What is the rolling radius of the wheel? Selection of tire size and calculation of rolling radius
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. Basic driving force is the traction force applied to the drive wheels. Traction force arises as a result of engine operation and is caused by the interaction of the drive wheels with the road. Traction force Pk is defined as the ratio of the moment on the axle shafts to the radius of the drive wheels during uniform motion of the vehicle. 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 while driving. In this regard, the following wheel radii are distinguished:
1.Nominal – radius of the wheel in a free state: r n =d/2+H, (6)
where d – rim diameter, m;
H – total height of the tire profile, m.
2. Static r c – the distance from the road surface to the axis of the loaded stationary wheel.
r with =(d/2+H)∙λ , (7)
where λ is the radial deformation coefficient of the tire.
3. Dynamic r d – distance from the road surface to the axis of a 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.
As the speed of the vehicle increases, under the influence of centrifugal forces, the tire stretches in the radial direction, as a result of which the radius r d increases. When a wheel rolls, the deformation of the rolling surface also changes compared to a stationary wheel. Therefore, the shoulder of application of the resultant tangential reactions of the road r d differs from r c. However, as experiments have shown, for practical traction calculations it is possible to take r c ~ r d.
4 Kinematic radius (rolling) of the wheel r k - the radius of such a conditional non-deformable ring that has the same angular and linear speeds with a given elastic wheel.
For a wheel rolling under the influence of torque, the tread elements that come into contact with the road are compressed, and the wheel at equal rotation speeds travels a shorter distance than during free rolling; on a wheel loaded with braking torque, the tread elements that come into contact with the road are stretched. Therefore, the brake wheel travels a slightly longer distance at equal speeds than a freely rolling wheel. Thus, under the influence of torque, the radius rк decreases, and under the influence of braking torque, it increases. To determine the value of r k using the “chalk prints” method, a transverse line is drawn on the road with chalk or paint, onto which a car wheel rolls, and then leaves prints on the road.
Measuring the distance l between the extreme prints, determine the rolling radius using the formula: r k = l / 2π∙n, (8)
where n is the wheel rotation speed corresponding to the distance l .
In case of complete wheel slip, the distance l = 0 and radius r to = 0. During sliding of non-rotating wheels (“SW”), rotation frequency n=0 and r to .
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 a wheel moves on a hard supporting surface at low speed in a 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 r co 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 torque is applied.
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 ,
To select tires and determine the wheel rolling radius based on their size, it is necessary to know the load distribution across the axles.
U passenger cars The distribution of the load from the total weight across the bridges 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 car's center of mass, 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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Good afternoon, dear readers. Today I want to immediately answer many questions related to wheel tire sizes. Many of my readers do not understand what they mean and why they are needed at all! Today I will try to explain in simple and understandable language what tire sizes on cars mean...
Wheel tire sizes contain a lot of useful information, you just need to be able to read it. Without this information, you will not be able to choose the right tires for your car; they simply will not fit in size. Although now the bodies of many brands have special plates with recommendations, you just read them and go to the store to buy the same ones. However, such signs are not always available and you need to determine the tire dimensions yourself! A small clarification, I will only talk about overall dimensions, there have already been many articles about other characteristics, the links will definitely be below.
I will tell you about my winter wheels, KAMA EURO 519, as an example; it should be noted that they are in no way inferior to their foreign counterparts. An informative read.
To begin with, the overall dimensions
My wheel size is R16 205/55 , these are the so-called overall dimensions. Rubber is considered low profile (more details).
The notorious letter R
Many people mistakenly think (to be honest, I thought so too) that the first english letter R stands for the abbreviation RADIUS! But that's not true! The letter R means radial tire, read the article -. This is a method of arranging rubber and metal cord during production. Of course, you can also see the letter D in front (diagonal), but such a designation is now really rare. In fact, this letter has nothing to do with size. Let's move on...
Disc diameter
The second number (in this case we have 16) indicates the diameter of the hole in the rubber, or which disc this rubber can be put on. We have 16, which means it’s 16 inches! Remember that this size is always indicated in inches (1 inch = 25.4 mm). If we calculate our size, it turns out - 16 X 25.4 mm = 406.4 mm. The disc cannot be larger or smaller than the diameter of the wheel; you simply won’t fit it. That is, if the tires are 16 (406.4 mm), then the disk should be 16 (406.4 mm).
Width
A large number almost always characterizes the width. In this case, this figure is 205. It is measured in millimeters, that is, the width of my wheel is 205 mm. The wider the rubber, the wider the track it has, which increases cross-country ability and traction.
Cord height
This is a smaller number that is applied through the fraction. In my case it is 55, measured as a percentage of the width (from the larger number). What does it mean? To find the height (in my case) you need to calculate 55% of 205 mm. Thus it turns out:
205 X 0.55 (55%) = 112.75 mm
This is the cord height of our rubber, also an important indicator, see the figure.
Overall wheel height
Let's calculate the total height of my wheel. What happens.
Rubber cord 112.75 X 2 (since the height is on both sides, top and bottom) = 225.5 mm
For a 16-inch disk = 406.4
Total - 406.4 + 225.5 = 631.9
Thus, my wheel is a little more than half a meter high, namely 0.631 meters
Let's look at the most common tires that are used by most cars, there are three of them - R13, R14 and R15
Tire sizesR13
The most common of all isR13175/70 such ones are installed on many models of the domestic VAZ (although they are now being phased out).
What happens:
R13 – diameter 13 inches (multiply by 25.4) = 330.2 mm
Width 175
Height - 70% of 175 = 122.5
Total - (122.5 X 2) + 330.2 = 574.2 mm
Tire sizesR14
One of the most common ones isR14175/65 are also installed on domestic VAZ models of more recent years of production, such as Priora, Kalina, Granta, as well as on some inexpensive (popular) foreign cars - for example Renault Logan, Kia RIO, Hyundai Solaris, etc.
What happens:
R14 – diameter 14 inches (multiply by 25.4) = 355.6 mm
Width - 175
Height – 65% of 175 = 113.75
Overall dimensions – (113.75 X 2) + 355.6 mm = 583.1 mm
Tire sizesR15
The most common example is -R15 195/65, installed on many foreign cars of the (popular) class, but in high trim levels.
What happens:
R15 – diameter 15 inches (multiply by 25.4) = 381 mm
Width 195
Height – 65% of 195 = 126.75
Total – (126.75 X 2) + 381 = 634.5 mm
As you can see, it is not that difficult to calculate the tire size.
Of course there is another one on the wheel useful information, I already wrote articles about her below. I’ll list them point by point for you, it’s useful and interesting to read:
In general, read the section - there is much more information there. As you can see, all this information can be read from a tire, sometimes you can’t even believe it!
According to this Rule Additional indices of speed and their load-bearing capacity are introduced into the marking of car tires. Some indices of speed and load-bearing capacity of car tires are presented in the table below.
Some indices of speed and load-bearing capacity of car tires:
k is the total weight of the car per wheel.Examples of tire designations according to UNECE Regulation 30:
175/80R16Q88 – tires for Niva;
175/80R16СN106 – tires for Gazelle.
Free wheel radius
Free radiusr 0 is the radius of the wheel in a free (unloaded) state. For example, for a low profile tire type 205/70-14 78 S(tire designation is given in accordance with UNECE Regulation 30) this radius will be found as:
r 0 = 0,5d+N= 0,5d+IN(N/A)10 -2 ; (100×N/V) – tire series; 1 inch equals 25.4 mm, that is:
r 0 = (0.5×14×25.4 + 205×0.7)×10 –3 = (177.8 + 143.5)×10 –3 = 0.321 m.
Static wheel radius
One of the determining factors when calculating the operational properties of a car is the value from the center of the wheel to the supporting surface of a stationary wheel loaded with a normal load (the weight of a stationary car). Strictly speaking, given that the tire is elastic and deforms when a load is applied, this value represents the distance from the center of the wheel to the chord, however, in car theory, this value is usually called the static radius ( r st). In technical data, the static radius value is often not given, and the tire marking is indicated instead. Obviously, if we designate the diameter of the rim - d, tire profile width - B, percentage ratio of the tire profile height to its width (tire series) - P, tire outer diameter - D, then the static radius is defined as:
For toroid tires:
;
For low profile tires:
;
For wide profile tires
.
Here: 
- coefficient of radial deformation of the tire. For passenger car tires with internal pressure in the range of 0.15 - 0.25 MPa as a first approximation we can take 
= 0.15, for truck tires with an internal pressure of 0.5 MPa
= 0,1.
Properties of pneumatic tire
Pneumatic tires are widely used due to their shock-absorbing properties. They significantly soften shocks from road unevenness.
The physical and mechanical properties of the tire determine such vehicle performance indicators as load capacity, efficiency, handling, cross-country ability, etc. Ultimately, all these indicators are determined by the value and type of deformation of the tire under the influence of external forces.
There are four types of deformation of a pneumatic tire: radial (normal), circumferential (tangential), transverse (lateral) and angular.
Radial tire deformation measured by its normal deflection h n, equal to the difference of free (r 0 ) and static ( r st) wheel radii:
h n =r 0 –r Art.
Under the influence of a static vertical load (the weight of a stationary vehicle), as a result of deformation of the elastic structure of the tire, the distance from the wheel axis to the supporting surface decreases.
Normal deflection– one of the most important characteristics of a tire, determining its load capacity and smooth ride. As the deflection increases, the stresses in the tire structural elements increase, and the fatigue strength and service life decrease. The highest permissible value of the normal load, at which, despite the radial deformation, the specified service life of the tire is ensured at a given air pressure in it, is usually called the load-carrying capacity of the tire. The normal load value is regulated by GOST or UNECE Rules 30 (for foreign-made vehicles).
The type and parameters of driving wheels for cars are selected (Table 1) in accordance with the normal load on them. The standard provides for several permissible loads on a tire depending on the air pressure in it. When choosing a tire for the machine you are calculating, you must be guided by the following rule. The normal load on the tire obtained by calculation should not exceed the maximum permissible according to the standard at the lowest air pressure in it from among the values provided by the standard.
When determining the load on the drive wheel, the maximum possible load during operation of the machine should be taken into account, taking into account its technological purpose.
With a uniform static distribution of the vehicle's weight along the axles, the maximum load on one wheel should be determined based on its possible redistribution during operation. In this case, the load on the drive wheel from the gravity of the vehicle and the cargo being transported, as well as from the vertical component of the traction force on the trailer hitch, is taken into account.
The parameters of the selected tire are checked against the type and parameters of the drive wheels of the prototype vehicle. When comparing the parameters of the selected wheel and the prototype wheel, it should be borne in mind that truck manufacturers sometimes use an increased tire size (if the requirements for the vehicle allow). “Oversized” tires are more durable, exert less pressure on the soil and give the vehicle higher traction properties. The use of such tires is most appropriate for trucks operating on unpaved or poorly paved roads.
Table 1.
Parameters of car tires (GOST 7463-89)
|
Automobile |
Wheel formula |
Tire designation |
Tire pressure, MPa: front/rear |
|
Normal tire deflection h n due to its deformation not only in the radial, but also in the circumferential and transverse directions. In this case, 40% of the total tire compression load is spent on deformation of its material and 60% on air compression.
Distinguish low, medium and high pressure tires. Low-pressure tires have an increased volume of air and fewer layers of cord. They absorb shocks from road unevenness more softly and have better shock-absorbing properties, but with a lower load capacity. For low and medium pressure tires, the permissible normal deformation of the tire is 15...20% of its height, and for high pressure tires - 10...12%.