A simplified approach to calculate the damping ratio matrix for multiple degree of freedom vibration systems

Abstract In modal analysis of damping systems, the criteria matrix of damping ratios is of importance for both experiments in identification of structure parameters and theoretical analysis of assessing dynamical response. In a single degree of freedom system, the damping ratio can be defined easily and obtained by some simple calculations. So the vibration response envelope can be predicted, which is important for passive isolation and active control of vibration in dynamical design. Likewise, in a multiple degree of freedom system, the concept can be defined similarly to the single degree of freedom,1 while it involves complicated manipulations and calculations to obtain the damping ratio matrix in terms of mass, damping and stiffness matrices. Some available stimulating references have discussed the damping ratio matrix and presented some alternative criteria matrices.1–4 However, the final formula to evaluate the damping ratios is quite complex and it needs several eigensolutions. If the mass matrix is formed by uniform interpolation to the stiffness, the calculations are even greater than that in the state space calculating eigensolutions directly. Regarding these, a new approach is presented in this paper in terms of modal parameters to calculate the damping ratio matrix. Therefore, the damping ratio matrix takes the parallel form to the single degree of freedom system, and the calculations are greatly reduced. Further, the paper also discusses the case of incomplete sets of eigenpairs, which is of practical importance since for most large structures the complete eigensolution is either unnecessary or computer‐timeconsuming and inaccurate for higher modes; in finite‐element modelling only lower‐order Ritz vectors or even static deformation modes in normal cases are employed.