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.

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 DISSIPATIVE OSCILLATORS’ POWER CHARACTERISTIC NON-SYMMETRY DYNAMIC EFFECT

2021 ◽  
pp. 65-75
Author(s):  
Vasiliy Olshanskiy ◽  
Stanislav Olshanskiy
Keyword(s):  
Differential Equation ◽  
Dry Friction ◽  
Elastic Characteristic ◽  
Approximation Error ◽  
Dynamic Effect ◽  
Viscous Friction ◽  
Equation Of Motion ◽  
Lambert Function ◽  
Coulomb Dry Friction ◽  
Non Linear Oscillator

The features of motion of a non-linear oscillator under the instantaneous force pulse loading are studied. The elastic characteristic of the oscillator is given by a polygonal chain consisting of two linear segments. The focus of the paper is on the influence of the dissipative forces on the possibility of occurrence of the elastic characteristic non-symmetry dynamic effect, studied previously without taking into account the influence of these forces. Four types of drag forces are considered, namely linear viscous friction, Coulomb dry friction, position friction, and quadratic viscous resistance. For the cases of linear viscous friction and Coulomb dry friction the analytical solutions of the differential equation of oscillations are found by the fitting method and the formulae for computing the swings are derived. The conditions on the parameters of the problem are determined for which the elastic characteristic non-symmetry dynamic effect occurs in the system. The conditions for the effect to occur in the system with the position friction are derived from the energy relations without solving the differential equation of motion. In the case of quadratic viscous friction the first integral of the differential equation of motion is given by the Lambert function of either positive or negative argument depending on the value of the initial velocity. The elastic characteristic non-symmetry dynamic effect is shown to occur for small initial velocities, whereas it is absent from the system when the initial velocities are sufficiently large. The values of the Lambert function are proposed to be computed by either linear interpolation of the known data or approximation of the Lambert function by elementary functions using asymptotic formulae which approximation error is less than 1%. The theoretical study presented in the paper is followed up by computational examples. The results of the computations by the formulae proposed in the paper are shown to be in perfect agreement with the results of numerical integration of the differential equation of motion of the oscillator using a computer.

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.

Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter

2014 ◽  
Vol 12 (05) ◽  
pp. 523-536 ◽  
Author(s):  
Chelo Ferreira ◽  
José L. López ◽  
Ester Pérez Sinusía
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Special Functions ◽  
Auxiliary Problem ◽  
Second Order ◽  
Large Parameter ◽  
Initial Value

We consider the second-order linear differential equation [Formula: see text] where x ∈ [0, X], X > 0, α ∈ (-∞, 2), Λ is a large complex parameter and g is a continuous function. The asymptotic method designed by Olver [Asymptotics and Special Functions (Academic Press, New York, 1974)] gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. We add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the idea may be applied also to nonlinear differential equations with a large parameter.

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.

Evolution of solitary waves in a two-pycnocline system

2009 ◽  
Vol 642 ◽  
pp. 235-277 ◽  
Author(s):  
M. NITSCHE ◽  
P. D. WEIDMAN ◽  
R. GRIMSHAW ◽  
M. GHRIST ◽  
B. FORNBERG
Keyword(s):  
Solitary Waves ◽  
Initial Conditions ◽  
Oscillation Period ◽  
Resonance Phenomenon ◽  
Numerical Code ◽  
Asymptotic Model ◽  
Narrow Region ◽  
Near Resonance ◽  
Long Time ◽  
The Waves

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results.

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.

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.

On New Aspects of Nested Carbon Nanotubes as Gigahertz Oscillators

2011 ◽  
Vol 133 (5) ◽  
Author(s):  
R. Ansari ◽  
B. Motevalli
Keyword(s):  
Carbon Nanotubes ◽  
Initial Velocity ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Interaction Force ◽  
Arbitrary Length ◽  
Oscillatory Frequency ◽  
Analytical Expression ◽  
System Parameters ◽  
The Core

Nested carbon nanotubes exhibit telescopic oscillatory motion with frequencies in the gigahertz range. In this paper, our previously proposed semi-analytical expression for the interaction force between two concentric carbon nanotubes is used to solve the equation of motion. That expression also enables a new semi-analytical expression for the precise evaluation of oscillation frequency to be introduced. Alternatively, an algebraic frequency formula derived based on the simplifying assumption of constant van der Waals force is also given. Based on the given formulas, a thorough study on different aspects of operating frequencies under various system parameters is conducted, which permits fresh insight into the problem. Some notable improvements over the previously drawn conclusions are made. The strong dependence of oscillatory frequency on system parameters including the extrusion distance and initial velocity of the core as initial conditions for the motion is shown. Interestingly, our results indicate that there is a special initial velocity at which oscillatory frequency is unique for any arbitrary length of the core. A particular relationship between the escape velocity (the minimum initial velocity beyond which the core will leave the outer nanotube) and this specific initial velocity is also revealed.

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