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Finding analytical solutions to the problems of thermal conductivity with variable physical properties of the medium by classical analytical methods is very complicated mathematically. The known expressions repre-senting complex infinite series including two types of Bessel functions and gamma-functions are, in fact, numerical as they require a numerical solution to complex transcendental equations with eigenvalues of the boundary problem. Such solutions can hardly be used in engineering applications, especially in cases when a solution to a certain problem is only an intermediate stage in other problems (such as thermoelasticity and control problems, inverse problems, etc.) which can be solved effectively only by finding analytical solutions to the initial problems. Therefore, an urgent problem now is to develop new methods of obtaining analytical solutions to the abovementioned problems, at least approximate ones. The study employed methods of additional boundary conditions and additional unknown functions in the integral method of heat balance. High-precision approximate analytical solutions to the transient heat conduction problem with nonhomogeneous physical properties of the medium for an infinite plate under symmetric boundary conditions of the first type have been obtained. The initial problem for partial differential equations is reduced to two problems in which ordinary differential equations are integrated. Additional boundary conditions are defined in such a way that their fulfillment in accordance with the new method is equivalent to the result of solving the initial partial differential equation at the boundary points and at the temperature perturbation front (for the first stage of the process). By combining methods with finite and infinite heat propagation rate we have been able to obtain high-precision analytical solutions for the whole time range of the unsteady process including its small and ultra small values. The solutions look like simple algebraical polynomials not including special functions (Bessel, Legendre, gamma-functions and others). Since it is not necessary to directly integrate the initial equations by the space variable and to reduce them to ordinary differential equations with additional unknown functions, the considered method can be used for solving complex boundary problems in which differential equations do not allow distinguishing between the variables (into nonlinear, with linear boundary conditions and heat sources, etc.).

Разработана методика получения приближённых аналитических решений квазистатических задач термоупругости (плоское напряжённое состояние, плоская деформация) для многослойных конструкций с переменными в пределах каждого слоя физическими свойствами среды. Использован рекуррентный метод построения систем координатных функций, точно удовлетворяющих граничным условиям сопряжения, заданным в виде равенства радиальных (нормальных) напряжений и перемещений в точках контакта слоёв. Ключевые слова: многослойные конструкции, аналитическое решение, задача термоупругости, переменные физические свойства среды, система координатных функций, ортогональный метод Бубнова Галёркина.

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531.76.66 Various aspects of the design of microprocessor systems for velocity, acceleration, and position measurements are considered. The velocity-measuring method described differs from known methods in that it has minimal errors in the range of velocities from fractions of a rotation to the maximum rotation frequency and requires a microprocessor. The position and acceleration are measured at the same time as is the velocity.One method of increasing the accuracy of modem electric drives of machine tools and industrial robots is by improving the dynamic and static characteristics of transducers in feedback circuits [1, 2]. The velocity of a controlled drive is generally measured by analog transducers, which reduces the control accuracy, especially in the range of low and ultralow values. Moreover, digital drives (to close the main feedback circuit) as well as analog drives (to close the velocity circuit) are used in autonomous follower drives.Coding transducers with magnetic or optical pickup are used to obtain information about the parameters of motion of an object. Essentially, the velocity is measured by two principal methods [2]: counting pulses from a coding transducer in a fixed interval of time and counting pulses from an auxiliary oscillator during coding transducer pulses. In the first case, the measurement is effective in the range of high velocities and in the second, low velocities. The position of the actuators is determined by directly counting pulses from coding transducers and then covering the count into the actual position.Both methods involve the use of equipment and are simple to apply. The quality of the measurements, however, is not always justified. We consider the salient features of the two methods in greater detail and estimate the data acquisition error.The essence of the direct method consists in measuring the number of sensor pulses in a given period of time. The rotational velocity of the sensor disks is calculated aswhere T is the measuring interval, N is the number of sensor pulses counted in the given time, and Z is the number of pulses per rotation. Even though the characteristic of the given method of measurements is linear over all ranges, measurement of slow velocities is a very complicated process. The main reason why the given method has an error of velocity measurement is that the number N is an integer since most often the pulses are counted by standard counters. As a rule, however, a fractional number of pulses fits in the measuring period T. Accordingly, that period must be increased to reduce the errors. For a wide-range sensor, the necessary measuring period is calculated for the minimum velocity since the sensor pulse frequency then is also a minimum.We consider the distribution of the error of velocity measurement by the given method, depending on the velocity and the measuring period (from Fig. 1).