A new exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness
The author presents a new exact approach for determining natural frequencies and mode shapes of one-step and multi-step nonuniform shear beams with concentrated masses and rectilinear springs. Using a transformation, the author reduces the governing equation of free vibrations to a second-order differential equation without first-order derivatives. An exact solution of the reduced differential equation is found, and it is shown that the basic solutions have a unit matrix property which is used to derive recurrent formulae. The mass distribution of the considered non-uniform shear beam can be arbitrary, whereas the stiffness distribution is a function of the mass distribution, and vice versa. The author presents numerical examples which demonstrate that the proposed procedure is an exact method, and there is no need to perform any matrix multiplication for free vibration analysis of a multi-step non-uniform beam with many translational springs.
scientific article; zbMATH DE number 1501291