Name: Group members: TAM 212 Worksheet 9: Cornering and banked turns The aim of this worksheet is to understand how vehices drive around curves, how sipping and roing imit the maximum speed, and how banking the road can increase the maximum speed. The photos above show two different views of a Setra S 411 HD bus cornering at high speed on the Mercedes- Benz test track at Untertürkheim, on the Neckar River just outside of Stuttgart, Germany. Beow is a top view of the track, showing the bus and its veocity and acceeration vectors. In this worksheet we wi aways assume a circuar track of fixed radius and a bus traveing at constant speed v. a v 1
Cornering on a fat road First we consider the bus driving around a curve on a fat horizonta road, as shown in the front-view free body diagram beow. We wi mode the bus as a rigid body that contacts the ground at two points (the whee centers) with norma forces N 1 and N 2, and we assume that the friction force F appies equay to each whee. Gravity g acts through the center of mass C. mg a C h F/2 F/2 ĵ N 1 N 2 î N 2 N 1 F/2 F/2 1. What is the acceeration a as a function of the bus speed v and radius of curvature of the road? a = v2 î 2
2. If the bus is not roing then it has zero anguar veocity about the ˆk direction, so the tota moment about the ˆk axis wi be zero. The kinetics wi thus satisfy F = m a and MC = 0. Use these two equations to find the forces F, N 1, and N 2 in terms of a other variabes. The rigid body equations are: F = m a F î + N 1 ĵ + N 2 ĵ mgĵ = m a = mv2 î MC = 0 Equating components gives the equations: and soving these gives: F hˆk N 1 ˆk + N 2 ˆk = 0. F = mv2 N 1 + N 2 mg = 0 F h N 1 + N 2 = 0 F = mv2 N 1 = mg 2 mv2 h 2 N 2 = mg 2 + mv2 h 2 3. The bus wi not side if F F max = µ(n 1 + N 2 ), where µ is the coefficient of friction for the tires on the road. For a track radius of = 50 m and a coefficient of friction µ = 0.3, what is the fastest that the bus can drive around the corner before it starts to side? Critica speed occurs for: F = F max = µ(n 1 + N 2 ) = µmg mv 2 = µmg v = µg 12.1 m/s 43 km/h 3
4. The bus wi not ro if N 1 0. For a track radius of = 50 m and bus geometry = 0.97 m and h = 0.87 m. What is the fastest the bus can drive around the corner before it ros over? Critica speed occurs for: mv 2 h 2 0 = N 1 = mg 2 mv2 h 2 = mg 2 g h 23.4 m/s 84.2km/h v = 5. If the bus tries to drive at high speed around a track as described in the ast two questions, wi siding or roing be the imiting factor? What is the maximum speed at which the bus can drive around this curve? Siding wi occur at a ower speed, so it is the imiting factor and the maximum speed is v 12.1 m/s 43 km/h. 6. Woud atering the dimensions and h of the bus change the speed at which it sides? Woud it change the speed at which it ros? How shoud and h be changed to make the bus harder to ro? From the answer to Question 3 we see that the siding speed is independent of and h. However, from the answer to Question 4 we see that the speed when roing occurs does depend on and h. To make this speed higher (so harder to ro) we shoud make arger and h smaer, as we see from the Question 4 soution. This woud tend to make the bus wide and ow, ike a tank. 4
Cornering on a banked turn We now consider the situation when the road is banked, as shown in the free body diagram beow. a mg C h F/2 F/2 ĵ N 2 N 2 N 1 î N 1 F/2 ê n F/2 ê t θ 7. The safest driving conditions are when there is no need for a friction force, so F = 0. Write Newton s equations in the ê t direction in this case and obtain a formua for the safest speed v to drive for a given bank ange θ. Ft = ma t mg sin θ = ma cos θ mg sin θ = mv2 cos θ v = g tan θ 5
8. For the bus shown on page 1, we have θ = 45 and = 50 m. Assuming the bus is driving at the safest speed (so F = 0), what is its speed? v = g tan θ 22.1 m/s 79.7 km/h 9. If we wanted to drive the bus at v = 40 m/s around the banked track with radius = 50 m, what bank ange woud be necessary (assuming we have the safest conditions of F = 0)? As a matter of interest, 40 m/s 144 km/h. v = g tan θ tan θ = v2 g θ 73.0 10. Is there any imit to how fast the bus can drive around the banked track, given that we can make the bank ange as cose to θ = 90 as we ike? Our rigid body mode does not show any imit to the speed, as we can just keep increasing θ to ensure that F = 0. In practice the norma acceeration wi be so high that the peope on the bus wi start to back out. Before this point the driver of the bus wi probaby ose contro, due to the high speed and sma margin for error at high bank anges. 6
Chaenge questions 11. For the banked curve, et us consider the genera case where F 0. Use F = m a and MC = 0 to derive expressions for F, N 1, and N 2 in terms of the other variabes. Hint: use the ê t,ê n basis. Ft = ma t F mg sin θ = mv2 cos θ Fn = ma n N 1 + N 2 mg cos θ = mv2 sin θ MC = 0 F h N 1 + N 2 = 0 Equating components gives the equations: and soving these gives: F = mv2 cos θ mg sin θ N 1 + N 2 = mg cos θ mv2 N 1 N 2 = F h = h (mv2 sin θ cos θ mg sin θ) F = mv2 N 1 = m {g(cos θ + 2 N 2 = m {g(cos θ 2 cos θ mg sin θ h v2 sin θ) + (sin θ h cos θ) h v2 sin θ) + (sin θ + h cos θ) } } 12. Using your answer to Question 11 and the same siding restriction as in Question 3, how fast can the bus drive around a track with bank ange θ = 30 before it starts to side? 7
mv 2 F F max = µ(n 1 + N 2 ) cos θ mg sin θ µ(mg cos θ + mv2 sin θ) sin θ+µ cos θ v g cos θ µ sin θ v 22.81(m/s) 46.19(km/h) 13. Using your answer to Question 11 and the same roing restriction as in Question 4, how fast can the bus drive around a track with bank ange θ = 30 before it starts to ro? N 1 = m 2 N 1 0 { } g(cos θ + h v2 sin θ) + (sin θ h cos θ) 0 v = g cos θ+ h sin θ h cos θ sin θ v 48.27(m/s) 173.8(km/h) 14. If the bus tries to drive at high speed around a track with bank ange θ = 30, as described in the ast two questions, wi siding or roing be the imiting factor? What is the maximum speed at which the bus can drive around this curve? Siding is a imiting factor. 8