Laminated Timoshenko beam without complementary dissipation

Abstract

In this study, the stability problem of a laminated beam with only structural damping is analyzed. The results obtained in this study improve the analysis of the problem by investigating stability without introducing additional dissipation. This is accomplished by considering only the usual assumption of equal wave velocities as the stability criterion.

Appendix

Our main proof of stability was based on an appeal to the well-known exponential stability characterization of Gearhart-Herbst-Pruss-Huang (See Theorem 1.3.2 in [15]) and the resolvent criterion of Borichev–Tomilov [4]. Here, we give a slightly different and concise version of their results.

Theorem A.11

Let $\{S_{\mathscr {A}}(t)\}_{t \ge 0}$ be a $C_{0}$-semigroup of contractions on a Hilbert space $\mathscr {H}$ with generator operator $\mathscr {A}: D(\mathscr {A}) \subset \mathscr {H} \rightarrow \mathscr {H}$, such that, $\bigl \{ i\lambda \,\big | \, \lambda \in R \bigl \}$ belongs to the resolvent set $\varrho (\mathscr {A})$ and $\Vert (i\lambda I-\mathscr {A})^{-1}\Vert _{\mathscr {L}(\mathscr {H})} \le C |\lambda |^\alpha , \, \forall \,\lambda \in \mathbb {R}$, for some $\alpha \ge 0$ and $C>0$.

$\mathrm{(i)}$ The semigroup $\{S_{\mathscr {A}}(t)\}_{t \ge 0}$ is exponentially stable if and only if $\alpha =0$;

$\mathrm{(ii)}$ If $\alpha >0$, the semigroup $\{S_{\mathscr {A}}(t)\}_{t \ge 0}$ satisfies the following rational decay rate

$$\begin{aligned} \Vert S_{\mathscr {A}}(t) {{\textbf {Y}}}_{0}\Vert _{\mathscr {H}}\le \frac{C}{t^{\frac{1}{\alpha }}} \Vert {{\textbf {Y}}}_0\Vert _{D(\mathscr {A})}, \, \forall \,{\textbf {Y}}_{0}\in D(\mathscr {A}). \end{aligned}$$