Semi-Analytical Approach to the Non-Linear Dynamic Response Problem of S–s and C–c Beams at Large Vibration Amplitudes Part I: General Theory and Application to the Single Mode Approach to Free and Forced Vibration Analysis

Azrar, L.; Benamar, Rhali; White, R. G.

doi:10.1006/jsvi.1998.1893

Cited by 166 publications

(98 citation statements)

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“…The top surface of the FG beam is ceramic rich (E c =322.03 GPa, ρ c =2370 Kg/m 3 ), whereas the bottom surface of the FG beam is metal rich (E m =207.08 GPa, ρ m =8166 Kg/m 3 ). In table 1, It is noted that the first nonlinear frequency ratios ω nl /ω l , calculated in the present work at various vibration amplitudes in the case of an isotropic clampedclamped beam, agree very well with the results obtained in [8], since the percentage error does not exceed 0.3%. However, the solutions given in [7] overestimate the frequencies of the clamped beams, especially for high values of dimensionless amplitude.…”

Section: Numerical Results and Discussionsupporting

confidence: 85%

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Large amplitude free vibration analysis of functionally graded beams using an homogenisation procedure

Bikri

Zerkane

Benamar³

2012

MATEC Web of Conferences

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Abstract. The purpose of the present paper is to show that the problem of geometrically non linear free vibrations of functionally graded (FG) beams with immovable ends can be reduced to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness parameters. The material properties of the functionally graded composites examined are assumed to be graded in the thickness direction and estimated through the rule of mixture. The theoretical model is based on the Euler-Bernouilli beam theory and the Von Kármán geometrical nonlinearity assumptions. An homogenization procedure is developed using the governing axial equation of the beam in which the axial inertia and damping are ignored. Hamilton's principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters, which are found to be in a good agreement with the published results. The non-dimensional curvatures associated to the nonlinear fundamental mode are also given in the case of clamped-clamped FG beams.

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Section: Numerical Results and Discussionsupporting

confidence: 85%

“…Expression (10) for the total strain energy is effective for replacing the FGM beam problem with an equivalent classical isotropic beam problem [8]. The kinetic energy, in which the axial and rotary inertia are ignored, is given by:…”

Section: Non Linear Free Vibration Analysismentioning

confidence: 99%

Large amplitude free vibration analysis of functionally graded beams using an homogenisation procedure

Bikri

Zerkane

Benamar³

2012

MATEC Web of Conferences

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“…The purpose of the present paper is to demonstrate the efficacy of the proposed method by applying to the large amplitude free vibrations of a uniform, shear flexible hinged-hinged and clamped-clamped beams in conjunction with the coupled displacement theory and the principle of conservation of total energy. The results are obtained in closed form expressions of the non-linear to linear radian frequency ratios for simply supported and clamped beams at various modes for various slenderness and maximum amplitude ratios, and are obtained using the harmonic balance method [10]. Comparison of the present results with those existing in literature for the first mode for shear flexible beams shows that the present results obtained from the concept of coupled displacement field analysis are accurate.…”

Section: Introductionmentioning

confidence: 48%

“…Using the Harmonic balance method (HBM) [10] as discussed in the next section can solve equation (23). …”

Section: Large Amplitude Vibrationsmentioning

confidence: 99%

Large amplitude free vibrations of Timoshenko beams at higher modes using coupled displacement field method

Bhaskar

Saheb

2015

J. Phys.: Conf. Ser.

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Abstract. A simple but accurate continuum solution for the shear flexible beam problem using the energy method involves in assuming suitable single term admissible functions for the lateral displacement and total rotation. This leads to two non-linear temporal differential equations in terms of the lateral displacement and the total rotation and are difficult, if not impossible, to solve to obtain the large amplitude fundamental frequencies of beams as a function of the amplitude and slenderness ratios of the vibrating beam. This situation can be avoided if one uses the concept of coupled displacement field where in the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation. In this paper the lateral displacement field is assumed and the field for the total rotation is evaluated through the coupling equation. This approach leads to only one undetermined coefficient which can easily be used in the principle of conservation of total energy of the vibrating beam at a given time, neglecting damping. Finally, through a number of algebraic manipulations, one gets a nonlinear equation of Duffing type which can be solved using any standard method. To demonstrate the simplicity of the method discussed above the problem of large amplitude free vibrations of a uniform shear flexible hinged beam at higher modes with ends immovable to move axially has been solved. The numerical results obtained from the present formulation are in very good agreement with those obtained through finite element and other continuum methods for the fundamental mode, thus demonstrating the efficacy of the proposed method. Also some interesting observations are made with variation of frequency Vs amplitude at different modes.

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“…The beams are also used to model rotating blades dynamics, for example blades of a helicopter rotor [4,5] or blades of turbines. Many authors use the classical FEM [6,7] and semi-analytical methods [8,9] to study this problem.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

2014

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Large amplitude vibrations of a Timoshenko beam under an influence of temperature are analysed in this paper. In the considered model the temperature increases instantly and the heat is uniformly distributed along the beams length and crosssection. The mathematical model, represented by partial differential equations takes into account thermal and mechanical loadings. Next, the problem is reduced by means of the Galerkin method, considering the first three natural vibration modes of a simply supported beam in the both ends. The influence of the temperature on amplitudes and localisation of the resonance zones and stability of the solutions is studied numerically and analytically by the multiple time scale method. The bifurcation points, existence of unstable lobes and transition from regular to chaotic oscillations are shown.

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Semi-Analytical Approach to the Non-Linear Dynamic Response Problem of S–s and C–c Beams at Large Vibration Amplitudes Part I: General Theory and Application to the Single Mode Approach to Free and Forced Vibration Analysis

Cited by 166 publications

References 0 publications

Large amplitude free vibration analysis of functionally graded beams using an homogenisation procedure

Large amplitude free vibration analysis of functionally graded beams using an homogenisation procedure

Large amplitude free vibrations of Timoshenko beams at higher modes using coupled displacement field method

Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

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