Saturation and Its Application in the Vibration Control of Nonlinear Systems

Volume 2 ◽  
2004 ◽  
Author(s):  
Mohammad Shoeybi ◽  
Mehrdaad Ghorashi
Keyword(s):  
Singular Point ◽  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Approximate Solutions ◽  
Stability Of Solutions ◽  
Nonlinear Controller ◽  
Point Analysis ◽  
Good Agreement

An investigation on the nonlinear vibrations of a system with two degrees of freedom when subjected to saturation is presented. The method is especially applied to a system that consists of a DC motor with a nonlinear controller and subjected to a harmonic forcing voltage. Approximate solutions are sought using the method of multiple scales. Also, singular point analysis is used to study stability of solutions. These findings are then compared with the corresponding numerical results. Good agreement between the two is observed.

Study on Free Oscillations of Systems with Even Nonlinearities by Means of the Method of Multiple Scales

2007 ◽  
Vol 10-12 ◽  
pp. 193-197
Author(s):  
J.M. Wen ◽  
Z.C. Cao
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Free Oscillations ◽  
Analytical Technique ◽  
Spring System ◽  
Mass Spring ◽  
Good Agreement

An analytical technique, namely the method of multiple scales, is applied to solve the differential equations of free oscillations with even nonlinearities in a mass-spring system. Unlike other perturbation methods, the method of multiple scales is effective in determining the transient response as well as determining the approximation to the frequency of the nonlinear system. In this paper, the periodic solutions of the even nonlinear differential equations have been obtained by using the method of multiple scales. Compared with the numerical examples, the approximate solutions are in good agreement with exact solutions. The numerical and analytical solutions have clearly shown that there exists the so-called drift phenomenon in the free oscillations of systems with even nonlinearities without any external excitation.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Hamiltonian systems with three degrees of freedom, singular-point analysis, and chaotic behavior

1986 ◽  
Vol 33 (3) ◽  
pp. 2131-2133 ◽  
Author(s):  
W.-H. Steeb ◽  
J. A. Louw ◽  
P. G. L. Leach ◽  
F. M. Mahomed
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Point Analysis ◽  
Three Degrees Of Freedom

scholarly journals Theoretical and Experimental Study of an Harmonically Forced Vibro-Impact Nonlinear Energy Sink

2013 ◽  
Author(s):  
Etienne Gourc ◽  
Guilhem Michon ◽  
Sébastien Seguy ◽  
Alain Berlioz
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Energy Sink ◽  
Small Mass ◽  
Mass Ratio ◽  
Strongly Coupled ◽  
Electrodynamic Shaker ◽  
Energy Sink ◽  
Nonlinear Energy ◽  
Good Agreement

Recently, it has been demonstrated that a Vibro-Impact type Nonlinear Energy Sink (VI-NES) can be used efficiently to mitigate vibration of a Linear Oscillator (LO) under transient loading. In this paper, the dynamic response of an harmonically forced LO, strongly coupled to a VI-NES is investigated theoretically and experimentally. Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equation of motion are analyzed using the method of multiple scales in the case of 1 : 1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit Strongly Modulated Response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR response regimes. A good agreement between theoretical and experimental results is observed.

Primary Resonance of the Current-Carrying Beam in Thermal-Magneto-Elasticity Field

2010 ◽  
Vol 29-32 ◽  
pp. 16-21 ◽  
Author(s):  
Xiao Yan Xi ◽  
Zhian Yang ◽  
Li Li Meng ◽  
Chang Jian Zhu
Keyword(s):  
Response Curve ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Response Curves ◽  
Bending Vibration ◽  
Approximation Solution ◽  
System Parameters ◽  
Practical Engineering

On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

scholarly journals Analyzing the Stability for the Motion of an Unstretched Double Pendulum Near Resonance

2021 ◽  
Vol 11 (20) ◽  
pp. 9520
Author(s):  
Tarek S. Amer ◽  
Roman Starosta ◽  
Adelkarim S. Elameer ◽  
Mohamed A. Bek
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Double Pendulum ◽  
Nonlinear Dynamical ◽  
Near Resonance ◽  
Wide Range ◽  
The Stability ◽  
The Impact

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

The Nonlinear Response of a Simply Supported Rectangular Metallic Plate to Transverse Harmonic Excitation

2000 ◽  
Vol 67 (3) ◽  
pp. 621-626 ◽  
Author(s):  
O. Elbeyli and ◽  
G. Anlas
Keyword(s):  
Critical Points ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Stability Of Solutions ◽  
Metallic Plate ◽  
Simply Supported

In this study, the nonlinear response of a simply supported metallic rectangular plate subject to transverse harmonic excitations is analyzed using the method of multiple scales. Stability of solutions, critical points, types of bifurcation in the presence of a one-to-one internal resonance, together with primary resonance, are determined. [S0021-8936(00)00603-6]

scholarly journals On the motion of a triple pendulum system under the influence of excitation force and torque

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
T. S. Amer ◽  
◽  
A. A. Galal ◽  
A. F. Abolila ◽  
◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Dynamical System ◽  
Vertical Plane ◽  
Approximate Solutions ◽  
Circular Path ◽  
Nonlinear Stability Analysis ◽  
Pendulum System ◽  
Triple Pendulum

In this article, a nonlinear dynamical system with three degrees of freedom (DOF) consisting of multiple pendulums (MP) is investigated. The motion of this system is restricted to be in a vertical plane, in which its pivot point moves in a circular path with constant angular velocity, under the action of an external harmonic force and a moment acting perpendicular to the direction of the last arm of MP and at the suspension point respectively. Multiple scales technique (MST) among other perturbation methods is used to obtain the approximate solutions of the equations of motion up to the third approximation because it is authorizing to execute a specific analysis of the system behaviour and to realize the solvability conditions given the resonance cases. The stability of the considered dynamical model utilizing the nonlinear stability analysis approach is examined. The solutions diagrams and resonance curves are drawn to illustrate the extent of the effect of various parameters on the solutions. The importance of this work is due to its uses in human or robotic walking analysis.

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