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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 278
Calculation of Eigenvalues of a Piezoelastic Beam with the Pseudospectral Method M. Kekana
Centre for Advanced Materials, Manufacturing and Prototyping Research, Department of Mechanical Engineering, Durban Institute of Technology, South Africa Full Bibliographic Reference for this paper
M. Kekana, "Calculation of Eigenvalues of a Piezoelastic Beam with the Pseudospectral Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 278, 2006. doi:10.4203/ccp.83.278
Keywords: Euler beam, eigenvalue problem, pseudospectral method, piezoelastic.
Summary
A number of mathematical models of piezoelastic structures (beams, plates and shells) derived using various theories have been reported. Two approaches in feedback control using piezoelectric sensor and actuator pair have been adopted. One assumes the measured signal to be the electric current, hence the closed circuit mode. The other assumes the measured signal to be the electric potential, hence the open circuit mode. This study is concerned with the latter approach; that is, the open circuit mode.
Analytical and approximate methods have been implemented with great success in solving the differential equation of motion of piezoelastic structures. These include the classical method [1], Fourier analysis [2], Green's method [3], finite element method [4], amongst others. However, the literature reviewed in the field of piezoelastic structures is devoid of the use of the pseudospectral method in solving the piezoelastic differential equations. Very few articles in solid mechanics use this method compared with those in fluid mechanics [5]. This study contributes by solving the same using the psuedospectral method. The method is elegant, highly accurate and does not require mathematical rigour of the methods mentioned above.
The study considers the free vibration problem of a piezoelastic beam. The differential equation of motion is derived based on the Euler beam theory. The kinematic behaviour is represented by the variable
where the basis functions ![]() ![]() ![]() ![]() ![]() ![]()
If the eigenmodes are needed, they may be evaluated by substituting the appropriate matrix eigenmode as the coefficients of
With a bit of care, the pseudospectral method is a very robust and efficient method for solving linear differential eigenvalue problems. The exponential rate of convergence with References
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