ANALYSIS OF MECHANICAL AND ELECTRICAL VIBRATIONS IN THE ENGINES OF MILITARY AND COMBAT VEHICLES ON THE BASISOF DIFFERENTIAL EQUATIONS

ABSTRACT

This article discusses the solution of one of the current problems of mechanical and electrical vibrations that occur in military and special equipment, using the section of mathematics of differential equations. The article presents a simple diagram of an electric circuit and gives a solution to the second derivative of current with respect to time with the right-hand side of the invariant coefficient of the second order using special differential equations.

АННОТАЦИЯ

В данной статье рассматривается решение одной из актуальных проблем механических и электрических вибраций, возникающих в военной и специальной технике, средствами раздела математики дифференциальных уравнений. В статье приведена простая схема электрической цепи и дано решение второй производной тока по времени с правой частью инвариантного коэффициента второго порядка с помощью специальных дифференциальных уравнений.

Ключевые слова: колебание, уравнение, дифференциальное уравнение, постоянный коэффициент.

Keywords: oscillation, equation, differential equation, constant coefficient.

It is known from the laws of mechanical motion that the vibration of a rigid body with mass can be free or forced. Free vibration equations are characterized by damping and consist of a homogeneous differential equation. Usually, in internal combustion engines, continuous oscillations are formed that are not necessarily damped. The reason for this is that external force periodically affects the system. In internal combustion engines, as a result of the explosion of fuel, mechanical vibrations (vibrations) are generated, and it is natural for electric vibrations to be generated in the electric circuit and electric motor (generator)[1]. Mechanical forced vibrations are represented by second-order inhomogeneous special differential equations with constant coefficients. Forced vibration is generated under the influence of an external force and its general formula is given in the form (1).

                                                (1)

where m- the mass of the body (object) (kg),

x- is the function representing the law of motion, the deviation of the points on the surface of the engine from a certain position (m), 

k- uniformity (stiffness) of the elastic system,

- proportionality coefficient,

- external force or driving force, - acceleration of the system,

- system speed.

This differential equation is derived from Newton's second law, one of the fundamental laws of dynamics[2]. Any moving object comes under the influence of this law. In turn, the rotational vibration of the flywheel on the axis is also expressed by equation (1). Only, instead of the law of motion, the angle of rotation of the flywheel (rad), instead of the mass, the moment of inertia of the flywheel is replaced, instead of the torque, the angular velocity of the axis is replaced, and as a result, the following equation (2) is formed.

                                            (2)

Equation (2) is called a special form equation with a second-order constant coefficient on the right side. In addition, it is possible to analyze the mechanical vibrations (vibrations) occurring in alternating synchronous three-phase generators (including asynchronous motors) using equation (1).

One of the factors (physical process) that causes the most damage to the techniques is vibration. The factors that create it are mechanical and electrical vibrations. We use the differential equations branch of mathematics to bring the mechanical or electrical vibrations in this phenomenon to the limit of human knowledge. Suppose a non-branched alternating current circuit connected in series is given (Fig. 1).

Figure 1. Unbranched alternating current

The main reason for this is that alternating current is produced in synchronous generators, which are considered the primary (main) source of electricity. The above simple electric circuit consists of L- inductive coil, R- resistance, C- capacity (capacitor), E- electric driving force. The current in the electric circuit, the capacitor charge, and the voltages are respectively denoted by ,  and. Since the elements in the electric circuit are connected in series, the total voltage is equal to. When evaluating any complex branched electric circuit, it is evaluated by its equivalent simple electric circuit. For this reason, our main goal is ,,to scientifically check a simple electric circuit formed by connecting the current elements in series[3]. We create the following differential equation from the equality of current, voltagees, and .

                                                 (3)

(3) by differentiating the differential equation once with respect to time and dividing both parts of the equation by L, we form the second-order non-homogeneous linear differential equation with the following constant coefficients.

                                      (4)

The differential equation (4) is the oscillation formula in which, , s are fixed numbers, for convenience , , by performing the substituton, we get the following equation

                                            (5)

(5) of the differential equation va we find a general solution that satisfies the initial conditions.

To solve the equation, first homogeneous     we find the general solution of .

We find the solution of a homogeneous differential equation with constant coefficients of the second order  we look for it. Taking first and second order derivatives

,

the private solution of the inhomogeneous part of the differential equation islooking in the view[4].first on  dan  bo‘yicha birinchi
we take the First-Order and second-order derivatives over dan (5) and take them to the differential equation.

(5) the total solution of the differential touch was equal to.

Using the boundary conditions and finding the values of the constants[5].Using the boundary conditions and finding the values of the constants[6-9].

(5) the general solution of the differential touch satisfying the initial condition  (6) is in view.

Conclusion

The idea of the study is that in the process of mechanical oscillation, an electrical signal arises with which you can diagnose the technical condition of the engine. Predict until the engine stops completely, excluding the non-maintainability of an expensive engine. It is possible to prevent a technical malfunction by partially replacing a specific unit, eliminating expensive replacement, which increases the life of the internal combustion engine. Modern diagnostic technology requires information for processing and solving where the address and location of the future problem are indicated.

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