Response of Hysteretic Multidegree-of-Freedom Systems Using Harmonic Balance Method

1999 ◽  
Author(s):  
Danilo Capecchi ◽  
Renato Masiani ◽  
Fabrizio Vestroni
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Elastic Model ◽  
Hysteretic Model ◽  
Response Curves ◽  
Periodic Response ◽  
Nonlinear Elastic ◽  
Force Displacement ◽  
Two Degree Of Freedom ◽  
System Characteristics

Abstract This article illustrates the nonlinear response of a hysteretic two degree of freedom system. The constitutive laws which define the force-displacement relation are based on a hysteretic model with Masing rules linked to a suitable nonlinear elastic model. Attention is focused on the periodic response, though an insight is also given to the non-periodic response. The method of analysis used is the harmonic balance with many components. Frequency-response curves are evaluated for different system characteristics. Ratios of small amplitude vibration frequency 3 and 2 are considered, with different hysteresis degree. Notwithstanding the dissipation due to hysteresis usually destroys most of the phenomena evidenced by the classical nonlinear oscillators, in the present analysis a rich behavior is found: IT symmetric and non symmetric, 2T periodic responses are found and so on.

Modified Harmonic Balance Method for Nonlinear Aeroelastic Analysis of Two Degree-of-Freedom Airfoils in Supersonic Flow

2018 ◽  
Vol 18 (06) ◽  
pp. 1871006 ◽  
Author(s):  
Yaobin Niu ◽  
Zhongwei Wang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Current Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Aerodynamic Load ◽  
Nonlinear Aeroelasticity ◽  
Unsteady Aerodynamic ◽  
Aeroelastic Analysis ◽  
Two Degree Of Freedom

In this paper, a new modified harmonic balance method is presented for the nonlinear aeroelastic analysis of two degree-of-freedom airfoils. Using this method, the nonlinear problem is first translated into a minimization problem, and the Particle Swarm Optimization which has high calculation efficiency is adopted to solve the problem. The proposed method is used to solve the nonlinear aeroelastic behavior of supersonic airfoil, with the unsteady aerodynamic load evaluated by the piston theory. Three examples of nonlinear aeroelasticity with significantly different coefficients are prepared, in which the frequencies and amplitudes of the limit cycles are obtained. The results show that the present current method is computationally more efficient.

RESONANCE RESPONSE OF A SIMPLY SUPPORTED ROTOR-MAGNETIC BEARING SYSTEM BY HARMONIC BALANCE

2012 ◽  
Vol 22 (06) ◽  
pp. 1250136 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
ZHONGJIN GUO
Keyword(s):  
Harmonic Balance ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Magnetic Bearing ◽  
Homotopy Continuation ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Resonance Response ◽  
Strongly Nonlinear System

Both the primary and superharmonic resonance responses of a rigid rotor supported by active magnetic bearings are investigated by means of the total harmonic balance method that does not linearize the nonlinear terms so that all solution branches can be studied. Two sets of second order ordinary differential equations governing the modulation of the amplitudes of vibration in the two orthogonal directions normal to the shaft axis are derived. Primary resonance is considered by six equations and superharmonic by eight equations. These equations are solved using the polynomial homotopy continuation technique to obtain all the steady state solutions whose stability is determined by the eigenvalues of the Jacobian matrix. It is found that different shapes of frequency-response and forcing amplitude-response curves can exist. Multiple-valued solutions, jump phenomenon, saddle-node, pitchfork and Hopf bifurcations are observed analytically and verified numerically. The new contributions include the foolproof multiple solutions of the strongly nonlinear system by means of the total harmonic balance. Some predicted frequency varying amplitudes could not be obtained by the multiple scales method.

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Hai-Tao Zhu ◽  
Siu-Siu Guo
Keyword(s):  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Solution Procedure ◽  
Balance Method ◽  
Periodic Response ◽  
Coupled Equations ◽  
The Mean ◽  
Closure Method ◽  
Gaussian Closure

This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

scholarly journals Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

2015 ◽  
Author(s):  
J. P. Noël ◽  
T. Detroux ◽  
L. Masset ◽  
G. Kerschen ◽  
L. N. Virgin
Keyword(s):  
Nonlinear System ◽  
Harmonic Balance ◽  
Global Analysis ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Basins Of Attraction ◽  
Response Curves ◽  
Restoring Force ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.

Dynamic Characteristics Research of Structure with Nonlinear TMD System

2013 ◽  
Vol 639-640 ◽  
pp. 812-817
Author(s):  
Zi Li Chen ◽  
Xiao Liang You ◽  
Chang Ping Chen ◽  
Ao Ling Ma
Keyword(s):  
Dynamic Characteristics ◽  
Primary Structure ◽  
Harmonic Balance ◽  
Degrees Of Freedom ◽  
Dynamic Control ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Response Curves ◽  
Dynamic Vibration Absorber ◽  
Dynamic Vibration

The dynamic control equations are derived for the mechanical model with two degrees of freedom that the soften spring is considered in the dynamic vibration absorber. In order to facilitate the computation to the equations, through the integral of the dynamic control equations which became a four order ordinary differential equation. The amplitude frequency response curves of the primary structure excited by a harmonic force were drawn with harmonic balance method, the influence of the dynamic characteristics of primary structure about this kind of nonlinear dynamic vibration absorber was discussed.

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