The n x n symmetric damping matrix [C] is formulated as a linear combination of the mass [M] and stiffness [K] matrices:

(Equation 1)

Alpha Coefficient: Sets the mass-proportional coefficient α.

Beta Coefficient: Sets the stiffness-proportional coefficient β.

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 c_{i} for the ith mode is calculated by:

(Equation 4),

and the viscous damping ratio ζ_{i} is expressed as

(Equation 5)

If the damping ratios for the ith and jth modes are ζ_{i} and ζ_{j}, then the Rayleigh coefficients α and β are calculated from the solution of the two algebraic equations:

(Equation 6)

If both modes have the same damping ratio ( ζ_{i} = ζ_{j} = ζ) , then the values of α and β are given by:

(Equation 7) (Equation 8)

The viscous damping ratio ζ for any other mode varies with frequency as shown in the figure:

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