Journal of Theoretical
and Applied Mechanics

14, 2, pp. 237-252, Warsaw 1976

Iteracyjna metoda wyznaczania częstości drgań własnych i amplitud układu o skończonej liczbie stopni swobody

Bohdan Kowalczyk, Tadeusz Ratajczak
The paper presents a new iterative method to determine the natural frequencies and modes of a multidegree-of-freedom system. The matrix equation (obtained from the second form Lagrange equation) is transformed to the standard form. Applying the Banachiewicz decomposition to the A-matrix, a series of matrices is generated. Every element of this series is represented as a product of two matrices: an orthogonal matrix and an upper triangular matrix. It is proved in the paper that the limit of the series is a product of orthogonal and diagonal matrices. The columns of the orthogonal matrix are the natural modes, and the non-zero elements of the diagonal matrix are the natural frequencies of the system. This method can be applied to calculate vibrations of structures by means of the finite element method, the rigid finite elernent method or the hybrid method. The method presented is a computer – oriented one. A new method of inverting triangular matrices is also presented; this method makes the transformation of equations to the standard form easy.