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Coupled vibration and natural frequency analysis of isotropic cylinders or disks of finite dimensions
Journal of Sound and Vibration (1995) 185(2), 193–199
COUPLED VIBRATION AND NATURAL FREQUENCY ANALYSIS OF ISOTROPIC CYLINDERS OR DISKS OF FINITE DIMENSIONS S.-Y. L Institute of Applied Acoustics, Shaanxi Teachers University, Xian, Shaanxi, 710062, People’s Republic of China (Received 30 September 1993, and in final form 8 April 1994) An analytical method is presented for studying the axisymmetric coupled vibration of isotropic cylinders and disks. When a mechanical coupling coefficient is introduced, the coupled vibration of the cylinder and disk is divided into two equivalent one-dimensional vibrations: one is the axial vibration and the other is the radial vibration and these two vibrations are coupled to each other by the mechanical coupling coefficient. The frequency equation is derived and the axial and radial natural frequencies are computed. The analysis shows that some decoupled vibration modes, such as the longitudinal vibration of slender rods and the radial vibration of thin disks, can be derived directly from the theory. Compared with the results of one-dimensional theory, the natural frequencies obtained by this method are in better agreement with the measured results. 7 1995 Academic Press Limited
1. INTRODUCTION
The vibration of finite-dimension isotropic cylinders and disks is an interesting subject [1–3]. When the height of the cylinder or disk is much larger or less than its radius, the vibration of the vibrator can be simplified to the longitudinal vibration of a slender rod or the radial vibration of a thin disk. In this case, the problem is simple and one-dimensional theory can be used. However, when the dimensions of the cylinder or disk do not meet the requirement of one-dimensional theory, the equations describing the coupled vibration of cylinders and disks of finite dimensions must be solved [4]. Because of the complexity of the equations, the analytical solutions are difficult to obtain. Rayleigh [5] and Love [6] studied the sound velocity of the longitudinal vibration in rods and obtained the corrected sound velocity and the dispersion equation. Mindlin and co-workers [7, 8] analyzed the axisymmetric vibration of disks using complex second order approximate theory and obtained the dispersion equation. With the development of computer technology, numerical methods are now widely used in the vibration analysis of elastic cylinders and disks [9–11], but the numerical computations are often timeconsuming and cumbersome. In this paper, a mechanical coupling coefficient is introduced and the coupled vibration of cylinders and disks of finite dimensions is studied by using an analytical method. In this analysis the shear stresses and strains are ignored. The natural frequency equation is derived and some decoupled vibration modes are analyzed. Compared with numerical methods, the method presented in this paper is simple and time-saving, and compared with the results of one-dimensional theory, the computed natural frequencies of the cylinders and disks are in good agreement with measured results. 193 0022–460X/95/320193 + 07 $12.00/0
7 1995 Academic Press Limited
.-.
194
2. THE COUPLED VIBRATION ANALYSIS OF THE CYLINDER AND DISK
Let the height and radius of the cylinder or disk be l and a. In the following analysis, the polar co-ordinate is used and the direction of the height of the cylinder or disk is along that of the Z-axis. Based on the theory of elastic dynamics, for the axisymmetrical coupled vibration of the cylinder or disk, when the shearing stresses and strains are ignored, the following equations can be obtained: err = [srr − n(suu + szz )]/E,
euu = [suu − n(srr + szz )]/E,
ezz = [szz − n(suu + srr )]/E, r1 2ur /1t 2 = 1srr /1r + (srr − suu )/r, err = 1ur /1r,
euu = ur /r,
r1 2uz /1t 2 = 1szz /1z, ezz = 1uz /1z.
(1–3) (4, 5) (6)
Here err , euu , ezz and srr , suu , szz are the radial, tangential and axial strains and stresses, ur and uz are the radial and axial displacement components. E, v and r are Young’s modulus, the Poisson ratio and the density of the cylinder material. Let a = szz /(suu + srr ), which is called the mechanical coupling coefficient. Then equations (1)–(3) can be reduced to err − euu = (1 + n)(srr − suu )/E,
err + euu = (1 − n − 2na)(srr + suu )/E,
ezz = (1 − n/a)szz /E.
(7–9)
It can be seen that when the mechanical coupling coefficient is introduced, the coupled vibration of the cylinder or disk can be reduced to two equivalent one-dimensional vibrations. One is the axial vibration which is described by equations (5) and (9), and the other is the planar radial vibration which is described by equations (4), (7) and (8). However, these two vibrations are not independent: they are coupled to each other by the mechanical coupling coefficient and constitute the coupled vibration of the cylinder or disk. In the following analysis, these two vibrations will be analyzed, respectively. 2.1. From equations (7) and (8) one has srr − suu = E(err − euu )/(1 + n),
srr + suu = E(err + euu )/(1 − n − 2na).
(10, 11)
Adding equations (10) and (11) yields srr = E[(err − euu )/(1 + n) + (err + euu )/(1 − n − 2na)]/2.
(12)
Substituting the expressions for srr and srr − suu into equation (4) yields r1 2ur /1t 2 = Er [1 2ur /1r 2 + (1ur /1r)/r − ur /r 2],
(13)
where Er = E(1 − na)/[(1 + n)(1 − n − 2na)] is called the equivalent radial elastic constant. In the derivation of equation (13), equation (6) is used. In the case of harmonic vibration, substituting the radial displacement component ur = ur0 exp (jvt) into equation (13) yields d2ur0 /dr 2 + (dur0 /dr)/r − ur0 /r 2 + Kr2 ur0 = 0,
(14)
where ur0 is a function only of position, Kr = v/Vr , Vr = (Er /r)1/2; Kr and Vr are called the equivalent radial wavenumber and sound velocity. It is obvious that equation (14) is Bessel’s equation of order one, and its solution is ur0 = Ar J1 (Kr r) + Br Y1(Kr r),
(15)
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where Ar and Br are constants, and J1 (Kr r) and Y1 (Kr r) are Bessel functions of the first and second kinds of order one. As Y1 (Kr r) becomes unbounded in the limit Kr r:0, this requires Br = 0 so that ur0 = Ar J1 (Kr r).
(16)
The boundary condition at the edge of a free disk is Fr = 0 at r = a, where Fr is the external force which can be expressed as Fr = −2palsrr =r = a .
(17)
Substituting equations (12), (16) and Fr = 0 into equation (17) yields the radial frequency equation of the equivalent radial vibration of the cylinder or disk: Kr aJ0 (Kr a)(1 − na) − (1 − n − 2na)J1 (Kr a) = 0.
(18)
It can be seen that when the radial vibration is coupled with the axial vibration of the cylinder or disk, the radial frequency equation is different from that of the thin disk in planar radial vibration. However, when the mechanical coupling coefficient becomes zero, equation (18) is the same as that of the thin disk. 2.2. From equation (9) one has szz = Ez ezz ,
(19)
where Ez = E/(1 − n/a) is called the equivalent axial elastic constant. Substituting equation (13) into the axial motion equation (5) yields r 1 2uz /1t 2 = Ez (1 2uz /1z 2).
(20)
For calculating normal modes it is conventional to assume the solution form uz = uz0 exp (jvt). Substitution into equation (20) yields d2uz0 /dz 2 + Kz2 uz0 = 0,
(21)
where Kz = v/Vz and Vz = (Ez /r)1/2; Kz and Vz are called the equivalent axial wavenumber and sound velocity of the equivalent axial vibration of the cylinder or disk. Solutions of equation (21) can be expressed as uz0 = Az sin (Kz z) + Bz cos (Kz z ),
(22)
where Az and Bz are constants which are determined by the boundary conditions. When the two ends of the cylinder or disk are free to move, there can be no internal elastic forces and external forces at the ends, and these conditions are equivalent to szz =z = 0 = 0,
szz =z = l = 0.
(23)
Substituting equations (19), (22) and (6) into equation (23) yields the axial frequency equation of the equivalent axial vibration of the cylinder or disk: sin (Kz l) = 0.
(24)
For the fundamental mode, the solution of equation (24) is Kz l = p.
(25)
It can be seen that equation (24) or (25) is similar to that of the slender rod in longitudinal vibration. However, as the equivalent wavenumber Kz depends on the mechanical coupling coefficient, it is impossible to obtain the axial normal frequency from
196
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equation (25). Similarly, it is also impossible to obtain the radial normal frequency from the radial frequency equation (18). Because equations (18) and (25) are coupled by the mechanical coupling coefficient, therefore, it is necessary to solve the two equations simultaneously to obtain the axial and radial frequencies. In brief, equations (18) and (25) are the frequency equations which determine the relation between the material parameter, geometrical dimensions, vibration modes and the normal frequencies of the cylinder or disk. As equation (18) is a transcendental one, it is impossible to find its analytical solutions, and therefore numerical methods must be used. By solving equations (18) and (25), it can be seen that two groups of solutions can be found from the frequency equations, which are denoted as f1 , a1 and f2 , a2 . Considering the natural vibration modes of the cylinder or disk, one concludes that the two vibration modes corresponding to the two groups of solutions are the axial and radial vibrations of the cylinder or disk, while the frequencies f1 and f2 are the axial and radial frequencies. For a cylinder, the axial vibration is the longitudinal one, while for a disk, the axial vibration is the thickness one. As mentioned above, the frequencies f1 and f2 from equations (18) and (25) are different from those of one-dimensional theory. When the material parameters and geometrical dimensions are given, the procedures to solve equations (18) and (25) are as follows. First choose a value of the mechanical coupling coefficient; then, two frequencies, f 1' and f 2' , can be found from equations (18) and (25). Second, change the mechanical coupling coefficient until the frequency f 1' is equal to f 2' ; this corresponds to one vibration mode of the cylinder or disk. Third, by using similar procedures, two groups of solutions can be found. From the above analysis, it can be seen that there are two frequencies for each coupled vibration of the cylinder or disk. This is different from the results of one-dimensional theory where only one frequency can be obtained for the slender rod or thin disk. On the other hand, it can be seen from the solutions of equations (18) and (25) that when the geometric dimensions satisfy certain conditions, such as l w a or l W a, the two frequencies from equations (18) and (25) are far away from each other. Therefore, the axial vibration is weakly coupled with the radial vibration, and for a given frequency range, the vibrations of the cylinder or disk, such as the longitudinal vibration, planar radial vibration or thickness vibration, can be regarded as decoupled. This is consistent with the results of one-dimensional theory where the vibration is approximated by a one-dimensional vibration mode. For example, the axial and radial frequencies for a thin disk are far away from each other, and the vibration can be approximated by the planar radial vibration (at low frequency) or thickness vibration (at high frequency). To sum up, the vibration of the cylinder or disk is a coupled one of different vibration modes. However, when the dimensions satisfy certain conditions, the vibration can be approximated by a one-dimensional mode. 3. ANALYSIS OF SOME DECOUPLED VIBRATION MODES
In the above analysis, the coupled vibrations of the cylinder or disk was studied. As the axial and radial vibrations are coupled to each other, the vibration analysis is complex. However, when the dimensions satisfy certain conditions, the vibrations can be regarded as decoupled. In the next section, some decoupled vibration modes will be discussed according to the theory of coupled vibration. 3.1. When the height is much larger than the radius, there are two extensional vibration modes for the rod. One is the longitudinal vibration and the other is the planar radial vibration of the slender rod.
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3.1.1. One-dimensional longitudinal vibration of the slender rod In this case, the axial stress and strain, and the radial and tangential strains exist, but the radial and tangential stresses disappear. Using equation (8) one finds that the mechanical coupling coefficient is a = a.
(26)
The equivalent axial elastic constant and sound velocity are Ez = E,
Vz = (E/r)1/2 .
(27)
It is obvious that when the height is much larger than the radius, one of the vibration modes of the rod is the longitudinal vibration of the slender rod. 3.1.2. The planar radial vibration of the slender rod In this case, the axial stress exists, but the axial strain disappears. From equation (19) one has Ez = a.
(28)
Using the expressions for Ez one can obtain the mechanical coupling coefficient as a = n,
(29)
Substituting equation (29) into the expression for the equivalent radial elastic constant Er yields Er = E(1 − n)/[(1 + n)(1 − 2n)].
(30)
The frequency equation (18) of the radial vibration of the cylinder or disk becomes that of the slender rod: Kr a J0 (Kr a)(1 − n) − (1 − 2n)J1 (Kr a) = 0.
(31)
When the Poisson ratio of the rod material is given, the solutions of equation (31) can be found. Let Kr a = X, then using the expressions for Kr and Er , the normal frequencies fr0 of the slender rod in planar radial vibration are found to be fr0 = (X/2pa)zE(1 − n)/r(1 + n)(1 − 2n).
(32)
It is obvious that equation (32) is the same as that of Airey’s work [12]. Therefore, the planar radial vibration is one of the vibration modes of the slender rod. 3.2. When the radius is much larger than the thickness of the disk, there are also two extensional vibration modes for the disk; one is the planar radial vibration of the thin disk and the other is the thickness vibration mode. 3.2.1. The planar radial vibration of the thin disk In this case, the axial stress disappears, but the radial and tangential stresses exist. Using the definition of the mechanical coupling coefficient one has a = 0. Therefore, the equivalent radial elastic constant and sound velocity are Er = E/(1 − n 2),
Vr = {E/[r(1 − n 2)]}1/2 .
(33)
The radial frequency equation (18) of the cylinder or disk in coupled vibration becomes Kr a J0 (Kr a) − (1 − n)J1 (Kr a) = 0.
(34)
.-.
198
It is obvious that equation (34) is consistent with that of the planar radial vibration of the thin disk. In other words, when the radius is much larger than the thickness, one of the vibration modes of the disk is the planar radial vibration of the thin disk. 3.2.2. The thickness vibration of the thin disk In this case, the axial stress and strain and the radial and tangential stresses exist, but the radial and tangential strains disappear. Using equation (8) one has a = (1 − n)/2n.
(35)
Substituting equation (35) into the expression for the equivalent axial elastic constant yields Ez = E(1 − n)/[(1 + n)(1 − 2n)],
(36)
and the sound velocity for the thickness vibration in the thin disk is Vz = {E/r · (1 − n)/[(1 + n)(1 − 2n)]}1/2 .
(37)
From equation (25) the fundamental frequency ft of the thin disk in thickness vibration is ft = (1/2l)zE(1 − n)/r(1 + n)(1 − 2n).
(38)
It can be seen that when the radius is much larger than the thickness, one of the vibration modes of the disk is the thickness vibration of the thin disk. From the above analysis, it can be seen that when the dimensions of the cylinder or disk satisfy certain conditions, some decoupled vibration modes, such as the longitudinal and planar radial vibrations of the slender rod, and the thickness and planar radial vibrations of the thin disk, can be derived directly from the analysis of this paper. In this case, the vibration analysis is simple, and one-dimensional theory can be used. 4. EXPERIMENTAL RESULTS
To validate the theory for the analysis of the coupled vibration of the cylinder or disk, some elastic cylinders and disks were designed and made. The cylinder and disk material used here was aluminium and its material parameters are as follows: one-dimensional longitudinal wave speed c = 5100 m/s, Poisson ratio n = 0·34. The normal frequencies of the cylinders and disks were measured. The results are shown in Table 1, where fr and fz are the radial and axial fundamental frequencies of the cylinders or disks computed from equations (18) and (25), while frm and fzm are the measured results, D1 = = frm − fr =/frm and D2== fzm −fz =/fzm . It can be seen that the computed frequencies are in good agreement with the measured results.
T 1 The computed and measured frequencies of the cylinders and disks l (mm)
a (mm)
fr (Hz)
fz (Hz)
frm (Hz)
fzm (Hz)
D1 (%)
D2 (%)
127·6 67·3 100·0 55·5
19·9 84·0 71·5 15·0
112 056 20 051 20 199 150 183
19 851 49 804 39 053 44 727
115 086 20 399 20 427 155 093
20 271 52 244 40 208 46 109
2·63 1·71 1·12 3·17
2·07 4·67 2·87 2·99
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5. CONCLUSIONS
An analytical method has been presented for the study of the coupled vibration of a cylinder or disk. The frequency equations were derived and some one-dimensional vibration modes were analyzed. To sum up the above analysis, the following conclusions can be drawn. Compared with numerical methods, the analytical method is simple and time-saving, and the explanation of the results is concise. Although the method is formally analytical, ultimately a numerical procedure must be used to calculate the frequencies and mode shapes, in general. There are two kinds of normal frequencies for the coupled vibration of the cylinder or disk; one is the axial frequency and the other is the radial frequency; and are different from those found by using one-dimensional theory. Strictly speaking a vibration mode of a cylinder or disk is always of a coupled type. However, when the dimensions satisfy certain conditions, the vibration can be regarded as decoupled, and one-dimensional theory can be used. The analysis can be used to study the normal frequencies of the coupled vibration of a cylinder or disk. Whether it can be used to study the displacement distribution and other vibration characteristics is a problem that still needs to be studied. REFERENCES 1. H. D. MN and D. C. P 1962 Journal of the Acoustical Society of America 34, 433–437. Axially symmetric waves in finite, elastic rods. 2. G. W. MM 1964 Journal of the Acoustical Society of America 36, 85–92. Experimental study of the vibrations of solid isotropic, elastic cylinders. 3. K. F. G 1975 Wave Motion in Elastic Solids. Oxford: Clarendon Press. 4. H. K 1963 Stress Waves in Solids. New York: Dover. 5. L R 1896 Theory of Sound. New York: Dover (1945 re-issue). 6. A. E. H. L 1944 A Treatise on the Mathematical Theory of Elasticity. Cambridge Cambridge University Press. 7. R. D. M and M. A. M 1959 Journal of Applied Mechanics 26, 511–569. Extensional vibration of elastic plates. 8. D. C. G and R. D. M 1960 Journal of Applied Mechanics 27, 541–547. Extensional vibration of waves in circular disk and a semi-infinite plate. 9. G. M. L. G and U. C. T 1972 Journal of Sound and Vibration 22, 143–157. Finite element analysis of axisymmetric vibrations of cylinders. 10. G. W. MM 1970 Journal of the Acoustical Society of America 48, 307–312. Finite difference analysis of the vibration of solid cylinders. 11. G. M. L. G and D. K. V 1975 Journal of Sound and Vibration 42, 137–145. Vibration analysis of axisymmetric resonators. 12. J. R. A 1913 Archiv der Mathematik und Physik 20, 289. The vibration of cylinders and cylindrical shells.
COUPLED VIBRATION AND NATURAL FREQUENCY ANALYSIS OF ISOTROPIC CYLINDERS OR DISKS OF FINITE DIMENSIONS S.-Y. L Institute of Applied Acoustics, Shaanxi Teachers University, Xian, Shaanxi, 710062, People’s Republic of China (Received 30 September 1993, and in final form 8 April 1994) An analytical method is presented for studying the axisymmetric coupled vibration of isotropic cylinders and disks. When a mechanical coupling coefficient is introduced, the coupled vibration of the cylinder and disk is divided into two equivalent one-dimensional vibrations: one is the axial vibration and the other is the radial vibration and these two vibrations are coupled to each other by the mechanical coupling coefficient. The frequency equation is derived and the axial and radial natural frequencies are computed. The analysis shows that some decoupled vibration modes, such as the longitudinal vibration of slender rods and the radial vibration of thin disks, can be derived directly from the theory. Compared with the results of one-dimensional theory, the natural frequencies obtained by this method are in better agreement with the measured results. 7 1995 Academic Press Limited
1. INTRODUCTION
The vibration of finite-dimension isotropic cylinders and disks is an interesting subject [1–3]. When the height of the cylinder or disk is much larger or less than its radius, the vibration of the vibrator can be simplified to the longitudinal vibration of a slender rod or the radial vibration of a thin disk. In this case, the problem is simple and one-dimensional theory can be used. However, when the dimensions of the cylinder or disk do not meet the requirement of one-dimensional theory, the equations describing the coupled vibration of cylinders and disks of finite dimensions must be solved [4]. Because of the complexity of the equations, the analytical solutions are difficult to obtain. Rayleigh [5] and Love [6] studied the sound velocity of the longitudinal vibration in rods and obtained the corrected sound velocity and the dispersion equation. Mindlin and co-workers [7, 8] analyzed the axisymmetric vibration of disks using complex second order approximate theory and obtained the dispersion equation. With the development of computer technology, numerical methods are now widely used in the vibration analysis of elastic cylinders and disks [9–11], but the numerical computations are often timeconsuming and cumbersome. In this paper, a mechanical coupling coefficient is introduced and the coupled vibration of cylinders and disks of finite dimensions is studied by using an analytical method. In this analysis the shear stresses and strains are ignored. The natural frequency equation is derived and some decoupled vibration modes are analyzed. Compared with numerical methods, the method presented in this paper is simple and time-saving, and compared with the results of one-dimensional theory, the computed natural frequencies of the cylinders and disks are in good agreement with measured results. 193 0022–460X/95/320193 + 07 $12.00/0
7 1995 Academic Press Limited
.-.
194
2. THE COUPLED VIBRATION ANALYSIS OF THE CYLINDER AND DISK
Let the height and radius of the cylinder or disk be l and a. In the following analysis, the polar co-ordinate is used and the direction of the height of the cylinder or disk is along that of the Z-axis. Based on the theory of elastic dynamics, for the axisymmetrical coupled vibration of the cylinder or disk, when the shearing stresses and strains are ignored, the following equations can be obtained: err = [srr − n(suu + szz )]/E,
euu = [suu − n(srr + szz )]/E,
ezz = [szz − n(suu + srr )]/E, r1 2ur /1t 2 = 1srr /1r + (srr − suu )/r, err = 1ur /1r,
euu = ur /r,
r1 2uz /1t 2 = 1szz /1z, ezz = 1uz /1z.
(1–3) (4, 5) (6)
Here err , euu , ezz and srr , suu , szz are the radial, tangential and axial strains and stresses, ur and uz are the radial and axial displacement components. E, v and r are Young’s modulus, the Poisson ratio and the density of the cylinder material. Let a = szz /(suu + srr ), which is called the mechanical coupling coefficient. Then equations (1)–(3) can be reduced to err − euu = (1 + n)(srr − suu )/E,
err + euu = (1 − n − 2na)(srr + suu )/E,
ezz = (1 − n/a)szz /E.
(7–9)
It can be seen that when the mechanical coupling coefficient is introduced, the coupled vibration of the cylinder or disk can be reduced to two equivalent one-dimensional vibrations. One is the axial vibration which is described by equations (5) and (9), and the other is the planar radial vibration which is described by equations (4), (7) and (8). However, these two vibrations are not independent: they are coupled to each other by the mechanical coupling coefficient and constitute the coupled vibration of the cylinder or disk. In the following analysis, these two vibrations will be analyzed, respectively. 2.1. From equations (7) and (8) one has srr − suu = E(err − euu )/(1 + n),
srr + suu = E(err + euu )/(1 − n − 2na).
(10, 11)
Adding equations (10) and (11) yields srr = E[(err − euu )/(1 + n) + (err + euu )/(1 − n − 2na)]/2.
(12)
Substituting the expressions for srr and srr − suu into equation (4) yields r1 2ur /1t 2 = Er [1 2ur /1r 2 + (1ur /1r)/r − ur /r 2],
(13)
where Er = E(1 − na)/[(1 + n)(1 − n − 2na)] is called the equivalent radial elastic constant. In the derivation of equation (13), equation (6) is used. In the case of harmonic vibration, substituting the radial displacement component ur = ur0 exp (jvt) into equation (13) yields d2ur0 /dr 2 + (dur0 /dr)/r − ur0 /r 2 + Kr2 ur0 = 0,
(14)
where ur0 is a function only of position, Kr = v/Vr , Vr = (Er /r)1/2; Kr and Vr are called the equivalent radial wavenumber and sound velocity. It is obvious that equation (14) is Bessel’s equation of order one, and its solution is ur0 = Ar J1 (Kr r) + Br Y1(Kr r),
(15)
195
where Ar and Br are constants, and J1 (Kr r) and Y1 (Kr r) are Bessel functions of the first and second kinds of order one. As Y1 (Kr r) becomes unbounded in the limit Kr r:0, this requires Br = 0 so that ur0 = Ar J1 (Kr r).
(16)
The boundary condition at the edge of a free disk is Fr = 0 at r = a, where Fr is the external force which can be expressed as Fr = −2palsrr =r = a .
(17)
Substituting equations (12), (16) and Fr = 0 into equation (17) yields the radial frequency equation of the equivalent radial vibration of the cylinder or disk: Kr aJ0 (Kr a)(1 − na) − (1 − n − 2na)J1 (Kr a) = 0.
(18)
It can be seen that when the radial vibration is coupled with the axial vibration of the cylinder or disk, the radial frequency equation is different from that of the thin disk in planar radial vibration. However, when the mechanical coupling coefficient becomes zero, equation (18) is the same as that of the thin disk. 2.2. From equation (9) one has szz = Ez ezz ,
(19)
where Ez = E/(1 − n/a) is called the equivalent axial elastic constant. Substituting equation (13) into the axial motion equation (5) yields r 1 2uz /1t 2 = Ez (1 2uz /1z 2).
(20)
For calculating normal modes it is conventional to assume the solution form uz = uz0 exp (jvt). Substitution into equation (20) yields d2uz0 /dz 2 + Kz2 uz0 = 0,
(21)
where Kz = v/Vz and Vz = (Ez /r)1/2; Kz and Vz are called the equivalent axial wavenumber and sound velocity of the equivalent axial vibration of the cylinder or disk. Solutions of equation (21) can be expressed as uz0 = Az sin (Kz z) + Bz cos (Kz z ),
(22)
where Az and Bz are constants which are determined by the boundary conditions. When the two ends of the cylinder or disk are free to move, there can be no internal elastic forces and external forces at the ends, and these conditions are equivalent to szz =z = 0 = 0,
szz =z = l = 0.
(23)
Substituting equations (19), (22) and (6) into equation (23) yields the axial frequency equation of the equivalent axial vibration of the cylinder or disk: sin (Kz l) = 0.
(24)
For the fundamental mode, the solution of equation (24) is Kz l = p.
(25)
It can be seen that equation (24) or (25) is similar to that of the slender rod in longitudinal vibration. However, as the equivalent wavenumber Kz depends on the mechanical coupling coefficient, it is impossible to obtain the axial normal frequency from
196
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equation (25). Similarly, it is also impossible to obtain the radial normal frequency from the radial frequency equation (18). Because equations (18) and (25) are coupled by the mechanical coupling coefficient, therefore, it is necessary to solve the two equations simultaneously to obtain the axial and radial frequencies. In brief, equations (18) and (25) are the frequency equations which determine the relation between the material parameter, geometrical dimensions, vibration modes and the normal frequencies of the cylinder or disk. As equation (18) is a transcendental one, it is impossible to find its analytical solutions, and therefore numerical methods must be used. By solving equations (18) and (25), it can be seen that two groups of solutions can be found from the frequency equations, which are denoted as f1 , a1 and f2 , a2 . Considering the natural vibration modes of the cylinder or disk, one concludes that the two vibration modes corresponding to the two groups of solutions are the axial and radial vibrations of the cylinder or disk, while the frequencies f1 and f2 are the axial and radial frequencies. For a cylinder, the axial vibration is the longitudinal one, while for a disk, the axial vibration is the thickness one. As mentioned above, the frequencies f1 and f2 from equations (18) and (25) are different from those of one-dimensional theory. When the material parameters and geometrical dimensions are given, the procedures to solve equations (18) and (25) are as follows. First choose a value of the mechanical coupling coefficient; then, two frequencies, f 1' and f 2' , can be found from equations (18) and (25). Second, change the mechanical coupling coefficient until the frequency f 1' is equal to f 2' ; this corresponds to one vibration mode of the cylinder or disk. Third, by using similar procedures, two groups of solutions can be found. From the above analysis, it can be seen that there are two frequencies for each coupled vibration of the cylinder or disk. This is different from the results of one-dimensional theory where only one frequency can be obtained for the slender rod or thin disk. On the other hand, it can be seen from the solutions of equations (18) and (25) that when the geometric dimensions satisfy certain conditions, such as l w a or l W a, the two frequencies from equations (18) and (25) are far away from each other. Therefore, the axial vibration is weakly coupled with the radial vibration, and for a given frequency range, the vibrations of the cylinder or disk, such as the longitudinal vibration, planar radial vibration or thickness vibration, can be regarded as decoupled. This is consistent with the results of one-dimensional theory where the vibration is approximated by a one-dimensional vibration mode. For example, the axial and radial frequencies for a thin disk are far away from each other, and the vibration can be approximated by the planar radial vibration (at low frequency) or thickness vibration (at high frequency). To sum up, the vibration of the cylinder or disk is a coupled one of different vibration modes. However, when the dimensions satisfy certain conditions, the vibration can be approximated by a one-dimensional mode. 3. ANALYSIS OF SOME DECOUPLED VIBRATION MODES
In the above analysis, the coupled vibrations of the cylinder or disk was studied. As the axial and radial vibrations are coupled to each other, the vibration analysis is complex. However, when the dimensions satisfy certain conditions, the vibrations can be regarded as decoupled. In the next section, some decoupled vibration modes will be discussed according to the theory of coupled vibration. 3.1. When the height is much larger than the radius, there are two extensional vibration modes for the rod. One is the longitudinal vibration and the other is the planar radial vibration of the slender rod.
197
3.1.1. One-dimensional longitudinal vibration of the slender rod In this case, the axial stress and strain, and the radial and tangential strains exist, but the radial and tangential stresses disappear. Using equation (8) one finds that the mechanical coupling coefficient is a = a.
(26)
The equivalent axial elastic constant and sound velocity are Ez = E,
Vz = (E/r)1/2 .
(27)
It is obvious that when the height is much larger than the radius, one of the vibration modes of the rod is the longitudinal vibration of the slender rod. 3.1.2. The planar radial vibration of the slender rod In this case, the axial stress exists, but the axial strain disappears. From equation (19) one has Ez = a.
(28)
Using the expressions for Ez one can obtain the mechanical coupling coefficient as a = n,
(29)
Substituting equation (29) into the expression for the equivalent radial elastic constant Er yields Er = E(1 − n)/[(1 + n)(1 − 2n)].
(30)
The frequency equation (18) of the radial vibration of the cylinder or disk becomes that of the slender rod: Kr a J0 (Kr a)(1 − n) − (1 − 2n)J1 (Kr a) = 0.
(31)
When the Poisson ratio of the rod material is given, the solutions of equation (31) can be found. Let Kr a = X, then using the expressions for Kr and Er , the normal frequencies fr0 of the slender rod in planar radial vibration are found to be fr0 = (X/2pa)zE(1 − n)/r(1 + n)(1 − 2n).
(32)
It is obvious that equation (32) is the same as that of Airey’s work [12]. Therefore, the planar radial vibration is one of the vibration modes of the slender rod. 3.2. When the radius is much larger than the thickness of the disk, there are also two extensional vibration modes for the disk; one is the planar radial vibration of the thin disk and the other is the thickness vibration mode. 3.2.1. The planar radial vibration of the thin disk In this case, the axial stress disappears, but the radial and tangential stresses exist. Using the definition of the mechanical coupling coefficient one has a = 0. Therefore, the equivalent radial elastic constant and sound velocity are Er = E/(1 − n 2),
Vr = {E/[r(1 − n 2)]}1/2 .
(33)
The radial frequency equation (18) of the cylinder or disk in coupled vibration becomes Kr a J0 (Kr a) − (1 − n)J1 (Kr a) = 0.
(34)
.-.
198
It is obvious that equation (34) is consistent with that of the planar radial vibration of the thin disk. In other words, when the radius is much larger than the thickness, one of the vibration modes of the disk is the planar radial vibration of the thin disk. 3.2.2. The thickness vibration of the thin disk In this case, the axial stress and strain and the radial and tangential stresses exist, but the radial and tangential strains disappear. Using equation (8) one has a = (1 − n)/2n.
(35)
Substituting equation (35) into the expression for the equivalent axial elastic constant yields Ez = E(1 − n)/[(1 + n)(1 − 2n)],
(36)
and the sound velocity for the thickness vibration in the thin disk is Vz = {E/r · (1 − n)/[(1 + n)(1 − 2n)]}1/2 .
(37)
From equation (25) the fundamental frequency ft of the thin disk in thickness vibration is ft = (1/2l)zE(1 − n)/r(1 + n)(1 − 2n).
(38)
It can be seen that when the radius is much larger than the thickness, one of the vibration modes of the disk is the thickness vibration of the thin disk. From the above analysis, it can be seen that when the dimensions of the cylinder or disk satisfy certain conditions, some decoupled vibration modes, such as the longitudinal and planar radial vibrations of the slender rod, and the thickness and planar radial vibrations of the thin disk, can be derived directly from the analysis of this paper. In this case, the vibration analysis is simple, and one-dimensional theory can be used. 4. EXPERIMENTAL RESULTS
To validate the theory for the analysis of the coupled vibration of the cylinder or disk, some elastic cylinders and disks were designed and made. The cylinder and disk material used here was aluminium and its material parameters are as follows: one-dimensional longitudinal wave speed c = 5100 m/s, Poisson ratio n = 0·34. The normal frequencies of the cylinders and disks were measured. The results are shown in Table 1, where fr and fz are the radial and axial fundamental frequencies of the cylinders or disks computed from equations (18) and (25), while frm and fzm are the measured results, D1 = = frm − fr =/frm and D2== fzm −fz =/fzm . It can be seen that the computed frequencies are in good agreement with the measured results.
T 1 The computed and measured frequencies of the cylinders and disks l (mm)
a (mm)
fr (Hz)
fz (Hz)
frm (Hz)
fzm (Hz)
D1 (%)
D2 (%)
127·6 67·3 100·0 55·5
19·9 84·0 71·5 15·0
112 056 20 051 20 199 150 183
19 851 49 804 39 053 44 727
115 086 20 399 20 427 155 093
20 271 52 244 40 208 46 109
2·63 1·71 1·12 3·17
2·07 4·67 2·87 2·99
199
5. CONCLUSIONS
An analytical method has been presented for the study of the coupled vibration of a cylinder or disk. The frequency equations were derived and some one-dimensional vibration modes were analyzed. To sum up the above analysis, the following conclusions can be drawn. Compared with numerical methods, the analytical method is simple and time-saving, and the explanation of the results is concise. Although the method is formally analytical, ultimately a numerical procedure must be used to calculate the frequencies and mode shapes, in general. There are two kinds of normal frequencies for the coupled vibration of the cylinder or disk; one is the axial frequency and the other is the radial frequency; and are different from those found by using one-dimensional theory. Strictly speaking a vibration mode of a cylinder or disk is always of a coupled type. However, when the dimensions satisfy certain conditions, the vibration can be regarded as decoupled, and one-dimensional theory can be used. The analysis can be used to study the normal frequencies of the coupled vibration of a cylinder or disk. Whether it can be used to study the displacement distribution and other vibration characteristics is a problem that still needs to be studied. REFERENCES 1. H. D. MN and D. C. P 1962 Journal of the Acoustical Society of America 34, 433–437. Axially symmetric waves in finite, elastic rods. 2. G. W. MM 1964 Journal of the Acoustical Society of America 36, 85–92. Experimental study of the vibrations of solid isotropic, elastic cylinders. 3. K. F. G 1975 Wave Motion in Elastic Solids. Oxford: Clarendon Press. 4. H. K 1963 Stress Waves in Solids. New York: Dover. 5. L R 1896 Theory of Sound. New York: Dover (1945 re-issue). 6. A. E. H. L 1944 A Treatise on the Mathematical Theory of Elasticity. Cambridge Cambridge University Press. 7. R. D. M and M. A. M 1959 Journal of Applied Mechanics 26, 511–569. Extensional vibration of elastic plates. 8. D. C. G and R. D. M 1960 Journal of Applied Mechanics 27, 541–547. Extensional vibration of waves in circular disk and a semi-infinite plate. 9. G. M. L. G and U. C. T 1972 Journal of Sound and Vibration 22, 143–157. Finite element analysis of axisymmetric vibrations of cylinders. 10. G. W. MM 1970 Journal of the Acoustical Society of America 48, 307–312. Finite difference analysis of the vibration of solid cylinders. 11. G. M. L. G and D. K. V 1975 Journal of Sound and Vibration 42, 137–145. Vibration analysis of axisymmetric resonators. 12. J. R. A 1913 Archiv der Mathematik und Physik 20, 289. The vibration of cylinders and cylindrical shells.