Analytical Method for Nonlinear Vibrations of Integral-Type Viscoelastic Beams Considering Longitudinal Inertia Coupling

Purpose: This paper investigates nonlinear vibrations of integral-type viscoelastic cantilever beams used in aerospace flexible structures, which involve complex coupling of viscoelastic integrals, geometric nonlinearity, and longitudinal inertia. Methods: A refined dynamic model is established considering longitudinal inertia, geometric nonlinearity, and integral-type viscoelasticity. The assumed mode method is used for discretization. An improved approximation converts viscoelastic integral forces into amplitude-dependent equivalent damping and restoring forces. Analytical solutions are derived via enhanced harmonic balance and modified stochastic averaging methods. Results: Numerical examples and Monte Carlo simulations validate the solutions. Longitudinal inertia regulates the beam’s nonlinear hardening/softening behavior, and viscoelastic parameters present coupled damping-stiffness effects. Conclusion: The proposed framework effectively analyzes nonlinear vibrations of integral-type viscoelastic beams, fills the theoretical gap, and provides a reliable tool for dynamic design of aerospace flexible structures.