scholarly journals Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1420
Author(s):  
Pilipchuk
Keyword(s):  
Closed Form ◽  
Free Vibrations ◽  
Nonlinear Case ◽  
Elementary Functions ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Periodic Processes ◽  
The Family ◽  
Linear And Nonlinear Systems ◽  
Strongly Nonlinear Oscillator

Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves.

AN ORDERING OF MEASURES OF THE WELFARE COST OF INFLATION IN ECONOMIES WITH INTEREST-BEARING DEPOSITS

2012 ◽  
Vol 16 (5) ◽  
pp. 732-751 ◽  
Author(s):  
Rubens Penha Cysne ◽  
David Turchick
Keyword(s):  
Closed Form ◽  
Relative Error ◽  
Welfare Cost ◽  
Opportunity Costs ◽  
Maximum Relative Error ◽  
Elementary Functions ◽  
Closed Form Solutions ◽  
Welfare Measure ◽  
Welfare Cost Of Inflation ◽  
Alternative Measures

This paper builds on Lucas [Econometrica 68 (2000), 247–274] and on Cysne [Journal of Money, Credit and Banking 35 (2003), 221–238] to derive and order six alternative measures of the welfare costs of inflation (five of them already existing in the literature) for any vector of opportunity costs. We provide examples and closed-form solutions for each welfare measure based both on log–log and on semilog money demands, whenever possible in terms of elementary functions. Estimates of the maximum relative error a researcher can incur when using any of these measures are given. Everything is done for economies with or without interest-bearing deposits.

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Themistoklis P. Sapsis ◽  
Alexander F. Vakakis
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Resonance Condition ◽  
Asymptotic Results ◽  
Strongly Nonlinear ◽  
Singular Perturbation Analysis ◽  
Local Bifurcation Analysis ◽  
The Family ◽  
Subharmonic Orbits ◽  
Strongly Nonlinear Oscillator

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

Dynamic Response of a Timoshenko Beam to a Moving Force

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Paweł Śniady
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Constant Velocity ◽  
Free Vibrations ◽  
Dynamical Response ◽  
Transverse Displacement ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Moving Force

We consider the dynamical response of a finite, simply supported Timoshenko beam loaded by a force moving with a constant velocity. The classical solution for the transverse displacement and the rotation of the cross section of a Timoshenko beam has a form of a sum of two infinite series, one of which represents the force vibrations (aperiodic vibrations) and the other one free vibrations of the beam. We show that one of the series, which represents aperiodic (force) vibrations of the beam, can be presented in a closed form. The closed form solutions take different forms depending if the velocity of the moving force is smaller or larger than the velocities of certain shear and bar velocities.

Analytical modal analysis of thin-film flat lenses

2005 ◽  
Vol 219 (1) ◽  
pp. 55-59 ◽  
Author(s):  
H R Hamidzadeh ◽  
L Moxey
Keyword(s):  
Thin Film ◽  
Analytical Solution ◽  
Dynamic Response ◽  
Modal Analysis ◽  
Closed Form ◽  
Material Properties ◽  
Mathieu Equation ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Closed Form Solutions

The free vibrations of circular and elliptical thin-film lens are investigated. In particular, linear closed-form solutions for free vibrations of these structures were achieved and modal analysis was performed. The vibration response of the thin-film membranes were mathematically modelled using the Mathieu equation. Numerical results for various nodal diameters were computed. For the limited case, when an elliptical lens becomes circular, an excellent comparison was established with the available analytical solution. Experimental analyses were conducted to determine the effects of various parameters, such as material properties, membrane pre-strain rate, and the geometry, on natural frequency and mode shapes of these structures. The comparison verified the adequacy of linear solutions to predict the dynamic response of thin-film lenses.

Analytical Modal Analysis of Thin-Film Flat Lenses

2003 ◽  
Author(s):  
L. Moxey ◽  
H. Hamidzadeh
Keyword(s):  
Thin Film ◽  
Analytical Solution ◽  
Dynamic Response ◽  
Modal Analysis ◽  
Closed Form ◽  
Material Properties ◽  
Mathieu Equation ◽  
Free Vibrations ◽  
Vibration Response ◽  
Closed Form Solutions

Dynamics of circular and elliptical thin-film lens are investigated. In particular, linear closed-form solutions for free vibrations of these structures were achieved and modal analysis was performed. The vibration response of the thin film membranes were mathematically modeled using Mathieu equation. Numerical results for various nodal diameters were computed. For the limited case when an elliptical lens becomes circular, an excellent comparison was established with the available analytical solution. Experimental analyses were conducted to determine the effects of various parameters such as material properties, membrane pre strain and the geometry on the dynamic response of these structures. The comparison verified the adequacy of linear solutions to predict the dynamic response of thin film lenses.

Sign in / Sign up

Export Citation Format

Share Document