In the present work, the Rayleigh—Ritz method (RRM) and the penalty function method (PFM) are used to solve vibration problems of shallow shells of rectangular planform with spherical, cylindrical, and hyperbolic paraboloidal geometries with classical boundary conditions. Problems with more complicated constraints can also be solved using the method presented in this work, which consists in developing the stiffness and mass matrices of a completely free shallow shell using the RRM and then all constraints of the shell are defined by the PFM through the use of either artificial stiffness or artificial inertia with either positive or negative values.

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SUMMARY In a recent paper, it was shown that for time domain analysis, the simultaneous use of inertial and stiffness type penalty parameters to enforce constraints was found to yield accurate and converging results without causing any stability problems. From a frequency domain perspective, this is somewhat unexpected because the solution converges from below when stiffness penalty parameters are used to model constraints, and the convergence is from above when inertial penalty parameters are used. The purpose of this paper is to explain the effect of the simultaneous use of stiffness and inertial penalty parameters on the natural frequencies of constrained systems. In this work, it is shown that if suitably tuned, the bipenalty approach works well for frequency domain analysis also, and that with two different tuned set of stiffness and inertial penalty parameters, bounded solutions to the natural frequencies of constrained systems may be obtained. The method is applicable for any linear eigenvalue problem. Copyright © 2012 John Wiley & Sons, Ltd.

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