# Vehicle Body 1DOF Longitudinal

Two-axle 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 degree-of-freedom (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

FExt

External force applied to vehicle CG in the vehicle-fixed frame.

External moments

MExt

External moment about vehicle CG in the vehicle-fixed frame.

Air temperature

AirTemp

Ambient air temperature. Consider this option if you want to vary the temperature during run-time.

Wind X,Y,Z

WindXYZ

Wind speed along earth-fixed X-, Y-, and Z-axes.

If you do not select this option, the block implements input port WindX — Longitudinal wind speed along the earth-fixed X-axis.

### 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 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). 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.

$\begin{array}{l}{F}_{b}=m\stackrel{¨}{x}\\ \\ {F}_{b}={F}_{xF}+{F}_{xR}-{F}_{d,x}+{F}_{ext,x}-mg\mathrm{sin}\gamma \end{array}$

Zero normal acceleration and zero pitch torque determine the normal force on each front and rear axles.

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.

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

By default, to calculate the wind speed along the vehicle-fixed x-axis, the block uses the longitudinal wind speed along the earth-fixed X-axis. If you select WindX,Y,Z, the block uses the wind speed along the earth-fixed X-, Y-, Z-axes.

### 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 force power

${P}_{FxExt}={F}_{xExt}\stackrel{˙}{x}$

PwrFwFx

Longitudinal force power applied at the front axle

${P}_{FwFx}={F}_{wF}\stackrel{˙}{x}$

PwrFwRx

Longitudinal force power 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

PwrFxDrag

Drag force power

${P}_{d}=-\frac{0.5{C}_{d}{A}_{f}{P}_{abs}{\left({\stackrel{˙}{x}}^{2}-{w}_{x}\right)}^{2}}{287.058T}\stackrel{˙}{x}$

PwrStored — Stored energy rate of change

• Positive signals indicate an increase

• Negative signals indicate a decrease

wrStoredGrvty

Rate change in gravitational potential energy

${P}_{g}=-mg\stackrel{˙}{Z}$

PwrStoredxdot

Rate in change of longitudinal kinetic energy

${P}_{\stackrel{˙}{x}}=m\stackrel{¨}{x}\stackrel{˙}{x}$

The equations use these variables.

 Fxf, Fxr Longitudinal forces on each wheel at the front and rear ground contact points, respectively Fzf, Fzr Normal load forces on each wheel at the front and rear ground contact points, respectively FwF, FwR Longitudinal force on front and rear axles along vehicle-fixed x-axis FxExt, FwR External force along the vehicle-fixed x-axis Fd,x, Fd,z Longitudinal and normal drag force on vehicle CG Md,y Torque due to drag on vehicle about the vehicle-fixed y-axis Fd Aerodynamic drag force Vx Velocity of the vehicle. When Vx > 0, the vehicle moves forward. When Vx < 0, the vehicle moves backward. Nf, Nr 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 Cd Frontal air drag coefficient Af Frontal area Pabs Absolute pressure ρ Mass density of air x, $\stackrel{˙}{x}$, $\stackrel{¨}{x}$ Vehicle longitudinal position, velocity, and acceleration along the vehicle-fixed x-axis wx Wind speed along the vehicle-fixed x-axis $\stackrel{˙}{Z}$ Vehicle vertical velocity along the vehicle-fixed z-axis

## 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 degrees-of-freedom (DOF), use the Vehicle Body 3DOF Longitudinal.

## Ports

### Input

expand all

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, Fxf, 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.

Longitudinal wind speed, Ww, along earth-fixed X-axis, in m/s.

#### Dependencies

To enable this port, clear Wind X,Y,Z components.

Wind speed, Ww, WwY, WwZ along inertial X-, Y-, and Z-axes, in m/s. Signal vector dimensions are [1x3] or [3x1].

#### Dependencies

To enable this port, select Wind X,Y,Z components.

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.

### 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)

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-axis0m
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-axis0m/s
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-axis0gn
ForcesBodyFxNet force on vehicle CG along the vehicle-fixed x-axis0N
FyNet force on vehicle CG along the vehicle-fixed y-axis0N
FzNet force on vehicle CG along the vehicle-fixed z-axis0N
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

0N
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

0N
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
MomentsBodyMxNet moment on vehicle CG about the vehicle-fixed x-axis0N·m
MyNet moment on vehicle CG about the vehicle-fixed y-axis0N·m
MzNet moment on vehicle CG about the vehicle-fixed z-axis0N·m
DragMxDrag moment on vehicle CG about the vehicle-fixed x-axis

Computed

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-axis

Computed

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 force power

Computed

W

PwrFwFx

Longitudinal force power applied at the front axleComputedW

PwrFwRx

Longitudinal force power applied at the rear axleComputedW

PwrNotTrnsfrd

PwrFxDrag

Drag force power

ComputedW

PwrStored

wrStoredGrvty

Rate change in gravitational potential energyComputedW

PwrStoredxdot

Rate in change of longitudinal kinetic energyComputedW

Vehicle body longitudinal velocity along the earth-fixed reference frame X-axis, in m/s.

Normal load force on the front axle, Fzf, along 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.

Specify to create input port WindXYZ.

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 ground, h, in m.

Air drag coefficient, Cd. The value is dimensionless.

Air lift coefficient, Cl. The value is dimensionless.

Pitch drag moment coefficient, Cpm. The value is dimensionless.

Effective vehicle cross-sectional area, A, to calculate the aerodynamic drag force on the vehicle, in m2.

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

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

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.