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.

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.

Modeling of a Vibration-Based Piezomagnetoelastic Energy Harvesting System by Using the Duffing Equation

2012 ◽  
Author(s):  
Henrik Westermann ◽  
Marcus Neubauer ◽  
Jörg Wallaschek
Keyword(s):  
Energy Harvesting ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Duffing Equation ◽  
Restoring Force ◽  
Piezoelectric Cantilever ◽  
Energy Harvesting System ◽  
Tip Mass

This article illustrates the modeling of a piezomagnetoelastic energy harvesting system. The generator consists of a piezoelectric cantilever with a magnetic tip mass. A second oppositely poled magnet is attached near the free end of the beam. Due to the nonlinear magnetic restoring force the system exhibits two symmetric stable equilibrium positions and one instable equilibrium position. The equation of motion is derived and it is shown that the system can be modeled as Duffing oscillator. An analytical approach is given to derive the Duffing parameters from the system parameters. The Duffing equation is solved for an oscillation around both equilibrium positions by using the harmonic balance method. For small orbit oscillations the equation of motion is solved by applying the fourth-order multiple scales method.

Piezoelectric energy harvester under parametric excitation: A theoretical and experimental investigation

2019 ◽  
Vol 31 (4) ◽  
pp. 612-631 ◽  
Author(s):  
Anshul Garg ◽  
Santosha K Dwivedy
Keyword(s):  
Experimental Investigation ◽  
Cantilever Beam ◽  
Parametric Resonance ◽  
Energy Harvester ◽  
Multiple Scales ◽  
Resonance Condition ◽  
Equation Of Motion ◽  
Attached Mass ◽  
Piezoelectric Patch ◽  
Piezoelectric Energy

In the present work, both theoretical and experimental investigation of a vertical cantilever beam–based piezoelectric energy harvester are carried out under principal parametric resonance condition. A piezoelectric patch is attached near the fixed end of the cantilever beam along with an attached mass positioned at an arbitrary location. The extended Hamilton’s principle is used to derive the spatio-temporal equation of motion, and generalized Galerkin’s approximation is used to obtain the temporal nonlinear electromechanical governing equation of motion. The method of multiple scales is used to find the reduced modulation equations. Due to large transverse deflection and effect of rotary inertia of the attached mass, the system exhibits cubic and inertial nonlinearities. An experimental setup with slider crank mechanism–based shaker and a harvester consisting of a cantilever beam with piezoelectric patch and attached mass is designed and developed. The challenges posed by parametric resonance in crack development in the PZT and in the beam are reported. The theoretical and experimental output voltage and the power obtained are found to be in good agreement. Furthermore, a qualitative and quantitative comparative study of 17 energy harvesters has been carried out, and the normalized power densities have been compared.

Closed-Form Solutions of Axisymmetrical Transverse Vibrations of Concave Parabolic Thickness Annular Plates

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Partial Differential ◽  
First Order ◽  
Annular Plates ◽  
Transverse Vibrations

This paper deals with transverse vibrations of axisymmetrical annular plates of concave parabolic thickness. A closed-form solution of the partial differential equation of motion is reported. An approach in which both method of multiple scales and method of factorization have been employed is presented. The method of multiple scales is used to reduce the partial differential equation of motion to two simpler partial differential equations that can be analytically solved. The solutions of the two differential equations are two levels of approximation of the exact solution of the problem. Using the factorization method for solving the first differential equation, which is homogeneous and includes a fourth-order spatial-dependent operator and second-order time-dependent operator, the general solution is obtained in terms of hypergeometric functions. The first diferential equation and the second differential equation (nonhomogeneous) along with the given boundary conditions give so-called zero-order and first-order approximations, respectively, of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Nonlinear Transverse Vibrations and Stability of Spinning Disks With Nonconstant Spinning Rate

1992 ◽  
Vol 114 (4) ◽  
pp. 506-513 ◽  
Author(s):  
T. H. Young
Keyword(s):  
Periodic Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Angular Speed ◽  
Equation Of Motion ◽  
Numerical Results ◽  
Transverse Vibrations ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper studies nonlinear transverse vibrations of spinning disks with nonconstant spinning rate. Here the angular speed of the disk is characterized as a small, periodic perturbation superimposed upon a constant speed. Due to this perturbation in angular speed, nonautonomous terms appear in the equation of motion, which results in the existence of parametric instability. In this paper, Galerkin’s method is first applied to yield a discretized system, and the method of multiple scales is used to obtain periodic solutions. All types of possible resonant combinations are investigated, and numerical results are shown for a simple harmonic speed perturbation.

NEW PHENOMENA IN THE NEIGHBORHOOD OF THE CODIMENSION-TWO BIFURCATION IN THE TWIN-WELL DUFFING OSCILLATOR

2000 ◽  
Vol 10 (06) ◽  
pp. 1367-1381 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
A. ZUBRZYCKI ◽  
E. TYRKIEL
Keyword(s):  
Bifurcation Point ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Codimension Two ◽  
Periodic Attractors ◽  
The Cross ◽  
Complex Phenomena ◽  
New Phenomena ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

A METHOD OF DESCRIPTION OF MAGNETIC OSCILLATIONS IN MESOSCOPIC SYSTEMS

1996 ◽  
Vol 10 (27) ◽  
pp. 1333-1338
Author(s):  
STANISLAW WALCERZ
Keyword(s):  
Experimental Investigation ◽  
Magnetic Properties ◽  
Translation Group ◽  
Mesoscopic Systems ◽  
Group Approach ◽  
Proposed Model ◽  
Planar Systems ◽  
Magnetic Translation ◽  
Onsager Theory ◽  
Magnetic Oscillations

A model for calculation of magnetic properties of small planar systems is proposed. The model combines the Onsager theory of de Haas-van Alphen oscillations with the magnetic translation group approach. The proposed model suggests a method for both theoretical and experimental investigation of magnetic properties of mesoscale systems.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

scholarly journals Dynamic analysis of a parametrically excited golden Muga silk embedded pneumatic artificial muscle

2018 ◽  
Vol 211 ◽  
pp. 02008 ◽  
Author(s):  
Bhaben Kalita ◽  
S. K. Dwivedy
Keyword(s):  
Dynamic Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Quadratic Function ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Time Response ◽  
Phase Portraits ◽  
Pneumatic Artificial Muscle ◽  
Pneumatic Muscle

In this work a novel pneumatic artificial muscle is fabricated using golden muga silk and silicon rubber. It is assumed that the muscle force is a quadratic function of pressure. Here a single degree of freedom system is considered where a mass is supported by a spring-damper-and pneumatically actuated muscle. While the spring-mass damper is a passive system, the addition of pneumatic muscle makes the system active. The dynamic analysis of this system is carried out by developing the equation of motion which contains multi-frequency excitations with both forced and parametric excitations. Using method of multiple scales the reduced equations are developed for simple and principal parametric resonance conditions. The time response obtained using method of multiple scales have been compared with those obtained by solving the original equation of motion numerically. Using both time response and phase portraits, variation of few systems parameters have been carried out. This work may find application in developing wearable device and robotic device for rehabilitation purpose.

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