Solving multi DOF damped system using block matrix and matrix exponential

Question

I'm trying to solve a multi DOF damped system for $U$. $[M]$ and $[K]$ are symmetric nxn matrices, but $[C]$ is not symmetric. Can I use the following approach?

1. Put in block matrix form
2. Solve the 1st order differential using exponential matrices

Is it true that because $[C]$ is not symmetrical then it can't have distinct eigenvalues/eigenvectors, making it not possible to diagonalize? But even if that's true, $e^{[A]t}$ can still be found without diagonalizing, right?

Answer 1

Your approach seems fine. By differentiating the candidate $X(t)$ we recover the correct differential equation.

A non-symmetric matrix can have distinct eigenvectors/ eigenvalues. Example $$M= \left(\begin{matrix} 1 & 1 \\ 0 & 2\end{matrix}\right)$$ Is not symmetric but has the following eigensystem $$\lambda_1=1, u_1=\left(\begin{matrix} 1 \\ 0 \end{matrix}\right)$$ $$\lambda_2=2, u_2=\left(\begin{matrix} 1 \\ 1 \end{matrix}\right)$$

There are many ways to compute the matrix exponential. The one you suggest is perfectly fine. The most direct is by using the power series of the exponential$^\dagger$

$$e^{At}=\sum_{n=0}^\infty \frac{A^n t^n}{n!}$$

Which is useful if your matrix has some nice property of powers, eg if $A^2=1$. A different useful method is via the Cayley-Hamilton theorem, for which you'll only need the eigenvalues. There's a worked example using this on the wiki page.

$\dagger$ Also useful because this definition of the matrix exponential is always well defined (the sum converges).

Answer to question in comment: "why is the damping approximated?" I can't comment on the engineering textbooks, but I assume the approximation is done in order to get a (set of) linear differential equations, which can then be cast into matrix form. Things are not so simple if your damping term is eg $ u\dot{u}$, the resulting equation is nonlinear and all the nice things done above won't work: the solution can't simply be cast into matrix form. The approximation could also be done to render your matrix $A$ time independent. If $A$ is time dependent (and the differential equations are linear), the method you described won't directly work, but the solution may still be cast in matrix form (see eg. A. Prosperetti: Advanced mathematics, ch18)