Rayleigh optimization concept and the use of sinusoidal co-ordinate functions

P. A. A. LAURA; D. V. BAMBILL; V. A. JEDERLINIC; K. RODRIGUEZ; P. DIAZ

JOURNAL OF SOUND AND VIBRATION

Año: 1997 vol. 200 p. 557 - 561

0022-460X

Lord Rayleigh suggested over a century ago the inclusion of an undetermined exponential parameter in the co!ordinate functions when employing his now famous method 0[ Since his method yields upper bounds for the eigenvalues one is able to optimize them by minimizing the characteristic values with respect to the undetermined parameter[ In the case of a _eld problem one minimizes the functional with respect to the exponential parameter as shown by Bert when solving a heat conduction problem 2[ Professor C[ W[ Bert "University of Oklahoma# and R[ Schmidt "University of Detroit# have contributed signi_cantly to the development of the method by solving numerous important applied mechanics problems[ Other research groups from Argentina "Institute of Applied Mechanics^ Universidad Nacional del Sur^ Facultad Regional Bahia Blanca UTN and Universidad Nacional de Mar del Plata# have also reported some research performed on the subject matter based on Schmidt and Bert|s work[ During the past ten years the approach suggested by Rayleigh "essentially a non!linear optimization procedure# has been applied to a variety of problems^ column buckling beam vibration plate buckling and vibration elastic torsion etc[ It is important to point out that apparently unaware of Lord Rayleigh|s suggestion well known authors such as Stodola 3 Pauling and Bright!Wilson 4 and Timoshenko and Goodier 5 also made use of Rayleigh|s optimization concept[ Exponential co!ordinate functions containing an undetermined exponential parameter have been employed in references 46[ A thorough discussion on application of other de~ection functions with undetermined parameters namely functions with real exponential and trigonometric terms is due to Schmidt in the context of buckling problems 7[ The present note reports some numerical experiments performed on the determination of the fundamental frequency vibration of a rectangular plate with three simply supported edges while the fourth is free^ see Figure 0[ The optimization parameter is contained in the argument of the sinusoidal terms of a truncated Fourier series[ An interesting and rather novel feature of the approach is the fact that the {{base|| function allows for very good engineering accuracy when a single!term approximation is used and constitutes the exact solution of the problem when the optimization parameter is taken equal to unity and the plate is simply supported at its four edges