A Globally Convergent Fixed Point Framework for Linear and Nonlinear Mechanical Vibration Systems

A rigorous fixed-point framework is developed for linear and nonlinear second-order mechanical vibration systems governed by ordinary differential equations with damping, stiffness, external forcing, and nonlinear restoring forces. The governing equation is reformulated as a Volterra operator equation in an appropriate Banach space, yielding a convenient integral formulation for constructing a Picard fixed-point iteration. Global existence, uniqueness, and convergence are established under suitable contraction conditions, with explicit bounds on the contraction constant and the corresponding geometric convergence rate of the iterative sequence. Energy-based stability estimates are derived for both forced and unforced systems, providing bounds on the mechanical energy and clarifying the dissipative behavior of the underlying dynamics. A discrete quadrature-based implementation of the Volterra formulation is developed and analyzed in terms of convergence order and stability properties under appropriate conditions. Numerical experiments for representative linear and nonlinear vibration problems are presented to validate the theoretical findings and to demonstrate the accuracy, convergence, and effectiveness of the proposed fixed-point approach.

کلیدواژه ها:

Fixed point iteration ، Mechanical vibrations ، Nonlinear systems ، Banach contraction ، Energy stability ، Numerical analysis