To select tires and determine wheel rolling radii based on their sizes, 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 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.
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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Due to the wide variety of types of deformation of a pneumatic tire, its radius does not have one specific value, like that of a wheel with a rigid rim.
The following rolling radii of wheels with pneumatic tires are distinguished: 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 a wheel free from external load. It is equal to the distance from the surface of the treadmill to the wheel axis.
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 static radius values 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 as normal load increases and tire pressure decreases. Dependence of dynamic radius on torque 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 vea, transmitted by the wheel, its dynamic radius decreases. This is explained by the fact that the vertical distance between the wheel axis and its supporting surface decreases due to the torsional deformation of the tire sidewall. In addition, under the influence 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 torque: b - change in the kinematic radius of the wheel under the influence of driving and braking torques
The magnitude of the dynamic radius also depends on the depth of the rut when moving on deformable ground or soil. The greater the rut depth, the smaller the dynamic radius. The dynamic radius of the wheel is the shoulder of application of the tangential reaction of the soil pushing the drive wheel. Therefore, the dynamic radius is also called the power radius.
Kinematic radius or rolling radius of a 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 velocity of rotation and translational speed as the real wheel:
where v K is the forward rolling speed of the wheel; с к - angular speed of rotation of the wheel; S K- wheel path per revolution, taking into account slipping or sliding.
From expression (5) it follows that with full wheel slip (v K = 0) the radius g to= 0, and with full sliding (with k = 0) the kinematic radius is equal to ©о.
In Fig. 9, A(schedule 2) presented by E.A. Chudakov, the dependence of the change in the kinematic radius of the wheel on the action of the torque M led on it. It follows from the figure that the magnitude of the change in the dynamic and kinematic radii depending on the action of the moment is different. 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. Firstly, the kinematic radius decreases by the same amount by which the dynamic radius decreases from the action of the driving moment, as shown in Fig. 9, i, schedule /. Secondly, the driving or braking torque applied to the tire causes compressive or tensile deformation of the running part of the tire. The processes accompanying these deformations are easy to trace if you imagine the wheel in the form of a cylindrical elastic spiral with uniform winding of turns. As shown in Fig. 10, a, under the influence 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 becomes smaller per revolution. The greater the compression deformation of the tire in the running part, the greater the reduction in distance SK, which, in accordance with (5), is proportional to the decrease in the kinematic radius g k.
When the braking torque is applied, the opposite phenomenon occurs. The stretched elements of the tire fit the supporting surface
(Fig. 10, b). Tire perimeter and wheel path SK, traveled per revolution increase as the braking torque increases. Therefore, the kinematic radius increases.
Rice. 10. Scheme of tire deformation from the action of moments M led (a) and M t(b)
In Fig. 9, b shows the dependence of the change in the radius of the wheel on the action of the active torque and brake on it 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 the wheel:
where g to 0 is the rolling radius of the wheel in free rolling mode, when the driving moment and the moment of rolling resistance are equal to each other; A, t is the 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 set air pressure and maximum load on it is usually used as the dynamic and kinematic radii. It is assumed that the wheel moves on an indestructible surface.
When driving along a rut, the static radius is the distance from the wheel axis to the bottom of the rut. However, when the wheel moves along a track, the point of application of the resultant of the elementary reactions of the soil, which forms the torque (driving or resistance), will be located 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
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.
