Rayleigh Damping
The n x n symmetric damping matrix [C] is formulated as a linear combination of the mass [M] and stiffness [K] matrices:
(Equation 1)
The type of damping described by (Equation 1) is known as Rayleigh or proportional damping.
This form of [C] is orthogonal with respect to the system eigenvectors.
By applying the modal coordinate transformation, the modal damping matrix [c] becomes diagonal:
(Equation 2)
You can define Rayleigh damping for linear and nonlinear dynamic studies.
Relation of Rayleigh Coefficients and Modal Damping Ratio
The modal damping matrix [c] is given by:
(Equation 3)
The coefficient of viscous damping ci for the ith mode is calculated by:
(Equation 4),
and the viscous damping ratio zi is expressed as
(Equation 5)
If the damping ratios for the ith and jth modes are z
i and z
j, then the Rayleigh coefficients a and b are calculated from the solution of the two algebraic equations:
(Equation 6)
If both modes have the same damping ratio ( z
i = z
j = z) , then the values of a and b are given by:
(Equation 7)
(Equation 8)
The viscous damping ratio z for any other mode varies with frequency as shown in the figure:
Related topics
Global Damping