Naslov
Damping optimization of linear vibrational systems with a singular mass matrix

Autori
Truhar, Ninoslav ; Petrač, Maja

Vrsta, podvrsta
Radovi u časopisima, znanstveni

Izvornik
Mathematics (2022)

Status rada
Prihvaćen

Ključne riječi
Damping optimization ; Mechanical system ; Singular mass matrix ; Lyapunov equation ; Trace minimization

Sažetak
We present two novel results on small damped oscillations, described by the vector differential equation $M \ddot{;x}; + C \dot{;x}; + K x = 0$, where the mass matrix $M$ can be singular, but standard deflation techniques cannot be applied. %For example, $\mathcal{;N};(M) \cap \mathcal{;N};(C) = \emptyset$. The first result is the novel formula for the solution ${;X};$ of the Lyapunov equation ${;A};^T {;X}; + {;X}; {;A}; = -I$, where ${;A};={;A};(v)$ is obtained from $M, C(v) \in \mathbb{;R};^{;n \times n};$ and $K \in \mathbb{;R};^{;n \times n}; $ so-called mass, damping, and stiffness matrices, respectively and $\rank(M)=n-1$. %In addition, we assume that $K$ is positive definite and Here $C(v)$ is positive semidefinite with $\rank({;C};(v))=1$.% and no internal damping. Using the obtained formula, we propose a very efficiently way for computation of the optimal damping matrix.% $C_{;opt};=v_{;opt}; d_{;opt}; d_{;opt};^T$. The second result was obtained for a different structure where we assume that $\dim(\mathcal{;N};(M))\geq 1$ and internal damping exists (usually a small percentage of the critical damping). For this structure, we will introduce a novel linearization, i.e., a novel construction of the matrix $A$ in the Lyapunov equation $A^T{;X}; + {;X};{;A}; = - {;I};$, and the novel optimization process. The proposed optimization process computes the optimal damping $C(v)$ that minimizes a function $v\mapsto{;\rm trace};({;Z};{;X};)$ (where ${;Z};$ is a chosen symmetric positive semidefinite matrix) using the approximation function $g(v) = c_v + \frac{;a};{;v}; + bv$, for the trace function $f(v) \doteq {;\rm trace};({;Z};{;X};(v))$. The results obtained in both parts are illustrated with several corresponding numerical examples.

Izvorni jezik
Engleski

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