Proportional damping (Rayleigh damping)

In the course of modal analysis, the terms in the equation of motion are transformed using the (Modal Analysis:modal matrix). As the eigenvectors are orthogonal with respect to the mass and stiffness matrix, this transformation leads to a diagonal mass $//<![CDATA[ \begin{array}{l}\mathbf{M^*}\end{array} //]]>$ and stiffness matrix $//<![CDATA[ \begin{array}{l}\mathbf{K^*}\end{array} //]]>$. Hence, the equations of motion for all $//<![CDATA[ \begin{array}{l}n\end{array} //]]>$ degrees of freedom are decoupled.

For an arbitrary damping matrix $//<![CDATA[ \begin{array}{l}\mathbf{C}\neq 0\end{array} //]]>$, the transformation using the undamped eigenmodes does not yield a diagonal generalized damping matrix $//<![CDATA[ \begin{array}{l}\mathbf{C^*}\end{array} //]]>$ and the $//<![CDATA[ \begin{array}{l}n\end{array} //]]>$ equations of motion do not decouple. However, if we consider a damping matrix proportional to stiffness and mass, i.e.

the damping matrix is diagonalizable with respect to the mode shapes $//<![CDATA[ \begin{array}{l}\pmb{\phi}_j\end{array} //]]>$:

This special form of damping is known as proportional damping or Rayleigh-damping. The relationship between the Rayleigh damping coefficients $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\beta\end{array} //]]>$ and the modal damping ratio $//<![CDATA[ \begin{array}{l}D_j\end{array} //]]>$ of the $//<![CDATA[ \begin{array}{l}j\end{array} //]]>$-th eigenmode is given by:

In the Rayleigh-damping model the free parameters $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\beta\end{array} //]]>$have to be chosen. A possibility is to choose desired damping values for two of the mode shapes $//<![CDATA[ \begin{array}{l}\pmb{\phi}_i\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\pmb{\phi}_j\end{array} //]]>$ with eigenfrequencies $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\omega_j\end{array} //]]>$, respectively and adjust the values for $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\beta\end{array} //]]>$ such that

The stiffness proportional part increases for increasing eigenfrequencies $//<![CDATA[ \begin{array}{l}\omega_j\end{array} //]]>$, whereas the mass proportional part decreases for increasing eigenfrequencies $//<![CDATA[ \begin{array}{l}\omega_j\end{array} //]]>$,  as it is proportional to its inverse $//<![CDATA[ \begin{array}{l}\frac{1}{\omega_{j}}\end{array} //]]>$.