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A METHOD FOR EXPERIMENTALLY DETERMINING ROTATIONAL MOBILITIES by STANLEY SIMONV,ATTINGER B.M.E., Georgia Institute of Technology (1962) M.S., Cornell Univeristy (1964) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY August, 1978 Signature redacted Signature of Author ................... ... ........ r ....... ...... Department of Mechanifl Engine ing' August 11, 1978 Signature redac ted Certified By...................... Thes'is Supevisor - A Signature redacted ....................... Accepted By......... .. Chairman, Department Committee on Graduate Students ARCHIVES OCT 2 7 1 7 8 A METHOD FOR EXPERIMENTALLY DETERMINING ROTATIONAL MOBILITIES by Stanley Simon Sattinger Submitted to the Department of Mechanical Engineering on August 11, 1978 in partial fulfillment of the requirements for the Degree of Master of Science. ABSTRACT Mobility functions involving rotational velocities and moment excitations must be determined for the prediction of the responses of certain types of structures in dynamic analyses. Previous investigators have approached the difficult task of experimentally measuring such mobilities with the use of special fixturing attached to the structures. It is shown that rotational mobilities of structures are equivalent to spatial derivatives of their translational mobilities. The method of finite differences is adapted to the approximation of these derivatives. By this approach the rotational mobilities are derived from sets of conventionally measured translational mobilities, eliminating the need for special fixturing. This method of determining rotational mobilities is demonstrated in a set of experiments on a free-free beam. Good agreement is obtained between experimentally and theoretically generated versions of two rotational velocity/force mobilities. An experi- mentally derived rotational velocity/moment mobility is found to - - 2 give reasonably good indications of resonances, but exhibits large amounts of scatter in some frequency bands. This scatter is attri- buted to the subtraction of translational mobility quantities which are nearly equal in magnitude with resultant magnification of minor Further investigation is recommended irregularities present in them. to determine an effective method of smoothing the translational mobility data before the differencing calculations to eliminate this scatter. The finite difference method of determining rotational mobilities is seen to accommodate considerable variation in the spacings of the points where the constituent translational mobilities are measured. Thesis Supervisor: Richard H. Lyon Title: Professor, Department of Mechanical Engineering - - 3 ACKNOWLEDGEMENTS I wish to thank Professor Richard H. Lyon for his guidance and suggestions and for his willingness to support me in an area of study which was of a great deal of personal interest to me. I am also grateful to Professor Emmett A. Witmer of the Department of Aeronautics and Astronautics for his aid in connection with the method of finite differences. Many thanks go to fellow student Charles Gedney for his pointers on the operation of the Acoustics and Vibration Laboratory minicomputer; to Dr. Richard Dedong of Cambridge Collaborative, Inc. for sharing some of his vibration testing experience; and to Mary Toscano for her diligence in the typing of this thesis. I owe a debt of gratitude to my employer, Westinghouse Electric Corporation, for having awarded me a B.G. Lamme Graduate Scholarship enabling me to pursue a course of study in vibration and acoustics at MIT. I am especially thankful to my wife, Jerry, and to our daughters, Julia and Allison, for the encouragement they gave me and the many hours of family time they sacrificed throughout the year of my studies at MIT. - - 4 TABLE OF CONTENTS Page ABSTRACT........................................................ 2 ACKNOWLEDGEMENTS................................................ 4 TABLE OF CONTENTS............................................... 5 LIST OF FIGURES................................................. 7 NOMENCLATURE.................................................... 10 II. III. IV. INTRODUCTION............................................. 14 A. Background........................................... 14 B. Scope................................................ 17 DERIVING ROTATIONAL MOBILITIES FROM TRANSLATIONAL MOBILITIES............................................... 18 A. Rotational Velocity/Force Mobility................... 18 B. Translational Velocity/Moment Mobility............... 20 C. Rotational Velocity/Moment Mobility.................. 21 D. Summary of Derivative Relationships.................. 22 E. Implementation of Finite Difference Method........... 23 EXPERIMENTAL MOBILITY MEASUREMENTS ON A FREE-FREE BEAM... 28 A. Test Specimen and Test Equipment..................... 28 B. Test Procedure....................................... 30 C. Measured Vs. Theoretical Translational Mobilities.... 32 D. Experimentally Derived Vs. Theoretical Rotational Mobilities........................................... CONCLUSIONS.............................................. - 5 - I. 37 41 TABLE OF CONTENTS (CONTINUED) Page ..00. .. . ... REFERENCES.. . . ... .. . ... .. ..... 42 . FIGURES..... ............................ .... 69 APPENDIX A: THEORETICAL MOBILITIES OF A FREE-FREE BEAM. 71 APPENDIX B: COMPUTER PROGRAM THEOR..... ................ 81 APPENDIX C: COMPUTER PROGRAM TRANS..... ................ 85 APPENDIX D: COMPUTER PROGRAM ROTAT..... ................ 87 - - 6 LIST OF FIGURES No. Page Transfer Mobilities Involving Various Combinations of Translational and Rotational Effects.............. 42 Relationship of Rotational Velocity/Force Mobility to Translational Mobility at a Single Frequency...... 43 Relationship of Translational Velocity/Moment Mobility to Translational Mobility at a Single Frequency............................................ 44 3,1 Test Beam Details..................... 45 3.2 Matrix of Desired Beam Mobilities..... 46 3.3 Test System Schematic Diagram......... ............... 47 3.4 Test Beam Translational Mobility 4,1 Before and After Data Substitution.............. ............... 48 Test Beam Experimental and Theoretical Translational Mobility 1 ,2 . .. ' '' '. . .. . . . . . ' .. 49 Test Beam Experimental and Theoretical Transl1'ational Mobilityik3 ,.' . '. . . . . ' . . . . . . . . . . 50 3.7 Test Beam Experimental and Theoretical Translational Mobility 2,2'''''........ '...... 51 3.8 Test Beam Experimental and Theoretical Trans1'ati'onal Mobility 4 3 ,2 ''' .. . '''. .. . .. . . ' .. . 52 3.9 Test Beam Experimental and Theoretical Translational Mobility 4 ,2. . . '' . .. . '' . . . .. . .. . . 53 3,10 Test Beam Experimental and Theoretical Translational Transl1ati onal Mobility'P3 3. . '. . . . ''' . '' . '. . '. 54 Test Beam Experimental and Theoretical Mobility 44,3".....'...'.''..'.. 55 1.1 2.1 2.2 3.5 ' ' 3.6 - 7 - 3.11 LIST OF FIGURES (CONTINUED) No. Page 3.12 Test Beam Experimental and Theoretical Translational Mobility p4 ,4.... ...... ............ .... 56 3.13 Test Beam Experimental and Theoretical Rotational Velocity/Force Mobility Y F(W) ..................... 0BFA 3.14 Test Beam Experimental and Theoretical Rotational Velocity/Force Mobility Y. F(() 3.15 8.................. Test Beam Experimental and Theoretical Rotational Velocity/Moment Mobility Y (W).................... OBMB 3.16 3.17 Constituent Terms of Derived Experimental Mobility Y (,) Over a Frequency Band of Large Scatter: 0B MB Real Components...................................... 60 Constituent Terms of Derived Experimental Mobility Y 0 M () Over a Frequency Band of Large Scatter: Imaginary Components........................ ......... 61 3.18 Quadrature Components of the Derived Experimental Mobility YBMB() Over the Frequency Band of Figures 3.16 and 3.17................................62 3.19 Rotational Velocity/Force Mobility Y. F (W) Derived by Differencing Theoretical Translational Mobilities: An = AE = .044m......................... Rotational Velocity/Moment Mobility Y0 BMB () Derived by Differencing Theoretical Translational Mobilities: An = AC = .044m............. ......... .. - 8 - 3.20 63 64 LIST OF FIGURES (CONTINUED) Page No. 3.21 Rotational Velocity/Force Mobility Y 0 FB Derived by Differencing Theoretical Translational Mobilities: An = A = .088m........................65 3.22 Rotational Velocity/Moment Mobility Y MB Derived by Differencing Theoretical Translational Mobilities: 3,23 An = Ac = .088m......................... Rotational Velocity/Force Mobility Y 66 F (W) Derived by Differencing Theoretical Translational Mobilities: An = AC = .022m........................67 ( Rotational Velocity/Moment Mobility Y0BMB Derived by Differencing Theoretical Translational Mobilities: An = A = .022m. ....................... - 9 - 3.24 68 NOMENCLATURE A Uniform beam cross-section area; also the complex amplitude of acceleration Constant coefficient of the ith term in the expression for W(x) 4. Arbitrary multiplier of the rth eigenfunction of a free vibration problem Modulus of elasticity of beam material Base of natural logarithms ) (no; Complex amplitude of sinusoidally varying force on a structure. applied at position Complex amplitude of the sinusoidally varying force f(t) W6 Cyclic frequency, Hz f Concentrated force applied at Point A on a structure Distributed loading applied to a beam, including damping forces f(X) fX (X Distributed loading applied to beam, exclusive of damping forces Cross spectral density of stationary random acceleration and random force (complex function of cyclic frequency) Power spectral density of stationary random force (real function of cyclic frequency) Uniform beam cross-section area moment of inertia Length of beam )) Complex amplitude of sinusoidally varying moment on a structure appli ed at position g - 10 - -I NOMENCLATURE (Continued) Complex amplitude of the sinusoidally varying moment Mr Modal mass of the rth mode of vibration /?7 Total mass of beam Concentrated moment applied at Point A on a structure Mode number at which infinite series of modal mobilities is truncated Complex amplitude of the sinusoidally varying force l(t) One member of a force couple equivalent to moment /7nA(t) Pr The rth eigenvalue of a free vibration problem Modal force of the rth mode of vibration t) rZ Modal force of the rth mode exclusive of damping forces Generalized coordinate or generalized displacement of the rth mode of vibration Time W(X) Complex amplitude of the sinusoidally varying displacement W(X) The rth eigenfunction of a free vibration problem 4(x) - X2 Complex amplitude of sinusoidally varying translational velocity measured at position 1 on a structure Complex amplitude of sinusoidally varying translational velocity measured at Point A on a structure NOMENCLATURE (Continued) w(Xjt) X74(w) F ".WA ~AWX Transverse displacement at location X. on a beam Displacement measured at Point A on a structure Complex amplitude of the sinusoidally varying generalized displacement (t Coordinate of axial position on a structure Translational velocity/force mobility: measured at A, excitation applied at B Translational velocity/moment mobility: measured at A, excitation applied at B Rotational velocity/force mobility: at A, excitation applied at B velocity velocity velocity measured Rotational velocity/moment mobility: velocity measured at A, excitation applied at B Coordinate of transverse position on a structure Dirac delta function of position coordinate x 6 'A Spacing of the members p(t) of a force couple Equivalent viscous damping ratio of the rth mode of vibration Axial coordinate of point of velocity measurement on a structure e(y) Spacing between adjacent velocity measurement locations Complex amplitude of sinusoidally varying rotational velocity measured at position 1 on a structure Complex amplitude of sinusoidally varying rotational velocity at Point A on a structure Rotation occurring at Point A on a structure Axial coordinate of point of excitation on a structure - - 12 NOMENCLATURE (Continued) -\ $Spacing between adjacent excitation points on a structure Mass density of beam material Mobility phase angle AF 4X X) Acceleration/force cross spectral density phase angle (function of cyclic frequency) The portion of the rth eigenfunction WrX)exclusive of the multiplerDr One of a set of translational velocity/force mobilities from which one or more rotational motilities will be derived 4L) Angular frequency, rad/sec OJM A particular value of frequency - - 13 I. INTRODUCTION A. Background The application of mobility functions* and their inverse quantities, mechanical impedances,to practical problems in vibration, shock, and acoustics has been treated extensively in the literature. Mobility and impedance concepts are readily adaptable to dynamic response predictions for assemblages of two or more component structures. A mobility function is a transfer function relating the complex amplitude of motion at some point on a structure in response to the complex amplitude of an excitation force applied at any point on the same structure. In the most commonly discussed type of mobility the response motion is a translational component of velocity, and the excitation is a translational force as illustrated in the transfer mobility example of Figures 1.1(a). However, the concept of mobility can be extended to rotational velocities and moment excitations as shown in the examples of Figures 1.1(b) through 1.1(d). A matrix relationship involving the transfer mobilities thus defined between the two points is shown in Figure 1.1(e). The matrix formulation shown in Figure 1.1(e) can be * specialized to the case where the response measurement point, B, Also denoted as admittances or receptances. - - 14 is coincident with the excitation point, A, i.e., each matrix element is a driving point mobility; or it can be generalized to include the existence of motions and excitations at both points. In the latter instance the second order mobility matrix shown would be expanded to the fourth order. Generalizing still further, the transfer mobility matrix shown in Figure 1.1(e) could be extended to the six possible senses of motion and applied excitation in a three-dimensional application, attaining order 6. The mobility matrix would be enlarged still further as additional locations for responses and excitations would be considered. In many instances the mobility or impedance quantities are determined experimentally. In cases such as the applications of impedance methods to vibration testing described in References (1) and (2), the motions and forces involved are limited to translational effects directed along a single axis. In the studies described in References (3), (4), and (5), the applications are broadened to treat the interconnection of components which may sustain rotational and translational components of motion, but are assumed to have only translational interaction effects. For an assemblage to be accurately modeled by such an approach, there must be negligibly small moment reactions among components at each interface in the actual system by virtue of joint configuration, symmetry, or other factors. - - 15 Cantilevered assemblages can be readily conceived wherein the most important interactions are rotational; in such cases there must be compatibility of rotations at connections, and moment reactions are far more significant than force interactions. Ex- tensions of mobility and impedance methods to response predictions in these cases have been hampered by difficulties in experimentally measuring the rotational mobilities. Whereas the measurement of translational velocities and forces is presently a routine process, apparatus for the measurement of rotational velocities and moments in structural dynamics applications is not commercially available. Noiseux and Meyer (6) suggest that the lack of a general measurement technique has retarded the application of mobility concepts and in some cases has distorted the applications by mandating the use of what can be measured rather than what should be measured. Explorations of methods for the measurement of moment excitations and rotation responses are described in References (7), (8) and (9). In each of these studies a special fixture has been attached to the structure being measured, and conventional linear force gages and accelerometers have been, in turn, mounted at By appropriate algebraic operations various locations on the fixture. on the data gathered in each of the various measurement configurations, rotational mobilities have been obtained with varying degrees of success. Corrections for the dynamic influences of the fixturing - 16 - have been required in each case. B. Scope The objective of this thesis is to improvise and demon- strate a method of generating experimental rotational mobility functions using conventional measurement techniques without a requirement for the use of special fixturing. The approach taken has been to represent mobilities involving rotational velocities and moment excitations as spatial derivatives of conventional translational mobilities; the derivatives are approximated as finite difference sums of sets of these translational mobilities. In Section II, the theoretical basis and calculational methods for these representations are developed. Section III describes the experimental and theoretical determination of mobilities of a free-free beam demonstrating these methods. the conclusions drawn. - - 17 Section IV presents II. A. DERIVING ROTATIONAL MOBILITIES FROM TRANSLATIONAL MOBILITIES Rotational Velocity/Force Mobility Figure 2.1(a) depicts a segment of a structure which is being driven by a sinusoidal translational force applied to Point A and in which the resultant sinusoidal translational velocity is being measured at Point B. Considering momentarily that the excitation is at a particular frequency&4)M, the value of the translational mobility at that frequency is the complex quantity 4)/zM M). Now suppose that the velocity measurement is made in turn at each point of a set of points adjacent to Point B with the excitation maintained at Point A as shown in Figure 2.1(b). ratios coordinate, 0;Je40) The resulting complex amplitude could be plotted as functions of the position , of the measuring point as shown in Figure 2.1(c). The real and imaginary mobility data, if carefully measured, would be found to lie on smooth curves by virtue of the continuity of the wave fields comprising the vibration of the structure.. Tangent lines could be drawn to these curves at the coordinate X of Point B. The slopes of these tangents would have the following significance. The instantaneous angular displacement of the structure at a relative to its rest position would be given by: 0w J - 18 - location C/ d:ow iJL4~ (2.1) 7 But, because the excitation is sinusoidal, this angular velocity could be expressed as (2.2) ) = By combining Eq. (2.2) and the relation v with Eq. (2.1), it is found that d-) = 0'~4 141tIZ), (2.3) from which is formed the ratio of complex amplitudes k41 t allC' k/s-'(4 e ) M4 ,(a4) 1[i~dW)~l) (2.4) If the indicated measurements and calculations are performed at intervals over a band of driving frequencies,60) , of interest, the rotational velocity/force mobility function is thus derived from the translational mobility symbolically as (2,5) I'&.SF A -19 - =t The time rate of change of this slope would be given by: B. Translational Velocity/Moment Mobility In Figure 2.2(a) the structure is again shown with translational excitation and response vectors at Points A and B, respectively. Again with the excitation frequency set at AIM suppose that the excitation force is applied in turn at each of a set of points adjacent to as shown in Figure 2.2(b). A 4M The resulting complex amplitude ratios could be plotted as functions of the position ;6 coordinate, A with responses measured at Point B , of the excitation point as shown in Figure 2.2(c). The real and imaginary mobility components would again be found to lie on smooth curves to which tangent lines could be drawn at the coordinate X of Point A. The significance of the tangent slopes to these curves is explained as follows. With reference to Figure 2.2(d) it is seen that an instantaneous moment applied to the structure at Point A could be equivalently represented as a pair of equal and opposite parallel forces separated a small distance, E . The response of the structure at Point B to a sinusoidally varying moment at Point A could then be expressed as follows: F( 2 20 -(26 (2o T hen WB 20, Av B 7ra),44; ci = 49 Rrakm) kme k(cdM) A4 A @ 4)M) 7~FtWA4~) I (2.7) from which is formed the ratio of complex amplitudes - d W) ) A4A ( Cc/T~T4~a60) 0 .;4*ZmfkIMW)7 (2.8) If this ratio is evaluated over a band of driving frequencies, 4) of interest, the translational velocity/moment mobility function is thus derived from the translational mobility symbolically as ci YjF (a; (2.9) ) (4A C. Rotational Velocity/Moment Mobility The structure will again be envisioned as being excited by a sinusoidal translational force of frequency 4OM at Point A. If the location coordinate 5 of the force application point is then made to vary about X , the resultant derivative of translational mobility relates the complex amplitude of translational velocity - 21 - at Point B to the complex amplitude of applied moment at Point A in accordance with Eq. (2.7). If Eq. (2.3) is written for the case q=X and is combined with Eq. (2.7), the result O/ZF 4 YA(WA44) &19m; 5) (2.10) is obtained. If this ratio is evaluated over a band of driving frequencies of interest, the rotational velocity/moment mobility function is thus derived from the translational mobility symbolically as: GB(A D. ) eD (Z ~~2g4/) (2.11) Summary of Derivative Relationships The mobility matrix relating the translational and rotational velocity amplitudes at the response measuring Point B to the amplitudes of force and moment at the excitation Point A on a structure was shown in Figure 1.1(e). In accordance with Eqs. (2.5), (2.9) and (2.11) each element of this mobility matrix is a function which can be re-expressed in terms of the translational velocity/ force mobility function, yielding the following: - - 22 < AA(a14? - F~~ ~~~ Y4(4~ C~f~ (2.12) or, using more compact notation, YWB 0 (2.13) It is emphasized that each element in the mobility matrix represents a function of angular frequency ) defined over some band of interest. E. Implementation by Finite Difference Method The calculation of spatial derivatives of translational velocity/force mobilities is the essence of the above described approach to determining rotational mobilities. For application of this approach to the experimental determination of mobilities, these derivatives must be approximated from conventional mobility measurements made at a limited number of discrete locations on a structure. The method of finite differences, (11), is used for this purpose. 23 - - References (10) and f Let the symbol denote a translational velocity/force mobility function to identify it as one of a set of conventional mobilities from which one or more rotational mobilities will be derived. These conventional mobilities will be determined with re- sponse velocity measurements made at locations .. spaced - z apart, and with force excitations applied at locations spacedAJ -/)$/..... Thus the notation apart. will represent the mobility function V P), If ( were a function of only the single spatial coordinate , each of the following expressions would approximate the continuous ordinary first derivative of that function, correct to within truncation errors of the order of Central Difference: C&& _ ;. _ -.._ _ (2.14) Forward Difference::-/ Backward Difference: 2 The choice of the approximation which would be used from among these three would depend on whether the location where the derivative is desired happens to be an inboard location, and if an end location, whether at the positive end or the negative end of the7 interval. For the mobility function O(a-;A ) )in which two spatial coordinates are involved, the first partial derivative approximations have - similar form to the ordinary derivative approximation: - 24 Central Difference: 07I-v fiml - &,0? 2417 _5M Cao~j JZA =L 5- Forward Difference: Backward Difference: -- " ___________ -4oi2/, 43 7 - -%-- 42A~ The latter expressions are directly usable in evaluating the rotational velocity/force mobility and the translational velocity/moment mobility as indicated in Eqs. (2.12) and (2.13) given that or n m and 15= 4) p The mixed second partial derivative can be approximated by one of the following expressions: Central Difference: - 2AZ1/ 2?z Forward Difference: +3~fr /--/F J2VI nQ - 25- 077- #-Al (2.16) i~A4 ' Backward Difference: cont'd) The above central difference expression is usable for evaluating any rotational velocity/moment mobility in which the hypothetical moment excitation is applied at a point inboard of the ends and the rotational velocity response is at the same point or any other inboard point. The forward difference expression applies only to the case in which both 1 and 5 coincide with the negative ends of their ranges; i.e., the hypothetical moment excitation is applied and the rotational velocity response is measured at the left end of the structure. Similarly, the backward difference expression applies only in the case where both the hypothetical moment excitation and the rotational velocity response locations are at the right (positive) end of the structure. Such end-located rotational mobilities would be of main importance in analyzing the dynamic response of cantilevered structures. However, in instances where the moment excitation would be applied at an end point and the rotational velocity response would be measured at some other location, or vice-versa, none of the above difference Reference (11) contains other expressions would be applicable. finite difference formulations which would apply in these instances. In summary, the evaluation of a particular rotational 26 - - mobility using the above finite difference approximation methods requires the prior determination of between two and nine conventional translational mobilities, the quantity depending on the location and type of rotational mobility desired. For driving point rotational velocity/moment mobilities the number of translational mobilities needed may be cut almost in half by resort to the use of the , ' reciprocal theorem for dynamic loads, Reference (12); because as a consequency of this theorem, either of these mobilities may be substituted for the other. It is further noted that several dif- ferent rotational mobilities can be evaluated using a common set of translational mobilities. The selection of response measurement and excitation location spacingsl 7 andA f , must achieve a balance between resolution and proper approximation of derivatives across the number of natural modes of vibration encompassed in the band of frequencies. Some analytically or experimentally obtained knowledge of mode shapes is desirable for use in the determination of the point spacings. The results of Section III.D demonstrate that this balance is achievable with latitude in the selectionat least in cases where a limited number of resonances are included in the frequency band. - - 27 III. A. EXPERIMENTAL MOBILITY MEASUREMENTS ON A FREE-FREE BEAM Test Specimen and Test Equipment Figure 3.1 depicts the beam which was prepared from cold rolled steel rectangular bar stock for experiments to demonstrate the previously described approach to obtaining rotational mobilities. Excitation point and motion monitoring point locations were established for experimental measurements of all the conventional translational mobilities needed to generate the rotational mobilities identified in Figure 3.2 by the methods of backward differences. The mobilities included therein would be among those required to predict the translational motion at Point A due to cantilever attachment of the beam to a moving foundation or other component at Point B. The particular set of beam cross section dimensions was chosen such that the off-axis (stiff direction) natural vibration frequencies would not coincide with the drive direction (flexible direction) natural frequencies. The tapped holes shown in Figure 3.1 were added to enable stud attachment of an impedance head for force measurements at each drive point location in turn. Because most of the required translational mobilities were to be transfer mobilities, all motion measurements were made by attaching the accelerometer to the opposite side of the beam from the impedance head using beeswax. Accurate placement of the accelerometer was facilitated by lines scribed on the surface coincident with the - - 28 driving stud hole centers. It is seen in Figure 3.1 that the outer- most drive points were located as close to the beam ends as possible with assurance of proper seating of the instrumentation at these end locations. The .044m spacing of the driving and measuring points at End B was established by first sketching the mode shape of the expected highest resonance within the planned test frequency band of 0-2000 Hz. The three driving points were then spaced at the widest distance where the backward difference method could be expected to approximate the slope of this mode shape reasonably well. This spacing was chosen as wide as possible to provide resolution for accuracy in the approximation of slopes at the frequency of the lowest resonance. The mobility tests were conducted using broad band The force and acceleration signals stationary random excitation. were recorded and processed by a minicomputer using the fast Fourier transform coherence/cross spectral density program COHER previously developed for the Acoustics and Vibration Laboratory in conjunction with the Reference (5) ScD dissertation. The overall test system with identification of the test equipment used is shown in Figure 3.3. The test beam, which weighed 5.64 kg, was suspended yertically from one end by means of elastic bands and was driven horizontally to effect the intended free-free boundary conditions. which The horizontally oriented shaker, was capable of generating a maximum force amplitude of about 25 nt, was connected to the stud29 - - mounted impedance head by means of a .05m long by .002m diameter shaft capble of accommodating minor misalignments between shaker and beam. The beeswax-mounted accelerometer was of 2 grams mass, and the total mass of the impedance head was 60 grams. B. Test Procedure Calibration of the accelerometer signal channel was performed by temporarily mounting the accelerometer on a General Radio Model 1557A calibrator. Subsequently, the force signal channel was calibrated by connecting a rigid disk of known mass to the impedance head and exciting it sinusoidally; the calibrated accelerometer signal and the known mass were used to establish the actual force amplitude represented by a given force signal. calibration values obtained were found to be close to( The the trans- The proper functioning of the entire ducer manufacturers' ratings, test system was later verified by driving the rigid disk with random force input; the mobility data generated by the system were matched very closely by the theoretical mobility of a pure mass of the same value. To maximize the dynamic range of the measurement system, it was necessary to make the frequency spectrum of both channels simultaneously as flat as possible across the 0-2000 Hz band of interest, The test beam was instrumented at typical driving and response locations, and the signal spectra were monitored in real 30 - - time using the spectrum analyzer. It was found that adequate flatness could be obtained with the use of one signal generator output bandpass A 63 Hz high pass corner frequency filter as shown in Figure 3.3. setting and a 1600 Hz low pass corner frequency setting were used for this filter throughout the beam mobility testing. These settings provided the required flatness of signal spectra from 0 to 2000 Hz while providing desired roll-off in driving force above 2000 Hz and precluding large-amplitude,low-frequency rigid body motions of the beam. Prior to the start of each mobility data acquisition run the signal channel gains were adjusted until the signal levels, monitored on the oscilloscope, seldom exceeded the 5 volt maximum input level of the analog to digital converters. The channel gain values and transducer sensitivity values were then specified as input data to the computer along with the desired number of averages (400. for each run). Also specified was the maximum frequency value (one-half the sampling rate), which was 2560 Hz for all runs. The force and acceleration signal channel bandpass filters were accordingly set at corner frequencies of 2 Hz (high pass) and 2000 Hz (low pass) for both channels. The latter setting was con- sistent with the Reference (5) recommendation that the high frequency roll-off point be set at 0.4 times the sampling rate to - 31 - eliminate aliasing effects, C. Measured Vs. Theoretical Translational Mobilities For each mobility test run, the minicomputer calculated power spectral densities and cross spectral densities of the force and acceleration signals along with their relative phase and coherence These outputs were generated at discrete frequencies spaced values. 10 Hz apart over a band extending from 10 Hz to 2000 Hz and were converted into translational mobilities as explained below. In the definition of a translational mobility as a transfer function relating sinusoidal force and velocity quantities, the acceleration corresponding to the velocity W, = 44C) L) = Aew irit is given as (3, ) W * Then the translational mobility is related to the acceleration/force , and the power spectral density of the cross spectral density,G-force, G, _ T in accordance with i_ / A / - -- - A A (3.2) Separating these complex quantities into their magnitude and phase components, we obtain / -.--.- (3.3) =..-.- 32 - - w here :XAF. Ais the phase angle of the complex cross spectral density The, magnitude portion of Eq.(3.3)can be re-expressed in terms of the frequency and spectral density quantities in the form 0 utput by the COHER program: (3.4) /64!), F (1/6 which yields 2O/oy4 /o/ // 1 0 1A A Fo -D/T7 (3.5) The phase portion of (3.3) is simply 5 = - Frequently the force and acceleration signals in mobility measurements are corrected for the mass and flexibility effects of the portion of the impedance head below the force gage. Such cor- rections are described in Reference (5), but the corrections therein pertain to driving point mobilities only. In transfer mobility measurements these effects cannot be determined with the test system described, as the measured accelerations are different from those sustained by the impedance head. Because the bulk of the mobilities measured were transfer mobilities, no impedance head - 33 - 490 (3.6) a corrections were made in any of the runs. The computer program TRANS, listed in Appendix C,.was written to convert the spectral density output data from COHER into translational mobilities per Eqs.(3.5) and (3.6) and to create plots and store the results in quadrature form on disk for later manipulation. Data input to the TRANS program is via punched cards. A separate pro- gram, THEOR, listed in Appendix B,was written to generate theoretical translational and rotational mobility functions for Bernoulli-Euler beams and to store them on disk in magnitude and phase form for use in comparison with the experimental results. The derivation of the equations programmed in THEOR is presented in Appendix A. By virtue of the reciprocal theorem for dynamic loads, Reference (12), mobility matrices such. as given in Figure 3.2 are symmetric. Thus all elements of the matrix shown would be established if only the upper or lower triangular portion were evaluated. If arbitrarily the lower triangular portion is chosen to be evaluated, the translational mobilities needed to establish this matrix are as follows: () A Y (4t))3)1)BA2)1 4: -223 - 34 - Y Z2 A further consequence of the reciprocal theorem is the symmetry of translational mobilities, i.e., . Also, by Combining these com- geometric symmetry it is seen that monalities, the entire nine-element mobility matrix shown in Figure 3.2 would be established by measurement of the following nine translational mobilities or their reciprocals: Theoretical and experimental versions of these translational mobilities were generated as explained above. A tendency toward erratic results was observed in the experimental magnitude data in regions of resonances. It was found that these erratic results occur- red at frequencies where the coherence values fell to low levels (less than .50). The low coherences were attributable to the force signal spectra having decayed to the level of the background noise floor at the resonances; this tendency is more pronounced with items having low damping such as the test beam. In an attempt to obtain the best possible translational mobility data for subsequent use in deriyjng rotational mobilities, replacenent magnitude data for the more noticeably erratic regions were obtained using sinusoidal excitation, A sine wave generator was substituted for the random signal generator and bandpass filter, and the acceleration and force signal peak values were read from the oscilloscope without the use - - 35 No revised phase measurements were made. of the computer. A typical the substituted-data version of the same mobility is shown for in Figure 3.4. / comparison of the original random excitation mboility results with Magnitude data substitutions- were made in the experi- mental mobilities as follows: Frequency Range(s) of Substitution Mobility /" 2 None 330-530, 770-960 300-530 , 750-950 None None None 130-220, 380-520 93 330-520 430-540, 790-960 The substitutions were made over wide enough bands of frequencies so that the sinusoidally generated data merged with the randomexcitation data with minimal discontinuities. Figures 3.5 through 3.12 show plots of the substituted data versions of the remaining experimental translational mobilities. All theoretical mobility plots are comprised of straight line segments connecting data points at 10 Hz intervals and are calculated based on an assumed equivalent viscous damping ratio of .005 for In general, the agreement between the experi- 36 - each elastic mode. mental and the theoretical results is good up to the third resonance (approximately 850 Hz); however, there are noticeable discrepancies in frequency at the fourth resonance. The reason for the discrepancies is not clear; possibly the impedance head rotational inertia became significant in this higher mode, where rotational kinetic energy acquired its greatest proportion of the total kinetic energy. It was concluded that the portions of the experimental data below 1000 Hz could be considered "good" data for generating rotational mobilities, but that satisfactory results could not be expected above 1000 Hz, D. Experimentally Derived Vs. Theoretical Rotational Mobilities The computer program ROTAT listed in Appendix D was written to perform the backward difference calculations indicated in Eqs. (2.15) and (2.16), which generate right-end rotational mobilities such as those indicated in Figure 3.2 from an appropriately chosen set of translational mobilities, The translational mobilities are read by the computer from storage on disk in quadrature component form over a set of discrete frequencies. The output rotational mobilities are plotted in magnitude and phase form and can be stored on disk for subsequent manipulation if desired. The previously discussed experimental translational mobilities were read by this program for calculation of the test beam 37 - - rotational mobilities F, and Y 14. The results are shown plotted in Figures 3.13, 3.14 and 3.15, respectively, along with corresponding theoretical mobility functions generated with the use of the previously mentioned computer program THEOR. Again a damping ratio value of .005 was used for each elastic mode in calculating the theoretical mobilities. The agreement between the experimental and theoretical versions of the rotational velocity/force mobilities and over the previously cited 0-1000 Hz band of "good" translational mobility data is reasonably close. Although the experi- mentally derived rotational velocity/moment mobility gives, reasonably clear and accurate, indications of resonances, it exhibits a great deal of scatter in both magnitude and phase in some regions, This latter mobility function and its constituent translational mobilities were examined closely in the frequency band 230 to 300 Hz, where there was a marked degree of scatter in both the magnitude and phase plots, The translational mobility data in this band were generated entirely by random excitation with no substitution of sinusoidally generated data. The scatter in the derived mobility data in this band seems at first glance to be inconsistent with the smoothness of the translational nobility data, Figures 3.4 through 3.12, within the same band. The quadrature components of the constituent trans- - lational mobilities over this band are plotted on expanded scales in - 38 Figures 3.16 and 3.17, and the quadrature components of the resultant rotational mobility are shown in Figure 3.18. It is seen that the translational mobilities had been nearly purely imaginary, but the algebraic summation of these numbers gave a resultant imaginary component which was much smaller than most of the individual constituents, magnifying the minor degrees of irregularity present in them. The scatter in the real components of the constituents, Figure 3.16, had been present due to minor deviations in measured The scatter in the quadrature com- phase from the ideal value, -90'. ponents of the resultant mobility Y, Figure 3.18,is the source of the scatter in the magnitude and phase noted in Figure 3.15. This examination of scatter in the Y45t 4W mobility shows that the stability of derived rotational mobilities, would be enhanced by performing smoothing operations on the translational mobility data before the differencing calculations. An effective approach to smoothing might be to fit analytical mobility expressions to a number of data points in each experimental mobility as described in References (4) and (9), Examples of the resultant rotational mobilities which can be derived by the differencing method from translational mobilities which are smooth and accurate are shown in Figures 3,19 and 3,20, The THEOR program was temporarily modified to establish quadrature versions of the theoretical translational mobiliti es of Figures 3.4 through 3.12 on disk files, and the - - 39 mobility data points in .g8A4) and 3F (4) Figures 3.19 and 3.20 were generated by having the ROTAT program process these files in the same manner as it had processed the experimental data. With the exception of small deviations in anti- resonant frequencies seen in Figure 3.20, the agreement between the theoretical and derived mobilities is excellent. Figures 3.21 through 3.24 show Y49 f C ) and rotational mobili'ty results similarly derived from theoretical translational mobilities which were calculated at locations corresponding toA8.038MandA 2 .02z%/ or twice and one-half the spacing of the experimental measurement points. The results for the wide spacing, Figures 3.21 and 3,22, show additional degrees of the antiresonant frequency deviation noted in Figure 3.20, but the more important matching of resonant frequencies is again achieved. The results for close spacing, Figures 3.23 and 3.24, show excellent agreement throughout, Thus the latitude of the differencing method of deriving rotational mobilities in accommodating variation in measurement location - 40 - spacings is demonstrated. IV. CONCLUSIONS Rotational mobilities of structures are equivalent to spatial derivatives of their translational mobilities and can be determined experimentally by finite difference approximations involving sets of measured translational mobilities. Good agreement was ob- tained between experimentally and theoretically generated versions of two rotational velocity/force mobilities of a free-free beam. An experimentally derived rotational velocity/moment mobility gave reasonably good indications of resonance, but exhibited large amounts of scatter in some frequency bands. This scatter was found to result from the subtraction of nearly equal translational mobility quantities in the differencing operation, magnifying minor irregularities present in them. It is believed that this scatter in the rotational mobilities can be eliminated by smoothing operations on the translational mobility data such as curve fitting before the differencing calculations. However, further investigation should be conducted to determine an efficient algorithm for performing the smoothing and to evaluate its effectiveness in reproducing the magnitudes and trends that characterize the experimental data. It has been shown that the differencing method of determining rotati onal mobilities can accommodate considerable variation in the spacings of the points where the constituent translational - 41 - mobilities are measured, Excitation: B A 4w Response: Mobility: (a) r4wA Y1I-,F Y(a) Translational Velocity/Force Mobility Excitation: Response: ( Mobility: (b) Rotational Velocity/Force Mobility 7A(t) Excitation: An.- Response: Mobility: (c) 72~ BA AW Translational Velocity/Moment Mobility 4() ei Excitation: AB Response: e w Mobility: (d) Rotational Velocity/Moment Mobility FT,4o) 8,A50 = C9 8(w~) j14 Yc7 V, A Matrix Equation Involving Combined Mobilities FIGURE 1.1: Transfer Mobilities Involving Various Combinations of Translational and Rotational Effects - 42 - (e) Kaidoj)1JAI Exci tation tiw (a) Response fA(t) () , 4 Fixed Response Measurement Location Excitation 2 W Re nse ~~7( i()M) e ia)44t WM) 2 (b) Mo. '4C 4 4UM ara) ( 44 ; 1 )a Varying Response Measurement Location. Imaginary Component ,4 Real Component X8 Plot of Resultant Complex Amplitude Ratios FIGURE 2.1: Relationship of Rotational Velocity/Force Mobility to Translational Mobility at a Single Frequency - 43 - (c) f4 1' Excitation ~2W ) Response ) A I (a) 0 A) e 64f-u Xz I Fixed Excitation Location Excitation iYt) ) A Response w ZZ) NI)e 0 I ) (b) 24-ICA Varying Excitation Location K% N I i i (c) Plot of Resultant Comolex ~? 1 Real Component Imaginary Component I Amolitude Ratios A /19? A Substitution of Equivalent Force Pair for Moment FIGURE 2.2: Relationship of Translational Velocity/Moment Mobility to Translational Mobility at a Single Frequency - 44 - (d) N -A.0318m .90m .009m typ-+ .L, 10-24NC x .25 in deep thd. 4 places Excitation pt. #4 .inaeep t. 4pc Excitation pt. #3 Excitation tt. #2 Excitation pt. #1 .0254m Light scribe lines coincident with hole centerlines Measurement pt. #4 Measurement pt. #3 Measurement pt. #2 Measurement pt. #1 Material: 1.00 in. x 1.25 in. cold rolled steel FIGURE 3.1: Test Beam Details An = 6 .044m typ. = I W3 AV I-* T ;F3 () W-4da) HA) Y B I~f wwJ3 YOS a) ynk1 Matrix of Desired Beam Mobilities -. 46 , FIGURE 3.2: WA A , YA, MB 4 Random Signal Generator General Radio 1390-A I Accelerometer Bandpass Filter Ithaco 4213 Impedance B&K 4344 Head WilI co xon Power Amplifier McIntosh MC40 Shaker Ling 203 ------- Z602 Beam Buffer/Attenuator Homemade Bandpass Filter Krohn-Hite 3550 I Spectrum Analyzer Federal Scientific UA-15A [ Two Channel A/D Converter I II Minicomputer Interdata M70 H- Teletype Test System Schematic Diagram - 47 - FIGURE 3.3: Bandpass Filter Krohn-Hite 3550 I X-Y Plotter Oscilloscope Acceleration Signal Preamplifier Ithaco 432 -30 LJ -35 _* Theoretical Magnitude Experimental Magnitude -40 Ui -4b -50 -55 -70 - 70 7 0 -75 -80 0 1500 1000 500 FREOUENCY, (a) 2000 HZ Original Mobility Obtained with Random Excitation -30 1 U LJ S -35 - * z -45 0 -50 - * z -60 * wI Li Theoretical Magnitude Experimental Magnitude -40 -65 X -70CO 0 95- E Respons - - > 0 FREQUENCY, (b) 1500 1000 500 2000 HZ After Substitution of Sinusoidally Generated Data . 200 LO - -100 X 0 CL 7 -100 -Theoretical Phase 4A Experimental Phase m -200 0 FREOUENCY, 2000 HZ Mobility Phase Plot FIGURE 3.4: Test Beam Translational Mobility *4 After Data Substitution 48 - (c) 1500 1000 500 Before and -40 LU Lfl -X -50 H- z -60 X- -70 -80 z -90 H- -- Theoretical -100 -Experimental Magnitude Magnitude F-4 oJ * tResponse -110 gniud Exeimta - 0 -120 0 500 -Tereia 1500 1000 FREQUENCY, Phs 2000 HZ 200 --- Theoretical Phase ID LU 6 Experimental Phase 100 0 H- 0A W a_ -100 AA -200 0 500 FREOUENCY, FIGURE 3.5: 1500 1000 HZ Test Beam Experimental and Theoretical Transl ational Mobility 1,2 49 2000 -30 --- z * -40 Theoretical Magnitude Experimental Magnitude Response 50 Excitation -60 -70 -80 F-1 CD o -90 -100 500 0 1000 FREOUENCY, 1500 2000 1500 2000 HZ 200 Theoretical Phase L-- A Experimental Phase A 100 0~ H -100 -200 0 500 1000 FREOUENCY, FIGURE 3.6: HZ Test Beam Experimental and Theoretical Translational Mobility *3,l 50 -40 LJ -50.) * Theoretical Magnitude Experimental Magnitude z -60 -* co -70 - -. -x -80 -* z -50 LD F-4 ED 0 -100 Excitation Response - -110 ~- -120 0 1500 1000 500 FREOUENCY, 2000 HZ 200 LO wj - Theoretical Phase A Experimental Phase 100 wn 0 U, - I 1000 1500 -100 -200 0 500 FREQUENCY, HZ Test Beam Experimental and Theoretical Translational Mobility ' , 2 2 - 51 - FIGURE 3.7: 2000 -40 Li w- Theoretical Magnitude Experimental Magnitude * -50 z Response w1 0 0Excitation -70 z -80 M -00 -100 0 500 1000 FREQUENLY, 1500 2000 HZ 200 Theoretical Phase - LD w0J A 100 Experimental Phase A A -Ru20 7 -100' :3 0 500 FREOUENLY, FIGURE 3.8: 1500 1000 HZ Test Beam Experimental and Theoretical Translational Mobility V3,2 - 52 2000 -40 - * U Theoretical Magnitude Experimental Magnitude - -50 Li -70 z S OD -80 X -90 tResponse Excitation 1X -100 0 500 1000 FREOUENLY, 1500 2000 1500 2000 HZ 200 -- LiA Theoretical Phase Experimental Phase 100 0F- 3 2 - -100 -200 0 500 1000 FREOUENCY, Test Beam Experimental and Theoretical Translational Mobility *4,2 - 53 - FIGURE 3.9: HZ -40 Li -X- Experimental Magnitude - Theoretical Magni tude - z *- -60 Excitation, Response I -70 z -80 C - * -90 100 0 500 1000 FREOUENCY, 1500 2000 HZ 200 Li - A 100 Theoretical Phase Experimental Phase Lfl 0 0 -100 :- -200 0 500 1000 FREOUENCY, FIGURE 3.10: 1500 HZ Test Beam Experimental and Theoretical Translational Mobility *3,3 - 54 2000 -30 LiJ Ln - Theoretical Magnitude -X- Experimental Magnitude -40 Response CD 50 E Exci tation z Lii M 60 -X- 70 CD *X - 80 90 100 1000 500 0 1500 2000 FREOUENEY, HZ 200 -- Theoretical Phase Li 100 & 0 Phase Experimental A A 0 A -100 ::- -200 0 1000 500 1500 FREQUENEY, HZ FIGURE 3.11: Test Beam Experimental and Theoretical Translational Mobility 4,3 -55 2000 LJ -- S-X- Theoretical Magnitude Experimental Magnitude 40 Exci tation, Response - -50 6- *X ED -70 >_ -80 - z X -X_ * H4 mX X0010020 X0 0 -100 0 1500 1000 500 FREQUENLY, 2000 HZ 200 -- A 100 Li n >_ A 7 - 0 Experimental Phase -N- 0 n Theoretical Phase -100' -200 0 FREQUENLY, HZ Test Beam Experimental and Theoretical Translational Mobility 4 4 - 56 - FIGURE 3.12: 1500 1000 500 2000 -15 I - Theoretical Magnitude X Experimentally Derived U~ wL Magnitude x x MX x 2- 30 xx Xx xX L:7 T x x xx x x x x X XX X x -45 0 500 1i000 FrE0JENCY, 1500 2000 1500 E000 HZ LID 200 - Response Excitation IF 100 IL 0 -j 1-4 -100 - Theoretical Phase v Experimentally Derived Phase -200 0 1000 500 FR n FIGURE 3.13: LE NE Y, HZ Test Beam Experimental and Theoretical Rotational Velocity/ Force Mobility YB FA - 57 0 Response f Excitation Li w -- Theoretical Magnitude X Experimentally Derived Magnitude 10 Ln z X Lii fm Kx I -30 X xX XX Ix SXX X -40C - XX X X X Mi x X x I X FT- -EO X X X X X LL x X x -70 x x - BO 0 1000 500 FRECU ENEY, 1500 2000 1500 2000 HZ Li 200 VI/, Li Tj .0 t-i CD -100 v -200 n Theoretical Phase Experimentally Derived Phase 1000 500 R7PEOUENCY, ( Test Beam Experimental and Theoretical Rotational Velocity/Force Mobility YOFB - 58 - FIGURE 3.14: HZ LP 0 z x x xX X x X -10 xY X - z F- -J -20 XXX -30 x X X %X X x X -40 -50 0' ii - X -60 Excitation Response Theoretical Magnitude Experimentally Derived Magnitude - w x x X -70 FREQUENCY, 2000 1500 1000 500 0 HZ 200 Experimentally Derived Phase LI v 100 L1- 0 1V -100 I CD -- Theoreti cal Phase -200 0 FREQUENCY, FIGURE 3.15: 1500 1000 500 2000 HZ Test Beam Experimental and Theoretical Rotational Velocity/ Moment Mobility YOM () - 59 - ?XX X ~ 10 Li I I i . .0006 I .0004 F- .0002 I0 a, (A *1-~ .0000 imt- dMP- ak. - WP PO - E -Q 0 '4- -. 0002 0 4-, C a, 0 0~ E 0 -. 0004 1- O---O 2x(-4)x"3,2 L2~-~ ~2x3x$ 4, 2 0-lx16x , a, 3 3 7 -. 0006 -. 0008 7 0-cD---O I 220 I 2x(-12)x 4,3 lx9x$ 44 1xl x 2 , 2 I 260 Frequency, | 240 I 280 Hz FIGURE 3.16: Constituent Terms of Derived Experimental Vbbility Y MB(w) Over a ftrequency Band - 60 - of Large Scatter: Real omponents II 300 .020 - I II_ I p 0 I 2x(-12 )x 4,93 .015 I0 aw 0l 4--) E .010 I- -o .005 F0 2x(-4)xiP3 2 0 4-) .000 lxlx E- 2 ,2 -. 005 1x9x M4 -. 010 1- lx16x -. 015 3 ,3 I 240 220 I 260 I 280 I 300 Frequency, Hz Constituent Terms of Derived Experimental Mobility Y0 BM (w) Over a Frequency Band of Scatter: Large - 61 - FIGURE 3.17: Imaginary Components I I I iI I i I .120 I- .080 1Imaginary Component E W%- .040 - a-) .000 C, -- -. 040 1-0 Real Component -.080- -. 120 |I 220 240 260 Frequency, I 300 Hz Quadrature Components of the Derived Experimental Mobility YBMB (w) Over the Frequency Band of Figures 3.16 and 3.17 - 62 - FIGURE 3.18: I 280 0 II A= AE Lfl I = .044m Magnitude [---Theoretical -10 Derived Magnitude -)--Response z ! Excitation -20 x- - 'N wf -40 -50 -70 -BO 0 500 1000 1500 FREOUENEY, 2000 HZ w 200 -J 100 Theoretical Phase v Derived Phase 0 m -100 11 -200 0 500 1000 FREOUENCY, HZ Rotational Velocity/Force Mobility YB F (w) Derived by Differencing Theoretical Translational Mobilities: An = AE = .044m - 63 - FIGURE 3.19 1500 2000' 10 X X U LF) 0 z -10 -20 M F-l xx -40 zx < X -50 -80 = T KExcitation, --Theoretical at= .044m Magnitude x Derived Magnitude )Response I -70 0 500 1000 FRE0UEN[Y, 1500 2000 HZ L 200 < I 100 -100 D - Theoretical Phase Derived Phase Ax K -200 I 0 500 1000 FREOUENCY, FIGURE 3:20: 1500 2000 HZ Rotational Velocity/Moment Mobility Y0B (w) Derived by Mobil ities: Di fferenci ng Theoretical Translational a= at= .044m - 64 0 An=AE Li wL Lfl =.088m -10 z Theoretical Magnitude X Derived Magnitude Response Excitation L- ---- x X -20 -30 x -40 CD -50 -S0 x x - -70 -80 0 500 1500 1000 FREOUENCY, 2000 HZ 200 - L: Ln 71C Theoretical Phase V Derived Phase 100 a- 0 CD HA =5 -100 -200 0 500 1500 1000 FREOUENCY, HZ FIGURE 3.21: Rotational Velocity/Force Mobility Y BFB(w) Derived by 0BFB Differencing Theoretical Translational Mobilities: An =A = .088m . 65 2000 X X , 10 Li AX 0 z -10 LiJ -20 C13 -30 - xX xx xMxn xd x x -40 x x -50 E Theoretical Magnitude -.088n X [j Excitation, - - -F0 X Derived Magnitude Response -70 0 1500 1000 500 FREOLENCY, 2000 HZ Li 200 100 CD 0 T -.100 Theoretical Phase v Derived Phase 0 IFREOUENEY, 2000 HZ Rotational Velocity/Moment Mobility YEB (40) Derived by Differencing Theoretical Translational Mobilities: An = A = .088m - 66 - FIGURE 3.22: 15Ij00 1000 500 u ----- Theoretical Magnitude Derived Magnitude A022AC Response E Excitation| -10 -20 0 frn - z -30 -30 -40 -9- 500 0 1000 1500 FREOL-ENEY, 2000 HZ CL 200 - -Theore ti cal Phase Derived Phase -100 -0 0 500 1000 FREUENCY, HZ Rotational Velocity/Force Mobility YOBF B(w) Derived by Differencing Theoretical Translational Mobilities: An = A = .022m _ 67 - FIGURE 3.23: 1500 2000 10 Li 0 FY Mu -10 - 20 Lu -30 -40 X -50 An = A 9 022m. --- Excitation, Response Theoretical Magnitude X Derived MagnitLude - -70C 7O 0 1500 1000 500 FREOUENEY, 2000 HZ Lu 200 LJ Lu n <_ 100 0Q 0 100 - -00 0 Theoretical Phase Derived Phase 1500 1000 500 FYEQLFNEY, 2000 HZ Rotational Velocity/Moment Mobility Y B (W) Deriv ed by Differencing Theoretical Translational Mobilities: An = AE = .022m - 68 - FIGURE 3.24: v REFERENCES 1. W. C. Ballard, S. L. Casey, and J. D. Clausen, "Vibration Testing with Mechanical Impedance Methods," Sound and Vibration, January, 1969, pp 10-21. 2. J. V. Otts and C. E. Nuckolls, "A Progress Report on ForceControlled Vibration Testing," J. Environmental Science, December, 1965, pp 24-28. 3. R. M. Mains, "The Application of Impedance Techniques to a Shipboard Vibration Absorber", Shock and Vibration Bulletin, 33, 4, March, 1964. 4. A. L. Klosterman and J. R. Lemon, "Dynamic Design Analysis Via the Building Block Approach", Shock and Vibration Bulletin, 42, 1, January, 1972. 5. R. DeJong, "Vibration Energy Transfer in a Diesel Engine", ScD Thesis, MIT, Dept. of Mech. Eng., 1976. 6. D. U. Noiseux and E. B. Meyer, "Application of Impedance Theory and Measurements to Structural Vibration, U.S. Air Force Flight Dynamics Laboratory Tech. Rept. AFFDL-TR-67-182. 7. F. J. On, "Preliminary Study of an Experimental Method in Multidimensional Mechanical Impedance Determination", Shock and Vibration Bulletin, 34, 3, December, 1964. 8. J. E. Smith, "Measurement of the Total Structural Mobility Matrix," Shock and Vibration Bulletin, 40, 7, December, 1969. 9. D. J. Ewins and P. T. Gleeson, "Experimental Determination of Multidirectional Mobility Data for Beams", Shock and Vibration Bulletin, 45, 5, June, 1975. 10. E. Isaacson and H. B. Keller, Analysis of Nwnerical Methods, John Wiley & Sons, Inc., New York, 1966. 11. J. W. Leech, L. Morino, and E. A. Witmer, "PETROS 2: A New Finite-Difference Method and Program for the Calculation of Large Elastic-Plastic Dynamically-Induced Deformations of General Thin Shells," U. S. Army Ballistic Research Laboratories, ASRL TR152, Contract Report No. 12, December, 1969. - - 69 REFERENCES (Continued) 12. S. Timoshenko, D. H. Young and W. Weaver, Jr., Vibration Problems in Engineering, 4th Ed., John Wiley & Sons, Inc., New York, 1974. 13. L. Meirovitch, Analytical Methods in Vibrations, Macmillan Company, 1967. R. D. Cavanaugh and J. E. Ruzicka, "Vibration Isolation of Non-Rigid Bodies", Colloquium on Mechanical Impedance Methods, ASME, New York,1958. - 70 - 14. APPENDIX A THEORETICAL MOBILITIES OF A FREE-FREE BEAM I The governing partial differential equation for the free vibration of an undamped uniform beam is given by Eq. (5.82) of Ref. (12) as 42 T 0-&(A.1) The derivation of this equation, referred to as the Bernoulli-Euler beam equation, is based on the assumption that the effects of rotary inertia and shearing deformations are negligible in comparison with the effects of translational inertia and flexural deformations (i.e., the beam is slender). The free vibration mode shapes and frequencies are obtained by first assuming a harmonic solution of the form - 71 - (A.2) Substitution of this expression into Eq. (A.1) results in the ordinary differential equation B~I,",~a (A.3) where the prime notation indicates differentiation with respect to x. Setting .10oa.~) ) (A.4) the general solution of Eq. (4.3) can be written Tex) -/s>, 4 1pR +GCOSCpx+C 3 s-- px+C4 cospx A.5) for which the first three derivatives are: h(?x') =? ( ) SkzA lK +COZ COsyx -p 6 4 s/fX osCh~w 71x' 4p CO ? e COS/?x 35/''Px->C -p 7Lh IX- CO5XS P .6) (A S/f The boundary conditions for a free-free beam are as follows: - 72 - W&I() = Z- 2. oy, L0 =L=0 4J (A.7) or, simplifying slightly, W/o) = 0 ,W()=0 (A. 8) W"'()=O Inserting the Eq. ( W'"(O)= A.8) boundary conditions into Eq. (A.5) and (A.6) gives the system of equations 5ifA cos4ifA si;7p1 - -C 5ky2~/ 0 Ca. - / 0 cas/i C 0 0 =0 Cl -coy/v 1 0 0 O (A.9) For nontrivial results, the determinant of the above 4 x 4 matrix must be equal to zero; effecting this condition yields the characteristic equation co&4, (A.10) / COSg>d The roots of this characteristic equation are the eigenvalues of the problem and are given in p. 165 of Ref. (13) as follows: - - 73 (rigid body modes) ,zI IP, 73= 0 ,Vz? 3 = A 2 (?r/4 4763 (first elastic mode) (A.ll) (second elastic mode) r >4 The natural frequencies are given by: - r4 *EI(Pr~)4 ~.AA or 64-4 (A.12) The eigenfunctions are found by arbitrarily setting and using the first two of Eqs. (A.9) to determine that and 6c=0. When these results are inserted into the third of Eq. (A.9) it is determined that: Cas pI / - C0 IZ 3 Then, the rth eigenfunction of the problem is: 4z[cos/A + COS~rY ~p - 74 - (A .13) for each eigenvalue ,- r ... , where.4.- is an arbitrary For the special case of the rigid body modes multiplier. the eigenfunctions are: () = (rigid body translation) (rigid body rotation). W ).. Let (A.14) ,A(Y)denotethe bracketed quantity in Eq. (A.13) for modes r=,23 .. ,, Per Appendix B of Ref.(14) the functions have the orthogonality properties el (y4 0 g /V = dX ny#? (A.15) Also, per p. 164 of Ref.i(13), the functions $/ are orthogonal to each other and to the functions these functions also and 4.X) have the properties 3 (A.16) At this point the forced vibration response for the undamped beam can be evaluated in terms of the preceding eigenfunctions. - - 75 Let w(x~,zt) (A.17) (x4 Ar where the %,() quantities are time-varying generalized coordinates Placing this expression into the governing partial to be determined. differential equation with forcing term, $f~4A-f4 =( - 7 ) (A,18) yields 19;0 1, 2I~- 74- 0 Now each term is multiplied by (A.19) 4x) and the resultant expression is integrated with respect to x: r=O Y I 444Z~- 1JSX,t-=O If the eigenpair 4 AX4. . satisfies the undamped homogeneous differential Eq. (A.), then -a P 4-=0 - 76 - z1~r,"" (A.20) and 5 C 4-, (A.21) J. substitutingjq. (A.21) into Eq. (A.20) t en yields , DOA 4-~ 1~~ f0 (A.22) Applying the orthogonality properties, this result reduces to the normal mode equations of motion, S))= ~~WO4 It1p = where .. is the modal force of the rth mode. X Damping of the elastic modes can be taken into account by modifying Eq. (A.23) to the form +o__)__+A (A.23) is th-e modal mass of the rth mode rX and ,/ )Z(A.24) is the revised modal force, where is the prescribed excitation loading exclusive of damping forces, and 2$, is the equivalent viscous damping ratio of the rth mode. To obtain mobilities we will apply loads i(t)C(_ I - - 77 denotes the Diract delta function at where 4Wr location X- X The pertinent response for each mode will be the steady-state sinusoidal generalized displacement >) ot -f-t) = XPO 'I,-6 where amplitude X, (W) (A.25) is complex. We will examine this response mode by mode: r= 0 a* 0 #1.m 7..) - (A.26) r= J2 /Z /2- 4) 40t',6i -a)( - 78 - 0' I 2aj)e '5 Z'XL7) (A. 27) r = 2,3,... Or 4'(>) Mr"fdx-)4,()<d~41-n = aeo't#,g P(),4V .-. X () (A. 28) As a consequence of Eqs. (A.17) and (A.25), it follows that (&t-) .A =E r=0O A ( c.40).X(a)e i l!t (A. 29) The mobility is then the ratio of the complex amplitude of the transverse velocity at / to the amplitude of the transverse force at 9 : F>) Using Eqs. &0v 047(/) _01>) r=O (A.26), (A.27) and (A.28), we obtain the resulting expression for translational mobility: - - 79 (A. 30) / _ W(W) 9 - ) a() IZTI1.A/ jWb7 (A. 31) In accordance with Eqs. (2.5), (2.9) and (2.11), the rotational Tca>) c? - ) mobilities are given by: iw nI /2 ~~ ' 10-2 .(v) &f&) 0() Mw) "7 r-=2 /A =? aZkIw 4~ 4(i) 2 (A. 32) &32--w U 2? .S',~i4-io Z ~$g(/7)4iv 4rgOgZZa> iWr~ 2 W l~,LLZ SrWrW The series terms in the above expressions are truncated to highest mode numbers r = N for computation, where , <9 . N is determined such that The latter condition, in turn, ensures that the exact mobilities are approximated by the truncated series results within much less than 0.5 dB deviation. - - 80 APPENDIX B COMPUTER PROGRAM THEOR C C *****.*******pIRevRqM THEOP ************ C PDvGRAm TO CAICUIATr, STORF, E PLOT TBEOPETICAT C NOnTTITTFS (rAfrNTTUDP & PHASE) INTEGEP*2 PUTNTD1(40), CASETD(40) INTFGWT*2 YLA4), XLP(40) 'lTVNSI0N TRANSLATIONAL AND ROTAlTONAt OnECR(25),ANUF1(2c),ANU?2(25), ANUM3(25), ANUM4(25) AMM1B(1,210), PgrTB(1,210), 75(4) CV!PLFX kDEND, AMOP D-"BTF OPFCTSTON 7,DPPL,A1PHA,PP,PRT,P3XT DnTIPLF rP1RCISION PHY, PHXT, PHPPX, PHDP7I,RCSH,DBSNH C D7FTN7 DOUPE PRECTSIWN SIWH , COSH ARTTHMWTIC STATEMENT FTNCTTONS DT1FNSION DCSH(Z)= (DEXP(Z)+rEXP(-7))/2. nl SvH(7)= (DETP(Z)-DEXP(-7))/2. DATA XF/ 0.,2000.,-400.,400./ DErTNF rITP l (15,420,U,NPP) C )'AI PPTPEPTES: r=(N/**2), AT= I (***4), "), 7ETA= AFSTUED VAVPING RATTO CELCU- AM= TOTAL MASS (rG), Al= IF GT4 ( C NqP7QZ= Nn. 07 FORCTNM. FPEQS. (INTECER) FOP WITCH Mf)PILITIFS WIII BE LATED, !PACED TNIFOPMLY, UP TO F1AX (FLOATING) C CHECW DIVENSTON STATEMENT FOP ARRAY SIZE C CAUTIOnRVAD (P,8r') RIIRTD1 P AD (8,95) NCASES D-AD (9,130) vrPEQS, FvAX or -1. (0,100) 7,kTI,AM, lT,,FT A Q F "rAT(40A2) 95 F')RvAT(T10) 130 FOUAT (I10, F10.0) 1lrv F0nP'ATr (F10.1, F10.2, 3F10.3) WQTTE (5,qC) UNTNID1 WRTpF (5,140) E,AI,AM,AL,7FTA WTrTE (5,160) NqREQS,FVAX. 90 F"R?4AT (21 ,4042) 14V FOPVAT(3H0P=,E10.3,4H I C I=,10.3,4F M=,F5.2,4 L=,F5.3, ZE"A=,F7.4) vAy rpEO =,F5.O///) Dl F000 NCA17E= 1,NCASES X= 10CATION OF VElOCITY,'XT= lCfATION OF FORCE OR MOMENT, KWF THRU XTHV ARE CONTPOLS OF WHICH TYPEF. OF WCRILITIES ARE GENERATED (1 FOR YES, 0 FCR NO), NSWF, ETC. ARE NOV. OF PLCTS CF EACR TYPE DESIPED: 0= NONF,1= MAGN. ONTY, 2= MAGN. C DRASE 160 C c 7P FnPMAT(14HOWO Or FPEQS =,I4,12H 81 - C C CAUTTIN- PIOT CONTROl DATA AND IARFL CARDS PUST BE PROVIDFD CCNSISTFNT WTTH C ABOVE INPUTS C LCWr, ETC. ARE DISY lOCATIONS FOR STORACE OF RPSULTS- SUPPIY0'' WHEN MCMLITTES APE NOT TO BE STCRED C READ (R,80) CASEIT PEAD (p,110) X,XI,KVF,YWF,YTHF,KTffM READ (P,120) VPWF,NDWM,NPTHF,NPTHM AAD (R,120) ISWF,LSWW,LSTHF,LSTHM 110 FODRAT (2F10.2, 4110) 120 FCORAT(4I10) PDT= 3.141593 R!I0= AM/AL ,PDAD (F*RT/(pHOA*AL**4))**Q*I5 Rv.AY= C.54(1./PTE)*(4.*PTF*r~WAX/PPAP))**0.5 ?qv'WX= RMAX+ 1. qRITF'(c~r0) CASFTD WPTTE (5,150) XXI, KWFrXW~rKTHVFTHN W'vTTE(fi,155) 'FPVFrNDW!rNPTVFoNPTP! WPTTF( 5 ,156) Tl-wF,1c~wLTFISTVMJ KT1P'=t KW=,12r7H 150C F rI.wAT (3V Y= , 6 .3 5F YT=,6.3,6H YWF=,T2,6H 'I T 2, 74 TWO! =, T 2) WPT'I=,I1) NPWT=,T,9H, T'P TF=,T1,9H, 155 Fn?'IAT(6H0VJPWF=,I1,%IH, LSTH=,pT1) TSTHr= ,T I 9R, 15)( FrP JT(6PCTWF,1,PV,, LSWV=,T1,0H, C'ILCUIATE WfDRL PARPFTFPS OF PFCU1RPTNG USF WPTTr (5,17f)) VPFQUJERCTFS OF VTPPATrPY MODES,14%/) 110 r',PI!AT(i4P0 WJUAI Dn 400 VR= 2, NRMAX IF M~-3) 210,220,230 210 PT=4.730 t7 ' TO 0 7 W 214n nm7Vp(NP)= (PPL)**2*PPA) WD.TT7 3:0 F0P!4AT (c',3n) r'pEox (20 y,F10.1) ?djP"A= (DJz'SNq(pDDp)+DSTV(DPPt))/(!Ml3CSP(DPRT)-DCoS(DPrRL)) ) FT 0= D PI F'( P t/A PPXT= r~PtF(PRL*X/A) PIXYDPCS(PP)+DCINPR)-AIPHA*(DPSH(PP!)4DSIM(PRX!)) 'OUYT=D' 7rS1J(PPXT +~SPY)AtH*DSPPTT+SFPY) PTIPRXI=PR*(DB5FW14(PRXT)-DSTV(PPXT ))--!LPPA*PD* 1(DPC'SH(DvXT)+DC09(PRXT)) T VITJ?02(wP)= 1.NLPR*"R ANT1V3(VR)= SN!PPXDFT AkU?44(NR)= SV17h(PPPRX*P4PPXT) CALCUIA"E W/F VOBTLTTTFS TF S"r-CTFTED TN TNPUT DATA 1000 II' (KWV-1) 2000,101001010 11,10 D!1 15CC NPT= 1,FFREOS 0m~rA= 2.*T*vAX*rCATIPT)/FLOIT(NFPF0S) BNU! = (X-A/2.)*(XT-AL/2.) BI)Fg r~ECA*Av*(Al**2)/12. Aw)+CVPI ( Ulf0 0)/CwPIX Awoq= p ly ( 1 .0 ,0 .0) /CVP IX(0. 0 ,OFA D!' 140fi NRP= 2, NRM'AX PDEN~= (0MEGRUJRR))**2 -rOI'RGA**2 C~vWV 2.0*ZFTA*0ME9R(NRP)*!MEGA UD1 "D= CP'DtYUNUTM1(WRP) ,0.0)/CPL(PDW,CDF'f) CCnvEW= ONEGA/Am 14,10 A~nR= AT0R+CNLX(0.fl,CCOEF)*ADWD A~MO(1,NT)=20 .*AL01~10 (CABS (ANO0B)) C MVr4ITI"DE Tn DR RE 1 M~/ NT SEC -82 Es - - Avwd'( dih Qvt1E UC soad~lbN 'L =&Wt iii SaILI1I~a0W avlaiil 'DOSE ;VQ OLG a . t.lvcI .fldAI ki1 GadI3~ds ~Fl2V,)X) &dIX (OYS'b) Cli'?z0 i O9~ oOYz'OOOE'CdJOE (L-Aridi4) at W/ =OAhcad aI XU j1% V Y VLMi (L-dc~oN) 0* ), X('a'Xit) d da V k i (. d -Z Jc~) GLd" s (kXoAA)ZN*J1V = AlI1 *dk)5i OLO~~2OLO atiaa9'O(Qw'UO I a - dk) d, ti ) l akUO) V a*, a i O = ~.))ti^ ;t .* I C D i d ( ) IuII 176 k iiid0 j Z 0 ikfli0t Xrla U ViY s it t afit")IIvtto3 cQQOS 009ti'000'000 (L-khldM.) (sLcjt7'c.3) (iihl~i lsO ilOA a.L1'am N ID-Li? i SLiTII.t L -I. i hi a flIN V Z- iSt ,i (L-AniLA) E (o'/ aI avihi 1 I=Qb u.L id, ElfVl Xl1111'(d OLS17 0im4)D1t.a =,kA /4id t717.T /~C; (Gt'O)~d/O'0dhLAiaA.. OLO00i04Oot1'u ildt t[ G~i~X~S ,YE s o z~ dc rJI flLI t of iu-a OECOGtt (Md'G 3LS Appfi1 C COmPVT C C C**********PRGRA PINM TRMS TRANS************ PROGRAM TO CALWULATE, STORE, - PLOT TRANSLATIONAL MOBILITY FUNCTIONS FFON CCRER PROCRAP ETP*TAt SPECTRAl DATA- ALSO PLrTS STORED THEOR. NOBIIITIES INTEGFR*2 IDTAPE(40),XLA(40),TLPP(40) DTrENSICN AR!AY(7,210),AWSCL(4),PHSCL(4) C C C C C C C C C C C - 85 - C DATA PHSCL/ O.,2000.,-400.,400./ DEFINE rTIE 10(20,420,U,NRP) DEFINE FILE 11(15,420,U,NRQ) READ NO. OF TAPES TO BE PROCESSED IN THIS RUW READ(8,100) NTAPES 100 FORMAT(M2) DO 1000 NT=1,NTAPES READ IN CONTROL DATA FOP EACH TAPE TAvSCL=1 FOR AUTOSCALED PAGNITUDE PLOT; -2 FOP SCALING PER AMSCL DATA LLAR=-4 FCR EXP'TAL DATA TO BE CCNNECTED PY LINES; -4004 FOR DATA SYMF#LS; THEOR. DATA CONNECTED RY LINES IN EITHER CASF IAMCCN=-10 TO PLOT EXP'TAL 9 THEOR. WAGNITUDE5; -8 FOR EXPITAL ONLY FOR EXP*TAL ONLY TPHCnN=-68 TO PLOT EXP*TAL & THEOR. PHASE; -6 LSM,NWFRFQSNPLOTS,NLISTNPRLSTHEOFNAX IDTAPE, P"AD (P,200) READ (q,205) (AMSCL(I),Izl,4),IANSCL,LLAP,IANCOW,IPHCON 200 FORMAT( 40A2/6110,F10.O) 205 FORMAT (4F10.0,4I10) READ PLOT 1ABELS IF APPLICABLE IF (NPLOTS-1) 240,210,210 210 READ (9,220) XLA 220 FORnMAT(40A2) IF (NP!OTS-1) 240,240,230 230 READ (8,220) YLPH PFAD ONE TAPE'S DATA FIOR CARDS 240 PEAD(P,300)((ARRAY(TJ),If1,4),J=,NFREOS) 300 FORMAT (F6.OF6.1,6X,2F6.1) I (NILTST) 400,400,350 LTST INPUT DATA I? SPECIFIED (NLIST = 1) 350 WRITE (5,360) (fARRAY(I,J),I=1,4),J=1,NFRErS) PHIAF$ CPSDAF PSDF FEQ 360 FORMAT (' 1//(71,F5.0,3E15.4)) REDUCE COHERENCE PROGRAM DATA NOTE- NO IMPEDANCE READ MASS OR FLEXIBILITY CORRECTIONS INCLUDED 4)0 DO 500 I=1,NFREQS W= 6.2R318*ARRAY(,I) ALP = 2.*(ARRAY(3,I)-ARRAY(2,I)) -20.*ALOG10(W) PHASE = ARRAY(41,I) - 90. ARFAY(4,I) = ALM IF (PHASE - IO.) 420,420,410 410 PHASE = PHASE - 360. 420 IF (PHASE + 180.) 430,440,440 430 PHASE = PHASE + 360. 440 ARRAY(7,I) = PHASE AM9N = 10.**(ALM/20.) PHASE = PHASE / 57.29578 ARRAY(2,I) = AWGN * COS(PHASE) ARRAY(3,I) = ANGI * SIN (PHASE) 500 CCfININD ITr (t5v) 565,565,550 CWPTTE WOBIITY CO C QUPP COMPI~'NTS ONTO DISY IF tSN>o 550 WPTTF(10LSI-Z) U(ARRAY(IJ),,T2,3),J,NFES) WPITF (5,o560) 560 FOR*AT (tryPEPTNENTAL !NCBTIITY FTLED') r,65 It' (tSTuEC) 575,575,57M c P1'AD THF'ORETTCAL TMOBTITTY FROr DTSK IF LSTPFr>r 570 READ (11'tSTHEO)((ARRAY(I,,J),I=2,3),J=1,NFPEQS) 572 POPPAT ('TT~rOPFTTCA1 14(R)'TITY PEAD FpOr FIFI) 57F WPTTE(c-,60f)) TDTA PF,LS14 f 'rFrQS M LOTStFNR XLSTHFO FlqAY=ItFc5.oo 13,' NPL0TS=',I,Tl TV (NPP) 650,650,625i (NPR=1) C PT'T OUJTPUT1 n;ATA IF SrrCTFIF) 625 WPTTF(5,640) (~PYIJ,=,)J1N'E~ CO t'RFO 6L40 FORWAT(* 3E15.LI,,F1O.2)) 1 'PSFIf/(3Y, F5.O, 650 Tr, (NPT"'TS-1) 1000,700,700 CPIOT MOS'ILITY ~ICNITUDE IF NPLCTS1l OR 2 1 LSTPEO=o,12///)' oQuTAq 1 It' WNPOTS-1) 1000,1000v9Z0 C t'IIT !qniJl.ITY PHASE IF NPIOTS= 2 900 C'LL PICTP(PRPY7YLPPPSCIPCONN'RFO-r,O,-1LBr,2,F'AX,1) 1%0 C"IIVTI~ur STI"P F I' r -86 APPEIIX D CUWviER PSNM ROTAT C * *** * * ** ****P1GV ROTAT************ TO CAtCULATf, STORF, 9 PLOT ROTATIONAL FOBILITIFS FROM STfRFD IRANSLAC PROGRK C TI'NAL MOBILITIES USIN' BACKWARD DIFFERENCEr INTrGEP*2 PUNTD(40), tANAG(40),LAfPH(40) DT"ENSION SYMN(2,210),SYNNI(2,210),SYPN2(2,210) nTvWNSIN SYVIN(2,210),SY'INI(2,210),SY1WN2(2,210) D)TMENSION SYM2N(2,210),SYW2N1(2,210),SYw2N2(2,210) DIVENSIIN PHS(4), WrS(4), WNSM4, THFS(4), THMS(4) DATA P45/O.,2000.,-400.,40C./ DEFINF VILE 10 (20,420,U,NRP) DEPINE FT1Y 11 (6,470,U,WRC) C RSAD RUN IDENTIFICATION DATA RPAD (P,90) RUNID q0 FCRNAT(40A2) t EXCITATION LOCATION NOS. AND WPICP TYPES OF MOPILITY ELtFENTS C RrAD NOTIC C 'RA To 3E CREATED: READ(R,100) NY,XI,KWFEX,KWWEX,KTHFEX,KTPPFX 1)0 FOPFMAT(AT5) 1 RFAD mASUR1?WENT POTNT NUMPERS W, ?-I, M-2, pNr DRIVING POINT NUMBERS %, N-1, N-2; 1TPPIY 1O' WHERE NUMPBR IS N/A: C P7AD (P,1rO) ,,2,,,2 C RFAr POINT SPACING VALUES (UNITS- M.), NO. OF FPEOS., MAX. FREO.: C CAUTI'N: CPECK DIWENSION STATFMENTS FOP APPAY 'IZFS VS. INPUT NO. OF FFEOS. priD (p,225) PELTX,DELTXI,NFRQS,FrAX 225 FCR!AT(2F5.3,T5,F5.0) C R'AD DTSK ICCATIOW NUMPEFRS OF TPAVSLAT. MOPTIITY FILES TO RE READ- SU;PIY C 'O' WHER7 NUMBER IS N/A: READ (8,25O)LSMN,LSV1N,LSM2N,LSNM1 ,ISW1N1,ISR2N1,LSNN2,tS1I1N2, 1 TSW2W2 WPITE(5,45O)LSNN,LSW1N,LSN2N,LSMN1,tSW1N1,1SN2N1,LSN2,LSW1N ltSr2W2 WPITF(9,50) tSWFX,LSWFX,LSTHFX,LSTRlX W6ITE(5,525) NPWF,NPWM,NPTHF,NPTHW - - 87 , 25C FCRWAT(915) WHFRE NUMBER IS N/A: C READ DISK IOCATIOVS FCP STORAGE OF RESULTS- SUPPLY 0 READ (8,300) ISWFX,LSWMX,LSTHFX,LSTHNI 300 FCRMAT(4IS) C RrAD NO. OF PLOTS OF EACH TYPE OF MOBTLITY DESTRED: O-NONE; 1= MAGN. CNLT; MAGN. AND PPASE 2 C Rt"AD (A,300) NPWF,NPW!,NPTHF,NPTPN C READ SCALE DATA FOR MAGNITUDE PLOTS PFAD (0,320) ( WFS(I), 1-1,4) RrAD (9,320) ( WNS(T), T=1,4) READ (8,320) (THFS(I), 1=1,4) READ (P,320) (TRMS(I), I=1,4) 320 FOPMAT (4F10.O) C PRINT OUT INPUT CONTROL DATA WPITE(!,325) RUNID 32S FORMAT(IH1,402) WRITE(5,350)NX,WXI,KWFEV,KWMEX,KTHFEX,TPMrX WTITE(5,400) M,M1,M2,N,NI,N2 WPITE(5,425) DELTX,DELTXI,NFREQS,FNAX 2 3%, FrnPAT(O0COPUTF MBIlITTIES FOR MOTTON LCCATION' ,T2,' AND FXCTTAT I TON i'"ATT'Y' ,12//*P KWFFX=',I1,', vNmFX=',Ii,', 'THFEX=' KTPM-tY=',T'/) ?2,T 1,*,1 4*-'C PCRMAT(20 ' EASUBrMENT PT!7.: 19 IS ',T2,0, til T!;.',T2,', wl2 1I ',T2/9 rORCTN(' ?TS.: N IS ',12,', Ni IS ',T2,', N2 IS 2, T2/) 42r FnrTMhAT('O POTNT SPACTNq VALUES: DFLT=',F5.3,", ELTYT =', N. OF FPEQS.=',13,', MIX. FPEC.=',F5.0/) 4%0 FCP'AT(OI4NPUT TRANSL. MnOBILITY DTS'3 STOPA-F LOCATIONS:'/20X,'(",N 1) T! ',I2,, (VP-1) IS ',T2,', (r,N-2) IS ',I2/20X,'(1-1,N) I 2 -,2,', ( P1,N-1) IS ',T2,', (T-1,N-2) IS f,12/20X,'(M-2,V) 3 ',T2,', (V-2,o-1) IS O,2,', (W-2,N-2) IS ',T2/) q'O PPMAT('0CJTPTJT MOPTIITY TDISY STORAGE LOCA IONS:'/* W/F IS ', 1?,', W/W IF ',12,', TM/F IS *,T2,', TR/M IS ',12/) 52r FP AT('CvPETRIMENTAL PICTS TO BF MADF: W/F:',I1,', W/M:',Ti,', 1 TJ/F:I, T1,I Tq/14:',Tl) C P-An T.INFI. MnPIITTY rITA FPOX DISK nVTO APPAYS: TT (KTHMEX-1) 60,550o,550 55 r AD (10'LSVN ((SYIIN (IJ ),oT=1,2 ),J=l ,'NFPEQS) TA (1!1 1 9 ) (- V V V N (IJ),T=1,2 ,J 1,FREOS) P% PFM (10*1720 P107 (17'7 A,? (1 ) ((SY MN (I,J),T=1,2),J=I, FREoS) rSMIN (SYVIN1 (1,JI=I,2),J= 1,FPE5S) 2 "1I1) (SY N 1 1 IJ), =I ,2),J= 1, vFR EOS DAn ) v121 ) P6CCIt (19~ (KT'c~Y-1 7((SYMN21(T,J),1=1,2),J=1,vrREoS) ,6O, CCD (1 1,,; 1 2 (17 )(SY ! 1;2 ( 7,1J),T=, p2 ),J =I,WF REQS) T)A 1 1 -'m2w,2) ((SY!02N2(T,J),T=1,7),J=1,fwrREOS) 1 0 r Cr I'F (KT'PTri-1l) 700,6T0,650 6 10 PrAp ( 1011,17NE ((SY*N T ,J ),T=1,2),J=1I, ' FPEOS) v) n ( )N(('PY=N (1,J),T=1,2),J=1,EFREQS) (- TO PrEAD (10'17F2n) ((cYy2N (TJ),I=1,2),J=1,vFREQS) 7 I0 TF (KWvrEX-1) PO00,175,75 71C PAD (10'ISyN ) (()3Y1 (T,J),T=1,2),J=1, Y (FREQS) P2AD ( 1Of ')S'N 1 ) ( )(1NI (TvJ)=1f2)J=IvFE S A(1**SVN 2N ) ((SYM2 (,J),T=1(,),J=1,FE() 80 P -A D ( 10 ' "M N ) (rM (IJ),T= I,2),J=1I, VFP 1QS) RUAT OTATIONAL MILTIES: C 10'0 nDY 2N2 IF 11(O C NFP=1,NrPEQ (KTP4EI-1) 1400,1100,1l0 =(.*SYN(1,RFR)-12.*SYN(1,NFP)+.*N22(1,NF P)-2.*ELT!) 2(2,NFP)+SYw!2N2(2,FlFR))/(4.*rELTX*DFtTXT) SY M 7N7(1,FR)= CO SYM2N2(2,NFP)= QUAD 14'j IF (KTPFEY-1) 1800,1500o,10O 1500 Cn = (3.*SYNN(1,?FR)-4.*SYNN(i,NFP)+SYW2(1,WFR))/(2.*DELT SYIA2N(1,NFP)= Cl) SYNf2N(2,MFl)= OUAD 1190 TF (KWNrX-1) 2000,1q0o,1900 3*YN1.F)4*YN(,NP+YN(,F)/2*ETI i9oo Cn - - 88 ) 2(1,NIFP)+SY,12N2(1,NFP))/(4.*DE.LTX*DFT-TXT) OUAD=(.**YmN(2,NFP)-12.*SYNW1(2,NFP)+3.*S(MN2(2NFR)-12.*SMLN(2, - ,68 ) . aA.j^ Zh Ilk~i. Zii~( i L-4 o I soAdih uia aio) sz*'~ s 17~I 0 iEi1 0hc 'wa i t- 0 L L - i H I Ji li J ~l z-'s'vLi tLhGiid odt* 0q'04!UkY 'Si :)3S IN AAO ~id hii Aok1'I/m 1.i GN.At 'Gi - 3k L+dk (..~1L0CtUo . 0UIu )J ;,LlIa>uW Al* L~li Zkj. hA)J~V O'OO'OOWUE (L-)%hidI) aI u1 I''L),w 03ajv ',0,LAMAG,0,80) *Tq/F MCTITTY MAGN., DE PE 1/ NT SEC FPEQUENCT, HZ CALL VCVE ( ',0,tAPf,0,R0) DEC. TH/F MOBTLTTY PHASE, * CALL PICTP(SY2N,1,LAMAC,THFS,-1,NEPEQS,0,-1,-1)O4,-2,FMX,1) IF (NPTHF-1) 3500,3500,3460 3460 CALL PTCTP(SYMN1,2,LAPH,PHS,-2,NFPEQS,0,-1,-2004,-2,FA,1) 3500 Ir (NPTqM-1) 3700,3600,3600 365r NFP= 1,WFREQS 36^0 T) AmnR2=SYF2N2(1,NFR)**2 +SYF2N2(2,NFP)**2 AmAGN= 10.*ALOG10(AqOP2+1.E-30) SYm2N2(2,NFP)= ATAN2(STM2N2(2,NFP),SYM2N2(1,NFP))*57.29578 3650 SYM1N2(1,NFP)= A'AGN FPEQUENCY, HZ 9 CALL VCVE ( ',rLARAG,0,80) *TT/r qC'TTTTY FAGNr., ,P PE 1/ NT M SEC FDEQUENCY, H? CALL %0VE ( DEC. fptpH,0,0) * TH/M MOPTLITY PHASE, - 90 - CALL PTCTP(SYM1N2,1,LAMAG ,THMS,-1,NFPEQS,0,-1,-100(,-2,FRA!,1) TV (NPTHM-I) 37m0,370r,3660 3660 CALL PTCTP(SYM2N2,2,LAPH,37r% CALL rXTT FND