Ackermann principle and steering linkages

Why it matters

When a car turns slowly, each wheel should roll along its own circle about one common centre; if the front wheels are steered through the wrong angles, at least one tyre is dragged sideways, which wears tyres, makes steering heavy and wastes energy. The Ackermann principle tells you what the inner and outer wheel angles must be, and the steering linkage is the mechanism that approximately produces them — both are standard GATE and university questions.

Key ideas

Condition for true rolling. For pure rolling at low speed (no tyre slip angles), the axes of all four wheels must meet at one point, the instantaneous centre (I). With a rigid rear axle that is not steered, I lies on the line of the rear axle. Because the inner front wheel is nearer to I, it must be steered through a larger angle (θ) than the outer wheel (φ). This is called toe-out on turns.

Correct steering condition. Taking the kingpin centres a distance c apart and wheelbase b, geometry gives cot φ − cot θ = c / b. A mechanism that satisfies it at every angle gives correct steering.

Davis steering gear — uses sliding pairs and satisfies the condition exactly at all angles in theory. The sliding pairs wear and add friction, so it is not used in practice. Its design condition is tan α = c / (2b), α being the angle of the steering arms to the longitudinal axis.

Ackermann steering gear (Ackermann linkage) — uses turning pairs only: the two steering (track-rod) arms and the track rod form a trapezoid four-bar linkage. The arms are angled so that, straight-ahead, their extensions meet approximately at the centre of the rear axle. It satisfies the correct-steering condition exactly at only three positions — straight ahead and one angle to each side — and approximately elsewhere. Being simple and durable, it is what vehicles use.

Steering linkage for a rigid (beam) axle. Steering wheel → column → steering gearbox → drop arm (pitman arm) → drag link (fore-and-aft) → steering arm on one stub axle. The two stub axles are tied together by the track-rod arms and a one-piece track rod, whose length (adjustable by threaded ends) sets the toe.

Linkage for independent front suspension. A one-piece track rod would cause each wheel to steer as it moves up and down (bump steer), so the track rod is split: either a rack-and-pinion gear with a tie rod to each wheel, or a centre (relay) link with an idler arm and two outer tie rods driven from a recirculating-ball box. Tie-rod inner pivots are placed so the tie-rod arcs match the suspension arcs, minimising bump steer.

Ball joints at every link end allow motion in all directions; their wear is the main cause of steering free play.

Beyond low speed. At speed the tyres run at slip angles, so racing cars often use less than full Ackermann (parallel or even anti-Ackermann). Road cars usually use partial Ackermann.

Formulas

cot φ − cot θ = c / b

• φ = outer front wheel angle, θ = inner front wheel angle (θ > φ), c = distance between kingpin (pivot) centres (m), b = wheelbase (m). Correct-steering (Ackermann) condition; rigid, unsteered rear axle; no slip angles.

tan α = c / (2b)

• α = angle of each steering arm to the vehicle's longitudinal axis in the straight-ahead position (Davis gear design condition; also the approximate Ackermann trapezoid layout with arm lines meeting at the rear axle centre).

R_of = b / sin φ + (a − c) / 2, R_if = b / sin θ − (a − c) / 2

• R_of, R_if = turning radii of the outer and inner front wheels (m), a = wheel track (m); (a − c)/2 = distance from kingpin centre to wheel centre along the stub axle (m).

R_or = b · cot φ + (a − c) / 2, R_ir = b · cot θ − (a − c) / 2

• Turning radii of the outer and inner rear wheels (m).

δ ≈ b / R (radians) — mean steer angle of a low-speed "bicycle model", R = turning radius of the rear axle centre (m), valid when R ≫ b.

Worked examples

Example 1 (standard) — outer wheel angle. A car has wheelbase b = 2.6 m and kingpin centres c = 1.3 m apart. The inner wheel is steered through 35°. Find the outer wheel angle for correct steering.

1. cot φ − cot θ = c / b → cot φ = cot θ + c / b.
2. cot 35° = 1.4281; c / b = 1.3 / 2.6 = 0.5.
3. cot φ = 1.4281 + 0.5 = 1.9281 → tan φ = 0.5186.
4. φ = arctan 0.5186 = 27.41°. φ ≈ 27.4° — smaller than the inner angle, as it must be.

Example 2 (GATE level) — maximum lock and turning circle. A vehicle has wheelbase 2.8 m, track 1.4 m and kingpin centres 1.2 m apart. The maximum inner-wheel lock is 40°. Assuming correct steering, find the outer-wheel lock and the turning radius of the outer front wheel.

1. cot φ = cot θ + c / b = cot 40° + 1.2 / 2.8 = 1.1918 + 0.4286 = 1.6203.
2. φ = arctan(1 / 1.6203) = 31.68°.
3. Distance from I to the outer kingpin: b / sin φ = 2.8 / sin 31.68° = 2.8 / 0.5252 = 5.331 m.
4. (a − c) / 2 = (1.4 − 1.2) / 2 = 0.1 m.
5. R_of = b / sin φ + (a − c)/2 = 5.331 + 0.1 = 5.431 m. φ ≈ 31.7°, R_of ≈ 5.43 m (turning-circle diameter about 10.9 m). For comparison, the inner front wheel runs on R_if = 2.8 / sin 40° − 0.1 = 4.26 m.

Common mistakes

• Writing the condition as cot θ − cot φ = c/b, which makes the outer wheel steer more — always check that the inner angle comes out larger.
• Using the wheel track a in place of the kingpin-centre distance c (they differ by twice the kingpin-to-wheel offset).
• Thinking the Ackermann linkage is exact at all angles; only the Davis gear is (in theory), and it is not used because of its sliding pairs.
• Calling rack-and-pinion or recirculating ball a "linkage" — they are steering gears; the linkage is the arms, links and rods.
• Forgetting that a one-piece track rod on independent suspension causes bump steer.

For GATE ME

This is mostly a theory-of-machines topic: expect the correct-steering condition (find the outer angle, the inner angle or c/b), the Davis gear condition tan α = c/2b, and the turning radius of a particular wheel. Practise drawing the instantaneous centre on the rear-axle line and reading every radius off the triangle.

Quick check

1. For correct steering, which front wheel turns through the larger angle?
2. Why is the Davis gear not used on vehicles?
3. At how many positions is the Ackermann linkage exact?
4. With c = 1.2 m and b = 3.0 m, what is cot φ − cot θ?
5. For a Davis gear with c/b = 0.4, what is tan α?

Answers: 1. The inner wheel. 2. Its sliding pairs wear and add friction. 3. Three (straight ahead and one on each side). 4. 0.4. 5. 0.2.