Experimental and Theoretical Investigation on the Nonlinear Mathieu Equation Applied to a Belt-Pulley System

2007 ◽  
Author(s):  
Guilhem Michon ◽  
Lionel Manin ◽  
Robert G. Parker ◽  
Regis Dufour
Keyword(s):  
Experimental Investigation ◽  
Harmonic Balance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Pulley System ◽  
Transverse Vibrations ◽  
Instability Regions ◽  
Set Up

This paper is devoted to the theoretical and experimental investigation of a sample automotive belt-pulley system subjected to tension fluctuations. The equation of motion for transverse vibrations leads to a nonlinear Mathieu equation. The analyzes are performed via either the harmonic balance method for establishing the instability regions or the multiple scales approach for predicting the nonlinear response. An experimental set-up gives rise to non-linear parametric instabilities. The experimental investigation shows that the model is satisfactory.

scholarly journals Duffing Oscillator With Parametric Excitation: Analytical and Experimental Investigation on a Belt-Pulley System

2008 ◽  
Vol 3 (3) ◽  
Author(s):  
Guilhem Michon ◽  
Lionel Manin ◽  
Robert G. Parker ◽  
Régis Dufour
Keyword(s):  
Experimental Investigation ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Equation Of Motion ◽  
Viscous Damping ◽  
Pulley System ◽  
Transverse Vibrations ◽  
Proposed Model ◽  
Complex Phenomena

This paper is devoted to the theoretical and experimental investigation of a sample automotive belt-pulley system subjected to tension fluctuations. The equation of motion for transverse vibrations leads to a Duffing oscillator parametrically excited. The analysis is performed via the multiple scales approach for predicting the nonlinear response, considering longitudinal viscous damping. An experimental setup gives rise to nonlinear parametric instabilities and also exhibits more complex phenomena. The experimental investigation validates the assumptions made and the proposed model.

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.

Influence of Tensioner Dry Friction on the Vibration of Belt Drives With Belt Bending Stiffness

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Farong Zhu ◽  
Robert G. Parker
Keyword(s):  
Dry Friction ◽  
Harmonic Balance ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Friction Torque ◽  
Stick Slip ◽  
Pulley System ◽  
Coulomb Damping ◽  
Belt Drives ◽  
Linear Subsystem

A model of dry friction tensioner in a belt-pulley system considering transverse belt vibration is developed, and the influence of the dry friction on the system dynamics is examined. The discretized formulation is divided into a linear subsystem including linear coordinates and a nonlinear subsystem addressing tensioner arm vibration, which reduces the dimension of the iteration matrices when employing the harmonic balance method. The Coulomb damping at the tensioner arm pivot mitigates the tensioner arm vibration but not necessarily the vibrations of other system components. The extent of the mitigation varies for different excitation frequency ranges. The critical amplitude of the dry friction torque beyond which the system operates with a locked arm is determined analytically. Superharmonic resonances are observed in the responses of the generalized span coordinates, but their amplitudes are small. The energy dissipation at the tensioner arm hub is discussed, and the stick-slip phenomena of the arm are reflected in the velocity reversals near the arm extreme location. Dependence of the span tension fluctuations on Coulomb torque is explored.

Influence of Tensioner Dry Friction on the Vibration of Belt Drives With Belt Bending Stiffness

2007 ◽  
Author(s):  
Farong Zhu ◽  
Robert G. Parker
Keyword(s):  
Dry Friction ◽  
Harmonic Balance ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Friction Torque ◽  
Stick Slip ◽  
Pulley System ◽  
Coulomb Damping ◽  
Belt Drives ◽  
Linear Subsystem

A model of dry friction tensioner in a belt-pulley system considering transverse belt vibration is developed, and the influence of the dry friction on the system dynamics is examined. The discretized formulation is divided into a linear subsystem including linear coordinates and a nonlinear subsystem addressing tensioner arm vibration, which reduces the dimension of the iteration matrices when employing the harmonic balance method. The Coulomb damping at the tensioner arm pivot mitigates the tensioner arm vibration but not necessarily the vibrations of other system components. The extent of the mitigation varies for different excitation frequency ranges. The critical amplitude of the dry friction torque beyond which the system operates with a locked arm is determined analytically. Superharmonic resonances are observed in the responses of the generalized span coordinates but their amplitudes are small. The energy dissipation at the tensioner arm hub is discussed, and the stick-slip phenomena of the arm are reflected in the velocity reversals near the arm extreme location. Dependence of the span tension fluctuations on Coulomb torque is explored.

Floquet-Based Analysis of General Responses of the Mathieu Equation

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Gizem Acar ◽  
Brian F. Feeny
Keyword(s):  
Infinite Series ◽  
Harmonic Balance ◽  
Series Solution ◽  
Floquet Theory ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Periodic Part ◽  
Floquet Solution ◽  
Exponential Part ◽  
Stability Of The Solution

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

A Study of Dynamic Instability of Plates by an Extended Incremental Harmonic Balance Method

1985 ◽  
Vol 52 (3) ◽  
pp. 693-697 ◽  
Author(s):  
C. Pierre ◽  
E. H. Dowell
Keyword(s):  
Harmonic Balance ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Equation System ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Nonlinear Equations Of Motion

The dynamic instability of plates is investigated with geometric nonlinearities being included in the model, which allows one to determine the amplitude of the parametric vibrations. A modal analysis allowing one spatial mode is performed on the nonlinear equations of motion and the resulting nonlinear Mathieu equation is solved by the incremental harmonic balance method, which takes several temporal harmonics into account. When viscous damping is included, a new algorithm is proposed to solve the equation system obtained by the incremental method. For this purpose, a new characterization of the parametric vibration by its total amplitude—or Euclidian norm—is introduced. This algorithm is particularly simple and convenient for computer implementation. The instability regions are obtained with a high degree of accuracy.

Nonlinear Parametric Vibrations of an Elastic Structure With a Cylindrical Liquid Container (in the Case of an Anti-Symmetric Sloshing Mode)

2001 ◽  
Author(s):  
Takashi Ikeda ◽  
Shin Murakami
Keyword(s):  
Natural Frequency ◽  
Harmonic Balance ◽  
Small Deviation ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Elastic Structure ◽  
Coupled Vibration ◽  
Sloshing Mode ◽  
Resonance Curves ◽  
Fft Analysis

Abstract The nonlinear-coupled vibrations of an elastic structure and liquid sloshing in a cylindrical container are investigated. Since the structure is vertically subjected to a sinusoidal excitation, the behavior of the liquid surface is governed by a kind of the Mathieu equation. Modal equations governing the coupled motions are derived, when the natural frequency of the structure is equal to twice the natural frequency of an anti-symmetric mode of sloshing. The theoretical resonance curves are also presented by using an FFT analysis and the improved harmonic balance method. The influences of a liquid level and a detuning parameter on the theoretical resonance curves are shown. A small deviation of the tuning condition can cause amplitude-modulated motions and separate the occurrence region of the coupled vibration into two regions. In the experiments, the theoretical resonance curves were qualitatively in agreement with the experimental data. In addition, amplitude-modulated motions were observed.

2:1:1 Resonance in the Quasiperiodic Mathieu Equation

2005 ◽  
Author(s):  
Richard Rand ◽  
Tina Morrison
Keyword(s):  
Asymptotic Expansion ◽  
Numerical Integration ◽  
Time Scales ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Mathieu Equation ◽  
Instability Region ◽  
Algebraic Form ◽  
Instability Regions ◽  
Good Agreement

We present a small ε perturbation analysis of the quasiperiodic Mathieu equation: x¨+(δ+εcost+εcosωt)x=0 in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales including terms of O(ε2) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows good agreement for ε = 0.1. Then we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine the size of instability regions as we go from one resonance to another in the δ-ω parameter plane.

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