The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

Abstract

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique

“…In this problem, we consider a beam with uniform length l and mass per unit length, hinged at the bottom of a rotational spring with stiffness , as shown in Figure 15 . The thickness of the beam is considered to be significantly smaller than the length of the beam; therefore, the effects of shear deformation and rotational inertia are neglected [ 20 ]. Inclination in the beam due to lumped mass in denoted by and displacement of the beam is given by .…”

“…Razzak [ 18 ] studied the phenomena of Duffing oscillators with rational and irrational forces, while A. Koochi [ 19 ] investigated the nonlinear oscillations of a CNT Nano-resonator based on nonlocal electricity using the energy balance method. Qian and Liu [ 20 , 21 ] used the residue harmonic method to study the vibrations of a system of cantilever beams carrying an intermediate lumped mass. All the above-discussed techniques were developed to analyze the mathematical models under different scenarios.…”

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.…”

Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method

“…In addition, the variational iteration method, [15][16] and the harmonic balance method. [17][18][19] Often, such traditional perturbation methods are not feasible in the case of the non-conservative oscillation, in which case it must deal with new methods. [20][21][22] One of the important types of nonlinear oscillators is the so-called conservative oscillator, whose restoring force is not conditional by time and whose amplitude is always constant.…”