Studying the Dynamic Properties of a Distributed Thermomechanical Controlled Plant with Intrinsic Feedback. II

Abstract

The dynamic properties of the response of a one-dimensional elastic mechanical system to an external mechanical action are examined. Transfer functions are calculated in two channels: from the force action at one of the system boundaries to the displacement of the medium sections and to the temperature. The asymptotic behavior of the transfer function is analyzed for each channel in the neighborhood of the origin on the complex plane. The case of no heat exchange between the system and the environment is considered separately.

Appendices

APPENDIX A

Proof of Theorem 1. (1) We apply the Laplace transform to Eqs. (2.1) with respect to the spatial coordinate x considering the first boundary condition in (2.2). (For details, see [12, item 80, formulas (6) and (7)].) As a result,

$$\left\{ \begin{gathered} ({{c}^{2}}{{q}^{2}} - {{p}^{2}})\bar {\bar {\varphi }}(q) - \beta q\bar {\bar {\theta }}(q) = {{z}_{1}}(q), \hfill \\ - {{\beta }_{{{\text{therm}}}}}qp\bar {\bar {\varphi }}(q) + (a{{q}^{2}} - p)\bar {\bar {\theta }}(q) = {{z}_{2}}(q), \hfill \\ \end{gathered} \right.$$

(A.1)

where z₁(q) = ${{c}^{2}}q\bar {\varphi }(0)$ – $\beta \bar {\theta }(0)$ and z₂(q) = $aq\bar {\theta }(0)$ + $a\frac{{\partial \bar {\theta }}}{{\partial x}}(0)$ – ${{\beta }_{{{\text{therm}}}}}p\bar {\varphi }(0)$.

In view of these expressions for z_i(q), the solution of system (A.l) in ($\bar {\bar {\varphi }}$(q), $\bar {\bar {\theta }}$(q)) is given by

$$\bar {\bar {\varphi }}(q) = \frac{{a{{c}^{2}}{{q}^{3}}\bar {\varphi }(0) + {{B}_{1}}q + \beta p\bar {\theta }(0)}}{{\Delta (q)}},$$

(A.2)

$$\bar {\bar {\theta }}(q) = \frac{{a{{c}^{2}}{{q}^{3}}\bar {\theta }(0) + a{{c}^{2}}{{q}^{2}}\frac{{\partial \bar {\theta }}}{{\partial x}}(0) + {{B}_{2}}{{p}^{2}} - {{b}_{2}}qp\bar {\theta }(0)}}{{\Delta (q)}},$$

(A.3)

where

$$\Delta (q) = a{{c}^{2}}\left( {{{q}^{2}} - {{\rho }_{1}}} \right)\left( {{{q}^{2}} - {{\rho }_{2}}} \right),\quad {{\rho }_{{1,2}}} = \frac{{p(ap + {{b}_{1}}) \pm R}}{{2a{{c}^{2}}}},$$

$$R = ap\sqrt {{{{(p + \mu )}}^{2}} + {{y}^{2}}} ,\quad \mu = \frac{{\beta {{\beta }_{{{\text{therm}}}}} - {{c}^{2}}}}{a},\quad y = 2\frac{c}{a}\sqrt {\beta {{\beta }_{{{\text{therm}}}}}} ,$$

$${{b}_{1}} = \beta {{\beta }_{{{\text{therm}}}}} + {{c}^{2}},\quad {{b}_{2}} = \beta {{\beta }_{{{\text{therm}}}}} + ap,$$

$${{B}_{1}} = \beta a\frac{{\partial \bar {\theta }}}{{\partial x}}(0) - {{b}_{1}}p\bar {\varphi }(0),\quad {{B}_{2}} = {{\beta }_{{{\text{therm}}}}}p\bar {\varphi }(0) - a\frac{{\partial \bar {\theta }}}{{\partial x}}(0).$$

Based on the relation

$$\frac{1}{{\Delta (q)}} = \frac{1}{R}\left( {\frac{1}{{{{q}^{2}} - {{\rho }_{1}}}} - \frac{1}{{{{q}^{2}} - {{\rho }_{2}}}}} \right),$$

(A.4)

from (A.2) and (A.3) we pass to the original functions with respect to the coordinate x:

$$\bar {\varphi }(x) = a{{c}^{2}}{{D}_{3}}(x)\bar {\varphi }(0) + {{D}_{1}}(x){{B}_{1}} + \beta p{{D}_{0}}(x)\bar {\theta }(0),$$

(A.5)

$$\bar {\theta }(x) = a{{c}^{2}}{{D}_{3}}(x)\bar {\theta }(0) + a{{c}^{2}}{{D}_{2}}(x)\frac{{\partial \bar {\theta }}}{{\partial x}}(0) - {{b}_{2}}p{{D}_{1}}(x)\bar {\theta }(0) + {{p}^{2}}{{D}_{0}}(x){{B}_{2}},$$

(A.6)

where

$${{D}_{j}}(x) = \frac{1}{R}\left( {\rho _{1}^{{(j - 1)/2}}\sinh \left( {x\sqrt {{{\rho }_{1}}} } \right) - \rho _{2}^{{(j - 1)/2}}\sinh \left( {x\sqrt {{{\rho }_{2}}} } \right)} \right)\quad (j = 0;2),$$

$${{D}_{j}}(x) = \frac{1}{R}\left( {\rho _{1}^{{(j - 1)/2}}\cosh \left( {x\sqrt {{{\rho }_{1}}} } \right) - \rho _{2}^{{(j - 1)/2}}\cosh \left( {x\sqrt {{{\rho }_{2}}} } \right)} \right)\quad (j = 1;3).$$

(The details can be found in [12, item 80, formula (4)].)

Due to (A.5) and the first condition in (2.3), the second boundary condition in (2.2) takes the form

$${{p}^{2}}\left[ {\left( {a{{c}^{2}}{{D}_{3}}(l) - {{b}_{1}}p{{D}_{1}}(l)} \right)\bar {\varphi }(0) + \beta (p{{D}_{0}}(l) + a\kappa {{D}_{1}}(l))\bar {\theta }(0)} \right] = \bar {u}.$$

(A.7)

According to (A.6), the second condition in (2.3) reduces to

$$\begin{gathered} a{{c}^{2}}({{D}_{4}}(l) + \kappa {{D}_{3}}(l))\bar {\theta }(0) + a{{c}^{2}}\kappa ({{D}_{3}}(l) + \kappa {{D}_{2}}(l))\bar {\theta }(0) \\ - \;{{b}_{2}}p({{D}_{2}}(l) + \kappa {{D}_{1}}(l))\bar {\theta }(0) + {{p}^{2}}({{D}_{1}}(l) + \kappa {{D}_{0}}(l))\left( {{{\beta }_{{{\text{therm}}}}}p\bar {\varphi }(0) - \kappa a\bar {\theta }(0)} \right) = 0, \\ \end{gathered} $$

(A.8)

where

$${{D}_{4}}(l) = \frac{1}{R}\left( {\rho _{1}^{{3/2}}\sinh \left( {x\sqrt {{{\rho }_{1}}} } \right) - \rho _{2}^{{3/2}}\sinh \left( {x\sqrt {{{\rho }_{2}}} } \right)} \right).$$

Conditions (A.7) and (A.8) make up the following system of equations in the vector $\left( \begin{gathered} \bar {\varphi }(0) \\ \bar {\theta }(0) \\ \end{gathered} \right)$:

$$A\left( \begin{gathered} \bar {\varphi }(0) \\ \bar {\theta }(0) \\ \end{gathered} \right) = \left( \begin{gathered} {\bar {u}} \\ 0 \\ \end{gathered} \right),$$

(A.9)

where A = (a_jk; j, k = 1, 2), a₁₁ = p²(ac²D₃(l) – b₁pD₁(l)),

$${{a}_{{12}}} = \beta {{p}^{2}}(\kappa a{{D}_{1}}(l) + p{{D}_{0}}(l)),\quad {{a}_{{21}}} = {{\beta }_{{{\text{therm}}}}}{{p}^{3}}({{D}_{1}}(l) + \kappa {{D}_{0}}(l)),$$

$${{a}_{{22}}} = a{{c}^{2}}{{D}_{4}}(l) + 2\kappa a{{c}^{2}}{{D}_{3}}(l) + \left( {{{\kappa }^{2}}a{{c}^{2}} - {{b}_{2}}p} \right){{D}_{2}}(l)$$

$$ - \;\kappa p(2ap + \beta {{\beta }_{{{\text{therm}}}}}){{D}_{1}}(l) - {{\kappa }^{2}}a{{p}^{2}}{{D}_{0}}(l).$$

The solution of system (A.9) is given by

$$\left( \begin{gathered} \bar {\varphi }(0) \\ \bar {\theta }(0) \\ \end{gathered} \right) = \left( \begin{gathered} {{a}_{{22}}} \\ - {{a}_{{21}}} \\ \end{gathered} \right)\frac{{\bar {u}}}{{{{\Delta }_{A}}}},$$

(A.10)

where Δ_A = a₁₁a₂₂ – a₁₂a₂₁.

Substituting (A.10) into (A.5) and (A.6) and using the formula for B_i and the first condition in (2.3), we finally arrive at the following expressions for $\bar {\varphi }$(x) and $\bar {\theta }$(x):

$$\bar {\varphi }(x) = \left\{ {{{a}_{{22}}}\left[ {a{{c}^{2}}{{D}_{3}}(x) - {{b}_{1}}p{{D}_{1}}(x)} \right] - \beta {{a}_{{21}}}\left[ {\kappa a{{D}_{1}}(x) + p{{D}_{0}}(x)} \right]} \right\}\frac{{\bar {u}}}{{{{\Delta }_{A}}}},$$

(A.11)

$$\bar {\theta }(x) = \left\{ {{{a}_{{22}}}{{\beta }_{{{\text{therm}}}}}{{p}^{3}}{{D}_{0}}(x) + {{a}_{{21}}}\left[ { - a{{c}^{2}}{{D}_{3}}(x) - \kappa a{{c}^{2}}{{D}_{2}}(x) + {{b}_{2}}p{{D}_{1}}(x) + \kappa a{{p}^{2}}{{D}_{0}}(x)} \right]} \right\}\frac{{\bar {u}}}{{{{\Delta }_{A}}}}.$$

(A.12)

APPENDIX B

Proof of Theorem 2. (1) In the neighborhood of the origin on the plane C, the functions R and ρ_j ( j = 1, 2; see the explanations for (2.4) and (2.5)) can be represented as

$$R = {{b}_{1}}p(1 + O(p)),\quad {{\rho }_{1}} = \frac{{{{b}_{1}}p}}{{a{{c}^{2}}}}(1 + O(p)),\quad {{\rho }_{2}} = O\left( {{{p}^{2}}} \right).$$

(B.1)

(2) In the neighborhood of the origin on the plane C, the functions D_j(x) ( j = 0$\div$4) can be represented as follows:

$${{D}_{0}}(x) = \frac{{{{x}^{3}}}}{6}\frac{{{{\rho }_{1}} - {{\rho }_{2}}}}{R} + O\left( {{{p}^{2}}} \right) = \frac{{{{x}^{3}}}}{{6a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right),$$

(B.2)

$${{D}_{1}}(x) = \frac{{{{x}^{2}}}}{2}\frac{{{{\rho }_{1}} - {{\rho }_{2}}}}{R} + O\left( {{{p}^{2}}} \right) = \frac{{{{x}^{2}}}}{{2a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right),$$

(B.3)

$${{D}_{2}}(x) = x\frac{{{{\rho }_{1}} - {{\rho }_{2}}}}{R} + O\left( {{{p}^{2}}} \right) = \frac{x}{{a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right),$$

(B.4)

$${{D}_{3}}(x) = \frac{{{{\rho }_{1}} - {{\rho }_{2}}}}{R} + O\left( {{{p}^{2}}} \right) = \frac{1}{{a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right),$$

(B.5)

$${{D}_{4}}(x) = x\frac{{\rho _{1}^{2} - \rho _{2}^{2}}}{R} + O\left( {{{p}^{2}}} \right) = x\frac{{{{\rho }_{1}} + {{\rho }_{2}}}}{{a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right) = x\frac{{{{b}_{1}}}}{{{{a}^{2}}{{c}^{4}}}}p + O\left( {{{p}^{2}}} \right).$$

(B.6)

(3) In the neighborhood of the origin on the plane C, the functions a_jk ( j, k = 1, 2; see the explanations for (A.9)) can be represented as follows:

$${{a}_{{11}}} = {{p}^{2}}\left( {1 - \frac{{{{b}_{1}}{{l}^{2}}}}{{2a{{c}^{2}}}}p + O\left( {{{p}^{2}}} \right)} \right) = {{p}^{2}}(1 + O(p)),$$

(B.7)

$${{a}_{{12}}} = \beta {{p}^{2}}\left( {\frac{{{{l}^{3}}}}{{6a{{c}^{2}}}}p + \kappa \frac{{{{l}^{2}}}}{{2{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right) = \beta \kappa \frac{{{{l}^{2}}}}{{2{{c}^{2}}}}{{p}^{2}}(1 + O(p)),$$

(B.8)

$${{a}_{{21}}} = {{\beta }_{{{\text{therm}}}}}{{p}^{3}}\left( {\frac{{{{l}^{2}}}}{{2a{{c}^{2}}}} + \kappa \frac{{{{l}^{3}}}}{{6a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right) = {{\beta }_{{{\text{therm}}}}}{{p}^{3}}\frac{{{{l}^{2}}}}{{2a{{c}^{2}}}}\left( {1 + \kappa \frac{l}{3}} \right)\left( {1 + O\left( {{{p}^{2}}} \right)} \right),$$

(B.9)

$$\begin{gathered} {{a}_{{22}}} = \frac{{{{b}_{1}}l}}{{a{{c}^{2}}}}p + 2\kappa + l\left( {{{\kappa }^{2}} - \frac{{{{b}_{2}}}}{{a{{c}^{2}}}}p} \right) - \kappa p\frac{{{{l}^{2}}}}{{{{c}^{2}}}}\left( {p + \frac{{\beta {{\beta }_{{{\text{therm}}}}}}}{{2a}}} \right) \\ - \;{{\kappa }^{2}}{{p}^{2}}\frac{{{{l}^{3}}}}{{6{{c}^{2}}}} + O\left( {{{p}^{2}}} \right) = \kappa (2 + \kappa l)(1 + O(p)). \\ \end{gathered} $$

(B.10)

(4) In the neighborhood of the origin on the plane C, the function Δ_A (see the explanations for (A.10)) and the ratios $\frac{{{{a}_{{2j}}}}}{{{{\Delta }_{A}}}}$ ( j = 1, 2; see (2.4) and (2.5)) can be represented as

$$\begin{gathered} {{\Delta }_{A}} = {{p}^{2}}\kappa (2 + \kappa l)(1 + O(p)) - {{p}^{5}}\kappa \frac{{\beta {{\beta }_{{{\text{therm}}}}}{{l}^{4}}}}{{4a{{c}^{4}}}}\left( {1 + \kappa \frac{l}{3}} \right)(1 + O(p)) \\ = {{p}^{2}}\kappa (2 + \kappa l)(1 + O(p)), \\ \end{gathered} $$

(B.11)

$$\frac{{{{a}_{{21}}}}}{{{{\Delta }_{A}}}} = p\frac{{{{\beta }_{{{\text{therm}}}}}{{l}^{2}}(1 + \kappa l{\text{/}}3)}}{{2\kappa a{{c}^{2}}(2 + \kappa l)}}(1 + O(p)) = p\frac{{{{\beta }_{{{\text{therm}}}}}{{l}^{2}}(3 + \kappa l)}}{{6\kappa a{{c}^{2}}(2 + \kappa l)}}(1 + O(p)),$$

(B.12)

$$\frac{{{{a}_{{22}}}}}{{{{\Delta }_{A}}}} = \frac{1}{{{{p}^{2}}}}(1 + O(p)).$$

(B.13)

(5) In the neighborhood of the origin on the plane C, the transfer functions ${{W}_{{u \to \varphi (x)}}}$ and ${{W}_{{u \to \theta (x)}}}$ can be represented as

$$\begin{gathered} {{W}_{{u \to \varphi (x)}}} = \frac{1}{{{{p}^{2}}}}\left( {1 - p\frac{{{{b}_{1}}}}{{2a{{c}^{2}}}}{{x}^{2}}} \right)\left( {1 + O\left( {{{p}^{2}}} \right)} \right) \\ - \;p\frac{{\beta {{\beta }_{{{\text{therm}}}}}{{l}^{2}}(1 + \kappa l{\text{/}}3)}}{{2a{{c}^{2}}\kappa (2 + \kappa l)}}\left( {\kappa \frac{{{{x}^{2}}}}{{2{{c}^{2}}}} + p\frac{{{{x}^{3}}}}{{6a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right) = \frac{1}{{{{p}^{2}}}}(1 + O(p)), \\ \end{gathered} $$

(B.14)

$$\begin{gathered} {{W}_{{u \to \theta }}}_{{(x)}} = p{{\beta }_{{{\text{therm}}}}}\left[ {\frac{{{{x}^{3}}}}{{6a{{c}^{2}}}} - \frac{{{{l}^{2}}(3 + \kappa l)}}{{6a{{c}^{2}}\kappa (2 + \kappa l)}}\left( {1 + \kappa x - p{{x}^{2}}\frac{{{{b}_{2}}}}{{2a{{c}^{2}}}} - {{p}^{2}}{{x}^{3}}\frac{\kappa }{{6{{c}^{2}}}}} \right)} \right](1 + O(p)) \\ = p\frac{{{{\beta }_{{{\text{therm}}}}}}}{{6a{{c}^{2}}}}\left[ {{{x}^{3}} - {{l}^{2}}\frac{{3 + \kappa l}}{{2 + \kappa l}}\left( {x + \frac{1}{\kappa }} \right)} \right](1 + O(p)). \\ \end{gathered} $$

(B.15)

APPENDIX C

Proof of Theorem 3. (1) Due to (3.5) and (3.6), in the neighborhood of the origin on the plane C, the functions a_jk ( j, k = 1, 2) for κ = 0 can be represented as follows:

$${{a}_{{11}}} = {{p}^{2}}(1 + O(p))\quad ({\text{coincides with (B}}{\text{.7)}}),$$

(C.1)

$${{a}_{{12}}} = \beta {{p}^{3}}\left( {\frac{{{{l}^{3}}}}{{6a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right) = \beta {{p}^{3}}\frac{{{{l}^{3}}}}{{6a{{c}^{2}}}}\left( {1 + O\left( {{{p}^{2}}} \right)} \right),$$

(C.2)

$${{a}_{{21}}} = {{\beta }_{{{\text{therm}}}}}{{p}^{3}}\left( {\frac{{{{l}^{2}}}}{{2a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right) = {{\beta }_{{{\text{therm}}}}}{{p}^{3}}\frac{{{{l}^{2}}}}{{2a{{c}^{2}}}}\left( {1 + O\left( {{{p}^{2}}} \right)} \right),$$

(C.3)

$${{a}_{{22}}} = p\frac{l}{{a{{c}^{2}}}}({{b}_{1}} - {{b}_{2}}) + O\left( {{{p}^{2}}} \right) = p\frac{l}{{a{{c}^{2}}}}\left( {{{c}^{2}} - ap} \right) + O\left( {{{p}^{2}}} \right) = p\frac{l}{a}(1 + O(p)).$$

(C.4)

(2) Consequently,

$${{\Delta }_{A}} = {{p}^{3}}\frac{l}{a}(1 + O(p)) - \beta {{\beta }_{{{\text{therm}}}}}{{p}^{6}}\frac{{{{l}^{5}}}}{{12{{a}^{2}}{{c}^{4}}}}\left( {1 + O\left( {{{p}^{2}}} \right)} \right) = {{p}^{3}}\frac{l}{a}(1 + O(p)),$$

(C.5)

$$\frac{{{{a}_{{21}}}}}{{{{\Delta }_{A}}}} = {{\beta }_{{{\text{therm}}}}}\frac{l}{{2{{c}^{2}}}}(1 + O(p)),\quad \frac{{{{a}_{{22}}}}}{{{{\Delta }_{A}}}} = \frac{1}{{{{p}^{2}}}}(1 + O(p)).$$

(C.6)

(3) As a result, we obtain

$$\begin{gathered} {{W}_{{u \to \varphi }}}_{{(x)}} = \frac{{1 + O(p)}}{{{{p}^{2}}}}\left[ {1 + O\left( {{{p}^{2}}} \right) - {{b}_{1}}p\left( {\frac{{{{x}^{2}}}}{{2a{{c}^{2}}}} + O\left( {{{p}^{2}}} \right)} \right)} \right] \\ - \;p\beta {{\beta }_{{{\text{therm}}}}}\left( {\frac{{l{{x}^{3}}}}{{12a{{c}^{4}}}} + O\left( {{{p}^{2}}} \right)} \right) = \frac{1}{{{{p}^{2}}}}(1 + O(p)), \\ \end{gathered} $$

(C.7)

$$\begin{gathered} {{W}_{{u \to \theta }}}_{{(x)}} = p{{\beta }_{{{\text{therm}}}}}\frac{{{{x}^{3}}}}{{6a{{c}^{2}}}}(1 + O(p)) \\ + \;{{\beta }_{{{\text{therm}}}}}\frac{l}{{2{{c}^{2}}}}(1 + O(p))\left[ {p{{b}_{2}}\frac{{{{x}^{2}}}}{{2a{{c}^{2}}}} - 1 + O\left( {{{p}^{2}}} \right)} \right] = - {{\beta }_{{{\text{therm}}}}}\frac{l}{{2{{c}^{2}}}}(1 + O(p)). \\ \end{gathered} $$

(C.8)