tribution of the pressure fluctuations is concentrated more at the low end of the spectrum. The greater the magnitude of this buf- feting, the greater the intensity of the lower frequency components which in turn tend to excite the primary structural modes of the vehicle. Finally, such things as hypersonic buzz (a hinge moment oscillation produced by high Speed flight) and flap flutter also lead to vibration effects. The third item considered as a primary source of vibration is the internal operational components. These, of course, could be such items as motors, hydraulic pumps, inverters, blowers and the like. It has been shown, in missile experience, that the vibration environment for the vehicle arises principally from acoustic pres- sures impinging on the vehicle surface. A typical curve of rms acceleration at some location on the re-entry vehicle, e.g., the guidance truss, as a function of flight time is given in Figure 5. At launch, the sound field from the booster engines envelops the re-entry vehicle and excites the structure and its contents. As the 12 vehicle clears the pad and begins to accelerate, the noise level drops until the excitation is due mostly to thrust pulsations being fed through the structure. As Mach 1 is approached, the buffeting forces build up and the vibration levels exceed those at launch. After Mach 1, the dynamic pressure continues to increase but the flow is less disturbed so that, in general, an intensity plateau is maintained until after maximum dynamic pressure is reached. As the dynamic pressure decreases and the vehicle leaves the atmosphere, the noise again drops to the structural-path contribution level. Mach 1 Maximum dynamic pro. owe 50f O O 1 L aunch w . I Q I ‘ I I # l I J J 1 , 0 20 60 80 80 100 Time of Flight (sec) FIGURE 5* (d O L i I g Acceleration (3 rm.) N O NOISE LEVELS FOR BOOST FLIGHT In producing such vehicles, the design provision for re- duction of equipment vibration level utilizes a well-tested proce- dure for frequency separation called the octave rule. The octave * Reprinted from "Astronautics and Aerospace Engineering" with permission. 13 rule requires the basic frequency of each component to be an octave higher than that of the structure on which it is mounted. In this way, the possibility of the resonant frequency of one equipment being the same as that of another is avoided. A typical vibration environment for an inertial measurement unit for a missile is a power spectral density of 0.07g2/cps over a frequency range of 20 to 2000 cps. From a system standpoint, this is a white noise input to the IMU, which is briefly described in the next section. 3.0 DESCRIPTION OF AN INERTIAL MEASUREMENT UNIT Having established the vibration environment, it is appro- priate to consider the IMU per se. This section presents a brief description of an IMU, its function in relation to the missile, and the major components. An inertial measurement unit is a device capable of measur- ing any changes in rotational or translational motion with respect to an inertial frame of reference. As implied, the means of opera- tion is based upon Newton's laws of motion and consequently an inertial guidance system is sometimes referred to as a "self- contained system." From the basics of kinematics it is known that the trajectory of a body may be defined by an appropriate combination of transla- tional and rotational measurements. Thus, to describe the path of a missile, an inertial measurement unit is used to provide such information (see Figure 6). This information is then supplied to a guidance computer in which it is utilized as the present position of the missile. Stored in the guidance computer is a specific program indicating where the missile should be at a particular time. The two pieces of information are compared and the difference then serves as the basis for applying corrections to the propulsion system and/or the control system. Hence, the accuracy of the tra- jectory is primarily dependent upon the accuracy of the inertial measurement unit. 14 15 The function of an IMU can be provided in two ways. In either case the basis of operation is the gyro and the accelerometer. One method, known as the stable platfonm approach, has a mechanical gimbal system which operates as part of a stabilization loop to maintain the accelerometers (located on the stable element) fixed to some reference (see Figure 7A). The second method, known as a strapped-down system, has the accelerometer and gyros located di- rectly on the missile airframe. Hence, a computer is required to convert from the missile body coordinate system to an inertial coordinate system (see Figure 7B). Consequently, in examining the effects of vibration on an inertial measurement unit, the gyros, accelerometers and the gimbal system must be analyzed. The first thing to be analyzed is the gimbal system which is treated in the next two sections. l6 2m omNHqm zzoo ommmb 02% m mmDUHm Kuhn-a 200 .quod mwx< wozm Zmomh<4m mamo ucapuacpm cmkaazoo . w 3.3.: asses... arm”: .3: 342.303 - _ .< mom>o 3225.584 _ 3:2 44:52. H.534 Scour—kn...— w..o<._.m 4.0 STATE MODEL OF A GIMBAL SYSTEM In establishing the effects of vibration on an inertial measurement unit, it is first necessary to develop a methematical model of the gimbal system. This model may then be utilized with specific vibrations present at the missile airframe as described in Section 2.0 to determine the type of vibration seen by the IMU in- struments located on the stable platform. A schematic diagram of a gimbal system can be represented by a combination of spring, mass and damper elements as shown in Figure 8. Three degrees of freedom are indicated since the gimbal system is assumed to consist of three gimbals. Additional degrees of freedom are sometimes introduced in the form of a fourth gimbal or a vibration isolator. However, this is a design parameter and only a slight modification of the state model is required to provide for this. The analysis herein considers translational motion only since the platform is assumed to be independent of rotations. This is a realistic assumption since the stabilization loops have a capability of maintaining the rotation to a few arc-seconds of motion. In the mechanical system, each mass element represents the total lumped mass parameter of each gimbal and the particular in- struments located on that gimbal. The spring element represents the 18 19 stiffness coefficient of the combination of the particular gimbal and the bearings used in the mechanization. The damper element represents the structural damping and windage effects. Figure 9 illustrates a typical IMU with roll, yaw, and pitch gimbals. The development of the state model for the gimbal system will now be considered. The terminal equations for the components in the system can be written in the form F. T C 1 “- I- P 61(t) - MT. 0 0 - fICt) d 0 1 ET:- 52(t) = 0 M—2- 0 f2(t) . 1 0 0 ——- f t 53(t) M3 3( ) L .4 L. a L. .4 'fi r' “ r- . '1 ffhft) K4 0 0 01+(t) d - 0 K o 6 (t) -d—t- f6(t) " 6 6 f8(t) 0 0 K8 580:) L. .4 L— -J L. .4 q r- “ . H H5 (t) 05 0 0 I656) f7(t) = 0 07 0 576) f9 (t) o 0 [)9 égm L. .4 L. a b .4 610(t) a specified across driver 20 Imm 3 Embm>m qm q 1). The torque is also a function of the pendulum swing which for this type of accelerometer is a constant. Reduction in the threshold level (pendulum swing) produces a proportional decrease in the vibropendulous torque. The final difference to be noted is the dependency of the torque on 0 rather than 2 O. Therefore, the maximum torque will be produced by an applied vibration lying essentially along a line perpendicular 49 (O < 90°) to the sensitive axis of the accelerometer. The require- ment for O to be less than 90° stems from the need for a small com- ponent of acceleration along the X axis which is necessary to drive the pendulum off of the null position. 6.1.2 Nonlinearities Another characteristic of the accelerometer that is impor- tant when considering vibration is the nonlinearity in gain that may exist. It will be shown that an error is produced by the non- linear terms due to rectification effects. Consider the output of an accelerometer to be described by the following expression: V = Ko + K1 A + K2 A2 + K3 A3 + ... where V = accelerometer output in volts A = acceleration input KO = bias terms K1 = scale factor K2 = coefficient of second order nonlinearity K3 = coefficient of third order nonlinearity 50 Assume the acceleration input consists of thrust acceleration and vibration. This can be represented by A = AT + AV Sln wt Substitution into the above expression with consideration given only to the second order nonlinearity (assuming the higher terms are negligible) and terms including vibrations yields < l 2K2 AT Av sin wt + K2 Av2 sin2 wt 01‘ K2 2 V = 2K2 A A sin wt +--——- (l-COS 2 wt) T V 2 Taking the average and considering only complete cycles results in the error produced by vibration as However, an important aspect of the two terms drOpped should not be overlooked. Although the average value of each term can be seen to be essentially zero, the amplitude of the error contribution can be significant. Therefore this could lead to saturation of the acceler- ometer and additional errors. For purposes of this thesis, it is assumed that the accelerometer is not saturated and that the above error equation holds. 51 6.1.3 Scale Factor Error The particular characteristic of an accelerometer defined as scale factor was noted previously. Here it will be shown how such a characteristic of a pulse-rebalance accelerometer generates an error under sinusoidal vibration. Consider the expression for the pulse-rebalance acceler- ometer as follows: P = K At where P = accelerometer output in pulses K = scale factor pulses/sec/g A = acceleration input g = gravity t = time in sec. Since the accelerometer is designed such that each pulse is equi- valent to a specific value of velocity, the total velocity can be found by performing an algebraic summation, or where N = total number of increments of velocity 52 The point under consideration here is that the scale factor is dif- ferent for the positive and negative sides of the accelerometer as shown in the figure. Consequently under a sinusoidal vibration the difference between the two slopes will result in an error. + PULSE 101K £1100 - PULSE CONT/5E0 This can be seen by considering the above expression with two different slopes as P = KlAlt1_K2A2t2 where the subscripts refer to the polarity. Assuming a sinusoidal vibration, the following holds: then P = (K1 - K2) At S3 The velocity error produced by this is Therefore, in spite of the fact that the input vibration has a zero mean, an extensive error can be built up due to vibration. This aspect is particularly important in inertial guidance systems since a common guidance technique utilized is to fly a tra- jectory such that the cross-axis acceleration is held to zero. Under this condition the cross-track accelerometer will be subjected to vibration inputs and could produce significant errors. The discussion of the effects of sinusoidal vibration on accelerometers is now complete. The next section will present the specific considerations of gyros under vibration. 6.2 GYROS The drift in gyro performance is categorized into two major groupsg: non-g-sensitive drift, and g-sensitive drift. The first category is unaffected by vibration, and consequently this section will be concerned only with g-sensitive drift as produced by sin- usoidal vibration. 54 The g-sensitive drift is further divided into two groups: 1) drift due to mass unbalance, and 2) drift due to anisoelastic effects. Each of these will be considered in relation to vibration. 6.2.1 Mass Unbalance Drift One effect of sinusoidal vibration on gyros can be seen by considering the expression for gyro drift due to mass unbalance: T . K1 0 = 'f— Adt o where O = drift rate in deg/hr K1 = error coefficient due to mass unbalance deg/hr/g A = total acceleration, g's If the total acceleration is again considered to be composed of the thrust acceleration and a sinusoidal vibration then: T $= ? (AT+Avsinwt)dt O I: ;1[ (ATt--:—Vcoswt)]: or 55 The first term is the error produced by the thrust acceler- ation and is normally considered in error analyses. The second term can be seen to have a zero average contribution. Hence, it can be concluded that sinusoidal vibration will produce no steady-state error from the mass unbalance coefficient. It is interesting to consider if this conclusion is upheld by applying an equation for a pendulous mass of an accelerometer. That is, the mass unbalance of the gyro can be viewed as a pendulous mass about the output axis; therefore, it is reasonable to ask if 3 vi- bropendulous error is also incurred. This may be answered by con- sidering the expression for vibropendulous torque of a pendulous force-balance accelerometer as presented in Section 6.1.1: P2 A2 6 sin 2 ¢ cosgp M = 2 where M = vibropendulous torque dyne-cm P = pendulosity A = rms vibration 6 = pendulum deflection rad/dyne-cm ¢ = line of applied vibration with respect to the sensitive axis 0 = phase lag of the pendulum deflection to the applied vibration 56 The particular parameter that must be determined is the pen- dulum deflection. The relationship between torque and the gyro output axis angle can be seen from the transfer function a = 1/0 T s(r s + l) where r = time constant, I/D p = gyro angular displacement T = torque input For most gyros the time constant is approximately equal to one, and in addition, the damping coefficient is much greater than one. Therefore, from these considerations it can be seen that any sinusoidal input signals would be greatly attenuated. Hence, under these conditions, the vibropendulous torque due to mass unbalance is insignificant and corroborates the previous analysis. 6.2.2 Anisoelastic Drift The anisoelastic drift due to sinusoidal vibration can be seen by considering the following equation: K Sin 2 9 2 A2 dt .9. "-1 57 where E = drift rate in deg/hr K2 = error coefficient due to anisoelastic effect in deg/hr/g2 e = angle the input axis makes with the applied vibration A = total acceleration, g's For the case where the total acceleration acting on the gyro is composed of the thrust acceleration and sinusoidal vibration, the drift error is found to be T 0 K2 sin 26 ¢ = ——T———— [52+2ATAvsinwt4’Av2 sin2 wt]dt o The first term is the error contributed by the thrust ac- celeration and shall not be considered further. The second term can be seen to result in an average contribution of zero. The third term results in the following . K2 A12 sin 29 1 . 1 T m I T [- ECOSthIDwt‘Fft O or . K2 sz sin 26 ¢ = 2 58 It should be noted that - due to the sine term — the drift rate of the gyro is sensitive to the direction of applied vibration. In some cases, where the direction is changing, some cancellation can occur . This completes the analysis of the major platform instruments in relation to sinusoidal vibration. The system error due to the combination of each of the platform instrument errors is treated in the next section. 7.0 SYSTEM ERROR DUE TO SINUSOIDAL VIBRATION The ultimate objective of the analyses carried out herein is the formulation of the equation for the total system error due to a sinusoidal vibration present on the case of the IMU. This section presents the development of such an equation. From the standpoint of vibration effects the IMU has been considered to consist of three major components: 1. a gimbal structure 2. accelerometers 3. gyros Each of these has been treated in the preceding sections but it remains to establish their relationships to the system error. Refer to Figure 15. The system error is seen to be the error existing in the accelerometer output, and is dependent upon the three major com- ponents. With a given vibration on the case of the IMU the gimbal structure serves to shape the vibration seen by both the gyros and the accelerometers. Consequently the susceptibility to vibration of the gyros and accelerometers,in turn, produce errors in performance. The accelerometer error adds directly to the system error whereas the gyro errors have an indirect effect. The errors in the gyros cause the stable platform to lose its orientation (tilt) there- by introducing coupling errors in the accelerometers. The errors 59 O 6 coax. zmpm»m _¢m.wzomm.moo< ‘Ar / / Hz. :zo..<.m Sumo: mo... Emewmw m. mmDUHm pzmxm.m m.m mw (I) II accelerometer scale factor error, ft. G = accelerometer error due to gyro drift, ft. This expression assumes that the acceleration and velocity errors defined by the individual error equations are appropriately in- tegrated over the time of flight of the missle. This completes the analyses associated with sinusoidal vi- bration. The next two sections are concerned with random vibration. 8.0 EFFECT OF RANDOM VIBRATION ON PLATFORM INSTRUMENTS Whereas sinusoidal vibrations can be described over all time by a particular function, a random vibration can only be defined on the basis of probability. This section will consider random vibra- tion and its effect on accelerometers and gyros. 8.1 ACCELEROMETERS Three possible sources of error in an accelerometer produced by a random vibration -- and known as vibropendulous torque, non- linearity, and scale factor variation -- will be presented here. The type of accelerometer to be considered is the pulse—rebalance accelerometer described in Section 6.1. 8.1.1 Vibropendulous Error Within the trigger levels of a pulse-rebalance accelerometer (refer to Section 6.1) the pendulum is essentially free with the only restraining force being produced by the damping fluid and the inertia of the pendulum. The vibropendulous error produced by a random vibration acting on such an accelerometer will be developed here. Consider a pendulum as illustrated in the figure. An ap- plied acceleration (A) is shown along a line at an angle 0 with the 68 69 accelerometer input axis. The components of the applied accelera- tion are seen to be ll X: Ax = A sin 6 Ay = A cos ¢ = y A A ‘ ACCELERONEIER OUTPUIAXIS I y mm 1005 (OUT 0r PAPER) 7 Each of the components of acceleration will produce a reaction torque due to the pendulosity of the accelerometer. The pendulosity is P = mr where m is the mass of the pendulum and r is the length. Then, the equation of motion is - P §(t) cos 0 + P §(t) sin e = I e + C 6 Since 6 is small, then cos 6 can be replaced by l and sin 6 = 6, so - P §(t) + P §(t) e = I e + c 6 This is a nonlinear equation which can be linearized by considering the magnitude of 6. The maximum value of 6 in a pulse-rebalance 70 -4 accelerometer is 2 x 10 radians. Therefore the contribution by the P x(t) 6 term is small compared to the P y(t) term and may be disregarded without any significant loss. Then the expression be- comes: - P §(t) = I 0 + c 0 Taking the Laplace transform (assuming all the initial conditions are zero) yields - P r(s) = Is2 0 + Cs 0 Solving for 6 gives P - - -Y(s) 6(5) f SES+1 The time solution can be obtained using 6(t) = - %- f1(T) f2(t-t)dr O 71 Let _T_ ' I c f1(T) = 1 - e f2(t-T) = §(t-i) then t - _T_ P [. .. I/C ] 6(t) = - C' y(t-T) - y(t-T)e dr 0 or t P o. 6(t) = - C. y(t-t)dr o t dT 72 Since the applied vibration is considered to be wide-sense ergodic, then the first term integrates to zero or T - T7E' y(t-t)e dT (urn 6(t) As pointed out in Section 6.1 the vibropendulous torque is the product of pendulosity, the cross-axis acceleration (in this case x(t)), and the swing of the pendulum. Therefore the vibro- pendulous torque is given by M(t) = P566) em Substituting from above yields t _..L_ p2 u . I/C M(t) = E—- x(t) y(t-T)e dr 0 Since §(t) = A(t) sin 0 in) = Mt) cos 0 73 and since the autocorrelation function of A(t) is R ) = E[A(t) A(t-r)] AA(T the expected torque can be written as 2 ' ' ME(t) a E[M(t)] = P Slnci cos I RAA(I)e T76. dt 0 Since the average value of the expected torque is desired, where t _ lim 1 avg - T + w T ME“)dt 0 then T t - T _ P2 sin 9 cos ¢ lim 17C Mavg - C T + w RAAU)e det o o Interchanging the order of integration and carrying out the inside integration results in two terms; one reduces to zero with the 74 substitution for RAA(T) as noted below. Considering only the remaining term, gives M = P2 sin 6 c0549 avg C If it is assumed that the input vibration has a constant power Spectrum over the bandwidth ml to wz and zero elsewhere, the autocorrelation function can be found to be A o . . R(r) = fir- Sln mgr - Sln wlr Substituting into the above yields r P2 Ao sin 0 cos 6 sin wZT - sin wlr ' T7C' M = e dr avg C T o or 5 __T_ I C P2 Ao sin 0 cos 0 sin wzr e / = (IT avg C r o L . m _..L. Sin wlr e I/C - dt T O .4 75 Let ¢1 = £———-- Sin ml T dT Differentiating under the integral sign gives r 3 4’1 ' I/C = 6 cos ml T dT 8 ml 0 or 1 d T1 I/C d ml — 1 2 __ + ml2 I/C Since - _i_ we — I/C the above becomes d 01 we d ml - w 2 + w12 C 76 Then (.0 C N w 2 + ml C Carrying out the integration gives From above it was found that P2 Ao Sin ¢ cos ¢ Mavg : C (T2 " T1) or P2 AO sin 6 cos ¢ -1 wz _1 (ml) - tan avg C m or 77 Since A0 is in g2/cps this needs to be corrected to rad/sec P2 AO sin O cos ¢ -1 w2 -1 ml M = tan -—— - an ——- avg 20C we we where Mavg = average vibrOpendulous torque, dyne-cm P = pendulosity, gm-cm A0 = power Spectral density, gZ/cps C = damping coefficient, dyne-cm-sec ¢ = angle the axis of applied vibration makes with the accelerometer input axis wl = lower frequency of vibration input, rad/sec mg = upper frequency of vibration input, rad/sec me = accelerometer corner frequency, rad/sec It can be seen that the vibropendulous torque is a function of the square of the pendulosity, the power Spectral density of the applied vibration, the direction of the applied vibration and the relationship of the accelerometer corner frequency to the range of frequencies of the applied vibration. 78 8.1.2 Nonlinearities As noted previously the output of an accelerometer can be described as follows: v = K0 + K1 A + K2 A2 + K3 A3 + ~-- where V = accelerometer output in volts K0 = bias term K1 = scale factor K2 = coefficient of second order nonlinearity K3 = coefficient of third order nonlinearity To determine the effect of random vibration on the output of the accelerometer a simplified approach may be takenlo, It is a reasonable assumption that the random vibration can be described by a finite number of discrete frequency components consisting of a series of sinusoidal vibrations having constant amplitudes, and random phase relationships such that the rms value over any reason- able time period remains constant. Let all the vibration in a Strip Aw be represented by a single discrete frequency. Let the magnitude of this discrete component be N. Since the assumed vibration is assumed to range from 0 to an 79 upper frequency of G then there will be G/Aw discrete components. The approximation for the vibration can then be written G/Aw A(t) = Z n sin (n Amt + 0) n=1 where ¢ = random phase angle AS Am + 0 this approximation becomes very good. For use here let Aw = l which results in the approximation for the vibration as A(t) n sin (n t + O) 1 II II M0 Tl Considering only the second order nonlinearity and making the sub- stitution into the above expression for the output of the acceler- ometer results in the term G K2 2 N sin (n t + 0) n=1 80 Carrying out the multiplication results in terms of the form K2 N2 sin2 (nt + 01) + K2 N2 sin (nt + 0“) sin (mt + 0m) The rms value is Therefore, the error in terms of the rms value is I‘IIIS where A = rms value of the applied sinusoidal vibration Thus, with the rectification effect of the accelerometer, each frequency contributes to the error. In addition, the error is also related to the bandwidth of the accelerometer which certainly is reasonable since frequencies above the bandwidth of the accelero- meter would not be expected to add to the error. From this consider— ation, it can be seen that two ways of minimizing the error are by reduction in the error coefficient and decreasing the accelerometer bandwidth. 81 8.1.3 Scale Factor Error As with sinusoidal vibration, a difference in the scale factor over the positive and negative range of operation of a pulse- rebalance accelerometer produces an error when subjected to random vibration. This section derives the error equation for such a con- dition. Here, again, the random vibration is assumed to be wide- sense ergodic. Consider the input vibration applied to the accelerometer as shown in Figure 17. The vibration is assumed to have a normal dis- tribution with a zero mean. For an ideal accelerometer the scale factor is a constant over the range of operation. Therefore, for a given input, the output would also be a normal distribution having a zero mean. In this case there is no error introduced by the ac- celerometer. Now consider the non-ideal condition where the scale factor is different for the positive and negative regions. With the given input the output amplitudes can be found to be yfor-°°§_>'5.0 Z = yGO') ky for o §.y g-w 82 PULSE COUNT ISEC [ACTUAL SCALE FACTOR mm / / / / I / ‘ . / / sum IN / I NEAR mus / - -A I! -|000 001901 slcmmsmaunon / m>\ _ 1 4. INPUT SICIAI. DISTRIBUTION FIGURE 17 EFFECT OF ACCELEROMETER SCALE FACTOR ERROR ON SIGNAL DISTRIBUTION 83 where Z = output amplitude y = input amplitude G(y) = transfer characteristics of the accelerometer (scale factor) k = ratio of non-ideal to ideal scale factors 'The output probability density function, po(y), would then be p10) for - miyio pi(Y) = l- X- for o < < w kpi k —"— and must have a value such that 00 o oo .00 —CD 0 The input probability density function can be seen to be 1 P10) - s 5.76 84 The probability density function associated with the non-ideal con- dition can be found to be It is desired to determine the effect on the mean of the output. The mean value, 9, is equal to the first moment of the density function. Then 0 0 [2 - 1 2 02 d _ _£L. y1= ypimdy = ‘7—0 2,, ye Y "/75; and °° °° _z.2__ - -1 X-—-—1——-— 21(202 =_1_<_0__ yz'k ypiIk)‘ko/§; ye dy .5? O 0 Since the mean is equal to the sum of the individual means, then - - o Y = Y1+Y2 = /2_1I (k‘l) 85 Thus it can be seen that the non-ideal condition results in a shift in the mean to some non-zero value. Since the expression for the output of a pulse-rebalance accelerometer is P = K At where P = accelerometer output, pulses K = scale factor, pulses/sec/g A = acceleration input, g's g = gravity t = time in sec the error due to the random vibration can be found to be P = i. (k-l) Kt E 5; The resulting velocity error is N v = Z .51. (k-l) Kt. E i=1 /2F 1 where o = rms value of random vibration number of seconds 2 ll Hence, the importance of maintaining the scale factor to a constant value has been shown. 86 8.2 GYROS The gyro drift errors due to mass unbalance and anisoelastic effects caused by random vibration will be considered in this section. 8.2.1 Mass Unbalance Drift As already shown for sinusoidal vibrations, no error is pro- duced by random vibrations due to gyro mass unbalance. This may be readily Shown by considering the expression for drift error due to mass unbalance: T . K1 ¢ = Tr' A dt 0 where O = drift rate in deg/hr Kl = error coefficient due to mass unbalance, deg/hr/g A = total acceleration, g's If the input acceleration is presumed to consist of a ran- dom vibration having a zero mean, the integral is equivalent to zero and consequently no drift error is contributed by the mass 87 unbalance term. This may be further verified by again referring to the transfer function for the gyro as: 1/0 s (I S + 1) "110 where .4 II time constant I/D If the input torque is assumed to be a wide-sense ergodic random process and the input power Spectral density is represented by ST(w), then the output power Spectral density is represented by 2 ST 0») $00») = 'HUw) where H (jm) = complex frequency response Since the overall magnitude of H (jw) is less than one, the output power spectral density will be small with respect to one. Therefore the motion of the mass unbalance due to the random vibra- tion will be small and in turn the vibrOpendulous error will be in- significant. 88 8.2.2 Anisoelastic Drift A gyro has elastic deflections that take place under vibra- tion. Since the metal used in the fabrication of the gyro is not perfect, variations in the deflections occur producing anisoelastic torque. The literature8 has already treated such a condition under random vibration and the pertient equations will be presented here. The mean torque about the gyro output axis due to random vibration is given by (gsofim sin 20 2 0-3l 0, 0AA(f) Re X, (if) - Re K, (if) df 0 where TOA = average anisoelastic torque about the gyro 0A (output axis) due to random vibration, dyne-cm m = mass of gyro rotor, grams 6 = angle between the line of applied vibration and the gyro spin reference axis OAA(f) = power spectral density of applied vibration acceleration transmitted to gyro case, Gz/cps 89 f2 - f2 n. RK.(jf)= 1 1 e 1 4n2 (f2 _ f2 )2 + 4§2 f2 f2 n. i n. 1 1 with fnl, fn = undamped natural frequencies of 2 rotor elasticity along the SRA and IA respectively, cps Ci = damping ratio SRA = spin reference axis IA = input axis From the above expression for the mean torque it can be seen that if the elastic characteristics along the spin reference axis and the input axis of the gyro are identical, the average torque is zero. However, if a difference exists then a mean torque is develOped which is a function of the power spectral density, the mass of the rotor and the direction of applied vibration. The drift due to the anisoelastic effects can be found through the use of the gyro equation 90 where E II gyro drift, rad/sec '-1 II mean anisoelastic torque, dyne-cm .— uh. ll gyro angular momentum, gm-cmz/sec This, together with the previous expression for mean torque, pro- vides a means of determining the anisoelastic drift of a gyro due to random vibration. 9.0 SYSTEM ERROR DUE TO RANDOM VIBRATION In considering the system error due to random vibration, the same system model applies here as already presented in Section 7.0. Based on this the system error equation - consisting of the error contributions from the gimbal structure, accelerometers, and gyros - is presented. Assuming a vibration on the case of the IMU, the gimbal structure serves to shape the environment for the accelerometers and gyros. The state model was developed in Section 4.0 with an example given in Section 5.0 and will not be repeated here. Stand- ard approaches applicable to random vibration can be used to find the vibration at the stable element. For random vibration, also, it was shown that an acceler- ometer had three types of errors - vibr0pendulous error, non- linearity, and scale factor error. From Section 8.1 the vibrOpend- ulous error equation was found to be r- q r- q F' '1 A A 0 0 A X X OX A = 0 A 0 A Y Y OY LA. 0 0 A A 2.4 L Z_J ,_ 02 J 91 92 where Ax’ Ay’ Ay are acceleration errors along the X, Y, Z axes, A , A , A are the power spectral density level in the ox oy oz X, Y, Z axes, GZ/cps and . -1 (1)2 -1 (1)1 A P Sln O cos O tan .__ _ tan ___ Zn w w c c where P = pendulosity, dyne-cm O = angle the line of applied vibration makes with accelerometer input axis ml = lower frequency limit of applied vibration wz = upper frequency limit of applied vibration w = accelerometer corner frequency The error equation for the nonlinearity is A i5 0 0 A2 x K nus IX (X) K2 Ay = o k—l 0 Aims 1y (y) K 2 AZ () () 2 THIS 12 L (z) \— ..J L ..4 The equation for scale factor error can be found to be N l k-l Kt. 0 A N l A o o 121 73: (k-1)Xti Anus 94 As pointed out in Section 7.0, the gyro errors produce a misorientation of the platform such that the accelerometers sense a component of gravity or the thrust vector or the combination of the two. Hence, the accelerometer output is in error by an amount de— pending upon the gyro drift. The general expressions for the ac- celerometer error produced by the gyros are AX=gGX A = G y gy A = TG y y A = TG Z 2 where Ax, Ay, AZ are the accelerometer error outputs along the X, Y, Z axes, gravity 00 II '-1 II thrust in g's and Gx’ Gy’ and G2 are terms associated with the X, Y, Z axes and are defined as follows: oA :1: 95 where t = time in seconds H = gyro angular momentum, gm-cmZ/sec and - (980)2 m sin 26 . _ . T0A 2 PMCf) Re K2 (Jf) Re K1 (Jf) df o where -oA = average anisoelastic torque about the gyro output axis due to random vibration, dyne-cm m = mass of gyro rotor, grams OAA(f) = power spectral density of applied vibration acceleration transmitted to gyro case, gZ/cps f2 - f2 . _ 1 1 _ Re Ki (jf) - 3:50 , 1 - l, 2 96 with f , f = undamped natural frequencies of rotor along the spin reference axis and the input axis respectively, cps. Each of the individual error forms has been develOped for the accelerometer and the gyros. It remains to establish the system error due to the combination of errors. In pursuing this, it is reasonable to make the assumption that each of the indiviual errors has a normal distribution and each is independent of all others. Then the system error can be found2 to be 5 s. = [Aiwiwswi] where (D II system error, ft ("1 = accelerometer vibropendulous error, ft 2”? accelerometer nonlinearity error, ft (03> u accelerometer scale factor error, ft C II accelerometer error due to the gyros, ft (‘1 97 An example of the errors produced by the platform instru- ments under random vibration was considered. Here, also, a 50-mile flight was assumed and the results are shown in Figure 18. It can be seen that the contribution to the system error can be signi- ficant. The vibration was assumed to have a constant power spec- tral density over the frequency range from 20 cps to 2000 cps. Therefore, the filtering action of the gimbal structure would tend to reduce the effects as indicated. In all, however, the importance of paying careful attention to vibration has been illustrated. 98 ASSONEO INAJECTONY so MILE FLIGHT VIBRATION PRESENT zoo sccowos ONLY DURING BOOST l0 stc. aoosr rmws‘r Io 0's I I I 56 -I 7 1r—