3DOF rigid vehicle body to calculate longitudinal, vertical, and pitch motion
Powertrain Blockset / Vehicle Dynamics
Vehicle Dynamics Blockset / Vehicle Body
The Vehicle Body 3DOF Longitudinal block implements a three degreesoffreedom (3DOF) rigid vehicle body model with configurable axle stiffness to calculate longitudinal, vertical, and pitch motion. The block accounts for body mass, aerodynamic drag, road incline, and weight distribution between the axles due to acceleration and the road profile.
You can specify the type of axle attachment to the vehicle:
Grade angle — Vertical axle displacement from road surface to axles remains constant. The block uses tabular stiffness and damping parameters to model the suspension forces acting between the vehicle body and axles.
Axle displacement — Axles have inputprovided vertical displacement and velocity with respect to the road grade. The block uses tabular stiffness and damping parameters to model the suspension forces acting between the vehicle body and axle.
External suspension — Axles have externally applied forces for coupling the vehicle body to custom suspension models.
If the weight transfer from vertical and pitch motions are not negligible, consider using this block to represent vehicle motion in powertrain and fuel economy studies. For example, in studies with heavy breaking or acceleration or road profiles that contain larger vertical changes.
The block uses rigidbody vehicle motion, suspension system forces, and wind and drag forces to calculate the normal forces on the front and rear axles. The block resolves the force components and moments on the rigid vehicle body frame:
$$\begin{array}{l}{F}_{x}={F}_{wF}+{F}_{wR}{F}_{d,x}{F}_{sx,F}{F}_{sx,R}+{F}_{g,x}\\ {F}_{z}={F}_{d,z}{F}_{sz,F}{F}_{sz,R}+{F}_{g,z}\\ {M}_{y}=a{F}_{sz,F}b{F}_{sz,R}+h\left({F}_{wF}+{F}_{wR}+{F}_{sx,F}+{F}_{sx,R}\right){M}_{d,y}\end{array}$$
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). Depending on the inclined angle, the weight pulls the vehicle to the ground and either forward or backward. Whether the vehicle travels forward or backward, aerodynamic drag slows it down. For simplicity, the drag is assumed to act through the CG.
The Vehicle Body 3DOF Longitudinal implements these equations.
$$\begin{array}{l}\ddot{x}=\frac{{F}_{x}}{m}qz\\ \ddot{z}=\frac{{F}_{z}}{m}qx\\ \dot{q}=\frac{{M}_{y}}{{I}_{yy}}\\ \dot{\theta}=q\end{array}$$
If you configure the block with the Ground interaction
type parameter Grade angle
or Axle
displacement, velocity
, the block uses nonlinear stiffness
and damping parameters to model the suspension system.
The front and rear axle suspension forces are given by:
$$\begin{array}{l}F{s}_{F}={N}_{F}\left[F{k}_{F}+F{b}_{F}\right]\\ F{s}_{R}={N}_{R}\left[F{k}_{R}+F{b}_{R}\right]\end{array}$$
The block uses lookup tables to implement the front and rear suspension stiffness. To account for kinematic and material nonlinearities, including collisions with endstops, the tables are functions of the stroke.
$$\begin{array}{l}F{k}_{F}=f(d{Z}_{F})\\ F{k}_{R}=f(d{Z}_{R})\end{array}$$
The block uses lookup tables to implement the front and rear suspension damping. To account for nonlinearities, compression, and rebound, the tables are functions of the stroke rate.
$$\begin{array}{l}F{b}_{F}=f(d{\dot{Z}}_{F})\\ F{b}_{R}=f(d{\dot{Z}}_{R})\end{array}$$
The stroke is the difference in the vehicle vertical and axle positions. The stroke rate is the difference in the vertical and axle velocities.
$$\begin{array}{l}d{Z}_{F}={Z}_{F}{\overline{Z}}_{F}\\ d{Z}_{R}={Z}_{R}{\overline{Z}}_{R}\\ d{\dot{Z}}_{F}={\dot{Z}}_{F}{\dot{\overline{Z}}}_{F}\\ d{\dot{Z}}_{R}={\dot{Z}}_{R}{\dot{\overline{Z}}}_{R}\end{array}$$
When the Ground interaction type parameter is Grade
angle
, the axle vertical positions ($${\overline{Z}}_{F},{\overline{Z}}_{R}$$) and velocities ($${\dot{\overline{Z}}}_{F},{\dot{\overline{Z}}}_{R}$$) are set to 0
.
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:
$$\begin{array}{l}{F}_{d,x}=\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{(}^{\dot{x}}\\ {F}_{d,z}=\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{(}^{\dot{x}}\\ {M}_{d,y}=\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{(}^{\dot{x}}(a+b)\end{array}$$
For the power accounting, the block implements these equations.
Bus Signal  Description  Equations  



 Externally applied longitudinal force power  ${P}_{FxExt}={F}_{xExt}\dot{x}$ 
 Externally applied longitudinal force power  ${P}_{FzExt}={F}_{zExt}\dot{z}$  
 Externally applied pitch moment power  ${P}_{MzExt}={M}_{zExt}\dot{\theta}$  
 Longitudinal force applied at the front axle  ${P}_{FwFx}={F}_{wF}\dot{x}$  
 Longitudinal force applied at the rear axle  ${P}_{FwRx}={F}_{wR}\dot{x}$  

 Internal power transferred between suspension and vehicle body at the front axle  $${P}_{Fs,F}={P}_{FwFx}+{P}_{FsbF}+{P}_{Fsk,F}+{F}_{xF}{\dot{x}}_{F}+{F}_{zF}{\dot{z}}_{F}$$  
 Internal power transferred between suspension and vehicle body at the rear axle  ${P}_{Fs,R}={P}_{FwRx}+{P}_{Fsb,R}+{P}_{Fsk,R}+{F}_{xF}{\dot{x}}_{F}+{F}_{zF}{\dot{z}}_{F}$  
 Longitudinal drag force power  ${P}_{d,x}={F}_{d,x}\dot{x}$  
 Vertical drag force power  ${P}_{d,z}={F}_{d,z}\dot{z}$  
 Drag pitch moment power  ${P}_{d,My}={M}_{d,y}\dot{\theta}$  
 Total suspension damping power  ${P}_{Fsb}={\displaystyle \sum}_{i=F,R}^{}{F}_{sb,i}{\dot{z}}_{i}$  

 Rate change in gravitational potential energy  ${P}_{g}=mg\dot{Z}$  
 Rate of change of longitudinal kinetic energy  ${P}_{\dot{x}}=m\ddot{x}\dot{x}$  
 Rate of change of longitudinal kinetic energy  ${P}_{\dot{z}}=m\ddot{z}\dot{z}$  
 Rate of change of rotational pitch kinetic energy  ${P}_{\dot{\theta}}={I}_{yy}\ddot{\theta}\dot{\theta}$  
 Stored spring energy from front suspension  ${P}_{FskF}={F}_{sk,F}{\dot{z}}_{F}$  
 Stored spring energy from rear suspension  ${P}_{FskF}={F}_{sk,R}{\dot{z}}_{R}$ 
The equations use these variables.
F_{x}  Longitudinal force on vehicle 
F_{z}  Normal force on vehicle 
M_{y}  Torque on vehicle about vehiclefixed yaxis 
F_{wF}, F_{wR}  Longitudinal force on front and rear axles along vehiclefixed xaxis 
F_{d,x}, F_{d,z}  Longitudinal and normal drag force on vehicle CG 
F_{sx,F}, F_{sx,R}  Longitudinal suspension force on front and rear axles 
F_{sz,F}, F_{sz,R}  Normal suspension force on front and rear axles 
F_{g,x},F_{g,z}  Longitudinal and normal gravitational force on vehicle along vehiclefixed frame 
M_{d,y}  Torque due to drag on vehicle about vehiclefixed yaxis 
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 along vehiclefixed zaxis 
Fs_{F}, Fs_{R}  Front and rear axle suspension force along vehiclefixed zaxis 
Z_{wF}, Z_{wR}  Front and rear vehicle normal position along earthfixed zaxis 
Θ  Vehicle pitch angle about vehiclefixed yaxis 
m  Vehicle body mass 
N_{F}, N_{R}  Number of front and rear wheels 
I_{yy}  Vehicle body moment of inertia about the vehiclefixed yaxis 
x, $$\dot{x}$$, $$\ddot{x}$$  Vehicle longitudinal position, velocity, and acceleration along vehiclefixed xaxis 
$$z\text{,}\dot{z}\text{,}\ddot{z}$$  Vehicle normal position, velocity, and acceleration along vehiclefixed zaxis 
Fk_{F}, Fk_{R}  Front and rear wheel suspension stiffness force along vehiclefixed zaxis 
Fb_{F}, Fb_{R}  Front and rear wheel suspension damping force along vehiclefixed zaxis 
Z_{F}, Z_{R}  Front and rear vehicle vertical position along earthfixed Zaxis 
$${\dot{Z}}_{F},{\dot{Z}}_{R}$$  Front and rear vehicle vertical velocity along vehiclefixed zaxis 
$${\overline{Z}}_{F},{\overline{Z}}_{R}$$  Front and rear wheel axle vertical position along vehiclefixed zaxis 
$${\dot{\overline{Z}}}_{F},{\dot{\overline{Z}}}_{R}$$  Front and rear wheel axle vertical velocity along earthfixed zaxis 
dZ_{F}, dZ_{R}  Front and rear axle suspension deflection along vehiclefixed zaxis 
$$d{\dot{Z}}_{F},d{\dot{Z}}_{R}$$  Front and rear axle suspension deflection rate along vehiclefixed zaxis 
C_{d}  Frontal air drag coefficient acting along vehiclefixed xaxis 
C_{l}  Lateral air drag coefficient acting along vehiclefixed zaxis 
C_{pm}  Air drag pitch moment acting about vehiclefixed yaxis 
A_{f}  Frontal area 
P_{abs}  Environmental absolute pressure 
R  Atmospheric specific gas constant 
T  Environmental air temperature 
w_{x}  Wind speed along vehiclefixed xaxis 
[1] Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers, 1992.
[2] Vehicle Dynamics Standards Committee. Vehicle Dynamics Terminology. SAE J670. Warrendale, PA: Society of Automotive Engineers, 2008.
[3] Technical Committee. Road vehicles — Vehicle dynamics and roadholding ability — Vocabulary. ISO 8855:2011. Geneva, Switzerland: International Organization for Standardization, 2011.