scholarly journals OSCILLATION OF PULSE-LOADED OSCILLATOR WITH DEGREE POSITIONAL FRICTION

2021 ◽  
Vol 3 (1) ◽  
pp. 37-46
Author(s):  
V. Olshanskiy ◽  
◽  
M. Slipchenko ◽  
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Equation Of Motion ◽  
Asymptotic Formulas ◽  
Force Characteristic ◽  
Half Cycle ◽  
The Cauchy Problem ◽  
Analytical Relations

Nonstationary oscillations of the oscillator with nonlinear positional friction caused by an instantaneous force pulse are described. The power dependence of the positional friction force on the displacement of the system, which generalizes the known models, is accepted. The corresponding dynamics problems were solved precisely by the method of adding and approximated by the method of energy balance. In the study, using periodic Ateb-functions, an exact analytical solution of the nonlinear differential equation of motion was constructed. Compact formulas for calculating oscillation ranges and half-cycle durations are derived. It is shown that the decrease in the amplitude of oscillations, as well as under the action of the force of linear viscous resistance, follows the law of geometric progression. The denominator of the progression is less than one and depends on the positional friction constants, in particular on the nonlinearity index. Thus, we have not only a decrease in the amplitude of oscillations, but also an increase in the durations of half-cycles, which is characteristic of nonlinear systems with a rigid force characteristic. Approximate displacement calculations use Pade-type approximations for periodic Ateb-functions. The error of these approximations is less than one percent. From the obtained analytical relations, as separate cases, the known dependences covered in the theory of oscillations for linear positional friction follow. It is shown that even in the case of nonlinear positional friction the process of oscillations caused by an instantaneous momentum has many oscillations and is not limited in time. In the case of power positional friction, the oscillation ranges of the pulse-loaded oscillator can be calculated by elementary formulas. The calculation of displacements in time is associated with the use of periodic Ateb-functions, the values of which are not difficult to determine by known asymptotic formulas. Calculations confirm that the obtained approximate formula does not give large errors. In order to verify the adequacy of the obtained analytical solutions, numerical computer integration of the original nonlinear differential equation of motion was performed. The results of the calculation, which lead to analytical and numerical solutions of the Cauchy problem, are well matched.

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.

scholarly journals An analytical approximation of the transient response of a voltage dependent supercapacitor model

2019 ◽  
Vol 39 (3) ◽  
pp. 310-319
Author(s):  
Tomislav Barić ◽  
Hrvoje Glavaš ◽  
Ružica Kljajić
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Nonlinear Differential Equation ◽  
Exact Analytical Solution ◽  
Initial Response ◽  
Analytical Approximation ◽  
Weight Functions ◽  
Transient Phenomenon ◽  
Voltage Dependent ◽  
Analytical Expressions

Supercapacitors are well known for their voltage dependent capacity. Due to this, it is not possible to obtain the exact analytical solution of the nonlinear differential equation which describes the transient charging and discharging. For this reason, approximations of differential equations must be carried out in order to obtain an approximate analytical solution. The focus of this paper is on a different approach. Instead of approximating the differential equation and obtaining analytical expressions for such approximations, an intuitive approach is chosen. This approach is based on the separation of the initial response from the rest of the transient phenomenon. Both parts of the transient phenomenon are described with adequate functions. Using appropriate weight functions, both functions are combined into a single function that describes the whole transient phenomenon. As shown in the paper, such an approach gives an excellent description of the whole transient. Also, it provides simpler expressions compared to those obtained by approximation of the nonlinear differential equation. With respect to their accuracy, these expressions do not lag behind the aforementioned approach. The validity of the presented analytical expressions was confirmed by comparing their results with those obtained by numerically solving the nonlinear differential equation.

Load-Supporting Fluid-Filled Cylindrical Membranes

1985 ◽  
Vol 52 (4) ◽  
pp. 913-918 ◽  
Author(s):  
V. Namias
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Entire Range ◽  
Exact Analytical Solution ◽  
Asymptotic Expressions ◽  
Flexible Membranes ◽  
Approximate Expressions

When long cylindrical flexible membranes are filled with a fluid and used to support external weights, the shape they assume and the relevant geometrical and dynamical quantities are governed by a nonlinear differential equation subject to particular boundary conditions. First, a complete and exact analytical solution is obtained for an unloaded membrane. Very accurate approximate expressions are derived directly from the exact solution for the entire range of applied pressures and fluid densities. Next, the nonlinear differential equation is solved exactly under boundary conditions corresponding to the loading of the membrane. Simple asymptotic expressions are also obtained in the limit of large loads.

scholarly journals Oscillator with a Sum of Noninteger-Order Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
L. Cveticanin ◽  
T. Pogány
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Equations Of Motion ◽  
Dominant Term ◽  
Advantages And Disadvantages ◽  
Polynomial Type ◽  
Polynomial Nonlinearity ◽  
Non Linear ◽  
Linear Differential

Free and self-excited vibrations of conservative oscillators with polynomial nonlinearity are considered. Mathematical model of the system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the case when only one nonlinear term exists, the exact analytical solution of the differential equation is determined as a cosine-Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non-linear order. Exact solutions of the averaged differential equations of motion are obtained. The obtained results are compared with “exact” numerical solutions and previously obtained analytical approximate ones. Advantages and disadvantages of the suggested procedure are discussed.

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.

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 The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Sixing Shi
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Efficient Method ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Integrodifferential Equations ◽  
Ordinary Nonlinear Differential Equation ◽  
Weakly Singular ◽  
Adomian Decomposition

The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solventh-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the givennth-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solventh-order nonlinear Volterra integro-differential equations.

Vibration Analysis of Axially Compressed Nanobeams and its Critical Pressure Using a New Nonlocal Stress Theory

2011 ◽  
Vol 105-107 ◽  
pp. 1788-1792 ◽  
Author(s):  
Cheng Li ◽  
C.W. Lim ◽  
Zhong Kui Zhu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Bending Moment ◽  
Nonlocal Elasticity Theory ◽  
Equation Of Motion ◽  
Basic Element ◽  
Mode Shapes

The transverse vibration of a nanobeam subject to initial axial compressive forces based on nonlocal elasticity theory is investigated. The effects of a small nanoscale parameter at molecular level unavailable in classical mechanics theory are presented and analyzed. Explicit solutions for natural frequency, vibration mode shapes are derived through two different methods: separation of variables and multiple scales. The respective numerical solutions are in close agreement. Validity of the models and approaches presented in the work are verified. Unlike the previous studies for a nonlocal nanostructure, this paper adopts the effective nonlocal bending moment instead of the pure traditional nonlocal bending moment. The analysis yields an infinite-order differential equation of motion which governs the vibrational behaviors. For practical analysis and as examples, an eight-order governing differential equation of motion is solved and the results are discussed. The paper presents a complete nonlocal nanobeam model and the results may be helpful for the application and design of various nano-electro-mechanical devices, e.g. nano-drivers, nano-oscillators, nano-sensors, etc., where a nanobeam acts as a basic element.

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