Nonlinear analysis of the forced response of structural elements (original) (raw)
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Non-linear response of buckled beams to 1:1 and 3:1 internal resonances
International Journal of Non-Linear Mechanics, 2013
The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integralpartial-differential equation and associated boundary conditions and obtain four first-order ordinarydifferential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency-response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitudemodulated response.
Advances in nonlinear vibration analysis of structures. PartI. Beams
Sadhana-academy Proceedings in Engineering Sciences, 2001
The development of nonlinear vibration formulations for beams in the literature can be seen to have gone through distinct phases — earlier continuum solutions, development of appropriate forms, extra-variational simplifications, debate and discussions, variationally correct formulations and finally applications. A review of work in each of these phases is very necessary in order to have a complete understanding of the
Journal of the Brazilian Society of Mechanical Sciences
This paper applies the Multi-Harmonic Nonlinear Receptance Coupling Approach (MUHANORCA) (Ferreira 1998) to evaluate the frequency response characteristics of a beam which is clamped at one end and supported at the other end by a nonlinear cubic stiffness joint. In order to apply the substructure coupling technique, the problem was characterised by coupling a clamped linear beam with a nonlinear cubic stiffness joint. The experimental results were obtained by a sinusoidal excitation with a special force control algorithm where the level of the fundamental force is kept constant and the level of the harmonics is kept zero for all the frequencies measured.
Journal of Sound and Vibration, 1999
In a previous series of papers [1-3], a general model based on Hamilton's principle and spectral analysis was developed for non-linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates. As an introduction to the present work, concerned with the forced non-linear response of CC and S-S beams, the above model has been derived using spectral analysis, Lagrange's equations and the harmonic balance method. Then, the forced case has been examined and the analysis led to a set of non-linear partial differential equations which reduces to the classical modal analysis forced response matrix equation when the non-linear terms are neglected. On the other hand, if only one mode is assumed, this set reduces to the Duffing equation, very well known in one mode analyses of non-linear systems having cubic non-linearities. So, it appeared sensible to consider such a formulation as the multidimensional Duffing equation. In order to solve the multidimensional Duffing equation in the case of harmonic excitation of beam like structures, a method is proposed, based on the harmonic balance method, and a set of non-linear algebraic equations is obtained whose numerical solution leads in each case to the basic function
Communications in Nonlinear Science and Numerical Simulation, 2014
Please cite this article in press as: Claeys M et al. Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.04.008 methods have been developed to compute a nonlinear frequency response . Yet the first applications of nonlinear simulation to industrial structures are very recent. In particular Renson and Kerschen [3] applied a time-integration method with a shooting algorithm [4] to compute the nonlinear vibrations of the SmallSat spacecraft, and focused on internal resonances between linear modes. Sinou [5] applied a Harmonic Balance Method to a nonlinear model of an industrial rotor and computed a multi-harmonic frequency response.
Study on the frequency – amplitude relation of beam vibration
International Journal of the Physical Sciences, 2011
New analytical work on the well-known preload nonlinearity using the innovative equivalent function (EF) is presented in this paper. The nonlinear vibration of cantilever beam with nonlinear boundary condition in the presence of preload spring with cubic nonlinearity is studied. The powerful analytical method, called He's Parameter Expanding Method (HPEM) is used to obtain the exact solution of dynamic behavior of the mentioned system. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution. Finally, we successfully compare our analysis with numerical solutions.
On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation
Nonlinear Dynamics, 2004
We investigate the nonlinear response of a clamped-clamped buckled beam to a primary-resonance excitation of its first vibration mode. The beam is subjected to an axial force beyond the critical load of the first buckling mode and a transverse harmonic excitation. We solve the nonlinear buckling problem to determine the buckled configurations as a function of the applied axial load. A Galerkin approximation is used to discretize the nonlinear partial-differential equation governing the motion of the beam about its buckled configuration and obtain a set of nonlinearly coupled ordinary-differential equations governing the time evolution of the response. Single-and multi-mode Galerkin approximations are used. We found out that using a single-mode approximation leads to quantitative and qualitative errors in the static and dynamic behaviors. To investigate the global dynamics, we use a shooting method to integrate the discretized equations and obtain periodic orbits. The stability and bifurcations of the periodic orbits are investigated using Floquet theory. The obtained theoretical results are in good qualitative agreement with the experimental results obtained by Kreider and Nayfeh (Nonlinear Dynamics 15, 1998, 155-177).
Nonlinear vibrations of thin rectangular plates are considered, using the von kármán equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.
Finite element analysis of nonlinear vibration of beam columns
AIAA Journal, 1973
The influence of finite amplitudes on the free flexural vibration response of moderately thick laminated plates is investigated. For this purpose, a simple higher order theory involving only four unknowns and satisfying the stress free conditions at the top and bottom surface of the composite plate is proposed. The proposed theory eliminates the use of shear correction factors which are otherwise required in Mindlin's plate theory. A rectangular four-node C 1 continuous finite element is developed based on this theory. The non-linear finite element equations are reduced to two non-linear ordinary differential equations governing the response of positive and negative deflection cycles. Direct numerical integration method is then employed to obtain the periods or non-linear frequencies. The finite element developed and the direct numerical integration method employed are validated for the case of isotropic rectangular plates. It is found that unsymmetrically laminated rectangular plates with hinged-hinged edge conditions oscillate with different amplitudes in the positive and negative deflection cycles. Furthermore, such plates would oscillate with a frequency less than the fundamental frequency for finite small amplitudes of oscillation. It is shown that this behaviour is strongly influenced by the boundary conditions. Results are presented for many configurations of composite plates.