irq 7T7, exp ( - fpt) { [1/2 - f [ j ( l - r 2 ) ' / ! + r] 1

wo ( i - r-r X exp [i2irg - ipt( 1 - f 2 ) ' A - 47rgf(l - W^jE,

+ [1/2 - j f [ - j ( i - r-y/! + m exP ^ + iPi - n , A

- 47T?f(i - r-y/!iEi*

- [1/2 - ( - 7 ( 1 - nV'- + f ] exp [-i2w7 COnVA

(12)

(13)

For most purposes the envelope of the response function rather than the particular time-dependent response is desired. The following upper bound on the envelope, which greatly simplifies the results, may be readily found after reducing equation (11) to a trigonometric form.

V_ yo

7r , { ( N + N ) e x p [ - 2 t r s f l w / p + d - f 2 ) ' / ; ] ] 2(1 _ $)/.

+ + \I2\) exp [-27rryf[co/p - (1 - r 2 ) ' / 2 ] ]

+ 2f(2?i2 + L2 ) l /*[ f + (1 - $!)'/]

X exp [ - a ^ f l w / p + (1 - f2)1 /2]]

+ 2f(Z?22 + 722) 'A[f + (1 - f 2 ) 'A]

X exp [ - 2 t t f f f [ / p - (1 - f2) ' /=]]} (14)

Results Equation (14) is the principal result of this study. This ex-

pression may be directly compared with the exact results of Lewis [1] for the case of zero damping where the two forcing functions are identical. The response envelopes obtained from (14) with f = 0 are everywhere indistinguishable from Lewis' curves indi-cating that this upper bound is quite close to the exact response envelope if the damping is small. In both analyses, after passing through resonance the response envelope oscillates with decaying amplitude around an amplification factor of 7rtf''!. Note that increasing acceleration rates result in a smaller value for q and that q = co represents a steady forced vibration.

The influence of damping on the response envelope is evident from Fig. 1 where the acceleration rates have been held constant. The attenuation of peak response at the resonant point is ap-proximately proportional to the attenuation achieved with the same damping ratio when the excitation is stationary. In gen-eral, the effect of an accelerating forcing function is to increase the resonant frequency and reduce the amplitude of the response envelope in comparison with equivalent .stationary forced vibra-tions.

References 1 F. M. Lewis, "Vibration During Acceleration Through a Critical

Speed," TRANS. A S M E , vol. 54, A P M , 1932, pp. 253-261. 2 C. W. Martz, "Tables of the Complex Fresnel Integrals," Na-

tional Aeronautics and Space Administration Report SP-3010, Wash-ington, D. C 1964.

3 W. J. Stronge, "Vibrations of a Mechanical System Traveling Over a Stationary Wave Form," Master's Thesis, University of California at Los Angeles, Los Angeles, Calif., 1964, p. 40.

BRIEF NOTES

On the N o n l i n e a r O s c i l l a t i o n of an Axial ly M o v i n g S t r i n g

C. D. M O T E , JR.1

THE linear vibration of axially moving strings has been exten-sively studied; for example, see [14],2 In all of these investiga-tions the writers correct!}' state or imply that the linear analysis is applicable to small-amplitude transverse vibrations. Just how small is small for an axially moving string? Since this question has not been investigated, one usually associates linearity condi-tions of the axially moving string with the familiar linearity conditions of the stationary string. The purpose of this Note is to show that this association is incorrect. Where a tension-amplitude relationship characterizes the linear oscillation of a stationary string, a tension-amplitude-axial velocity relationship characterizes the linear oscillation of the axially moving string.

The analysis consists of the numerical solution of the funda-mental period of oscillation of stationary and axially moving strings. The maximum amplitude of oscillation is specified small O/2 percent of the length) so that an amplitude-dependent ten-sion is the only nonlinearity considered in each case. The string fundamental period is determined for various combinations of initial tension and axial velocity. The strings are homogeneous with fixed ends.

Formulation The transverse equation of motion is easily determined from

variational principles. String kinetic and potential energies are

T = bpA f [(, + cx)2 + (it, + c)2]cfe ( la ) Jo

Nomenclature A = string cross-sectional area c = string constant axial velocity E = elastic modulus 1 = string free length between supports

P = string initial tension t = time u = axial displacement with respect to coordinates translating

at velocity c v = transverse displacement with respect to fixed coordinates

tv = nondimensional transverse displacement x = fixed axial coordinate a = nondimensional initial tension ft = nondimensional axial velocity e = strain 77 = nondimensional time = nondimensional axial coordinate r = nondimensional period

-1 V = | Pedx + iEA I e2dx ( lb) f Pedx + iEA f

Jo Jo

where v and u are transverse and axial displacements, v, = bv/dt, c is the string axial velocity, and the remaining terms are as expected. The action then becomes

2cut + c2) /( nil fi I Ldt = I (ip.4{t>,2 + 2cvxv, + c V + 7(,2 +

Jt 1 J li Jo

- P {[(1 + m,)s + ^ 2 ] ' / 2 - 1} - iEA {[1 + ux]* + *' + 1 - 2 [ ( l + i Lxy + vX'*})dxdt (2)

For ux

BRIEF NOTES

In terms of the following nondimensional numbers: 2.4

v " l

= ( AE) < 1 V = K I <

0 = P \V: p i ,

the equation of motion and the boundary conditions are

w + 20w(l - (1 - 13* + ) wi( = 0 (4)

and w(0, rj) = !(1, j;) = 0. The equation of motion (4) is seen to be always hyperbolic for velocities less than critical (/S < 1). From a practical standpoint, the factor (3 will always be less than 1. Numerical solutions of the quasi-linear equation (4) can best be determined by the method of characteristics. A detailed dis-cussion of the method of characteristics as applied to this problem will not be given because no special techniques are required (for a clear discussion of the procedure see [6, 7]). Differences be-tween the stationary and moving string analyses, both linear and nonlinear, are easily seen in the characteristics in the ij-tj plane. In terms of standard notation, the slopes of the characteristics are given by

1 / =

0), is zero and the initial displacement corresponds to the particular f3 linear fundamental mode with wmax = 0.005; (b) the characteristics are constructed; (c) the string displacement and transverse velocity are determined on -q = constant lines in the

plane. Initially, the determination of one period was based upon observing when the string returned to its original configura-tion. Since the string never exactly returns to its initial con-figuration, and since the string spends a long time in close proximity of the initial configuration, this method of period de-termination was abandoned. Instead, the period was determined by observing when the RMS transverse velocit3r was minimum and, at the same time, checking the final configuration. Modern computing facilities are requisite for these calculations.

The linear nondimensional period, obtained by a periodic solu-tion substitution w(, 77) = IF exp [i(wr] + ?i)] in the linearized equation (4), is

7 = dV* (6) The linear fundamental periods can be compared with the non-

linear periods in Fig. 1. The reduced period observed in the figure

o o

<

< o z 3

(5)

INITIAL TENSION

Fig. 1 Fundamental period of oscil lat ion for w m a x = 0.005 as a funct ion of nondimensional ini t ial tension a . (Period predicted by l inear theory is indicated by a tick on the ordinate at the part icular sir ing veloci ty /3.)

is, of course, expected, but the figure also illustrates the increase.! significance of the nonlinear correction term with increasing string velocity /3. The maximum string axial velocity examined was (3 = 0.4 since numerical error increases with f3. The velocity 13 = 0.4 is sufficiently large to show the importance of nonlinear terms in the small-amplitude, axially moving string vibration.

Conclusions The relationship between smallness of displacement and linear-

ity for the stationary string cannot be extrapolated to the axially moving string. As the string axial velocity increases, the effect of tension variation during oscillation becomes increasingly sig-nificant. As the velocity approaches critical, i.e., as (3 * 1, linear oscillations are not meaningful even for small amplitudes.

The foregoing observations indicate that linear analyses of axially moving strings are of limited value, limited to the high-tension, low-velocity regime. Note that the high-tension, low-velocit}' linear natural frequency analysis has been verified ex-perimentally for a neighboring problem [5], but the linear analysis in general has received no experimental attention.

Acknowledgment The author sincerely thanks the National Science Foundation

for partially funding this study and the Carnegie Institute of Technology Computation Center for the use of computation facilities. The author also thanks Miss E. J. Stiles for aiding in the preparation of the manuscript.

References 1 Rudolf Skutch, "Ueber die Bewegung eines gespannten Fadens,

welcher gezwungen ist, durch, zwei feste Punkte, mit einer constanten Geschwindigkeit zu gehen, und Zwischen denselben in Transversal-Schwingungen von gerlinger Amplitude versetzt wird," Annalen der Physik iind Chemie, vol. 61, 1897, pp. 190-195.

2 R. A. Sack, "Transverse Oscillations in Traveling Strings," British Journal o] Applied Physics, vol. 5, 1954, pp. 224-226.

3 F. R. Archibald and A. G. Emslie, "The Vibration of a String Having a Uniform Motion Along Its Length," TRANS. ASME, vol. 80, 1958, pp. 347-34S.

4 R. D. Swope and W. F. Ames, "Vibration of a Moving Thread-line," The Journal of the Franklin Institute, vol. 275, 1963, pp. 36-55.

5 C. D. Mote, Jr., and S. Naguleswaran, "Theoretical and Ex-perimental Band Saw Vibrations," Journal of Engineering for Indus-try. TRANS. ASME, vol. 88, Series B, 1966, pp. 163-168.

6 W. Fliigge, Handbook of Engineering Mechanics, McGraw-Hill Book Company, Inc., New York, N. Y., 1962, chapter 11, pp. 8-10.

7 Modern Computing Methods, Her Majesty's Stationary Office, London, England, 1961, Chapter 11, pp. 101-111.

4 6 4 / J U N E 1 9 6 6 Transactions of the ASiVIE

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