Vehicle Body 1DOF Longitudinal
Twoaxle vehicle in forward and reverse motion
 Library:
Powertrain Blockset / Vehicle Dynamics
Vehicle Dynamics Blockset / Vehicle Body
Description
The Vehicle Body 1DOF Longitudinal block implements a one degreeoffreedom (1DOF) rigid vehicle body with constant mass undergoing longitudinal (that is, forward and reverse) motion. Use the block:
In powertrain and fuel economy studies to represent the vehicle inertial and drag loads when weight transfer from vertical and pitch motions are negligible.
To determine the engine torque and power required for the vehicle to follow a specified drive cycle.
You can select block options to create input ports for external forces, moments, air temperature, and wind speed.
Block Option Setting  External Input Ports  Description 

External forces 
 External force applied to vehicle CG in the vehiclefixed frame. 
External moments 
 External moment about vehicle CG in the vehiclefixed frame. 
Air temperature 
 Ambient air temperature. Consider this option if you want to vary the temperature during runtime. 
Wind X,Y,Z 
 Wind speed along earthfixed X, Y, and Zaxes. If
you do not select this option, the block implements input port

Vehicle Body Model
The vehicle axles are parallel and form a plane. The longitudinal direction lies in this plane and is perpendicular to the axles. If the vehicle is traveling on an inclined slope, the normal direction is not parallel to gravity but is always perpendicular to the axlelongitudinal plane.
The block uses the net effect of all the forces and torques acting on it to determine the vehicle motion. The longitudinal tire forces push the vehicle forward or backward. The weight of the vehicle acts through its center of gravity (CG). The grade angle changes the direction of the resolved gravitational force acting on the vehicle CG. Similarly, the block resolves the resistive aerodynamic drag force on the vehicle CM.
The Vehicle Body 1DOF Longitudinal block implements these equations.
$$\begin{array}{l}{F}_{b}=m\ddot{x}\\ \\ {F}_{b}={F}_{xF}+{F}_{xR}{F}_{d,x}+{F}_{ext,x}mg\mathrm{sin}\gamma \end{array}$$
Zero normal acceleration and zero pitch torque determine the normal force on each front and rear axles.
$$\begin{array}{l}{F}_{zF}=\frac{{M}_{ext,y}{M}_{d,y}+b\left({F}_{d,z}+{F}_{ext,z}+mg\mathrm{cos}\gamma \right)h\left({F}_{ext,x}+{F}_{d,x}+mg\mathrm{sin}\gamma +m\ddot{x}\right)}{{N}_{F}(a+b)}\\ \\ {F}_{zR}=\frac{{M}_{ext,y}+{M}_{d,y}+a\left({F}_{d,z}+{F}_{ext,z}+mg\mathrm{cos}\gamma \right)+h\left({F}_{ext,x}+{F}_{d,x}+mg\mathrm{sin}\gamma +m\ddot{x}\right)}{{N}_{R}(a+b)}\end{array}$$
The wheel normal forces satisfy this equation.
$${N}_{F}{F}_{zF}+{N}_{R}{F}_{zR}{F}_{ext,z}=mg\mathrm{cos}\gamma $$
Wind and Drag Forces
The block subtracts the wind speeds from the vehicle velocity components to obtain a net relative airspeed. To calculate the drag force and moments acting on the vehicle, the block uses the net relative airspeed.
$$\begin{array}{l}{F}_{d,x}=\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{(}^{\dot{x}}\\ {F}_{d,z}=\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{(}^{\dot{x}}\\ {M}_{d,y}=\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{(}^{\dot{x}}(a+b)\end{array}$$
By default, to calculate the wind speed along the vehiclefixed xaxis, the block uses the longitudinal wind speed along the earthfixed Xaxis. If you select WindX,Y,Z, the block uses the wind speed along the earthfixed X, Y, Zaxes.
Power Accounting
For the power accounting, the block implements these equations.
Bus Signal  Description  Equations  



 Externally applied force power  ${P}_{FxExt}={F}_{xExt}\dot{x}$ 
 Longitudinal force power applied at the front axle  ${P}_{FwFx}={F}_{wF}\dot{x}$  
 Longitudinal force power applied at the rear axle  ${P}_{FwRx}={F}_{wR}\dot{x}$  

 Drag force power  ${P}_{d}=\frac{0.5{C}_{d}{A}_{f}{P}_{abs}{\left({\dot{x}}^{2}{w}_{x}\right)}^{2}}{287.058T}\dot{x}$  

 Rate change in gravitational potential energy  ${P}_{g}=mg\dot{Z}$  
 Rate in change of longitudinal kinetic energy  ${P}_{\dot{x}}=m\ddot{x}\dot{x}$ 
The equations use these variables.
F_{xf}, F_{xr}  Longitudinal forces on each wheel at the front and rear ground contact points, respectively 
F_{zf}, F_{zr}  Normal load forces on each wheel at the front and rear ground contact points, respectively 
F_{wF}, F_{wR}  Longitudinal force on front and rear axles along vehiclefixed xaxis 
F_{xExt}, F_{wR}  External force along the vehiclefixed xaxis 
F_{d,x}, F_{d,z}  Longitudinal and normal drag force on vehicle CG 
M_{d,y}  Torque due to drag on vehicle about the vehiclefixed yaxis 
F_{d}  Aerodynamic drag force 
V_{x}  Velocity of the vehicle. When V_{x} > 0, the vehicle moves forward. When V_{x} < 0, the vehicle moves backward. 
N_{f}, N_{r}  Number of wheels on front and rear axle, respectively 
$\gamma $  Angle of road grade 
m  Vehicle body mass 
a,b  Distance of front and rear axles, respectively, from the normal projection point of vehicle CG onto the common axle plane 
h  Height of vehicle CG above the axle plane 
C_{d}  Frontal air drag coefficient 
A_{f}  Frontal area 
P_{abs}  Absolute pressure 
ρ  Mass density of air 
x, $$\dot{x}$$, $$\ddot{x}$$  Vehicle longitudinal position, velocity, and acceleration along the vehiclefixed xaxis 
w_{x}  Wind speed along the vehiclefixed xaxis 
$$\dot{Z}$$  Vehicle vertical velocity along the vehiclefixed zaxis 
Limitations
The Vehicle Body 1DOF Longitudinal block lets you model only longitudinal dynamics, parallel to the ground and oriented along the direction of motion. The vehicle is assumed to be in pitch and normal equilibrium. The block does not model pitch or vertical movement. To model a vehicle with three degreesoffreedom (DOF), use the Vehicle Body 3DOF Longitudinal.