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

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.


INTRODUCTION
Thermomechanical systems with mechanical vibrations and heat transfer processes are widely used in modern engineering. Therefore, it is necessary to study the dynamic properties of such systems in mathematical terms and develop control methods for them.
The literature on the thermoelasticity phenomenon is quite extensive. The early works [1][2][3] were followed by [4], where thermoelasticity was investigated as part of general elasticity effects. In the recent literature, we mention the publications [5][6][7] devoted to various properties of thermoelastic media. The book [8] developed a modern theory of thermomechanics of elastoplastic deformation. A coupled dynamic thermoelasticity problem for a one-dimensional medium was stated in [9].
In this paper, we analyze the dynamic properties of a one-dimensional distributed elastic thermomechanical system. The mathematical model of processes in such a system is based on the classical work [4]. In contrast to [10], the system is subjected to a mechanical (force) action at one of its boundaries instead of a thermal action. The system dynamics equations have the form where t ≥ 0, 0 ≤ x < l, and a, c, β, and β therm are positive constants. (For details, we refer, e.g., to [4].) In these equations, ϕ(x)(t) denotes the displacement of the section located at a distance l -x from the point of application of the force action; θ(x)(t) is the temperature of the medium in the section x.
The initial conditions with respect to the time variable are assumed to be zero. The boundary conditions are as follows: (a) for the function ϕ, where u is the control action (with the physical sense of a mechanical (force) action applied to the system);  where α and λ are positive constants.

CALCULATION OF THE VECTOR TRANSFER FUNCTION
We perform the Laplace transform of Eqs. (1.1) with the boundary conditions ((1.2), (1.3)) to obtain the system of ordinary differential equations (2.1) where κ = . In this system, the pair of unknown functions ( , ) consists of the Laplace images of the desired functions (ϕ(x)(t), θ(x)(t)).
Solving the boundary-value problem (2.1)-(2.3) yields the following expressions for the transfer functions in the channels u → ϕ(x) and u → θ(x). Theorem 1. The transfer functions of the system in the channels u → ϕ(x) and u → θ(x) are given by (2.4) and (2.5) respectively. In these formulas, Δ A = a 11 a 22 -a 12 a 21 ,  3. ASYMPTOTIC BEHAVIOR OF TRANSFER FUNCTIONS AS p → 0 We study the dynamic properties of the system, beginning with the asymptotic behavior of its transfer functions in the neighborhood of the origin on the complex plane C. Theorem 2. In the neighborhood of the origin on the plane C, the transfer function can be represented as (3.1) and the transfer function as Here, O(p) denotes a function f(p) (p ∈ C) with a bounded ratio as p → 0.
Thus, the system has the double integrating property in the channel u → ϕ(x) and the differentiating property in the channel u → θ(x).
Remark. According to (3.2), the asymptotic formula for the transfer function as p → 0 includes the ratio = . Therefore, the case κ = 0 (no heat exchange with the environment) should be considered separately; see the next section.
In this case, the functions a jk ( j, k = 1, 2) have the form Thus, in the case κ = 0, the transfer function of the system in the channel u → θ(x) has a finite nonzero limit as p → 0. This property can be called static.
The proofs of Theorems 1-3 are given in Appendices A-C, respectively.

CONCLUSIONS
As has been demonstrated by this study, the thermomechanical controlled plant subjected to a mechanical (force) external action possesses the following dynamic properties: double integration in the channel from the force action to the displacement of the one-dimensional medium and (but only under heat exchange with the environment) differentiation in the channel from the force action to the temperature.
t h e r m t h e r m , The resulting conclusions should be considered when designing control systems for thermomechanical plants with dynamic properties described by (1.1)-(1.3).
According to the results of [10] and this paper, the intrinsic feedback of the plant (from the displacement of the sections to the temperature) complicates the description of its dynamic properties compared to the case of no feedback, which was investigated in [11].    (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: