scholarly journals Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified Harmonic Balance Approach and Vieta’s Substitution

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Y. Y. Lee
Keyword(s):  
Harmonic Balance ◽  
Reasonable Agreement ◽  
Solution Method ◽  
Analytic Solution ◽  
Integration Method ◽  
Numerical Integration Method ◽  
Algebraic Equations ◽  
Beam Vibration ◽  
Nonlinear Algebraic Equations ◽  
Balance Solution

This paper presents a modified harmonic balance solution method incorporated with Vieta’s substitution technique for nonlinear multimode damped beam vibration. The aim of the modification in the solution procedures is to develop the analytic formulations, which are used to calculate the vibration amplitudes of a nonlinear multimode damped beam without the need of nonlinear equation solver for the nonlinear algebraic equations generated in the harmonic balance processes. The result obtained from the proposed method shows reasonable agreement with that from a previous numerical integration method. In general, the results can show the convergence and prove the accuracy of the proposed method.

scholarly journals A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

2020 ◽  
pp. 146134842092343 ◽  
Author(s):  
M W Ullah ◽  
M S Rahman ◽  
M A Uddin
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Vibration Problems ◽  
Nonlinear Forced Vibration

In this paper, a modified harmonic balance method is presented to solve nonlinear forced vibration problems. A set of nonlinear algebraic equations appears among the unknown coefficients of harmonic terms and the frequency of the forcing term. Usually a numerical method is used to solve them. In this article, a set of linear algebraic equations is solved together with a nonlinear one. The solution obtained by the proposed method has been compared to those obtained by variational and numerical methods. The results show good agreement with the results obtained by both methods mentioned above.

scholarly journals Chaotic Phenomena and Nonlinear Responses in a Vibroacoustic System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Numerical Integration ◽  
Governing Equation ◽  
Harmonic Balance ◽  
Integration Method ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Numerical Integration Method ◽  
Chaotic Phenomena ◽  
Nonlinear Responses ◽  
Flexible Panel

This study addresses the chaotic phenomena and nonlinear responses in a vibroacoustic system. It is the first study about the chaotic phenomena in a vibroacoustic system, which is formed by a flexible panel coupled with a cavity. A multimode formulation is developed from the acoustic governing equation and nonlinear structural governing equation. The chaotic and various nonlinear responses are computed from the multimode formulation using a numerical integration method. The results obtained from the proposed method and classical harmonic balance method are generally consistent. A set of modal convergence studies is performed to check the proposed method. The effects of various parameters on triggering the nonchaotic responses to chaotic responses in a vibroacoustic system are studied in detail.

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu ◽  
J. L. Huang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Response Surfaces ◽  
Algebraic Equations ◽  
Incremental Harmonic Balance Method ◽  
Calculation Time ◽  
High Order Harmonic ◽  
Nonlinear Algebraic Equations ◽  
Incremental Harmonic Balance ◽  
Ihb Method

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

A Second-Order Scheme for Nonlinear Fractional Oscillators Based on Newmark-β Algorithm

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Q. X. Liu ◽  
J. K. Liu ◽  
Y. M. Chen
Keyword(s):  
Fractional Derivatives ◽  
Solution Method ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Second Order ◽  
Algebraic Equations ◽  
Order Accuracy ◽  
Hybrid Solution ◽  
Nonlinear Algebraic Equations ◽  
Order Scheme

This paper presents an accurate and efficient hybrid solution method, based on Newmark-β algorithm, for solving nonlinear oscillators containing fractional derivatives (FDs) of arbitrary order. Basically, this method employs a quadrature method and the Newmark-β algorithm to handle FDs and integer derivatives, respectively. To reduce the computational burden, the proposed approach provides a strategy to avoid nonlinear algebraic equations arising routinely in the Newmark-β algorithm. Numerical results show that the presented method has second-order accuracy. Importantly, it can be applied to both linear and nonlinear oscillators with FDs of arbitrary order, without losing any precision and efficiency.

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

2016 ◽  
Vol 16 (02) ◽  
pp. 1450100 ◽  
Author(s):  
Y. Y. Lee
Keyword(s):  
Differential Equations ◽  
Harmonic Balance ◽  
Solution Method ◽  
Transmission Loss ◽  
Mass Movement ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Multi Level

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.

scholarly journals Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

2021 ◽  
Vol 12 (2) ◽  
pp. 701-714
Author(s):  
Xigui Wang ◽  
Siyuan An ◽  
Yongmei Wang ◽  
Jiafu Ruan ◽  
Baixue Fu
Keyword(s):  
Frequency Response ◽  
Harmonic Balance ◽  
Integration Method ◽  
Numerical Integration Method ◽  
Superharmonic Resonance ◽  
Response Characteristics ◽  
Multi Scale ◽  
Incremental Harmonic Balance ◽  
Scale Perturbation ◽  
Planetary Transmission

Abstract. This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

scholarly journals The modified Lindstedt–Poincare method for solving quadratic nonlinear oscillators

2020 ◽  
pp. 146134842097975
Author(s):  
Ismot A Yeasmin ◽  
MS Rahman ◽  
MS Alam
Keyword(s):  
Analytical Solution ◽  
Small Parameter ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Algebraic Equations ◽  
Related Parameter ◽  
Nonlinear Algebraic Equations

Recently, an analytical solution of a quadratic nonlinear oscillator has been presented based on the harmonic balance method. By introducing a small parameter, a set of nonlinear algebraic equations have been solved which usually appear among unknown coefficients of several harmonic terms. But the method is not suitable for all quadratic oscillators. Earlier, introducing a small parameter to the frequency series, Cheung et al. modified the Lindstedt–Poincare method and used it to solve strong nonlinear oscillators including a quadratic oscillator. But due to some limitations of both parameters, a changed form of frequency-related parameter (introduced by Cheung et al.) has been presented for solving various quadratic oscillators.

A Hybrid Numerical Integration Method for Machine Dynamic Simulation

1986 ◽  
Vol 108 (2) ◽  
pp. 211-216 ◽  
Author(s):  
T. W. Park ◽  
E. J. Haug
Keyword(s):  
Mechanical System ◽  
Dynamic Simulation ◽  
Error Control ◽  
Integration Method ◽  
Differential Algebraic Equations ◽  
Numerical Integration Method ◽  
Algebraic Equations ◽  
Stable Method ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

An efficient and stable method for solving mixed-differential algebraic equations of constrained mechanical system dynamics is presented. The algorithm combines constraint stabilization and generalized coordinate partitioning methods, taking advantage of their attractive speed and error control characteristics, respectively. Three examples are studied to demonstrate efficiency and stability of the method.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Ye Ding ◽  
Jinbo Niu ◽  
LiMin Zhu ◽  
Han Ding
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Numerical Integration ◽  
Integration Method ◽  
Transcendental Equation ◽  
Spindle Speed ◽  
Machining Parameters ◽  
Numerical Integration Method ◽  
Stability Diagrams ◽  
Time Period

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

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