Analysis of vertical force in steering system

Condition for Ackerman-

δ_o = tan^-1 [L/(R+t/2)]

δ_i = tan^-1 [L/(R-t/2)]

 

Error in steering

Toe change-

For an ideal steering system relay arm (tie rod) should follow the same arc as the suspension follows. Practically it is impossible to achieve due to nonlinearities in the motion of suspension. When the wheel moves in the jounce and rebound, relay linkage will follow an arc which will pull the steering arm to the right and causing a left hand steer when located behind the centre of the wheel.

Roll type steer-

When inboard joint located below the wheel centre and linkage also located at rear of front wheel centre, at jounce a left hand steer will be produced and right hand when it goes into rebound.

Steering system forces and moment

Three types of forces are normally seen in vehicle tire.

1. Normal force (aligning torque) z-direction
2. Tractive force (Rolling resistance moment) y direction
3. Lateral force (Overturning moment) x direction

The reaction in the steering system is due to the moment about steering axis, which must be reduce to control the wheel steer angle.

1. Vertical force

It has inclusion of two forces

• Due to lateral inclination angle (left side of equn)
• Caster angle (right side of equn)

M_v = -(F_zl + F_zr ) d sin λ sin δ + (F_zl – F_zr ) d sin γ cos δ

M_v = Total moment from left and right wheels

F_zl, F_zr = Vertical load on left and right wheel

d = lateral offset on ground or scrub radius

λ = lateral inclination angle or king pin angle

δ = Steer angle

γ = caster angle

Due to lateral inclination angle

• F_zr sin λ = Sine angle of force component acting laterally and parallel to king pin axis.
• d sin δ = moment arm of above force
• The moment is zero when no steering. When steering, because of this force vehicle tends to lift, Increasing the steering effort and also self-centring force.
• Axle lift when steered
• Unaffected by right left load differences
• Torque gradient depends upon wheel offset at ground, Inclination angle, axle load

Due to caster angle.

• F_zr sin γ = Force component in the direction parallel to caster angle seen in side view.
• d cos δ = moment arm forward to force
• Moment due to both wheel is opposite in direction. This force balance the left right wheel load. This may result into wheel toe-in and asymmetry of tie rod resulting in its push or pull.
• Axle rolls with steered
• Sensitive to left right load imbalance
• Torque gradient depends upon wheel offset at the ground, caster angle, left right load difference in cornering, front and rear suspension roll stiffness, Suspension roll centre height, centre of gravity height, lateral acceleration level.