Mathematical modelling of vertical and longitudinal-angular vibrations in the composition of the rigidity and damping of the hierarchical parts of the tractor taking into account road impact

The article contains generalized systems of equations describing the vertical and longitudinal-angular vibrations of the main parts of the tractor with the mathematical model of the hierarchical vertical linear oscillation of the tractor, the irregularities of the roads, taking into account the rigidity and damping. It includes systems: equations describing the vertical vibrations of the skeleton over the propellers of a tractor on an uneven road; equations describing the vertical vibrations of the cab of the rear of the tractor, taking into account the walking of the tractor on uneven roads. Equation systems have also been compiled that describe the vertical oscillations of the uppermost hierarchical link the seat of the tractor operator, taking into account the walking of the tractor on an uneven road. The use of this original mathematical model will make it possible to find random unwanted oscillations of the new designed machine design. It will help to optimize their geometric and dynamic parameters.


Introduction
The calculation method of the lower rubber damper of the cab allows you to determine the geometric dimensions of the mechanical sub-cab damper.
Using the definition of the vibration of a linear system with one degree of freedom of the under-cab shock absorber of a tractor, we can find the coefficient of efficiency of vibration isolation, which notifies the presence or absence of vibration of the operator's cab.
By numerically solving these equations, we can find the angular and vertical displacement and angular velocity of the cab and the main parts of the tractor. This means that it will be found how much the deflation of the main parts is displaced during vibration.
The article [1] describes theoretical methods for damping low-frequency vibrations on the driver's seat of a wheeled tractor based on the development of mathematical models with various elastic dissipative characteristics of the suspension.
Methods of incorporating damping relaxation elements into the suspension for shifting resonant vibrations to the frequency range greater than 1 ... 8 Hz, as well as methods allowing the practical implementation of the inclusion of Creating an effective vibration protection system for a wheeled tractor driver is an urgent task [2] aimed at improving the driver's work and ensuring safety.
Seat acceleration and noise levels in the driver's cab of a wheeled vehicle are strictly regulated by international standards. Therefore, global manufacturers of wheeled vehicles are constantly improving vibration protection systems and introducing new absorbing materials in the design of driver's cabins. Thus, studies [2] have determined the frequency ranges of natural vibrations of various human organs and established the distribution of vibration frequencies over the structural elements of a wheeled tractor.
F. Deine and M. Michke pay special attention to low-frequency vibrations (1 ... 8 Hz), which have the most harmful effect on the driver and cause him occupational diseases.
Studies have established that the natural frequency of human vibrations on the seat of modern wheeled tractors is 1 3 Hz, and the permissible values of the root-mean-square accelerations and amplitudes of vertical movements of the seat should not exceed 8.5 m / s2 and 0.02 m, respectively, which is stipulated by GOST 12.1. 012-90 and Council Directive No. 78/764.

Research Methods
The equations describing vertical and longitudinal-angular oscillations are written according to the Lagrange equation of the second kind.
In the mathematical modelling of the hierarchical vertical linear oscillation of the tractor, taking into account the unevenness of the roads, the stiffness and damping coefficient is considered.

Results and discussion
The power load and the speed of the tractors are constantly increasing, which leads to an increase in the dynamic load of the main parts and an increase in the level of oscillations generated by them. Vibration loads have a negative effect on the components and parts of the tractor, on the environment, and on the cab with the operator.
The main parts of the tractor are described by the flat, created at the first stage, and at the subsequent stage, the spatial model of the suspension systems [1].

b) equations describing longitudinal-angular oscillations
Here mi is the i-th sprung mass; Ji-moment of inertia of the i-th sprung mass; sprung mass, z i , z i, z ï are respectively vertical displacement, speed, acceleration, i-th sprung mass; dz i are corresponding speeds of vertical displacements; φ i ,φ i,φ ï are angular displacement, speed, acceleration, i-th sprung mass; c i i-th bond stiffness; k i i-th damping factor; a, b, c, d, e, f, h, k, n are geometric parameters of elements; g is the acceleration of gravity.
Based on a flat model, when using an algorithmic program, it is possible to obtain the oscillatory systems of the frame, cab, and seat and graphs of displacements, speeds, and accelerations of these tractor parts.
Mathematical modeling of the hierarchical vertical linear oscillation of the tractor, taking into account the unevenness of the roads, considering the coefficient of stiffness and damping.
Oscillations in machines are often undesirable, as they reduce the reliability of machines, cause noise and have a harmful effect on the human body. The dynamic equivalent total design of the main parts of the tractor is shown in Fig. At the same time, for the convenience of analytical calculation and creation of the possibility of implementing ideas, we take the following assumptions:  takes only the stiffness coefficient in the calculation; -the damping factor is neglected; if we take into account the mathematical model, it becomes too complicated; -various forced and natural vibrations are considered absent, etc. Let us consider the effect of oscillation on a mechanical system using a model depicting the movement of a mass of 1/3 m and 2/3 m; we assume that a third of the mass falls on the front of the tractor and two of a third of the mass that is elastically suspended on the wheel falls on the rear of the tractor, which rolls along a rigid path with cosine-shaped irregularities ( Figure 3). It is obvious that the system will perform forced oscillations.
and substitute it into equation (2). As a result, we get  As is known, the solution to a homogeneous equation can be represented as: Then the general solution of equation (2) is represented as The origin of time (t = 0) in this system can be taken for such a moment when z = 0. In this case, substituting t = 0 and z = 0 into (4), we obtain This will be the general solution to our equation under the initial conditions adopted above.
As ω approaches ν, the period T 1 , T 2 increases. When the quantities ω and ν.θ coincide exactly, the phenomenon of resonance occurs. b) equations describing the vertical vibrations of the cab located at the rear of the tractor, taking into account the walking of the tractor on an uneven road.
The cab is located at the rear of the tractor. To simplify the analytical calculation, we assume that the vertical vibrations of the front and engine vibrations do not affect the vibrations of the cab and the tractor operator. We believe that the rocking of the rear of the tractor affects the vibrations in the cab and the operator's workplace.
If we take 16 from equation (6) and substitute it into equation (13). As a result, we get Then the general solution of equation (13) is represented as c) equations describing the vertical vibrations of the tractor operator's seat, taking into account the walking of the tractor on an uneven road. Let us first find a particular solution to the equation. Imagine that

Conclusions
For the first time, the hierarchical link of the tractor was mathematically modeled, taking into account the unevenness of the roads based on the coefficient of stiffness and damping. The equations describe the vertical and longitudinal-angular vibrations of the main parts of the tractor. The hierarchical vertical linear oscillation of the tractor is mathematically modeled, taking into account the unevenness of the roads, the coefficient of stiffness and damping, which serve in the design of wheeled machines, is considered. The use of this mathematical model will make it possible to determine in advance the possible fluctuations of the new designed design of machines. It will help to optimize their design and dynamic parameters.