I have used the Galerkin method in order to acquire the Mass and Stiffness matrices and by calculation of the natural frequencies and comparing them with FEM software, the matrices seem to be valid. I have to say that the assumed mode shapes of the problem are of the Bessel kind. by the use of foresaid matrices and 'ode45' solver of MATLAB when it comes to time dynamic response to the mentioned load, the solution always diverges. the force matrix is already validated by checking the static response of the plate. I have reduced the time step, time span, relative and absolute tolerances and also set the 'NormControl' option to 'on' in order to reduce to solution's errors but unfortunately nothing changed. so what the problem could be? should I change the solver to another one such as Newmark-Beta ? by the way, the Mass matrix determinant is about 0.006 and is a little close to singularity but it haven't caused any trouble in calculating the natural frequencies where we need the inverse of this matrix the same as calculating the dynamic response, however in calculation of time response I have used 'pinv' function for the inversion of the Mass matrix, which is used in such cases to increase the accuracy.
Second order dynamic systems that present structure dynamic are usually ill conditioned!
Try using the Newmark Method for time response, it helped me a lot before.
there is also the Wilson-Theta method (I never used it) which is quite popular for the time response of structures.
What are your conditions numbers (rcond(M), rcond(K))?
Also, are you working in the state-space? This usually helps a lot if you are trying to integrate equations of motion. Moreover, since there is no damping in your eqution of motion, you should consider using ode15s (for stiff problems) or ode23s (might be more effective than ode15s for equation with crude error tolerances).
Note (transition to the state-space): Let me assume that your equation of motion is of the form
M . \ddot{q} + K . q = f(q,t).
After introducing the following identity
M . \dot{q} - M . \dot{q} = 0,
you can transform resulting system of equations into the form
\dot{u} = A . u(t) + b(q, t), (1)
where
A = [-M-1 . B, -M-1 . K; E, 0] = [0, -M-1 . K; E, 0]
u = [\dot{q}; q]
b = [M-1 * f(q,t); 0]
and E is the identity matrix. Equation (1) is usually well-conditioned and can be solved by ode directly.
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