scholarly journals Numerical study of coupled oscillator system using the classical Euler-Lagrange equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
H. Shanak ◽  
H. Khalilia ◽  
R. Jarrar ◽  
J. Asad
Keyword(s):  
Coupled Oscillators ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Lagrangian Method ◽  
Order Equation ◽  
Kutta Method ◽  
Lagrange Equations ◽  
Coupled Oscillator ◽  
Oscillator System ◽  
Runge Kutta Method

Abstract Problems involving vibrations (mechanical or electrical) can be reduced to problems of coupled oscillators. For this, we consider the motion of coupled oscillators system using Lagrangian method. The Lagrangian of the system was initially constructed, and then the Euler-Lagrange equations (i.e., equations of motion of the system) have been obtained. The obtained equations of motion are a homogenous second-order equation. These equations were solved numerically using the ode45 code, which is based on Runge-Kutta method.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

The Effects of Bearing Stiffness on Nonlinear Dynamic Behaviors of Multi-Mesh Gear Train

2012 ◽  
Author(s):  
T. N. Shiau ◽  
T. H. Young ◽  
J. R. Chang ◽  
K. H. Huang ◽  
C. R. Wang
Keyword(s):  
Chaotic Motion ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
Gear Train ◽  
Time Varying ◽  
Kutta Method ◽  
Dynamic Behaviors ◽  
Runge Kutta Method ◽  
Bearing Stiffness

In this study, the nonlinear dynamic analysis of the multi-mesh gear train with elastic bearing effect is investigated. The gear system includes the three rigid shafts, two gear pairs and elastic bearings. The stiffness and damper coefficient of elastic bearing are considered. The equations of motion of nonlinear time-varying system are derived using Lagrangian approach. The Runge-Kutta Method is employed to determine the system dynamic behaviors including the bifurcation and chaotic motion. The results show that the periodic motion, quasi-periodical motion and chaos can be excited with the elastic bearing effect. Especially, the results also indicate the dynamic response will go from periodic to quasi-periodical before the chaotic motion when the bearing stiffness is increased.

scholarly journals The Dynamics of Coupled  Oscillators

2021 ◽  
Author(s):  
◽  
Nigel Lawrence Holland
Keyword(s):  
Heart Rate ◽  
Differential Equations ◽  
Cardiovascular System ◽  
Coupled Oscillators ◽  
Dynamical Process ◽  
Coupled Oscillator ◽  
Sinus Arrhythmia ◽  
Oscillator System ◽  
Respiration Frequency ◽  
Human Cardiovascular System

<p>The subject is introduced by considering the treatment of oscillators in Mathematics from the simple Poincar´e oscillator, a single variable dynamical process defined on a circle, to the oscillatory dynamics of systems of differential equations. Some models of real oscillator systems are considered. Noise processes are included in the dynamics of the system. Coupling between oscillators is investigated both in terms of analytical systems and as coupled oscillator models. It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 dimensions due to the dependence of the dynamics on the phase difference of the oscillators. This means that the dynamics are easily able to be modelled by a 1D or 2D map. The analysis of N coupled oscillator systems is also described. The human cardiovascular system is studied as an example of a coupled oscillator system. The heart oscillator system is described by a system of delay differential equations and the dynamics characterised. The mechanics of the coupling with the respiration is described. In particular the model of the heart oscillator includes the baroreceptor reflex with time delay whereby the aortic fluid pressure influences the heart rate and the peripheral resistance. Respiration is modelled as forcing the heart oscillator system. Locking zones caused by respiratory sinus arrhythmia (RSA), the synchronisation of the heart with respiration, are found by plotting the rotation number against respiration frequency. These are seen to be relatively narrow for typical physiological parameters and only occur for low ratios of heart rate to respiration frequency. Plots of the diastolic pressure and heart interval in terms of respiration phase parameterised by respiration frequency illustrate the dynamics of synchronisation in the human cardiovascular system.</p>

Computer Simulation of the Response of a Railway Vehicle Carbody

2002 ◽  
Author(s):  
M. Senthil Kumar ◽  
P. M. Jawahar
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Railway Vehicle ◽  
Lateral Stiffness ◽  
Lateral Vibration ◽  
Kutta Method ◽  
Nonlinear Mathematical Model ◽  
Rail Vehicle ◽  
Runge Kutta Method

In this paper, a nonlinear mathematical model has been constructed by deriving the equations of motion of a Rail Vehicle carbody using Newton’s law. The nonlinear formula is used to evaluate the wheel rail contact forces. The nonlinear profile of wheel and rail are taken into account. Also the lateral stiffness of the track is taken into consideration. The equations of motion are derived for (a) Carbody with conventional wheelset (b) Carbody with unconventional wheelset (independently rotating wheels). For lateral vibration, 17 degrees of freedom are considered. The degrees of freedom represent lateral and yaw movements of 4 wheelsets and lateral, yaw and roll movements of the bogie and carbody. These equations of motion are transformed into a form suitable for numerical differential equation by Runge Kutta method. In the interest of computing economy, certain approximations have been introduced for calculating the creep forces. Sample results are given for a model of a typical railway vehicle used by the Indian Railways. The lateral dynamic response of the railway vehicle carbody for both conventional and unconventional wheelset has been analysed.

A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena1

2016 ◽  
Vol 83 (11) ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Thomas Joseph Impelluso
Keyword(s):  
Angular Momentum ◽  
Numerical Study ◽  
The Body ◽  
Moment Of Inertia ◽  
Kutta Method ◽  
Principal Moment ◽  
Tennis Racket ◽  
Angular Momenta ◽  
Runge Kutta Method ◽  
Principal Axes

This paper presents a complete explanation of the Dzhanibekov and the tennis racket phenomena. These phenomena are described by Euler's equation for an unconstrained rigid body that has three distinct moment of inertia values. In the two phenomena, the rotations of a body about the principal axes that correspond to the largest and the smallest moments of inertia are stable. However, the rotation about the axis corresponding to the intermediate principal moment of inertia becomes unstable, leading to the unexpected rotations that are the basis of the phenomena. If this unexpected rotation is not explained from a complete perspective which accounts for the relevant physical and mathematical aspects, one might misconstrue the phenomena as a violation of the conservation of angular momenta. To address this, the phenomenon is investigated using more precise mathematical and graphical tools than those employed previously. The torque-free Euler equations are integrated using the fourth-order Runge–Kutta method. Then, a recovery equation is applied to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments are shown to preserve the conservation of angular momentum.

Vibration of a Crane System Superimposed upon its Nominal Motion

2012 ◽  
Vol 430-432 ◽  
pp. 1847-1850
Author(s):  
Jin Fu Zhang ◽  
Qi Ren Luo
Keyword(s):  
Fourth Order ◽  
Equations Of Motion ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Actual Motion ◽  
Runge Kutta ◽  
The Relationship

The equations of motion for a crane system with considering the elasticity of the hoisting cable are derived. Using such equations and the relationship between the actual motion and the nominal motion of the crane system, the equations of vibration of the crane system superimposed upon its nominal motion are established. The responses of the vibration can be determined by numerically integrating the equations using the fourth order Runge–Kutta method. Based on the analysis of responses of the vibration, some conclusions concerning the vibration are obtained.

On the Design of Spring-Actuated Mechanism for 69KV SF6 Gas Insulated Circuit Breaker

2003 ◽  
Vol 125 (4) ◽  
pp. 840-845 ◽  
Author(s):  
Fu-Chen Chen
Keyword(s):  
Design Method ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Circuit Breaker ◽  
Total Duration ◽  
Kutta Method ◽  
Circuit Breakers ◽  
Loop Method ◽  
Runge Kutta Method ◽  
The Times

This paper studies the design of a spring-actuated mechanism of 69KV SF6 Gas insulated circuit breakers. The creative mechanism design method is first used to synthesize all the feasible mechanisms that satisfy the requirements for the circuit breaker. The kinematics of the mechanism are then analyzed using the vector-loop method. Subsequently, the equations of motion are derived with the Lagrange equation and solved by the Runge-Kutta method. The duration of individual operation, and hence the total duration to complete the full cycle of the mechanism has also been calculated. The times taken for the closing and opening operations were found to be 0.116 and 0.076 sec, respectively, comparable with the experiments.

scholarly journals Investigation of a spatial double pendulum: an engineering approach

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
S. Bendersky ◽  
B. Sandler
Keyword(s):  
Fourier Transformation ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Double Pendulum ◽  
Kutta Method ◽  
Free Oscillations ◽  
Frequency Spectra ◽  
Runge Kutta Method ◽  
Dynamic Investigation

The behavior of a spatial double pendulum (SDP), comprising two pendulums that swing in different planes, was investigated. Movement equations (i.e., mathematical model) were derived for this SDP, and oscillations of the system were computed and compared with experimental results. Matlab computer programs were used for solving the nonlinear differential equations by the Runge-Kutta method. Fourier transformation was used to obtain the frequency spectra for analyses of the oscillations of the two pendulums. Solutions for free oscillations of the pendulums and graphic descriptions of changes in the frequency spectra were used for the dynamic investigation of the pendulums for different initial conditions of motion. The value of the friction constant was estimated experimentally and incorporated into the equations of motion of the pendulums. This step facilitated the comparison between the computed and measured oscillations.

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