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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



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