Ackermann-linkage Geometry

Ackermann-linkage Geometry

In parallel-set steering arms layout (Fig. 27.26A), the track-rod dimensions yi, xi andy0, x0, remain equal for all angles of turn. With the inclined arms (Fig. 27.26B and C), the inner-wheel track-rod end dimension yi, is always smaller than the outer wheel dimension y0, while negotiating a curve. On the other hand, there is very little variation between xi and x0 for small angular movements. For small steering angles about the king-pin up to say 10 degrees, there is very little difference between yi and y0 and between the inner and outer wheel turning angles.



Figure 27.26B illustrates that for a 10 degrees set track rod arms if the outer wheel is turned at 20 degrees, then the corresponding inner wheel is shown to rotate 23 degrees. Similarly for the same set, for 40 degrees outer wheel turn, the inner wheel rotates 51 degrees (Fig. 27.26 C).



 Therefore, for a given angular movement of the stub axles, the inner-wheel track-rod arm and track-rod are more effective than the outer-wheel linkage in turning the steered wheel. For a given amount of transverse track-rod movement with inclined track-rod arms, the least effective angular displacement of stub-axle pivot occurs in the straight-ahead region, and the most effective angular displacement takes place as the stub-axles move away from the mid-position.



Thus, the angular movement of the inner wheel relative to the outer wheel becomes much greater as both wheels approach movement of the inner wheel relative to the outer wheel becomes much greater as both wheels full lock (Fig. 27.27). With modern radial tyres, the difference between front and back-lock steering angles is sometimes reduced.

Fig. 27.27. Front and back lock steering-angle curves

Fig. 27.28. Analytical solution diagram for Ackermann-linkage.

Analytical Solution.

If the slight inclination of the track rod (Fig. 27.28) is neglected, the movements of M and N in the direction parallel to the axle beam PQ can be considered as the same, say z. Let M', N' represent the correct steering position and, r, denote the cross-arm radius.

Example 27.2. A track has pivot pins 1.37 m apart, the length of each track arm is 0.18 m and the track rod is behind front axle and 1.27 m long. Determine the wheel base which will give true rolling for all wheels when the car is turning so that the inner wheel stub axle is 60° to the centre line of the car. A geometrical construction may be used.

Example 27.3. The distance between the king-pins of a car is 1.3 m. The track arms are 0.1525 m long and the length of the track rod is 1.2 m. For a track of 1.42 m and a wheel base of 2.85 m, find the radius of curvature of the path followed by the near-side front wheel at which correct steering is obtained when the car is turning to the right

Fig. 27.30. Graphical solution diagram for Ackermann-linkage

Fig. 27.31. Deflection of outer wheels for assumed deflection of inner wheel.

Thus for the graphical representation of the steering mechanism on a paper of reasonable size with a sufficient degree of accuracy, it can be assumed that the steering arms are of definite length and make a certain angle with the longitudinal axis of the vehicle.

Now determine graphically the deflection on the outer wheel, <|> for various assumed deflection of the inner wheels, say 0 = 10°, 15°, 20°, 25°, 30°, 35°, 40° and 45° for these steering arms as shown in Fig. 27.31.

After knowing these values of 0, the correspond­ing angles 9 and $ are laid off on the opposite ends of the line C as shown in Fig. 27.32. A curve is then drawn through the intersection of lines describing the angle 9 and <j) correspondingly. The curve drawn is called steering error curve, because their devia­tion from true steering curve indicates the error in the steering angles. The angles 9 and § correspond­ing to the intersection of these two curves determine the correct steering angle for a particular a and c/b. If a is changed for the same c/b another steering error curve is obtained. Thus it can be concluded that the most advantageous angle of the knuckle arms, i.e. a depends upon the turning range of the inner wheel.

Fig. 27.32. Steering error curve.