A Lyapunov–Schmidt Analysis of Forced Oscillations in an Inhomogeneous Linear Oscillator Chain

Longitudinal oscillations of an inhomogeneous chain of linear oscillators coupled by springs are investigated. Both outer springs of the chain are rigidly fixed to immovable supports. The system is subjected to external periodic forces. The inhomogeneity of the chain (the perturbed system) is due to the different stiffness coefficients of the springs. These coefficients deviate slightly from a certain nominal value and depend on dimensionless deviation parameters. Zero values of these parameters correspond to a homogeneous (unperturbed) system. The resonant case is considered when the frequency of the external periodic force coincides with one of the eigenfrequencies of the unperturbed system. To construct an exact periodic solution of the perturbed system, the Lyapunov–Schmidt method is applied. As the problem is linear, this method allows to reduce it to a finite-dimensional algebraic problem of constructing a generalized Jordan chain for a degenerate linear operator. Necessary and sufficient conditions on the dimensionless deviation parameters are obtained, under which the length of such a chain is equal to 1 or 2. For each case, explicit exact formulas for the chain are derived, providing a complete description of the periodic solution. It is shown that for a generalized Jordan chain of length 1, the periodic solution of the perturbed system continuously transforms into a certain periodic solution of the unperturbed system as the small parameter $\varepsilon$ tends to zero. If the length of the generalized Jordan chain is $2$, the periodic solution of the perturbed system possesses a first-order pole at $\varepsilon=0$ and, reduces to a one-parameter family of periodic solutions of the unperturbed system. Numerical simulation was performed for a chain of eight oscillators. Plots of periodic solutions and phase trajectories of the perturbed system are constructed for various values of the small parameter.