COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCES OF A BAR H. Jeong, D. Beshers To cite this version: H. Jeong, D. Beshers. COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDI- NAL RESONANCES OF A BAR. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-317-C8-322. 10.1051/jphyscol:1987846. jpa-00227150 HAL Id: jpa-00227150 https://hal.archives-ouvertes.fr/jpa-00227150 Submitted on 1 Jan 1987 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C8, supplément au n°12, Tome 48, décembre 1987 C8-317 COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCE OF A BAR H.S. JEONG and D.N. BESHERS Henry Krumb School of Mines, Columbia University, New York, NY 10027, U.S.A. Résumé - Nous avons ajoute un terme quadratique peu important I l'équation du modèle du solide anélastique standard et nous avons étudie son effet dans l'approximation du premier ordre. La résonance fondamen­ tale ne change pas, en première approximation, mais uneharmonique de deuxième ordre apparaît. En fait, la. deuxième harmonique résulte de deux ondes spatiales à même fréquence. Comme l'amortissement linéaire tend vers zéro, la forme mathématique de la solution change, les deux ondes dégénèrent en une seule qui n'est pas exactement sinusoïdale. Abstract - The addition of a small quadratic term to the standard anelas- tic solid model has been investigated to first order approximation. The fundamental resonance does not change, to first order, but a second har­ monic wave appears. The second harmonic actually consists of two spatial waves with one frequency. As the linear damping goes to zero, the mathe­ matical form of the solution changes, the two waves degenerating into one that is not entirely sinusoidal. I - INTRODUCTION The longitudinal resonance of a bar has often been exploited for the measurement of internal friction. The analysis of the experiments has usually been con­ ducted in an elementary fashion, and often that suffices. However, when the internal friction is nonlinear, that is to say dependent on the amplitude of oscillation, then there is a need for more careful consideration of the problem, a need universally neglected for lack of a suitable theory. A first step towards such a theory, an approximate treatment for the case of a small nonli­ near effect co-existing with a linear damping, is presented in this paper. II - THE MODEL Because nonlinear internal friction usually occurs in materials that also exhi­ bit linear damping, it is important to treat the linear and nonlinear aspects together. The basic model for linear damping is the standard anelastic solid (SAS) /1/. The longitudinal resonance of a bar made from a SAS has been treated already /2/. We consider here a bar of homogeneous material obeying a consti^ tutive equation based on the standard anelastic solid, but augmented by one nonlinear term: * ~ 2 a + ba - ae + ce + de , (1 ) where a is the longitudinal stress, e the corresponding strain, and a, b, c, d are material constants; in the usual notation, a = 1/(JU+6J), c = t a> and b - J c. The coefficient d is assumed subsequently to be small. For a long, thin bar the equations of stress equilibrium reduce to Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987846 JOURNAL DE PHYSIQUE where p denotes the density of the material and u = u(x,t) the displacement of a cross-section of the bar. Strain and displacement are related by We eliminate stress in favor of u by differentiating (1) with respect to x, then using (2), and replacing E with u by (3), obtaining the equation of motion to be solved: putt + bputtt = au + cu + 2duxuxx. (4) XX xxt The bar is considered to be driven at one end, x = 0, by a periodic stress of amplitude 20 and circular frequency w and to be free at the other end, x = L. The boundaryOconditions are then, choosing the phase appropriately, o(~,t)= ooeiWt + 0 e-iwt atx=O (5) 0 We seek a steady-state solution of this problem, expecting to find resonant modes, and to examine the behavior in the vicinity of resonance. By steady- state we mean that every term in the solution must repeat in time with the fun- damental frequency; that is we assume that harmonics of the fundamental will occur, but not subharmonics. This assumption is consistent with experimental results in this laboratory. The nonlinear term necessitates care in the use of complex exponential func- tions, which are otherwise the simplest representation with which to work. The real part of any complex number Z is given by but R, as an operator, does not commute with multiplication. We shall therefore in effect require that R operate first by writing the trial solution in the form m u(x,t) = z [un(x)einwt + u *(x) e-inwt]. (8) n=O In this way, all our expressions are real, although sometimes we will not bother to write out the conjugate part. I11 - THE SOLUTION Substitution of (8) in (4) gives a very lengthy equation. Because exp(inwt) and exp(-inwt) are linearly independent functions, we require that the coefficient of each be zero, which has the effect of converting our partial differential equation into an infinite set of coupled ordinary differential equations. Each of these ordlnary equations may be viewed as consisting of two parts. One part contains four terms which correspond to the four terms of the standard linear solid at the frequency nw and the other part is an infinite series which has the coefficient d of the nonlinear term in (1). These ordinary equations are there- fore inhomogeneous. Now the solution to an inhomogeneous equation is the gem- era1 solution of the corresponding homogeneous equation plus any particular solution of the inhomogeneous equation. We can obtain the solutions to the homogeneous equations immediately, and then approximate the infinite series in terms of these known solutions, after which particular solutions can be found. This strategy is facilitated by the fact that the terms of the infinite series, while nonlinear, become products of trigonometric functions of x and therefore expressible as linear combinations of trigonometric functions of other periodic- i ties: the equations become linearized. The homogeneous equations The reduced equation for any n has the form where U' indicates dU/dx, and another equation which is the complex conjugate of (9). Equation (9) has the well-known form O" + K 2~n0= 0 "n (10) where the complex propagation constant K = kn + ian is determined by The general solution of (10) is unO = C sinK x + DncosK x (12) where the constants C and D are to be determined eventually by the boundary conditions. For the Boundary conditions given by (5) and (6). the solution is and u(x,t) = ulo(x)eiwt + C.C. (14) where C.C. stands for complex conjugate. This is the solution given previously /2/ for the resonance of a bar made of a standard linear solid. The Inhomogeneous equations Returning to the inhomogeneous equation, we start the process of approximating to the nonlinear terms by assuming that and that all other Un are small of order d, at least, and proceed to sort the terms by the powers of d, in the end neglecting all terms containing d to a power higher than 1. In the process of approximation we are also helped by assuming that 6J/J is small so that the strain amplitude at resonance is orders of magnitude less than unity. The only equations that survive this process are those for n = 0,1,2. For n = 1, the nonlinear term vanishes in this approxima.- tion. Therefore the present treatment will not, in fact, extend to the case of nonlinear damping. Rather, the approximation is one in which the fundamental is unaffected to first order, but small harmonic generation occurs. Our approxirnw tions have then reduced the infinite set of equations envisioned in (8) to three only. We focus our attention on the equations for n = 2, putting aside the equation for n = 0 which will not affect the results. Thereby, we discuss here a system with only two states, a closed system not an open one. We have then to solve the equations for n = 2 0 and the complex conjugate of (16). With U, given by (131, and denoting the JOURNAL DE PHYSIQUE coefficient of cosK1!L-x) in (13) by A, the particular solution of (16) is by the method of variation of parameters where 2 2 D = [(2K1) - K2 1 ( a + ic2w). The boundary conditions, (5) and (6), imply that the stress corresponding to U2 shall be zero at both ends of the bar, forcing the other terms in U to be proportional to the coefficient of the particular solution. Let P senote the coefficient of sin2K1 (L-x) in (17). Then the general solution of (16) is U2 = C sinK (L-x) + C2cosK2(L-x) + Psin2K1(L-x) 12 (19) where Cl = (,-K2/2K1)P, and C2 = P(K~/~K,)[CO~~K~L- cosK2Ll/sinK2L. Equation (19) shows that our solution is more complicated than we might have expected.