THEORY AND ALGORITHM OF THE EXACT METHOD FOR FREE VIBRATION PROBLEMS OF SKELETAL STRUCTURES

The solution of free vibration problem of skeletal structures provides important data for engineering design. This paper surveys a new exact method which uses exact dynamic stiffnesses and therefore results in a nonlinear eigenvalue problem in contrast to the traditional finite element method which reduces the problem into a linear algebraic eigenvalue problem by using approximate shape functions. The well-established Wittrick-Williams algorithm can be used to calculate the frequencies of the nonlinear problem elegantly. Recently the authors presented a new Newton-type method for this nonlinear eigenvalue problem, incorperating many optimum techniques. The recursive use of the proposed method employs the Wittrick-Williams algorithm to guide and guard each Newton correction and therefore gives secure second order convergence on both natural frequencies and mode vectors. Numerical examples show that this method is notonly exact and reliable but also very economical and efficient, and therefore is well applicable to large-scale engineering problems.