Non-linear vibration of a constant-tension string

Journal of Sound and

Vibration (1990) 143(3), 455-460

NON-LINEAR VIBRATION OF A CONSTANT-TENSION STRING H. I’. w. Ckx-rLIEBt Department

of Mathematics,

(Received

University of Queensland,

St. Lucia, Queensland

4067, Australia

1989, and in jinal form 20 February 1990)

14 November

The effect of non-linearity due to the purely geometrical property of curvature on the vibrations of a constant-tension string is investigated. The method of harmonic balance is used to find one- and two-term analytical approximations to the fundamental frequency which give good agreement with numerical results over appropriate ranges of the amplitude parameter.

1.

INTRODUCTION

In their book on waves, Coulson and JefErey [l] (cf. [2]) present the derivation of the non-linear wave equation for transverse vibrations of a flexible string under constant tension. They obtain the non-linear partial differential equation c2 a2u/ax2 = [l +(du/ax)‘]*

d2u/r3t2

(1.1)

for the transverse amplitude u(x, I), where c* = TV/po, with 7. the tension and p. the mass per unit length. While they indicate that the condition for constant tension is not too large a disturbance, inspection shows that they have in fact taken full account of the exact curvature of the string: they neglect squares of small quantities such as 6x in proceeding to the differential equation, but allow the slope au/ax to be not small throughout. Thus equation (1.1) is valid for large-amplitude vibrations of a string stretched under a constant tension which is the equilibrium tension. The assumption is therefore made that there is a tension-maintaining mechanism. In the case of a string of finite vibration length between end-points which are a fixed distance apart, this could be considered to take the form of a frictionless pulley at either end. Mathematically, this corresponds to the fact that in the mass term p. ds in the derivation in reference [ 11, where ds is the arc-length element, p. indeed represents the constant density since the tension is constant, and the full exact expression for ds may be used. In this paper modal solutions of Coulson’s equation (1.1) [2] are investigated for non-linear free vibrations of a constant-tension string. The right side of equation (1.1) is devoid of any material property parameters. Thus this study concerns the effects of non-!inearity on the (lowest) eigenfrequency due to the purely geometrical property of curvature, which is incorporated exactly. The boundary conditions for the string vibrating between fixed end-points length L apart are u(0, t) =o= tPresent (permanent) address: Queensland

Division

of Science

u(L, 1).

and Technology,

(1.2) Griffith

University,

Nathan,

Brisbane,

4111, Australia. 455

0022-460X/90/240455+06

%03.00/O

0

1990 Academic

Press Limited

H. P.

456

W. GOTTLIEB

The Coulson equation is quite different from the various non-linear equations which have been derived for a string with time-varying tension. General derivations may be found in references [3,4]. There, when constitutive relations are used, the Young’s modulus E appears in specific model equations. Early investigations of the non-linear vibration period of such strings were carried out by Kirchoff [S] and Carrier [6,7]. More recent studies, which have incorporated damping terms, include those by Narasimha [8] (for a derivation of the equations), Anand [9] and Gough [lo]. Forced motion has also been investigated [ll-13 and references therein]. In these papers the weakly non-linear equations of motion [8] are solved. By contrast, equation (1.1) includes all the non-linearities relevant to the present constant-tension problem. When the tension is not constant, the cited references show that those motions may in general be non-planar, especially in the case of forced response. Equation (1.1) deals with transverse planar autonomous motion. It is emphasized that the non-linearities in equation (1.1) need not be small; moreover, there are no small numerical parameters in this equation. A major contrast between the present work and the previous results based on the variable-tension string is that for the latter the frequency in general increases with amplitude (cf., e.g., reference [6]), whereas for the constant-tension case here the frequency is found to decrease with amplitude. This is not unexpected, since the vibrating length (with constant density) is increased by the curvature. As an aside, mention may also be made here of a situation involving constant tension for a non-linear wave as described by Morse and Ingard [ 14, p. 8591 but where the tension involved is the new tension, greater than the equilibrium tension. Then a distortionless waveform can propagate on the (unbounded) string with an appropriate wave speed. In section 2, the governing partial differential equation is reduced to an ordinary differential equation in the space or time variable through averaging techniques. Oneand two-term approximations to the fundamental frequency are obtained in section 3 by using the method of harmonic balance, and are compared with the exact numerical integrations of the temporal equation for a range of amplitude parameters in section 4. The second approximation gives remarkable agreement, even for amplitudes which are not small. 2. REDUCTION TO AN ORDINARY DIFFERENTIAL EQUATION In order to proceed with the analysis of the partial differential equation (l.l), reduction to an ordinary differential equation is effected, by utilizing averaging techniques. These may be over either the time variable (Ritz method) or the space variable (Galerkin method) to yield alternative forms more amenable to solution. (Thus the situation here is quite different from the “standard”, variable-tension, weakly non-linear case [ 11, 15, p. 11 I], where a quadratic term is subject to a definite spatial integral, so the separation of variables approach is then applicable.) 2.1.

ASSUMED

The

HARMONIC

FREQUENCY

amplitude here is written in the harmonic form u(x, t) = cos wtU(x).

(2.1)

Then averaging over one harmonic cycle, i.e. multiplying equation (1 .l) through by cos wt and integrating with respect to wt from 0 to 2~ (a Ritz procedure), yields an ordinary non-linear differential equation for the x-dependent function U(x), U”= -(o/c)‘U[1+(3/2)(

U’)2+(5/8)(

U’)“],

OSXSL,

(2.2a)

NON-LINEAR

VIBRATION

OF

A STRING

457

subject to U(0) = 0 = U(L). Since U”= d[f(U’)2]/dU,

a first integral of equations (2.2) is

(dU/dx)‘=(2/5) 2.2.

ASSUMED

LINEAR

(2.2b)

tan [constant-a(w/c)*U*]-(6/5).

(2.3)

MODE

The amplitude here is written under the assumption that the modal shape may be approximated by the fundamental shape for the linearized wave equation obtained from equation (1.1) subject to the boundary conditions (1.2). This provides the Ansatz

u(x, r)=sin

(7rx/L)T(t).

(2.4)

Substitution of expression (2.4) into equation (1.1) and averaging over the length of the string, i.e., multiplying through by the assumed mode shape and integrating with respect to x from 0 to L (a Galerkin procedure), yields the ordinary non-linear differential equation for the time-dependent function T(t): ~[l+5(~/L)2T*+b(7T/L)4T4]=-(~c/L)ZT. Since f = d($F*)/dT,

(2.5)

a first integral may be written down: (dT/dt)2=constant-4c2arctan[~(~/L)2T2+l].

(2.6)

However (as with equation (2.3)), this is not particularly helpful from a solution point of view, and the direct equation (2.5) will be solved approximately in a subsequent section. 3. SOLUTIONS FOR THE FUNDAMENTAL FREQUENCY The ordinary differential equations (2.2) or (2.5) are non-linear, and must be solved by approximate methods. Since the coefficients of the non-linear terms do not involve small parameters, the usual perturbation techniques [ 161 are inappropriate. Mickens [ 171 has indicated that the only generally applicable technique in such situations is the method of harmonic balance. which is now utilized. 3.1. SOLUTION OF THE A first approximation obtained by writing

SPATIAL

EQUATION

to the solution of equation (2.2) for the fundamental U(x) = A sin (TX/~),

mode is (3.1)

where A is the amplitude. Substitution of expression (3.1) into equation (2.2) and equating the coefficients of sin ( ~TX/L) (or, equivalently, using a Galerkin procedure), after use of trigonometric identities to re-express all quantities in terms of sin (nrx/ L), n = 1,3, 5, yields [wL/(7~)]~=1/[1+&rA/L)‘+&rA/L)~].

(3.2)

In contrast to conventional fixed-ends non-linear strings, the frequency therefore decreases with increase in amplitude, as discussed in the introduction. EQUATION 3.2. SOLUTION OF THE TEMPORAL Equation (2.5) may be written in the form

9[l+;s’+$s‘+]

= --Lys,

(3.3)

a = (7rc/ L)‘.

(3.4)

where S=(?r/L)T,

458

H. P. W. GOT-I-LIEB

3.2.1. First approximation The function S is written in the form S(t) = a cos ut,

(3Sa)

where A = (L/r)a

is the amplitude of balance [17], which neglecting the terms must, since equation (3.5).

(3. 5b)

T in equation

(2.4). Then invocation of the method of harmonic involves equating the coefficients of cos ot in equation (3.3) and in cos 3ot, cos 5wt, results in the expression (3.2) for o again, as it (2.1) with equation (3.1) is the same as equation (2.4) with equation

3.2.2. Second approximation to the fundamental frequency A second approximation to the frequency W, and a check on the reasonableness method of harmonic balance in this case (cf. [17]), is now achieved by writing S(t) = a cos ot + b cos 3wt.

of the (3.6)

After extensive use of trigonometric identities and application of the method of harmonic balance to retain only terms involving cos wt and cos 3ot, two simultaneous equations for w in terms of the amplitudes a and b are obtained. The assumption (to be checked subsequently) is now made that the ratio b/a is small, so coefficient terms involving higher powers b*, b3,. . . . , etc., can be neglected. There results ~*/a = 1/[1+ia2+&a4+yab+$a3b],

(3.7)

(cf. equation (3.2) when b = 0), with b/a = -(a2/128)(16+5a2)/{8+(a2/32)(76+

17a*)}.

(3.8)

This actually satisfies 0 < (-b/a)

< 5168 = O-07353

(3.9)

for all values of a, attaining zero as a + 0 and tending to 5/68 as a + 00. By equation (3.9), the ratio b/a for this solution (3.6) is thus indeed small, whatever a is. Thus the first approximation (3.2) with equations (3.5) was a good one, with corrections as given here by equations (3.7) and (3.8) with equation (3.6). This is confirmed in the next section in which numerical solution of equation (3.3) is described. 4. NUMERICAL DETERMINATION OF THE FUNDAMENTAL PERIOD Equation (3.3), corresponding in dimensionless form as

to the assumed fundamental

d*S/dr*=

-S/[1+$S2+$4],

mode (2.4), may be written (4.la)

where 7 = Iret/ L. This ordinary second-order non-linear by using a fourth order Runge-Kutta amplitude constants. The results of obtained by the method of harmonic

(4.lb)

differential equation may be integrated numerically technique to determine the period, for a range of this are now compared with the analytical forms balance in section 3.2.

NON-LINEAR

VIBRATION

The first approximation (3Sa), equations (3.2) and (4.lb))

OF

459

A STRING

S = a cos wt yields the dimensionless

period (from

P ‘“‘=2P[l+~a*+~a4]“2.

The second approximation

(4.2)

(3.6), S = a cos or+ b cos 3w? yields (from equation (3.7))

P ‘“.b)=2~[l+~a2+~u4+~ub+~

65 a3bl”‘,

(4.3)

with (b/u) given by equation (3.8). These were evaluated for a = 0.1, 0*2,0.5, 1, 2, 5 and 10 in each case. Equation (4.la) was integrated numerically with initial conditions S(0) = a, S’(0) = 0 for comparison with equation (4.2), and initial conditions S(0) = a + b and S’(0) = 0 for comparison with equation (4.3). Results are displayed in Table 1. The first approximation (4.2) gives good agreement for small amplitude, and is reasonable even for a = 1. The second approximation, equations (4.3) and (3.8), gives seven-figure accuracy for small a = 0.1, 0.2, and gives three-figure agreement right up to a = 5; even as high as a = 10 there is two-figure agreement. The first approximation always overestimates the exact value. The numerical results show that the second approximation is not uniform, but performs remarkably well for the wide range of amplitude parameters investigated. 1

TABLE

Fundamental dimensionless periods obtained by method of harmonic balance for lowest harmonic coeficient u, compared with exact petiods p obtained by numerical integration of equation (4.la) (a +O: p-+27r=6.283185) a=O.l (a)

a =0.2

a=2

a=5

a=10

0.1

0.2

0.5

1.0

6.294980

6.33052

6.58576

7.57411

12.1673

48.345

179.9

6.294917

6.33047

6.58379

7.54147

11.7148

41.918

148.8

(b) 2nd approximation-p’“.” S(O) 0.099984373

P(a.hI P

a=1

S(0) = a, S’(0) = 0

1st approximation-p’“‘(4.2):

S(O) P10, P

a =0.5

(4.3:

2.0

5.0

10.0

S(0) = a + b, S’(O) = 0

0.19987494

0.49804732

0.98495702

6.294973

6.330415

6.581367

7.499928

Il.1700

37.145

132.0

6.294973

6.330415

6.581404

7.502130

11.2321

37.144

128.9

1.9134615

4.6552500

9.2746941

5. DISCUSSION

Mickens [ 171 has emphasized the value of the method of harmonic balance for obtaining approximate analytical solutions of non-linear differential equations when the nonlinearities are not small. Such is the case with the dimensionless temporal equation (4.la) corresponding to the fundamental mode of the constant-tension string described by equation (1.1). While the first approximation (4.2) already gave good results for initial amplitude a up to about 1, the two-term approximation (4.3) was found to give excellent agreement even for larger a. Thus this problem provides another vindication of the harmonic balance method. Because of the nature of the tension-maintaining mechanism, as discussed in the introduction, a further refinement to this problem could be the inclusion of the effect of axial displacements. This will be the subject of future work.

460

H. P. W. GO-ITLlEB ACKNOWLEDGMENTS

I should like to express my thanks to the staff of the Department the University of Queensland, where this work was carried out during Program visit.

of Mathematics at an Outside Studies

REFERENCES 1. C. A. COULSON and A. JEFFREY 1977 Waves, 2nd edition. London: Longman. See para. 17. 2. C. A. COULSON 1955 Waves, 7th edition. Edinburgh: Oliver and Boyd. See para. 13. 3. H. F. WEINBERGER 1965 A First Course in Partial Dtflerential Equations with Complex Variables and Transform Methods. Lexington: Xerox College Publishers. See chapter 1. 4. S. S. ANTMAN 1980 American Mathematical Monthly 87, 359-370. The equations for large vibrations of strings. 5. G. KIRCHHOFF 1883 Vorlesungen iiber Mathematische Physik. Leipzig: B. G. Teubner, p. 445. 6. G. F. CARRIER 1945 QuarterZy of Applied Mathematics 3, 157-165. On the non-linear vibration problem of the elastic string. 7. G. F. CARRIER 1949 Quarterly of Applied Mathematics 7,97- 101. A note on the vibrating string. 8. R. NARASIMA 1968 Journal of Sound and Vibration 8, 134-146. Non-linear vibration of an elastic string. 9. G. V. ANAND 1969 Journal of the Acoustical Society of America 45, 1089-1096. Large-amplitude damped free vibration of a stretched string. 10. C. GOUGH 1984 Journal of the Acoustical Society of America 75, 1770-1776. The nonlinear free vibration of a damped elastic string. 11. D. W. OPLINGER 1960 Journal of the Acoustical Society of America 32, 1529-1538. Frequency response of a nonlinear stretched string. 12. G. S. SRINIVASA MURTHY and B. S. RAMAKRISHNA 1965 Journal of the Acoustical Society of America 38, 461-471. Nonlinear character of resonance in stretched strings. 13. J. MILES 1984 Journal of the Acoustical Society of America 75, 1505-1510. Resonant, nonplanar motion of a stretched string. 14. P. M. MORSE and K. U. IMGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. 15. W. F. AMES 1965 Nonlinear Partial Diferential Equations in Engineering. New York: Academic Press. 16. R. E. MICKENS 1981 An Introduction to Nonlinear Oscillations. Cambridge: Cambridge University Press. 17. R. E. MICKENS 1984 Journal of Sound and Vibration 94, 456-460. Comments on the method of harmonic balance.