9.6 Proportional Damping

The equation of motion for the shaft element with damping for free vibrations can be written as

where is the element mass matrix, is the element damping matrix, is the element stiffness matrix, and is the element vector of nodal degrees of freedom (DOFs). The damping matrix of the beam element can be expressed in terms of the element mass and stiffness matrices as (Clough and Penzien, 1993)

..

          

where and are the system fundamental and highest natural frequencies of interest. For most of the engineering structures the damping ratio is of the order of 0.01. However, it is advisable to estimate experimentally the modal damping ratios for relevant modes from the real system.