Schauder fixed point theorem based existence of periodic solution for the response of Duffing’s oscillator (original) (raw)

Dynamic Response of a Simply Supported Plate Due to Excitation at Different Points

Second International Conference on Advances in Mechanical, Aeronautical and Production Techniques - MAPT 2014, 2014

The rectangular plate vibration has been of immense interests to the researchers for a long time. In this literature, an investigation is performed to measure the dynamic response at the different points on a rectangular plate due to excitation at different points. The possibility of occurring different modes of vibration at the different points on the plate according to the position of excitation is demonstrated and explained from the modal shape of the plate obtained from the modal analysis. A conventional approach is performed to obtain the modes and the modes are compared with the FEM results. An analytical equation is used to obtain the dynamic deflection of harmonic excitation for free vibration to observe the occurring of different modes and also used to validate the results obtained from the finite element method. Then a constant damping condition is simulated to understand properly the variation of amplitude at different points of plate for different excitation conditions.

On the Dynamic Analysis of a Simply Supported Rectangular Plate

Journal of the Nigerian Association of Mathematical Physics, 2011

The dynamic behaviour of a simply supported rectangular plate is studied. This research work is ‎based on the theory of the orthotropic plate simply supported on two sides and free on two other ‎sides. The plate is excited by a moving load while the dynamic response of the structure was ‎obtained using the classical double Fourier series expansion technique, which satisfies the ‎boundary conditions at the four edges. In the absence of the external excitation, the vibration ‎yields free frequencies, otherwise, forced frequency is produced. The results obtained from the ‎numerical example are in agreement with the ones in the existing literatures. In addition, the ‎effects of vibrations in flexural rigidity and that of the frequencies of vibrations are also presented.‎

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

Volume 4: Dynamics, Control and Uncertainty, Parts A and B, 2012

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

Periodic Solutions for Damped Vibration Problems

Journal of Advances in Applied & Computational Mathematics, 2014

In this paper we are concerned with the following damped vibration problem u!!(t) + g(t)u!(t) = !V(t,u(t)), a.e. t ! [0,T ] u(0) = u(T), u!(0) = u!(T), " # % $ $ where T > 0 , g ! L!(0,T; R) with G(t) = 0 t ! g(s)ds and G(T) = 0 , V(t,u) = 1 2 (L(t)u,u) !W(t,u) is T -periodic in t such that L ! C(R, Rn2 ) is a T -periodic, positive definite symmetric matrix and W satisfies the global AmbrosettiRabinowitz condition or is subquadratic at infinity. By use of the Mountain Pass Theorem or the genus properties in the critical theory, we establish some new criteria to guarantee the existence and multiplicity of periodic solutions. Recent results in the literature are generalized and significantly improved

An Evaluation of the Fixed Point Method of Vibration Analysis for a Particular System With Initial Damping

Journal of Engineering for Industry, 1963

The maximum steady-state response of a particular linear damped two-degree-of-freedom vibratory system is minimized by determining the optimum damping constant for a single damper. This is accomplished by both a well-known approximate method and by an exact numerical method. Since the approximate method does not take into account the damping which is initially in the system, attention in this analysis is directed to determining the influence of the initial damping on the optimum value for the single damper. In order to make direct comparison of the methods, a system was chosen in which an exact numerical determination of the optimum damping was possible. The results of the investigation show for the particular case considered that, although the value of the damping constant for the optimum damper increases considerably as initial damping is included in the system, use of the value obtained for the initially undamped case would give values of the maximum steady-state response within ...

Nonlinear Vibration of Plates and Higher Order Theory for Different Boundary Conditions

Volume 5: 22nd International Conference on Design Theory and Methodology; Special Conference on Mechanical Vibration and Noise, 2010

Numerous applications of plate structures may be found in aerospace and marine engineering. The present study is a continuation of the work by Amabili and Sirwan [1] extending their investigation to laminate composite rectangular plates with different boundary conditions subjected to an external point force. The excitation frequency lies within the neighbourhood of the fundamental mode of the plate. The analysis is performed using three different nonlinear plate theories, namely: i) the classical Von Kárman theory, ii) first-order shear deformation theory, and iii) third-order shear deformation theory. The plates are tested using three sets of boundary conditions: a) classical clamped boundary conditions, b) simplysupported ends with immovable edges, and c) simply-supported ends with movable boundaries. The results discuss the limitations associated with using lower order theory to describe the large-amplitude oscillations of plates, investigate the effect of boundary conditions highlighting the different responses obtained from using isotropic or laminate composite rectangular plates and indicate chaotic oscillations observed for specific values of the excitation force.

On the motion of an oscillator with a periodically time-varying mass

Nonlinear Analysis-real World Applications, 2009

The stability of the motion of an oscillator with a periodically timevarying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [9] to the forced case; second, for a weakly forced Duffing's oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero. relevant example, where the formation of a water rivulet along the upper part of the cable may have a resonant effect. When the mass flow incoming on the cable is different from the mass flow shaken off then the mass of raindrops attached to the oscillator varies in time. According with the deduction made in [10, p.152], this model can be described by the scalar differential equation

Study of Non-Linear Behavior of Vibrating System

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. In the case of the real structures a linear model will be insufficient to describe the dynamic behavior correctly. It thus appears natural to introduce non-linear models of structures which are able to predict the dynamic behavior of the real structures. This seminar includes study of non-linear vibrations, it different types and various applications. Here the vibratory behavior of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters is optimised. The total system is modeled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. one proposes to optimize the vibratory behavior of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modelled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional diagram. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces.

Vibrations of a simply supported plate carrying an elastically mounted concentrated mass

Ocean Engineering, 1993

A large number of papers and technical reports are available on the technically important problem of structural elements executing transverse vibrations and carrying concentrated masses. In general it is assumed that the attachment is perfectly rigid. On the other hand, a rather limited amount of published work is available when the mass is elastically connected to the structure. The system then exhibits a more complex behavior than in the case of rigid attachment. The present study deals with the solution of the title problem using the well-known normal mode, sinusoidal eigenfunction expansions.