Vehicle dynamics

1
Vehicle dynamics

• Vehicle resistance
• Power
• Acceleration
• Motion equations

2
Vehicle resistance (1)

• Components
• Ideal
• Rolling resistance
• Way resistance
• Air resistance
• Non-ideal
• Gradient
• Curvature
• Rolling resistance
• Deformation of wheel and way
• Sucking effect
• Slippage

• Way resistance
• Track position
• Riding surface
• Joints
• Flanges
• Sway and oscillation
• Concussion

3
Vehicle resistance (2)

• Rolling resistance
• Ro (C1C2.V)W
• Ro Rolling and way resistance (N)
• C 1 and C 2 coefficients (N/kN)
• V Speed of the vehicle (km/h)
• W Weight of the vehicle (kN)
• Beware of different dimensions of terms
• Ro (C1C2.V C3 /Q)W
• Q is axle loading (kN).

• Air resistance
• Ra ( /2g).Ca!.A.(V/3.6)2
• Ra Ca.A.V2
• Ra Air resistance (N)
• A Frontal area of the vehicle (m2)
• V Relative speed of the vehicle (relative to
  the air speed) (km/h)
• Ca Coefficient of air resistance

4
Vehicle resistance (Road)

• R (C1 C2 V)W Ca AV2
• Typical USA values
• C1 7,6
• C2 0,056
• Ca 0,020 to 0,025 for cars and 0,011 to
  0,025 for buses
• A 1,6 to 2,4 for passenger cars
• 2,7 to 4,0 for vans (minibus-taxis),
• 6,5 to 7,0 for single-decker buses and
• 9,0 to 10,0 for double-decker buses

• R (7,6 0,056 V)W 0,018 AV2.

km/hr
kN
5
Vehicle resistance (e.g.1)

• What is the resistance of a vehicle travelling at
  
• 60km/h,
• 40km/h and
• 20 km/h
• given that
• C1 7,6
• C2 0,056
• Ca 0,022
• A 2,3
• Weight 20kN

• R (C1 C2 V) W Ca A V2
• _at_ 40 km/h ?
• _at_ 20 km/h ?

• (7,6 0,056 60) 20 0,022 2,3 602
• (7,6 3,36) 20 182.16
• 219.2 182.16
• 401,4 N

6
Vehicle resistance (Rail)

• R (C1C2V C3/Q)W CaV2
• Typical USA values
• C1 0,65 to 0,70
• C2 0,0009
• C3 120 to 130
• Ca 0,016 to 0,024 for
• streamlined diesel and
• electric vehicles
• 0,012 for regional rail
• and 0,013 for rapid rail
• A 7,0 to 8,0 for LRT and
• 7,5 to 10,0 for
• conventional rail.

• R (0,65 0,0009 V 129/Q)W 0,0716 V2
• (Note A is missing).
• London underground
• R 2,7.W 0,88 V2
• Leningrad
• R (3,20,034V 0,0005V2.).W
• Moscow
• R (5,0 0.005V2).W

7
Vehicle resistance (All 1)
8
Vehicle resistance (Grade)

• R ? W.sin ? 1000
• R ? Resistance (N)
• W Weight of the vehicle (kN)
• ? Angle of gradient

• When ? is small,
• sin ? tan ?. grade
• R ? 10.i.W
• i is the grade

9
Vehicle resistance (e.g. 2)

• What is the resistance of a vehicle travelling at
  
• 60km,
• 40km and
• 20 km
• given that
• C1 7,6
• C2 0,056
• Ca 0,022
• A 2,3
• Weight 20kN
• Grade 5

• Rg W i 10
• 20510 1000N
• (Actually 20 0005/100)
• SR _at_ 60 km/h ?
• SR _at_ 40 km/h ?
• SR _at_ 20 km/h ?

10
Vehicle resistance (Curvature)

• Curvature resistance
• Complex
• Empirically derived
• Factors
• Curvature
• Wheel spacing
• Axle width
• Rail width

11
Power for propulsion (1)

• Power
• Source
• At shaft to wheels
• At perimeter of wheels
• TE k P/V
• P power generated by the motor (kW)
• (Watts Nm/sec)
• V Speed (km/h)
• k Factor to account for the losses

• Given
• P 50kW (Power)
• V 20 km/h (Speed)
• k 70 (Efficiency)
• Find TE (Tractive effort)
• TE

50000/(20/3,6)0,70 6300 N
12
Power for propulsion (2)

• Internal combustion engine

13
Power for propulsion (3)

• Electric motor

14
Power for propulsion (4)

• Adhesion
• F m W
• F Force in N
• m coefficient of friction
• W normal force between the two surfaces (N)
• m(road) between 0,6 and 0,7
• m(rail) between 0,2 and 0,35

• Given
• m 0,5
• W 20 kN
• Find F
• F

• 0,5 20000
• 10 000 N

15
Motion equations (1)

• Acceleration
• F m.a
• F TE - R
• Distance (S) f(t)
• Speed (v) dS/dt
• Acceleration (a) dv/dt d2S/dt2

• If speed at the t0 is u and speed at t t is v,
  then
• v u a.t
• S u.t 1/2.a.t2
• S (V2 -U2)/2a

16
Motion equations (2)
17
Motion equations (e.g. 3)

• Given
• Speed 10km/h
• Vehicle weight 20kN
• Power 50kN
• Efficiency 70
• Grade 5
• C1 7,6 C2 0,056
• Ca 0,022
• A 2,3 m2
• Rc
• Rg
• SR

• TE
• Nett TE
• a
• For time intervals of 1 sec
• V
• S
• BUT at low speeds check adhesion limit
• Max TE
• Nett TE
• a
• V
• S
• Adhesion limit

18
Motion equations (e.g. 3)

• Given
• Speed 10km/h
• Vehicle weight 20kN
• Power 50kW
• Efficiency 70
• Grade 5 m 0,5
• C1 7,6 C2 0,056
• Ca 0,022
• A 2,3 m2
• Rc
• Rg
• SR

• TE
• Nett TE
• a
• For time intervals of 1 sec
• V
• S
• Nett TE
• a
• V
• S

50000 / (10 / 3,6) 0,7 12600 N
12600-1168 11432 N
11432 / (20000 / 9.81) 5,61 m/s2
10/3,65,611 8,39 m/s or 30,2 km/h
10 / 3,6 1 5,612 / 2 5,59 m

• BUT at low speeds check adhesion limit
• Max TE 0,5 20000 10000 N

10000 -1168 8832 N
(7,6 0,05610)20 0,0222,3102 168 N
8832 / (20000 / 9,81) 4,33 m/s2
10/3,64,331 7,11 m/s or 25,6 km/h
5/10020000 1000 N
10/3,6 14,332/2 4,95 m
1168 N

• Adhesion limit 50000/(V/3.6)0,7 10000

• V 12,6 km/h