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.

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 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.

scholarly journals MATHEMATICAL MODELLING OF VIBRATIONS OF NON-UNIFORM RING-SHAPED ULTRASONIC WAVEGUIDES

2021 ◽  
Vol 3 (56) ◽  
pp. 90-96
Author(s):  
Dmitry A. STEPANENKO ◽  
◽  
Ksenija A. BUNCHUK ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Central Line ◽  
Balance Method ◽  
Vibration Modes ◽  
Constant Sign ◽  
Algebraic Equations ◽  
Bending Vibrations ◽  
Singular Matrices ◽  
Uniform Ring

The article describes technique for modelling of ultrasonic vibrations amplifiers, which are implemented in the form of non-uniform ring-shaped waveguides, based on application of harmonic balance method. Bending vibrations of the waveguide are described by means of non-uniform integral and differential equations equivalent to Euler–Bernoulli equations in order to simplify calculation of amplitude-frequency characteristics of vibrations, particularly, to exclude the need of working with singular matrices. Using harmonic balance method, equations of vibrations are reduced to overdetermined non-uniform linear system of algebraic equations, which least-squares solution is determined by means of pseudo-inverse matrix. On the basis of analysis of numerical example possibility of existence of variable-sign and constant-sign vibration modes of the waveguide is shown and it is determined that for realization of amplifying function it is necessary to use waveguide at constant-sign vibration mode. The constant-sign vibration modes are combinations of bending defor-mation and extensional deformation of central line of the waveguide and they are detected due to accounting extensibility of the central line in equations of vibrations. Validity of the obtained results is confirmed by comparing them to the results of modelling by means of finite element method.

scholarly journals HARMONIC BALANCE THEORY FOR SCHEME TECHNICAL DESIGN

T-Comm ◽  
2020 ◽  
Vol 14 (11) ◽  
pp. 21-32
Author(s):  
Svetlana F. Gorgadze ◽  
◽  
Anton A. Maximov ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Projection Methods ◽  
Algebraic Equations ◽  
Electronic Circuits ◽  
Large Signal ◽  
Linear Algebraic Equations ◽  
Methods Of Synthesis ◽  
Signal Mode

The analysis and generalization of the main publications on the methods of synthesis and analysis of non-linear active microwave circuits based on the use of the harmonic balance method are presented. As a result of some classification of mathematical approaches and techniques used in the context of this method, a selection and review of basic algorithms was made, the sequential application of which makes it possible to obtain the final result for a scheme of any complexity. The principles of drawing up the initial system of differential equations for electronic circuits and reducing it to a system of linear algebraic equations are considered. A detailed and, at the same time, simplified interpretation of the approaches involving the use of projection methods and Krylov subspaces is given in order to make them easier to understand. Both the complete and the restart generalized method of minimal residuals are considered, in which the desired solution is obtained in the course of an iterative process, at each stage of which subspaces of lower dimension are constructed. The possibilities of simulators and application packages intended for circuit design of electronic circuits are considered. The problem of matching a power amplifier in large signal mode using the APLAC simulator, which is NI AWR technology for designing high-frequency circuits, is discussed.

Harmonic Balance Method for Nonlinear Vibration of Compound Planetary Gear Sets

2015 ◽  
Author(s):  
Weilin Zhu ◽  
Shijing Wu ◽  
Xiaosun Wang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Response Characteristic ◽  
Transmission Error ◽  
Balance Method ◽  
Time Varying ◽  
Algebraic Equations ◽  
Gear Backlash ◽  
Planetary Gear Sets

In this paper, a new nonlinear time-varying dynamic model for compound planetary gear sets, which incorporates the time-varying meshing stiffness, transmission errors and gear backlash, has been presented. The harmonic balance method (HBM), which is an analytical approach widely used for nonlinear oscillators, is employed to investigate the dynamic characteristics of the gear sets. The matrix form iteration algebraic equations has been established and solved by HBM and single rank inverse Broyden method to reveal the effect of transmission error and gear backlash on the frequency response characteristic of the system. Sub-harmonic resonant, super-harmonic resonant and jump phenomenon have been illustrated by several examples.

RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM

2013 ◽  
Vol 06 (02) ◽  
pp. 1350019 ◽  
Author(s):  
M. Daeichin ◽  
M. A. Ahmadpoor ◽  
H. Askari ◽  
A. Yildirim
Keyword(s):  
Energy Balance ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Problems ◽  
Nonlinear Oscillators ◽  
Initial Guess ◽  
Second Order ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Novel Approach

In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method.

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.

Analysis of Large-Amplitude Oscillations in Triple-Well Non-Natural Systems

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. K. Lai ◽  
X. Yang ◽  
F. B. Gao
Keyword(s):  
Large Amplitude ◽  
Harmonic Balance ◽  
Structural Engineering ◽  
Equilibrium Points ◽  
Harmonic Balance Method ◽  
Integral Form ◽  
Homoclinic Orbits ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Linear Algebraic Equations

In this paper, the large-amplitude oscillation of a triple-well non-natural system, covering both qualitative and quantitative analysis, is investigated. The nonlinear system is governed by a quadratic velocity term and an odd-parity restoring force having cubic and quintic nonlinearities. Many mathematical models in mechanical and structural engineering applications can give rise to this nonlinear problem. In terms of qualitative analysis, the equilibrium points and its trajectories due to the change of the governing parameters are studied. It is interesting that there exist heteroclinic and homoclinic orbits under different equilibrium states. By adjusting the parameter values, the dynamic behavior of this conservative system is shifted accordingly. As exact solutions for this problem expressed in terms of an integral form must be solved numerically, an analytical approximation method can be used to construct accurate solutions to the oscillation around the stable equilibrium points of this system. This method is based on the harmonic balance method incorporated with Newton's method, in which a series of linear algebraic equations can be derived to replace coupled and complicated nonlinear algebraic equations. According to this harmonic balance-based approach, only the use of Fourier series expansions of known functions is required. Accurate analytical approximate solutions can be derived using lower order harmonic balance procedures. The proposed analytical method can offer good agreement with the corresponding numerical results for the whole range of oscillation amplitudes.

scholarly journals Analytical and Dynamic study of Pulled Mass Nonlinear Vibration by Two Cables using Newton's Harmonic Balance Method

2020 ◽  
Vol 2 (2) ◽  
pp. 79-86
Author(s):  
Masoud Rahmani ◽  
Ionut Cristian Scurtu ◽  
Amin Moslemi Petrudi
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Vibrations ◽  
Nonlinear Differential Equations ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Good Match ◽  
Engineering Research ◽  
Solution Methods ◽  
Vibration Problems

In recent years, much research has been done on nonlinear vibrations, and analytical and numerical methods have been used to solve complex nonlinear equations. The behavior of nonlinear oscillating equations is discussed until the second order is approximated. Harmonic balance method, which itself has limitations in application. This method continues to be able to study a wider range of nonlinear differential equations. In general, nonlinear vibration problems are of great importance in physics, mechanical structures, and other engineering research. First, the equation of nonlinear vibrations governing the mass of the particle mass connect to the drawn cable is calculated and then the Newton Harmonic Balance Method is used to study the nonlinear vibrations of the set and obtain the answer and its frequency. The method (NHBM) is done with Maple software and a comparison between the results of this method with the solution methods used by other researchers is shown to be a good match.  

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