Name: Group members: TM 212 Worksheet 5: ar steering This worksheet aims to understand how cars steer. The #avs webpage on Steering geometry illustrates the basic ideas. On the diagram below, the kingpins at and are a distance g apart (this is almost the same as the track distance between the front wheel tire centers), while the wheelbase distance is F = E = l. Modern cars use ball joints instead of actual pins at the kingpin joints. θ L θ R g = 2 m direction of turn l = 3 m M F P E 1. onsider the four-wheeled car configuration shown above. The left-front wheel is turned at an angle of θ L, and the turning radius of the car is ρ, measured from the center P of the rear axle to the instantaneous center M. erive a formula for ρ in terms of θ L, leaving measurements g and l in symbolic form. We see MF = θ L, so tan θ L = F/MF = l/(ρ g/2), giving: ρ = l tan θ L + g 2 2. Similarly to the previous question, derive a formula for ρ in terms of the angle θ R of the right-front wheel. Similar to the previous question: ρ = l tan θ R g 2 1
3. While trying to park our car in a tight spot, we want to drive our car around a counter-clockwise curve with a radius of curvature of ρ = 5 m. t what angles θ L and θ R should we ideally set our wheels, in order to make this turn? Give your answers in numeric form. Solving the Q1 and Q2 equations: θ L = tan 1 (3/4) 36.9 θ R = tan 1 (1/2) 26.6 4. ckermann steering geometry, shown in the figure below, uses a four-bar linkage to constrain the wheel angles θ L and θ R. The tie rod has length = f, while the steering arms have lengths = a and = b. simple rule of thumb for designing ckermann steering sets the linkage geometry so that the steering arms point to the center P of the rear axle, as shown. Given lengths a = b = 0.2 m, what is the angle γ and the appropriate length f of the tie rod? g = 2 m a f b γ l = 3 m F P E tan γ = g/2 l γ 18.4 (g f)/2 sin γ = a f = g 2a sin γ 1.87 m 2
5. The initial and turned state of front wheels of ckermann steering geometry case are drawn as below. On this figure, indicate the turning angles of right and left wheel (θr and θ L ), and γ. 3
6. θ L and θ R we found from Q3 are ideal angles during the turn without the tie rod. We want to see how well the ckermann steering geometry we designed in the Q4 works. onsider the turn from Q3 with ρ = 5 m, and set the left-front wheel angle θ L is equal to the value θ L found in Q3 (θ L = θ L). lso, the geometric results from Q5 should be helpful. What right-front angle θr is now determined by the linkage? Use the diagram below to start with θ L and work your way across the diagram to find θr. The law of cosines will be helpful for determine angles on general triangles, for example c 2 = a 2 + g 2 2ag cos g a c f b Figure Quantity Value c = + = θ R Using the law of cosines: = 90 γ θ L = 34.7 c 2 = a 2 + g 2 2ag cos c 1.84 m cos = g2 + c 2 a 2 2gc 3.55 cos = c2 + b 2 f 2 2cb 96.9 = + 100.4 = 90 γ + θ R θ R = + γ 90 28.8 4
7. How close is the ckermann value of θ R from Q6 to the ideal value θ R from Q3? Is this ckermann steering geometry acceptable for real-world usage? The ideal value θ R and the ckermann angle θ R are: θ R 26.6 θ R 28.8 θ R θ R 2.2 This is a small enough difference that the ckermann steering system would perform very close to ideal in practical applications, even on such sharp turns as ρ = 5 m. On more gradual turns the difference will be even smaller. 5
onus Questions 8. While making the turn in the above question, we measure the speed of point P to be v P = 2 m/s (we are parking very quickly!). What is the angular velocity ω of the car during the turn? ω = v = 0.4 rad/s ρ 9. While turning as above, what are the speeds of the four wheel joints v, v, v E, and v F? The distances from the instantaneous center M are: r = (ρ g/2) 2 + l 2 = 5 m r = (ρ + g/2) 2 + l 2 6.71 m r E = ρ + g/2 = 6 m r F = ρ g/2 = 4 m. The speeds are thus: v = r ω = 2 m/s v = r ω 2.68 m/s v E = r E ω = 2.4 m/s v F = r F ω = 1.6 m/s. 10. onsidering the velocities of the four wheel joints you found in Q9, would it be reasonable to build a rear-wheel-drive car with a rear driveshaft consisting of a single rod bolted to each wheel? Why or why not? What might an alternative be? It would not be reasonable to use a solid rear driveshaft bolted to both wheels, as the outer wheel (rear right) is moving 50% faster than the inner wheel (rear left), and so must also be rotating 50% faster. If a single driveshaft was used then one or both wheels would be forced to slip. Instead cars use a differential to allow the rear wheels to turn at different rates. 6