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Response of a Nonlinear String to Random Loading1 discussion the barreling of the specimen were not large enough to be noticea- Since ble. The authors believe the effects of lateral inertia are small and 77" ^ DL much less serious than the end conditions. Referring again to ™ = g = Wo Fig. 4, the 1.5, 4.5, 6.5, and 7.5 fringes are nearly uniformly dis- tributed over the width of the specimen. The other fringes are and not as well distributed, but it is believed that the conditions at DL the ends of the specimen produced the irregularities rather than 7U72 = VV^ tr,2 = =-, lateral inertia effects. f~1 3ft 7'„ + AT) If the material were perfectly elastic and if one neglected lateral one has inertia effects, the fringes would build up as a consequence of wave propagation and reflection. Thus a difference in the fringe t/7W = (i + at/To)-> = (i + ffff./v)-1 (5) order over the length of the specimen woidd always exist. Inspec- tion of Fig. 4 shows this wave-propagation effect for the first which is Caughey's result for aai.^N small, which it must be for 6 frames as the one-half fringe propagates down the specimen the analysis to be consistent. It is obvious that the new linear Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/27/2/369/5443489/370_1.pdf by guest on 27 September 2021 and reflects from the right-hand pendulum. However, from system with tension To + AT will satisfy equipartition. frame 7 on there is little evidence of wave propagation and the The work just referred to does not use equation (1) but rather fringe order at both ends of the specimen is equal. Any local more general equations similar to those developed by Carrier.4 differences in the fringe order at the ends of the specimen may be Although the assumption is made that EA 7'0, the result does attributed to frictional effects. not reduce to (5) but rather It is the authors belief that lateral inertia and wave propaga- tion effects will not influence the distribution of the stress in a o-.Vo-.-.o2 = (1 + 2Q,AT/To)~i (6) specimen of a low-modulus material to any appreciable extent. where Q, is the quality factor for the ith mode. Since for a large Frictional constraint is the greatest factor which introduces ex- number of systems Q, 1, the reduction in displacement is perimental error and it should be the first concern of the inves- stronger than (5) would indicate. This result, when traced tigator. back, occurs by an interaction between longitudinal and trans- verse motions of the string. Author's Closure Response of a Nonlinear String The author wishes to thank Dr. Lyon for his interesting com- 1 ments on the paper. The author would like to refute the claim to Random Loading that equation (1) of the above paper is invalid. To this end, con- sider Carrier's equations for the string,4 modified to include damp- R. H. LYON.2 In his paper, Caughey has presented calcula- ing and external forces. tions for the mean square displacement of a string governed by the equation of motion (his notation) f(x, I) = pu„ + ru, - bx[T sin 0] (1) 2 L 2 2 (d u a bu\ T AE C ( bu\ "1 b u o = pV„ + vV, - [T cos 6] (2) where sin 6 = —— (3) +/(*,<) (i) 1 + e where/(x, I) is governed by the correlation function and the local strain e is given by fix, hMn, fa) = 4D8(x - i?)(sin cocr)/r. (2) e = ((1 + Vx)2 + ut2)'/> - 1 (4) It is the purpose of this discussion to show that his results may T = To + EAe (5) be obtained by elementary considerations and to suggest that recent work of the present author3 has discounted the validity of If the following assumptions are made: using equation (1) in this problem. (а) Vx = 0{u2)<T 1 The mean square displacement of the ilh linear mode may be (б) ToCEA easily found to be [see Caughey's equation (21)] (c) wc < n, o-.-.o2 = 2DL/i2w2f3To. (3, where we is the highest frequency excited in transverse vibration The effect of the total motion is merely to increase the average and fii is the lowest eigen value for longitudinal vibrations. tension by an amount If assumption (c) is satisfied, the time derivative terms in (2) may be neglected. Hence, — EAir2 EAD , ^ = g = ^ W bx [T cos 6] = 0 (6) /. T cos 6 = g(t) (7) 1 By T. K. Caughey, published in the September, 1959, issue of the where g{t) is independent of x. JOURNAL OF APPLIED MECHANICS, vol. 26, TRANS. ASME, series E, Using assumption (a) and equations (3) and (4), vol. 81, pp. 341-344. 2 Associate Professor of Electrical Engineering, University of Minnesota, Minneapolis, Minn. Currently on leave as National T ^ To + EA (f, + j uA (8) Science Foundation Postdoctoral Fellow at the Department of Mathematics, The University, Manchester, England. 3 R. H. Lyon, "The Random Vibration of Elastic Strings— 4 G. F. Carrier, "On the Nonlinear Vibration Problem of the Elastic Theoretical," WADC Technical Report 58-570, September, 1958. String," Quarterly of Applied Mathematics, vol. 3, 1945, pp. 157-165. 370 / j u n e 19 6 0 Transactions of the ASME Copyright © 1960 by ASME discussion 2 The perturbation scheme used by Lyon perturbs the trans- cos 6 1 - (9) verse and longitudinal displacements; however, the frequency is left unperturbed. This is highly unusual in nonlinear problems T cos 6 ^ 7'0 + AVI Fx + (B/l - 7T0)'Ai^2 = g(t) (10) unless one retains all the secular terms as Carrier did.1 For further discussion of this point, the reader is referred to J. J. Now the boundary conditions require that the axial displacement Stokes.6 V vanish at both ends of the string. It would therefore appear that the methods used by Lyon are somewhat questionable. VjIX = 0 (11) Thus, integrating (10), EA - To g(t) ^ T0 + Mx (12) 21 Nonlinear Creep Deformations of r.* 1 Using assumption (6), Columns of Rectangular Cross Section Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/27/2/369/5443489/370_1.pdf by guest on 27 September 2021 EA rl H. G. McCOMB, JR.2 The contribution of this paper is 'AO ~ T, + hlx (13) ~2l Jo " the development of easily calculated upper and lower bounds for the creep collapse time of rectangular section columns. In Substituting (7) into (1), recent work at the Langley Research Center, NASA [l],3 the variational theorem referred to by the authors has been utilized f(x, I) = pu„ + ru, - g(l)dx (tan 8) (14) with direct methods to make some very accurate calculations for Using assumption (a) and equation (3), this problem. In [1) the power creep law was used as in the pres- ent paper. In addition, the stress distribution through the thick- i<i tan $ (15) ness was allowed to take on a nonlinear shape. The results of 1 + Vx [1 ], therefore, represent a refinement of the previous work men- tioned by the authors [2] which utilized the variational theorem Hence, equation (14) becomes but in conjunction with a linear stress distribution. An interest- ing comparison between the results of [1] and the authors' results f(x, t) = pit,, (16) is shown in Fig. 1 of this discussion. This figure is like Fig. 4 of + "" - + I" Jo the paper where TCR is plotted against zo/h, the initial deflection Equation (16) is identical to equation (1) of Caughey.1 This re- parameter. Only the case k = 3 is shown. The dashed and solid sult was first proved by Carrier5 in a somewhat different manner. lines are the upper and lower bounds for TCR from the authors' Having established the validity of equation (1) of the paper, paper. The dash-dot lines are from [1]; the upper one is for it is necessary to explain the differences between the results ob- oaltsE = 0.2 and the lower for Co/crE = 0.8. The results of [1] tained in Caughey1 and those obtained by Lyon.3 If equation lie between the authors' bounds, and therefore are consistent with (2.11) of Lyon3 is examined in the light of the assumptions (a), them. The figure gives some idea of the accuracy of the authors' (b), and (c) above, it may easily be shown that: conjecture that for oa/oK < 0.2 the upper bound should be a good approximation to the collapse time and for tr0/(rB > 0.8 the lower bound should be a good approximation. Fx(2) = -- + b(t) (17) where UPPER BOUND ON TCR l LOWER BOUND ON T,C R L C u 2«W x (18) 6W " 21¥ Jo VARIATIONAL If equations (17) and (18) are substituted into equation (2.11) of Lyon,3 there results the equation: 'CR Pu »<3) + n«„<3' = J^ !(12<i)rfxj <i) (19) 2Z This equation is exactly what one would expect by applying Lyon's perturbation scheme to equation (16). Hence, the dif- ferences between the two theories cannot be explained by the coupling between the transverse and longitudinal motions as sug- gested by Lyon. The difference must therefore be due to the per- O I I I 11 i i I I I I I 11 turbation method used by Lyon. .005 .01 .02 05 .10 .20 Examination of the perturbation scheme used by Lj'on reveals the following disturbing facts: Fig. 1 Comparison between variational results and bounds on Tc 1 Examination of equation (2.10) of Lyon3 reveals that the quantities are not all of the same magnitude.
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