Analytical approximations to primary resonance response of harmonically forced oscillators with strongly general nonlinearity

2020 ◽  
Vol 87 ◽  
pp. 534-545
Author(s):  
Yang Zhou ◽  
Baisheng Wu ◽  
C.W. Lim ◽  
Weipeng Sun
Keyword(s):  
Primary Resonance ◽  
Analytical Approximations ◽  
Forced Oscillators ◽  
Resonance Response ◽  
General Nonlinearity

Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

2011 ◽  
Vol 110-116 ◽  
pp. 2277-2283 ◽  
Author(s):  
Xiang Meng Zhang ◽  
Ben Li Wang ◽  
Xian Ren Kong ◽  
A Yang Xiao
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Effective Technique ◽  
Duffing System ◽  
First Order ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Case Based

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

scholarly journals Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Approximate Solution ◽  
Steady State ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Elliptic Functions ◽  
Dynamical Response ◽  
Response Curves ◽  
Resonance Response ◽  
Steady State Responses ◽  
Transient And Steady State

We study the dynamical response of an asymmetric forced, damped Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of the Helmholtz-Duffing oscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves. The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes.

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.

scholarly journals Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Liu-Yang Xiong ◽  
Guo-Ce Zhang ◽  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Mode Shapes ◽  
Analytical Approximations ◽  
Buckled Beam ◽  
Validity Range ◽  
The Galerkin Method

Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated for the first time. For appropriate choice of system parameters, the natural frequency of the second mode is approximately twice that of the first providing the condition for 2 : 1 internal resonance. The ordinary differential equations of the two mode shapes are established using the Galerkin method. The problem is replaced by two coupled second-order differential equations with quadratic and cubic nonlinearities. The multiple scales method is applied to derive the modulation-phase equations. Steady-state solutions of the system as well as their stability are examined. The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction. The double-jump, the saturation phenomenon, and the nonperiodic region phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal resonance are explored via numerical simulations.

Resonance Analysis of Fractional-Order Mathieu Oscillator

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Jiangchuan Niu ◽  
Hector Gutierrez ◽  
Bin Ren
Keyword(s):  
Fractional Order ◽  
Order Term ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Resonance Analysis ◽  
Approximate Analytical Solutions ◽  
Resonance Response ◽  
Resonant Behavior

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

scholarly journals Nonlinear Vibration Mitigation of a Beam Excited by Moving Load with Time-Delayed Velocity and Acceleration Feedback

2020 ◽  
Vol 10 (11) ◽  
pp. 3685
Author(s):  
Yiwei Tang ◽  
Jian Peng ◽  
Luxin Li ◽  
Hongxin Sun
Keyword(s):  
Feedback Control ◽  
Multiple Scales ◽  
Moving Load ◽  
Primary Resonance ◽  
Resonance Region ◽  
Excitation Amplitude ◽  
Main Resonance ◽  
Control Gain ◽  
Resonance Response ◽  
Acceleration Feedback

The time-delayed velocity and acceleration feedback control are provided to mitigate the resonances response of a nonlinear dynamic beam. By use of the method of multiple scales, the primary resonance and the 1/3 subharmonic resonance response of the controlled beam are analyzed. The excitation amplitude response peak and critical expression are obtained, and numerical simulations are also given. The effect of the feedback gains and time delayed on the steady-state response of the two types of resonances are investigated. The result show that time-delayed acceleration feedback control can effectively mitigate amplitude, and the main resonance response is affected periodically. Selecting reasonable control gain and time delay quantity can avoid the main resonance region and unstable multi-solutions, and can improve the efficiency of the vibration control.

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