Analyzing the dynamics of a single car wheel

. The paper presents an analytical solution to the equation of dynamic energy spending for rectilinear uniform rousset of a drive wheel with an elastic tire when driving on a solid support surface. To that end, the paper proposes different initial calculation charts to analyze the dynamics of the drive wheel; it also finds the efficiency and the additional energy spending in wheel rousset.


Introduction
Various approaches and therefore different resulting calculation charts are used when studying car wheels. This results in certain contradictions. Attempts to solve those from the standpoint of classical mechanics are mostly fruitless, as a tire is a deforming body rather than a solid. The paper considers two different representations of the initial drive-wheel calculation diagram for uniform rousset, for which it finds the relevant efficiency expressions. To that end, the study takes into account the parameters that characterize the yielding of tires and the rolling resistance.
State of the Art. Wheel rolling is a process that defines the dynamics of a car. A car wheel that interacts with the support surface is exposed to forces that keep it on the road, move it, stop it, or cause it to change the direction. When exposed to a vertical load, the tire is deformed in the area where it contacts the support surface. The distance from the wheel axis to the support surface becomes smaller than the free radius. In papers [1,2,3] E.A. Chudakov considers the drive-wheel dynamics in accordance with the force and moments chart shown in Figure 1 [1]. Fig. 1. Forces, moments, and reactions affecting the drive wheels of a car when rolling on a stiff horizontal surface; Pz is the normal load per wheel, Rz is the normal road-to-wheel reaction, Rx is the tangent road-to-wheel reaction, Px is the car frame to wheel axis reaction, ωk is the angular velocity of the wheel, rд is the dynamic radius of the wheel, Mw ; Pw are the moment and the force of air resistance, Mj ; Pj is the moment and force of inertia as applied to the wheel, a is the rolling friction coefficient (shift of the normal reaction Rz).
When studying uniform motion of a wheel, Mj = 0 и Pj = 0 The following studies have shown that the air resistance encountered in wheel rolling is negligible, and therefore the values Mw; Pw can be ignored. Papers [4,5] present equations of force and power planace for a uniformly rolling drive wheel, represented as follows: where Nf is the power the wheel loses in rolling (rolling resistance power). There is a correlation between the linear wheel-axis velocity (Va) and the angular velocity (ωk) of the wheel.
where is the kinematic radius of the wheel. Note that by introducing the concept opf dynamic ( д ) and kinematic ( ) wheel radii [1,2,3] one can effectively reduce an elastic system (which a wheel with a pneumatic tire is) to a rigid system. By definition, the dynamic radius д is the distance from the axis to the support surface, see Figure 1; therefore, it is directly measurable. The dynamic radius д is smaller than the free radius св , which means an increase in traction force and decrease in the linear velocity as the tire is increasingly deformed when exposed to the increasing force Pz. The kinematic radius can be found by indirect management provided the known values and from the equation (2).
Therefore, one can argue that losses in energy in a tire depend on the losses in the axis motion speed . This is typical of yielding systems. A solid has no such losses. Therefore, both the dynamic radius д and the kinematic radius of a wheel depend on the deformation and the slippage of the tire in the wheel-to-road contact patch and can be used to find the efficiency of the wheel. In papers [4,5] dividing the left-hand and the right-hand parts off the equation (1) by к while taking into account the expression (3) produces the force balance equation Since Px = | Rx |, what makes the above force balance equation (5) different from the equation (2) is the use of the radius instead of д . This is not an option, as the equation (5) is a dynamic rather than kinematic equation. The difference in the moments obtained from (2) and (5) respectively equals For a rigid zero-slippage wheel, ΔMf = 0, as rд = rк = rсв. Therefore, it is necessary to refine the force balance equation (1) and the power balance equation (2) for the drive wheel with due account of losses caused by tire yielding.

Research goal and statement of problem
The research goal consists in finding the dynamic energy spending for rectilinear uniform motion of a drive wheel with an elastic tire when driving on a solid support surface. To that end, one has to solve the following problems: -propose output calculation charts to analyze the dynamics of the drive wheel; -find the efficiency and the additional energy spending of wheel motion.

Core contents of research
The wheel of a car is an element of the car-road system; when finding its efficiency, one has to clearly classify all the forces into wheel-external and wheel-internal forces. Paper [7] considers a kinematic diagram of the car chassis, see Figure 2.

Fig. 2.
Reducing the car frame to a replacement mechanism: 1 is the drive rocker (drive wheel), 2 is the idler rocker (idler wheel), 3 is the rod, 4 is the holder.
When a kinematic chain with higher pairs is reduced to a kinematic chain with lower pairs [8], it becomes clear that the torque is a force external to the mechanism [7]. This is contrary to the popular belief [1,2,3,6] that the drive-wheel torque is an internal force, while the tangent reaction in the wheel-to-road contact patch is an external one. There emerges a question whether the moment of rolling resistance is external or external to the wheel as a link of a dual-rocker mechanism, see Figure 2. Two approaches are possible here. Approach 1. By defining the wheel as the initial link in a dual-rocker mechanism, see Figure 2, we consider the kinematic wheel-holder pair an ideal pair [8]. In that case, the rolling resistance moment Mf can be considered an external force. Then the instantaneous (power) efficiency can be defined as Given the equation (4) as well as the dynamic-radius expression We obtain the final instantaneous-efficiency expression мгн = (9) In that case, the power-balance equation (1) is represented as By dividing the left-hand and the right-hand parts of the equation (10) by and taking into account the equation (9), we obtain We then refinre and rewrite the power balance equation (2) as where сил is the force efficiency of the wheel. From the equation (12), we find From the equation (11), we find the wheel torque By substituting the equation (14) into (13), we obtain It is clear from (15) that the force efficiency of a wheel depends on the rolling resistance moment , which is a force external to the wheel. At = 0, сил = 1.
Approach 2. Consider the kinematic wheel-road pair a non-ideal pair. In this case, the rolling resistance moment is the moment of friction in the holder, i.e. an internal force. In this approach, the instantaneous wheel efficiency The expression in brackets in the right-hand part of the equation (16) is a wheel efficiency component that takes into account the loss of energy to overcome the rolling resistance Therefore, the power balance equation can be defined as By dividing the left-hand and the right-hand parts of the equation (18) Analysis of results obtained by both approaches has shown that aside from energy spent to overcome the rolling resistance of the drive wheels, some losses are due to the tire yielding. This component can be defined as In Approach 1 Additional yielding-caused energy loss factor