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 un and is under application of m damping forces. Each force can be expressed as follows:
f D i = c i 1 u 1 ˙ + c i 2 u 2 ˙ + ⋯ + c i n u n ˙ = ∑ j = 1 n c i , j u j ˙ {\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;
F D = C U ˙ {\displaystyle F_{D}=C{\dot {U}}}
where C is the damping matrix composed by the damping coefficients:
C = ( c i , j ) 1 ≤ i ≤ n , 1 ≤ j ≤ m {\displaystyle C=(c_{i,j})_{1\leq i\leq n,1\leq j\leq m}}