STATIONARY OSCILLATIONS OF THERMOELASTIC ROD UNDER ACTION OF EXTERNAL DISTURBANCES

ABSTRACT The dynamics of a thermoelastic rod of finite length are studied under action of periodic external force and thermal sources. The model of connected thermoelasticity are used, taking into account the effect of temperature on the elastic deformation and stresses and as well as the effect of elastic deformation speed on the temperature field in the rod. Based on the method of generalized functions the analytical solutions of boundary value problems of the stationary vibration of a thermoelastic rod for various types of boundary conditions are constructed. The computer implementation of the boundary value problems are performed. The calculation results of movement and temperature of the rod at different frequencies, and a comparative analysis of solutions are presented.


INTRODUCTION
The core design is widely used in engineering as coupling and transmission units for the structural elements for different machines and mechanisms. During the operation they are subjected to variable mechanical and thermal influences that create a complex stress-strain state in designing elements, depending on their temperature, which affect at their reliability and durability. Therefore, the determination of thermoelastic stresses in rod structures with regard to their mechanical properties (especially the thermoelasticity) refers to actual scientific and technical problem.
Mathematical modeling of thermodynamic processes in rods leads to solving the boundary problems for thermoelastic media. There are various models of thermoelastic media. In the study of slow dynamic processes the unconnected thermoelasticity model are frequently used, which does not take into account the influence of elastic properties of medium on its temperature field. But fast vibratory processes in designs affect at the temperature field. In the study of such processes the coupled thermoelasticity model should be used, which is considered here to simulate the dynamics of thermoelastic rods.
In [1] there were constructed the fundamental and generalized solutions of thermoelasticity equations in a spatially one-dimensional case. They determine the thermoelastic stress-strain state of infinite thermoelastic rod under action of various periodic forces and heat sources. They may be described by distributions, both regular and singular, allowing to research the effect of concentrated sources of various types. Here we consider the boundary value problems (BVPs) of the dynamics of a thermoelastic rod of finite length at stationary oscillations. Four types of boundary conditions at each end of the rod have been considered. Based on the method of generalized functions the analytical solutions of BVPs are constructed and investegated.

THE MOTION EQUATIONS OF A THERMOELASTIC ROD HARMONIC OSCILLATIONS
Let consider a thermoelastic rod with the length 2L . The longitudinal displacements of a rod u and rod temperature  are described by the mixed hyperbolicparabolic equations of second order in the form [2]: EJ c   . The symbol after the comma denotes the partial derivative with respect to the specified index variable (for example   2   , , , Let assume that the rod is subjected to the actions of periodic longitudinal force  is the frequency of oscillation.
Thermoelastic stresses in the rod   , xt  are defined by the Duhamel-Neumann low: The boundary conditions at the ends of the rod may be different. Here we formulate them for four boundary value problems (BVP) which are taken in the classical theory of thermoelasticity [1]: Also we can consider the boundary value problems with the one type of boundary conditions on the left end of the rod (any from (4)) and other type of boundary conditions on the right end.
By virtue of the harmony of acting forces (2) and boundary conditions (41-44), the solution can be presented in the same form: where the complex amplitudes satisfy the following system of differential equations: We define the complex amplitudes of solutions satisfying (5) and one of the conditions (4), respectively to solved BVP.

METHOD OF GENERALIZED FUNCTIONS. ANALYTICAL SOLUTION
To solve the BVP we used the method of generalized functions [3,4]. On this basis, using a matrix of fundamental , , . Here the convolution for regular forces and heat sources are calculated by formulae: H(x) is Heaviside function. For singular 12, FF you should use the definition of convolution (see [3]).
Their asymptotic properties are described in detail in [1].

THE RESOLVING EQUATIONS FOR BOUNDARY VALUE PROBLEMS
In the paper [4] it was shown, that the unknown amplitudes of the boundary functions of all BVPs satisfy to resolving system of equations, which can be represented in the matrix form: where components of matrix A1 and A2 are equal to Components of vector b depend on acting external forces and heat sources:

BOUNDARY VALUE PROBLEM 1 AND ITS SOLUTION
We consider here the first boundary value problem by known temperatures and the movement at the ends of the rod (41),. In this case, the resolving system of equations has the form: where the matrices A and B are expressed through the elements of the matrices A1 and A2 so 11  13  11  13  12  14  12  14   21  23  21  23  22  24  22  24   31  33  31  33  32  34  32  34   41  43  41  43  42  44  42  44   1  1  2  2  1  1  2  2   1  1  2  2  1  1  2  2   1  1  2  2  1 1 2 2 Solving this system we determine unknown stresses and heat flow on the ends of a rod. Then, using (61), (62), we calculate displacement and temperature in any point of a rod.
On the figures 3, 5, 7,9 (a, b) you see the real and imaginary parts of the complex amplitudes of displacement and temperature, which describe the real state of the rod at