"Вестник МГСУ"

Operating conditions of uneven non-stationary heating can cause changes in physical and mechanical properties of materials. The awareness of the value and nature of thermal stresses is needed to perform a comprehensive analysis of structural strength.
The authors provide their solution to the problem of identification of natural frequencies of vibrations of rectangular plates, whenever a thermal factor is taken into account.
In the introductory section of the paper, the authors provide the equation describing the thermoelastic vibration of a plate and set the initial and boundary conditions. Furthermore, the authors provide a frequency equation derivation for the problem that has an analytical solution available (if all edges are simply supported at zero temperature). The equation derived by the authors has no analytical solution and can be solved only numerically.
In the middle of the paper, the authors describe a method of frequency equation derivation for plates exposed to special boundary conditions, if the two opposite edges are simply supported at zero temperature, while the two other edges have arbitrary types of fixation and arbitrary thermal modes.
For this boundary condition derived as a general solution, varying fixation of the two edges makes it possible to obtain transcendental trigonometric equations reducible to algebraic frequency equations by using expending in series. Thus, the obtaining frequency equations different from the general solution becomes possible for different types of boundary conditions.
The final section of the paper covers the practical testing of the described method for the problem that has an analytical solution (all edges are simply supported at zero temperature) as solved above. An approximate equation provided in the research leads to the analytical solution that is already available.

The operating conditions of uneven warming can cause changes in physical and mechanical
properties of the material. Awareness of the intensity and nature of thermal stresses is required to
perform a comprehensive analysis of the structural strength.
The authors provide their solution to the problem of identifi cation of natural frequencies of
vibrations of rectangular plates, if a thermal factor is taken into account.
The introductory section of the paper covers the equation of the thermoelastic vibration of a
plate and formalizes initial and boundary conditions.
The middle section of the paper covers the method of frequency equation derivation for plates
exposed to special boundary conditions, if the two opposite edges of a plate are pinned and their
surface temperature is equal to zero, while the two other edges have an arbitrary type of fixation
and an arbitrary thermal mode.
A general solution is developed for the boundary conditions of pinned edges, while any alternative
types of fixation of the two other edges require derivation of transcendental trigonometric
equations reducible to algebraic frequency equations expendable in series. Thus, derivation of
frequency equations on the basis of the general solution becomes possible for different types of
boundary conditions.
The final section of this paper covers the derivation of the solution for a selected problem
through the application of the method proposed by the authors. The results demonstrate that a thermoelastic
plate with two pinned and two rigidly fixed edges has five natural frequency patterns, two
of which represent the frequencies produced by the plate, if it is free from any temperature influence.

The problem of forced vibrations of plates exposed to the thermal impact is interesting both
as a theoretical implication and an issue of practical importance. A thermal impact causes formation
of a non-steady temperature field. Thereafter, some materials turn fragile and cannot withstand the
exposure to the impact of a thermal field.
The authors propose a solution to the problem of influence of a thermal impact onto an isotropic
plate that demonstrates special boundary conditions, if its two opposite edges are simply
supported, and the surface temperature is equal to zero, while the two other edges might have an
arbitrary type of fixation and an arbitrary thermal mode.
In the first part of the paper, the authors provide their derivation of the elastic plate vibration
equation, if the plate is exposed to the thermal impact under the pre-set boundary conditions.
In the second part of the paper, the authors provide their solution to the aforementioned problem
based on a strictly mathematical approach. Their solution is presented as an integral function
of the plate deflection. The solution in question may be reduced to algebraic frequency equations
by using the method of expansion of trigonometric functions. Thus, it is possible to identify natural
frequencies of the plate vibration caused by the thermal impact.

Operating conditions of uneven heating can cause changes in the physical and mechanical properties of materials. Awareness of the values and nature of thermal stresses are required for a comprehensive structural strength analysis. The authors propose their solution to the problem of identification of natural frequencies of vibrations of rectangular plates using a thermal factor.
The introductory part of the paper covers the derivation of equations of (a) the thermoelastic vibration of a plate, (b) initial and boundary conditions.
In the next part of the paper, the authors describe a method of frequency equation derivation for plates exposed to special boundary conditions, if the two opposite edges of the plate are simply supported, the temperature of the plate surface is equal to zero degrees Celsius, while the two other edges have an arbitrary type of fixation and an arbitrary thermal mode.
The authors have derived a general solution for the above boundary conditions, and by altering the method of fixation of the two edges of a plate, the authors obtain transcendental trigonometric equations reducible to algebraic frequency equations by using expanding in series. Thus, derivation of frequency equations different from the general solution is feasible for various types of boundary conditions.
The final part of the paper contains a derivation of the solution to the selected problem using the proposed method. The results demonstrate that the thermoelastic plate has four natural frequencies, two of them being equal to the frequencies of a plate free from the temperature influence, while the other two are close to the frequency of free vibrations of a plate.