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The response of a non-uniformly tensioned circular string to a moving load
Journal of Sound and Vibration (1995) 182(3), 415–426
THE RESPONSE OF A NON-UNIFORMLY TENSIONED CIRCULAR STRING TO A MOVING LOAD A. P G. G. A Department of Mechanical Engineering, Northeastern University, Boston, Massachusetts 02115, U.S.A. (Received 21 September 1993, and in final form 28 February 1994) A non-uniformly tensioned circular string on a damped elastic foundation is acted upon by a moving concentrated load. This configuration can be viewed as a very simplified representation of a spinning rectangularly orthotropic disk acted upon by a spatially fixed force, or of a stationary orthotropic disk subjected to a circumferentially moving load. The natural frequencies and normal modes of vibration of the string on elastic foundation are determined for a range of values of the tension inhomogeneity. The response of the string to a moving load is then determined. The results indicate that there are many speeds, below the critical speed of the uniformly tensioned string, for which the forced response is large. It is anticipated that the response of this simple model to a moving load can provide insight into the behavior of the mathematically more complicated rectangularly orthotropic circular disk.
1. INTRODUCTION
The determination of the response of spinning/stationary disks to stationary/moving loads has been the subject of numerous investigations due to its relevance to turbine rotor dynamics, circular saw blades and flexible (floppy) disks used in magnetic recording applications. The first analysis of the spinning disk was by Lamb and Southwell [1], who determined the natural frequencies and mode shapes of a solid circular disk. Results are included for a very flexible disk (negligible bending stiffness), a slowly rotating disk (negligible membrane forces), and an approximate solution for the case in which both bending stiffness and membrane forces are included. Benson and Bogy [2] investigated the spinning disk problem using linear membrane theory. They showed that without the effect of bending stiffness in the analysis, the eigenvalue problem for the determination of the natural frequencies of vibration is singular. The point-load problem is then solved with bending stiffness included. A critical speed of the spinning disk can be interpreted as a rotational velocity at which the propagation speed of a transverse wave is equal to the rotational speed of the disk. Hence such a back-travelling wave will appear fixed in space, and a space-fixed load can cause an unbounded response. Alternatively, a critical speed may be viewed as a speed of rotation at which the natural frequency, in space-fixed co-ordinates, becomes zero. Tobias and Arnold [3] investigated the effects of imperfections on the free and forced vibrations of a spinning disk. In a perfect disk, each natural frequency has two modes which are phase shifted through an arbitrary angle; this allows a travelling wave to be formed as an appropriate combination of these two modes [3]. When an imperfection is 415 0022–460X/95/180415 + 12 $08.00/0
7 1995 Academic Press Limited
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. . .
present, the angular positions of the modes cease to be arbitrary and their frequencies are no longer equal. Thus, the presence of an imperfection distorts the travelling waves, including the back-travelling wave which causes a critical speed of the perfect disk. The stability of a circular plate subjected to peripheral moving loads was investigated by Mote [4] under two different loading conditions. The moving load has either a harmonically varying amplitude or a harmonically varying speed. Iwan and Stahl [5] and Iwan and Moeller [6] also investigated the stability of a flexible disk. In the former paper the disk is stationary and subjected to a circumferentially moving massive load, while in the latter investigation the disk is spinning under the action of a space-fixed massive load. In each case, approximations of the critical speeds are determined and other regions of instability are identified. A simple approximation for the critical speeds of a rotating disk was found by Chonan [7]. Adams [8] determined the critical speeds of rotation of a spinning disk with the effect of an elastic foundation parameter included. The behavior of a rotating circular string with a fixed elastic restraint was investigated by Schajer [9]. That simple model is intended qualitatively to represent the more complicated problem of a spinning disk with a fixed elastic restraint which was later treated by Yano and Kotera [10]. A parametric excitation is one in which the forcing function produces a time varying modification of a system parameter. The response of these systems is characterized by resonances at various frequencies, each of which is related through an integer, or fractional multiple, to the excitation frequency. Campbell [11] recognized the role of parametric vibrations in his study of turbine disks. Further descriptions and analyses of parametrically excited systems can be found in the monographs of Yakubovich and Starzhinskii [12] and Bolotin [13]. A spinning disk, such as a floppy disk used for storing digital information in a personal computer, is rectangularly orthotropic due to the rolling process by which the substrate material (polyethylene terepthalate) is manufactured (Bogy and Talke [14]). The combination of circular geometry with rectangular orthotropy complicates the analysis considerably. A stationary annular disk with rectangular orthotropy was treated by Bianchi, Avalos and Laura [15]. Estimates of the natural frequencies are given, but only for modes which do not have any angular variation. In the present work, the response of a circular string, which has a tension varying continuously along its length and rests on a damped elastic foundation, is investigated. This simple model has qualitative similarities with the orthotropic disk. As a load moves circumferentially with respect to a rectangularly orthotropic disk, the force acts on portions of the disk whose radial and circumferential stiffnesses change continuously. Due to rectangular orthotropy, after 90° of rotation the load travels into a region in which the radial and circumferential bending stiffnesses have interchanged their values from the previous location. Thus the load goes through two complete stiffness cycles for each complete disk/load rotation. In the string problem, the load is subject to a variation in tension stiffness which also goes through two cycles for each revolution of the load. Thus, the orthotropy of the disk corresponds to tension variation in the string. The natural frequencies and normal modes of vibration of the string are determined. The forced response of the string to a moving concentrated force is then found. It is discovered that there are many speeds, below the critical speed of the uniformly tensioned string, for which large amplitude motion results. It is expected that the investigation of the behavior of this simple model can provide insight into the type of phenomena to be encountered in the considerably more complex analysis of a rectangularly orthotropic circular disk.
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2. PROBLEM DESCRIPTION
Now consider a circular string on a damped elastic foundation, subject to a transverse load P moving with constant speed uˆ (Figure 1). The string tension (T ) is inhomogeneous, i.e. T varies with the arc-length co-ordinate sˆ according to T (sˆ ) = T 0 (1 + e cos (2sˆ /Rˆ )),
(1)
where T 0 is the average tension, e defines the degree of tension inhomogeneity, and R is the string radius. The partial differential equation which describes the transverse motion of the string is given by 1 1sˆ
−
0 1 T
1wˆ 1 2wˆ + rˆ = pˆ , 1tˆ 2 1sˆ
0 E sˆ E 2pR .
(2)
In equation (2), wˆ is the transverse string deflection, tˆ is time, rˆ is the mass per unit length, and pˆ is the force per unit length exerted on the string. Acting on the string is a concentrated force P which moves with speed uˆ in the direction of increasing sˆ . In addition, the elastic foundation resists deflections and velocities by exerting a force per unit length on the string which is opposite in direction and proportional to the transverse string deflection and transverse string velocity. Thus pˆ = P d(sˆ − uˆtˆ ) − k wˆ − hˆ
1wˆ , 1tˆ
(3)
where k is the elastic foundation constant, hˆ is the foundation viscous damping coefficient, and d( ) is the Dirac delta function. Furthermore, it can be readily shown that in-plane equilibrium of the string requires radial and circumferential body forces which vary as functions of the arc-length sˆ . It is noted that the presence of these body forces is artificial, and that the use of an inhomogeneous mass density would be more realistic if the only interest was the string problem. However, the ultimate objective of this investigation is a better understanding of the orthotropic disk which has a stiffness variation due to material anisotropy. Thus a stiffness variation, rather than a density variation, is chosen for the string. The partial differential equation (2) is combined with equation (3) and is written in dimensionless form as
0 1
1w 1w 1 2w 1 T + kw + h + 2 = Pd(u − ut), 1u 1t 1t 1u
−
T(u) = T (sˆ )/T 0 = 1 + e cos 2u.
Figure 1. A circular string on a damped elastic foundation subjected to a moving concentrated force.
(4)
418 . . . In equation (4), w, t, k, h, P and u are the dimensionless forms of wˆ , tˆ , k , hˆ , P and uˆ respectively, i.e., w = wˆ /R , u = sˆ /R , t = tˆ uˆ0 /R , k = k R 2/T 0 , h = hˆ R uˆ0 /T 0 , P = P /T 0 , u = uˆ /uˆ0 , uˆ0 = zT 0 /rˆ , (5) where u is the angular co-ordinate and uˆ0 is the wave speed of the homogeneous string in the absence of an elastic foundation. It is noted that the use of a co-ordinate u*, moving with the load P at constant speed (u), will transform equation (4) into
0
1
0
1
1 1w* 1w* 1w* 1 2w* 1 2w* 1 2w* T* + kw* + h −u + 2 − 2u + u2 = Pd(u*), (6) 1u* 1u* 1t 1u* 1t 1t 1u* 1u*2
−
where u* = u − ut, w*(u*, t) = w(u, t), T*(u*, t) = 1 + e cos 2(u* + ut). (7) The periodic tension term T*(u*, t), defined by equation (7) and appearing in equation (6), allows the parametric nature of the excited system to become explicit. The frequency of parametric excitation is seen to be twice the frequency of the moving load. Note that the moving force P does not directly produce the time-varying coefficient of the partial differential equation (4) usually associated with a parametric excitation. However, the presence of the moving load suggests the use of a moving co-ordinate u* from which the tension term exhibits a time variation. Thus, it is the combination of the moving load and the inhomogeneous tension which causes this system to behave as one which is parametrically excited. The undamped natural frequencies and normal modes of free vibration will now be determined. Assume a solution of equation (4) in the form of w(u, t) = W(u)F(t). Standard methods of separation of variables yield [(1 + e cos 2u)W'(u)]' + (v 2 − k)W(u) = 0, 0 E u E 2p, F(t) = C cos (vt − f), (8a, b) where the prime (') denotes differentiation with respect to u, and the square root of the separation constant is equal to the natural frequency (v). The function W(u) is subject to the continuity conditions W(0) = W(2p), W'(0) = W'(2p), (9) which can be satisfied automatically through the use of a Fourier series representation, i.e., a
W (c)(u) = a0 + s an cos nu,
W(u) = W (c)(u) + W (s)(u),
n=1 a
W (s)(u) = s bn sin nu.
(10)
n=1
Substitution of equations (10) into equation (8a), followed by multiplication by cos mu and integration over the interval (0, 2p), results in e ka0 = [v (c)]2a0 , (k + 1)a1 + (−a1 + 3a3 ) = [v (c)]2a1 , 2 (k + 4)a2 + 4ea4 = [v (c)]2a2 , me [(m − 2)am − 2 + (m + 2)am + 2 ] = [v (c)]2am , m = 3, 4, 5, . . . , a. (11) (k + m 2)am + 2
419
A similar set of equations can be obtained for bm by substitution of equations (10) into equation (8a), multiplication by sin mu, and integration over (0, 2p). The result is e (k + 1)b1 + (b1 + 3b3 ) = [v (s)]2b1 , (k + 4)b2 + 4eb4 = [v (s)]2b2 , (12) 2 me (k + m 2)bm + [(m − 2)bm − 2 + (m + 2)bm + 2 ] = [v (s)]2bm , m = 3, 4, 5, . . . , a. 2 In order to obtain equations (11) and (12), the trigonometric relations F Gp , =m − n= = 2, (m $ 0, n $ 0), G2 Gp , m = 1, n = 1, 2p cos 2u cos mu cos nu du = g 2
g
G Gp, G f0,
0
g
(m = 0, n = 2)
or
(m = 2, n = 0),
(13)
otherwise,
2p
cos 2u sin mu cos nu du = 0,
for all m, n,
(14)
0
p F G 2, G 2p cos 2u sin mu sin nu du = g−p , 0 G 2 G f 0,
g
=m − n= = 2,
(m $ 0, n $ 0),
m = 1, n = 1,
(15)
otherwise, have been used. These identities were obtained by first expressing terms such as cos mu cos nu in terms of sin (m + n)u, sin (m − n)u, cos (m + n)u and cos (m − n)u, followed by integration over (0, 2p). Equations (11) and (12) are truncated at m = M and constitute two uncoupled eigenvalue problems which can be solved by standard methods. The nth eigenvalue of the solution of equation (11) is the square of vn(c) and the nth eigenvector is amn (m = 0, 1, 2, . . . , M). Similarly, the nth eigenvalue of the solution of equation (12) is the square of vn(s) and the nth eigenvector is bmn (m = 1, 2, . . . , M). The M + 1 eigenfunctions Wn(c) (u) and the M eigenfunctions Wn(s) (u) are then given by M
Wn(c) (u) = s amn cos mu,
n = 0, 1, 2, . . . , M,
m=0
M
Wn(s) (u) = s bmn sin mu,
n = 1, 2, . . . , M,
(16)
m=1
which represent the natural modes of vibration. It can be readily demonstrated that these eigenfunctions are orthogonal and can be normalized, i.e.,
g g g
2p
Wm(c) (u)Wn(c) (u) du = dmn ,
m = 0, 1, 2, . . . , M,
Wm(s) (u)Wn(s) (u) du = dmn ,
m = 1, 2, . . . , M,
n = 1, 2, . . . , M,
m = 0, 1, 2, . . . , M,
n = 1, 2, . . . , M,
n = 0, 1, 2, . . . , M,
0 2p
0 2p
0
Wm(c) (u)Wn(s) (u) du = 0,
(17)
420
. . .
where dmn is the Kronecker delta. Substitution of equations (16) into equations (17) gives the normalization conditions on amn and on bmn , i.e., M
2a0m a0n + s ajm ajn = dmn /p,
m, n = 0, 1, 2, . . . , M,
j=1
M
s bjm bjn = dmn /p,
m, n = 1, 2, . . . , M.
(18)
j=1
Note that v0(c) = zk and am0 = dm0 /z2p satisfy equations (11) automatically. Thus, the first equation (11) decouples from the others; i.e. a0n = 0, n $ 0. It is also observed that the equations for the even-numbered modes can be separated from those of the odd-numbered modes. It is noted that for homogeneous string tension, equations (11) and (12) are each a diagonal set of equations. Thus vn(c) = vn(s) = zk + n 2, amn = dmn /zp (m $ 0) and bmn = dmn /zp. Therefore, each eigenfunction of the uniformly tensioned string corresponds to a single Fourier component. For non-uniform tension, equations (11) and (12) become each a banded system of equations and so the eigenfunctions contain contributions from many Fourier components. Now consider the forced motion of the string with damping, i.e. equation (4) with h $ 0 and P $ 0. Write the deflection w(u, t) as a modal expansion, M
w(u, t) = a0 (t)W0(c)(u) + s [an (t)Wn(c) (u) + bn (t)Wn(s) (u)].
(19)
n=1
Substitution of equation (19) into equation (4), multiplication by Wm(c) (u), integration over the interval (0, 2p) and use of orthogonality (17), yields the uncoupled equations M
a¨ n + ha˙ n + [vn(c) ]2an = P s amn cos mut,
n = 0, 1, 2, . . . , M.
(20)
m=0
The coefficients an (t) are then Pamn cos (mut − fm(c) ) , (c) 2 2 2 2 m = 0 z{[vn ] − (mu) } + (hmu) M
an (t) = s
(21)
where hmu , 0 E fm(c) E p. (22) [vn(c) ]2 − (mu)2 The bn (t) coefficients are obtained in a similar manner to that used for an (t). Substitution of equation (19) into equation (4), multiplication by Wm(s) (u), integration over the interval (0, 2p) and use of orthogonality yields the uncoupled equations fm(c) = tan−1
M
b n + hb n + [vn(s) ]2bn = P s bmn sin mut,
n = 1, 2, . . . , M.
(23)
m=0
The coefficients bn (t) are given by Pbmn sin (mut − fm(s) ) , (s) 2 2 2 2 m = 0 z{[vn ] − (mu) } + (hmu) M
bn (t) = s
(24)
where fm(s) = tan−1
hmu , [vn(s) ]2 − (mu)2
0 E fm(s) E p.
(25)
421
Figure 2. The normalized square of the natural frequency versus the inhomogeneity parameter (e) for the cosine modes (C) and for the sine modes (S), with k = 0·1 and different values of the mode number n.
Thus the forced response due to the moving load is given by equation (19), with an (t) defined by equation (21) and bn (t) given by equation (24). 3. RESULTS AND DISCUSSION
Consider first the results of the free vibration problem. The ratio of the eigenvalues of the string with inhomogeneous tension to those of the uniformly tensioned string, as a function of the inhomogeneity parameter (e), are shown in Figure 2. Results are given for the squares of the normalized natural frequencies [vn(c) ]2/(k + n 2) (labelled C1 , C2 , and C3 for n = 1, 2, 3 respectively) and for [vn(s) ]2/(k + n 2) (labelled S1 , S2 and S3 ). The C2 frequency increases with e, whereas all of the other frequencies decrease with e. It is noted that C2 corresponds to the mode the diagonal component of which is cos 2u and thus matches the string tension variation. In Figure 3 is shown the variation of amn with e for various values of m and n. Analogous information is given in Figure 4 for bmn . In each graph, the diagonal components of the modes (ann , bnn ) are dominant, especially at low
Figure 3. The mode parameters zpamn versus the inhomogeneity parameter (e) for k = 0·1 and various values of m and n.
422
. . .
Figure 4. The mode parameters zpbmn versus the inhomogeneity parameter (e) for k = 0·1 and various values of m and n.
Figure 5. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for z1 = 0·01, k = 0·1 and various values of the parameter e.
Figure 6. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for z1 = 0·01, e = 0·4 and different values of the foundation stiffness (k).
423
values of e. Furthermore, as was noted earlier, the even numbered components (m) of amn and bmn decouple from the corresponding odd components. Thus, amn = 0, =m − n= = 2j + 1, j = 0, 1, 2, . . . , bmn = 0, =m − n= = 2j + 1, j = 0, 1, 2, . . . . (26) Furthermore, as =m − n= increases, the values of amn and bmn decrease. The results of Figures 3 and 4 are very similar. From equations (11) and (12) it is observed that the equations which determine amn and bmn for a given mode differ only in the coefficients of ea1n and eb1n of the m = 1 equation. Thus, in view of the previous discussion related to even and odd components of amn and bmn , the results for even-numbered m of amn and bmn are identical and the odd-numbered components differ slightly as indicated in Figures 3 and 4. In Figure 5 is shown the maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u). Results are given for the dimensionless damping h = 0·02098 which corresponds to a damping ratio in the first mode of z1 = h/2v1 = 0·01, with e = 0. Curves are for e = 0·0, 0·2 and 0·6, all with k = 0·1. It is noted that the orthotropic floppy disks used in magnetic recording applications [14] can be shown to have degrees of orthotropy as high as the value of 0·6 used here for the string. Also, the first mode damping ratio is taken as the measure of damping rather than the corresponding higher value of h. Figure 6 is a similar graph with e = 0·4 and two values of the elastic foundation constant (k). It is seen that the higher k gives greater sensitivity to speed than does lower k. Note, however, that these values of deflection have been normalized by ws , which is also affected by k. The actual values of the deflection are less for k = 10 than for k = 0·1. Greater values of the foundation stiffness, along with higher speeds, tend to cause the deflection to depend more heavily on the local tension then it otherwise would. Hence the effect of an increase in k is to amplify the effect of larger values of e. The string deflection is shown versus speed with e = 0·4 and k = 0·1 in Figure 7, but for two values of the first mode damping ratio (z1 = 0·001, 0·05). The results of Figures 5–7 show that the overall behavior of the non-uniformly tensioned string is somewhat similar to that of the uniformly tensioned one. However, at many speeds less than unity, the deflection versus speed curves have sharp peaks. Note that these peaks occur at speeds less than the classical critical speed, as well as at u = 1. Furthermore, these peaks increase both in their number of occurrences and in their magnitude as e increases and/or z1 decreases.
Figure 7. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for e = 0·4, k = 0·1 and different values of the first mode damping ratio (z1 ).
424
. . .
It is seen from equations (21) and (24) that resonance occurs at a speed u* if mu* = vn for any values of m and n. Thus, the critical speeds (u*) are given by u* = vn /m 2 zk/m 2 + (n/m)2,
(27)
where vn represents either vn(c) or vn(s) , and the 2 sign becomes an equality when e = 0. For small e, the sensitivity of vn to e is small, as was found from the numerical solution of the free vibration problem. For any given speed u, it is observed from equation (27) that it is possible to find a critical speed arbitrarily close to u by selecting an appropriate pair of values for (m, n). Furthermore, these critical speeds become very small for large m. However, the damping term is proportional to mh, and hence the large m corresponding to low critical speeds results in an increased effect of damping. Therefore, low speeds are less susceptible to large deflections than are higher speeds. With absolutely no damping (h = 0), virtually any speed resonates. The results of Figure 7 indicate this behavior as the z1 = 0·001 curve has much higher peaks than does that for z1 = 0·05. It is noted that the results described here show the features of a parametrically excited system. Thus the resonances given by equation (27) in the context of forced vibrations can also be interpreted as occurring when the parametric excitation frequency (2u) is related through an integer, or fractional multiple, to a natural frequency (vn ) of the system. It is now interesting to consider the string with uniform tension and investigate why it does not have critical speeds other than at u = 1. Write the moving load term as a Fourier series in u, i.e., a
Pd(u − ut) = P/2p + (P/p) s cos mut cos mu,
(28)
m=1
and note that for each forcing frequency (mu) there is a corresponding spatial component (cos mu). Furthermore, the eigenfunctions for the uniformly tensioned string each have only one Fourier term, i.e., 1 1 cos nu, Wn(s) (u) = sin nu. (29) zp zp Thus the modal forcing function is orthogonal to each of the eigenfunctions except for the nth component (i.e., the modal force is zero for m $ n) and resonance occurs only when Wn(c) (u) =
u* = vn /n = zk/n 2 + 1 q 1.
(30)
Thus the fact that the uniformly tensioned string does not have critical speeds for u Q 1 is a consequence of perfectly uniform tension. Alternatively, there is no longer parametric excitation through equation (7) when e = 0. It is further noted that the higher modes (n:a) give critical velocities approaching unity because the influence of the foundation diminishes in comparison with the effect of string tension for high wavenumbers. 4. CONCLUSIONS
The free and forced vibrations of a non-uniformly tensioned circular string on an elastic foundation have been investigated. The variation of the natural frequencies and normal modes of vibration with the degree of tension variation was determined. Each natural mode was found to contain contributions from more than one Fourier component. The forced vibration due to a moving concentrated load was then investigated. It was observed that large deflections occurred at many speeds below the critical speed of a uniformly tensioned string. This phenomenon is a consequence of the fact that each of the modes contain contributions from many Fourier components. An alternative description has also been given in terms of parametric excitations.
425
The effects of the tension inhomogeneity, damping and foundation stiffness were investigated. The magnitudes of the large deflections, as well as the number of such peaks, increased with greater tension non-uniformity and/or decreased damping. Greater foundation stiffness amplified the effect of tension inhomogeneity. This problem is qualitatively similar to the rectangularly orthotropic spinning disk, and is thus expected to provide insight into that particularly complex problem. REFERENCES 1. H. L and R. V. S 1921 Proceedings of the Royal Society of London Series A99, 272–280. The vibrations of a spinning disk. 2. R. C. B and D. B. B 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 636–642. Deflection of a very flexible spinning disk due to a stationary transverse load. 3. S. A. T and R. N. A 1957 Proceedings of the Institute for Mechanical Engineers 171, 669–690. The influence of dynamical imperfection on the vibration of rotating disks. 4. C. D. M J. 1970 Journal of the Franklin Institute 290, 329–333. Stability of circular plates subjected to moving loads. 5. W. D. I and K. J. S 1973 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 445–451. The response of an elastic disk with a moving mass system. 6. W. D. I and T. L. M 1976 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 43, 485–490. The stability of a spinning elastic disk with a transverse load system. 7. S. C 1987 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 54, 967–968. On the critical speed of a rotating circular plate. 8. G. G. A 1987 International Journal of Mechanical Sciences 29, 525–531. Critical speeds for a flexible spinning disk. 9. G. S. S 1984 Journal of Sound and Vibration 92, 11–19. The vibration of a rotating circular string subject to a fixed end restraint. 10. S. Y and T. K 1991 Archive of Applied Mechanics 61, 110–118. Instability of the vibrations of a rotating thin disk due to an additional support. 11. W. C 1924 Transactions of the American Society of Mechanical Engineers 46, 31–160. The protection of stream-turbine disk wheels from axial vibration. 12. V. A. Y and V. M. S 1975 Linear Differential equations with constant coefficients. New York: John Wiley and Sons. 13. V. V. B 1964 The Dynamic Stability of Elastic Systems, San Francisco: Holden-Day. 14. D. B. B and F. E. T 1984 Tribology and Mechanics of Magnetic Storage Systems 1, ASLE SP-16, 115–125. Creep of a rotating orthotropic plate. 15. A. B, D. R. A and P. A. A. L 1985 Journal of Sound and Vibration 99, 140–143. A note on the transverse vibrations of annular circular plates of rectangular orthotropy. APPENDIX: NOMENCLATURE am amn bm bmn F(t) k M P R
sˆ t T(u) T 0 u
mth component of an eigenvector of a cosine mode mth component of the nth eigenvector of a cosine mode mth component of an eigenvector of a sine mode mth component of the nth eigenvector of a sine mode time-varying part of the string deflection dimensionless elastic foundation constant number of terms used in series expansions dimensionless transverse concentrated force string radius arc-length co-ordinate dimensionless time dimensionless inhomogeneous string tension average string tension dimensionless speed of moving load
426 u* uˆ0 w(u, t) w*(u*, t*) W(u) Wn(c) (u) Wn(s) (u) an (t) bn (t) d( ) dmn e h u u* rˆ fm(c) fm(s) v vn(c) vn(s) ˆ ' ·
. . . dimensionless critical speed string wave speed without elastic foundation dimensionless transverse string deflection dimensionless transverse string deflection in moving co-ordinates circumferentially varying part of string deflection nth cosine natural mode nth sine natural mode time-varying coefficients of Wn(c) in the modal expansion time-varying coefficients of Wn(s) in the modal expansion Dirac delta function Kronecker delta string inhomogeneity parameter dimensionless viscous damping coefficient angular co-ordinate moving angular co-ordinate mass per unit length phase angle of the response of the mth component of a cosine mode phase angle of the response of the mth component of a sine mode natural frequency of vibration natural frequency of the nth cosine mode natural frequency of the nth sine mode dimensioned counterpart of a dimensionless variable differentiation with respect to u differentiation with respect to time
THE RESPONSE OF A NON-UNIFORMLY TENSIONED CIRCULAR STRING TO A MOVING LOAD A. P G. G. A Department of Mechanical Engineering, Northeastern University, Boston, Massachusetts 02115, U.S.A. (Received 21 September 1993, and in final form 28 February 1994) A non-uniformly tensioned circular string on a damped elastic foundation is acted upon by a moving concentrated load. This configuration can be viewed as a very simplified representation of a spinning rectangularly orthotropic disk acted upon by a spatially fixed force, or of a stationary orthotropic disk subjected to a circumferentially moving load. The natural frequencies and normal modes of vibration of the string on elastic foundation are determined for a range of values of the tension inhomogeneity. The response of the string to a moving load is then determined. The results indicate that there are many speeds, below the critical speed of the uniformly tensioned string, for which the forced response is large. It is anticipated that the response of this simple model to a moving load can provide insight into the behavior of the mathematically more complicated rectangularly orthotropic circular disk.
1. INTRODUCTION
The determination of the response of spinning/stationary disks to stationary/moving loads has been the subject of numerous investigations due to its relevance to turbine rotor dynamics, circular saw blades and flexible (floppy) disks used in magnetic recording applications. The first analysis of the spinning disk was by Lamb and Southwell [1], who determined the natural frequencies and mode shapes of a solid circular disk. Results are included for a very flexible disk (negligible bending stiffness), a slowly rotating disk (negligible membrane forces), and an approximate solution for the case in which both bending stiffness and membrane forces are included. Benson and Bogy [2] investigated the spinning disk problem using linear membrane theory. They showed that without the effect of bending stiffness in the analysis, the eigenvalue problem for the determination of the natural frequencies of vibration is singular. The point-load problem is then solved with bending stiffness included. A critical speed of the spinning disk can be interpreted as a rotational velocity at which the propagation speed of a transverse wave is equal to the rotational speed of the disk. Hence such a back-travelling wave will appear fixed in space, and a space-fixed load can cause an unbounded response. Alternatively, a critical speed may be viewed as a speed of rotation at which the natural frequency, in space-fixed co-ordinates, becomes zero. Tobias and Arnold [3] investigated the effects of imperfections on the free and forced vibrations of a spinning disk. In a perfect disk, each natural frequency has two modes which are phase shifted through an arbitrary angle; this allows a travelling wave to be formed as an appropriate combination of these two modes [3]. When an imperfection is 415 0022–460X/95/180415 + 12 $08.00/0
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. . .
present, the angular positions of the modes cease to be arbitrary and their frequencies are no longer equal. Thus, the presence of an imperfection distorts the travelling waves, including the back-travelling wave which causes a critical speed of the perfect disk. The stability of a circular plate subjected to peripheral moving loads was investigated by Mote [4] under two different loading conditions. The moving load has either a harmonically varying amplitude or a harmonically varying speed. Iwan and Stahl [5] and Iwan and Moeller [6] also investigated the stability of a flexible disk. In the former paper the disk is stationary and subjected to a circumferentially moving massive load, while in the latter investigation the disk is spinning under the action of a space-fixed massive load. In each case, approximations of the critical speeds are determined and other regions of instability are identified. A simple approximation for the critical speeds of a rotating disk was found by Chonan [7]. Adams [8] determined the critical speeds of rotation of a spinning disk with the effect of an elastic foundation parameter included. The behavior of a rotating circular string with a fixed elastic restraint was investigated by Schajer [9]. That simple model is intended qualitatively to represent the more complicated problem of a spinning disk with a fixed elastic restraint which was later treated by Yano and Kotera [10]. A parametric excitation is one in which the forcing function produces a time varying modification of a system parameter. The response of these systems is characterized by resonances at various frequencies, each of which is related through an integer, or fractional multiple, to the excitation frequency. Campbell [11] recognized the role of parametric vibrations in his study of turbine disks. Further descriptions and analyses of parametrically excited systems can be found in the monographs of Yakubovich and Starzhinskii [12] and Bolotin [13]. A spinning disk, such as a floppy disk used for storing digital information in a personal computer, is rectangularly orthotropic due to the rolling process by which the substrate material (polyethylene terepthalate) is manufactured (Bogy and Talke [14]). The combination of circular geometry with rectangular orthotropy complicates the analysis considerably. A stationary annular disk with rectangular orthotropy was treated by Bianchi, Avalos and Laura [15]. Estimates of the natural frequencies are given, but only for modes which do not have any angular variation. In the present work, the response of a circular string, which has a tension varying continuously along its length and rests on a damped elastic foundation, is investigated. This simple model has qualitative similarities with the orthotropic disk. As a load moves circumferentially with respect to a rectangularly orthotropic disk, the force acts on portions of the disk whose radial and circumferential stiffnesses change continuously. Due to rectangular orthotropy, after 90° of rotation the load travels into a region in which the radial and circumferential bending stiffnesses have interchanged their values from the previous location. Thus the load goes through two complete stiffness cycles for each complete disk/load rotation. In the string problem, the load is subject to a variation in tension stiffness which also goes through two cycles for each revolution of the load. Thus, the orthotropy of the disk corresponds to tension variation in the string. The natural frequencies and normal modes of vibration of the string are determined. The forced response of the string to a moving concentrated force is then found. It is discovered that there are many speeds, below the critical speed of the uniformly tensioned string, for which large amplitude motion results. It is expected that the investigation of the behavior of this simple model can provide insight into the type of phenomena to be encountered in the considerably more complex analysis of a rectangularly orthotropic circular disk.
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2. PROBLEM DESCRIPTION
Now consider a circular string on a damped elastic foundation, subject to a transverse load P moving with constant speed uˆ (Figure 1). The string tension (T ) is inhomogeneous, i.e. T varies with the arc-length co-ordinate sˆ according to T (sˆ ) = T 0 (1 + e cos (2sˆ /Rˆ )),
(1)
where T 0 is the average tension, e defines the degree of tension inhomogeneity, and R is the string radius. The partial differential equation which describes the transverse motion of the string is given by 1 1sˆ
−
0 1 T
1wˆ 1 2wˆ + rˆ = pˆ , 1tˆ 2 1sˆ
0 E sˆ E 2pR .
(2)
In equation (2), wˆ is the transverse string deflection, tˆ is time, rˆ is the mass per unit length, and pˆ is the force per unit length exerted on the string. Acting on the string is a concentrated force P which moves with speed uˆ in the direction of increasing sˆ . In addition, the elastic foundation resists deflections and velocities by exerting a force per unit length on the string which is opposite in direction and proportional to the transverse string deflection and transverse string velocity. Thus pˆ = P d(sˆ − uˆtˆ ) − k wˆ − hˆ
1wˆ , 1tˆ
(3)
where k is the elastic foundation constant, hˆ is the foundation viscous damping coefficient, and d( ) is the Dirac delta function. Furthermore, it can be readily shown that in-plane equilibrium of the string requires radial and circumferential body forces which vary as functions of the arc-length sˆ . It is noted that the presence of these body forces is artificial, and that the use of an inhomogeneous mass density would be more realistic if the only interest was the string problem. However, the ultimate objective of this investigation is a better understanding of the orthotropic disk which has a stiffness variation due to material anisotropy. Thus a stiffness variation, rather than a density variation, is chosen for the string. The partial differential equation (2) is combined with equation (3) and is written in dimensionless form as
0 1
1w 1w 1 2w 1 T + kw + h + 2 = Pd(u − ut), 1u 1t 1t 1u
−
T(u) = T (sˆ )/T 0 = 1 + e cos 2u.
Figure 1. A circular string on a damped elastic foundation subjected to a moving concentrated force.
(4)
418 . . . In equation (4), w, t, k, h, P and u are the dimensionless forms of wˆ , tˆ , k , hˆ , P and uˆ respectively, i.e., w = wˆ /R , u = sˆ /R , t = tˆ uˆ0 /R , k = k R 2/T 0 , h = hˆ R uˆ0 /T 0 , P = P /T 0 , u = uˆ /uˆ0 , uˆ0 = zT 0 /rˆ , (5) where u is the angular co-ordinate and uˆ0 is the wave speed of the homogeneous string in the absence of an elastic foundation. It is noted that the use of a co-ordinate u*, moving with the load P at constant speed (u), will transform equation (4) into
0
1
0
1
1 1w* 1w* 1w* 1 2w* 1 2w* 1 2w* T* + kw* + h −u + 2 − 2u + u2 = Pd(u*), (6) 1u* 1u* 1t 1u* 1t 1t 1u* 1u*2
−
where u* = u − ut, w*(u*, t) = w(u, t), T*(u*, t) = 1 + e cos 2(u* + ut). (7) The periodic tension term T*(u*, t), defined by equation (7) and appearing in equation (6), allows the parametric nature of the excited system to become explicit. The frequency of parametric excitation is seen to be twice the frequency of the moving load. Note that the moving force P does not directly produce the time-varying coefficient of the partial differential equation (4) usually associated with a parametric excitation. However, the presence of the moving load suggests the use of a moving co-ordinate u* from which the tension term exhibits a time variation. Thus, it is the combination of the moving load and the inhomogeneous tension which causes this system to behave as one which is parametrically excited. The undamped natural frequencies and normal modes of free vibration will now be determined. Assume a solution of equation (4) in the form of w(u, t) = W(u)F(t). Standard methods of separation of variables yield [(1 + e cos 2u)W'(u)]' + (v 2 − k)W(u) = 0, 0 E u E 2p, F(t) = C cos (vt − f), (8a, b) where the prime (') denotes differentiation with respect to u, and the square root of the separation constant is equal to the natural frequency (v). The function W(u) is subject to the continuity conditions W(0) = W(2p), W'(0) = W'(2p), (9) which can be satisfied automatically through the use of a Fourier series representation, i.e., a
W (c)(u) = a0 + s an cos nu,
W(u) = W (c)(u) + W (s)(u),
n=1 a
W (s)(u) = s bn sin nu.
(10)
n=1
Substitution of equations (10) into equation (8a), followed by multiplication by cos mu and integration over the interval (0, 2p), results in e ka0 = [v (c)]2a0 , (k + 1)a1 + (−a1 + 3a3 ) = [v (c)]2a1 , 2 (k + 4)a2 + 4ea4 = [v (c)]2a2 , me [(m − 2)am − 2 + (m + 2)am + 2 ] = [v (c)]2am , m = 3, 4, 5, . . . , a. (11) (k + m 2)am + 2
419
A similar set of equations can be obtained for bm by substitution of equations (10) into equation (8a), multiplication by sin mu, and integration over (0, 2p). The result is e (k + 1)b1 + (b1 + 3b3 ) = [v (s)]2b1 , (k + 4)b2 + 4eb4 = [v (s)]2b2 , (12) 2 me (k + m 2)bm + [(m − 2)bm − 2 + (m + 2)bm + 2 ] = [v (s)]2bm , m = 3, 4, 5, . . . , a. 2 In order to obtain equations (11) and (12), the trigonometric relations F Gp , =m − n= = 2, (m $ 0, n $ 0), G2 Gp , m = 1, n = 1, 2p cos 2u cos mu cos nu du = g 2
g
G Gp, G f0,
0
g
(m = 0, n = 2)
or
(m = 2, n = 0),
(13)
otherwise,
2p
cos 2u sin mu cos nu du = 0,
for all m, n,
(14)
0
p F G 2, G 2p cos 2u sin mu sin nu du = g−p , 0 G 2 G f 0,
g
=m − n= = 2,
(m $ 0, n $ 0),
m = 1, n = 1,
(15)
otherwise, have been used. These identities were obtained by first expressing terms such as cos mu cos nu in terms of sin (m + n)u, sin (m − n)u, cos (m + n)u and cos (m − n)u, followed by integration over (0, 2p). Equations (11) and (12) are truncated at m = M and constitute two uncoupled eigenvalue problems which can be solved by standard methods. The nth eigenvalue of the solution of equation (11) is the square of vn(c) and the nth eigenvector is amn (m = 0, 1, 2, . . . , M). Similarly, the nth eigenvalue of the solution of equation (12) is the square of vn(s) and the nth eigenvector is bmn (m = 1, 2, . . . , M). The M + 1 eigenfunctions Wn(c) (u) and the M eigenfunctions Wn(s) (u) are then given by M
Wn(c) (u) = s amn cos mu,
n = 0, 1, 2, . . . , M,
m=0
M
Wn(s) (u) = s bmn sin mu,
n = 1, 2, . . . , M,
(16)
m=1
which represent the natural modes of vibration. It can be readily demonstrated that these eigenfunctions are orthogonal and can be normalized, i.e.,
g g g
2p
Wm(c) (u)Wn(c) (u) du = dmn ,
m = 0, 1, 2, . . . , M,
Wm(s) (u)Wn(s) (u) du = dmn ,
m = 1, 2, . . . , M,
n = 1, 2, . . . , M,
m = 0, 1, 2, . . . , M,
n = 1, 2, . . . , M,
n = 0, 1, 2, . . . , M,
0 2p
0 2p
0
Wm(c) (u)Wn(s) (u) du = 0,
(17)
420
. . .
where dmn is the Kronecker delta. Substitution of equations (16) into equations (17) gives the normalization conditions on amn and on bmn , i.e., M
2a0m a0n + s ajm ajn = dmn /p,
m, n = 0, 1, 2, . . . , M,
j=1
M
s bjm bjn = dmn /p,
m, n = 1, 2, . . . , M.
(18)
j=1
Note that v0(c) = zk and am0 = dm0 /z2p satisfy equations (11) automatically. Thus, the first equation (11) decouples from the others; i.e. a0n = 0, n $ 0. It is also observed that the equations for the even-numbered modes can be separated from those of the odd-numbered modes. It is noted that for homogeneous string tension, equations (11) and (12) are each a diagonal set of equations. Thus vn(c) = vn(s) = zk + n 2, amn = dmn /zp (m $ 0) and bmn = dmn /zp. Therefore, each eigenfunction of the uniformly tensioned string corresponds to a single Fourier component. For non-uniform tension, equations (11) and (12) become each a banded system of equations and so the eigenfunctions contain contributions from many Fourier components. Now consider the forced motion of the string with damping, i.e. equation (4) with h $ 0 and P $ 0. Write the deflection w(u, t) as a modal expansion, M
w(u, t) = a0 (t)W0(c)(u) + s [an (t)Wn(c) (u) + bn (t)Wn(s) (u)].
(19)
n=1
Substitution of equation (19) into equation (4), multiplication by Wm(c) (u), integration over the interval (0, 2p) and use of orthogonality (17), yields the uncoupled equations M
a¨ n + ha˙ n + [vn(c) ]2an = P s amn cos mut,
n = 0, 1, 2, . . . , M.
(20)
m=0
The coefficients an (t) are then Pamn cos (mut − fm(c) ) , (c) 2 2 2 2 m = 0 z{[vn ] − (mu) } + (hmu) M
an (t) = s
(21)
where hmu , 0 E fm(c) E p. (22) [vn(c) ]2 − (mu)2 The bn (t) coefficients are obtained in a similar manner to that used for an (t). Substitution of equation (19) into equation (4), multiplication by Wm(s) (u), integration over the interval (0, 2p) and use of orthogonality yields the uncoupled equations fm(c) = tan−1
M
b n + hb n + [vn(s) ]2bn = P s bmn sin mut,
n = 1, 2, . . . , M.
(23)
m=0
The coefficients bn (t) are given by Pbmn sin (mut − fm(s) ) , (s) 2 2 2 2 m = 0 z{[vn ] − (mu) } + (hmu) M
bn (t) = s
(24)
where fm(s) = tan−1
hmu , [vn(s) ]2 − (mu)2
0 E fm(s) E p.
(25)
421
Figure 2. The normalized square of the natural frequency versus the inhomogeneity parameter (e) for the cosine modes (C) and for the sine modes (S), with k = 0·1 and different values of the mode number n.
Thus the forced response due to the moving load is given by equation (19), with an (t) defined by equation (21) and bn (t) given by equation (24). 3. RESULTS AND DISCUSSION
Consider first the results of the free vibration problem. The ratio of the eigenvalues of the string with inhomogeneous tension to those of the uniformly tensioned string, as a function of the inhomogeneity parameter (e), are shown in Figure 2. Results are given for the squares of the normalized natural frequencies [vn(c) ]2/(k + n 2) (labelled C1 , C2 , and C3 for n = 1, 2, 3 respectively) and for [vn(s) ]2/(k + n 2) (labelled S1 , S2 and S3 ). The C2 frequency increases with e, whereas all of the other frequencies decrease with e. It is noted that C2 corresponds to the mode the diagonal component of which is cos 2u and thus matches the string tension variation. In Figure 3 is shown the variation of amn with e for various values of m and n. Analogous information is given in Figure 4 for bmn . In each graph, the diagonal components of the modes (ann , bnn ) are dominant, especially at low
Figure 3. The mode parameters zpamn versus the inhomogeneity parameter (e) for k = 0·1 and various values of m and n.
422
. . .
Figure 4. The mode parameters zpbmn versus the inhomogeneity parameter (e) for k = 0·1 and various values of m and n.
Figure 5. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for z1 = 0·01, k = 0·1 and various values of the parameter e.
Figure 6. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for z1 = 0·01, e = 0·4 and different values of the foundation stiffness (k).
423
values of e. Furthermore, as was noted earlier, the even numbered components (m) of amn and bmn decouple from the corresponding odd components. Thus, amn = 0, =m − n= = 2j + 1, j = 0, 1, 2, . . . , bmn = 0, =m − n= = 2j + 1, j = 0, 1, 2, . . . . (26) Furthermore, as =m − n= increases, the values of amn and bmn decrease. The results of Figures 3 and 4 are very similar. From equations (11) and (12) it is observed that the equations which determine amn and bmn for a given mode differ only in the coefficients of ea1n and eb1n of the m = 1 equation. Thus, in view of the previous discussion related to even and odd components of amn and bmn , the results for even-numbered m of amn and bmn are identical and the odd-numbered components differ slightly as indicated in Figures 3 and 4. In Figure 5 is shown the maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u). Results are given for the dimensionless damping h = 0·02098 which corresponds to a damping ratio in the first mode of z1 = h/2v1 = 0·01, with e = 0. Curves are for e = 0·0, 0·2 and 0·6, all with k = 0·1. It is noted that the orthotropic floppy disks used in magnetic recording applications [14] can be shown to have degrees of orthotropy as high as the value of 0·6 used here for the string. Also, the first mode damping ratio is taken as the measure of damping rather than the corresponding higher value of h. Figure 6 is a similar graph with e = 0·4 and two values of the elastic foundation constant (k). It is seen that the higher k gives greater sensitivity to speed than does lower k. Note, however, that these values of deflection have been normalized by ws , which is also affected by k. The actual values of the deflection are less for k = 10 than for k = 0·1. Greater values of the foundation stiffness, along with higher speeds, tend to cause the deflection to depend more heavily on the local tension then it otherwise would. Hence the effect of an increase in k is to amplify the effect of larger values of e. The string deflection is shown versus speed with e = 0·4 and k = 0·1 in Figure 7, but for two values of the first mode damping ratio (z1 = 0·001, 0·05). The results of Figures 5–7 show that the overall behavior of the non-uniformly tensioned string is somewhat similar to that of the uniformly tensioned one. However, at many speeds less than unity, the deflection versus speed curves have sharp peaks. Note that these peaks occur at speeds less than the classical critical speed, as well as at u = 1. Furthermore, these peaks increase both in their number of occurrences and in their magnitude as e increases and/or z1 decreases.
Figure 7. The maximum deflection directly under the moving load (max {w(ut, t)}) divided by the maximum static deflection under the load (ws ) versus speed (u), for e = 0·4, k = 0·1 and different values of the first mode damping ratio (z1 ).
424
. . .
It is seen from equations (21) and (24) that resonance occurs at a speed u* if mu* = vn for any values of m and n. Thus, the critical speeds (u*) are given by u* = vn /m 2 zk/m 2 + (n/m)2,
(27)
where vn represents either vn(c) or vn(s) , and the 2 sign becomes an equality when e = 0. For small e, the sensitivity of vn to e is small, as was found from the numerical solution of the free vibration problem. For any given speed u, it is observed from equation (27) that it is possible to find a critical speed arbitrarily close to u by selecting an appropriate pair of values for (m, n). Furthermore, these critical speeds become very small for large m. However, the damping term is proportional to mh, and hence the large m corresponding to low critical speeds results in an increased effect of damping. Therefore, low speeds are less susceptible to large deflections than are higher speeds. With absolutely no damping (h = 0), virtually any speed resonates. The results of Figure 7 indicate this behavior as the z1 = 0·001 curve has much higher peaks than does that for z1 = 0·05. It is noted that the results described here show the features of a parametrically excited system. Thus the resonances given by equation (27) in the context of forced vibrations can also be interpreted as occurring when the parametric excitation frequency (2u) is related through an integer, or fractional multiple, to a natural frequency (vn ) of the system. It is now interesting to consider the string with uniform tension and investigate why it does not have critical speeds other than at u = 1. Write the moving load term as a Fourier series in u, i.e., a
Pd(u − ut) = P/2p + (P/p) s cos mut cos mu,
(28)
m=1
and note that for each forcing frequency (mu) there is a corresponding spatial component (cos mu). Furthermore, the eigenfunctions for the uniformly tensioned string each have only one Fourier term, i.e., 1 1 cos nu, Wn(s) (u) = sin nu. (29) zp zp Thus the modal forcing function is orthogonal to each of the eigenfunctions except for the nth component (i.e., the modal force is zero for m $ n) and resonance occurs only when Wn(c) (u) =
u* = vn /n = zk/n 2 + 1 q 1.
(30)
Thus the fact that the uniformly tensioned string does not have critical speeds for u Q 1 is a consequence of perfectly uniform tension. Alternatively, there is no longer parametric excitation through equation (7) when e = 0. It is further noted that the higher modes (n:a) give critical velocities approaching unity because the influence of the foundation diminishes in comparison with the effect of string tension for high wavenumbers. 4. CONCLUSIONS
The free and forced vibrations of a non-uniformly tensioned circular string on an elastic foundation have been investigated. The variation of the natural frequencies and normal modes of vibration with the degree of tension variation was determined. Each natural mode was found to contain contributions from more than one Fourier component. The forced vibration due to a moving concentrated load was then investigated. It was observed that large deflections occurred at many speeds below the critical speed of a uniformly tensioned string. This phenomenon is a consequence of the fact that each of the modes contain contributions from many Fourier components. An alternative description has also been given in terms of parametric excitations.
425
The effects of the tension inhomogeneity, damping and foundation stiffness were investigated. The magnitudes of the large deflections, as well as the number of such peaks, increased with greater tension non-uniformity and/or decreased damping. Greater foundation stiffness amplified the effect of tension inhomogeneity. This problem is qualitatively similar to the rectangularly orthotropic spinning disk, and is thus expected to provide insight into that particularly complex problem. REFERENCES 1. H. L and R. V. S 1921 Proceedings of the Royal Society of London Series A99, 272–280. The vibrations of a spinning disk. 2. R. C. B and D. B. B 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 636–642. Deflection of a very flexible spinning disk due to a stationary transverse load. 3. S. A. T and R. N. A 1957 Proceedings of the Institute for Mechanical Engineers 171, 669–690. The influence of dynamical imperfection on the vibration of rotating disks. 4. C. D. M J. 1970 Journal of the Franklin Institute 290, 329–333. Stability of circular plates subjected to moving loads. 5. W. D. I and K. J. S 1973 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 445–451. The response of an elastic disk with a moving mass system. 6. W. D. I and T. L. M 1976 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 43, 485–490. The stability of a spinning elastic disk with a transverse load system. 7. S. C 1987 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 54, 967–968. On the critical speed of a rotating circular plate. 8. G. G. A 1987 International Journal of Mechanical Sciences 29, 525–531. Critical speeds for a flexible spinning disk. 9. G. S. S 1984 Journal of Sound and Vibration 92, 11–19. The vibration of a rotating circular string subject to a fixed end restraint. 10. S. Y and T. K 1991 Archive of Applied Mechanics 61, 110–118. Instability of the vibrations of a rotating thin disk due to an additional support. 11. W. C 1924 Transactions of the American Society of Mechanical Engineers 46, 31–160. The protection of stream-turbine disk wheels from axial vibration. 12. V. A. Y and V. M. S 1975 Linear Differential equations with constant coefficients. New York: John Wiley and Sons. 13. V. V. B 1964 The Dynamic Stability of Elastic Systems, San Francisco: Holden-Day. 14. D. B. B and F. E. T 1984 Tribology and Mechanics of Magnetic Storage Systems 1, ASLE SP-16, 115–125. Creep of a rotating orthotropic plate. 15. A. B, D. R. A and P. A. A. L 1985 Journal of Sound and Vibration 99, 140–143. A note on the transverse vibrations of annular circular plates of rectangular orthotropy. APPENDIX: NOMENCLATURE am amn bm bmn F(t) k M P R
sˆ t T(u) T 0 u
mth component of an eigenvector of a cosine mode mth component of the nth eigenvector of a cosine mode mth component of an eigenvector of a sine mode mth component of the nth eigenvector of a sine mode time-varying part of the string deflection dimensionless elastic foundation constant number of terms used in series expansions dimensionless transverse concentrated force string radius arc-length co-ordinate dimensionless time dimensionless inhomogeneous string tension average string tension dimensionless speed of moving load
426 u* uˆ0 w(u, t) w*(u*, t*) W(u) Wn(c) (u) Wn(s) (u) an (t) bn (t) d( ) dmn e h u u* rˆ fm(c) fm(s) v vn(c) vn(s) ˆ ' ·
. . . dimensionless critical speed string wave speed without elastic foundation dimensionless transverse string deflection dimensionless transverse string deflection in moving co-ordinates circumferentially varying part of string deflection nth cosine natural mode nth sine natural mode time-varying coefficients of Wn(c) in the modal expansion time-varying coefficients of Wn(s) in the modal expansion Dirac delta function Kronecker delta string inhomogeneity parameter dimensionless viscous damping coefficient angular co-ordinate moving angular co-ordinate mass per unit length phase angle of the response of the mth component of a cosine mode phase angle of the response of the mth component of a sine mode natural frequency of vibration natural frequency of the nth cosine mode natural frequency of the nth sine mode dimensioned counterpart of a dimensionless variable differentiation with respect to u differentiation with respect to time