Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

Kadiri, M. El; Benamar, Rhali

doi:10.1016/s0022-460x(02)01162-8

Cited by 32 publications

(15 citation statements)

References 27 publications

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“…If the time and space functions are supposed to be separable and harmonic motion is assumed, the normal displacement function can be written as: (5) Using the expressions for the axial displacements commonly adopted [6][7][8], the corresponding functions U(x,y,t) and V(x,y,t) are assumed to be:…”

Section: Solution Proceduresmentioning

confidence: 99%

“…The dimensional generalized force f i c corresponding respectively to the concentrated force F c at point (x 0 , y 0 ) for a fully clamped rectangular plate having the characteristics a, b, H, and D, is [6]:…”

Section: Case Of Fcr Plates Excited Harmonically By a Concentrated Forcementioning

confidence: 99%

“…Applying the explicit solution procedure proposed previously in [6] to equations (26) and (27) leads to:…”

Section: Multimode Approach In the Vicinity Of The First Non-linear mentioning

confidence: 99%

“…The expressions for the mass tensor m ij , and the linear rigidity tensor k ij presented in [4,5] are unchanged, but the expression for the non-linear rigidity tensor b ijkl is affected in the present theory. Thus, the general features of the model remain the same and the mathematical treatment of the set of non-linear algebraic equations giving arise to solution of the problem of large vibration amplitudes of rectangular plates is carried out using an explicit analytical solution procedure proposed previously [6]. …”

Section: Introductionmentioning

confidence: 99%

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Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

Merrimi

Bikri

Benamar

2012

MATEC Web of Conferences

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Abstract. The objective of this work is to study the geometrically non-linear steady state periodic forced response of fully clamped rectangular plates (FCRP) with immovable in-plane conditions, taking into account the effect of the in-plane displacements. A complete formulation has been proposed first, reducing the equations of motion to a system of coupled non-linear algebraic equations, which are decoupled once the in-plane inertia is omitted. An averaging technique has then been developed, in order to simplify the first method and to develop an engineering complete theory. The forced response is given in the case of a concentrated harmonic excitation force with various intensities. The numerical results obtained here with the two formulations, using an explicit analytical solution, were compared with those obtained previously using a formulation in which the in-plane displacements have been neglected, showing an "over-stiff" effect..

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Section: Solution Proceduresmentioning

confidence: 99%

Section: Case Of Fcr Plates Excited Harmonically By a Concentrated Forcementioning

confidence: 99%

“…Applying the explicit solution procedure proposed previously in [6] to equations (26) and (27) leads to:…”

Section: Multimode Approach In the Vicinity Of The First Non-linear mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

Merrimi

Bikri

Benamar

2012

MATEC Web of Conferences

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“…Systems of this type, sometimes with a large number of degrees of freedom, typically occur in the analysis of structures constituted by beams and plates, which are often modelled by ordinary differential equations that result from the application of Galerkin, finite-element or similar methods. 10,21,22,[30][31][32][33][34][35][36][37][38][39] In the first method here reviewed, the perturbation to the periodic solution is written as a product of an exponential and a periodic function given by a Fourier series. The HBM is then applied and an eigenvalue problem solved in order to determine the characteristic exponents that define the stability.…”

Section: Introductionmentioning

confidence: 99%

Stability of multi-degree-of-freedom Duffing oscillators

Ribeiro¹

2009

Proceedings of the Institution of Civil Engineers - Engineering

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Different methods of examining the stability of periodic solutions of non-linear multi-degree-of-freedom systems are compared in this paper. The methods are particularly suited for solutions computed with the harmonic balance method and the non-linear systems considered are represented by systems of coupled Duffing-type equations; that is, with cubic non-linear expressions. The latter systems appear frequently in models of thin-walled structures, particularly in straight beams and flat plates. In the first method reviewed in this paper the harmonic balance procedure is applied to define an eigenvalue problem, from which the characteristic exponents are determined. If the real part of any of these characteristic exponents is greater than zero, then the solution is unstable. The second method relies on the sign of the determinant of a Jacobian matrix; a matrix that is also often used to solve the equations of motion. The last method is based on a perturbation procedure and with it a closed-form expression for stable regions is achieved. The advantages and shortcomings of the different methods are discussed. INTRODUCTIONA non-linear dynamic system excited by a harmonic force can have periodic and non-periodic solutions. With regard to the periodic solutions, it is possible that more than one solution exists for a single excitation frequency and the initial conditions determine to which solution the system converges. An unstable steady-state solution does not occur in practice: if a solution is unstable, the system will evolve to another attractor. Hence, it is important to determine whether or not a solution is stable. The local stability can be determined by investigating the evolution of a perturbed solution that results from the addition of a small disturbance. A perturbation near an unstable equilibrium condition leads to a departure from this condition and the inverse occurs near a stable equilibrium condition. Several types and orders of instability can occur and this is a subject widely analysed 1-33 but still of considerable interest.

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The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses

Babahammou

Benamar

2021

Applied Condition Monitoring

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Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

Cited by 32 publications

References 27 publications

Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

Stability of multi-degree-of-freedom Duffing oscillators

The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses

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