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Damping in vacuo

This small chapter considers thermomechanical damping of parametric oscillations in a thin rod {\it in vacuo}. The mathematical model represents an extension of evolution equations describing triple-mode resonant interactions, see section \ref{barraux}. When the intense high-frequency longitudinal wave is in phase with a pair of small-amplitude low-frequency bending waves which can be interpreted in their turn as infinitesimal perturbations the energy dissipation rate of this longitudinal wave would approach dramatically to some maximum in contrast to that damping that we obtained in section \ref{bar-therm}. Indeed, small perturbations cause breakup instability the bending modes being in phase with the primary longitudinal wave to rise. The damping rate of bending waves always exceeds that of longitudinal waves. Therefore the longitudinal wave decays much faster at the expense of comparatively fast damping of the secondary flexural waves. Perhaps, this interesting fact can be used in the practice when designing micromechanical sensor and devices. Let us trace the influence of thermal effects on three-wave resonant interactions. In the deformed rod, the length of the longitudinal $x$ and transverse $z$ axes, the heat balance equation has the following form \begin{equation} \begin{array}{l} \displaystyle C{{\rho }_{0}}{{T}_{t}}+\frac{E}{3}\alpha \left( {{T}_{0}}+T \right){{\varepsilon }_{t}}=\kappa \left( {{T}_{x,x}}+{{T}_{z,z}} \right), \end{array} \label{therm-01} \end{equation} where denotes the specific heat capacity per unit weight of an elastic material, ${{\rho }_{0}}$ is its mass density, ${{T}_{0}}$ is absolute temperature, $\alpha $ is the linear thermal expansion coefficient, is Young’s modulus, $\kappa $ stands for the thermal conductivity coefficient, ${{\varepsilon }_{t}}$ is the velocity of deformation distributed longwise the $z$-axis as follows: $\varepsilon ={{u}_{x}}+\tfrac{1}{2}{{w}_{x}}^{2}-z{{w}_{x,x}}$\footnote{ For simplification, the temperature is supposed to be the same longwise the axis $y$.}. In turn, the stress in the rod depends on the temperature , i.e., $\sigma =E\left( \varepsilon -\alpha T \right)$. Let us assume that in the rod has a square cross-section of the area $F=bh$. The temperature obeys the simple heat equation with insulated boundary conditions\footnote{The simple heat equation results from Eq.~\eqref{therm-01} when $\alpha=0.$}: \[T={{\theta }_{e}}\left( x,t \right)+\sin \left( \frac{\pi z}{h} \right){{\theta }_{o}}\left( x,t \right).\] Thus, we represent the Lagrangian\index{Lagrangian} function as follows \begin{equation} \begin{array}{l} \displaystyle \mathcal{L}=\frac{{{\rho }_{0}}F}{2}\left( u_{t}^{2}+w_{t}^{2} \right)-\frac{EF}{2}u_{x}^{2}-\frac{EF}{24}{{h}^{2}}w_{x,x}^{2}-\frac{EF}{2}{{u}_{x}}w_{x}^{2}-\frac{EF}{8}w_{x}^{4}+ \\ \\ \phantom{\mathcal{L}=} \displaystyle \frac{EF}{2}\alpha {{\theta }_{e}}w_{x}^{2}+EF\alpha {{\theta }_{e}}{{u}_{x}}-\frac{2EF}{{{\pi }^{2}}}\alpha h{{\theta }_{o}}{{w}_{x,x}}. \end{array} \label{therm-02} \end{equation} Here, ${{\theta }_{e}}$ and ${{\theta }_{o}}$ are the even and odd parts of temperature that depends on the variable $z$. This function produces a pair of equations for determining the components of displacements $u$ and $w$: \begin{equation} \begin{array}{l} \displaystyle{{u}_{t,t}}={{c}^{2}}{{\partial }_{x}}\left( {{u}_{x}}+\tfrac{1}{2}w_{x}^{2}-\alpha {{\theta }_{e}} \right); \\ \\ \displaystyle{{w}_{t,t}}+\frac{{{c}^{2}}{{h}^{2}}}{12}{{w}_{x,x,x,x}}+{{c}^{2}}{{\partial }_{x}}\left( {{u}_{x}}{{w}_{x}} \right)+\frac{{{c}^{2}}}{2}{{\partial }_{x}}\left( w_{x}^{3} \right)= \\ \\ \phantom{\displaystyle{{w}_{t,t}}+} \displaystyle{{c}^{2}}\alpha {{\partial }_{x}}\left( {{w}_{x}}{{\theta }_{e}} \right)+\frac{2{{c}^{2}}\alpha h}{{{\pi }^{2}}}{{\theta }_{o;x,x}}. \end{array} \label{therm-03} \end{equation} By integrating Eq.~\eqref{therm-01} from $-h/2$ to $h/2$ by the coordinate $z$, we obtain one more pair of equations for determining the temperatures ${{\theta }_{e}}$ and ${{\theta }_{o}}$: \begin{equation} \begin{array}{l} \displaystyle C{{\theta }_{e;t}}-\frac{\kappa }{{{\rho }_{0}}}{{\theta }_{e;x,x}}= \\ \\ \phantom{C{{\theta }_{e;t}}-} \displaystyle-\frac{{{c}^{2}}}{3}\alpha \left( {{T}_{0}}+{{\theta }_{e}} \right){{\partial }_{t}}\left( {{u}_{x}}+\tfrac{1}{2}w_{x}^{2} \right)+\frac{2{{c}^{2}}}{3}\frac{\alpha h}{{{\pi }^{2}}}{{\theta }_{o}}{{w}_{x,x,t}}; \\ \\ \displaystyle C{{\theta }_{o;t}}-\frac{\kappa }{{{\rho }_{0}}}{{\theta }_{o;x,x}}+\frac{{{\pi }^{2}}\kappa {{\theta }_{o}}}{{{\rho }_{0}}{{h}^{2}}}= \\ \\ \phantom{C{{\theta }_{e;t}}-} \displaystyle \frac{h{{c}^{2}}{{\pi }^{2}}\alpha }{72}\left( {{T}_{0}}+{{\theta }_{e}} \right){{w}_{x,x,t}}-\frac{{{c}^{2}}}{3}\alpha {{\theta }_{o}}{{\partial }_{t}}\left( {{u}_{x}}+\frac{1}{2}w_{x}^{2} \right), \end{array} \label{therm-04} \end{equation} where $c=\sqrt{E/{{\rho }_{0}}}$ is the typical propagation velocity\footnote{The second equation of the set \eqref{therm-04} appears after multiplying Eq.~\eqref{therm-01} on $z$ and integrating the result from $-h/2$ to $h/2$ by $z$.}. The linear part of Eqs.~\eqref{therm-04} defines the pair of independent dispersion relations \begin{equation} \begin{array}{l} \displaystyle\frac{3i\left( E{{k}^{2}}-{{\rho }_{0}}{{\omega }^{2}} \right)}{E\alpha k}=\frac{\alpha {{T}_{0}}E\omega k}{i{{\rho }_{0}}\omega C+{{k}^{2}}\kappa }; \\ \\ \displaystyle\frac{{{\pi }^{2}}\left( {{h}^{2}}{{k}^{4}}E-12{{\rho }_{0}}{{\omega }^{2}} \right)}{24E\alpha h{{k}^{2}}}=-\frac{{{\pi }^{2}}\alpha E{{h}^{3}}{{T}_{0}}\omega {{k}^{2}}}{72\left( {{\rho }_{0}}{{h}^{2}}C\omega -i\kappa \left( {{h}^{2}}{{k}^{2}}+{{\pi }^{2}} \right) \right)}, \end{array} \label{therm-05} \end{equation} where $\omega$ and $k$ denote, correspondingly, the frequency and the wavenumber of a linear infinitely small signal. In particular, the first equation from this set describes the spectrum state of longitudinal waves weakly decaying due to the thermomechanical coupling while the second refers to bending waves. Both equations entering this set reduce to cubic polynomials having three roots. Two of them are complex conjugated, and one more is imaginary. The complex-conjugate roots of both equations \eqref{therm-05} describe the pairs of weakly decaying quasi-harmonic waves travelling in opposite directions. The imaginary roots of Eqs.~\eqref{therm-05} refer to heat diffusion caused, respectively, by quasistatic longitudinal and flexural deformations of the rod. The real parts of complex-conjugate roots of Eqs.~\eqref{therm-05} are almost the same as defined in section~\ref{bar_dsp}. Figure~\ref{therm-bar-01} displays a monotonic dependence of the imaginary parts of the complex-conjugate roots of longitudinal waves on the wavenumbers, in accord with results obtained in section \ref{bar-therm}. In turn, the rate of diffusion of quasistatic flexural deformations approaches the absolute maximum for bending waves in the long-wave limit, as shown in Fig.~\ref{therm-bar-02}\footnote{The system parameters need for plotting just mentioned figures see below. }. Infinitesimal amplitudes of longitudinal and bending quasi-harmonic waves depend on the temperature. The thermomechanical coupling of a longitudinal wave $U\left( \tau ,\xi \right)=A{{\text{e}}^{-\delta \tau }}{{\text{e}}^{i\left( \Omega \tau +k\xi \right)}}+\left( ^{*} \right)$ with the related temperature wavefield ${{\Theta }_{e}}\left( \tau ,\xi \right)=B{{\text{e}}^{-\delta \tau }}{{\text{e}}^{i\left( \Omega \tau +k\xi \right)}}+\left( ^{*} \right)$ describes the interrelation coefficient \begin{equation} {{\chi }_{l}}=\frac{A}{B}=\frac{i{{\rho }_{0}}\left( E{{k}^{2}}-{{\left( \Omega +i\delta \right)}^{2}} \right)}{E\alpha k}, \label{therm-06} \end{equation} where $\Omega =\operatorname{Re}\left( \omega \right)$ and $\delta =\operatorname{Im}\left( \omega \right)$.

The analogous interrelation coefficient associated with the bending wave $W\left( \tau ,\xi \right)=A{{\text{e}}^{-\delta \tau }}{{\text{e}}^{i\left( \Omega \tau +k\xi \right)}}+\left( ^{*} \right)$ and the related temperature ${{\Theta }_{o}}\left( \tau ,\xi \right)=B{{\text{e}}^{-\delta \tau }}{{\text{e}}^{i\left( \Omega \tau +k\xi \right)}}+\left( ^{*} \right)$ reads \begin{equation} {{\chi }_{b}}=\frac{A}{B}=\frac{{{\pi }^{2}}\left( {{h}^{2}}{{k}^{4}}E-12{{\rho }_{0}}{{\left( \Omega +i\delta \right)}^{2}} \right)}{24E\alpha h{{k}^{2}}}. \label{therm-07} \end{equation} After performing the change of variables $t={L\tau }/{c}$; $x=L\xi$; $u\left( x,t \right)=hU\left( \tau ,\xi \right)$; $w\left( x,t \right)=hW\left( \tau ,\xi \right)$;${{\theta }_{e}}\left( x,t \right)={{T}_{0}}{{\Theta }_{e}}\left( \tau ,\xi \right)$; ${{\theta }_{o}}\left( x,t \right)={{T}_{0}}{{\Theta }_{o}}\left( \tau ,\xi \right)$, Eqs.~\eqref{therm-04} rewrite in the dimensionless coordinates: \begin{equation} \begin{array}{l} \displaystyle {{U}_{\tau ,\tau }}-{{U}_{\xi ,\xi }}+\frac{\alpha {{T}_{0}}}{\varepsilon }{{\Theta }_{e;\xi }}=\frac{\varepsilon }{2}{{\partial }_{\xi }}\left( W_{\xi }^{2} \right); \\ \\ \displaystyle {{W}_{\tau ,\tau }}+\frac{{{\varepsilon }^{2}}}{12}{{W}_{\xi ,\xi ,\xi ,\xi }}+\alpha {{T}_{0}}{{\partial }_{\xi }}\left( {{\Theta }_{e}}{{W}_{\xi }} \right)+\frac{2}{{{\pi }^{2}}}\alpha {{T}_{0}}{{\Theta }_{o;\xi ,\xi }}= \\ \\ \phantom{{{W}_{\tau ,\tau }}+} \varepsilon {{\partial }_{\xi }}\left( {{U}_{\xi }}{{W}_{\xi }} \right)+\frac{{{\varepsilon }^{2}}}{2}{{\partial }_{\xi }}\left( {{W}_{\xi }}^{3} \right), \end{array} \label{therm-08} \end{equation} and \begin{equation} \begin{array}{l} \displaystyle C{{\Theta }_{e;\tau }}-\frac{\kappa }{L\sqrt{{{\rho }_{0}}E}}{{\Theta }_{e;\xi ,\xi }}= \\ \\ \phantom{C{{\Theta }_{e;\tau }}-} \displaystyle \frac{2\alpha {{c}^{2}}{{\varepsilon }^{2}}}{3{{\pi }^{2}}}{{W}_{\tau ,\xi ,\xi }}{{\Theta }_{o}}-\frac{\alpha {{c}^{2}}\varepsilon }{3}\left( 1+{{\Theta }_{e}} \right){{\partial }_{\tau }}\left( {{U}_{\xi }}+\frac{\varepsilon }{2}W_{\xi }^{2} \right); \\ \\ \displaystyle C{{\Theta }_{o;\tau }}+\frac{{{\pi }^{2}}\kappa }{h\varepsilon \sqrt{{{\rho }_{0}}E}}{{\Theta }_{o}}-\frac{\kappa }{L\sqrt{{{\rho }_{0}}E}}{{\Theta }_{o;\xi ,\xi }}= \\ \\ \phantom{C{{\Theta }_{e;\tau }}-} \displaystyle \frac{{{c}^{2}}{{\varepsilon }^{2}}{{\pi }^{2}}\alpha }{72}\left( 1+{{\Theta }_{e}} \right){{W}_{\tau ,\xi ,\xi }}-\frac{\alpha }{3}\varepsilon {{c}^{2}}{{\Theta }_{o}}{{\partial }_{\tau }}\left( {{U}_{\xi }}+\frac{\varepsilon }{2}W_{\xi }^{2} \right), \end{array} \label{therm-09} \end{equation} where $\varepsilon =h/L$. Let the system parameters be $C=439 {\bf{J}}{\bf{kg}^{-1}\bf{K}^{-1}}$, $E=1.88\times {{10}^{11}}\bf{Pa}$, $L=0.01\bf{m}$, $\alpha =0.150\times {{10}^{-3}}{{\bf{K}}^{-1}}$, $h=0.976\times {{10}^{-4}}\bf{m}$, $\kappa =54.4\bf{W}{\bf{m}^{-1}}{\bf{K}^{-1}}$, ${{T}_{0}}=290\bf{K}$, ${{\rho }_{0}}=7800\bf{kg}{\bf{m}^{-3}}$. Here, the length and thickness $h$ of a rod we chose so to satisfy requirements described in subsection \ref{bar_resonator}, turning the rod into a beam resonator tuned for the triple-mode resonant coupling. Indeed, if supposing $h={\frac {2\,\sqrt {3}}{113\,\pi }}\bf{m}\approx 0.976\times {{10}^{-4}}\bf{m}$ and ${{k}_{1}}=\pi \,\bf{rad /m}$, then the wavenumbers of the secondary pair of resonantly interacting waves would be equal to ${{k}_{2}}=8\pi \,\bf{rad /m}$, ${{k}_{3}}=-7\pi \,\bf{rad /m}$ that guarantee fulfilment the phase matching conditions $k_1=k _2+k _3$ and $\omega _1=\omega _2+\omega _3$ in the absence of thermomechanical coupling. We look for a three-wave resonant solution to Eqs.~\eqref{therm-08}--\eqref{therm-09} in the following form \begin{equation} \begin{array}{l} U\left( \tau ,\xi \right)=\mu {{A}_{1}}\left( t \right){{\text{e}}^{i\left( {{\Omega }_{1}}\tau +{{k}_{1}}\xi \right)}}{{\text{e}}^{-{{\delta }_{1}}\tau }}+(^*); \\ \\ {{\Theta }_{e}}\left( \tau ,\xi \right)=\mu {{\chi }_{1}}{{A}_{1}}\left( t \right){{\text{e}}^{i\left( {{\Omega }_{1}}\tau +{{k}_{1}}\xi \right)}}{{\text{e}}^{-{{\delta }_{1}}\tau }}+(^*), \end{array} \label{therm-10} \end{equation} and \begin{equation} \begin{array}{l} W\left( \tau ,\xi \right)=\mu \sum\limits_{q=2}^{3}{{{A}_{q}}\left( t \right){{\text{e}}^{i\left( {{\Omega }_{q}}\tau +{{k}_{q}}\xi \right)}}{{\text{e}}^{-{{\delta }_{q}}\tau }}}+(^*); \\ \\ {{\Theta }_{o}}\left( \tau ,\xi \right)=\mu \sum\limits_{q=2}^{3}{{{\chi }_{q}}{{A}_{q}}\left( t \right){{\text{e}}^{i\left( {{\Omega }_{q}}\tau +{{k}_{q}}\xi \right)}}{{\text{e}}^{-{{\delta }_{q}}\tau }}}+(^*), \end{array} \label{therm-11} \end{equation} where ${{A}_{p}}\left( t \right)$ are the amplitudes of waves slowly varying in the timescale $t=\mu \tau$, $\mu$ is the small parameter determined by physical restrictions of the problem, Eqs.~\eqref{therm-06}--\eqref{therm-07} define the coefficients ${\chi }_{p}$. The averaging procedure yields the following set of truncated evolution equations \begin{equation} \begin{array}{l} 3{{\pi }^{2}}{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{1}}\left( 1+{{\left| {{\chi }_{1}} \right|}^{2}} \right)-{{\delta }_{1}} \right){{\dot{A}}_{1}}=-{{\eta }_{1}}{{A}_{2}}{{A}_{3}}{{\text{e}}^{i \Delta \omega \tau}}; \\ \\ 72{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{2}}\left( 1+{{\left| {{\chi }_{2}} \right|}^{2}} \right)-{{\delta }_{2}} \right){{\dot{A}}_{2}}={{\eta }_{2}}{{\bar{A}}_{3}}{{A}_{1}}{{\text{e}}^{-i\Delta \omega \tau }}; \\ \\ 72{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{3}}\left( 1+{{\left| {{\chi }_{3}} \right|}^{2}} \right)-{{\delta }_{3}} \right){{\dot{A}}_{3}}={{\eta }_{3}}{{\bar{A}}_{2}}{{A}_{1}}{{\text{e}}^{-i\Delta \omega \tau }}, \end{array} \label{therm-12} \end{equation} where $\Delta \omega=\omega _1 - \omega _2 -\omega _3$ is the detuning. The coefficients of Eqs.~\eqref{therm-12} depend on the parameters of the system: \begin{equation*} \begin{array}{l} {{\eta }_{1}}=\alpha {{h}^{2}}{{\bar{\chi }}_{1}}E{{\Omega }_{1}}\left( {{\pi }^{2}}{{k}_{2}}{{k}_{3}}{{\omega }_{1}}-2{{\chi }_{3}}k_{2}^{2}{{\omega }_{2}}-2{{\chi }_{2}}k_{3}^{2}{{\omega }_{3}} \right)+ \\ \\ \phantom{{{\eta }_{1}}=-} 3i{{\pi }^{2}}{{\rho }_{0}}{{k}_{1}}{{k}_{2}}{{k}_{3}}hLC; \\ \\ {{\eta }_{2}}=\alpha hE{{\Omega }_{2}}{{\bar{\chi }}_{2}}\left( 24i{{k}_{1}}{{{\bar{\chi }}}_{3}}{{\omega }_{1}}L+{{\pi }^{2}}k_{3}^{2}h{{\chi }_{1}}{{{\bar{\omega }}}_{3}} \right)+ \\ \\ \phantom{{{\eta }_{1}}=-} 72{{\rho }_{0}}{{k}_{2}}{{k}_{3}}LC\left( i{{k}_{1}}h-\alpha {{\chi }_{1}}{{T}_{0}}L \right); \\ \\ {{\eta }_{3}}=\alpha hE{{\Omega }_{3}}{{\bar{\chi }}_{3}}\left( 24i{{k}_{1}}{{{\bar{\chi }}}_{2}}{{\omega }_{1}}L-{{\pi }^{2}}k_{2}^{2}h{{\chi }_{1}}{{{\bar{\omega }}}_{1}} \right)+ \\ \\ \phantom{{{\eta }_{1}}=-} 72{{\rho }_{0}}{{k}_{2}}{{k}_{3}}LC\left( i{{k}_{1}}h-\alpha {{\chi }_{1}}{{T}_{0}}L \right). \end{array} \end{equation*} Using the blowup transform of amplitudes; $A_{{p}} \left( t \right) =B_{{p}} \left( t \right) {{\rm e}^{\delta_{{p}}t}}$, the set ~\eqref{therm-12} rewrites in more convenient for analysis form \begin{equation} \begin{array}{l} 3{{\pi }^{2}}{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{1}}\left( 1+{{\left| {{\chi }_{1}} \right|}^{2}} \right)-{{\delta }_{1}} \right){{\dot{B}}_{1}}-{{\delta }_{1}}{{B}_{1}}=-{{\eta }_{1}}{{B}_{2}}{{B}_{3}}{{\text{e}}^{i\Delta \Omega \tau }}; \\ \\ 72{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{2}}\left( 1+{{\left| {{\chi }_{2}} \right|}^{2}} \right)-{{\delta }_{2}} \right){{\dot{B}}_{2}}-{{\delta }_{2}}{{B}_{2}}={{\eta }_{2}}{{\bar{B}}_{3}}{{B}_{1}}{{\text{e}}^{-i\Delta \Omega \tau }}; \\ \\ 72{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{3}}\left( 1+{{\left| {{\chi }_{3}} \right|}^{2}} \right)-{{\delta }_{3}} \right){{\dot{B}}_{3}}-{{\delta }_{3}}{{B}_{3}}={{\eta }_{3}}{{\bar{B}}_{2}}{{B}_{1}}{{\text{e}}^{-\Delta \Omega \tau }}. \end{array} \label{therm-13} \end{equation} where $\Delta \Omega=\Omega _1 - \Omega _2 -\Omega _3$. These equations cannot be solved analytically because of the absence of first integrals that is a natural consequence of the thermomechanical coupling making the system nonconservative. Nonetheless, we can try to study the set \eqref{therm-13} by tracing the processes of destroying the invariants of the related conservative system, for instance, the evolution of destruction of the Manly-Rowe relations \eqref{bar-eq-17} when applying the procedure described in subsection \ref{Energy dissipation}. One more we change the variables ${{B}_{n}}={{b}_{n}}(t)\exp i{{\varphi }_{n}}\left( t \right)$ with the real-valued amplitudes and phases ${{b}_{n}}$ and ${{\varphi }_{n}}$, respectively. As a result, the set~\eqref{therm-13} reduces to four ordinary differential equations for three amplitudes ${{b}_{n}}$ and the combinational phase $\Phi ={{\varphi }_{1}}-{{\varphi }_{2}}-{{\varphi }_{3}}$. These expressions are too long. We write here a numerical result using the system parameters defined above: \begin{equation} \begin{array}{l} \displaystyle {{\dot{b}}_{1}}=5.1 \,{{b}_{3}}{{b}_{2}}\cos \Phi +\text{1}\text{.55}\times {{10}^{-7}}{{b}_{3}}{{b}_{2}}\sin \Phi -1.70\times {{10}^{-7}}{{b}_{1}}; \\ \\ \displaystyle {{\dot{b}}_{2}}=-9.01\,{{b}_{1}}{{b}_{3}}\cos \Phi -8.60\times {{10}^{-3}}{{b}_{1}}{{b}_{3}}\sin \Phi -1.80\times {{10}^{-3}}{{b}_{2}}; \\ \\ \displaystyle {{\dot{b}}_{3}}=-11.8\,{{b}_{2}}{{b}_{1}}\cos \Phi -1.46\times {{10}^{-2}}{{b}_{2}}{{b}_{1}}\sin \Phi -1.79\times {{10}^{-3}}{{b}_{3}}; \\ \\ \displaystyle\dot{\Phi }=-8.60\times {{10}^{-3}}\frac{{{b}_{1}}{{b}_{3}}\cos \Phi }{{{b}_{2}}}-1.46\times {{10}^{-2}}\frac{{{b}_{2}}{{b}_{1}}\cos \Phi }{{{b}_{3}}}+ \\ \\ \phantom{\dot{\Phi }=-} \displaystyle1.55\times {{10}^{-7}}\frac{{{b}_{3}}{{b}_{2}}\cos \Phi }{{{b}_{1}}}+9.01\,\frac{{{b}_{1}}{{b}_{3}}\sin \Phi }{{{b}_{2}}}+ \\ \\ \phantom{\dot{\Phi }=-} \displaystyle11.8\,\frac{{{b}_{2}}{{b}_{1}}\sin \Phi }{{{b}_{3}}}-5.1 \,\frac{{{b}_{3}}{{b}_{2}}\sin \Phi }{{{b}_{1}}}-2\times {{10}^{-9}}. \end{array} \label{therm-14} \end{equation} In the absence of nonlinearity, or when oscillations are infinitely small, the analytical solution, corresponding to the initial conditions: ${{b}_{1}}\left( 0 \right){{=10}^{-2}}$, ${{b}_{2}}\left( 0 \right){{=10}^{-4}}$, ${{b}_{3}}\left( 0 \right){{=10}^{-4}},\Phi \left( 0 \right)=0$, is expressed through the following exponents $\Phi \left( t \right)=-2\times {{10}^{-9}}t$, ${{b}_{1}}\left( t \right){{=10}^{-2}}{{\text{e}}^{-1.70\times {{10}^{-7}}t}}$, ${{b}_{2}}\left( t \right){{=10}^{-4}}{{\text{e}}^{-1.80\times {{10}^{-3}}t}}$, ${{b}_{3}}\left( t \right){{=10}^{-4}}{{\text{e}}^{-1.79\times {{10}^{-3}}t}}$. So, the resonant triad of waves is almost in phase. The longitudinal mode decays much slower than the bending satellites. Figure~\ref{therm-bar-03} displays a numerical solution to Eqs.~\eqref{therm-14} with the same initial conditions, as previously.

We can see that this is a typical exponentially decaying backward chirp. The oscillations disappear when reaching the threshold defined analytically from the eigenvalue problem formulated for the linearised slightly perturbed subset of Eqs.~\eqref{therm-13}. We use the following change of the variables: ${{B}_{1}}\left( t \right)={{b}_{1}}\left( t \right){{\text{e}}^{-i\Delta \Omega t}}$, ${{B}_{2,3}}\left( t \right)={{b}_{2,3}}\left( t \right)$, to obtain these eigenvalues. This transform leads to an autonomous set of equations. Then, we look for the solution to this set in the form ${{b}_{1}}\left( t \right)={{x}_{1}}$, ${{b}_{2,3}}\left( t \right)={{x}_{2,3}}{{\text{e}}^{\lambda t}}$. Supposing that ${{x}_{1}}$ is much greater than ${{x}_{2,3}}$, we obtain a pair of quadratic equations $\left( \lambda +{{\delta }_{2}} \right)\left( \lambda +{{\delta }_{3}} \right)={{\left| {{x}_{1}} \right|}^{2}}{{z}_{2}}{{\bar{z}}_{3}}$ and $\left( \lambda +{{\delta }_{2}} \right)\left( \lambda +{{\delta }_{3}} \right)={{\left| {{x}_{1}} \right|}^{2}}{{\bar{z}}_{2}}{{z}_{3}}$ for determining four eigenvalues $\lambda $, where \begin{equation*} \begin{array}{l} \displaystyle {{z}_{2}}=-\frac{{{\omega }_{2}}\alpha Eh{{{\bar{\chi }}}_{2}}\left( 24i{{\omega }_{1}}{{k}_{1}}{{{\bar{\chi }}}_{3}}L-{{\pi }^{2}}k_{3}^{2}{{\omega }_{3}}{{\chi }_{1}}h \right)}{72{{\rho }_{0}}C{{L}^{2}}\left( i{{\Omega }_{2}}\left( 1+{{\left| {{\chi }_{2}} \right|}^{2}} \right)-{{\delta }_{2}} \right)}- \\ \\ \phantom{ {{z}_{2}}=-} \displaystyle \frac{{{k}_{2}}{{k}_{3}}\left( i{{k}_{1}}h-\alpha {{\chi }_{1}}L{{T}_{0}} \right)}{L\left( i{{\Omega }_{2}}\left( 1+{{\left| {{\chi }_{2}} \right|}^{2}} \right)-{{\delta }_{2}} \right)}; \\ \\ \displaystyle {{z}_{3}}=-\frac{{{\omega }_{3}}\alpha Eh{{{\bar{\chi }}}_{3}}\left( 24i{{\omega }_{1}}L{{k}_{1}}{{{\bar{\chi }}}_{2}}-{{\pi }^{2}}k_{2}^{2}{{\omega }_{2}}{{\chi }_{1}}h \right)}{72C\left( i{{\Omega }_{3}}\left( 1+{{\left| {{\chi }_{3}} \right|}^{2}} \right)-{{\delta }_{3}} \right){{L}^{2}}{{\rho }_{0}}}- \\ \\ \phantom{ {{z}_{2}}=-} \displaystyle \frac{{{k}_{2}}{{k}_{3}}\left( i{{k}_{1}}h-\alpha {{\chi }_{1}}L{{T}_{0}} \right)}{L\left( i{{\Omega }_{3}}\left( 1+{{\left| {{\chi }_{3}} \right|}^{2}} \right)-{{\delta }_{3}} \right)}. \end{array} \end{equation*} When an eigenvalue $\lambda$ passes zero coming from positive values to negatives, the nonlinearity of the system~\eqref{therm-13} disappears. When the eigenvalue is zero, the following expression \[{{\left| {{x}_{1}} \right|}^{2}}=\frac{{{\delta }_{2}}{{\delta }_{3}}}{{{z}_{2}}{{{\bar{z}}}_{3}}}=\frac{{{\delta }_{2}}{{\delta }_{3}}}{{{{\bar{z}}}_{2}}{{z}_{3}}}\] defines the threshold by the amplitude ${{z}_{2}}$ and ${{z}_{3}}$ represent complex-conjugated values. For the given system parameters, this threshold corresponds to the longitudinal wave: $\left| {{B}_{1}} \right|=1.74\times {{10}^{-4}}$. Figure \ref{therm-bar-04} displays a fragment of the pass to the linear exponential decaying from the nonlinear oscillatory regime through this threshold.

(Tutorial ends here.)

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