scholarly journals Dynamics of an oscillator with a quadratic nonlinearity in the expression of the elastic force loaded with a step pulse

2021 ◽  
pp. 21-26
Author(s):  
Maksym Slipchenko ◽  
Vasil Olshanskiy ◽  
Stanislav Olshanskiy
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Analytical Solution ◽  
Nonlinear Oscillator ◽  
Elliptic Functions ◽  
Elastic Force ◽  
Quadratic Nonlinearity ◽  
Jacobi Elliptic Functions ◽  
Dynamic Coefficient ◽  
Elasticity Characteristic

The unsteady oscillations of an oscillator with a quadratic nonlinearity in the expression of the elastic force under the action of an instantaneously applied constant force are described. The analytical solution of a second-order nonlinear differential equation is expressed in terms of periodic Jacobi elliptic functions. It is shown that the dynamic coefficient of a nonlinear system depends on the value of the instantaneously applied force and the direction of its action, since the elasticity characteristic of the system is asymmetric. If the force is directed towards positive displacements, then the characteristic of the system is "rigid" and the dynamic coefficient is in the interval , that is, it is smaller than that of a linear system. In the case when the force is directed towards negative displacements, the elasticity characteristic of the system is «soft» and the dynamic coefficient falls into the gap (2, 3), that is, it is larger than in the linear system. In the second case of deformation, there are static and dynamic critical values of the force, the excess of which leads to a loss of stability of the system. The dynamic critical force value is less than the static one. Since the displacement of the oscillator is expressed in terms of the Jacobi functions, the proposed formula for their approximate calculation using the table of the full elliptic integral of the first kind. The results of calculations are given, which illustrate the possibilities of the stated theory. For comparison, in parallel with the use of analytical solutions, numerical computer integration of the differential equation of motion was carried out. The convergence of the calculation results in two ways confirmed the adequacy of the derived formulas, which are also suitable for analyzing the motion of a quadratically nonlinear oscillator with a symmetric elastic characteristic. Thus, the considered nonlinear problem has an analytical solution in elliptic functions, and the process of motion depends on the direction in which the external force acts. In addition, when a force is applied towards a lower rigidity, a loss of system stability is possible. Keywords: nonlinear oscillator, quadratic nonlinearity, stepwise force impulse, Jacobi elliptic functions.

scholarly journals A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Kim Johannessen
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Nonlinear Differential Equation ◽  
Elliptic Function ◽  
Polar Angle ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Functions ◽  
Poisson Equations ◽  
Poisson Boltzmann

A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions.

scholarly journals Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1692
Author(s):  
Chaudry Masood Khalique ◽  
Oke Davies Adeyemo
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Elliptic Functions ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Direct Integration ◽  
Hyperbolic Functions ◽  
Soliton Equation ◽  
Closed Form Solutions ◽  
Expansion Technique

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G′/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem.

scholarly journals Numerical solution for time period of simple pendulum with large angle

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 25-30
Author(s):  
Sarkew Abdulkareem ◽  
Ali Akgül ◽  
Viyan Jalal ◽  
Bawar Faraj ◽  
Omed Abdulla
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Present Method ◽  
Large Angle ◽  
Elliptic Functions ◽  
Simple Pendulum ◽  
Time Period ◽  
Quadrature Methods ◽  
Relative Errors

In this study, the numerical solution of the ordinary kind of differential equation for a simple pendulum with large-angle of oscillation was introduced to obtain the time period. The analytical solution is obtained in terms of elliptic functions, and numerical solution of the problem was achieved by using two numerical quadrature methods, namely, Simpson?s 3/8 and Boole?s method. The period of a simple pendulum with large angle is presented. A comparison has been carried out between the analytical solution and the numerical integration results. In the case of error analysis, absolute and relative errors of the problem have been presented. A numerical algorithm has been developed by MATLAB software 2013R and used for analyzing the result. It is established that the results of the comparison guaranty the ability and the accuracy of the present method.

scholarly journals A New Approach for Solving Fractional Partial Differential Equations in the Sense of the Modified Riemann-Liouville Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Functions ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Liouville Derivative

Based on a fractional complex transformation, certain fractional partial differential equation in the sense of the modified Riemann-Liouville derivative is converted into another ordinary differential equation of integer order, and the exact solutions of the latter are assumed to be expressed in a polynomial in Jacobi elliptic functions including the Jacobi sine function, the Jacobi cosine function, and the Jacobi elliptic function of the third kind. The degree of the polynomial can be determined by the homogeneous balance principle. With the aid of mathematical software, a series of exact solutions for the fractional partial differential equation can be found. For demonstrating the validity of this approach, we apply it to solve the space fractional KdV equation and the space-time fractional Fokas equation. As a result, some Jacobi elliptic functions solutions for the two equations are obtained.

scholarly journals Closed form integration of the rotating plane pendulum nonlinear equation

2003 ◽  
Vol 34 (4) ◽  
pp. 327-350 ◽  
Author(s):  
Giovanni Mingari Scarpello ◽  
Daniele Ritelli
Keyword(s):  
Differential Equation ◽  
Nonlinear Equation ◽  
Closed Form ◽  
Nonlinear Differential Equation ◽  
Vertical Plane ◽  
Elliptic Functions ◽  
Angular Speed ◽  
Jacobi Elliptic Functions

The article deals with the nonlinear differential equation of the frictionless motion of a heavy pendulum swinging in a vertical plane which rotates at a fixed angular speed. The authors focused on its closed form integration by means of the Jacobi elliptic functions. This research took its origin by an autonomous work of the authors; this subject was also developed by [3], who did a treatment by far different from ours.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

2011 ◽  
Vol 3 (6) ◽  
pp. 689-701
Author(s):  
Malik Mamode
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Elliptic Functions ◽  
Oscillatory System ◽  
Balance Method ◽  
Closed Form Expression ◽  
Form Expression ◽  
Polynomial Potential ◽  
Exact Analytical Expression

AbstractThe exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

Minimum Energy Control of a Unicycle Model Robot

2021 ◽  
Author(s):  
Youngjin Kim ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Minimum Energy ◽  
Kinematic Model ◽  
Elliptic Functions ◽  
Terminal Point ◽  
Jacobi Elliptic Functions ◽  
Differential Drive ◽  
Terminal Constraints ◽  
Differential Drive Robot ◽  
Point To Point

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

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