Mazda

Tire rolling. Dynamic and kinematic radii of the wheel Tire rolling radius

To select tires and determine the rolling radius of the wheel based on their size, it is necessary to know the load distribution across the axles.

For passenger cars, the distribution of the load from the total weight across axles depends mainly on the layout. With a classic layout, the rear axle accounts for 52...55% of the load of the total weight, for front-wheel drive vehicles 48%.

The rolling radius of the wheel rк is selected depending on the load on one wheel. The greatest load on the wheel is determined by the position of the center of mass of the car, which is established according to a preliminary sketch or prototype of the car.

G2=Ga*48%=14000*48%=6720N

G1=Ga*52%=14000*52%=7280N

Consequently, the load on each wheel of the front and rear axles of the car, respectively, can be determined by the formulas:

P1=7280/2=3360 N

P2=6720/2=3640 N

We find the distance from the front axle to the center of mass using the formula:

L-base of the car, mm.

a= (6720*2.46) /14000=1.18m.

Distance from center of mass to rear axle:

h=2.46-1.18=1.27m

Tire type (according to the GOST table) - 165-13/6.45-13. Using these dimensions, you can determine the radius of the wheel in a free state:

Where b is the width of the tire section (165 mm)

d - tire rim diameter (13 inches)

1inch=25.4mm

rc=13*25.4/2+165=330 mm

The rolling radius of the wheel rk is determined taking into account the load-dependent deformation:

rk=0.5*d+ (1-k) *b (9)

where k is the radial deformation coefficient. For standard and wide-profile tires k is taken to be 0.3

rk=0.5*330+ (1-0.3) *165=280mm=0.28m

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To select tires and determine wheel rolling radii based on their sizes, it is necessary to know the load distribution across the axles.

The rolling radius of the wheel rk is selected depending on the load on one wheel. The greatest load on the wheel is determined by the position of the center of mass of the car, which is established according to a preliminary sketch or prototype of the car.

Consequently, the load on each wheel of the front and rear axles of the car, respectively, can be determined by the formulas:

P 1 = G 1 / 2, (6)

P 2 = G 2 / 2. (7)

where G 1, G 2 are the loads from the total mass on the front and rear axles of the car, respectively.

We find the distance from the front axle to the center of mass using the formula:

a=G 2 *L/G a , (8)

where G a is the vehicle’s gravity module (N);

L – car base.

Distance from center of mass to rear axle

We select tires based on the load on each wheel according to Table 1.

Table 1 – Car tires

Tire designation	Tire designation

155-13/6,45-13		240-508 (8,15-20)
165-13/6,45-13		260-508P (9.00P-20)
5,90-13		280-508 (10,00-20)
155/80 R13		300-508 (11.00R-20)
155/82 R13		320-508 (12,00-20)
175/70 R13		370-508 (14,00-20)
175-13/6,95-13		430-610 (16,00-24)
165/80 R13		500-610 (18,00-25)
6,40-13		500-635 (18,00-25)
185-14/7,35-14		570-711 (21,00-78)

175-16/6,95-16		570-838 (21,00-33)
205/70 R14		760-838 (27,00-33)
6,50-16
8,40-15
185/80 R15
220-508P (7.50R-20)
240-508 (8,25-20)
240-381 (8,25-20)

For example: 165-13/6.45-13 with a maximum load of 4250 N, 165 and 6.45 - profile width mm and inches, respectively, rim seat diameter 13 inches. From these dimensions you can determine the radius of the wheel in a free state.

r c = + b, (10)

where b – tire profile width (mm);

d – tire rim diameter (mm), (1 inch = 25.4 mm)

The rolling radius of the wheel r k is determined taking into account the deformation depending on the load

r k = 0.5 * d + (1 - k) * b, (11)

where k is the radial deformation coefficient. For standard and wide-profile tires, k is taken as 0.1…0.16.

Calculation of external characteristics of the engine

The calculation begins with determining the power Nev required to ensure movement at a given maximum speed Vmax.

When the vehicle is in steady motion, the engine power, depending on road conditions, can be expressed by the following formula (kW):

N ev = V max * (G a * + K in * F * V ) / (1000 * * K p), (12)

where - the coefficient of total road resistance for passenger cars is determined by the formula:

0.01+5*10 -6 * V . (13)

K in – streamlining coefficient, K in = 0.3 N*s 2* m -4 ;

F – frontal area of the car, m2;

Transmission efficiency;

K p – correction coefficient.

Total road resistance coefficient for trucks and road trains

=(0.015+0.02)+6*10 -6 * V . (14)

We find the frontal area for passenger cars from the formula:

F A = 0.8 * B g * H g, (15)

where B g – overall width;

H g – overall height.

Frontal area for trucks

F A = B * H g, (16)

Engine speed

The engine crankshaft speed n v corresponding to the maximum vehicle speed is determined from the equation (min -1):

n v = Vmax * , (17)

where is the engine speed coefficient.

For existing passenger cars, the engine speed ratio is in the range of 30...35, for trucks with a carburetor engine - 35...45; for trucks with a diesel engine – 30…35.

All forces acting on the car from the road are transmitted through the wheels. The radius of a wheel equipped with a pneumatic tire may vary depending on the weight of the load, driving mode, internal air pressure, and tread wear.

Wheels have the following radii:

1) free; 3) dynamic;

2) static; 4) kinematic.

Free radius(r св) is the distance from the axis of a stationary and unloaded wheel to the most distant part of the treadmill. For the same wheel, the value of Rst depends only on the value of the internal air pressure in the tire.

The free radius of the wheel is indicated in the technical specifications of the tire. If the specified characteristic is not in the reference data, then its value can be determined by the tire marking.

Static radius(r st) - this is the distance from the center of a stationary wheel, loaded only by normal force, to the reference plane. The value of the static radius is less than the free radius by the amount of radial deformation:

r st = r st - h z = r st - R z /С sh, (5.1)

where h z = R z /С Ш - radial (normal) deformation of the tire, m;

R z - normal road reaction, N;

C w - radial (normal) tire stiffness, N/m.

The normal road reaction acting on one wheel can be determined by the formula:

R z = G O / 2, (5.2)

where G O is the weight of the car on a specific axle.

From formula (1) we find the value of the radial stiffness of the tire:

S w = R z / r st - r st, (5.3)

The radial stiffness of a tire depends on its design and internal air pressure p w. If the dependence of Cw on pw is known, then the amount of tire deformation can be determined at any internal air pressure. At rated air pressure and load, the static radius of the wheel can be found using the formula:

r st = 0.5d o + (1 - l w)N w, (5.4)

where d o - wheel rim diameter, m;

N w - height of the tire profile in the free state, m;

l w - coefficient of radial deformation of the tire.

For regular profile tires, as well as wide-profile tires, l w = 0.10 - 0.15; for arched and pneumatic rollers l w = 0.20 - 0.25.

The nominal value of rst of the wheel in relation to the rated load and internal air pressure is indicated in the technical specifications of the tire.

Dynamic radius(r d) is the distance from the center of the rolling wheel to the reference plane. The value of r d depends mainly on the internal air pressure in the tire, the vertical load on the wheel and its speed. As the vehicle speed increases, the dynamic radius increases slightly, which is explained by the stretching of the tire by centrifugal inertial forces.

Kinematic radius(r к) is the radius of a conditional non-deforming rolling wheel without sliding, which has the same angular and linear speeds with a given elastic wheel:

r k = V x /w k. (5.5)

The value of r k is determined empirically by measuring the path S covered by the car in n k full revolutions:

r k = V x /w k = V x * t /w k* t = S/2p n k, (5.6)

where V x is the linear speed of the wheel;

w k - angular speed of the wheel;

t - movement time.

The difference between the radii r d and r k is due to the presence of slipping in the area of contact of the tire with the road.

In the case of complete wheel slip, the path traveled by the wheel is zero S = 0, and therefore r k = 0. During sliding of braked, non-rotating (locked) wheels, i.e. when moving in a skid, n k = 0 and r k ® ¥.

When driving a car on roads with a hard surface and good grip, approximately r k = r d = r c = r.

When rolling, the tire is subject to centrifugal forces. The magnitude of centrifugal forces depends on the rolling speed, mass and size of the tire. Under the influence of centrifugal sieves, the tire slightly increases in diameter. Tests have shown that when the tire rolls at a speed of 180-220 km/h, the profile height increases by 10-13% (results of tire tests in road-circuit motorcycle racing).

At the same time, the action of centrifugal forces causes (due to an increase in the radial stiffness of the tire) a slight increase in the distance from the wheel axis to the supporting surface (road plane) with a simultaneous decrease in the contact area of the tire with the road. This distance is called the dynamic tire radius Ro, which is greater than the static radius Rc, i.e. Ro>Rc.

However, at operating speeds Ro is practically equal to Rс.

The rolling radius is the ratio of the linear speed of the wheel to the angular speed of rotation of the wheel:

where Rк - rolling radius, m;
V - linear speed, m/s;
w - angular velocity, rad/s.

Rolling resistance

Rice. Tire rolling on a hard surface

When a wheel rolls on a hard surface, the tire frame is subject to cyclic deformations. When entering contact, the tire is deformed and bends, and when leaving contact, it restores its original shape. The deformation energy of the tire, generated when the elements come into contact with the surface, is spent on internal friction between the layers of the carcass and slipping in the contact zone. Some of this energy is converted into heat and transferred to the environment. Due to the loss of mechanical energy, the rate of restoration of the original shape of the tire when the tire elements leave the contact is less than the rate of deformation of the tire when the elements enter the contact. Due to this, normal reactions in the contact zone are somewhat redistributed (compared to a stationary wheel) and the distribution diagram of normal forces takes the form as shown in the figure. The resultant of normal reactions, equal in magnitude to the radial load on the tire, moves forward relative to the vertical passing through the wheel axis by a certain amount a (“drift” of the radial reaction).

The moment created by the radial reaction relative to the wheel axis is called the moment of rolling resistance:

Under the condition of steady motion (at a constant rolling speed) of the driven wheel, a moment acts that balances the moment of rolling resistance. This moment is created by two forces - pushing
force P and horizontal reaction of the road X:

M = XRd = PRd,
where P is the pushing force;
X - horizontal reaction of the road;
Rd - dynamic radius.

PRd = Qa - condition of steady motion.

The ratio of the pushing force P to the radial reaction Q is called the rolling resistance coefficient k.

In addition to the tire, the rolling resistance coefficient is significantly influenced by the quality of the road surface.

The power Nk spent on rolling the driven wheel is equal to the product of the rolling resistance force Pc and the linear rolling speed V:

Expanding this equation, we can write:

Nк = N1 + N2 + N3 - N4,
where N1 is the power expended on tire deformation;
N2 is the power expended on the tire slipping in the contact zone;
N3 - power expended on friction in wheel bearings and air resistance;
N4 is the power developed by the tire when restoring the shape of the tire at the moment the elements leave contact.

Power losses due to wheel rolling increase significantly with increasing rolling speed, since in this case the deformation energy increases and, consequently, most of the energy is converted into heat.

As the deflection increases, the deformation of the tire carcass and tread increases sharply, i.e., energy loss due to hysteresis.

At the same time, heat generation increases. All this ultimately leads to an increase in the power expended on rolling the tire.

Tests have shown that rolling a motorcycle tire under driven wheel conditions (on a smooth drum) requires power from 1.2 to 3 liters. With. (depending on tire size and rolling speed).

Thus, the total losses from tires are quite significant and are comparable to the power of the motorcycle engine.

It is clear that addressing the issue of reducing the power expended on rolling motorcycle tires is of utmost importance. Reducing these losses will not only increase the durability of the tires, but will significantly increase the service life of the engine and motorcycle components, and will also have a positive effect on the fuel efficiency of engines.

Research carried out during the creation of P-type tires has shown that power losses during rolling of tires of this type are significantly less (30-40%) than those of standard tires.

In addition, losses are reduced when converting tires to a two-layer carcass made of cord 232 KT.

It is especially important to minimize power losses when rolling tires for racing motorcycles, since when they move at high speeds, losses in tires amount to up to 30% in relation to the total power consumption for movement. One of the methods for reducing these losses is the use of 0.40 K nylon cord in the carcass of racing tires. By using such a cord, the thickness of the carcass was reduced, the weight of the tire was reduced, and it became more elastic and less susceptible to heating.

The nature of the tread pattern has a great influence on the tire's rolling resistance coefficient.

To reduce the energy generated when elements come into contact with the road, the tread weight of racing tires is reduced as much as possible. While road tires have a tread depth of 7-9 mm, racing tires have a tread depth of 5 mm.

In addition, the tread pattern of racing tires is designed in such a way that its elements provide the least resistance when the tire rolls.

As a rule, the tread pattern of the tires of the front (driven) and rear (drive) wheels of a motorcycle is different. This is explained by the fact that the purpose of the front wheel tire is to ensure reliable handling, and the purpose of the rear wheel is to transmit torque.

The presence of ring lugs on the front tires helps reduce rolling losses and improves handling and stability, especially when cornering.

Rice. Curves of power loss versus rolling speed: 1 - tire size 80-484 (3.25-19), model L-130 (road); 2 - tire size 85-484 (3.25-19) model L-179 (for the rear wheel of road-ring motorcycles)

The zigzag tread pattern of the rear wheel ensures reliable transmission of torque and also reduces rolling losses. All of the above measures make it possible, in general, to significantly reduce power losses when tires roll. The graph shows the power loss curves at different speeds for road and racing tires. As can be seen from the figure, racing tires have lower losses compared to road tires.

Rice. The appearance of a “wave” when the tire rolls at a critical speed: 1 - tire; 2 - test bench drum

Critical tire rolling speed

When the rolling speed of a tire reaches a certain limit value, rolling power losses increase sharply. The rolling resistance coefficient increases approximately 10 times.

A “wave” appears on the surface of the tire tread. This “wave,” while remaining motionless in space, moves along the tire frame at the speed of its rotation.

The formation of a “wave” leads to rapid destruction of the tire. In the tread-carcass area, the temperature increases sharply, as the internal friction in the tire becomes more intense, and the strength of the bond between the tread and the carcass decreases.

Under the influence of centrifugal forces, which are significant in magnitude at high rolling speeds, sections of the tread or pattern elements are torn off.

The rolling speed at which the “wave” appears is considered the critical rolling speed of the tire.

As a rule, when rolling at a critical speed, the tire is destroyed after a run of 5-15 km.

As tire pressure increases, the critical speed increases.

However, practice shows that during the SSC, the speed of motorcycles in some areas is 20-25% higher than the critical tire speed determined at the stand (when the tire rolls on a drum). In this case, the tires are not destroyed. This is explained by the fact that when rolling on a plane, the deformation of the tire is less (under the same conditions) than when rolling on a drum, and therefore the critical speed is higher. In addition, the time it takes for a motorcycle to move at a speed exceeding the critical speed of the tires is negligible. At the same time, the tire is well cooled by the oncoming air flow. In this regard, the technical characteristics of sports motorcycle tires intended for GCS allow short-term overspeeding within certain limits.

Tire rolling under driving and braking conditions. Tire rolling under driving wheel conditions occurs when torque Mkr is applied to the wheel.

The diagram of the forces acting on the drive wheel is shown in the figure.

Rice. Diagram of forces acting on the drive wheel tire when rolling

A torque Mkr is applied to a wheel loaded with a vertical force Q.

The road reaction Qp, equal in magnitude to the load Q, is displaced relative to the wheel axis by a certain distance a. Force Qp creates a moment of rolling resistance Mc:

The torque Mkr creates the traction sieve RT:

Рт = Мкр/Rк

where Rк is the rolling radius.

When the tire rolls under the driving wheel conditions, under the influence of torque, a redistribution of tangential forces in contact occurs.

In the front part of the contact in the direction of movement, the tangential forces increase, in the rear part they decrease. In this case, the resultant of the tangential forces X is equal to the traction force Рт.

The power expended on rolling the drive wheel is equal to the product of the torque Mkr and the angular speed Wk of rotation of the wheel:

This equation is valid only when there is no slip in the contact.

However, tangential forces cause the tread pattern elements to slip relative to the road.

Due to this, the actual value of the speed of translational movement of the wheel Ud is slightly lower than the theoretical Vt.

The ratio of the actual forward speed Vd to the theoretical Vt is called the wheel efficiency, which takes into account the speed loss due to the tire slipping relative to the road.

The amount of slippage a can be estimated using the following formula:

Obviously, the value of the actual speed Vd can vary from Vt to 0, i.e.:

The intensity of slipping depends on the magnitude of the tangential forces, which in turn are determined by the magnitude of the torque.

Previously shown:

Mkr = XRk;
X = Рт = Qv,
where v is the coefficient of adhesion of the tire to the road.

When the torque increases to a certain value exceeding the critical value, the magnitude of the resultant tangential forces X becomes higher than permissible and the tire completely slips relative to the road.

Existing motorcycle tires in the operating load range can transmit a torque of 55-75 kgf*m without complete slipping (depending on the tire size, load, pressure, etc.).

When a motorcycle brakes, the forces acting on the tire are similar in nature to the forces that arise when the tire operates under driving wheel conditions.

When a braking torque Mt is applied to the wheel, a redistribution of tangential forces occurs in the contact zone. The greatest tangential forces occur at the rear of the contact. The resultant of the tangential forces coincides in magnitude and direction with the braking force T:

When the braking torque Mt increases above a certain critical value, the braking force T becomes greater than the adhesion force of the tire to the road (T>Qv) and complete slipping begins in contact, the phenomenon of skidding occurs.

When braking from a skid in the contact zone, the tread temperature rises, the adhesion coefficient drops, and the wear of the tread pattern sharply increases. Braking efficiency decreases (braking distance increases).

The most effective braking occurs when the braking force T is close in magnitude to the adhesion force of the tire to the road.

Consequently, when the driver uses the dynamic qualities of a motorcycle in order to reduce tire wear, a torque must be supplied to the drive wheel that ensures the least slip of the tire relative to the road.

In general, a car wheel consists of a rigid rim, elastic sidewalls and a contact print. The tire contact mark represents the tire elements in contact with the supporting surface at the time in question. Its shape and dimensions depend on the type of tire, load on the tire, 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 hard surface road);

A 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 movement exceeds the resistance to rectilinear movement. This is especially true for three-axle vehicles with a balanced rear bogie. Thus, 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 from the exhaust pipe, and fuel consumption increases sharply. All this is a consequence of the increase in resistance to curvilinear movement several times compared to rectilinear movement.

Below we consider the radii of an elastic wheel for a special case - with the rectilinear movement of the wheel on a non-deformable supporting surface.

There are four radii of a car wheel:

1) free; 2) static; 3) dynamic; 4) wheel rolling radius.

Free wheel radius - characterizes the size of the wheel in an unloaded state at the 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 r c– free radius of the wheel in m;

D n– outer diameter of the wheel in m, which is determined experimentally in the absence of contact of the wheel with 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 during its kinematics.

The static radius of a wheel is the distance from the supporting surface to the axis of rotation of the wheel in place. Determined experimentally or calculated using the formula

r st = 0.5 d + l z H,

Where r st– static radius of the wheel in m;

d– landing diameter of the wheel rim in m;

l z- coefficient of vertical deformation of the tire. Accepted for toroid tires l z =0.85…0.87; for adjustable pressure tires l z=0,8…0,85;

H – tire profile height in m.

Dynamic wheel radius r d– the distance from the supporting surface to the axis of rotation of the wheel during movement. When the wheel moves on a hard supporting surface at low speed in the driven mode, it is assumed

r st » r d .

The rolling radius of the wheel r k is the path traversed by the center of the wheel when it turns by one radian. Determined by the formula

r to = ,

Where S– the distance covered by the wheel per revolution in m;

2p is the number of radians in one revolution.

When a wheel rolls, it can be subject 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 moves in a skid, 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. Consequently, the rolling radius of the wheel varies from zero to infinity.

The experimental dependence of the rolling radius on the applied moments is presented in Fig. 3.1. We highlight five characteristic points on the graph: 1,2,3,4,5.

Point 1 – corresponds to the skidding movement of the wheel when braking torque is applied. The rolling radius at this point tends to infinity. Point 5 corresponds to wheel slipping in place when 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 rko when the wheel rolls in driven mode.

Fig.3.1.Dependency r k = f (M).

The rolling radius of the wheel in this linear section is determined by the formula

r k = r k ± l T M,

Where l t – coefficient of tangential elasticity of the tire;

M- moment applied to the wheel in N.m.

Take the “+” sign if a braking torque is applied to the wheel, and the “-” sign if a torque is applied to the wheel.

In sections 1-2 and 4-5 there are no dependencies for determining the rolling radius of the wheel.

For the convenience of presenting the material, we will further introduce the concept of “wheel radius” r to, bearing in mind the following: if the parameters of the kinematics of the car are determined (path, speed, acceleration), then the radius of the wheel refers to the rolling radius of the wheel; if the dynamics parameters are determined (force, moment), then this radius is understood as the dynamic radius of the wheel r d. Taking into account what is accepted in the future, the dynamic radius and rolling radius will be denoted r to ,