Matlab：将二阶矩阵微分方程化简为标准特征问题

Question

I want to obtain the natural frequencies of a simple mechanical system with mass matrix M and stiffness matrix K (.mat-file -> Download ):

M x''(t)+K x(t)=0 (x= Position).

It means basically, that I have to solve det(Kw^2*M)=0. But how can I solve it in Matlab (or if necessary reduce it to a standard eigenvalue problem and solve it then)? The matrices are definitely solvable with Abaqus (FEM Software), but I have to solve it in Matlab.

I tried the following without success: det(Kw^2*M)=0 => det(M^-1*Kw^2*I)=0 (I := unity matrix) But solving this eigenvalue problem with

sqrt(eigs(K*M^-1)) 

delivers wrong values and the warning:

"Matrix is singular to working precision.

In matlab.internal.math.mpower.viaMtimes (line 35)"

Other wrong values can be obtained via det(Kw^2*M)=0 => det(I/(w^2)-M*K^-1)=0:

1./sqrt(eigs(M*K^-1))

Any hint would help me. Thanks in advance.

Answer 1

As @Arpi mentioned, you actually want to solve the generalized eigenvalue problem:

K*x = w^2*M*x

Since your matrices K and M are apparently singular (or just one of them), it is not possible to use eigs , but you have to use eig :

V = eig(K,M);

w = sqrt(V);