Extension of the modal superposition method for general damping applied in railway dynamics

Abstract

[EN] The frequency response function (FRF) permits to characterise in the frequency domain the systems governed by linear dynamics by means of a relationship between an excitation applied at one degree of freedom (dof) and the consequent output response in a particular dof. A modal approach is widely extended in the engineering fields as efficient method of computing the FRF of matrix second-order linear equations of motion derived from the application of the Finite Element Method (FEM) [1]. This approach is based on the truncation of the number of vibration modes that conform the base of the new modal coordinates. The criterion for the truncation is linked to the frequency range of the dynamic study, ordering the vibration modes with respect to the eigenvalues (the square of natural frequencies) associated. The natural frequency associated with the last vibration mode selected establishes the maximum frequency that can describe the time response of the system. The truncation permits to reduce the dimension of the system from N number of dofs in physical coordinates to m truncated vibration modes in modal coordinates. The fundamental numerical problem derived from the truncation is the resulting non-square vibration modes matrix, used as transformation matrix in the physical to modal change of variable [1,2]. This change should allow the diagonalisation of the matrices involved in the equation of motion: mass, stiffness and damping matrices. The diagonalisation is essential to decouple the system in m second-order linear differential equations that can be solved analytically in the time domain. Nevertheless, the inverse of vibration modes matrix required for the diagonalisation cannot be applied from its non-square nature and it can only be replaced by the transpose matrix if both mass and stiffness matrices are symmetric. The complexity increases in a case of general damping instead of proportional or spectral ones [3], in which the damping matrix must be included in the eigenproblem in order to diagonalise the whole modal system. This work proposes a methodology to overcome the issues abovementioned and applies this in the field of railway dynamics in order to compute the modal properties of a railway wheel modelled by using FEM [4]. The paper includes a study of the numerical performance of this method and its comparison with other numerical procedures to find the FRF of the wheel.