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MODAL ANALYSIS OF A THICK-WALLED CIRCULAR CYLINDER
Mechanical Systems and Signal Processing (2002) 16(1), 141}153 doi:10.1006/mssp.2001.1420, available online at http://www.idealibrary.com on
MODAL ANALYSIS OF A THICK-WALLED CIRCULAR CYLINDER R. K. SINGHAL Canadian Space Agency, David Florida Laboratory, P.O. Box 11490, Station H, Ottawa, Ontario, Canada K2H 8S2
W. GUAN National Research Council, Institute for Research in Construction, Structures Laboratory, Ottawa, Ontario, Canada K1A 0R6 AND
K. WILLIAMS Department of Mechanical Engineering, College of Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0 (Received 10 March 2000, accepted 18 April 2001) The objective of this paper is to "nd true descriptors for the mode shapes of circular cylinders. The vibrational behaviour of a thick-walled circular cylinder is of considerable engineering importance as such elements have numerous applications in electrical machines, sti!ened cylinders, and gears. Mode shapes are also very important sources of information for understanding and controlling the vibration of a structure. Theoretical and experimental modal analyses were carried out using a thick-walled circular cylinder model to obtain its natural frequencies and mode shapes. The theoretical modal analysis was done using the "nite element method. The results for the frequency range from 20 Hz to 20 kHz were veri"ed using experimental modal analysis. The correlation between the analytical and the experimental results is very good. The largest error for all frequencies is 4.05%, and less than 2% for most frequencies. 2002 Elsevier Science Ltd.
1. INTRODUCTION
A number of theories for the prediction of the natural frequencies of cylinders have been developed and used over the years, as mentioned in reference [1]. Because the solutions for the vibrational behaviour of a cylinder cannot be easily obtained by the full linear elastic theory, people have created various approximate theories using various assumptions for the displacement components. Some of the theories are capable of dealing with "nite-length free hollow cylinders, while others are appropriate only for solid cylinders of in"nite length. Among these researchers, only a few give a complete description of the mode shapes. Bancroft [2] explored the problem numerically. More recent work on the subject began with a paper by Mindlin and Herrmann [3], and was formed into the well-known &Three-mode theory' in a long series of papers by McNiven and his associates, beginning with Mindlin and McNiven [4]. Many authors have considered approximate solutions, for the limiting cases in which the cylinder approaches either a thin disk or a slender rod. These approximate solutions were discussed at length in reference [1], and compared with the experimental results. 0888}3270/02/010141#13 $35.00/0
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Solutions for the axisymmetric vibration of free cylinders have been presented by several authors. In 1962, McNiven and Perry [5] presented a solution based on approximate equations which take into account the coupling between longitudinal, axial shear, and radial modes of propagation in a rod of in"nite length. In 1971, Rumerman and Raynor [6] developed a spectrum of frequencies using the Rayleigh}Ritz procedure with displacement functions corresponding to pure radial and axial modes of in"nite cylinder. In 1972, Hutchinson [7] presented a solution for this problem using the method of three-dimensional series solutions; he later [8] used this method for a similar problem and in 1980, developed a series solution for the general three-dimensional equations of linear elasticity [9]. The solution was used to "nd accurate natural frequencies for the vibration of solid elastic cylinders with traction-free surfaces, and gave accurate frequencies with the use of very few terms. Results were given for height-to-diameter ratios from zero (disc) to two. These analytical results agreed well with previous experimental results. The vibration of in"nitely long traction-free hollow circular cylinders was investigated on the basis of the linear three-dimensional theory of elasticity by Greenspon [10] and Gazis [11]. McNiven et al. [12] developed the &Three-mode theory', as mentioned before, for axisymmetric vibrations of hollow cylinders. McNiven and Shah [13] used it to investigate the end mode. Gladwell and Tahbildar [14] solved the axisymmetric problem using a "nite element analysis. The general problem of three-dimensional vibrations of a "nite-length circular solid or hollow cylinder with traction-free surfaces was "rst investigated by Gladwell and Vijay [15], using "nite elements. In 1986, by using the same technique as in his papers [7, 9], Hutchinson investigated the vibrations of free hollow circular cylinders [16]. The method of solution developed in this paper involves combining the exact solutions of the governing equations in "ve series which term by term satisfy four of the nine boundary conditions. The remaining "ve boundary conditions are satis"ed by orthogonalisation on the boundaries. A series of frequencies for di!erent sizes of hollow cylinders was calculated, and results showed good agreement with the "nite element results of Gladwell and Vijay [15]. The most recent work on the thick-walled cylinder is contained in the paper of Singal and Williams [17]. Based on the three-dimensional theory of elasticity, the well-known energy method was used in the derivation of the frequency equation of the cylinder. The frequency equation yields natural frequencies for all the circumferential modes of vibration, including the breathing and beam-type modes. Experimental investigations were carried out on several models and showed a very close agreement with the theoretical natural frequencies. So far, most of the work mentioned was dedicated to discovering a theoretical or numerical way of determining the frequencies of cylinders accurately and concisely. Few of the authors give a description of the mode shapes of the cylinders, and none of them provides a unique way for describing all the modes shapes. McMahon [1] gave some mode charts showing sand patterns on the plane surface of a solid cylinder and the approximate form of the vibration at a diametrical cross-section. In references [18}20], the mode shapes of thin-walled cylinders were also presented. Singal and Williams [17] gave a description for the mode shapes of thick-walled hollow cylinders. However, all these descriptions are based on two parameters, namely, the numbers of circumferential and axial nodes. Such a description is not su$cient for describing the mode shapes of a three-dimensional structure uniquely, as we know from reference [17]. In reference [17], the well-known energy method, based on the three-dimensional theory of elasticity, has been shown to be ideally suited to the determination of the natural frequencies of both thick-walled hollow cylinders and rings; however, it does not provide much information on eigenvectors. The technique provides only the magnitude of n (where
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Figure 1. (a) Illustration of parameter n. (b) Illustration of parameter m.
2n"the number of circumferential cross points in the radial displacement shape), and whether m (the number of cross points in the radial displacement shape along any axial generatrix) is an even or odd integer. The meanings of m and n are illustrated in Fig. 1. The actual magnitudes of the integer m were obtained experimentally in reference [17]. Some of the results from reference [17] are listed in Table 1; it shows that there are indeed two or three frequencies which produce the same combination of the parameters n and m (for example, for the combination of n"2 and m"2 the frequencies are 6395, 12 591 and 18 879 Hz). It is impossible for several frequencies to have exactly the same mode shape. It must be concluded that the descriptors m and n alone are not adequate for describing such a three-dimensional problem. The objective of this paper is to "nd true descriptors for the mode shapes of thick-walled circular cylinders. The actual mode shapes are to be determined both analytically and experimentally. In detail, this includes: (i) a theoretical modal analysis using the "nite element method to obtain the natural frequencies and mode shapes for the same model used in reference [17]; (ii) an experimental modal analysis to verify the results of the theoretical analysis; (iii) using the postprocessors, in both the "nite element program and the experimental test software package, to examine the mode shape results; (iv) understanding of the
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TABLE 1 Results for a thick cylinder having inside diameter"152.4 mm, outside diameter"228.6 mm and axial length"250.4 mm [2] Experimental Mode n 0
Calculated (Hz)
(Hz)
(a) Symmetric longitudinal modes 8109 8149 8817 8886 11 268 * 15 026 15 030
m
% Error
0 2
!0.49 !0.67
4
!0.03
1
7037 9803 12 666 15 596 16 867
7104 9859 12 809 15 201 16 911
2 2 0 4 4
!0.95 !0.57 !1.13 2.53 !0.26
2
2555 6395 12 591 16 862 18 879
2577 6429 12 653 16 326 19 015
0 2 2 4 2
!0.86 !0.53 !0.49 3.18 !0.72
3
6592 9232 16 245
6618 9210 16 056
0 2 2
!0.39 0.24 1.16
4
11 385 13 473 19 615
11 417 13 532 19 683
0 2 4
!0.28 !0.44 !0.35
5
16 594 18 402
16 597 18 390
0 2
!0.02 0.07
(b) Antisymmetric longitudinal modes 8550 8615 1 10 719 10 874 3
!0.76 !1.45
0 1
6231 10 770 12 533 18 938
6286 10 803 12 616 18 979
1 3 1 3
!0.88 !0.31 !0.66 !0.22
2
2946 9521 11 848 17 350
2962 9588 * 17 441
1 3
!0.54 !0.70
1
!0.52
3
7064 12 584 15 622
7100 12 717 15 693
1 3 3
!0.51 !1.06 !0.45
4
11 835 16 435 20 463
11 861 16 391 20 471
1 3 3
!0.22 0.27 !0.04
5
16 961 20 801
16 966 20 737
1 3
!0.03 0.31
*Frequency could not be traced.
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Figure 2. The "nite element model with ADINA.
mode shapes in general. ADINA "nite element analysis programme (produced by R & D Inc.) was used in the analytical determination. For the experimental work, the STAR modal analysis software package (produced by Structural Measurement Systems) was used to process and analyse the measured data.
2. THE CYLINDER MODEL
The model chosen in the present study is the same as the "nite-length, thick-walled, smooth cylindrical circular shell used in reference [17] so that the results could be compared with each other. Both the theoretical and experimental modal analyses were carried out on this model. The model is made of mild steel and its dimensions are: inside diameter"152.4 mm, outside diameter"228.6 mm, and axial length"250.4 mm.
3. THE FINITE ELEMENT MODEL
The material constants used in the calculations were consistent with the work in reference [17], i.e. modulus of elasticity"207.0 GPa; density"7860 kg/m; and Poisson's ratio "0.28. The "nite element was chosen to be the three-dimensional isoparametric solid element in the calculations with the ADINA programme. An element with 20 nodes was chosen. There were three degrees of freedom at each node, which corresponded to the three translational x-, y- and z-direction respectively, as shown in Fig. 2. There were 1280 nodes, 160 elements, and the total number of degrees of freedom was 3840. The natural frequencies and the mode shapes of the model were veri"ed by experimental modal testing. Experiments included the measurement of the frequency response functions (FRFs) at 144 di!erent points all over the model, plus the processing of the measured data. The FRFs were analysed by the STAR system.
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4. RESULTS
Both the theoretical and experimental modal analyses for the model were performed for the frequency range from 20 Hz to 20 kHz. Extended "nite element calculations for the range from 20 to 30 kHz were also performed. 4.1. THEORETICAL RESULTS Because of the large amount of calculations required, the frequency range from 20 Hz to 20 kHz was divided into ten sections. Table 2 lists the frequency results and the descriptors n and m. In this paper, all the mode shape results will be described by these two descriptors and the criteria for assigning values of n and m will be discussed later in this section. 4.2. ANALYSIS OF THEORETICAL MODE SHAPE RESULTS To get a better understanding of the mode shapes, numerical analysis was performed on the results. First, the numerical results of the displacements on each node in the global coordinate system were transferred from the global x, y, z coordinate system into the local r, t, z coordinate system of each node. The r-, t-, and z-directions correspond to the radial, tangential, and longitudinal directions of the node, respectively. Then the rms values for directions r, t, and z were calculated,
u " uG P P G u " uG R R G
where
u " uG X X G u "u cos #u sin P V W u "u sin #u cos R V W
u "u X X and i is for all the nodes on the middle layer of the model thick wall. The rms values for each displacement direction represent the energy in that direction. Finally, the percentage distribution of energy in each direction was computed. Table 2 lists the results for the energy distribution as a percentage; the largest component direction is italicised. Some typical theoretical mode shapes are shown in Fig. 3. 4.3. EXPERIMENTAL RESULTS An experimental veri"cation was performed on the model for the frequency range from 20 Hz to 20 kHz. The excitation was "xed at one point, and the responses were picked up through 144 di!erent points all over the outer surface of the model. To ensure that none of the resonant frequencies was missed and to get the best data quality for each mode, the measurements were "rst performed with the excitation in the radial direction then repeated with longitudinal excitation. To get better resolution, the whole frequency range was divided into three sections (6.4 kHz each) for the radial excitation, and two sections (12.8 kHz each) for the longitudinal excitation. The measured frequency results are listed in
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MODAL ANALYSIS OF CYLINDER
TABLE 2 Comparison of theoretical and experimental results for the frequency range from 20 Hz to 20 kHz Energy (rms) distribution Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
f $# (Hz) 2538 2993 6264 6405 6557 6699 7151 7217 8141 8552 8838 9414 9603 9884 11 007 11 026 11 390 11 634 12 118 12 120 12 594 12 688 12 711 12 830 13 008 13 868 15 800 15 823 15 932 16 180 16 854 17 008 17 053 17 269 17 425 17 485 18 921 18 928 19 187 19 357 19 728 20 047 20 598
f
(Hz) 2576 2960 6320 * 6536 6632 7183 7064 8200 8672 8920 9328 9648 9944 10 945 11 024 11 536 11 440 11 979 11 904 12 688 12 744 * * 12 850 13 616 15 728 15 296 15 440 16 176 16 496 16 627 17 098 16 593 17 006 17 600 18 456 * 19 408 * 18 960 19 840 20 472
Error (%) 0.27 0.57 !0.89 * 0.32 1.01 !0.45 1.41 !0.72 !0.93 !0.92 1.27 !0.47 !0.6 0.57 0.11 1.27 1.69 1.16 1.81 !0.74 !0.63 * * 1.23 1.85 0.25 3.45 0.25 0.02 2.17 2.29 !0.21 3.88 2.46 !0.65 2.52 * !1.14 * 4.05 1.04 0.62
n
m
2 0 2 1 1 1 Torsion Torsion 2 2 3 0 1 2 3 1 0 0 0 1 0 2 3 2 2 3 1 2 1 3 0 3 0 0 4 0 2 3 4 1 1 1 2 2 1 0 Torsion Torsion 3 3 4 2 3 3 0 4 1 4 3 2 4 3 5 0 1 4 2 4 5 1 2 1 5 2 2 0 1 5 Torsion Torsion 3 4 4 4 4 3
Radial 0.67 0.54 0.40 0.00 0.54 0.75 0.55 0.64 0.59 0.88 0.93 0.57 0.28 0.32 0.79 0.91 0.33 0.79 0.65 0.70 0.34 0.29 0.39 0.00 0.44 0.60 0.45 0.93 0.66 0.25 0.49 0.80 0.33 0.76 0.73 0.22 0.62 0.27 0.17 0.00 0.69 0.29 0.35
Longitudinal Tangential 0.00 0.18 0.41 0.00 0.20 0.01 0.18 0.12 0.41 0.12 0.07 0.18 0.52 0.49 0.04 0.09 0.67 0.01 0.21 0.10 0.22 0.52 0.21 0.00 0.31 0.17 0.44 0.07 0.12 0.50 0.26 0.02 0.27 0.06 0.09 0.41 0.16 0.12 0.33 0.00 0.16 0.44 0.49
0.33 0.28 0.19 1.00 0.26 0.25 0.26 0.24 0.00 0.00 0.00 0.24 0.20 0.19 0.16 0.00 0.00 0.20 0.15 0.21 0.45 0.19 0.41 1.00 0.24 0.23 0.11 0.00 0.22 0.24 0.25 0.18 0.40 0.18 0.19 0.37 0.21 0.61 0.49 1.00 0.15 0.27 0.16
* Frequency could not be traced in the experiment.
Table 2. Applying excitation in more than one direction will produce a duplication in the results. One set of results can be used as a veri"cation of the other, or both sets of results can be used alternately. The selection of results from the di!erent measurement frequency sections is based on the joint consideration of the smoothness of the obtained mode shape
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Figure 3. Some typical theoretical mode shapes.
and the MAC. Extended analytical calculations for the frequency range from 20 to 30 kHz were also performed. The frequency results are listed in Table 3. 4.4. CRITERIA FOR ASSIGNING VALUES OF n AND m A pair of descriptors, n and m, have been assigned to every mode shape in the table. Several criteria were used in assigning values of n and m to a certain mode shape. Parameter n can be easily assigned to any mode shape result. Some examples are illustrated in Fig. 1(a). For most of the mode shapes, the parameter m can be uniquely assigned without di$culty. There is only one possible m which can be assigned to this kind of mode shape. Figure 1(b) gives some examples of them. For some other mode shapes, the assignment of m may be unclear. When in doubt, the following two criteria were used in assigning a value of m to a mode shape. (i) If the di!erences between the magnitudes of peaks in the radial displacements along the generatrix are large, only the peaks with large magnitude are considered in counting the number of nodes. The small peaks are ignored. These small peaks may arise as a consequence of Poisson's e!ect, or because of the accumulated numerical error of the theoretical analysis. This applied to modes 7, 30, 36, 44, 47, 48, 58, 59, and 75. Figure 4 gives an example of this case. (ii) If the magnitudes of peaks in the radial displacements along the axial generatrix are of the same magnitude, all the peaks are considered in counting the number of nodes. This applied to modes 39, 46, 64, and 65. Figure 5 shows an example to which this criterion was applied.
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MODAL ANALYSIS OF CYLINDER
TABLE 3 ¹heoretical results for the frequency range from 20 to 30 kHz Energy (rms) distribution Mode 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78
f
$#
(Hz)
21 194 21 376 21 476 21 826 22 085 22 611 22 672 22 955 22 982 23 369 24 026 24 248 24 252 24 297 24 988 25 410 25 525 26 079 26 145 26 304 26 502 27 276 27 787 27 948 28 153 28 225 28 331 28 602 28 788 29 106 29 179 29 659 29 718 29 728 30 113
n
m
Radial
Longitudinal
Tangential
3 5 0 2 2 6 0 6 1 4 1 5 6 2 4 3 5 3 Torsion 6 1 3 5 3 2 2 7 7 6 4 5 4 0 7 1
1 3 3 1 2 0 5 1 5 4 5 4 2 5 1 2 3 5 Torsion 3 6 4 4 5 5 6 0 1 4 5 5 2 6 2 6
0.21 0.51 0.12 0.27 0.19 0.80 0.93 0.74 0.72 0.61 0.28 0.33 0.63 0.75 0.21 0.19 0.31 0.70 0.00 0.51 0.16 0.19 0.56 0.30 0.19 0.24 0.79 0.73 0.35 0.70 0.23 0.20 0.88 0.62 0.73
0.50 0.25 0.88 0.25 0.36 0.03 0.07 0.09 0.16 0.25 0.43 0.39 0.16 0.08 0.52 0.36 0.49 0.08 0.00 0.24 0.22 0.27 0.31 0.19 0.43 0.26 0.05 0.09 0.36 0.10 0.52 0.47 0.12 0.17 0.12
0.29 0.25 0.00 0.48 0.45 0.17 0.00 0.18 0.11 0.14 0.29 0.28 0.21 0.17 0.27 0.45 0.20 0.22 1.00 0.25 0.63 0.54 0.13 0.50 0.38 0.51 0.16 0.18 0.28 0.19 0.25 0.33 0.00 0.21 0.15
5. DISCUSSION OF RESULTS
5.1. COMPARISON OF FEA AND EXPERIMENTAL RESULTS One of the purposes of experimental modal analysis is to verify the theoretical results. The comparison of the frequency results is listed in Table 2. The experimental results are treated as a true estimation for the physical model. The relative errors in Table 2 are calculated as error"( f !f )/f . The error results in Table 2 show that the agreement $# between the analytical and experimental results is very good. The largest error is 4.05%, and, for most of the modes, the errors are less than 2%. There are several frequencies which correspond to pure torsional vibrations which cannot be found by experiment due to limitations in the excitation and measurement equipment. Based upon the descriptors n and m, a comparison of the mode shapes shows that the agreement between the analytical and
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R. K. SINGHAL E¹ A¸.
Figure 4. An example of mode shape of Criterion 1 (mode 36, f"17 485 Hz, n"2, m"1; dashed lines indicate undeformed shape).
experimental results is also very good: the descriptors n and m are in all cases identical to each other. In order to provide more information for examining the vibrational behaviour of the cylinder, the extended theoretical analysis was performed for the additional frequency range 20}30 kHz. However, because of limitation in the measurement equipment, modal tests could not be conducted at such high frequencies. Because of the excellent agreement shown for the lower frequency range, it is likely that analytical results alone are enough to re#ect the true vibrational behaviour of the physical model for these higher frequencies. The following discussion is based on the theoretical results only. 5.2. MODE SHAPE RESULTS In order to have a better understanding of the mode shape results, all the frequencies from 20 Hz to 30 kHz are rearranged in terms of the descriptors n and m listed in Table 4. The table shows that there exists at most three modes which correspond to the same combination n and m. By carefully examining these modes it can be seen that they are di!erent from each other. This is also proven by the numerical analysis of the mode shape results as listed in Tables 2 and 3. Every mode has its unique mode shape, although they share the same n and m combination. In Table 4, the direction corresponding to the largest energy component is also indicated for each mode. Further, there are values of n and m (n"2, m"1; n"2, m"2 and n"3, m"2) for which all three combinations can be found in the frequency range from 20 to 30 kHz. By examining these modes in conjunction with the numerical analysis results for the energy distributions, it can be seen that the three modes for any one combination of
MODAL ANALYSIS OF CYLINDER
151
Figure 5. An example of mode shape of Criterion 2 (mode 65, f"27 276 Hz, n"3, m"4; dashed lines indicate undeformed shape).
n and m are predominantly radial, longitudinal, and tangential, respectively. Other supportive evidence for this is that, in those other combinations which have less than three modes within the observed frequency range, each mode has its own predominant direction without overlap. Note that although there is a dominant direction in each vibration mode, the cross e!ect between the di!erent displacement directions cannot be neglected. For some modes, such an e!ect can distort the energy distribution signi"cantly. The combination of n"3 and m"3 is a good example. For the second mode in this combination, f"15 800 Hz, the percentage energy for the radial direction exceeds that for the longitudinal direction by 1%, thus 45% of the energy is radial and 44% is longitudinal. In some modes, this e!ect could in#uence the displacement distribution, so that it becomes di$cult to decide on the descriptor m without careful examination.
6. CONCLUSIONS
Extensive modal analysis has been conducted on a "nite-length, thick-walled circular cylinder. This was done by theoretical modal analysis using the "nite element method for the frequency range from 20 Hz to 30 kHz, and was veri"ed by experimental modal testing for the frequency range from 20 Hz to 20 kHz. Numerical analysis has been performed on the mode shape results obtained. The results for the natural frequencies and mode shapes of the cylinder from 20 Hz to 30 kHz are presented in this paper. As a result of the observation
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R. K. SINGHAL E¹ A¸.
TABLE 4 All frequencies listed in terms of mode shapes n
m
0
0 1 2 3 4 5 6
8141r 8552r 8838r 11 026r 15 823r 22 672r 29 718r
11 390l
0 1 2 3 4 5 6
2583r 2993r 6557r 9603l 17 269r 24 297r 28 225t
18 928t 17 485l 12 688l 12 118r
0 1 2 3 4 5
11 634r 12 120r 13 868r 16 854r 20 047l 29 106r
0 1 2 3 4
22 611r 22 955r 24 252r 26 304r 28 788l
2
4
6
Frequency* (Hz)
n
m
1
0 1 2 3 4 5 6
12 711t 6264l 7151r 11 007r 15 932r 22 982r 26 502t
12 594t 9884l 19 187t 17 053t 24 026l 30 113r
0 1 2 3 4 5
6699r 7217r 9414r 13 008r 19 728r 26 079r
21 194l 16 180l 15 800l 27 276t 27 948t
0 1 2 3 4 5
17 008r 17 425r 18 921r 21 376r 24 248l 29 179l
0 1 2
28 311r 28 603r 29 728r
21476l
3 21 826t 22 085t
28 153l 5 24 988l 29 659l 20 598l 23 369r 7
Frequency* (Hz)
25 410t
25 525l 27 787r
* r, l, t indicates that the largest energy component is in the radial, longitudinal, or tangential direction, respectively.
presented in this paper, the following conclusions are drawn. 1. Theoretical modal analysis using the "nite element method provides very accurate frequency results for a thick-walled circular cylinder: there is close agreement between theoretical and experimental results. 2. The "nite element analysis results are enough to describe the vibrational behaviour of the shell. 3. Each frequency has a unique mode shape even though three di!erent frequencies might share the same mode shape descriptors n and m. 4. For each mode the vibrational behaviour is dominated by one of the radial, longitudinal, or tangential directions. Those modes which correspond to the same n and m combination are predominantly radial, longitudinal, or tangential, respectively. 5. The cross e!ect between the di!erent displacement directions cannot be neglected, sometimes such an e!ect could distort the energy distribution signi"cantly.
ACKNOWLEDGEMENTS
The "nancial support provided by the National Science and Engineering Research Council, Grant No. 0GP0037980, is gratefully acknowledged.
MODAL ANALYSIS OF CYLINDER
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REFERENCES 1. G. W. MCMAHON 1964 Journal of the Acoustical Society of America 36, 85}92. Experimental study of the vibrations of solid, isotropic elastic cylinders. 2. D. BANCROFT 1941 ¹he Physical Review 59, 588}593. The velocity of longitudinal waves in cylindrical bars. 3. R. D. MINDLIN and G. HERRMANN 1951 Proceedings of First ;.S. National Congress of Applied Mechanics, 187}191. One dimensional theory of compressional waves in elastic rod. 4. R. D. MINDLIN and H. D. MCNIVEN 1960 ¹ransactions of the American Society of Mechanical Engineers Series E, Journal of Applied Mechanics 27, 141}151. Axially symmetric waves in elastic rods. 5. H. D. MCNIVEN and D. C. PERRY 1962 Journal of the Acoustical Society of America 34, 433}437. Axially symmetric waves in "nite, elastic rods. 6. M. RUMERMAN and S. RAYNOR 1971 Journal of Sound and
MODAL ANALYSIS OF A THICK-WALLED CIRCULAR CYLINDER R. K. SINGHAL Canadian Space Agency, David Florida Laboratory, P.O. Box 11490, Station H, Ottawa, Ontario, Canada K2H 8S2
W. GUAN National Research Council, Institute for Research in Construction, Structures Laboratory, Ottawa, Ontario, Canada K1A 0R6 AND
K. WILLIAMS Department of Mechanical Engineering, College of Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0 (Received 10 March 2000, accepted 18 April 2001) The objective of this paper is to "nd true descriptors for the mode shapes of circular cylinders. The vibrational behaviour of a thick-walled circular cylinder is of considerable engineering importance as such elements have numerous applications in electrical machines, sti!ened cylinders, and gears. Mode shapes are also very important sources of information for understanding and controlling the vibration of a structure. Theoretical and experimental modal analyses were carried out using a thick-walled circular cylinder model to obtain its natural frequencies and mode shapes. The theoretical modal analysis was done using the "nite element method. The results for the frequency range from 20 Hz to 20 kHz were veri"ed using experimental modal analysis. The correlation between the analytical and the experimental results is very good. The largest error for all frequencies is 4.05%, and less than 2% for most frequencies. 2002 Elsevier Science Ltd.
1. INTRODUCTION
A number of theories for the prediction of the natural frequencies of cylinders have been developed and used over the years, as mentioned in reference [1]. Because the solutions for the vibrational behaviour of a cylinder cannot be easily obtained by the full linear elastic theory, people have created various approximate theories using various assumptions for the displacement components. Some of the theories are capable of dealing with "nite-length free hollow cylinders, while others are appropriate only for solid cylinders of in"nite length. Among these researchers, only a few give a complete description of the mode shapes. Bancroft [2] explored the problem numerically. More recent work on the subject began with a paper by Mindlin and Herrmann [3], and was formed into the well-known &Three-mode theory' in a long series of papers by McNiven and his associates, beginning with Mindlin and McNiven [4]. Many authors have considered approximate solutions, for the limiting cases in which the cylinder approaches either a thin disk or a slender rod. These approximate solutions were discussed at length in reference [1], and compared with the experimental results. 0888}3270/02/010141#13 $35.00/0
2002 Elsevier Science Ltd.
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R. K. SINGHAL E¹ A¸.
Solutions for the axisymmetric vibration of free cylinders have been presented by several authors. In 1962, McNiven and Perry [5] presented a solution based on approximate equations which take into account the coupling between longitudinal, axial shear, and radial modes of propagation in a rod of in"nite length. In 1971, Rumerman and Raynor [6] developed a spectrum of frequencies using the Rayleigh}Ritz procedure with displacement functions corresponding to pure radial and axial modes of in"nite cylinder. In 1972, Hutchinson [7] presented a solution for this problem using the method of three-dimensional series solutions; he later [8] used this method for a similar problem and in 1980, developed a series solution for the general three-dimensional equations of linear elasticity [9]. The solution was used to "nd accurate natural frequencies for the vibration of solid elastic cylinders with traction-free surfaces, and gave accurate frequencies with the use of very few terms. Results were given for height-to-diameter ratios from zero (disc) to two. These analytical results agreed well with previous experimental results. The vibration of in"nitely long traction-free hollow circular cylinders was investigated on the basis of the linear three-dimensional theory of elasticity by Greenspon [10] and Gazis [11]. McNiven et al. [12] developed the &Three-mode theory', as mentioned before, for axisymmetric vibrations of hollow cylinders. McNiven and Shah [13] used it to investigate the end mode. Gladwell and Tahbildar [14] solved the axisymmetric problem using a "nite element analysis. The general problem of three-dimensional vibrations of a "nite-length circular solid or hollow cylinder with traction-free surfaces was "rst investigated by Gladwell and Vijay [15], using "nite elements. In 1986, by using the same technique as in his papers [7, 9], Hutchinson investigated the vibrations of free hollow circular cylinders [16]. The method of solution developed in this paper involves combining the exact solutions of the governing equations in "ve series which term by term satisfy four of the nine boundary conditions. The remaining "ve boundary conditions are satis"ed by orthogonalisation on the boundaries. A series of frequencies for di!erent sizes of hollow cylinders was calculated, and results showed good agreement with the "nite element results of Gladwell and Vijay [15]. The most recent work on the thick-walled cylinder is contained in the paper of Singal and Williams [17]. Based on the three-dimensional theory of elasticity, the well-known energy method was used in the derivation of the frequency equation of the cylinder. The frequency equation yields natural frequencies for all the circumferential modes of vibration, including the breathing and beam-type modes. Experimental investigations were carried out on several models and showed a very close agreement with the theoretical natural frequencies. So far, most of the work mentioned was dedicated to discovering a theoretical or numerical way of determining the frequencies of cylinders accurately and concisely. Few of the authors give a description of the mode shapes of the cylinders, and none of them provides a unique way for describing all the modes shapes. McMahon [1] gave some mode charts showing sand patterns on the plane surface of a solid cylinder and the approximate form of the vibration at a diametrical cross-section. In references [18}20], the mode shapes of thin-walled cylinders were also presented. Singal and Williams [17] gave a description for the mode shapes of thick-walled hollow cylinders. However, all these descriptions are based on two parameters, namely, the numbers of circumferential and axial nodes. Such a description is not su$cient for describing the mode shapes of a three-dimensional structure uniquely, as we know from reference [17]. In reference [17], the well-known energy method, based on the three-dimensional theory of elasticity, has been shown to be ideally suited to the determination of the natural frequencies of both thick-walled hollow cylinders and rings; however, it does not provide much information on eigenvectors. The technique provides only the magnitude of n (where
MODAL ANALYSIS OF CYLINDER
143
Figure 1. (a) Illustration of parameter n. (b) Illustration of parameter m.
2n"the number of circumferential cross points in the radial displacement shape), and whether m (the number of cross points in the radial displacement shape along any axial generatrix) is an even or odd integer. The meanings of m and n are illustrated in Fig. 1. The actual magnitudes of the integer m were obtained experimentally in reference [17]. Some of the results from reference [17] are listed in Table 1; it shows that there are indeed two or three frequencies which produce the same combination of the parameters n and m (for example, for the combination of n"2 and m"2 the frequencies are 6395, 12 591 and 18 879 Hz). It is impossible for several frequencies to have exactly the same mode shape. It must be concluded that the descriptors m and n alone are not adequate for describing such a three-dimensional problem. The objective of this paper is to "nd true descriptors for the mode shapes of thick-walled circular cylinders. The actual mode shapes are to be determined both analytically and experimentally. In detail, this includes: (i) a theoretical modal analysis using the "nite element method to obtain the natural frequencies and mode shapes for the same model used in reference [17]; (ii) an experimental modal analysis to verify the results of the theoretical analysis; (iii) using the postprocessors, in both the "nite element program and the experimental test software package, to examine the mode shape results; (iv) understanding of the
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R. K. SINGHAL E¹ A¸.
TABLE 1 Results for a thick cylinder having inside diameter"152.4 mm, outside diameter"228.6 mm and axial length"250.4 mm [2] Experimental Mode n 0
Calculated (Hz)
(Hz)
(a) Symmetric longitudinal modes 8109 8149 8817 8886 11 268 * 15 026 15 030
m
% Error
0 2
!0.49 !0.67
4
!0.03
1
7037 9803 12 666 15 596 16 867
7104 9859 12 809 15 201 16 911
2 2 0 4 4
!0.95 !0.57 !1.13 2.53 !0.26
2
2555 6395 12 591 16 862 18 879
2577 6429 12 653 16 326 19 015
0 2 2 4 2
!0.86 !0.53 !0.49 3.18 !0.72
3
6592 9232 16 245
6618 9210 16 056
0 2 2
!0.39 0.24 1.16
4
11 385 13 473 19 615
11 417 13 532 19 683
0 2 4
!0.28 !0.44 !0.35
5
16 594 18 402
16 597 18 390
0 2
!0.02 0.07
(b) Antisymmetric longitudinal modes 8550 8615 1 10 719 10 874 3
!0.76 !1.45
0 1
6231 10 770 12 533 18 938
6286 10 803 12 616 18 979
1 3 1 3
!0.88 !0.31 !0.66 !0.22
2
2946 9521 11 848 17 350
2962 9588 * 17 441
1 3
!0.54 !0.70
1
!0.52
3
7064 12 584 15 622
7100 12 717 15 693
1 3 3
!0.51 !1.06 !0.45
4
11 835 16 435 20 463
11 861 16 391 20 471
1 3 3
!0.22 0.27 !0.04
5
16 961 20 801
16 966 20 737
1 3
!0.03 0.31
*Frequency could not be traced.
MODAL ANALYSIS OF CYLINDER
145
Figure 2. The "nite element model with ADINA.
mode shapes in general. ADINA "nite element analysis programme (produced by R & D Inc.) was used in the analytical determination. For the experimental work, the STAR modal analysis software package (produced by Structural Measurement Systems) was used to process and analyse the measured data.
2. THE CYLINDER MODEL
The model chosen in the present study is the same as the "nite-length, thick-walled, smooth cylindrical circular shell used in reference [17] so that the results could be compared with each other. Both the theoretical and experimental modal analyses were carried out on this model. The model is made of mild steel and its dimensions are: inside diameter"152.4 mm, outside diameter"228.6 mm, and axial length"250.4 mm.
3. THE FINITE ELEMENT MODEL
The material constants used in the calculations were consistent with the work in reference [17], i.e. modulus of elasticity"207.0 GPa; density"7860 kg/m; and Poisson's ratio "0.28. The "nite element was chosen to be the three-dimensional isoparametric solid element in the calculations with the ADINA programme. An element with 20 nodes was chosen. There were three degrees of freedom at each node, which corresponded to the three translational x-, y- and z-direction respectively, as shown in Fig. 2. There were 1280 nodes, 160 elements, and the total number of degrees of freedom was 3840. The natural frequencies and the mode shapes of the model were veri"ed by experimental modal testing. Experiments included the measurement of the frequency response functions (FRFs) at 144 di!erent points all over the model, plus the processing of the measured data. The FRFs were analysed by the STAR system.
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4. RESULTS
Both the theoretical and experimental modal analyses for the model were performed for the frequency range from 20 Hz to 20 kHz. Extended "nite element calculations for the range from 20 to 30 kHz were also performed. 4.1. THEORETICAL RESULTS Because of the large amount of calculations required, the frequency range from 20 Hz to 20 kHz was divided into ten sections. Table 2 lists the frequency results and the descriptors n and m. In this paper, all the mode shape results will be described by these two descriptors and the criteria for assigning values of n and m will be discussed later in this section. 4.2. ANALYSIS OF THEORETICAL MODE SHAPE RESULTS To get a better understanding of the mode shapes, numerical analysis was performed on the results. First, the numerical results of the displacements on each node in the global coordinate system were transferred from the global x, y, z coordinate system into the local r, t, z coordinate system of each node. The r-, t-, and z-directions correspond to the radial, tangential, and longitudinal directions of the node, respectively. Then the rms values for directions r, t, and z were calculated,
u " uG P P G u " uG R R G
where
u " uG X X G u "u cos #u sin P V W u "u sin #u cos R V W
u "u X X and i is for all the nodes on the middle layer of the model thick wall. The rms values for each displacement direction represent the energy in that direction. Finally, the percentage distribution of energy in each direction was computed. Table 2 lists the results for the energy distribution as a percentage; the largest component direction is italicised. Some typical theoretical mode shapes are shown in Fig. 3. 4.3. EXPERIMENTAL RESULTS An experimental veri"cation was performed on the model for the frequency range from 20 Hz to 20 kHz. The excitation was "xed at one point, and the responses were picked up through 144 di!erent points all over the outer surface of the model. To ensure that none of the resonant frequencies was missed and to get the best data quality for each mode, the measurements were "rst performed with the excitation in the radial direction then repeated with longitudinal excitation. To get better resolution, the whole frequency range was divided into three sections (6.4 kHz each) for the radial excitation, and two sections (12.8 kHz each) for the longitudinal excitation. The measured frequency results are listed in
147
MODAL ANALYSIS OF CYLINDER
TABLE 2 Comparison of theoretical and experimental results for the frequency range from 20 Hz to 20 kHz Energy (rms) distribution Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
f $# (Hz) 2538 2993 6264 6405 6557 6699 7151 7217 8141 8552 8838 9414 9603 9884 11 007 11 026 11 390 11 634 12 118 12 120 12 594 12 688 12 711 12 830 13 008 13 868 15 800 15 823 15 932 16 180 16 854 17 008 17 053 17 269 17 425 17 485 18 921 18 928 19 187 19 357 19 728 20 047 20 598
f
(Hz) 2576 2960 6320 * 6536 6632 7183 7064 8200 8672 8920 9328 9648 9944 10 945 11 024 11 536 11 440 11 979 11 904 12 688 12 744 * * 12 850 13 616 15 728 15 296 15 440 16 176 16 496 16 627 17 098 16 593 17 006 17 600 18 456 * 19 408 * 18 960 19 840 20 472
Error (%) 0.27 0.57 !0.89 * 0.32 1.01 !0.45 1.41 !0.72 !0.93 !0.92 1.27 !0.47 !0.6 0.57 0.11 1.27 1.69 1.16 1.81 !0.74 !0.63 * * 1.23 1.85 0.25 3.45 0.25 0.02 2.17 2.29 !0.21 3.88 2.46 !0.65 2.52 * !1.14 * 4.05 1.04 0.62
n
m
2 0 2 1 1 1 Torsion Torsion 2 2 3 0 1 2 3 1 0 0 0 1 0 2 3 2 2 3 1 2 1 3 0 3 0 0 4 0 2 3 4 1 1 1 2 2 1 0 Torsion Torsion 3 3 4 2 3 3 0 4 1 4 3 2 4 3 5 0 1 4 2 4 5 1 2 1 5 2 2 0 1 5 Torsion Torsion 3 4 4 4 4 3
Radial 0.67 0.54 0.40 0.00 0.54 0.75 0.55 0.64 0.59 0.88 0.93 0.57 0.28 0.32 0.79 0.91 0.33 0.79 0.65 0.70 0.34 0.29 0.39 0.00 0.44 0.60 0.45 0.93 0.66 0.25 0.49 0.80 0.33 0.76 0.73 0.22 0.62 0.27 0.17 0.00 0.69 0.29 0.35
Longitudinal Tangential 0.00 0.18 0.41 0.00 0.20 0.01 0.18 0.12 0.41 0.12 0.07 0.18 0.52 0.49 0.04 0.09 0.67 0.01 0.21 0.10 0.22 0.52 0.21 0.00 0.31 0.17 0.44 0.07 0.12 0.50 0.26 0.02 0.27 0.06 0.09 0.41 0.16 0.12 0.33 0.00 0.16 0.44 0.49
0.33 0.28 0.19 1.00 0.26 0.25 0.26 0.24 0.00 0.00 0.00 0.24 0.20 0.19 0.16 0.00 0.00 0.20 0.15 0.21 0.45 0.19 0.41 1.00 0.24 0.23 0.11 0.00 0.22 0.24 0.25 0.18 0.40 0.18 0.19 0.37 0.21 0.61 0.49 1.00 0.15 0.27 0.16
* Frequency could not be traced in the experiment.
Table 2. Applying excitation in more than one direction will produce a duplication in the results. One set of results can be used as a veri"cation of the other, or both sets of results can be used alternately. The selection of results from the di!erent measurement frequency sections is based on the joint consideration of the smoothness of the obtained mode shape
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R. K. SINGHAL E¹ A¸.
Figure 3. Some typical theoretical mode shapes.
and the MAC. Extended analytical calculations for the frequency range from 20 to 30 kHz were also performed. The frequency results are listed in Table 3. 4.4. CRITERIA FOR ASSIGNING VALUES OF n AND m A pair of descriptors, n and m, have been assigned to every mode shape in the table. Several criteria were used in assigning values of n and m to a certain mode shape. Parameter n can be easily assigned to any mode shape result. Some examples are illustrated in Fig. 1(a). For most of the mode shapes, the parameter m can be uniquely assigned without di$culty. There is only one possible m which can be assigned to this kind of mode shape. Figure 1(b) gives some examples of them. For some other mode shapes, the assignment of m may be unclear. When in doubt, the following two criteria were used in assigning a value of m to a mode shape. (i) If the di!erences between the magnitudes of peaks in the radial displacements along the generatrix are large, only the peaks with large magnitude are considered in counting the number of nodes. The small peaks are ignored. These small peaks may arise as a consequence of Poisson's e!ect, or because of the accumulated numerical error of the theoretical analysis. This applied to modes 7, 30, 36, 44, 47, 48, 58, 59, and 75. Figure 4 gives an example of this case. (ii) If the magnitudes of peaks in the radial displacements along the axial generatrix are of the same magnitude, all the peaks are considered in counting the number of nodes. This applied to modes 39, 46, 64, and 65. Figure 5 shows an example to which this criterion was applied.
149
MODAL ANALYSIS OF CYLINDER
TABLE 3 ¹heoretical results for the frequency range from 20 to 30 kHz Energy (rms) distribution Mode 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78
f
$#
(Hz)
21 194 21 376 21 476 21 826 22 085 22 611 22 672 22 955 22 982 23 369 24 026 24 248 24 252 24 297 24 988 25 410 25 525 26 079 26 145 26 304 26 502 27 276 27 787 27 948 28 153 28 225 28 331 28 602 28 788 29 106 29 179 29 659 29 718 29 728 30 113
n
m
Radial
Longitudinal
Tangential
3 5 0 2 2 6 0 6 1 4 1 5 6 2 4 3 5 3 Torsion 6 1 3 5 3 2 2 7 7 6 4 5 4 0 7 1
1 3 3 1 2 0 5 1 5 4 5 4 2 5 1 2 3 5 Torsion 3 6 4 4 5 5 6 0 1 4 5 5 2 6 2 6
0.21 0.51 0.12 0.27 0.19 0.80 0.93 0.74 0.72 0.61 0.28 0.33 0.63 0.75 0.21 0.19 0.31 0.70 0.00 0.51 0.16 0.19 0.56 0.30 0.19 0.24 0.79 0.73 0.35 0.70 0.23 0.20 0.88 0.62 0.73
0.50 0.25 0.88 0.25 0.36 0.03 0.07 0.09 0.16 0.25 0.43 0.39 0.16 0.08 0.52 0.36 0.49 0.08 0.00 0.24 0.22 0.27 0.31 0.19 0.43 0.26 0.05 0.09 0.36 0.10 0.52 0.47 0.12 0.17 0.12
0.29 0.25 0.00 0.48 0.45 0.17 0.00 0.18 0.11 0.14 0.29 0.28 0.21 0.17 0.27 0.45 0.20 0.22 1.00 0.25 0.63 0.54 0.13 0.50 0.38 0.51 0.16 0.18 0.28 0.19 0.25 0.33 0.00 0.21 0.15
5. DISCUSSION OF RESULTS
5.1. COMPARISON OF FEA AND EXPERIMENTAL RESULTS One of the purposes of experimental modal analysis is to verify the theoretical results. The comparison of the frequency results is listed in Table 2. The experimental results are treated as a true estimation for the physical model. The relative errors in Table 2 are calculated as error"( f !f )/f . The error results in Table 2 show that the agreement $# between the analytical and experimental results is very good. The largest error is 4.05%, and, for most of the modes, the errors are less than 2%. There are several frequencies which correspond to pure torsional vibrations which cannot be found by experiment due to limitations in the excitation and measurement equipment. Based upon the descriptors n and m, a comparison of the mode shapes shows that the agreement between the analytical and
150
R. K. SINGHAL E¹ A¸.
Figure 4. An example of mode shape of Criterion 1 (mode 36, f"17 485 Hz, n"2, m"1; dashed lines indicate undeformed shape).
experimental results is also very good: the descriptors n and m are in all cases identical to each other. In order to provide more information for examining the vibrational behaviour of the cylinder, the extended theoretical analysis was performed for the additional frequency range 20}30 kHz. However, because of limitation in the measurement equipment, modal tests could not be conducted at such high frequencies. Because of the excellent agreement shown for the lower frequency range, it is likely that analytical results alone are enough to re#ect the true vibrational behaviour of the physical model for these higher frequencies. The following discussion is based on the theoretical results only. 5.2. MODE SHAPE RESULTS In order to have a better understanding of the mode shape results, all the frequencies from 20 Hz to 30 kHz are rearranged in terms of the descriptors n and m listed in Table 4. The table shows that there exists at most three modes which correspond to the same combination n and m. By carefully examining these modes it can be seen that they are di!erent from each other. This is also proven by the numerical analysis of the mode shape results as listed in Tables 2 and 3. Every mode has its unique mode shape, although they share the same n and m combination. In Table 4, the direction corresponding to the largest energy component is also indicated for each mode. Further, there are values of n and m (n"2, m"1; n"2, m"2 and n"3, m"2) for which all three combinations can be found in the frequency range from 20 to 30 kHz. By examining these modes in conjunction with the numerical analysis results for the energy distributions, it can be seen that the three modes for any one combination of
MODAL ANALYSIS OF CYLINDER
151
Figure 5. An example of mode shape of Criterion 2 (mode 65, f"27 276 Hz, n"3, m"4; dashed lines indicate undeformed shape).
n and m are predominantly radial, longitudinal, and tangential, respectively. Other supportive evidence for this is that, in those other combinations which have less than three modes within the observed frequency range, each mode has its own predominant direction without overlap. Note that although there is a dominant direction in each vibration mode, the cross e!ect between the di!erent displacement directions cannot be neglected. For some modes, such an e!ect can distort the energy distribution signi"cantly. The combination of n"3 and m"3 is a good example. For the second mode in this combination, f"15 800 Hz, the percentage energy for the radial direction exceeds that for the longitudinal direction by 1%, thus 45% of the energy is radial and 44% is longitudinal. In some modes, this e!ect could in#uence the displacement distribution, so that it becomes di$cult to decide on the descriptor m without careful examination.
6. CONCLUSIONS
Extensive modal analysis has been conducted on a "nite-length, thick-walled circular cylinder. This was done by theoretical modal analysis using the "nite element method for the frequency range from 20 Hz to 30 kHz, and was veri"ed by experimental modal testing for the frequency range from 20 Hz to 20 kHz. Numerical analysis has been performed on the mode shape results obtained. The results for the natural frequencies and mode shapes of the cylinder from 20 Hz to 30 kHz are presented in this paper. As a result of the observation
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R. K. SINGHAL E¹ A¸.
TABLE 4 All frequencies listed in terms of mode shapes n
m
0
0 1 2 3 4 5 6
8141r 8552r 8838r 11 026r 15 823r 22 672r 29 718r
11 390l
0 1 2 3 4 5 6
2583r 2993r 6557r 9603l 17 269r 24 297r 28 225t
18 928t 17 485l 12 688l 12 118r
0 1 2 3 4 5
11 634r 12 120r 13 868r 16 854r 20 047l 29 106r
0 1 2 3 4
22 611r 22 955r 24 252r 26 304r 28 788l
2
4
6
Frequency* (Hz)
n
m
1
0 1 2 3 4 5 6
12 711t 6264l 7151r 11 007r 15 932r 22 982r 26 502t
12 594t 9884l 19 187t 17 053t 24 026l 30 113r
0 1 2 3 4 5
6699r 7217r 9414r 13 008r 19 728r 26 079r
21 194l 16 180l 15 800l 27 276t 27 948t
0 1 2 3 4 5
17 008r 17 425r 18 921r 21 376r 24 248l 29 179l
0 1 2
28 311r 28 603r 29 728r
21476l
3 21 826t 22 085t
28 153l 5 24 988l 29 659l 20 598l 23 369r 7
Frequency* (Hz)
25 410t
25 525l 27 787r
* r, l, t indicates that the largest energy component is in the radial, longitudinal, or tangential direction, respectively.
presented in this paper, the following conclusions are drawn. 1. Theoretical modal analysis using the "nite element method provides very accurate frequency results for a thick-walled circular cylinder: there is close agreement between theoretical and experimental results. 2. The "nite element analysis results are enough to describe the vibrational behaviour of the shell. 3. Each frequency has a unique mode shape even though three di!erent frequencies might share the same mode shape descriptors n and m. 4. For each mode the vibrational behaviour is dominated by one of the radial, longitudinal, or tangential directions. Those modes which correspond to the same n and m combination are predominantly radial, longitudinal, or tangential, respectively. 5. The cross e!ect between the di!erent displacement directions cannot be neglected, sometimes such an e!ect could distort the energy distribution signi"cantly.
ACKNOWLEDGEMENTS
The "nancial support provided by the National Science and Engineering Research Council, Grant No. 0GP0037980, is gratefully acknowledged.
MODAL ANALYSIS OF CYLINDER
153
REFERENCES 1. G. W. MCMAHON 1964 Journal of the Acoustical Society of America 36, 85}92. Experimental study of the vibrations of solid, isotropic elastic cylinders. 2. D. BANCROFT 1941 ¹he Physical Review 59, 588}593. The velocity of longitudinal waves in cylindrical bars. 3. R. D. MINDLIN and G. HERRMANN 1951 Proceedings of First ;.S. National Congress of Applied Mechanics, 187}191. One dimensional theory of compressional waves in elastic rod. 4. R. D. MINDLIN and H. D. MCNIVEN 1960 ¹ransactions of the American Society of Mechanical Engineers Series E, Journal of Applied Mechanics 27, 141}151. Axially symmetric waves in elastic rods. 5. H. D. MCNIVEN and D. C. PERRY 1962 Journal of the Acoustical Society of America 34, 433}437. Axially symmetric waves in "nite, elastic rods. 6. M. RUMERMAN and S. RAYNOR 1971 Journal of Sound and