EXPERIMENTAL STUDY OF ELASTICA VIBRATION M. Hammer Rafael P.O. Box 2250 Haifa, Israel N. C. Perkins Assistant Professor Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, Michigan 48109 July, 1989 Report No. UM-MEAM-89-07

EXPERIMENTAL STUDY OF ELASTIC VIBRATION 1. Introduction This research is concerned with the design of a nonlinear suspension employing an elastica, as originally proposed by Shoup [2-4]. The following summary, from Shoup [2], states the key characteristics of his design. Shock and vibration problems in the aerospace and transportation industries arise from many causes, such as the isolation of instruments and controls or the protection of human occupants of vehicles. The usual solution to these problems involves the use of lightly damped flexible supports. These soft supports cause the natural frequency of the suspension system to be far below the disturbing frequency. This solution is effective for the isolation of steady-state vibration; however, when these suspensions encounter shock excitation their softness often leads to damagingly large deflection. It has been pointed out that this undesirable feature is not present in suspension systems utilizing symmetrically nonlinear springs that harden. These springs become progressively stiffer when subjected to large deflection from their "operation point". A number of ingenious ways have been developed to produce nonlinear spring devices, but unfortunately many of these are not symmetrical in behavior or are rather complex to construct. As weight, cost, and reliability requirements become more important, designers are forced to search for new ways to improve existing designs by reducing the number of moving parts in suspension systems. Shoup proposes using thin elastic strips in the form of an elastica, as the nonlinear restoring elements.

In his analysis, Shoup treats the elastica as massless spring with nonlinear characteristics. In the actual designs, however, the mass of the elastica element is often comparable to the mass of the suspended payload and the inertialess assumption remains suspect. The purpose of this research is to propose a dynamical model for the elastica which accounts for its inertia. The object of this project is to provide experimental evidence in support of the theoretical model. The specific goals of this project are as follows: 1. Develop a suitable test stand for studying elastica dynamics. 2. Perform an experimental modal analysis of the elastica to determine its vibration characteristics. 2. Design and Construction of Elastica Suspension Test Stand 2.1 Design Requirements Design a test stand that allows measurement of the natural frequencies and the mode shapes of a test rod. The ends of the rod must connect to support that allow the rod to rotate, with minimum friction, in the vertical plane. One support must also be able to slide horizontally with the motion of the payload. The supports should restrict out-of-plane motion of the elastica but permit a large range of test rod lengths up to 1.9m. to be used. The measurement equipment should be selected to detect motion in the vertical plane. The design should permit the attachment of a vibration shaker for future studies. 2

2.2 Description of the Design A schematic of the test stand is shown in figure 1 and a photograph of the test stand is provided in figure 2. The test rod (1) was cut to length L, and the accelerometer base (8) and the impact surface (9) were assembled on the rod. The ends of the test rod were connected to the holders (3) that freely pivot in the vertical plane about small shafts in the supports (2). The shafts were lubricated with graphite to reduce friction. The supports are mounted directly to the base at the left end of the rod (6) and to a bearing seat (2) and (4) at the right end. Two parallel ground shafts (10) are used to guide the linear bearings (5) horizontally. The ground shafts are connected to the base (1) of the test stand which consists of extruded aluminium channel having rectangular (7.6 cm. x 4.8 cm.) cross-section and length 2m. To prevent excessive deflection of the ground shafts, two clamps (7) provide additional support and may also lock the movable support at one location. The accelerometer base (8) can be adjusted to any angle to allow measuring the horizontal and vertical components of the rod acceleration. The accelerometers (12) are bonded to the accelerometer base with petro wax. The impact surface (9) provides a planer surface for hammer impacts. 3

Schematic drawing of Test Stand (part 1) Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 7 /. j=:e,L_ _I CJ \J Figure 1 4

Parts list (part 2) No. Name Quantity 1 Base 1 2 Support 2 3 Rod Holder 2 4 Bearing seat 1 5 Linear bearing 2 6 Shaft support 2 7 Clamp 2 8 Accelerometer base 1 9 Impact surface 1 10 Ground shaft 2 11 Test rod 1 12 Accelerometer 2 Table 1 5

Measuring equipment (part 3) No. Name Quantity 1 Piezoelectric Unixial, Accelerometer-PCB-309A 2 2 Amplifier-PCB-483A08 1 3 Structural Dynamics Analyzer - Gen-Rad 2515 2 4 Piezoelectric Impact hammer-PCB-086B03 1 (with soft, rubber tip) Table 2

Rod Material Properties (part 4) No. 1 Material - LEXAN 2 E = 2344 Mpa (Young's modulus) 3 y = 1190kg/m3. (specific gravity) 4 O.D. = 6.35mm. 5 L = 1.2m. Table 3 7

.r, +~s Figure 2 8

3. Experimental Modal Analysis 3.1 Purpose of Test A theoretical model of elastica vibration has recently been proposed in [1]. The purpose of the present project is to validate this model by providing companion experimental results. Figure 3 depicts the theoretical natural frequencies and mode shapes as functions of the applied end-load,n. The end-load, n, is related to the support separation Ix (1)I as observed in figure 4. Both of these quantities are nondimensional and are related to dimensional quantities as described in [1]. 3.2 Test Procedure and Results The test procedure was divided into two major parts. First, the natural frequencies of the first four modes of the test rod were found, and second, the mode shapes were determined. To find the natural frequencies of the test rod an accelerometer was bonded to the test rod in the normal direction to detect the nomal component of the acceleration. The amplified accelerometer signal was sampled by the structural dynamics analyzer and the frequency response of this signal was computed following on initial impact; see figure 6. The natural frequencies of the rod appear as local maxima in the frequency response plot. Five tests were conducted for each of nine endload values in the range n = 10-20. The average of these test results are shown in table 6 which also shows the corresponding theoretical frequencies. 9

The experimental frequencies were non-dimensionalized as shown below [1]: = = (rad/sec) C pL4 YEI where Q1 is the dimensional circular frequency (rad/sec.) w is the nondimensional circular frequency p = mass/length = Ay/g L = length of test rod I = area moment of inertia = id4/64 E = young's modulus A = cross section area = fd2/4 In this computation, p is the mass/length of the rod alone and does not include the accelerometer and base, the holders, and the impact surface. 10

E 91TnbTa OZ u'p-80- PU3 t -1 08 i- ozv L (9 1\ ~ ~ ~ ~ ~ ~ ~ ~ "9

10000 - -oo 100' 0o 10 U. 10 0 00.2 i 0.2 0,4 jx~l00 ~ Support Separation, Figure 4 12

Every mass attached to the rod shown in figure 5 was measured and the overall mass/length was computed using different combinations of these masses; see table 4. The effect this calculation has on the agreement between measured and theoretical frequencies is shown in table 5. The best overall agreement occurred when the mass of the accelerometer Ma, the accelerometer holder Mb, and the impact surface Mb were included. Ms Mx Mb Mr Ma Figure 5 13

Ms = 5.09 gr Mx = 2.89 gr Mb = 1.68 gr Mr = 44.95 gr Ma = 1 gr M1 = 2(Ms+Mx+Mb)+Mr+Ma M2 = 2(Mx+Mb)+Mr+Ma M3 = 2Mb+Mr+Ma M4 = Mr+Ma M5 = Mr No. Mi(gr) pi(kg/m) Ci 1 65.27 0.00303 0.6479 2 55.09 0.00255 0.654 3 49.31 0.00229 0.6782 4 45.95 0.00213 0.7156 5 44.95 0.00209 0.78 Table 4 14

Summary of the results The natural frequency for N = 12 Ci Cl C2 C3 C4 C5 Mode M1 4.27 4.6 4.81 4. 95 5.5 M2 13.87 14.94 15.36 16.09 17.87 M3 27.16 29.26 30.62 31.5 34.98 M4 44.4 47.52 49.73 51.16 56.84 Table 5 15

AVERAGE FREQUENCY N H/L N L | 1 | 2 3 4 110 0974 33. 188 82.928 152.133 241.01 W (TH) 10 0.974 29.454 78.595 124.966 227.653 W(EX) 11 0.8 26.95 77.353 146.866 253.813 11 0.8 25.721 75.893 140.263 230.415 12 0653 l22.4 72.77 142.464 231.387 12 0.653 22.158 71.393 147.993 237.497 18.873 68.929 138.717 227.567 13 0.53 18.54 67.481 143.135 237.556 16 65.659 135.49 224.236 14 0.433 15.93 64.302 140.34 232.937 13.58 62.841 32.672 221.305 15 0.349 13.313 61.677 134.971 228.446 11.947 60.16 129.97 218.463 16 0.274 11.173 58.976 132.056 215.406 7.899 56.327 126.075 214.31 18 0.154 6.982 55.175 118.429 211.358 4.567 53.114 122.785 210.74 20 0.06 4.696 52.626 113.741 216.949 C = 0.6782 Table 6 16

( I r 00E I a M\; ~~, "-'I I -7.oo E-oIL I.3E0 -i~~~0X I'~ ~ ~ ~~~~~~~~~~4 /! 8'~ J 1 r I~~~~~ i % F' I.-'r~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~/. 5WGE~ ~~~~ ~51- 1 5 1 Figure 6 17 i- 1I'~~~~~~~~~~~~~~~~~~~r i~ I' s.00E- 0:l:~,~ 6g.39E-01 G3.39E+0l Pt-e ss R7ETUlRN...: FreoReso-Bode ZOX- 19X- B ~ 042888-! 41812 0~5058-1 si S 16 Figure 6 17

A complete modal test was performed for a representative geoemetry defined by n = 15, H = 0.42m and H/L = 0.349. The modal test procedure was divided into three major parts. First the geometrical shape of the rod was entered into the structural dynamics program. Second, the force input and response output data were collected. And third, the modal parameters were extracted from the data using standard modal analysis procedures. The rod was divided into 23 equally spaced points including the ends. No measurements were taken at the ends. The rod was subjected to the end-load, n = 15, by adjusting H such that H/L = 0.349. The X - Y coordinates of every test point were measured and entered into the structural dynamics program. The impact surface was located at point 21 near the right support and the accelerometer base was placed at each point with accelerometers aligned with the X-Y directions. The accelerometer and force signals were sampled by the structural dynamic analyzer during the after the rod was struck by the hammer (see appendix 3). For each location of the accelerometer, 10 tests performed and the average frequency response was computed and stored. The natural frequencies, damping ratios, and modal participation factors were extracted from the measured data using a single degree of freedom, circle fit routine as seen in figure 7. The first two mode shapes determined in this way are given in figure 8, and compare very favorably with the theoretical predictions shown in figure 2. 18

: — r - _q -:3.258 j Damp= 3. 07638 - | — 4X- /1 ~0X- 4X-'"' / 1tode 3hape'): 3_cale 5.63 Mtode Coef-f ic ient Pe a!. 00000aE-01 I'maq. L.33E16E+00 imp _.:338 SE+00 i _ i. - o,".S:3.' —' 5-i': L. C,, Z S. E, I, RB, T l i I.- / / -i-' I! - Figure 7 19

Second Mode Fundamental ModeA /I 1 if / Equilibrium I \ I'I I I I I.I I S=O S=L I_ H-, Figure 8 20

4. Summary and Conclusions This project provides experimental evidence in support of a theoretical model for planar elastica vibration. A test stand was designed to permit vibration tests on wide range of elastica geometries. Results for the first four natural frequencies show that the natural frequencies predicted by the model are generally accurate to within 5% of those measured experimentally. Larger discrepancies occur when the ends of the elastica are close and these can be attributed to the influence of the rotary inertia of the supports which were not modeled. Experimental measurements of the first two vibration mode shapes for a representative goemetry are in superb agreement with the theory. 21

5. References 1. Perkins, N. C., 1989, "Planar Vibration of an Elastica Arch: Theory and Experiment," submitted to the ASME J. of Vibration and Acoustics. 2. Shoup, T. E., 1971, "Shock and Vibration Isolation Using a Nonlinear Elastic Suspension", AIAA Journal, Vol. 9, No. 8, pp. 1643-1645. 3. Shoup, T. E., 1972, "Experimental Investigation of a Nonlinear Elastic Suspension," AIAA Journal, Vol. 10, No. 4, pp. 559-560. 4. Shoup, T. E., 1977, "An Adjustable Spring Rate Suspension System," AIAA Journal, Vol. 15, No. 6, pp. 865-866. 22

Appendix 1 Measured Frequency

ROD FREQUENCY TEST DATE H/L N C 1 2-27-89.653 12 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 5.503762 17.87580 34.99596 56.83957 2 501.99 5295.5 20296.06 53539.87 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 5.23 16.83 34.57 55.74 ** RUN #2 5.08 16.64 34.77 55.74 RUN #3 5.23 16.64 34.77 56.39 RUN #4 5.23 16.83 34.77 55.1 RUN #5 5.23 16.83 34.77 55.7 21.16855 68.20345 141.3815 226.8861 w(nd) AVERAGE 5.2 16.754 34.73 55.734 f(Hz) ERROR % 5.519178 6.275555 0.759998 1.945076 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 2 3-1-89.53 13 j 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 4.636232 16.93214 34.07549 55.90128 2 w( 356.21 4751.16 19242.44 51786.83 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 4.35 15.8 33.59 55.74 ** RUN #2 4.35 15.8 33.59 55.78 RUN #3 4.352 15.89 33.59 55.74 RUN #4 4.352 15.89 33.59 55.74 RUN #5 4.352 15.8 33.59 55.74 17.71319 64.46638 136.7407 226.9431 w(nd) AVERAGE 4.3512 15.836 33.59 55.748 f(Hz) ERROR % 6.147933 6.473751 1.424773 0.274212 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 3 3-1-89 0.433 14 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 3.931049 16.12893 33.28180 55.08299 2 (0 256.09 4311.09 18356.48 50281.79 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 3.747 15.09 33.01 54.16 ** RUN #2 3.747 15.09 33.01 54.79 RUN #3 3.747 15.09 32.63 54.79 RUN #4 3.747 15.09 33.01 54.79 RUN #5 3.704 15.09 33.01 54.79 15.21856 61.42951 134.0702 222.5303 w(nd) AVERAGE 3.7384 15.09 32.934 54.664 f(Hz) ERROR % 4.900703 6.441436 1.045026 0.760654 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 4 3-1-89 0.349 15 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 3.337461 15.43674 32.59051 54.36296 2 0C0 a1 184.59 3949 17601.84 48975.84 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 3.117 14.49 31.53 51.72 ** RUN #2 3.117 14.49 31.71 54.16 RUN #3 3.117 14.41 31.71 53.85 RUN #4 3.153 14.49 31.71 54.16 RUN #5 3.117 14.49 31.71 54.16 12.71823 58.92185 128.9409 218.2396 w(nd) AVERAGE 3.1242 14.474 31.674 53.61 f(Hz) ERROR % 6.389925 6.236707 2.812205 1.385061 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 5 3-1-89 0.274 16 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 2.774510 14.77807 31.92687 53.66485 C0 127.57 3619.19 16892.29 47726.07 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 2.622 13.84 30.99 50.84 ** RUN #2 2.622 13.84 30.99 50.84 RUN #3 2.622 13.84 30.99 50.84 RUN #4 2.622 13.84 30.99 50.26 RUN #5 2.622 13.84 30.99 49.97 10.67383 56.34092 126.1564 205.7827 w(nd) AVERAGE 2.622 13.84 30.99 50.55 f(Hz) ERROR % 5.496855 6.347743 2.934440 5.804279 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 6 3-1-89 0.154 18 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 1.940458 13.83673 30.96995 52.64471 2 20 62.4 3172.8 15894.86 45928.82 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.743 12.99 30.11 47.17 ** RUN #2 1.608 12.99 28.92 50.26 RUN #3 1.617 12.92 27.46 47.99 RUN #4 1.608 12.92 26.99 52.32 RUN #5 1.617 12.92 25.48 50.26 6.670537 52.70969 113.1377 201.9154 w(nd) AVERAGE 1.6386 12.948 27.792 49.6 f(Hz) ERROR % 15.55605 6.422984 10.26139 5.783518 NOTE: * = MEASURED FROM FREQUENCY GRAPH * = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 6a 3-1-89 0.154 18 0.654 THEORETIC FREQUENCY MODES 1 2 3 4 1.922325 13.70766 30.68108 52.15368 2 0) 62.3978 3172.796 15894.86 45928.82 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.743 12.99 30.11 47.17 ** RUN #2 1.608 12.99 28.92 50..26 RUN #3 1.617 12.92 27.46 47.99 RUN #4 1.608 12.92 26.99 52.32 RUN #5 1.617 12.92 25.48 50.26 6.733340 53.20596 114.2029 203.8164 w(nd) AVERAGE 1.6386 12.948 27.792 49.6 f(Hz) ERROR % 14.75951 5.541894 9.416506 4.896467 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 6b 3-1-89 0.154 18 0.6782 THEORETIC FREQUENCY MODES 1 2 3 4 1.853732 13.21853 29.58630 50.29270 2 (0 62.3978 3172.796 15894.86 45928.82 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.743 12.99 30.11 47.17 ** RUN #2 1.608 12.99 28.92 50..26 RUN #3 1.617 12.92 27.46 47.99 RUN #4 1.608 12.92 26.99 52.32 RUN #5 1.617 12.92 25.48 50.26 6.982494 55.17474 118.4288 211.3583 w(nd) AVERAGE 1.6386 12.948 27.792 49.6 f(Hz) ERROR % 11.60535 2.046655 6.064640 1.377345 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 6c 3-1-89 0.154 18 0.7156 THEORETIC FREQUENCY MODES 1 2 3 4 1.756849 12.52768 28.04000 47.66421 2 w 1 62.3978 3172.796 15894.86 45928.82 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.743 12.99 30.11 47.17 RUN #2 1.608 12.99 28.92 50..26 RUN #3 1.617 12.92 27.46 47.99 RUN #4 1.608 12.92 26.99 52.32 RUN #5 1.617 12.92 25.48 50.26 7.367550 58.21741 124.9597 223.0138 w(nd) AVERAGE 1.6386 12.948 27.792 49.6 f(Hz) ERROR % 6.730744 -3.35507 0.884483 -4.06129 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 6d 3-1-89 0.154 18 0.78 THEORETIC FREQUENCY MODES 1 2 3 4 1.611796 11.49334 25.72491 43.72886 2 C)0 62.3978 3172.796 15894.86 45928.82 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.743 12.99 30.11 47.17 ** RUN #2 1.608 12.99 28.92 5.0.26 RUN #3 1.617 12.92 27.46 47.99 RUN #4 1.608 12.92 26.99 52.32 RUN #5 1.617 12.92 25.48 50.26 8.030589 63.45665 136.2053 243.0838 w(nd) AVERAGE 1.6386 12.948 27.792 49.6 f(Hz) ERROR % -1.662 96 -12. 6564 -8.0353 -13.4262 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 7 3-1-89 0.060 20 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 1.121939 13.04739 30.16188 51.76811 60 1 20.86 2821.13 15076.23 44412 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 1.119 12.77 27.46 51.43 RUN #2 1.119 12.19 28.92 51.43 RUN #3 1.136 12.19 24.61 51.43 RUN #4 1.134 12.41 28.42 51.72 RUN #5 1.003 12.19 24.05 48.55 4.486919 50.27531 108.6598 207.2564 w(nd) AVERAGE 1.1022 12.35 26.692 50.912 f(Hz) ERROR % 1.759378 5.345066 11.50421 1.653741 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 8 3-1-89 0.80 11 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 6.620276 19.00167 36.07740 57.92678 2 co r 726.32 5983.56 21569.81 55607.66 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 5.02 17.83 31.7 50.8 ** RUN #2 6.62 17.63 30.1 55.1 RUN #3 6.11 17.83 32.63 52.3 RUN #4 6.32 17.93 34.57 54.79 RUN #5 6.11 17.83 35.58 57.37 24.57180 72.50229 133.9969 220.1203 w (nd) AVERAGE 6.036 17.81 32.916 54.072 f(Hz) ERROR % 8.825565 6.271440 8.762843 6.654589 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

ROD FREQUENCY TEST DATE H/L N C 9 3-1-89 0.974 10 0.6479 THEORETIC FREQUENCY MODES 1 2 3 4 8.152533 20.37111 37.37106 59.20329 2 c( 1101.44 6877.1 23144.44 58085.47 MEASURED FREQUENCY MODES 1 2 3 4 RUN #1 6.93 18.46 30.63 51.72 ** RUN #2 6.93 18.46 26.37 53.85 RUN #3 6.21 17.63 29.76 55.1 RUN #4 7.91 18.67 29.76 53.85 RUN #5 6.58 19 30.11 52.6 28.13789 75.08323 119.3825 217.4824 w(nd) AVERAGE 6.912 18.444 29.326 53.424 f (Hz) ERROR % 15.21653 9.460041 21.52752 9.761784 NOTE: * = MEASURED FROM FREQUENCY GRAPH ** = MEASURED FROM SPECTRUM GRAPH

Appendix 2 The error [%] between the theoretical calculation and the test results as a function of the influence of the weight of the Test Rod and the Supports.

Error [%] N = 10 mode 1 2 3 4 -onst. 0.6479 15.216 9.460 21.527 9.761 0.654 14.418 8.607 20.788 8.912 0.6782 11.251 5.226 17.857 5.541 0.7156 6.357 -0.0006 13.327 0.332 0.780 -2.069 -9.0001 5.527 -8.6368

Error [% ] N = 11 \1 2 3 4 0.6479 8.8255 6.271 8.762 6.654 0.654 7.9671 5.388 7.9038 5.775 0.6782 4.561 1.888 4.496 2.2289 0.7156 -0.7014 -3.522 -0.7706 -3.0992 0.780 -9.763 -12.838 -9.839 -12.377

Error [%] N = 12 mode 1 2 3 4 const. 0.6479 5.519 6.275 0.7599 1.9456 0.654 4. 6296 5.393 -0.1743 1.0218 0.6782 1.1006 1.892 -3.881 -2.640 0.7156 -4.353 -3.517 -9.609 -8.302 0.780 -13.744 -12.833 -19.473 -18.047

Error [% ] N = 13 mode 1 2 3 4 const. 0.6479 6.1479 6.473 1.4247 0.2742 0.654 5.264 5.593 0.496 -0.664 0.6782 1.7588 2.099 -3.185 -4.389 0.7156 -3.658 -3.298 -8.875 -10.146 0.780 -12.987 -12.595 -18.673 -20.058

Error [% ] N = 14 mode 1 2 3 4 const. 0.6479 4.90 6.441 1.045 0.761 0.654 4.0053 5.5605 0.1133 -0.1736 0.6782 0.4532 2.066 -3.582 -3.880 0.7156 -5.0363 -3.3346 -9.2949 -9.669 0.780 -14.489 -12.6341 -19.13 -19.47

Error [ % ] N= 15 1 2 3 4 0.6479 9 6.389 6.236 2.8122 1.385 0.654 5.508 5.354 1.897 0.4566 0.6782 2.0121 1.8517 -1.7329 -3.2268 0.7156 -3.3915 -3.5607 -7.3431 -8. 9193 0.780 -12.596 -12.88 -17.003 -18.7214

Error [% ] N = 16 mode 1 2 3 4 const. 0.6479 5.4968 6.3477 2.9344 5.8042 0.654 4.6071 5.466 2.020 4.9174 0.6782 1.0772 1.968 -1.6049 1.399 0.7156 -4.378 -3.4381 -7.208 -4.0386 0.780 -13.771 -12.747 -16.856 -13.40 1~~~~~ ~ -16.86

Error [ % ] N = 18 mode 1 2 3 4 const. 0.6479 15.556 6.4229 10.2614 5.7835 0.654 14.759 5.5418 9.4165 4.8964 0.6782 4.6053 2.0466 6.0640 1.3773 0.7156 6.7307 -3.355 0.8844 -4.0612 0.780 -1.6629 -12.6564 -8.03535 -13.4262 -1. 2 J -12 I56I

Error [% ] N = 20 o 1 2 3 4 const. 0.6479 1.7593 5.345 11.504 1.653 0.654 0.8344 4.453 10.6710 0.7278 0.6782 -2.835 0.9283 7.3655 -2.9455 0.7156 -8.506 -4.545 2.2571 -8.6225 0.780 -18.2708 -13.954 -6.539 -18.398

Appendix 3 Mode Shape Data Aquisition Conditions

Data acquisition conditions: 1 Trigger Type 2 9 Trigger Delay 0 2 Trigger Level 15 10 Clear Freq L 2.0000 3 Coupling Code 0 11 Clear Freq U 60.000 4 Weighting Code 0 12 Minimum Freq 0.00000 5 Ensemble Size 10 13 Overlap Facto 0 6 Maximum Freq 64.000 14 Auxil Scale 1.0000 7 A-A Filters 80.000 15 Reference Count 2 8 Excitation 2 16 Response Count 2 19 Master Indent 0 Channel 20-Coord 21-Range 22-Scale 23-Signal 24-Bias 1 100X+ 8.0000 1.0000 4 0 2 20X- 1.0000 1.0000 4 0 3 20X- 2.0000 1.0000 3 0 4 20Y+ 0.5000 1.0000 3 0 5 1X+ 8.0000 1.0000 3 0 6 lX+ 8.0000 1.0000 3 0 7 1X+ 8.0000 1.0000 3 0 8 lX+ 8.0000 1.0000 3 0 Press RETURN... #

Modal Parameters, CONSOLIDATED Label Freq Dampling Amplitude Phase Ref Res Mode Flags 1 3.258 0.07638 1.483 1.571 20X- 19X- 1 0 0 0 1 1 2 14.620 0.02088 14.48 1.758 20X- 19X- 2 0 0 0 1 1 3 29.769 0.01973 43.83 1.596 20X- 19X- 3 0 0 0 1 1 4 50.152 0.01300 59.86 1.490 20X- 19X- 4 0 0 0 1 1 Enter the modal parameter label number #

/ - \:..1,.'''\'.,!',,, ('t f, Ni~~ ~~~~~~~~~~~~~~~~~~~ i!I'~' i am., J i._'\,,', I I I;! I I;~~~~~~~~~~~~~~~~~~~' 2_0X- meal:,F=:3.~S8 Hz,:'0.0: 0.0,7 100.0, 0.02)=~Y'iew First mode Figure 7

..' "!' —i;' I;i I' t':.1' 11 1l X ~ / iI I'i i'' I,, i i i! i i i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,' I'i; I tr! 1, 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ t! l ____ /\ I I i DGI FISE 2NDI rstuDE X t ~: Z@lX- PQ~a1,?F= i4. 62Q Hz: Q. 3. F tl t 0te.@, i3 )-ieui Sec ond mode ~Figure 6 I: xi~~~~~~~~~~~~~~~ DO!F~ - ~MDU rr MODE, 9jX-.~0;5'?O-~ R~ ~~ ealF = i4. 620 Hz,;.0 O.ECi.~ 100t.0 k].OLC)= VelW Second mode Figure 8