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.

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 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.

Dynamic Analysis of a Brushless D.C. Motor Using a Modified Harmonic Balance Method

1995 ◽  
Vol 117 (3) ◽  
pp. 283-291 ◽  
Author(s):  
Ming-ran Lee ◽  
Chandramouli Padmanabhan ◽  
Rajendra Singh
Keyword(s):  
Numerical Integration ◽  
Harmonic Balance ◽  
Broad Band ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Time Varying ◽  
Computationally Efficient ◽  
Dynamic Interactions ◽  
Torque Pulsations ◽  
Analytical Approaches

Analysis of brushless D.C. motor (BDCM) torque pulsations is an essential step in the diagnosis and control of vibration and noise generated by many electro-mechanical devices. The broad band spectral content of the torque pulsations, as predicted by a mathematical model which accounts for various complex effects, can often be obtained only by numerical integration which is time consuming while permitting little understanding of the dynamic interactions. Prior analytical approaches, such as the Fourier series technique or the d-q axis theory, are limited by the simplifying assumptions needed to compute the torque spectrum. This paper develops a new semi-analytical formulation for the analysis of nonlinear, time-varying BDCM’s which involve both spatial and temporal domains. A modified multi-term harmonic balance method, based on a transformation of the dual-domain problem to a spatial domain formulation, is developed here specifically to compute the magnitude of several harmonics of the pulsating torque. The interacting effects of key parameters, like dynamic eccentricity, magnetic saturation and open stator slots, on the time-varying inductances and rotor flux density distribution are included explicitly in the formulation. The predicted spectra compare very well with those obtained by direct time domain numerical integration. Yet, the proposed method is computationally efficient especially when the model dimension is reduced. It also provides better insight into the high frequency dynamics of the sample case.

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