scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

scholarly journals Modeling the motion of an oscillator with a soft elastic characteristic

2017 ◽  
pp. 113-124
Author(s):  
Vasiliy Olshanskii ◽  
Stanislav Olshanskii
Keyword(s):  
Differential Equation ◽  
Initial Velocity ◽  
Initial Conditions ◽  
Special Functions ◽  
Elastic Characteristic ◽  
Equation Of Motion ◽  
Oscillation Period ◽  
Free Oscillations ◽  
Initial Deviation ◽  
Incomplete Elliptic Integrals

The free oscillations of a system with one degree of freedom are considered under the assumption that the elasticity of a spring is proportional to the cubic root of its deformation. Two forms of the analytical solution of the nonlinear differential equation of motion of the oscillator are obtained. In the first displacement of the oscillator in time is expressed in terms of incomplete elliptic integrals of the first and second kind. In the second form, the solution is expressed in terms of periodic Ateb-functions. The tables of the involved functions are made, which simplify the calculation. Formulas are also derived for calculating the oscillation periods when the oscillator is signaled or the initial deviation from the equilibrium position or the initial velocity (instantaneous pulse) in this position. The dependence of the oscillation period on the parameters of the oscillator and the initial conditions is established. Examples of calculations of oscillations are presented with the use of compiled tables of special functions and using the proposed approximations of the Ateb-functions. Comparison of numerical results obtained by different methods is made.

Numerical Analysis of the Trajectory of a Basketball Considering Lift and Drag

2015 ◽  
Vol 798 ◽  
pp. 493-499
Author(s):  
T.R Nikhilesh ◽  
Prahlad Kulkarni
Keyword(s):  
Differential Equation ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Basketball Players ◽  
Magnus Effect ◽  
Lift And Drag

Basketball players are taught to release the ball with a backward spin. This causes a lift due to the Magnus effect. Added to this there is a drag on the ball which always acts opposite to the direction of motion. In this paper, the trajectory of a basketball considering the lift and drag is calculated using numerical methods and also the force required to shoot the ball with different initial conditions from a distance of 25 feet away from a basket which is at a height of 10 feet is analyzed. A differential equation of motion of the ball in air is framed which accounts for all the forces on the ball. It is solved by discretizing the equation and solved using a C++ code. The trajectory of the balls with different initial conditions is plotted and it is found that as the spin on the ball increases, the effort required to shoot decreases.

APPLICATION OF IMPROVED AMPLITUDE-FREQUENCY FORMULATION TO NONLINEAR DIFFERENTIAL EQUATION OF MOTION EQUATIONS

2009 ◽  
Vol 23 (28) ◽  
pp. 3427-3436 ◽  
Author(s):  
A. G. DAVODI ◽  
D. D. GANJI ◽  
R. AZAMI ◽  
H. BABAZADEH
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Oscillations ◽  
Equation Of Motion ◽  
Energy Balance Method ◽  
Approximate Analytical Solutions ◽  
Relativistic Oscillator ◽  
High Degree ◽  
General Motion ◽  
New Algorithms

This paper presents an approach for solving accurate approximate analytical solutions for strong nonlinear oscillators called improved amplitude-frequency formulation. For illustrating the accuracy of the method, we also solved equations with He's energy balance method and compared results. New algorithms offer promising approaches, which are useful for nonlinear oscillations. We find that these attained solutions not only benefit from a high degree of accuracy, but are also uniformly valid in the whole solution domain which is so simple to do and effective. The studied equations are the general motion equation and the non-dimensional nonlinear differential equation of motion for the relativistic oscillator, which their solution can be useful for researchers to extend this ability into their other works.

scholarly journals OSCILLATIONS DESCRIBED BY ANALOGUES OF VAN DER POL AND RAYLEIGH EQUATIONS

2021 ◽  
pp. 76-82
Author(s):  
Vasiliy Olshanskiy ◽  
Stanislav Olshanskiy
Keyword(s):  
Steady State ◽  
Initial Conditions ◽  
System Stability ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Steady State Mode ◽  
Van Der Pol ◽  
Differential Formulation ◽  
Special Cases ◽  
The Impact

The paper deals with the modes of steady-state quasilinear self-oscillations described by the analogues of Van der Pol and Rayleigh differential equations. The differential formulation of the energy balance method is applied for studying the motion. The conditions for the equations to describe quasilinear self-oscillations with the amplitude independent of the initial conditions are derived in the form of inequalities. The formulae for computing this amplitude using the table of gamma functions are proposed. The steady-state mode of the self-oscillations is proved to be stable in contrast to the static equilibrium which appears unstable. The inequalities are also obtained which guarantee the equations of the type considered to describe the damped free oscillations about the zero equilibrium or the oscillations which build up resulting in the loss of system stability. These forms of motion depend on the initial conditions. For small initial deflections, which are less than the threshold value, the oscillations decay whereas for large once they build up. The dynamical system, which is stable in small, is unstable in large. The impact of the constant component of the resistance force on the oscillatory process is also studied. It is shown to cause the shift of the position about which the steady-state self-oscillations occur but not to influence their amplitude or frequency, which is the result of the linear elasticity of the system. The special cases are separated, when the computational formulae proposed become the results previously known. The analytical studies are followed by numerical solution of the respective Cauchy problem. By comparing the results obtained by the two methods we substantiate the adequacy of the computational formulae obtained.

scholarly journals Properties of the solution set of a generalized differential equation

1972 ◽  
Vol 6 (3) ◽  
pp. 379-398 ◽  
Author(s):  
J.L. Davy
Keyword(s):  
Differential Equation ◽  
Additional Assumption ◽  
Initial Conditions ◽  
Upper Semicontinuity ◽  
The State ◽  
Solution Set ◽  
Initial Time ◽  
State Variable ◽  
Generalized Differential Equation ◽  
Generalized Differential

We prove that the solution set of a generalized differential equation is connected and that points on the boundary of the solution funnel are peripherally attainable. This is done without the additional assumption of continuity in the state variable required in previous results. The result on upper semicontinuity of the solution set with respect to initial conditions is extended to include variations of initial time.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

2014 ◽  
Vol 1030-1032 ◽  
pp. 223-227
Author(s):  
Lin Fan ◽  
Song Rong Qian ◽  
Teng Fei Ma
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Numerical Integration ◽  
Integration Method ◽  
Equation Of Motion ◽  
Numerical Integration Method ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Method Of Molecular Dynamics ◽  
Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

Magnetohydrodynamic Squeeze Film Characteristics Between Parallel Circular Plates Containing a Single Central Air Bubble in the Inertial Flow Regime

1999 ◽  
Vol 66 (4) ◽  
pp. 1021-1023 ◽  
Author(s):  
R. Usha ◽  
P. Vimala
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Bubble Radius ◽  
Fluid Film ◽  
Squeeze Film ◽  
Parallel Plates ◽  
Circular Plates ◽  
Air Bubble ◽  
Approximate Analytical Solutions

In this paper, the magnetic effects on the Newtonian squeeze film between two circular parallel plates, containing a single central air bubble of cylindrical shape are theoretically investigated. A uniform magnetic field is applied perpendicular to the circular plates, which are in sinusoidal relative motion, and fluid film inertia effects are included in the analysis. Assuming an ideal gas under isothermal condition for an air bubble, a nonlinear differential equation for the bubble radius is obtained by approximating the momentum equation governing the magnetohydrodynamic squeeze film by the mean value averaged across the film thickness. Approximate analytical solutions for the air bubble radius, pressure distribution, and squeeze film force are determined by a perturbation method for small amplitude of sinusoidal motion and are compared with the numerical solution obtained by solving the nonlinear differential equation. The combined effects of air bubble, fluid film inertia, and magnetic field on the squeeze film force are analyzed.

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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