Rectilinear car motions taking into account the elasticity and deformability of tires

. In this paper, the problem of developing a mathematical model of rectilinear motion of a car is solved, taking into account the elasticity and deformability of tires, as well as, for the same radii of the wheels of the car, and taking into account plane-parallel motion at a constant speed. The kinetic energy of the rectilinear motion of the car is found in the case of the same rotation of the front wheels around the axis of the pivots. Generalized forces are taken as sum of forces acting on the system and forces due to the deformation of the pneumatics. The forces acting on the system are found by virtual work method. The forces due to the deformation of the pneumatics are found as the generalized forces acting on the system under consideration, in the calculation of which all forces are taken into account, except for the tire deformation forces associated with angles and displacements. Using these kinetic energies and generalized force expressions, Lagrange's equations of the second type were used to obtain the differential equations of the rectilinear motion of the car. From the resulting mathematical model, it is possible to check and analyze the dynamics and stabilities of the system in various specific cases, i.e., under the influence of non-potential forces in the tire materials, and at large values of the tire kinematic parameters, and at high speed movements of the car. As a result of determining the stability field and stability borders at different values of constructive parameters, it will be possible to find the optimal parameters of the system and study its dynamics.


Introduction
The development of computer technologies associated with analytical transformations makes it possible to consider vehicle models with a large number of degrees of freedom.
At the present, a lot of experience has been accumulated in testing automotive equipment, methods have been developed for assessing the performance properties of a car as a whole, as well as individual systems, components and assemblies. Nevertheless, it is relevant to conduct full-scale tests to check the operability of the structure, compliance with the requirements stated in the design specification, as well as to solve other related problems in the works [1][2][3].
In the work [4], mathematical models of the movement of a small truck "Labo" were developed on the basis of the obtained equations of motion. The values of vertical and horizontal vibrations of the car during the process of moving over irregularities are determined. According to the given masses, the parameters of the spring are determined and the model of the vehicle movement is solved using the Runge-Kutta numerical method.
In the work [5], a mathematical model of car movement was studied. The motion model includes mathematical models of the engine, steering, transmission and wheels. The mathematical model is of interest for further improvement of the design parameters of multiaxle vehicles.
In the work [6], a model of car motion is considered, and models of car motion with all steerable wheels are developed.
Mathematical modeling of nonlinear mechanical systems, studying their dynamics, exploring the stability of their vibrations and reducing harmful vibrations at low and high frequencies were solved in the works [7][8][9][10][11][12][13]. The Stability behavior of the system was checked at different values of the parameters, conclusions were obtained as a result of numerical calculations.
In the works [14][15][16][17], the problem of mathematical modeling and exploring dynamics of curvilinear motion of the car is considered according to properties of the elastic and deformability.
The use of dynamic models makes it possible to assess the impact design parameters on the motion of the car, to develop effective algorithms for driving cars, to implement them in the form of so-called active safety means.

Material and methods
The problem of developing a mathematical model of the rectilinear motion of a car, taking into account the elasticity and deformability of tires, and also, for the same radius of the wheels of the car, plane-parallel motion at a constant speed. The kinetic energy of the rectilinear motion of the car in the case of the same rotation of the front wheels around the axis of the pivots. This problem is considered as a special case of curvilinear motion of the car.
Let us assume that for small deviations of the vehicle from rectilinear motion at a constant speed V along the axis Oy, there is no tire slip along the road, and the values i, i, i, i are sufficiently small.  We can express the kinematic relations of the system as following: Denote the product of matrices 1 2 3 4 A A A A through the matrix A 11 where the elements Аij (i,j=1,2,3) of this matrix in the linear approximation will have the following forms: (1) 33 a = -0 +1, 1 is angle of the rotation of the front wheel around the pivot, counted from the direction of the longitudinal axis of the vehicle. is the angle of the rotation of the front suspension axle together with the wheels around the longitudinal axis of the vehicle. 0 is angle of longitudinal inclination of the pin. It is positive when the upper end of the pin is moved back; 0 is the angle of the transverse inclination of the pin.
For the rear axle and rear wheels, the relations take place (1) , therefore, we will have: (2) 31 B = + 0 2; (2) 32 B = 0 -0 2; (2) 33 is angle of the rotation of the front wheel around the pivot, counted from the direction of the longitudinal axis of the vehicle.
Using relations (2) -(3), we find: x y z y x y z 2 2 x y z y and the median plane of the wheel, i is the angle between the Oy axis and the trace of the median plane of the wheel on the road, xi, yi are the coordinates of the meeting point of the straight line of greatest inclination passing in the median plane of the wheel through its center with the plane XOY roads. According to matrix (1)  is the angle of the transverse inclination of the rear wheels. From these relations we find (in a linear approximation): The quantities xi, yi are related to the generalized coordinates as follows:

Results and discussion
The dynamic system under consideration consists of 6 interconnected bodies: a front dependent suspension, a rear part of the car without a front wheel suspension, and four wheels. Therefore, the kinetic energy of the system is where T1 is the kinetic energy of the front suspension, T2 is the kinetic energy of the rear of the vehicle without front suspension and wheels, T3 is the kinetic energy of the front left wheel, T4 is the kinetic energy of the front right wheel, T5 is the kinetic energy of the rear left wheel, T6 is the kinetic energy of the rear right wheel.
Expressions ( 1,6) i Т i are defined as follows: m1 is the mass of the rear of the car without wheels and front suspension; m2i is mass of the i-th wheel; D is the moment of inertia of the rear of the car without front suspension and wheels relative to the axis passing through its center of mass; Ai is the moment of inertia of the i-th wheel with the hub and a brake drum relative to its diameter; B is the moment of inertia of the front suspension about an axis perpendicular to it and passing through the center of mass (central moment of inertia of the front suspension); Ci is axial moment of inertia of the i-th wheel; i instantaneous angular velocity of the system relative to its center of mass.
In accordance with Kőnig's theorem, we have T is the kinetic energy of translational motion with the velocity i V of the center of mass of the system, and '' i T is the kinetic energy of rotational motion with the instantaneous angular velocity i of the system relative to its center of mass. These projections of instantaneous angular velocities relative to the center of mass are determined by:   , , j x j y j z respectively from (11) and (16) into (14) we find the kinetic energies of the center of mass of the front and rear axles, as well as the left and right wheels of the front and rear axles:   Substituting the values ( 1,6) (17) into (13), we find the kinetic energy of the car: 2   2  2  2  3  1  21  22  23  24  3 1  1   2  2  2  22  1 2  1  21  1  1  2  22  1  1  3  23  2  1  4   2  2  2  2  2  24  2  1  1  21 3  1  2  22 3  2  1  21 1 )   2  2  2  2  2 2  1  1  3  2 3  2  1  1  1  3  2 3  2  1  2  2 2  1  1  3  2 3  2  22  1  1  3 Introducing the following notation:  Taking into account (18`), we rewrite the kinetic energy of the system: The first case. Let us assume that the wheel radii [ (  . Then the kinetic energy of the rectilinear motion of the car in the case of the same rotation of the front wheels around the axis of the pivots will take the form: When compiling the equations of vehicle motion, taking into account the deformability of tires, one should take into account the kinematic relationships imposed on the rolling of the wheels [18]. We assume that the wheel is dynamically and geometrically symmetrical about its median plane. Consider the line obtained by the intersection of the middle plane of the undeformed tire. This line is called the center line of the wheel. When a wheel rolls on a horizontal plane with a deformable tire, the deformation of the center line is associated with the deformation of the entire tire. We will assume that tire deformations are sufficiently small. According to the theory of rolling of an elastic tire [19], due to the smallness of the tire deformation, we will consider the transverse, longitudinal, angular and radial deformations of the tire. When the tire is deformed, a contact patch will be obtained, in which there is a whole segment of the center line. Further, from the assumption that the center of the contact area does not slip, the condition that the velocity is equal to zero for any point of this segment follows. In the case of rectilinear motion, the kinematic equations, which mean the conditions for rolling the wheel without sliding, with transverse and longitudinal deformation of the tires have the form 0 0, 0, , , , ..., x y z using formulas (12) and (12`). Substituting the values of these quantities in (23), we obtain the equations of kinematic constraints in the case under consideration:   r r r r r , the kinematic equations will have the form: The equations of motion of a vehicle, according to the theory of motion of rolling systems, are written as where j Q are the generalized forces acting on the system, j R are the generalized forces due to the deformation of the pneumatics, T is the kinetic energy of the system under consideration ,   1  2  3  4  5  6  1  7  2  8  1  9  2  1 0  3 1 1  4   ,  ,  ,  ,  ,  ,  ,  ,  , , q x q y q z q q q q q q q q are generalized coordinates. The generalized forces are calculated by the formula or in our case 11 1 sin cos sin cos where Fi is the transverse force (Fi = The projection of the generalized forces on the x, y, z, , , 1, 2, 1, 2, 3, 4 axes have the form:    2 2  3 3  4 4  51 1  52 2  53 3  54 4  1  1  1 1   1 1 1  1 1  0 1 1 1  2 1  0 2 1 1  1  1  2 2  1 2 2   2 2  0 2 2 2  22  0 22 2  2  3  3 3  13 3  3 3   2  4  4 (  h  h  h  h  h  h  h  h   1  5 2 2  5 3 3  5 4  52 2  53 3  1  5 2 2  5 3 3  5 4  52 2  53 3  1  1 1 1  3 3  13 3  3 3  3 3  13 3  3  3  1 3 3  13 3 ( 2 ) (       2  3  3 3  13  2  2  3  3 3  1 3  2  1  3  3 3  1  3 3   (24) and (32) are a mathematical model of the rectilinear motion of the car, taking into account the elasticity and deformability of tires, and also, for the same radii of the wheels of the car, plane-parallel movement at a constant speed. The kinetic energy of the rectilinear motion of the car in the case of the same rotation of the front wheels around the axis of the pivots.

Conclusion
Further refinement of the right parts of equations (32) depends on certain assumptions about the nature of the deformation by pneumatics, which is determined by carrying out computational experiments.
This obtained system of differential equations allows to estimate the dynamics and stability of the considered system at different values of structural parameters and elastic dissipative properties.