Damping matrix

In applied mathematics, a damping matrix is a matrix corresponding to any of certain systems of linear ordinary differential equations. A damping matrix is defined as follows. If the system has n degrees of freedom u_n and is under application of m damping forces. Each force can be expressed as follows:

${\displaystyle f_{Di}=c_{i1}{\dot {u_{1}}}+c_{i2}{\dot {u_{2}}}+\cdots +c_{in}{\dot {u_{n}}}=\sum _{j=1}^{n}c_{i,j}{\dot {u_{j}}}}$

It yields in matrix form;

${\displaystyle F_{D}=C{\dot {U}}}$

where C is the damping matrix composed by the damping coefficients:

${\displaystyle C=(c_{i,j})_{1\leq i\leq n,1\leq j\leq m}}$