# A note on the damped vibrating systems

Abstract: The presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated. The number of such eigenvalues is determined using the rank of a matrix which is directly related to the system matrices.

## Summary (1 min read)

### 1 Introduction and preliminaries

- In this note the authors are interested in the determination of the number of pure imaginary eigenvalues of the system without computing the zeros of the characteristic polynomial (4).
- The main result given in the next section (Theorem 1) is based on the well-known condition of asymptotic stability [5] , which coincides with the rank condition of controllability of a linear system (see [6] ).

### Introduce the n

- The result follows from Lemma 1 and the additional fact that eigenvectors associated with distinct eigenvalues of a symmetric matrix are orthogonal.
- Since the system (2) is stable, the multiple eigenvalue iω must be semi-simple, which means that the eigenvalue has k linearly independent eigenvectors.
- When the matrix K has distinct eigenvalues, and r its eigenvectors lie in the nullspace of the damping matrix, the decomposability of the system in modal coordinates was observed in [3] .
- Since D and K commute there exists an orthogonal matrix such that both D and K are orthogonally congruent to diagonal matrices [4] .

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### Cites background from "A note on the damped vibrating syst..."

...The main result given in section 3 (Theorem 2) recently derived in our paper [6]....

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3 citations

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##### References

28 citations

### "A note on the damped vibrating syst..." refers background in this paper

...The main result given in the next section (Theorem 1) is based on the well-known condition of asymptotic stability [5], which coincides with the rank condition of controllability of a linear system (see [6])....

[...]

...which plays key role in a test for asymptotic stability of the system [5]....

[...]

...Under the above assumptions it is clear that the (n− r) dimensional subsystem of (18) z̈ + D̂n−rż + K̂n−rz = 0, z ∈ <n−r (19) is asymptotically stable and, according to well-known result [5], we have...

[...]

18 citations

### "A note on the damped vibrating syst..." refers background in this paper

...Also, it should be mentioned that the paper [1] rediscovered an old criterion for asymptotic stability of the system [3], as was recently stressed in [4]....

[...]

...When the matrix K has distinct eigenvalues, and r its eigenvectors lie in the nullspace of the damping matrix, the decomposability of the system in modal coordinates was observed in [3]....

[...]

8 citations

### "A note on the damped vibrating syst..." refers background in this paper

...Also, it should be mentioned that the paper [1] rediscovered an old criterion for asymptotic stability of the system [3], as was recently stressed in [4]....

[...]

...Since D and K commute there exists an orthogonal matrix such that both D and K are orthogonally congruent to diagonal matrices [4]....

[...]

8 citations