A GENERALIZED APPROACH TO COUPLED NONLINEAR VIBRA TIONS OF CONTINUOUS SYSTEMS

Abstract A mathematical model covering many practical vibration problems of continuous systems has been proposed. The equations of motion consist of two nonlinearly coupled partial differential equations. The quadratic and cubic nonlinearities as well as the linear part of the equations are represented by arbitrary operators. A perturbation approach (method of multiple time scales) has been applied directly to the partial differential equations. The responses as well as the amplitude and phase modulation equations are found for the case of primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations. The coefficients of the amplitude and phase modulation equations are calculated in their most general form. Results are then applied to a nonlinear cable vibration problem having small sagto-span ratios.

A generalized system of nonlinear partial differential equations is treated.This general model covers many practical vibration problems of continuous systems such as beam, string and cable vibrations.The quadratic and cubic nonlinearities, which occur frequently in vibrations of continuous systems, are inserted into the equations in the form of arbitrary spatial operators.Also the linear part of the equation is left arbitrary.With the aid of these arbitrary operators, the equations have the advantage of modeling a wide range of nonlinear problems.A perturbation method (method of multiple time scales) is applied directly to the partial differential equations.This directly applying perturbations to the partial differential system (direct perturbation method) has, at least, the advantage of yielding spatial corrections to the linear normal modes [1][2][3][4][5][6][7].The responses and the amplitude and phase modulation equations are presented for this general system of equations.Because of the arbitrariness, the solution procedure yields an algorithm for solutions of many practical problems.Using this algorithm, solutions are presented for a specific problem of ponlinear cable vibrations.
W 2 + Ldw2) + 'u2W2+ Q4(WI,WI) + Q5(W2,W2) + Q6(WI,W2) + C4(W2,W2,W2) where n denotes time differentiation.,UJ and'u2 are the damping coefficients, and ft and n are the external excitation amplitude and frequency respectively.Li are the linear spatial differential and/or integral operators.Qi and C are the arbitrary spatial differential and/or integral operators producing quadratic and cubic nonlinearities respectively.The quadratic and cubic operators possess the property of being multilinear such that where Ci are some arbitrary constants.Instead of Q(w), a more common notation, the reason that we use Q(w.w) is that it gives the flexibility to represent the contributions of nonlinearity at higher orders of approximation correctly.
The associated boundary conditions for Eqs. ( 1) and ( 2) are assumed to be linear and homogenous and free from time derivatives.
There have been attempts of generalizing the direct perturbation method for a wide range of nonlinear vibration problems very recently.For a single arbitrary nonlinear partial differential equation, Pakdemirli [3] presented results for primary resonance case.For finite mode truncations, he [3] also compared results of direct approach method with discretization methods.The analysis is generalized to infinite modes by Pakdemirli and Boyacl [4].Using the same model of single differential equation, Pakdemirli and Boyacl [8] treated the subharmonic, superharmonic and combination resonance cases in detail.Nayfeh et al. [5] compared results of direct perturbation method with those of discretization methods by considering several examples from continuous systems.Solutions are also presented for an arbitrary cubic nonlinear spatial and temporal operator using the nonlinear normal mode concept.Two different versions of method of multiple scales are compared using general models, one with an arbitrary cubic operator and the other arbitrary quadratic and cubic operators [9].Finally a general odd-nonlinearity model was treated by Pakdemirli, Boyacl and Y llmaz [ 10].
In this section, we apply the method of multiple scales [ 11,12] directly to the partial differential equations (1) and (2).Contrary to the harmonic balance or discretization methods, this direct method requires no initial assumptions for the form of solutions.Due to cubic nonlinearities, we assume expansions up to third order for WI and W2 of the form where 8 is a small dimensionless measure of the amplitude of WI and W2, used as a book-keeping device, Win and W2n are 0(1).To = t is a fast-time scale characterizing changes occuring at the external excitation and natural frequencies; T2 = 8 2 t is a slow-time scale.The later analysis shows that there is no T I = E:t dependence, hence we omit it from the beginning.Assuming a weakly nonlinear system, damping and excitation coefficients are ordered such that their effects appear at the third order, where Dn = (} T .Substituting Eqs. ( 4)-( 8) into Eqs.( 1) and ( 2) and seperating the eqll.:tlOnsatn each order of £ ; we finally obtain In this section, we search for general solutions to the above equations at each order of approximation.At order £, the solution can be written as where cc stands for the complex conjugate of the preceeding terms.Y1 (x) and Y2 (x) satisfy the following equations where OJI and OJ2 are the eigenvalues and Y1 and Y2 are the corresponding eigenfunctions.Note that there are infinite number of eigenvalues for continuous systems.In the presence of damping, and for weakly nonlinear systems, the mode that is directly excited by an external excitation or indirectly excited by an internal resonance would survive and all other modes would decay with time [1].In the following analysis, we will assume that w, is directly excited by an external excitation and w] is indirectly excited by a one-to-one internal resonance with w, .Note that the boundary conditions for Eqs. ( 17) and ( 18) are homogenous and linear, as assumed previously.
At order &2 , the right hand sides of Eqs. ( 11) and ( 12) are known functions and we assume suitable solutions for W'2 and W22 of the form Substituting Eqs. ( 15), ( 16), ( 19) and (20) into Eqs.( 11) and ( 12), we find the equations determining the unknown functions ~lx) where (T and p are detuning parameters of 0 (1).Since the homogenous part of Eqs. ( 13) and ( 14) have a nontrivial solution, the inhomogenous Eqs. ( 13) and ( 14) have a solution, only if a solvability condition is satisfied [7].To find this condition, we express their solution in the form where WI and W2 are governed by Eqs. ( 13) and ( 14) with the terms proportional to exp(iOJITo) and exp(iOJ2TO) being deleted.Hence WI and W2 are unique and free of secular and small-divisor terms.
Substituting Eqs. ( 35), ( 36), ( 19), ( 20), ( 15) and ( 16) into Eqs.( 13) and ( 14), using Eqs.( 33  We now assume that the linear operators Li with the associated boundary conditions are selfadjoint.The solvability conditions for Eqs.(37) and (38) are + a6I1]j e-1pT1 + f f e iuT1 = 0 (39) iW2 (2B' + f.L2B) + a7I1]j + asA A B + a 9 A 2 ]j iipT 1 , a/oAB Jj e ipT1 + all B 2 A e-1pT2 + a/2A2 A e ,pT1 = 0 (40) where D is the domain of integration.Representing the solutions in this form makes it convenient to see explicitly the contributions of each operator to the coefficients.Note that, an extension of the above formulas for two or more spatial variables is trivial.In arriving at Eqs. ( 39) and (40), we normalized the following integrals Eqs. ( 39) and (40) determine the complex functions A(T JJ and B(T JJ.Note that, once the functions Y and ~are known either in closed form or numerically, ai coefficients can be calculated numerically for specific operators. To find the amplitude and phase modulation equations, we insert the following polar form into Eqs.( 39) and ( 40) and obtain the amplitude and phase modulation equations for the responses 2.
Once the numerical values of a coefficients are determined for specific problems, stability and bifurcation analysis of equations ( 56)-( 59) can be made.
In this section, we will apply the formalism derived in the previous sections to differential equations modelling the nonlinear vibrations of cables with small sag-to-span ratios.Following the previous analysis, we will investigate primary resonances of the excitation and one-to-one internal resonances between the natural frequencies of the in-plane and out-of-plane vibrations.The equations of motion, first derived by Lee and Perkins are [13] where x is the dimensionless arclength coordinate, and () / denotes differentiation with respect to x The constants VI and VI are the dimensionless propagation speeds of transverse and longitudinal waves respectively.WI is the in-plane and W2 is the out-of-plane displacements Assuming expansions (4) and (5) for the displacements, we obtain solutions (15) and ( 16) for the linear problem where the eigenfunctions YI (x) and Y2 (x) satisfY Eqs.(17) and (18) which can be written for this case as follows The symmetric in-plane solution can account for stretching of the cable whereas the anti-symmetric solution corresponds to zero stretching case[l3).We will consider only the symmetric solution in the following analysis.
The next step is to find the solutions at order &2.The form of the solutions are Eqs.( 19) and (20) where ~i satisfy Eqs. ( 21 75)-(86), using Eq. ( 72) when necessary, we finally obtain the solutions for c; i At order E! , we obtain the solvability conditions (39) and (40) where the coefficients a i are defined in Eqs.(41)-(52).Substituting the specific forms of the operators defined in Eqs.(67) into Eqs.( 41)-(52), we obtain the coefficients of the modulation equations where 1 f .a =--j.+ ab cOSYI 412 (x) } + ...
All results presented here are in full agreement with those given in references [2] and [4] provided that the following notation is adapted The formalism developed in this paper can be cast into a purely numerical algorithm for more involved problems where it becomes hard to find the functions explicitly.
A general nonlinear vibration problem has been proposed.The problem generalizes many vibration problems of continuous systems such as beam, string and cable vibrations.Solutions of this problem are presented in their most general form so that an algorithm can be constructed for solutions of a wide range of problems.As an illustration, the algorithm is used to solve a nonlinear cable vibration problem.
In this study, as an initial step, only the arbitrary linear and homogenous boundary conditions are considered.Nonlinear boundary conditions can be added as a next step.Subharmonic and superharmonic resonances of the external excitation as well as different internal resonances can be considered in a similar way.Numerical solutions can be searched when it is hard to find explicitly the functions appearing at the first and second orders of approximation.This will not involve any problem at the last level of approximation, since the coefficients of the modulation equations are presented in a suitable way to enable further numerical calculations.

Vr 2 + 1 C
(02 ; = Equation (68) with the boundary conditions (70) possess two types of solutions, namely the symmetric and the anti-symmetric in-plane solutions with respect to the mid-span of the cable.The symmetric in-plane solution is should be chosen such that J y/2 dx = 1.The anti-symmetric in-plane solution is o