A highly accurate and explicit analytical approximation of the frequency–amplitude relationship for symmetric non-natural oscillators

Abstract

Non-natural oscillators represent a class of nonlinear dynamical systems characterized by kinetic energy that is non-squared in velocity. Such systems commonly appear in engineered applications and in predictive models used to describe certain natural phenomena. The frequency–amplitude relationship is central to both qualitative and quantitative analyses of their dynamic behavior. This technique note establishes highly accurate and explicit analytical approximation of the frequency–amplitude relationship for symmetric non-natural oscillators where the effective inertia and generalized restoring force are odd functions of displacement. The proposed methodology comprises two key steps: (i) construction of the Jacobi integral associated with the oscillator motion; and (ii) application of variable transformations and two-node and three-node Gaussian–Legendre quadrature formulas. Accuracy and effectiveness of the proposed method are demonstrated through vibration analysis of a cantilever beam exhibiting both inertial effects and static nonlinearity. The resulting expression remains valid across the full spectrum ranging from small to large of oscillation amplitude. The ratio of approximate frequency to exact frequency is 1.0007142 as the oscillation amplitude tends to infinity.