Element spaces and their separation for determining bounds to eigenvalues in vibration problems

This paper introduces the concept of convergent eigenvalue transformations and elaborates on its use with the finite element method for obtaining close upper and lower bounds to the eigenvalues of continuous systems in free vibration. It is shown that, when incorporated with the finite element concept, convergent eigenvalue transformations attain a unique form defining an infinite number of convergent elements, whose totality is referred to as an element space. Criteria are derived for the determination of the elements which give upper and lower bounds to the eigenvalues and their applicability is discussed with reference to some vibration, problems of beams. The analysis presupposes the knowledge of a convergent element but creates useful possiblities such as the improvement of known bounds, generation of bounds that are opposite to the known bounds and assessment of the prevalent error laws. © 1978 Academic Press Inc. (London) Limited.