Multi-level Residue Harmonic Balance Method Research Articles

Multi-level residue harmonic balance method for nonlinear vibration of the beam

This study presents the nonlinear vibration and chaotic response of a beam subjected to harmonic excitation. The multi-level residue harmonic balance method is applied to solve the geometrically cubic nonlinear vibration of the simply supported beam. The obtained results agree well with those of the numerical integration method. The amplitude frequency response curves are presented to illustrate the nonlinear dynamic system response both for a damping and without damping model. Also, the chaotic response is examined for a simply supported beam with a nonlinear dynamic system.

Surface effects on the quasi-periodical free vibration of the nanobeam: semi-analytical solution based on the residue harmonic balance method

The study of surface effects on quasi-periodical vibration is significant because of its promising applications in sensors of nano/micro-electro-mechanical systems (N/MEMS). In this study, the surface effects parameters are adopted from the atomistic calculations. The governing equation of the nanobeam considering the nonlinear curvature and surface effects is established. Considering the first two modes, the Garlerkin method is used to translate the partial differential equation into ordinary differential equations. The multi-level residue harmonic balance method (RHBM) with two time variables is developed to solve these ODEs. The influences of surface and aspect ratio on the two frequencies and corresponding amplitudes are analyzed. Both of the frequency and amplitude discrepancies of the nanobeam to the counterparts of the classical beam become greater when the height of the beam shrinks at nanoscale. For the beam with the same height, both of the first- and second-frequencies decrease with the increase of the aspect ratio, and for the beams with the same aspect ratio, they decrease with the increase of the height. The solution of RHBM fits well with the molecular dynamics and Runge–Kutta numerical simulations. The study provides insight into the mechanism of the nonlinear dynamics of nanostructures, and shed light on quantitative design of the elements in N/MEMS, sensors, actuators, and resonators.

Modified Multi-level Residue Harmonic Balance Method for Solving Nonlinear Vibration Problem of Beam Resting on Nonlinear Elastic Foundation

Nonlinear vibration behavior of beam is an important issue of structural engineering. In this study, a mathematical modeling of a forced nonlinear vibration of Euler-Bernoulli beam resting on nonlinear elastic foundation is presented. The nonlinear vibration behavior of the beam is investigated by using a modified multi-level residue harmonic balance method. The main advantage of the method is that only one nonlinear algebraic equation is generated at each solution level. The computational time of using the new method is much less than that spent on solving the set nonlinear algebraic equations generated in the classical harmonic balance method. Besides the new method can generate higher-level nonlinear solutions neglected by previous multi-level residue harmonic balance methods. The results obtained from the proposed method compared with those obtained by a classical harmonic balance method to verify the accuracy of the method which shows good agreement between the proposed and classical harmonic balance method. Besides, the effect of various parameters such as excitation magnitude, linear and nonlinear foundation stiffness, shearing stiffness etc. on the nonlinear vibration behaviors are examined

Nonlinear structure-extended cavity interaction simulation using a new version of harmonic balance method

This study addresses the nonlinear structure-extended cavity interaction simulation using a new version of the multilevel residue harmonic balance method. This method has only been adopted once to solve a nonlinear beam problem. This is the first study to use this method to solve a nonlinear structural acoustic problem. This study has two focuses: 1) the new version of the multilevel residue harmonic balance method can generate the higher-level nonlinear solutions ignored in the previous version and 2) the effect of the extended cavity, which has not been considered in previous studies, is examined. The cavity length of a panel-cavity system is sometimes longer than the panel length. However, many studies have adopted a model in which the cavity length is equal to the panel length. The effects of excitation magnitude, cavity depth, damping and number of structural modes on sound and vibration responses are investigated for various panel cases. In the simulations, the present harmonic balance solutions agree reasonably well with those obtained from the classical harmonic balance method. There are two important findings. First, the nonlinearity of a structural acoustic system highly depends on the cavity size. If the cavity size is smaller, the nonlinearity is higher. A large cavity volume implies a low stiffness or small acoustic pressure transmitted from the source panel to the nonlinear panel. In other words, the additional volume in an extended cavity affects the nonlinearity, sound and vibration responses of a structural acoustic system. Second, if an acoustic resonance couples with a structural resonance, nonlinearity is amplified and thus the insertion loss is adversely affected.

New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem

In this study, a new modified multi-level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams. Although numerous nonlinear beam or linear double-beam problems have been tackled and solved, there have been few studies of this nonlinear double-beam problem. The geometric nonlinear formulations for a double-beam model are developed. The main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level. This heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method. The proposed method can generate the higher-level nonlinear solutions that are neglected by the previous modified harmonic balance method. The results from the proposed method agree reasonably well with those from the classical harmonic balance method. The effects of damping, axial force, and excitation magnitude on the nonlinear vibrational behaviour are examined.

Nonlinear Vibration Response of a Rectangular Tube with a Flexible End and Non-Rigid Acoustic Boundaries

This paper addresses the analysis for the nonlinear vibration response of a rectangular tube with a flexible end and non-rigid acoustic boundaries. The structural–acoustic modal formulations are developed from the Duffing differential equation and wave equation, which represent the large-amplitude structural vibration of a flexible panel coupled with a cavity. This problem considers both non-rigid acoustic boundary and structural cubic nonlinearity. The multi-level residue harmonic balance method is employed for solving the nonlinear coupled differential equations developed in the problem. The results obtained from the proposed method and numerical method are generally in good agreement. The effects of excitation magnitude, tube length, and phase shift parameter, etc., are examined.

Two modelling techniques for the vibration and transmission loss of a nonlinear vibro-acoustic system

Fahy’s and Pretlove’s modelling techniques for a vibro-acoustic system have been employed in many research works. This is the first article addressing the differences between these modelling techniques for a nonlinear vibro-acoustic system. Pretlove’s technique is used for vibro-acoustic systems with one flexible panel only (or 2D enclosures); Fahy’s is applied to vibro-acoustic systems with multiple flexible panels (or 3D enclosures). The main difference between them is that all acoustic boundary conditions can be satisfied in the 2D modelling technique, while only rigid acoustic boundary conditions can be satisfied in the 3D modelling technique. The multi-level residue harmonic balance method, which was recently developed and modified from the classical harmonic balance method, is employed here. The advantage of this method is that it requires less computation effort when solving the nonlinear equations generated in the entire solution process. The comparisons between various transmission loss and nonlinear vibration results generated from the two models are investigated.

Multi-Level Residue Harmonic Balance Method for the Transmission Loss of a Nonlinearly Vibrating Perforated Panel

This paper investigates the transmission loss of a nonlinearly vibrating perforated panel using the multi-level residue harmonic balance method. The coupled governing differential equations which represent the air mass movement at each hole and the nonlinear panel vibration are developed. The proposed analytical solution method, which is revised from a previous harmonic balance method for single mode problems, is newly applied for solving the coupled differential equations. The main advantage of this solution method is that only one set of nonlinear algebraic equations is generated in the zero level solution procedure while the higher level solutions to any desired accuracy can be obtained by solving a set of linear algebraic equations. The results obtained from the multi-level residue harmonic balance method agree reasonably with those obtained from a numerical integration method. In the parametric study, the velocity amplitude convergences have been checked. The effects of excitation level, perforation ratio, diameter of hole, and panel thickness are examined.

The effect of leakage on the sound absorption of a nonlinear perforated panel backed by a cavity

This study investigates the sound absorption of a nonlinear perforated panel backed by a cavity with leaking edges. Many other related works assume that the cavity is perfectly enclosed. This is the first study to address the effect of leakage on the sound absorption of a nonlinear perforated panel. The development of the absorption formula is based on the classical nonlinear thin plate theory, the three-dimensional wave equation and micro-perforation acoustic impedance. The recently developed multi-level residue harmonic balance method is introduced to compute the nonlinear vibration responses. Unlike previous harmonic balance methods, the proposed method generates nonlinear algebraic equations in the process of computing zero-level solutions only, and produces linear algebraic equations in the higher-level processes. The theoretical results obtained from the proposed method and the numerical integration method are in reasonable agreement. The effects of various parameters such as the excitation magnitude, damping ratio, cavity depth, perforation ratio and amount of leakage are examined.

Free vibration analysis of a nonlinear panel coupled with extended cavity using the multi-level residue harmonic balance method

This article addresses the free vibration analysis of a nonlinear panel coupled with extended cavity. In practice, the cavity length of a panel-cavity system is sometimes longer than the panel length. Therefore, this study examines the effect of cavity length on the natural frequency of a nonlinear panel coupled with extended cavity. The multi-level residue harmonic balance method, which was recently developed by the author and his research partners, is used to solve this nonlinear problem. The present harmonic balance solution agrees reasonably well with the results obtained from a previous classical solution and shows that the cavity length is a very important factor that significantly affects the panel vibration and should not be ignored in the modelling process. The natural frequency of a panel-cavity system is very sensitive to the cavity length and decreases significantly when the cavity is longer, due to its larger volume or weaker stiffness. Moreover, when the cavity is very long and its resonant frequencies are close to that of the panel, multiple frequency solutions are obtained.

Sound and vibration analysis of a nonlinear panel absorber mounted on an enclosure panel using the multi-level residue harmonic balance method

A thin metal panel absorber is usually mounted on an enclosure panel to enhance its acoustic absorption performance because of its light weight and good durability. Few studies have investigated the sound radiation from an enclosure panel mounted with a nonlinear panel absorber, although there have been many on nonlinear plates, linear structural–acoustic coupling problems, the sound absorption of various panel absorbers, and sound radiation from vibrating structures. In this paper, a multi-structural acoustic modal formulation is derived from two coupled partial differential equations representing the sound radiation from an enclosure panel subject to the acoustic pressure induced by a nonlinearly vibrating panel absorber. The results obtained from the multi-level residue harmonic balance method are in reasonable agreement with those obtained from a numerical integration method. In a parametric study, the panel displacement amplitude converged with an increasing number of modes. The effects of excitation level, damping, aspect ratio, cavity depth, and resonant frequency ratio are examined here. It can be concluded from some of the numerical results that the nonlinear resonances of a panel absorber can increase sound radiation if they are strongly coupled with the linear resonance of the enclosure panel. In other words, the panel absorber deteriorates the enclosure panel’s sound insulation performance.

The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation

In this paper, an efficient analytical solution method, namely, multi-level residue harmonic balance, is introduced and developed for the nonlinear vibrations of multi-mode flexible beams on an elastic foundation subject to external harmonic excitation. The main advantage of this solution method is that only one set of nonlinear algebraic equations is generated in the zero level solution procedure while the higher level solutions for any desired accuracy can be obtained by solving a set of linear algebraic equations. In other words, the computation effort to find more accurate nonlinear solutions is much less. In this paper, a multi-mode formulation, which represents the nonlinear beam vibration, is derived and set up. Then, the solution procedures are developed for obtaining the nonlinear multi-mode solution. The results from the multi-level residue harmonic balance method agree well with those from a numerical integration method. The effects of various parameters such as vibration amplitude, foundation modulus coefficient, damping factor and excitation level etc., on the nonlinear behaviors are examined. A convergence study is also performed to verify the solutions. The stability analysis is conducted using the virtue of Floquet theory and steady-state solutions are investigated.