Parametric modal analysis of mechanical systems with an application to a ball screw model

In control design for mechatronic systems, it is crucial to know the characteristic frequencies of the system. They are not necessarily constant but may vary in dependence on the environment conditions or the operating point. A method to obtain a parameter dependent representation of the characteristic frequencies of a second order system with negligible damping is presented. Starting from a polynomial representation of the mass and stiffness matrix, a Taylor series type representation of the characteristic frequencies is obtained. As a by-product, Taylor series expansions for the corresponding eigenmodes are obtained. The method is applied to a ball screw model, which are mechanical structures with high dynamics and significant resonance behavior. The resonance frequencies depend especially on slide position and load mass which vary during normal operation. It is crucial for controller design to know the resonance frequencies in dependence on the current operating conditions to avoid excitation at these frequencies. It is shown that the described method captures the resonance frequency change for a broad range of operating conditions.