Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

Abstract

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-offreedom (TDOF) mass-spring system with serial combined linear-nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt-Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge-Kutta solutions.

A Generalization of Lindstedt–Poincaré Perturbation Method to Strongly Mixed-Parity Nonlinear Oscillators

“…Hence, the exact frequency ( ) and the periodic solution ( ) to (1) may be obtained by combinatory piecing of the two solutions above [13] [14] (…”

“…For such oscillating systems with high nonlinearity and large parameters, there exist very few research works that present analytical or approximate solutions that are sufficiently accurate. Particular challenges are surfaced when the oscillating systems involve a strongly mixed-parity restoring force [11][12][13], and the recently much solicited, tunable optoelectronic meta-oscillators [2][3][4].…”

Exact solutions for oscillators with quadratic damping and mixed-parity nonlinearity

“…The analytical expressions for B, β and γ are the same as those given in equations ( 6), ( 9) and (10). Nevertheless, the value of m derived from equation ( 13) is non-negative and is thus more convenient.…”

“…• absence of quadratic damping and cubic nonlinearity, i.e. ε = γ = 0, α = 0 and β = 0; the equation arises in oscillations with restoring forces, such as the mechanical oscillations of the human eardrum [9] and mass-spring systems [10], and the exact solution is given by x(t) = A + B ep 2 (ωt, m) [11][12][13]; and…”

Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation

“…A new accurate analytical approach has been developed for solving the nonlinear TDOF oscillation systems with constant excitation. [4] In 2006, a smooth and discontinuous oscillator (SD oscillator) was investigated by Cao et al, [5−10] which can be used to study the transition from smooth to discontinuous dynamics depending on the value of the smoothness parameter 𝛼. The equation of motion in a dimensionless form [5−10] is…”