Analysis of Chaotic Characteristics of the Strong Nonlinear Vibration Isolation System Based on Harmonic Balance Method

2014 ◽  
Vol 635-637 ◽  
pp. 117-122
Author(s):  
Wei Min Mao ◽  
Hui Zhang ◽  
Jing Jun Lou
Keyword(s):  
Nonlinear Equations ◽  
Harmonic Balance ◽  
Vibration Isolation ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Isolation System ◽  
Runge Kutta Method ◽  
Chaotic Region ◽  
Response Range ◽  
Subharmonic Response

The key of studying the nonlinear system of the stationary period solutions by the HBM is solving a group of higher order multi-degree-of-freedom nonlinear equations. Aiming at this difficulty, this paper can easily get a higher harmonic balance truncation order expression of solution using a powerful symbolic computation software, and avoid the tedious derivation process. And the main subharmonic response range was accurately obtained by the EACM for solving the nonlinear equations. Meanwhile, the chaotic region was estimated through subharmonic cascade regions combined with the bifurcation of the Feigenbaum rule by solving the system. And, numerical results calculated by the Runge-Kutta method were given to verify the results of the period doubling bifurcations and chaotic region obtained by this method. It has been shown that they are in good agreement.

scholarly journals The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Weilei Wu ◽  
Bin Tang
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Vibration Isolation ◽  
Harmonic Balance Method ◽  
Elliptic Functions ◽  
Balance Method ◽  
Damping Force ◽  
Two Stage ◽  
Isolation System ◽  
Vibration Isolation System

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

Period Doubling Motions of a Nonlinear Rotating Beam at 1:1 Resonance

2014 ◽  
Vol 24 (12) ◽  
pp. 1450159 ◽  
Author(s):  
Fengxia Wang ◽  
Yuhui Qu
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Balance Method ◽  
Rotating Beam ◽  
Period 2 ◽  
Geometric Stiffening ◽  
Torsional Excitation ◽  
The Galerkin Method

A rotating beam subjected to a torsional excitation is studied in this paper. Both quadratic and cubic geometric stiffening nonlinearities are retained in the equation of motion, and the reduced model is obtained via the Galerkin method. Saddle-node bifurcations and Hopf bifurcations of the period-1 motions of the model were obtained via the higher order harmonic balance method. The period-2 and period-4 solutions, which are emanated from the period-1 and period-2 motions, respectively, are obtained by the combined implementation of the harmonic balance method, Floquet theory, and Discrete Fourier transform (DFT). The analytical periodic solutions and their stabilities are verified through numerical simulation.

Bifurcation Analysis of a Power System Model with Three Machines and Four Buses

2016 ◽  
Vol 26 (05) ◽  
pp. 1650082 ◽  
Author(s):  
Yu Chang ◽  
Xiaoli Wang ◽  
Dashun Xu
Keyword(s):  
Power Systems ◽  
Power System ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Bifurcation Phenomena ◽  
Fold Bifurcation ◽  
Approximate Expressions

The bifurcation phenomena in a power system with three machines and four buses are investigated by applying bifurcation theory and harmonic balance method. The existence of saddle-node bifurcation and Hopf bifurcation is analyzed in time domain and in frequency domain, respectively. The approach of the fourth-order harmonic balance is then applied to derive the approximate expressions of periodic solutions bifurcated from Hopf bifurcations and predict their frequencies and amplitudes. Since the approach is valid only in some neighborhood of a bifurcation point, numerical simulations and the software Auto2007 are utilized to verify the predictions and further study bifurcations of these periodic solutions. It is shown that the power system may have various types of bifurcations, including period-doubling bifurcation, torus bifurcation, cyclic fold bifurcation, and complex dynamical behaviors, including quasi-periodic oscillations and chaotic behavior. These findings help to better understand the dynamics of the power system and may provide insight into the instability of power systems.

scholarly journals High-Efficiency Vibration Isolation for a Three-Phase Power Transformer by a Quasi-Zero-Stiffness Isolator

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hao Cao ◽  
Yaopeng Chang ◽  
Jiaxi Zhou ◽  
Xuhui Zhao ◽  
Ling Lu ◽  
...  
Keyword(s):  
Vibration Isolation ◽  
High Efficiency ◽  
Power Transformer ◽  
Dynamic Stiffness ◽  
Harmonic Balance Method ◽  
Isolation System ◽  
Three Phase ◽  
Engineering Practices ◽  
Vibration Isolation Performance ◽  
Isolation Performance

The vibrations generated by a three-phase power transformer reduce the comfort of residents and the service life of surrounding equipment. To resolve this tough issue, a quasi-zero-stiffness (QZS) isolator for the transformer is proposed. This paper is devoted to developing a QZS isolator in a simple way for engineering practices. The vertical springs are used to support the heavy weight of the transformer, while the oblique springs are employed to fulfill negative stiffness to neutralize the positive stiffness of the vertical spring. Hence, a combination of the vertical and oblique spring can yield high static but low dynamic stiffness, and the vibration isolation efficiency can be improved substantially. The dynamic analysis for the QZS vibration isolation system is conducted by the harmonic balance method, and the vibration isolation performance is estimated. Finally, the prototype of the QZS isolator is manufactured, and then the vibration isolation performance is tested comparing with the linear isolator under real power loading conditions. The experimental results show that the QZS isolator prominently outperforms the existing linear isolator. This is the first time to devise a QZS isolator for three-phase power transformers with heavy payloads in engineering practices.

Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method

2006 ◽  
Vol 1 (3) ◽  
pp. 221-229 ◽  
Author(s):  
J. F. Dunne
Keyword(s):  
Stability Analysis ◽  
High Frequency ◽  
Harmonic Balance ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Equation Error ◽  
Subharmonic Response ◽  
Split Frequency ◽  
Frequency Harmonic

A split-frequency harmonic balance method (SF-HBM) is developed to obtain subharmonic responses of a nonlinear single-degree-of-freedom oscillator driven by periodic excitation. This method is capable of generating highly accurate periodic solutions involving a large number of solution harmonics. Responses at the excitation period, or corresponding multiples (such as period 2 and period 3), can be readily obtained with this method, either in isolation or as combinations. To achieve this, the oscillator equation error is first expressed in terms of two Mickens functions, where the assumed Fourier series solution is split into two groups, nominally associated with low-frequency or high-frequency harmonics. The number of low-frequency harmonics remains small compared to the number of high-frequency harmonics. By exploiting a convergence property of the equation-error functions, accurate low-frequency harmonics can be obtained in a new iterative scheme using a conventional harmonic balance method, in a separate step from obtaining the high-frequency harmonics. The algebraic equations (needed in the HBM part of the method) are generated wholly numerically via a fast Fourier transform, using a discrete-time formulation to include inexpansible nonlinearities. A nonlinear forced-response stability analysis is adapted for use with solutions obtained with this SF-HBM. Period-3 subharmonic responses are obtained for an oscillator with power-law nonlinear stiffness. The paper shows that for this type of oscillator, two qualitatively different period-3 subharmonic response branches can be obtained across a broad frequency range. Stability analysis reveals, however, that for an increasingly stiff model, neither of these subharmonic branches are stable.

DETECTION OF LIMIT CYCLE BIFURCATIONS USING HARMONIC BALANCE METHODS

2004 ◽  
Vol 14 (10) ◽  
pp. 3647-3654 ◽  
Author(s):  
FEDERICO I. ROBBIO ◽  
DIEGO M. ALONSO ◽  
JORGE L. MOIOLA
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Vector Fields ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Zero Eigenvalue ◽  
Complex Singularities

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities.

Study on the Mechanism for Line Spectrum Reduction in Nonlinear Vibration Isolation System

2007 ◽  
Author(s):  
Jing-Jing Wang ◽  
Shi-Jian Zhu ◽  
Shu-Yong Liu
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Isolation ◽  
Harmonic Balance Method ◽  
Chaotic Signal ◽  
Dominant Frequency ◽  
Line Spectrum ◽  
Isolation System ◽  
Vibration Isolation System ◽  
Chaotic Response ◽  
Frequency Components

The chaotic response and mechanism for line spectrum reduction in nonlinear vibration isolation system are studied. The harmonic balance method is applied to uncover the interaction between different harmonics. It is clear that the considerable energy transfers from the fundamental harmonic to the others by the nonlinear interactions, and thus the energy at the dominant frequency is reduced greatly. When the nonlinear vibration isolation system is in a chaotic state, the response is characteristic of the broadband spectrum, and thus the energy is distributed to all the frequency components. Chaotic attractor is different from the point, limit cycle and so on, and the fractal dimension can be applied to describe its characteristic. Furthermore, the chaotic signal is distinguished from the random one by the saturation of the correlation dimension. The former approaches to saturation with the increasing embedding dimension, but the latter does not. The phase space reconstruction based on wavelet transform can achieve the study of both the geometry and frequency characteristics of the chaos, so that provides a new way to study chaotic response.

TRANSITION CURVES AND BIFURCATIONS OF A CLASS OF FRACTIONAL MATHIEU-TYPE EQUATIONS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250275 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
ZHONGJIN GUO ◽  
H. X. YANG
Keyword(s):  
Fractional Order ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Duffing Equation ◽  
Response Amplitude ◽  
Transition Curve ◽  
Homotopy Continuation ◽  
Transition Curves ◽  
Fractional Orders

A general version of the fractional Mathieu equation and the corresponding fractional Mathieu–Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu–Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter.

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