Stabilization of coupled inverted pendula: From discrete to continuous case

2020 ◽  
pp. 107754632092343 ◽  
Author(s):  
Mikhail E Semenov ◽  
Andrey M Solovyov ◽  
Peter A Meleshenko
Keyword(s):  
Wave Motion ◽  
Solid Medium ◽  
Continuous Case ◽  
Nonlinear Material ◽  
Parameter Method ◽  
Nonlinear Coupling ◽  
Stabilization Problem ◽  
Small Parameter Method ◽  
Stabilization Process ◽  
Feedback Algorithm

This study is focused on the investigation of stabilization problem for the system of coupled inverted pendula. The corresponding algorithm of control is based on the feedback principles and demonstrates some features in the physical implementation of the stabilization process. In the discrete case, the wave-like motion appears and leads to an observable interference pattern. In the continuous case, the provided feedback algorithm allows to simulate the stabilization process for the initially unstable solid medium. In these case conditions, ensuring the stabilization process is presented in the form of corresponding physical restrictions on the wave motion. Also, within the small parameter method, we investigate the nonlinear oscillatory motion of the material (the solid medium is modeled by the proposed system with nonlinear coupling). Results of numerical simulation for the system under consideration are presented and discussed. Particularly, numerical results show that the nonlinear material exhibits greater stability than the linear one.

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

scholarly journals THE INVESTIGATION OF DYNAMICS OF MISALIGNED UNITS IN VEHICLE TRANSMISSION

Transport ◽  
2002 ◽  
Vol 17 (6) ◽  
pp. 226-229
Author(s):  
Vytautas Turla ◽  
Igor Iljin
Keyword(s):  
Radial Direction ◽  
Parameter Method ◽  
Free Oscillations ◽  
Small Parameter Method ◽  
Mathematical Pendulum ◽  
Rotational Movement ◽  
Double Frequency ◽  
Shaft System ◽  
Small Parameters ◽  
Moment Of Resistance

In the article the problems related to the dynamics of a mechanic system on misalignment of shafts in radial direction are presented. The object of the investigation is a two-shaft system connected with an elastic centrifugal ring coupling. Using equations of static equilibrium it was found that an internal moment of resistance to rotation appears in the coupling connecting the radially misaligned shafts. Using Dalamber's principle for the rotational movement the differential equation that describes the rotation of the second shaft has been developed. It was shown that after the perfonnance of the corresponding actions and the introduction of a new variable the said equation is transformed in to an equation which character virtually coincides with the equation describing free oscillations of a mathematical pendulum. Because the value of misalignment of shafts in the radial direction is small in comparison with other parameters, a small parameter method was used for the solution of this equation. The found solutions show that rotational vibrations with the double frequency of rotational movement are excited in the misaligned mechanical system with an elastic centrifugal ring coupling. The restrictions ofthe application of a small parameters method have been explored and the limits of its application have been found.

scholarly journals On the Solution of the Exterior Boundary Value Problem with the Aid of Series

1978 ◽  
Vol 41 ◽  
pp. 175-176
Author(s):  
M. S. Petrovskaya
Keyword(s):  
Gravitational Field ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Integral Equations ◽  
Boundary Value ◽  
Parameter Method ◽  
Small Parameter Method ◽  
Exterior Boundary ◽  
Exterior Boundary Value Problem ◽  
Molodensky Problem

AbstractThe exterior gravitational field depending on the Earth’s non-sphericity is usually determined from the analysis of satellite data or by the solution of the exterior boundary value problem. In the latter case some integral equations are solved which correlate the exterior potential with the known vector of gravity and the shape of the Earth’s surface (molodensky problem). In order to carry out the integration the small parameter method is applied. As a result, all the quantities which involve the equations should be expanded in powers of a certain small parameter, among these being the heights of the Earth’s surface points as well as the inclination α of the Earth’s physical surface. Since the angle α can be significant, especially in mountains, and in fact does not depend on any small parameter then the solution of integral equations is possible only for the Earth’s surface which is smoothed enough.

scholarly journals Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Small Parameter ◽  
Initial Conditions ◽  
Geometric Interpretation ◽  
Rigid Bodies ◽  
Parameter Method ◽  
Large Parameter ◽  
Small Parameter Method ◽  
Rotational Motions ◽  
The Stability

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

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