Nonlinear vibration of a two-mass system with nonlinear stiffnesses

Abstract

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-offreedom mass-spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a twomass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton's method and harmonic balance method. New and accurate higherorder analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newtonharmonic balance (NHB) 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 unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. 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 the classical harmonic balance methods. Two examples of nonlinear two-degree-offreedom mass-spring system are analyzed and verified with published result, exact solutions and numerical integration data.

Periodic Solution of Nonlinear Conservative Systems

“…In contrast to the perturbation methods, the nonperturbation methods do not use any small or artificial parameter. Examples include Adomian decomposition method [29], Homotopy analysis method [30], Variational iteration method [31], Energy balance method [2] and its modifications [32][33][34], He Chengtian's interpolation method [27] also called max-min approach, amplitude-frequency formulation [35], Hamiltonian approach [36], global error minimization method [37], Harmonic balance method [4] and its modifications [38][39][40][41][42], cubication methods [43][44][45][46], variational methods [47][48][49], differential transform method [50], and continuous piecewise linearization method [8]. Nonperturbation methods also have various limitations.…”

Analysis of nonlinear vibrations of two-degree-of-freedom systems

“…The motion of nonlinear two-degree-of-freedom (TDOF) oscillation system has been widely investigated in the past few decades [1,2,3,4,5,6]. TDOF systems are important in engineering because many practical engineering components consist of coupled vibrating systems that can be modeled by using TDOF systems such as elastic beams supported by two springs and vibration of a milling machine.…”

A magnetic levitation-based tristable hybrid energy harvester for scavenging energy from low-frequency structural vibration

“…To address this issue, nonlinear energy harvesting technologies have been introduced to broaden the usable bandwidth of linear counterparts, including mono-stable, bi-stable and multistable oscillating mechanisms [31]. In mechanical and structural engineering, nonlinear dynamics is useful for explaining complicated phenomena in many low-and high-dimensional problems [32][33][34][35][36]. By creating more potential wells in a nonlinear dynamic system, its potential energy can be distributed more uniformly, thereby providing a shallower potential well and resulting in a lower excitation threshold for inter-well motions [37].…”