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Steering calculations ACKERMANN ANGLE Ackerman Angle

A

Wheel base

TAN Angle B =

King Pin Center to Center Distance 2

C

B

king pin center to center distance / 2 Wheelbase

Angle B = ARCTAN (king pin center to center distance / 2) Wheelbase Angle B = ARCTAN

56” / 2 68”

Angle B =22.38 °

So the ackerman angle for new design is 22.38

LENGTH OF THE TIE ROD

°

Y SIN Ackerman Angle = Ackerman Arm Radius.

SIN 22.38º = Y/5” Y =1.904”

LT = DKC – 2Y Where: LT is the length of the tie rod and rack rod DKC is the distance between king pins center to center LT = 56” – 2*5”*SIN 22.38º LT =52.193”

TURNING RADIUS From Law of Sin a b c = = sinA sin ⁡B sin ⁡C turning radius wheel base = sin 90 sin ⁡( θi )

68 } over {sin (θi)} 3000 =¿ sin 90

90 ( 1727.330×sin ) 00

θi=arcsin ⁡

θi=35.15

◦

ACKERMANN STEERING GEOMETRY TO DETERMINING INNER AND OUTER WHEEL TURN ANGLE FOR 100% ACKERMAN Cot θo – Cot θi = L/B Where Θo= turn angle of the wheel on the outside of the turn Θi= turn angle of the wheel on the inside of the turn L= track width B= wheel base Wheel base =1727.2mm Track width=1422.4mm Substitute the W and L in above equation Cot θ o – Cot θi = 0.824 As we known the maximum wheel turning angle as 35.15 ° are calculated from ackerman geometry (turning radius).

Cot

θo

Cot

θo = 0.824+1.42

θo

– Cot (35.15) = 0.824

= arccot(2.244)

θo = 24.017

°

TO DETERMINING INNER AND OUTER WHEEL TURN ANGLE FOR ACTUAL DESIGN

Point B’s X coordinate = RAA * COS(AA + SAL)

Point B’s Y coordinate = RAA * SIN(AA + SAL) Assume if a car takes 30º left turn… Point B’s X coordinate = 5” * COS(22.38º + 35.15º) Point B’s X coordinate = 2.684” Point B’s Y coordinate = 5” * SIN(22.38º + 35.15º) Point B’s Y coordinate = 4.218” So, the coordinates of Point B at a 0º left turn are (2.684”, 4.218”)

TO DETERMINE ANGLE α

DE = AD – AE DE = 56”-4.218” DE = 51.78”

Now that we know EB and ED, we can find the length of BD because it is a hypotenuse of the triangle formed. Using Pythagorean Theorem:

BD =

BD =

(EB 2 + (DE)2

(51.78”) 2 + (2.684”)2

BD = 51.85”

we know the sides of the triangle we can determine angle α

TAN

α = EB/ED

ARCTAN (EB/ED) =

α

ARCTAN (2.684”/51.78”) =α

α=2.96” So now that we know angle k and the Ackerman angle TO FIND ANGLE(

γ)

Law of cosines for non-right triangles COS γ = A2 + B2 – C2 2AB ARCCOS A2 + B2 – C2

=γ

2AB ARCCOS (52)2 + (5)2 – (51.78)2

= γ

2(52)(5) γ =84.72°

Now if we add up angle α, γ and the Ackerman angle, we’ll have the tire’s steer angle from the line that connects the two kingpins. To get the steer angle, we have to subtract 90°. The formula is: Steer Angle = α + γ + Ackerman Angle - 90° Steer Angle = 2.96º + 84.72° + 22.38° - 90° Steer Angle = 20°

STEERING RATIO CALCULATION

Steering ratio =

Turn of steering

wheel Turn of wheel Maximum wheel turning angle =35.15

Steering ratio =

°

360

°

35.15 °

Steering ratio =11:1