Vehicle Dynamics
Longitudinal Acceleration
The acceleration of a vehicle depends on the net force acting on it. The basic equation for longitudinal acceleration is:
Where:
: Net force acting on the vehicle (N) : Total mass of the vehicle (kg) : Acceleration (m/s²)
Longitudinal acceleration is calculated as:
Resistive Forces to Motion
Aerodynamic Drag
Where:
: Coefficient of aerodynamic drag (dimensionless) : Frontal area of the vehicle (m²) : Air density (kg/m³) : Speed of the vehicle (m/s)
Rolling Resistance
Where:
: Coefficient of rolling resistance (dimensionless) : Acceleration due to gravity (9.81 m/s²)
Slope Resistance (or Gradient)
Where:
: Angle of inclination of the slope (rad)
Vehicle Motion Equation (Sum of Forces)
The sum of all forces acting on a vehicle gives the net force:
Maximum Speed of the Vehicle
The maximum speed can be found when the traction force
Solving for speed:
Transmission and Powertrain
Gear Ratio
The gear ratio for a gear system can be calculated as:
Where:
: Number of teeth on the driving gear : Number of teeth on the driven gear
Power at the Wheels
The power available at the wheels is related to torque and angular velocity:
Where:
: Power (W) : Torque (N·m) : Angular velocity (rad/s)
Torque
The torque at the wheels is related to the torque at the engine through the gear ratio:
Where:
: Efficiency of the transmission
Vehicle Aerodynamics
Aerodynamic Drag
Aerodynamic drag is the main resistive force acting on a vehicle at high speeds. The formula has been provided but is broken down into the following factors:
Drag Coefficient
Depends on the shape and aerodynamic design of the vehicle.
Frontal Area
Effective area facing air resistance.
Lift Force
Some vehicles generate lift or aerodynamic load due to their design. This force is calculated similarly to drag:
Where:
: Lift coefficient.
Braking
Braking Force
The total braking force that a vehicle can exert is given by the following equation:
Where:
: Coefficient of friction between the tire and the road. : Normal force on the tires, which equals the weight of the vehicle on flat surfaces.
Braking Distance
The distance required to stop a vehicle from an initial speed
Suspension and Tires
Natural Frequency of the Suspension
The natural frequency of a suspension is important for the comfort and stability of the vehicle. It is calculated as:
Where:
: Spring constant (N/m) : Suspended mass (kg)
Load on Tires
The vertical load on a tire can be calculated as:
Where:
: Distance from the center of mass to the front axle. : Distance from the center of mass to the rear axle. : Wheelbase.
Power and Fuel Consumption
Required Power
The power required to move a vehicle at constant speed is determined by the total resistive forces:
Where:
: Speed of the vehicle (m/s)
Specific Fuel Consumption (SFC)
Where:
: specific fuel consumption (kg/W·h) : mass flow rate of fuel (kg/h) : engine power (W)
Thermal Efficiency
Where:
: useful power (W) : chemical power of the fuel (W)
Range
Where:
: range of the vehicle (km) : energy stored in the tank (J or Wh) : fuel consumption per km (L/km)