Three-Axle Vehicle Body

Vehicle body with three axles

Since R2026a

Libraries:
Simscape / Driveline / Tires & Vehicles

Description

The Three-Axle Vehicle Body block represents a three-axle vehicle in longitudinal motion. You can specify the number of wheels for each axle. The block accounts for body mass, aerodynamic drag, road incline, pitch and heave, and weight distribution between axles due to acceleration and road profile. The vehicle mass and center of gravity remain constant.

Vehicle Model

The vehicle axles are parallel and form a plane. The local frame moves with the center of gravity. The x-axis is tangential to the road surface in the direction of forward motion, while the y-axis points to the center of gravity.

If the vehicle travels on an inclined road, the normal direction, y, is not parallel to gravity but is always perpendicular to the axle-longitudinal plane. θ is the relative pitch angle between the road inclination angle, β, and the angle of the vehicle. p = θ+β, where p is the pitch angle, measured against the X-axis of the world frame.

The y-axis of the local frame that defines the vehicle vertical position is normal to the road surface and points to the center of gravity of the vehicle.

Equations

The block balances force in the x and y-directions, such that

$\begin{array}{l} m {\ddot{x}}_{C G} = 0 = F_{r o a d} - F_{d r a g} - m g \sin (β) - m \dot{v} \\ m {\ddot{y}}_{C G} = (F_{f} + F_{m} + F_{r}) - m g \cos (β) - m \dot{v} β \end{array}$

where:

m is the mass of the vehicle.
x_CG and y_CG are the positions of the center of gravity along the x-axis and y-axis, respectively.
F_road is the force of the road acting on the vehicle in the x-direction.
F_drag is the force due to drag acting on the vehicle in the x-direction.
g is the gravitational acceleration.
v is the vehicle velocity tangential to the road surface.
F_f, F_m, and F_r are the forces applied to the front, middle, and rear suspension in the y-direction, respectively.

The block defines the horizontal locations of the center of gravity (CG), middle axis, and rear axle with respect to the location of the front axle. The figure shows the vehicle body with and without suspension deformation.

The left figure shows the vehicle body without any suspension deformations, and the right figure shows the vehicle body with arbitrary suspension deformation.

The block balances torque about the CG, such that

$τ_{C G} = \ddot{I} p = L_{C G} F_{f} - (L_{m} - L_{C G}) F_{m} - (L_{r} - L_{C G}) F_{r} + (y_{c} - s) F_{r o a d},$

where:

I is the pitch moment of inertia of the vehicle.
L_CG is the distance from the front axle to the CG along the x-axis.
L_m is the distance from the front axle to the middle axle along the x-axis.
L_r is the distance from the front axle to the rear axle along the x-axis.
y_c is the height of the CG with no deformation.
s is the change in position of the CG due to suspension deformation, such that s = y_c-y_CG.

The block defines the forces at the front, middle, and rear suspensions, such that

$\begin{array}{l} F_{f} = F_{f, s u s p e n s i o n} (_{s_{f}, {\dot{s}}_{f}) +} F_{f, h a r d s t o p} (s_{f}, {\dot{s}}_{f}) \\ F_{m} = F_{m, s u s p e n s i o n} (_{s_{m}, {\dot{s}}_{m}) +} F_{m, h a r d s t o p} (s_{m}, {\dot{s}}_{m}) \\ F_{r} = F_{r, s u s p e n s i o n} (_{s_{r}, {\dot{s}}_{r}) +} F_{r, h a r d s t o p} (s_{r}, {\dot{s}}_{r}) \end{array}$

where:

F_f,suspension, F_m,suspension, F_r,suspension are the forces transmitted by the front, middle, and rear suspension, respectively.
F_f,hardstop, F_m,hardstop, and F_r,hardstop are the forces applied by the hard stops at the front, middle, and rear suspensions, respectively.

The figure illustrates the geometric relationships.

The block leverages the small angle and rigid body assumptions to simplify calculations, such that

$\begin{array}{l} θ = \frac{s_{r} - s_{f}}{L_{r}} \\ \frac{s_{r} - s_{m}}{L_{r} - L_{m}} = \frac{s_{r} - s_{f}}{L_{r}} \end{array}$

The block calculates the height change of the CG due to deformation, such that

$s = s_{r} - θ (L_{r} - L_{C G}) .$

When you select the Enable hard stops parameter, the block uses a subcomponent implementation of the Translational Hard Stop block, where the setting of the Hard stop model parameter is Stiffness and damping applied smoothly through transition region, damped rebound.

Ports

Input

expand all

W — Headwind speed, m/s
physical signal

Physical signal input port associated with the headwind speed, in m/s.

CD — Drag coefficient, unitless
physical signal

Physical signal input port associated with the drag coefficient.

Dependencies

To enable this port, select Variable Drag Coefficient.

β — Road incline angle, rad
physical signal

Physical signal input port associated with the road incline angle, β, in rad.

Output

expand all

V — Longitudinal velocity, m/s
physical signal

Physical signal output port associated with the longitudinal velocity of the vehicle, in m/s.

NF — Front axle normal force, N
physical signal

Physical signal output port associated with the front axle normal force, in N.

NM — Middle axle normal force, N
physical signal

Physical signal output port associated with the middle axle normal force, in N.

NR — Rear axle normal force, N
physical signal

Physical signal output port associated with the rear axle normal force, in N.

Conserving

expand all

H — Horizontal motion
mechanical translational

Mechanical translational conserving port associated with the horizontal motion of the vehicle.

Parameters

expand all

Main

Mass — Vehicle mass
`5000` `kg` (default) | positive scalar

Mass of the vehicle.

Number of wheels per axle [front, middle, rear] — Wheel count
`[2, 4, 4]` (default) | three-element vector

Number of wheels on the front, middle, and rear axles. From left to right, the elements represent the number of wheels for the front, middle, and rear axles. For example, if the input is the array [2,4,4], then the front axle has two wheels, and the middle and rear axles have four wheels each.

Horizontal distance from front axle to CG — Center of gravity location
`3.0` `m` (default) | nonnegative scalar

Location of the center of gravity with respect to the front axle, LC_CG.

Horizontal distance from front axle to middle axle — Middle axle location
`4.0` `m` (default) | positive scalar

Location of the middle axle with respect to the front axle, L_m.

Horizontal distance from front axle to rear axle — Rear axle location
`5.0` `m` (default) | positive scalar

Location of the rear axle with respect to the front axle, L_r.

CG height with no suspension deformation — Distance between CG and ground
`1.5` `m` (default) | scalar

Distance, y_c, between the center of gravity of the vehicle and the ground.

Gravitational acceleration — Gravitational acceleration
`9.81` `m/s^2` (default) | nonnegative scalar

Acceleration due to gravitational force acting at the center of gravity of the vehicle.

Negative normal force warning — Option to generate warning for negative normal force
`off` (default) | `on`

Option to generate a warning for negative normal force.

Input filtering time constant — Time constant for input filtering
`1e-6` `s` (default) | positive scalar

Time constant to filter inputs.

Drag

Frontal area — Effective cross-sectional area
`8` `m^2` (default) | nonnegative scalar

Effective cross-sectional area, A, presented by the vehicle in longitudinal motion. The block uses this value to calculate the aerodynamic drag force on the vehicle.

Variable drag coefficient — Variable drag option
`off` (default) | `on`

Whether the drag coefficient is variable. Enable this parameter to expose port CD.

Drag coefficient — Aerodynamic drag coefficient
`0.6` (default) | nonnegative scalar

Aerodynamic drag coefficient, C_D. The block uses this value to calculate the aerodynamic drag force on the vehicle.

Dependencies

To enable this parameter, clear the Variable Drag Coefficient parameter.

Air density — Ambient air density
`1.18` `kg/m^3` (default) | nonnegative scalar

Density of the air that surrounds the vehicle.

Pitch

Pitch moment of inertia — Pitch moment of inertia
`15000` `kg*m^2` (default) | positive scalar

Pitch moment of inertia of the vehicle, I.

Suspension model — Suspension parameterization method
`Linear` (default) | `By table lookup`

Parameterization method to use to model the suspension. The setting determines how the block calculates F_f,suspension, F_m,suspension, and F_r,suspension. When you select Linear, you specify the front, middle, and rear stiffness and damping coefficients by using scalar values. When you select By table lookup, you specify the stiffness and damping forces and suspension deformation values as vectors.

Front suspension stiffness — Front suspension stiffness
`1e5` `N/m` (default) | positive scalar

Suspension stiffness at the front axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Front suspension damping — Front suspension damping
`1e4` `N*s/m` (default) | positive scalar

Suspension damping at the front axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Middle suspension stiffness — Middle suspension stiffness
`1e5` `N/m` (default) | positive scalar

Suspension stiffness at the middle axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Middle suspension damping — Middle suspension damping
`1e4` `N*s/m` (default) | positive scalar

Suspension damping at the middle axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Rear suspension stiffness — Rear suspension stiffness
`1e5` `N/m` (default) | positive scalar

Suspension stiffness at the rear axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Rear suspension damping — Rear suspension damping
`1e4` `N*s/m` (default) | positive scalar

Suspension damping at the rear axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Front suspension stiffness force vector — Front suspension stiffness force
`[-200000, -100000, 0, 100000, 200000]` `N` (default) | vector

Stiffness force at the front suspension. Each element in the vector corresponds to the elements in the Front suspension deformation vector parameter.