A car (tractor) moves as a result of the action of various forces on it, which are divided into driving forces and forces of resistance to movement. The main driving force is the traction force applied to the drive wheels. Traction is generated by the operation of the engine and is caused by the interaction of the drive wheels with the road. Traction force P to is defined as the ratio of the moment on the axle shafts to the radius of the driving wheels with a uniform movement of the car. Therefore, to determine the traction force, it is necessary to know the radius of the drive wheel. Since elastic pneumatic tires are installed on the wheels of the car, the radius of the wheel changes during movement. In this regard, the following wheel radii are distinguished:
1. Nominal - the radius of the wheel in the free state: r n \u003d d / 2 + H, (6)
where d is the rim diameter, m;
H is the total height of the tire profile, m.
2. Static r s is the distance from the road surface to the axis of the loaded stationary wheel.
r с =(d/2+H)∙λ , (7)
where λ is the tire radial deformation coefficient.
3. Dynamic r d is the distance from the road surface to the axis of the rolling loaded wheel. This radius increases with a decrease in the perceived load of the wheel G k and an increase in the internal air pressure in the tire p w.
With an increase in the speed of the car under the action of centrifugal forces, the tire is stretched in the radial direction, as a result of which the radius r d increases. When the wheel is rolling, the deformation of the rolling surface also changes in comparison with a stationary wheel. Therefore, the shoulder of the application of the resultant tangential reactions of the road r d differs from r s. However, as experiments have shown, for practical traction calculations, r s ~ r d can be taken.
4 Kinematic radius (rolling) of the wheel r k - the radius of such a conditional non-deformable ring, which has the same angular and linear speeds with a given elastic wheel.
At a wheel rolling under the action of a torque, the tread elements that come into contact with the road are compressed, and the wheel travels a shorter distance at equal speeds than during free rolling; for a wheel loaded with braking torque, the tread elements that come into contact with the road are stretched. Therefore, at equal speeds, the brake wheel travels a slightly longer distance than a freely rolling wheel. Thus, under the action of torque, the radius r to - decreases, and under the action of braking torque - increases. To determine the value of rk by the method of “chalk prints”, a transverse line is applied on the road with chalk or paint, on which the car wheel rolls, and then leaves prints on the road.
Measuring the distance l between the extreme prints, determine the rolling radius by the formula: r to = l / 2π∙n , (8)
where n is the wheel rotation frequency corresponding to the distance l .
In the event of complete wheel slip, the distance l = 0 and radius r to = 0. During the sliding of non-rotating wheels (“SW”) rotation frequency n=0 and r to .
When rolling an elastic (deformed) wheel under the influence of force factors, a tangential deformation of the tire occurs, in which the actual distance from the axis of rotation of the wheel to the supporting surface decreases. This distance is called dynamic radius r d wheels. Its value depends on a number of design and operational factors, such as tire stiffness and internal pressure in it, vehicle weight per wheel, speed, acceleration, rolling resistance, etc.
The dynamic radius decreases with increasing torque and decreasing tire pressure. Value r d slightly increases with increasing vehicle speed due to the growth of centrifugal forces. The dynamic radius of the wheel is the shoulder of the application of the pushing force. Therefore it is also called power radius.
The rolling of an elastic wheel on a solid support surface (for example, on an asphalt or concrete highway) is accompanied by some slippage of the wheel tread elements in the zone of its contact with the road. This is due to the difference in the lengths of the sections of the wheel and the road that come into contact. This phenomenon is called elastic slip tires, as opposed to slip(slip), when all elements of the tread are displaced relative to the supporting surface. There would be no elastic slip if these sections are absolutely equal. But this is possible only when the wheel and the road have contact in an arc. In reality, the bearing contour of the deformed wheel comes into contact with the flat surface of the undeformed road, and slippage becomes inevitable.
To account for this phenomenon, the concept is used in calculations. kinematic radius wheels ( rolling radius) r to. Thus, the calculated rolling radius r k is such a radius of the fictitious undeformed a wheel that, in the absence of slippage, has the same linear (translational) rolling speeds with a real (deformed) wheel v and angular rotation ω to. That is, the value r to characterizes conditional radius, which serves to express the calculated kinematic relationship between the speed of movement v vehicle and wheel speed ω to:
A feature of the rolling radius of the wheel is that it cannot be measured directly, but is determined only theoretically. If we rewrite the above formula as:
, (τ
- time)
then from the resulting expression it is clear that to determine the value r can be calculated. To do this, you need to measure the path S, passed by the wheel for n revolutions, and divide it by the angle of rotation of the wheel ( φ to = 2pn).
The amount of elastic slip increases with a simultaneous increase in the elasticity (compliance) of the tire and the rigidity of the road, or, conversely, with an increase in the rigidity of the tire and the softness of the road. On a soft dirt road, increased tire pressure increases ground deformation losses. Reducing the internal pressure in the tire allows, on soft soils, to reduce the movement of soil particles and the deformation of its layers, which leads to a decrease in rolling resistance and an increase in patency.
However, on a firm ground at low pressure, excessive tire deflection occurs with an increase in the rolling friction arm. A. A compromise solution to this problem is the use of tires with adjustable internal pressure.
In practical calculations, the wheel rolling radius is estimated by the approximate formula:
r k \u003d (0.85 ... 0.9) r 0 (here r 0 - free wheel radius).
For paved roads (wheel movement with minimal slippage) take: r k = rd.
Due to the large variety of types of deformation of a pneumatic tire, its radius does not have one specific value, as in a wheel with a rigid rim.
There are the following rolling radii of a wheel with a pneumatic tire: free g 0 , static r cv dynamic g a and kinematic g k.
free radius g 0- this is the largest radius of the treadmill of the wheel, free from external load. It is equal to the distance from the surface of the treadmill to the axis of the wheel.
The static radius r st is the distance from the axis of a stationary wheel loaded with a normal load to the plane of its support. The values of the static radius at maximum load are regulated by the standard for each tire.
dynamic radius g i- this is the distance from the axis of the moving wheel to the point of application of the resulting elementary soil reactions acting on the wheel.
Static and dynamic radii decrease with increasing normal load and decreasing tire pressure. The dependence of the dynamic radius on the moment load, obtained experimentally by E.A. Chudakov, shown in Fig. 9, A, schedule 1. It can be seen from the figure that with increasing torque M wea, transmitted by the wheel, its dynamic radius is reduced. This is because the vertical distance between the wheel axle and its bearing surface is reduced due to the twisting deformation of the tire sidewall. In addition, under the action of torque, not only a tangential force arises, but also a normal component, which tends to press the wheel to the road surface.
Rice. 9. Dependencies obtained by E.A. Chudakov: a - change in dynamic (I and kinematic ( 2) wheel radii depending on the driving moment: b - change in the kinematic radius of the wheel under the action of driving and braking torques
The value of the dynamic radius also depends on the depth of the track when driving on deformable ground or soil. The deeper the track, the smaller the dynamic radius. The dynamic wheel radius is the application arm of the tangential reaction of the soil pushing the drive wheel. Therefore, the dynamic radius is also called the force radius.
The kinematic radius or rolling radius of the wheel is divided by 2k the actual distance traveled by the wheel in one revolution. The kinematic radius is also defined as the radius of such a fictitious wheel with a rigid rim, which, in the absence of slipping and slipping, has the same angular rotation speed and translational speed as the actual wheel:
where v K is the forward rolling speed of the wheel; wk - angular speed of rotation of the wheel; S K- the path of the wheel in one revolution, taking into account slipping or slipping.
From expression (5) it follows that with full wheel slip (v K = 0) the radius g to= 0, and with full slip (co k = 0) the kinematic radius is equal to ©o.
On fig. 9, A(schedule 2) presented received by E.A. Chudakov, the dependence of the change in the kinematic radius of the wheel on the action of the torque M ved on it. It follows from the figure that the magnitude of the change in the dynamic and kinematic radii is different depending on the action of the moment. The steeper dependence of the kinematic radius of the wheel compared to the dependence of the dynamic radius can be explained by the action of two factors on it. First, the kinematic radius is reduced by the same amount that the dynamic radius is reduced by the driving moment, as shown in Fig. 9, I, schedule /. Secondly, the driving or braking torque applied to the tire causes a deformation of compression or tension of the rolling part of the tire. The processes accompanying these deformations can be easily traced if the wheel is represented as a cylindrical elastic spiral with a uniform winding of coils. As shown in fig. 10, a, under the action of the driving moment, the running part of the tire (front) is compressed, as a result of which the total perimeter of the tire tread circumference decreases, the wheel path S K decreases with one turn. The greater the compression deformation of the tire in the running part, the greater the reduction in path S K , which, in accordance with (5), is proportional to the decrease in the kinematic radius g k.
When braking torque is applied, the opposite occurs. Tensile tire elements approach the bearing surface
(Fig. 10, b). Tire circumference and wheel path S K , passed in one revolution, increase as the braking torque increases. Therefore, the kinematic radius increases.
Rice. 10. Scheme of tire deformation from the action of moments M ved (a) and M t(b)
On fig. 9, b the dependence of the change in the radius of the wheel on the action of the torsional active L / ved and the brake M 1 moments with stable adhesion of the wheel to the supporting surface. E.A. Chudakov proposed the following formula for determining the radius of a wheel:
where r to 0 is the rolling radius of the wheel in the free rolling mode, when the driving moment and the rolling resistance moment are equal to each other; A, m - coefficient of tangential elasticity of the tire, depending on its type and design, which is found from the results of experiments.
In engineering calculations, the static radius of a given tire given in the standard at a given air pressure and maximum load on it is usually used as dynamic and kinematic radii. It is assumed that the wheel moves on an indestructible surface.
When driving on a track, the static radius is the distance from the wheel axle to the bottom of the track. However, when the wheel moves along the track, the point of application of the resultant elementary reactions of the soil, which forms the torque (leading or resistance), will be above the bottom of the track and below the soil surface (see Fig. 17). The dynamic radius in this case depends on the depth of the track: the deeper it is, the greater the difference between the static and dynamic radii of the wheels, the greater the calculation error from the assumption g l = g st
In general, a car wheel consists of a rigid rim, elastic sidewalls, and a contact imprint. The contact footprint of the tire is the elements of the tire in contact with the supporting surface at the considered point in time. Its shape and dimensions depend on the type of tire, tire load, air pressure, deformation properties of the supporting surface and its profile.
Depending on the ratio of deformations of the wheel and the supporting surface, the following types of movement are possible:
Elastic wheel on a non-deformable surface (wheel movement on a paved road);
Rigid wheel on a deformable surface (wheel movement on loose snow);
A deformable wheel on a deformable surface (wheel movement on deformable soil, loose snow with reduced air pressure).
Depending on the trajectory, rectilinear and curvilinear movements are possible. Note that the resistance to curvilinear motion exceeds the resistance to rectilinear motion. This is especially true for three-axle vehicles with a balancing rear bogie. So, when a three-axle vehicle moves along a trajectory with a minimum radius on a road with a high coefficient of adhesion, tire marks remain, black smoke comes out of the exhaust pipe, and fuel consumption increases sharply. All this is a consequence of the increase in the resistance to curvilinear motion by several times compared to the rectilinear one.
Below we consider the radii of an elastic wheel for a particular case - with a rectilinear motion of the wheel on a non-deformable supporting surface.
There are four car wheel radii:
1) free; 2) static; 3) dynamic; 4) wheel rolling radius.
Free wheel radius - characterizes the size of the wheel in an unloaded state at nominal air pressure in the tire. This radius is equal to half the outer diameter of the wheel.
r c = 0.5 D n ,
Where rc is the free radius of the wheel in m;
D n- the outer diameter of the wheel in m, which is determined experimentally in the absence of contact between the wheel and the road and the nominal air pressure in the tire.
In practice, this radius is used by the designer to determine the overall dimensions of the car, the gaps between the wheels and the car body with its kinematics.
Static wheel radius - the distance from the bearing surface to the axis of rotation of the wheel in place. Determined experimentally or calculated by the formula
r st \u003d 0.5 d + l z H,
Where r st is the static wheel radius in m;
d- landing diameter of the wheel rim in m;
z- the coefficient of vertical deformation of the tire. Accepted for toroid tires l z =0.85…0.87; for adjustable pressure tires z=0,8…0,85;
H is the height of the tire profile in m.
Dynamic wheel radius rd- the distance from the supporting surface to the axis of rotation of the wheel during movement. When the wheel moves along a solid supporting surface at low speed in the driven mode,
r st » r d .
The rolling radius of the wheel r k is the path traveled by the center of the wheel when it rotates by one radian. Determined by the formula
r to = ,
Where S- the path traveled by the wheel in one revolution in m .;
2p is the number of radians in one revolution.
When the wheel is rolling, it can be subjected to torque M cr and brake M t moments. In this case, the torque reduces the rolling radius, and the braking moment increases it.
When the wheel is skidding, when there is a path and there is no rotation of the wheel, the rolling radius tends to infinity. If slipping occurs in place, then the rolling radius is zero. Therefore, the rolling radius of the wheel varies from zero to infinity.
The experimental dependence of the rolling radius on the applied moments is shown in Fig.3.1. Let's select five characteristic points on the graph: 1,2,3,4,5.
Point 1 - corresponds to the movement of the wheel skidding when applying the braking torque. The rolling radius at this point tends to infinity. Point 5- corresponds to the wheel slipping in place when a torque is applied. The rolling radius at this point approaches zero.
Section 2-3-4 is conditionally linear, and point 3 corresponds to the radius r ko when the wheel is rolling in driven mode.
Fig.3.1.Dependence r to = f (M).
The rolling radius of the wheel in this linear section is determined by the formula
r to = r to ± l T M,
Where l m is the coefficient of tangential elasticity of the tire;
M- torque applied to the wheel in N.m.
Take the “+” sign if braking torque is applied to the wheel, and the “-” sign if torque is applied.
In sections 1-2 and 4-5, there are no dependencies for determining the wheel rolling radius.
For the convenience of presenting the material, in the future we introduce the concept of "wheel radius" r to, bearing in mind the following: if the parameters of the kinematics of the car (path, speed, acceleration) are determined, then the radius of the wheel is understood as the radius of the rolling of the wheel; if dynamic parameters are determined (force, moment), then this radius is understood as the dynamic radius of the wheel rd. Taking into account the further adopted dynamic radius and rolling radius will be denoted r to ,
