Large Amplitude Free Vibrations of Mindlin Square Plates using Coupled Displacement Field Method

Abstract Complex structures used in various engineering applications are made up of simple structural members like beams, plates and shells. The fundamental frequency is absolutely essential in determining the response of these structural elements subjected to the dynamic loads. However, for short beams, one has to consider the effect of shear deformation and rotary inertia in order to evaluate their fundamental linear frequencies. In this paper, the authors developed a Coupled Displacement Field method where the number of undetermined coefficients 2n existing in the classical Rayleigh-Ritz method are reduced to n, which significantly simplifies the procedure to obtain the analytical solution. This is accomplished by using a coupling equation derived from the static equilibrium of the shear flexible structural element. In this paper, the free vibration behaviour in terms of slenderness ratio and foundation parameters have been derived for the most practically used shear flexible uniform Timoshenko Hinged-Hinged, Clamped-Clamped beams resting on Pasternak foundation. The findings obtained by the present Coupled Displacement Field Method are compared with the existing literature wherever possible and the agreement is good.

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Large Amplitude Free Vibration Analysis of Tapered Timoshenko Beams Using Coupled Displacement Field Method

International Journal of Applied Mechanics and Engineering ◽

10.2478/ijame-2018-0037 ◽

2018 ◽

Vol 23 (3) ◽

pp. 673-688 ◽

Cited By ~ 1

Author(s):

K. Rajesh ◽

K.M. Saheb

Keyword(s):

Fundamental Frequency ◽

Displacement Field ◽

Large Amplitude ◽

Amplitude Ratio ◽

Slenderness Ratio ◽

Maximum Amplitude ◽

Free Vibration Analysis ◽

Free Vibrations ◽

Timoshenko Beams ◽

Functional Requirements

Abstract Tapered beams are more efficient compared to uniform beams as they provide a better distribution of mass and strength and also meet special functional requirements in many engineering applications. In this paper, the linear and non-linear fundamental frequency parameter values of the tapered Timoshenko beams are evaluated by using the coupled displacement field (CDF) method and closed form expressions are derived in terms of frequency ratio as a function of slenderness ratio, taper ratio and maximum amplitude ratio for hinged-hinged and clamped-clamped beam boundary conditions. The effectiveness of the CDF method is brought out through the solution of the large amplitude free vibrations, in terms of fundamental frequency of tapered Timoshenko beams with axially immovable ends. The results obtained by the present CDF method are validated with the existing literature wherever possible.

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Concept of Coupled Displacement Field for Large Amplitude Free Vibrations of Shear Flexible Beams

Journal of Vibration and Acoustics ◽

10.1115/1.2159038 ◽

2005 ◽

Vol 128 (2) ◽

pp. 251-255 ◽

Cited By ~ 11

Author(s):

G. Venkateswara Rao ◽

K. Meera Saheb ◽

G. Ranga Janardhan

Keyword(s):

Displacement Field ◽

Large Amplitude ◽

Lateral Displacement ◽

Nonlinear Ordinary Differential Equation ◽

Free Vibrations ◽

Flexible Beam ◽

Flexible Beams ◽

Uniform Shear ◽

Single Term ◽

Admissible Functions

Continuum solutions for solving the large amplitude free vibration problem of shear flexible beams using the energy method involves assuming suitable admissible functions for the lateral displacement and the total rotation. Use of even, single-term admissible functions leads to two coupled nonlinear temporal differential equations in terms of the lateral displacement and the total rotation, the solution of which is rather involved. This situation can be effectively tackled if one uses the concept of a coupled displacement field wherein the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation of the shear flexible beam. This approach leads to only one undetermined coefficient, in the case of single-term admissible functions, which can easily be used in the principle of conservation of total energy, neglecting damping, to solve the problem. Finally, one gets a nonlinear ordinary differential equation of the Duffing type which can be solved using any available standard method. The effectiveness of the concept discussed above is brought out through the solution of the large amplitude free vibrations, in terms of the fundamental frequency, of uniform shear flexible beams, with axially immovable ends, using single-term admissible functions.

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