scholarly journals On the dynamics of a particle inside a U – shaped rotating thin tube

2018 ◽  
Vol 27 (2018) ◽  
pp. 41-46
Author(s):  
Dumitru Deleanu
Keyword(s):  
Energy Balance ◽  
Governing Equation ◽  
Equilibrium Points ◽  
Relative Equilibrium ◽  
Equation Of Motion ◽  
Energy Balance Method ◽  
Analytical Technique ◽  
Strongly Nonlinear ◽  
Thin Tube ◽  
Nonlinear Motion

In this paper the problem of the strongly nonlinear motion of a particle on a rotating parabola is generalized for an arbitrary U-shaped curve. The governing equation of motion is deducted and then particularized on three cases, namely quadratic parabola, quartic parabola and cosine curve. Each case is numerically investigated for various small and large parameters and the results are contrasted with those provided by a relatively new analytical technique called energy balance method. The importance of the relative equilibrium points on the particle’s dynamics is highlighted.

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.

An analytical technique for solving quadratic nonlinear oscillator

2017 ◽  
Vol 13 (3) ◽  
pp. 424-433 ◽  
Author(s):  
Md. Helal Uddin Molla ◽  
Md. Abdur Razzak ◽  
M.S. Alam
Keyword(s):  
Energy Balance ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Design Methodology ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Energy Balance Method ◽  
Content Type ◽  
Analytical Technique ◽  
Present Technique

Purpose The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the approximate solution of quadratic nonlinear oscillator (QNO). Design/methodology/approach This oscillator (QNO) is used as a mathematical model of the human eardrum oscillation. Findings It has been shown that the results by the present technique are very close to the numerical solution. Originality/value The results obtained in this paper are compared with those obtained by Hu (harmonic balance method) and Khan et al. The result shows that the method is more accurate and effective than harmonic balance as well as improved energy balance methods.

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.  

Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

2015 ◽  
Vol 7 (1) ◽  
pp. 43-57 ◽  
Author(s):  
H. Askari ◽  
D. Younesian ◽  
Z. Saadatnia
Keyword(s):  
Energy Balance ◽  
Closed Form ◽  
Natural Frequency ◽  
Governing Equation ◽  
Nonlinear Oscillations ◽  
Quadratic Function ◽  
Balance Method ◽  
Closed Form Expression ◽  
Energy Balance Method ◽  
Elevator Cable

AbstractThis paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian’s method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator’s drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.

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