ANALYSIS OF THE DYNAMICS OF VEHICLE MOTION IN CASE OF SUDDEN BRAKING CONNECTED WITH DAMAGE OF THE TIRE

One of the urgent problems to be solved in the analysis of road traffic accidents is the calculation of vehicle speed at the time of sudden braking. This makes it possible to determine the dynamics of further movement of the vehicle and to determine the technical ability of the driver to prevent a traffic collision. The article presents a method employed in calculating the speed of the car at the time of sudden braking and determines the dynamics of movement up to its complete standstill. The correct calculation of the speed of movement allows us to obtain the reliability of the restored co-existence, as well as data for an objective analysis of the road traffic accident.


INTRODUCTION
When investigating road accidents, one of the focal factors determining the possibility of analyzing the mechanism of the event and its individual elements is determining the speed of the vehicle at the time of the accident. By the magnitude of the speed, the technical ability of the driver to prevent an accident is determined reproduce the relative position of the elements, objects and parties of the event at the moment of occurrence a danger to traffic [1,4].
The existing methods for determining the vehicle speed are based on the calculation of the traces of braking, and in their absence, based on the observation of eyewitnesses.
However, an estimate of the vehicle speed by the tracks of braking does not always adequately present real events, and eyewitness reports can be rather vague. Appling scientific approaches based on the analysis of physical phenomena occurring in the contact of the wheel with the road surface is a more sound and reliable approach [5].

PROBLEM FORMULATION
As an example, let is consider a traffic accident in which a vehicle, as a result of sudden braking due to the destruction of the front left tire, sharply spins at an angle to the left with a sidewise skidding and simultaneous linear movement in the direction of the initial movement and rotation.
It is required to determine the speed of the vehicle at the time of the sidewise skidding, taking into account the linear movement of the center of mass of the vehicle.

PROBLEM SOLVING
To answer the question, let us assume as follows: the destruction of the tire occurred instantly; at the time of destruction of the tire of the front left wheel, the driver removed his foot from the accelerator pedal and braked; we will disregard the laws of the forces of rolling resistance, friction in the transmission of the car and air resistance. In addition, we will make allowance that at the time of tire destruction, the vehicle in the process of lateral deceleration simultaneously suffered a turn relative to the point of contact of the front left tire with the road. In this case, the stiffness of the front left wheel suspension during this process is so insignificant that it can be ignored in the calculations.
The vehicle sidewise skid at the point of going into the curve causes the excess of the linear velocity limits acceptable for riding stability. For this road traffic situation, we define the dynamics of the vehicle at the time of the sidewise skidding as follows: the sidewise skidding is followed by a linear movement of the vehicle in the direction of the initial movement and rotation. In this case, the total work of the tire's friction on the road is expressed as follows: (1) where Аt is the work of the tire's friction on the road during the vehicle rotation, kGm; Аfr is the work of the friction on the road at the linear displacement of the mass center of the vehicle, kGm. We take into account that (2) where Gа is the weight of the vehicle, kG; φsis the coefficient of side grip between the tire and the roadway; L is the vehicle base length ( fig. 1; [6]), m; and nrev is the number of revolutions performed by the vehicle (see fig. 1) on the path of its linear motion in the direction of its initial motion.
Kinetic energy of rotation of the car: where J is the inertial torque toward the О-point, kg•m 2 ; ω is the angular velocity rate (see fig.1), rad/s. Where (4) where mа is the car mass, kg; L is the car base length, m (see fig.1); and β is the angle between the longitudinal axis of the car and the line connecting points О and О1 (see fig. 1). Then, expression (3) will be written as follows: The angular velocity rate ω will be calculated as follows: where Vа is the linear speed of the car at the start of its rotation, m/s. Considering expression (6), expression (5) will be written as follows: The kinetic energy of rotation goes into the performans of the tire friction while turning during spinning around point О (8) where Gа is the car's weight, kG; nrev is the number of revolutions per time of linear displacement of the mass center.
Considering the equality Wt = Wt, 1 we will obtain (9) The linear translation of point О to elementary distance dS 1 over an infinitesimally small period of time dt (10) where V 1 is the speed of linear movement of the vehicle, m/s. At the movement of point O to an elementary distance dS1, point O1 will strike an infinitesimally small arc dS (11) where dγ is an infinitesimally small angle of the center of the rear axle (point О1; see fig. 1), rad.
By transformation of expression (25) considering jа, we obtain Considering expression (26), we obtain (27) where S 1 is the linear movement of the center of mass of the vehicle from the moment of its sidewise skidding down to complete standstill, m.  (27), we obtain (28) Tire friction work at a linear car traverse is expressed as follows: Considering expressions (28) and (29), expression (2) will be written as follows: From this, the dynamics of the car motion at the sidewise skid will be written as follows: The vehicle's braking distance from the sudden braking down to a complete standstill is calculated from the following formula: where tt is the turn time of the car with respect to point О (see fig. 1) and for angle α ( fig. 2) by braking force with the left front wheel, s; tl is the time of linear movement of the center of mass of the car from the moment of its sidewise skid down to complete standstill от, s; and ttl is the time spent on rotation of the car at its forward-rotation movement, s. In the picture (see fig.2  where Fi is the inertia force at the moment of braking with the left front wheel (see fig.1), kG; J is the moment of inertia with respect to point О (see fig.1), kgm 2 ; and ωа is the angular velocity of the car (see fig. 2), rad/s. Let us transform the right-hand side of expression (33); we will obtain (34) where Then, expression (33) will be written as follows: The arc length ld (see fig. 2) can be written as follows: where Vcv is the circular velocity of the center of the car's rear axle, m/s. Where where Jrp is the inertia of the rotating parts of the engine and related transmission parts, kg·m 2 ; ηtr is the transmission efficiency; kct is the transformation coefficient; ωн, ωт are the angular velocity of shaft and turbine rotations, respectively, с -1 ; rk is the wheel rolling radius, m; and mа is the car mass, kg; where ∑Jk is the total inertia moment of the car wheels, kg·m 2 .
Considering expression (41), expression (40) will be written as follows: (46) By substituting expression (46) into expression (38), we will obtain Then, the slew time considering expressions (45) and (48) will be written as follows: The time of linear movement of the center of mass of the car from the sidewise skid start moment down to complete standstill of the car will be obtained considering the linear velocity using the following formula: The table shows that the vehicle speeds, calculated with the suggested method, are in agreement with the speeds as determined by computer simulation.

CONCLUSIONS
The suggested method for calculating the vehicle speed at the time of sudden braking provides a supporting rationale for scientifically based restoration of the course of events of a road accident and to identify the data that can serve as probative evidence for a traffic accident.
The currently available methods for calculating the speed of vehicles in the case of sudden breaking are based either on the visual perception of an event by an accident witness [7,8] or on the data of the accident investigative experiment [9,10]. In the first case, the reliability of the event coverage is hinged on human psychophysical abilities [11,12], whereas in case of the simulation, it is complicated by different technical and operational properties of vehicles being used in the accident investigative experiment [13,14]. For this reason, accident modeling solely based on the laws of physics [15,16] is considered adequate and overriding.