2.4 Longitudinal Vehicle Dynamics Flashcards
What are the main assumptions for analyzing vehicle longitudinal dynamics in driving scenarios?
The main assumptions are:
- 1 degree of freedom (motion along a straight line)
- Neglect suspension deformations and sprung mass motions
- Flat road (no transversal inclination)
- Neglect aerodynamic downforce
- All wheels in small slip conditions
What is the formula for static load distribution on front wheels on a flat road?
$F_{ZF} = mg\frac{b}{L}$
Where:
- m is vehicle mass
- g is gravitational acceleration
- b is distance from center of gravity to rear axle
- L is wheelbase
What is the formula for static load distribution on rear wheels on a flat road?
$F_{ZR} = mg\frac{a}{L}$
Where:
- m is vehicle mass
- g is gravitational acceleration
- a is distance from center of gravity to front axle
- L is wheelbase
What is the formula for aerodynamic drag force?
$F_{Xa} = F_a = \frac{1}{2}\rho SC_X V^2$
Where:
- ρ is air density
- S is cross-section area
- C_X is aerodynamic drag coefficient
- V is vehicle speed
How is rolling resistance calculated?
$F_{roll} = fmg = (f_0 + f_2V^2)mg$
Where:
- f is rolling resistance coefficient
- f_0 is constant term of rolling resistance coefficient
- f_2 is speed-dependent term
- m is vehicle mass
- g is gravitational acceleration
What is the formula for climbing resistance?
$F_{cli} = mg\sin\alpha$
Where:
- m is vehicle mass
- g is gravitational acceleration
- α is slope angle
What are the components of total motion resistance?
Total motion resistance: $F_{res} = F_{aer} + F_{rol} + F_{cli}$
Where:
- F_aer is aerodynamic resistance
- F_rol is rolling resistance
- F_cli is climbing resistance
How is the power necessary for motion calculated?
$P_n = F_{res} \cdot V$
Where:
- F_res is total resistance force
- V is vehicle speed
What is the traction hyperbola?
The traction hyperbola represents the ideal available force at the wheels as a function of speed:
$F_X = \frac{\eta_t P_{max}}{V}$
Where:
- η_t is transmission efficiency
- P_max is maximum power
- V is vehicle speed
What are the three main disadvantages of internal combustion engines?
- Cannot produce torque from rest (zero engine speed)
- Maximum power only at a specific engine speed
- Fuel consumption is strongly dependent on the operating point
How do electric motor characteristics compare to internal combustion engines for vehicle traction?
Electric motors:
- Can produce maximum torque from zero speed
- Have a nearly constant power output over a wide speed range
- Match the ideal traction hyperbola much better than ICEs
- Often don’t require multi-speed transmissions due to their favorable torque-speed characteristics
How is vehicle acceleration calculated in relation to power?
$a = \frac{dV}{dt} = \frac{P_e\eta_t - P_{res}}{m_e V}$
Where:
- P_e is engine power
- η_t is transmission efficiency
- P_res is power required to overcome resistance
- m_e is equivalent mass
- V is vehicle speed
What is equivalent mass and why is it important for vehicle dynamics?
Equivalent mass accounts for both translational and rotational inertia:
$m_e = m + \frac{J_e\tau_g^2\tau_d^2}{R^2} + \sum_{4}\frac{J_w}{R^2}$
Where:
- m is vehicle mass
- J_e is engine inertia
- τ_g is gear ratio
- τ_d is differential ratio
- J_w is wheel inertia
- R is wheel radius
It’s important because it accounts for the additional energy needed to accelerate rotating components and varies with the selected gear.
How is the maximum theoretical slope a vehicle can climb calculated?
$i = \tan\alpha \cong \frac{\frac{P_e\eta_t}{V} - (F_{aer} + F_{rol})}{mg}$
Where:
- P_e is engine power
- η_t is transmission efficiency
- V is vehicle speed
- F_aer is aerodynamic resistance
- F_rol is rolling resistance
- m is vehicle mass
- g is gravitational acceleration
What physical factor ultimately limits a vehicle’s climbing ability?
The maximum slope is ultimately limited by tire-road friction. For an all-wheel drive vehicle with pneumatic tires:
$i \approx \tan\alpha = \mu$
Where μ is the coefficient of friction between tires and road surface.
What are the key differences between NEDC and WLTP driving cycles?
Key differences:
- Duration: WLTP: 1,800s vs NEDC: 1,180s
- Distance: WLTP: 23.253km vs NEDC: 11.007km
- Maximum speed: WLTP: 131.3km/h vs NEDC: 120km/h
- Average speed: WLTP: 46.5km/h vs NEDC: 33.6km/h
- Gear shifts: WLTP: vehicle-specific vs NEDC: fixed
- Idle time: WLTP: 13.1% vs NEDC: 21.8%
- Maximum acceleration: WLTP: 1.67m/s² vs NEDC: 1m/s²
How are vehicles classified in the WLTP system?
Vehicles are classified based on their Power-to-Mass Ratio (PMR):
- Class 3b: PMR > 34 W/kg, v_max ≥ 120 km/h
- Class 3a: PMR > 34 W/kg, v_max < 120 km/h
- Class 2: 22 < PMR ≤ 34 W/kg
- Class 1: PMR ≤ 22 W/kg
What is the equation for rotational wheel dynamics?
$\dot{\omega}_r = \frac{T_m - T_B - T_R - F_x R}{I_r}$
Where:
- ω_r is wheel angular velocity
- T_m is motor torque
- T_B is brake torque
- T_R is rolling resistance torque
- F_x is longitudinal force
- R is wheel radius
- I_r is wheel inertia
How is longitudinal slip ratio defined and calculated?
Longitudinal slip ratio (σ):
$\sigma_i = \frac{V - \omega_{r,i}R}{V}$
Where:
- V is vehicle speed
- ω_r,i is angular velocity of wheel i
- R is wheel radius
This parameter is crucial for determining the longitudinal force generated by the tire.
