Solution to the uncoupled system
The solution to the uncoupled equations is obtained readily as a particular integral for the loading and a solution to the homogeneous equation (with no right-hand side). These solutions can be combined and written in the general form
where aij and bij , i,j=1,2 are constants, since we have assumed that the loading only varies linearly over the time increment (that is, Δf/Δt is constant).
There are three cases of this solution for nonrigid body motion ( ω≠0 ), depending on whether the damping in the modal equilibrium equation is greater than, equal to, or less than critical damping (that is, depending on whether (ξ2-1) is positive, zero, or negative).