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 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.

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.

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 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 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.  

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

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