scholarly journals Numerical integration of the equation of motion of an oscillator with dry friction

2018 ◽  
Vol 444 ◽  
pp. 042006
Author(s):  
V M Dinu
Keyword(s):  
Numerical Integration ◽  
Dry Friction ◽  
Equation Of Motion

scholarly journals Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

2021 ◽  
pp. 48-54
Author(s):  
Vasil Olshanskiy ◽  
Stanislav Olshanskiy ◽  
Maksym Slipchenko
Keyword(s):  
Differential Equation ◽  
Numerical Integration ◽  
Nonlinear Differential Equation ◽  
Dry Friction ◽  
Equation Of Motion ◽  
Asymptotic Formulas ◽  
Viscous Resistance ◽  
Lambert Function ◽  
External Resistance ◽  
Number Of Cycles

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction. Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

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 Truck Ride Analysis

1974 ◽  
Vol 96 (2) ◽  
pp. 597-602 ◽  
Author(s):  
G. R. Potts ◽  
H. S. Walker
Keyword(s):  
Numerical Integration ◽  
Nonlinear Equations ◽  
Digital Computer ◽  
Dry Friction ◽  
Shock Absorber ◽  
Equations Of Motion ◽  
Deflection Curve ◽  
Force Velocity ◽  
Nonlinear Equations Of Motion ◽  
Dry Friction Damping

The nonlinear vibratory motions of a three-axle semitrailer truck were investigated via the use of a digital computer. The nonlinear equations of motion are presented and a method of numerical integration is discussed. The analysis allows any shape of suspension force-deflection curve (including wheel hop, suspension stops, and dry friction damping) and a similar liberality of shock absorber force-velocity characteristics. An experimental vibration study, performed on a model truck, is described and the results compare favorably with the calculated results of the numerical integration.

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.

scholarly journals Measurements of Friction Coefficients of Snow blocks

1989 ◽  
Vol 13 ◽  
pp. 40-44 ◽  
Author(s):  
G. Casassa ◽  
H. Narita ◽  
N. Maeno
Keyword(s):  
Dry Friction ◽  
Kinematic Viscosity ◽  
Coulomb Friction ◽  
Video Camera ◽  
Equation Of Motion ◽  
Friction Coefficients ◽  
Friction Component ◽  
Avalanche Dynamics ◽  
Time Distance ◽  
Natural Snow

Snow blocks were slid down natural snow slopes and filmed with a video camera. Friction coefficients were calculated from time-distance curves and the equation of motion. Dry-friction coefficients ranged from 0.57 to 0.84, and could be separated into Coulomb friction and a friction component proportional to the contact area of the blocks (adhesion). These values are greater than the values usually used in avalanche dynamics, but are consistent with previous coefficients obtained for snow blocks sliding over snow.When uniform ploughing occurred and a shear layer developed along the track the apparent friction coefficients increased with velocity, and could be modelled by considering the kinematic viscosity of the snow. The values of kinematic viscosity ranged from 10−3 to 10−4 m2/s and agreed well with those values obtained by other researchers.

scholarly journals FREE OSCILLATOR OSCILLATIONS IN THE PRESENCE OF QUADRATIC VISCOUS RESISTANCE AND DRY FRICTION

2020 ◽  
pp. 33-40
Author(s):  
Vasyl Olshanskiy ◽  
Maksym Slipchenko ◽  
Oleksandr Spolnik ◽  
Mykhailo Zamrii
Keyword(s):  
Differential Equation ◽  
Energy Balance ◽  
Nonlinear Differential Equation ◽  
Dry Friction ◽  
Equation Of Motion ◽  
Balance Method ◽  
Forced Oscillations ◽  
Viscous Resistance ◽  
Energy Balance Method ◽  
Damped Oscillations

The article is devoted to the derivation of formulas for calculating the ranges of free damped oscillations of a double nonlinear oscillator. Using the Lambert function and the first integral of the nonlinear differential equation of motion, formulas are derived for calculating the ranges of free damped oscillations of a linearly elastic oscillator under the combined action of the forces of quadratic viscous resistance and Coulomb dry friction. The calculations involve a table of the specified special function of the negative argument. It is shown that the presence of viscous resistance reduces the duration of free oscillations to a complete stop of the oscillator. The set dynamics problem is also approximately solved by the energy balance method, and a numerical integration of the nonlinear differential equation of motion on a computer is carried out. The satisfactory convergence of the numerical results obtained in various ways confirmed the suitability of the derived closed formulas for engineering calculations. In addition to calculating the magnitude of the oscillations, the energy balance method is also used for an approximate solution of the inverse problem of dynamics, by identifying the values of the coefficient of quadratic resistance and dry friction force in the presence of an experimental vibrogram of free damped oscillations. An example of identification is given. This information on friction is needed to calculate forced oscillations, especially under resonance conditions. It is noted that from the obtained results, in some cases, well-known formulas follow, where the quadratic viscous resistance is not associated with dry friction.

scholarly journals Four-dimensional Integral Model of Dry Friction on the Example of Wheel Movement

2021 ◽  
Vol 2096 (1) ◽  
pp. 012043
Author(s):  
M S Salimov ◽  
G R Saypulaev ◽  
I V Merkuriev
Keyword(s):  
Dry Friction ◽  
Rolling Friction ◽  
Equation Of Motion ◽  
Friction Model ◽  
The Body ◽  
Integral Model ◽  
Control Torque ◽  
Padé Approximations ◽  
Pade Approximations ◽  
Dimensional Integral

Abstract A four-dimensional model of dry friction in the interaction of a solid wheel and a horizontal rough surface is investigated. It is assumed that there is no separation between the wheel and the horizontal surface. The movement of the body occurs in conditions of combined dynamics, when in addition to the sliding movement, the body participates in spinning and rolling. The equation of motion of the wheel is compiled using the Appel equation. The resulting model of sliding, spinning, and rolling friction is given for the case where the contact area is a circle. The cumbersome integral expressions were replaced by fractional-linear Pade approximations. Pade approximations accurately describe the behavior of the components of the friction model. A mathematical model is proposed that describes the simultaneous sliding, spinning and rolling of a solid wheel. The dependences of the parallel and perpendicular components of the friction force and the torque of the spinning friction were ploted with respect to the parameter that characterizes the movement of the wheel. Comparisons of the integral friction model and the model based on Pade approximations are presented. The results of the comparison showed a qualitative correspondence of the models. After obtaining the equation of motion, the simulation of motion at a constant control torque of the wheel is carried out. The graphs allow you to match the logical behavior of the wheel movement.

scholarly journals Measurements of Friction Coefficients of Snow blocks

1989 ◽  
Vol 13 ◽  
pp. 40-44 ◽  
Author(s):  
G. Casassa ◽  
H. Narita ◽  
N. Maeno
Keyword(s):  
Dry Friction ◽  
Kinematic Viscosity ◽  
Coulomb Friction ◽  
Video Camera ◽  
Equation Of Motion ◽  
Friction Coefficients ◽  
Friction Component ◽  
Avalanche Dynamics ◽  
Time Distance ◽  
Natural Snow

Snow blocks were slid down natural snow slopes and filmed with a video camera. Friction coefficients were calculated from time-distance curves and the equation of motion. Dry-friction coefficients ranged from 0.57 to 0.84, and could be separated into Coulomb friction and a friction component proportional to the contact area of the blocks (adhesion). These values are greater than the values usually used in avalanche dynamics, but are consistent with previous coefficients obtained for snow blocks sliding over snow. When uniform ploughing occurred and a shear layer developed along the track the apparent friction coefficients increased with velocity, and could be modelled by considering the kinematic viscosity of the snow. The values of kinematic viscosity ranged from 10−3 to 10−4 m2/s and agreed well with those values obtained by other researchers.

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