Introduction:
The front wheels do not steer at the same angle in a turn. The inside wheel will always have a “sharper” turn than the outside wheel. The image shows why this is the case.
The image shows that the lines from the front wheels converge at point M. Point M is the common pivot point for both front wheels. If the wheels were to turn at the same angle (where both wheels are in exactly the same position), the lines from the wheels would run parallel to each other into infinity, never finding the common pivot point M. As a result, the steering characteristics would be very poor in this situation. This entire principle is called “toe-out in turns.” All modern cars are constructed with this feature.
On smooth surfaces, e.g., the floor of a parking garage, you can hear the tires squeal when steering. This is due to this principle. The inner wheel, which makes a sharper angle than the outer wheel, will experience some form of slip. This is referred to as a steering error. More information about the steering error (and a graph) can be found on the page steering error.
This page explains how to calculate the steering angles (in degrees) of both front wheels using a number of parameters.
Calculating the Steering Angles:
The following vehicle data is needed to calculate the steering angles:
- Track Width
- Wheelbase
- Turning Circle Diameter
- Kingpin Distance (on this page, we keep the kingpin distance equal to the track width)
- Tire Size (depending on the calculation. On this page, calculations are done with the tire size, but calculations up to the bumper corners can also be made, in which case more angles are involved).
| Track Width = 1600mm | Wheelbase = 3200mm |
| Turning Circle Diameter = 13.225m | Kingpin Distance = Track Width = 1600mm |
| Tire Size = 225 | L and L’ = unknown |
a
Explanation of the Symbols:
b1 = Alpha
b2 = Beta
b3 = Gamma
These letters are from the Greek alphabet and are often used for angle calculations.
L = length
L’ = L with “prime” as a supplement commonly used in mathematics. L2 could have been used just as well. A third L would have received two primes: L”.
The same applies to R”.
The angles Alpha, Beta, and Gamma are located at point M.
Angle Alpha + Gamma = angle Beta.
The entire turning circle is 13.225 meters. R is the radius, which is half the turning circle (6612.5). In the image, R’ is given. This R’ is not fixed. It must be calculated by subtracting half the tire width. Another method is subtracting the kingpin distance, but on this page, we adhere to: Track Width = kingpin distance. This results in a simple calculation:
R = 6612.5 mm
R’ = R – half tire width
R’ = 6612.5 – (225 ÷ 2)
R’ = 6612.5 – 112.5
R’ = 6500 mm
We fill R’ into the image. Then we calculate the angle sin b1 (sine Alpha) using the Sine Rule. Next, we use the Tangent and the Pythagorean Theorem to calculate the other angles.
Angle Calculation with the Sine:
Sin b1 = Opposite side : Hypotenuse
Sin b1 = Wb : R’
Sin b1 = 3200 : 6500
Sin b1 = 0.492
Inv Sin b1 = 29.576
Explanation of the Calculation: 0
We want to calculate Sin b1. Sine is opposite side divided by hypotenuse (mnemonic: SIN = SOS).
Wb = wheelbase = 3200mm. R’ was previously calculated = 6500mm.
We then divide them into each other, giving us Sin b1 = 0.492. To convert this number into an angle, press the sin-1 button on the calculator (usually pressing the Shift button first and then the Sin button), followed by 0.492, or the ANS button. The angle of 29.5 degrees will then be displayed.
Sin b1 is now known. We actually want to calculate tan b2, but first, we need the length L’. This must be calculated first, and the result will later be used to calculate Tan b2.
L’ = L – Track Width.
We calculate L using the Pythagorean Theorem. The 2 sides of the triangle are known (6500 and 3200). The other side of 1600 is the track width, which runs from tire to tire, so it is not included. We calculate the base side, which runs from the left rear tire to the common point M. The calculation involves the entire blue triangle.
The Pythagorean Theorem is as follows:
A^2 + B^2 = C^2. (The symbol ^ represents “power.” Hence, A squared + B squared = C squared. We formulate it slightly differently below.
The length 3200 is called A, 6500 is called B, and the bottom unknown side is called C:
C^2 = 6500^2 – 3200^2
C^2 = 42250000 – 10240000
C^2 = 32010000^2
0a
To eliminate the square, we take the square root of the number.
C^2 = 1a32010000
C = 5658mm.
Side C is actually length L.
Now L’ can be calculated. The full length L and the track width are known, so the two can be easily subtracted:
L’ = L – Track Width
L’ = 5658 – 1600
L’ = 4058mm
Now that Wb and L’ are known, two of the three sides of the triangle are known, so the third side can be calculated using the Tangent:
Angle Calculation with the Tangent:
Tan b2 = Opposite side : Adjacent side
Tan b2 = Wb : L’
Tan b2 = 3200 : 4058
Tan b2 = 0.789
Inv Tan b2 = 38.376
Explanation of the Calculation: 0
We want to calculate Tan b2. Tangent is the opposite side divided by the adjacent side (mnemonic: TAN = TOA).
Wb = wheelbase = 3200mm. L’ was previously calculated = 4058mm.
We then divide them into each other, giving us Tan b2 = 0.789. To convert this number into an angle, press the tan-1 button on the calculator (usually pressing the Shift button first and then the Tan button), followed by 0.789, or the ANS button. The angle of 38.3 degrees will then be displayed.
Now the steering angles of both front wheels have been calculated. The left front wheel is at an angle of 29.576 and the right front wheel at an angle of 38.376. This means that the steering angle has a difference of 8.876 between both wheels. In a left turn, the same steering deflection will result in the same steering angle.
On the page wheel geometry, several wheel positions are described.