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 CESSATION OF FREE VIBRATIONS OF A NONLINEAR ELASTIC SYSTEM DUE TO VISCOUS RESISTANCE

2021 ◽  
pp. 69-75
Author(s):  
Vasiliy Olshanskiy ◽  
Stanislav Olshanskiy
Keyword(s):  
Energy Balance ◽  
Finite Number ◽  
Power Law ◽  
Dry Friction ◽  
Free Vibrations ◽  
Balance Method ◽  
Viscous Resistance ◽  
Energy Balance Method ◽  
Number Of Cycles ◽  
The Impact

The paper deals with free vibrations of a system with power-law nonlinear elasticity subjected to power-law viscous resistance. The relation between the nonlinearity indices is determined when the impact of the viscous resistance force causes the vibrations to die away. In this case the vibrations are limited in time i.e. consist of a finite number of cycles analogous to a system with Coulomb dry friction. The research exploits the energy balance method. The periodic Ateb-functions are used to obtain an approximate formula for the work of dissipative force over a semi-cycle of vibrations. A recursive power-law equation for the vibration swings is derived from the condition of equality of the work to the potential energy change. By analyzing the change of the coefficient in the equation, which is related to the change of the semi-cycle number as well as the vibration swings, the condition for the equation to have no positive root is determined, which means that the vibrations die away. The condition is formulated in the form of an inequality. It is shown to generalize the results previously known. The theoretical inferences are verified by numerical integration of the nonlinear differential equation of motion. It is shown that under the conditions proposed in the paper the free vibrations consist of a finite number of cycles even if dry friction is absent from the system. Special cases are highlighted, when the approximate energy balance method results into exact computational formulae. The length of the cycles increases during the motion since it depends on the swing of damped vibrations in the essentially nonlinear system with rigid force characteristics considered.

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.

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 Nonlinear Vibration of Microbeams Based on The Elastics Foundation Using High-Order Energy Balance Method and Global Error Minimization Method

2018 ◽  
Vol 7 (2.23) ◽  
pp. 47
Author(s):  
D V. Hieu
Keyword(s):  
Differential Equation ◽  
Energy Balance ◽  
Nonlinear Vibration ◽  
Equation Of Motion ◽  
High Order ◽  
Error Minimization ◽  
Minimization Method ◽  
Energy Balance Method ◽  
Order Energy ◽  
Nonlinear Elastic

In this paper, nonlinear vibration of microbeams based on the nonlinear  elastic  foundation  is  investigated. The  equation  of motion of microbeams based on three-layered nonlinear elastic medium (shear, linear and nonlinear layers) is described by the partial differential equation by using the modified couple stress theory.  The equation of motion of microbeams is transformed  into the ordinary differential equation by using Galerkin method. The high-order Energy Balance  method and the high-order Global Error Minimization method are  used  to  get  the  frequency –  amplitude relationships  for  the  nonlinear  vibration  of  microbeams  with pinned-pinned  and  clamped-clamped  end  conditions. Comparisons between the present solutions and the privious solutions  show  the  accuracy  of  the  obtained  results.  

Dynamics research of Fangzhu’s nanoscale surface

2021 ◽  
pp. 146134842110527
Author(s):  
Pinxia Wu ◽  
Weiwei Ling ◽  
Xiumei Li ◽  
Xichun He ◽  
Liangjin Xie
Keyword(s):  
Energy Balance ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Fractal Model ◽  
Balance Method ◽  
Transform Method ◽  
Energy Balance Method ◽  
Collection Rate ◽  
Fractal Derivative ◽  
Approximate Analytical Solutions

In this paper, we mainly focus on a fractal model of Fangzhu’s nanoscale surface for water collection which is established through He’s fractal derivative. Based on the fractal two-scale transform method, the approximate analytical solutions are obtained by the energy balance method and He’s frequency–amplitude formulation method with average residuals. Some specific numerical experiments of the model show that these two methods are simple and effective and can be adopted to other nonlinear fractal oscillators. In addition, these properties of the obtained solution reveal how to enhance the collection rate of Fangzhu by adjusting the smoothness of its surfaces.

scholarly journals A distributed snowmelt prediction model in mountain areas based on an energy balance method

1994 ◽  
Vol 19 ◽  
pp. 107-113 ◽  
Author(s):  
Takeshi Ohta
Keyword(s):  
Energy Balance ◽  
Prediction Model ◽  
Deciduous Forest ◽  
Balance Method ◽  
Evergreen Forest ◽  
Snowmelt Runoff ◽  
Mountain Areas ◽  
Mountain Area ◽  
Energy Balance Method ◽  
Surface Cover

A distributed snowmelt prediction model was developed for a mountain area. Topography of the study area was represented by a digital map. Cells On the map were divided into three surface-cover types; deciduous forest, evergreen forest and deforested area. Snowmelt rates for each cell were calculated by an energy balance method. Meteorological elements were estimated separately in each cell according to topographical characteristics and surface-cover type. Distributions of water equivalent of snow cover were estimated by the model. Snowmelt runoff in the watershed was also simulated by snowmelt rates calculated by the model. The model showed thai the snowmelt period and snowmelt runoff after timber harvests would be about two weeks earlier than under the forest-covered condition.

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