Introducing a Green-Volterra series formalism to solve weakly nonlinear boundary problems: application to the Kirchhoff's string
Résumé
This paper introduces a formalism which extends that of the “Green’s function” and that of “Volterra series”. These formalisms are typically used to solve, respectively, linear inhomogeneous space-time differential equations in physics and weakly nonlinear time-differential input- to-output systems in automatic control. While the Green’s function is a space-time integral kernel which fully characterizes a linear problem, Volterra series expansions involve a sequence of multi-variate time integral kernels (of convolution type for time-invariant systems). The extension proposed here consists in combining the two approaches, by introducing a series expansion based on multi-variate space-time integral kernels. This series allows the representation of the space-time solution of weakly nonlinear boundary problems excited by an “input” which depends on space and time. This formalism is introduced on and applied to a nonlinear model of a damped string that is excited by a transverse mass force f(x,t). The Green-Volterra kernels that solve the transverse displacement dynamics are computed. The first order kernel exactly corresponds to the Green’s function of the linearized problem. The higher order kernels satisfy a sequence of linear boundary problems that lead to (both) analytic closed-form solutions and modal decompositions. These results lead to an efficient simulation structure, which proves to be as simple as the one based on Volterra series, that has been obtained in a previous work for excitation forces with separated variables f(x,t) = ?(x) ftot(t). Numerical results are presented.
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