# Vehicle Body 1DOF Longitudinal

Two-axle vehicle in forward and reverse motion

Libraries:
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

expand all

Bus signal containing these block values.

SignalDescriptionValueUnits
`InertFrm``Cg``Disp``X`Vehicle CG displacement along earth-fixed `X`-axis

Computed

m
`Y`Vehicle CG displacement along earth-fixed `Y`-axis`0`

m

`Z`Vehicle CG displacement along earth-fixed `Z`-axis

Computed

m
`Vel``Xdot`Vehicle CG velocity along earth-fixed `X`-axis

Computed

m/s

`Ydot`Vehicle CG velocity along earth-fixed `Y`-axis`0`m/s
`Zdot`Vehicle CG velocity along earth-fixed `Z`-axis

Computed

m/s
`Ang``phi`Rotation of vehicle-fixed frame about the earth-fixed `X`-axis (roll)`0`rad
`theta`Rotation of vehicle-fixed frame about the earth-fixed `Y`-axis (pitch)

`psi`Rotation of vehicle-fixed frame about the earth-fixed `Z`-axis (yaw)`0`rad
`FrntAxl``Disp``X`Front axle displacement along the earth-fixed `X`-axis

Computed

m
`Y`Front axle displacement along the earth-fixed `Y`-axis`0`m
`Z`Front axle displacement along the earth-fixed `Z`-axis

Computed

m
`Vel``Xdot`Front axle velocity along the earth-fixed `X`-axis

Computed

m/s
`Ydot`Front axle velocity along the earth-fixed `Y`-axis`0`m/s
`Zdot`Front axle velocity along the earth-fixed `Z`-axis

Computed

m/s
`RearAxl``Disp``X`Rear axle displacement along the earth-fixed `X`-axis

Computed

m
`Y`Rear axle displacement along the earth-fixed `Y`-axis`0`m
`Z`Rear axle displacement along the earth-fixed `Z`-axis

Computed

m
`Vel``Xdot`Rear axle velocity along the earth-fixed `X`-axis

Computed

m/s
`Ydot`Rear axle velocity along the earth-fixed `Y`-axis`0`m/s
`Zdot`Rear axle velocity along the earth-fixed `Z`-axis

Computed

m/s
`BdyFrm``Cg``Disp``x`Vehicle CG displacement along the vehicle-fixed `x`-axis

Computed

m
`y`Vehicle CG displacement along the vehicle-fixed `y`-axis`0`m
`z`Vehicle CG displacement along the vehicle-fixed `z`-axis`0`m
`Vel``xdot`Vehicle CG velocity along the vehicle-fixed `x`-axis

Computed

m/s
`ydot`Vehicle CG velocity along the vehicle-fixed `y`-axis`0`m/s
`zdot`Vehicle CG velocity along the vehicle-fixed `z`-axis`0`m/s
`AngVel``p`Vehicle angular velocity about the vehicle-fixed `x`-axis (roll rate)`0`rad/s
`q`Vehicle angular velocity about the vehicle-fixed y-axis (pitch rate)`0`rad/s
`r`Vehicle angular velocity about the vehicle-fixed z-axis (yaw rate)`0`rad/s
`Accel``ax`Vehicle CG acceleration along the vehicle-fixed `x`-axis

Computed

gn
`ay`Vehicle CG acceleration along the vehicle-fixed `y`-axis`0`gn
`az`Vehicle CG acceleration along the vehicle-fixed `z`-axis`0`gn
`Forces``Body``Fx`Net force on vehicle CG along the vehicle-fixed `x`-axis`0`N
`Fy`Net force on vehicle CG along the vehicle-fixed `y`-axis`0`N
`Fz`Net force on vehicle CG along the vehicle-fixed `z`-axis`0`N
`Ext``Fx`External force on vehicle CG along the vehicle-fixed `x`-axis`Computed`N
`Fy`External force on vehicle CG along the vehicle-fixed `y`-axis`Computed`N
`Fz`External force on vehicle CG along the vehicle-fixed `z`-axis`Computed`N
`FrntAxl``Fx`

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

`0`N
`Fy`

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

`0`N
`Fz`

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

ComputedN
`RearAxl``Fx`

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

`0`N
`Fy`

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

`0`N
`Fz`

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

ComputedN
`Tires``FrntTire``Fx`

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

`0`N
`Fy`

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

`0`N
`Fz`

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

ComputedN
`RearTire``Fx`

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

`0`N
`Fy`

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

`0`N
`Fz`

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

ComputedN
`Drag``Fx`Drag force on vehicle CG along the vehicle-fixed `x`-axis

Computed

N
`Fy`Drag force on vehicle CG along the vehicle-fixed `y`-axis

Computed

N
`Fz`Drag force on vehicle CG along the vehicle-fixed `z`-axis

Computed

N
`Grvty``Fx`Gravity force on vehicle CG along the vehicle-fixed `x`-axis

Computed

N
`Fy`Gravity force on vehicle CG along the vehicle-fixed `y`-axis`0`N
`Fz`Gravity force on vehicle CG along the vehicle-fixed `z`-axis

Computed

N
`Moments``Body``Mx`Net moment on vehicle CG about the vehicle-fixed `x`-axis`0`N·m
`My`Net moment on vehicle CG about the vehicle-fixed `y`-axis`0`N·m
`Mz`Net moment on vehicle CG about the vehicle-fixed `z`-axis`0`N·m
`Drag``Mx`Drag moment on vehicle CG about the vehicle-fixed `x`-axis

Computed

N·m
`My`Drag moment on vehicle CG about the vehicle-fixed `y`-axis

Computed

N·m
`Mz`Drag moment on vehicle CG about the vehicle-fixed `z`-axis

Computed

N·m
`Ext``Fx`External moment on vehicle CG about the vehicle-fixed `x`-axis`Computed`N·m
`Fy`External moment on vehicle CG about the vehicle-fixed `y`-axis`Computed`N·m
`Fz`External moment on vehicle CG about the vehicle-fixed `z`-axis`Computed`N·m
`FrntAxl``Disp``x`Front axle displacement along the vehicle-fixed `x`-axis

Computed

m
`y`Front axle displacement along the vehicle-fixed `y`-axis`0`m
`z`Front axle displacement along the vehicle-fixed `z`-axis

Computed

m
`Vel``xdot`Front axle velocity along the vehicle-fixed `x`-axis

Computed

m/s
`ydot`Front axle velocity along the vehicle-fixed `y`-axis`0`m/s
`zdot`Front axle velocity along the vehicle-fixed `z`-axis

Computed

m/s
`Steer``WhlAngFL`

Front left wheel steering angle

Computed

`WhlAngFR`

Front right wheel steering angle

Computed

`RearAxl``Disp``x`Rear axle displacement along the vehicle-fixed `x`-axis

Computed

m
`y`Rear axle displacement along the vehicle-fixed `y`-axis`0`m
`z`Rear axle displacement along the vehicle-fixed `z`-axis

Computed

m
`Vel``xdot`Rear axle velocity along the vehicle-fixed `x`-axis

Computed

m/s
`ydot`Rear axle velocity along the vehicle-fixed `y`-axis`0`m/s
`zdot`Rear axle velocity along the vehicle-fixed `z`-axis

Computed

m/s
`Steer``WhlAngRL`

Rear left wheel steering angle

Computed

`WhlAngRR`

Rear right wheel steering angle

Computed

`Pwr``PwrExt`Applied external power

Computed

W
`Drag`Power 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 vehicle-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

expand all

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.

## Version History

Introduced in R2017a