Vehicle Body 3DOF Longitudinal

3DOF rigid vehicle body to calculate longitudinal, vertical, and pitch motion

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• Powertrain Blockset / Vehicle Dynamics

Vehicle Dynamics Blockset / Vehicle Body

• Description

The Vehicle Body 3DOF Longitudinal block implements a three degrees-of-freedom (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 input-provided 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 rigid-body 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:

$\begin{array}{l}{F}_{x}={F}_{wF}+{F}_{wR}-{F}_{d,x}-{F}_{sx,F}-{F}_{sx,R}+{F}_{g,x}\\ {F}_{z}={F}_{d,z}-{F}_{sz,F}-{F}_{sz,R}+{F}_{g,z}\\ {M}_{y}=a{F}_{sz,F}-b{F}_{sz,R}+h\left({F}_{wF}+{F}_{wR}+{F}_{sx,F}+{F}_{sx,R}\right)-{M}_{d,y}\end{array}$ Rigid-Body Vehicle Motion

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 axle-longitudinal 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.

$\begin{array}{l}\stackrel{¨}{x}=\frac{{F}_{x}}{m}-qz\\ \stackrel{¨}{z}=\frac{{F}_{z}}{m}-qx\\ \stackrel{˙}{q}=\frac{{M}_{y}}{{I}_{yy}}\\ \stackrel{˙}{\theta }=q\end{array}$

Suspension System Forces

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 end-stops, the tables are functions of the stroke.

$\begin{array}{l}F{k}_{F}=f\left(d{Z}_{F}\right)\\ F{k}_{R}=f\left(d{Z}_{R}\right)\end{array}$

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.

$\begin{array}{l}F{b}_{F}=f\left(d{\stackrel{˙}{Z}}_{F}\right)\\ F{b}_{R}=f\left(d{\stackrel{˙}{Z}}_{R}\right)\end{array}$

The stroke is the difference in the vehicle vertical and axle positions. The stroke rate is the difference in the vertical and axle velocities.

$\begin{array}{l}d{Z}_{F}={Z}_{F}-{\overline{Z}}_{F}\\ d{Z}_{R}={Z}_{R}-{\overline{Z}}_{R}\\ d{\stackrel{˙}{Z}}_{F}={\stackrel{˙}{Z}}_{F}-{\stackrel{˙}{\overline{Z}}}_{F}\\ d{\stackrel{˙}{Z}}_{R}={\stackrel{˙}{Z}}_{R}-{\stackrel{˙}{\overline{Z}}}_{R}\end{array}$

When the Ground interaction type parameter is Grade angle, the axle vertical positions (${\overline{Z}}_{F},{\overline{Z}}_{R}$) and velocities (${\stackrel{˙}{\overline{Z}}}_{F},{\stackrel{˙}{\overline{Z}}}_{R}$) are set to 0.

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:

$\begin{array}{l}{F}_{d,x}=\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{\left(}^{\stackrel{˙}{x}}\\ {F}_{d,z}=\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{\left(}^{\stackrel{˙}{x}}\\ {M}_{d,y}=\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{\left(}^{\stackrel{˙}{x}}\left(a+b\right)\end{array}$

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal DescriptionEquations

PwrInfo

PwrTrnsfrd — Power transferred between blocks

• Positive signals indicate flow into block

• Negative signals indicate flow out of block

PwrFxExt

Externally applied longitudinal force power${P}_{FxExt}={F}_{xExt}\stackrel{˙}{x}$

PwrFzExt

Externally applied longitudinal force power${P}_{FzExt}={F}_{zExt}\stackrel{˙}{z}$

PwrMyExt

Externally applied pitch moment power${P}_{MzExt}={M}_{zExt}\stackrel{˙}{\theta }$

PwrFwFx

Longitudinal force applied at the front axle${P}_{FwFx}={F}_{wF}\stackrel{˙}{x}$

PwrFwRx

Longitudinal force applied at the rear axle${P}_{FwRx}={F}_{wR}\stackrel{˙}{x}$

PwrNotTrnsfrd — Power crossing the block boundary, but not transferred

• Positive signals indicate an input

• Negative signals indicate a loss

PwrFsF

Internal power transferred between suspension and vehicle body at the front axle${P}_{Fs,F}=-{P}_{FwFx}+{P}_{FsbF}+{P}_{Fsk,F}+{F}_{xF}{\stackrel{˙}{x}}_{F}+{F}_{zF}{\stackrel{˙}{z}}_{F}$

PwrFsR

Internal power transferred between suspension and vehicle body at the rear axle${P}_{Fs,R}=-{P}_{FwRx}+{P}_{Fsb,R}+{P}_{Fsk,R}+{F}_{xF}{\stackrel{˙}{x}}_{F}+{F}_{zF}{\stackrel{˙}{z}}_{F}$

PwrFxDrag

Longitudinal drag force power${P}_{d,x}={F}_{d,x}\stackrel{˙}{x}$

PwrFzDrag

Vertical drag force power${P}_{d,z}={F}_{d,z}\stackrel{˙}{z}$

PwrMyDrag

Drag pitch moment power${P}_{d,My}={M}_{d,y}\stackrel{˙}{\theta }$

PwrFsb

Total suspension damping power${P}_{Fsb}=\sum _{i=F,R}^{}{F}_{sb,i}{\stackrel{˙}{z}}_{i}$

PwrStored — Stored energy rate of change

• Positive signals indicate an increase

• Negative signals indicate a decrease

PwrStoredGrvty

Rate change in gravitational potential energy${P}_{g}=-mg\stackrel{˙}{Z}$

PwrStoredxdot

Rate of change of longitudinal kinetic energy${P}_{\stackrel{˙}{x}}=m\stackrel{¨}{x}\stackrel{˙}{x}$

PwrStoredzdot

Rate of change of longitudinal kinetic energy${P}_{\stackrel{˙}{z}}=m\stackrel{¨}{z}\stackrel{˙}{z}$

PwrStoredq

Rate of change of rotational pitch kinetic energy${P}_{\stackrel{˙}{\theta }}={I}_{yy}\stackrel{¨}{\theta }\stackrel{˙}{\theta }$

PwrStoredFsFzSprng

Stored spring energy from front suspension${P}_{FskF}={F}_{sk,F}{\stackrel{˙}{z}}_{F}$

PwrStoredFsRzSprng

Stored spring energy from rear suspension${P}_{FskF}={F}_{sk,R}{\stackrel{˙}{z}}_{R}$

The equations use these variables.

 Fx Longitudinal force on vehicle Fz Normal force on vehicle My Torque on vehicle about the vehicle-fixed y-axis FwF, FwR Longitudinal force on front and rear axles along vehicle-fixed x-axis Fd,x, Fd,z Longitudinal and normal drag force on vehicle CG Fsx,F, Fsx,R Longitudinal suspension force on front and rear axles Fsz,F, Fsz,R Normal suspension force on front and rear axles Fg,x,Fg,z Longitudinal and normal gravitational force on vehicle along the vehicle-fixed frame Md,y Torque due to drag on vehicle about the vehicle-fixed y-axis 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 vehicle-fixed z-axis FsF, FsR Front and rear axle suspension force along vehicle-fixed z-axis ZwF, ZwR Front and rear vehicle normal position along earth-fixed z-axis Θ Vehicle pitch angle about the vehicle-fixed y-axis m Vehicle body mass NF, NR Number of front and rear wheels Iyy Vehicle body moment of inertia about the vehicle-fixed y-axis x, $\stackrel{˙}{x}$, $\stackrel{¨}{x}$ Vehicle longitudinal position, velocity, and acceleration along the vehicle-fixed x-axis Vehicle normal position, velocity, and acceleration along the vehicle-fixed z-axis FkF, FkR Front and rear wheel suspension stiffness force along vehicle-fixed z-axis FbF, FbR Front and rear wheel suspension damping force along vehicle-fixed z-axis ZF, ZR Front and rear vehicle vertical position along earth-fixed Z-axis ${\stackrel{˙}{Z}}_{F},{\stackrel{˙}{Z}}_{R}$ Front and rear vehicle vertical velocity along vehicle-fixed z-axis ${\overline{Z}}_{F},{\overline{Z}}_{R}$ Front and rear wheel axle vertical position along vehicle-fixed z-axis ${\stackrel{˙}{\overline{Z}}}_{F},{\stackrel{˙}{\overline{Z}}}_{R}$ Front and rear wheel axle vertical velocity along earth-fixed z-axis dZF, dZR Front and rear axle suspension deflection along vehicle-fixed z-axis $d{\stackrel{˙}{Z}}_{F},d{\stackrel{˙}{Z}}_{R}$ Front and rear axle suspension deflection rate along vehicle-fixed z-axis Cd Frontal air drag coefficient acting along the vehicle-fixed x-axis Cl Lateral air drag coefficient acting along the vehicle-fixed z-axis Cpm Air drag pitch moment acting about the vehicle-fixed y-axis Af Frontal area Pabs Environmental absolute pressure R Atmospheric specific gas constant T Environmental air temperature wx Wind speed along the vehicle-fixed x-axis

Ports

Input

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External forces applied to vehicle CG, Fxext, Fyext, Fzext, in vehicle-fixed frame, in N. Signal vector dimensions are [1x3] or [3x1].

Dependencies

To enable this port, select External forces.

External moment about vehicle CG, Mx, My, Mz, in the vehicle-fixed frame, in N·m. Signal vector dimensions are [1x3] or [3x1].

Dependencies

To enable this port, select External moments.

Longitudinal force on the front axle, FwF, along vehicle-fixed x-axis, in N.

Longitudinal force on the rear axle, FwR, along vehicle-fixed x-axis, in N.

Road grade angle, $\gamma$, in deg.

Suspension force on front axle, FsF, along the vehicle-fixed z-axis, in N.

Dependencies

To enable this port, for the Ground interaction type parameter, select External suspension.

Suspension force on rear axle, FsR, along the vehicle-fixed z-axis, in N.

Dependencies

To enable this port, for the Ground interaction type parameter, select External suspension.

Wind speed, WX, WY, WZ along earth-fixed X-, Y-, and Z-axes, in m/s. Signal vector dimensions are [1x3] or [3x1].

Ambient air temperature, Tair, in K. Considering this option if you want to vary the temperature during run-time.

Dependencies

To enable this port, select Air temperature.

Forward and rear axle positions along the vehicle-fixed z-axis, ${\overline{Z}}_{F},{\overline{Z}}_{R}$, in m.

Dependencies

To enable this port, for the Ground interaction type parameter, select Axle displacement, velocity.

Forward and rear axle velocities along the vehicle-fixed z-axis, ${\stackrel{˙}{\overline{Z}}}_{F},{\stackrel{˙}{\overline{Z}}}_{R}$, in m/s.

Dependencies

To enable this port, for the Ground interaction type parameter, select Axle displacement, velocity.

Output

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Bus signal containing these block values.

SignalDescriptionValueUnits
InertFrmCgDispXVehicle CG displacement along earth-fixed X-axis

Computed

m
YVehicle CG displacement along earth-fixed Y-axis0

m

ZVehicle CG displacement along earth-fixed Z-axis

Computed

m
VelXdotVehicle CG velocity along earth-fixed X-axis

Computed

m/s

YdotVehicle CG velocity along earth-fixed Y-axis0m/s
ZdotVehicle CG velocity along earth-fixed Z-axis

Computed

m/s
thetaRotation of vehicle-fixed frame about the earth-fixed Y-axis (pitch)

Computed

FrntAxlDispXFront axle displacement along the earth-fixed X-axis

Computed

m
YFront axle displacement along the earth-fixed Y-axis0m
ZFront axle displacement along the earth-fixed Z-axis

Computed

m
VelXdotFront axle velocity along the earth-fixed X-axis

Computed

m/s
YdotFront axle velocity along the earth-fixed Y-axis0m/s
ZdotFront axle velocity along the earth-fixed Z-axis

Computed

m/s
RearAxlDispXRear axle displacement along the earth-fixed X-axis

Computed

m
YRear axle displacement along the earth-fixed Y-axis0m
ZRear axle displacement along the earth-fixed Z-axis

Computed

m
VelXdotRear axle velocity along the earth-fixed X-axis

Computed

m/s
YdotRear axle velocity along the earth-fixed Y-axis0m/s
ZdotRear axle velocity along the earth-fixed Z-axis

Computed

m/s
BdyFrmCgDispxVehicle CG displacement along the vehicle-fixed x-axis

Computed

m
yVehicle CG displacement along the vehicle-fixed y-axis0m
zVehicle CG displacement along the vehicle-fixed z-axisComputedm
VelxdotVehicle CG velocity along the vehicle-fixed x-axis

Computed

m/s
ydotVehicle CG velocity along the vehicle-fixed y-axis0m/s
zdotVehicle CG velocity along the vehicle-fixed z-axisComputedm/s
qVehicle angular velocity about the vehicle-fixed y-axis (pitch rate)

Computed

AccelaxVehicle CG acceleration along the vehicle-fixed x-axis

Computed

gn
ayVehicle CG acceleration along the vehicle-fixed y-axis0gn
azVehicle CG acceleration along the vehicle-fixed z-axis

Computed

gn
ForcesBodyFxNet force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyNet force on vehicle CG along the vehicle-fixed y-axis0N
FzNet force on vehicle CG along the vehicle-fixed z-axis

Computed

N
ExtFxExternal force on vehicle CG along the vehicle-fixed x-axisComputedN
FyExternal force on vehicle CG along the vehicle-fixed y-axisComputedN
FzExternal force on vehicle CG along the vehicle-fixed z-axisComputedN
FrntAxlFx

Longitudinal force on front axle, along the vehicle-fixed x-axis

Computed

N
Fy

Lateral force on front axle, along the vehicle-fixed y-axis

0N
Fz

Normal force on front axle, along the vehicle-fixed z-axis

ComputedN
RearAxlFx

Longitudinal force on rear axle, along the vehicle-fixed x-axis

Computed

N
Fy

Lateral force on rear axle, along the vehicle-fixed y-axis

0N
Fz

Normal force on rear axle, along the vehicle-fixed z-axis

ComputedN
TiresFrntTireFx

Front tire force, along the vehicle-fixed x-axis

0N
Fy

Front tire force, along the vehicle-fixed y-axis

0N
Fz

Front tire force, along the vehicle-fixed z-axis

ComputedN
RearTireFx

Rear tire force, along the vehicle-fixed x-axis

0N
Fy

Rear tire force, along the vehicle-fixed y-axis

0N
Fz

Rear tire force, along the vehicle-fixed z-axis

ComputedN
DragFxDrag force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyDrag force on vehicle CG along the vehicle-fixed y-axis

Computed

N
FzDrag force on vehicle CG along the vehicle-fixed z-axis

Computed

N
GrvtyFxGravity force on vehicle CG along the vehicle-fixed x-axis

Computed

N
FyGravity force on vehicle CG along the vehicle-fixed y-axis0N
FzGravity force on vehicle CG along the vehicle-fixed z-axis

Computed

N
MomentsBodyMxBody moment on vehicle CG about the vehicle-fixed x-axis0 N·m
MyBody moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
MzBody moment on vehicle CG about the vehicle-fixed z-axis0 N·m
DragMxDrag moment on vehicle CG about the vehicle-fixed x-axis0 N·m
MyDrag moment on vehicle CG about the vehicle-fixed y-axis

Computed

N·m
MzDrag moment on vehicle CG about the vehicle-fixed z-axis0 N·m
ExtFxExternal moment on vehicle CG about the vehicle-fixed x-axisComputedN·m
FyExternal moment on vehicle CG about the vehicle-fixed y-axisComputedN·m
FzExternal moment on vehicle CG about the vehicle-fixed z-axisComputedN·m
FrntAxlDispxFront axle displacement along the vehicle-fixed x-axis

Computed

m
yFront axle displacement along the vehicle-fixed y-axis0m
zFront axle displacement along the vehicle-fixed z-axis

Computed

m
VelxdotFront axle velocity along the vehicle-fixed x-axis

Computed

m/s
ydotFront axle velocity along the vehicle-fixed y-axis0m/s
zdotFront axle velocity along the vehicle-fixed z-axis

Computed

m/s
SteerWhlAngFL

Front left wheel steering angle

Computed

WhlAngFR

Front right wheel steering angle

Computed

RearAxlDispxRear axle displacement along the vehicle-fixed x-axis

Computed

m
yRear axle displacement along the vehicle-fixed y-axis0m
zRear axle displacement along the vehicle-fixed z-axis

Computed

m
VelxdotRear axle velocity along the vehicle-fixed x-axis

Computed

m/s
ydotRear axle velocity along the vehicle-fixed y-axis0m/s
zdotRear axle velocity along the vehicle-fixed z-axis

Computed

m/s
SteerWhlAngRL

Rear left wheel steering angle

Computed

WhlAngRR

Rear right wheel steering angle

Computed

PwrPwrExtApplied external power

Computed

W
DragPower loss due to drag

Computed

W
PwrInfo

PwrTrnsfrd

PwrFxExt

Externally applied longitudinal force power

Computed

W

PwrFzExt

Externally applied longitudinal force power

Computed

W

PwrMyExt

Externally applied pitch moment power

Computed

W

PwrFwFx

Longitudinal force applied at the front axle

Computed

W

PwrFwRx

Longitudinal force applied at the rear axle

Computed

W

PwrNotTrnsfrd

PwrFsF

Internal power transferred between suspension and vehicle body at the front axle

Computed

W

PwrFsR

Internal power transferred between suspension and vehicle body at the rear axle

Computed

W

PwrFxDrag

Longitudinal drag force power

Computed

W

PwrFzDrag

Vertical drag force power

Computed

W

PwrMyDrag

Drag pitch moment power

Computed

W

PwrFsb

Total suspension damping power

Computed

W

PwrStored

PwrStoredGrvty

Rate change in gravitational potential energy

Computed

W

PwrStoredxdot

Rate of change of longitudinal kinetic energy

Computed

W

PwrStoredzdot

Rate of change of longitudinal kinetic energy

Computed

W

PwrStoredq

Rate of change of rotational pitch kinetic energy

Computed

W

PwrStoredFsFzSprng

Stored spring energy from front suspension

Computed

W

PwrStoredFsRzSprng

Stored spring energy from rear suspension

Computed

W

Vehicle CG velocity along the vehicle-fixed x-axis, in m/s.

Normal force on front axle, FzF, along the vehicle-fixed z-axis, in N.

Normal force on rear axle, FzR, along the vehicle-fixed z-axis, in N.

Parameters

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Options

Specify to create input port FExt.

Specify to create input port MExt.

Specify to create input port AirTemp.

Longitudinal

Number of wheels on front axle, NF. The value is dimensionless.

Number of wheels on rear axle, NR. The value is dimensionless.

Vehicle mass, m, in kg.

Horizontal distance a from the vehicle CG to the front wheel axle, in m.

Horizontal distance b from the vehicle CG to the rear wheel axle, in m.

Height of vehicle CG above the axles, h, in m.

Air drag coefficient, Cd. The value is dimensionless.

Effective vehicle cross-sectional area, Af to calculate the aerodynamic drag force on the vehicle, in m^2.

Vehicle body longitudinal initial position along earth-fixed x-axis, xo, in m.

Vehicle body longitudinal initial velocity along earth-fixed x-axis, ${\stackrel{˙}{x}}_{0}$, in m/s.

Vertical

Lift coefficient, Cl. The value is dimensionless.

Initial vertical CG position, zo, along the vehicle-fixed z-axis, in m.

Initial vertical CG velocity, zdoto, along the vehicle-fixed z-axis, in m.

Pitch

Vehicle body moment of inertia about body z-axis.

Pitch drag moment coefficient. The value is dimensionless.

Suspension

Front axle stiffness force data, FkF, in N.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Front axle displacement data, in m.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Front axle damping force, in N.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Front axle velocity data, in m/s.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Rear axle stiffness force data, in N.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Rear axle displacement data, in m.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Rear axle damping force, in N.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Rear axle velocity data, in m/s.

Dependencies

To enable this parameter, for the Ground interaction type parameter, select Grade angle or Axle displacement, velocity.

Environment

Environmental air absolute pressure, Pabs, in Pa.

Ambient air temperature, Tair, in K.

Dependencies

To enable this parameter, clear Air temperature.

Gravitational acceleration, g, in m/s2.

 Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers, 1992.

 Vehicle Dynamics Standards Committee. Vehicle Dynamics Terminology. SAE J670. Warrendale, PA: Society of Automotive Engineers, 2008.

 Technical Committee. Road vehicles — Vehicle dynamics and road-holding ability — Vocabulary. ISO 8855:2011. Geneva, Switzerland: International Organization for Standardization, 2011.

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