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Tire

Linear and Magic Formula tire model component

Note:  This component is used with the Longitudinal Vehicle Dynamics component.

 

 

The Tire component models the longitudinal dynamics of a two tires together with the wheels and axle. The tire longitudinal forces are modeled either by a linear formulation (with saturation) or by the Magic formula. Wheel/axle inertia is included but no compliance is considered.

 

Equations

The tire translation and rotation velocities are given as

 

vx = ⅆⅆ tsx 

 

ω = ⅆⅆ tϕ  ,    α = ⅆⅆ tω 

 

 

Where  sx  is the tire translation along the road and ϕ  is the tire rotation.

 

The equation of motion is

 

 

J · α  = τ  f · r

 

Where  J  is the tire inertia, τ  is the axle torque,  f  is the traction/brake force, and r  is the tire radius.    

 

The slip ratio is approximated by

 

 

k = r · ω  vxmax  r · ω, vx, ϵv

 

   

Where  ϵv  is a small positive number added to avoid singularity.

 

The tire contact force, f, is calculated as

 

 

 

 

 

f  &equals;  &lcub;0               fz <0 &lcub;fz · Fx0Fz0 · kk0k< k0 fz · Fx0Fz0 · kkk k0            formulation &equals; Linear K·fz · D · sinC · tan1B · k  E · &lpar;B · k  tan1B · k&rpar;&plus;Sv                formulation &equals; Magic

 

where parameters K and Sv are added to the standard Magic formula to adjust the tire force and facilitate fixed step integration, respectively. Also,

 

fx &plus; f &equals; 0

 

Connections 

Name

Condition

Description

ID

drive shaft

-

Drive shaft rotational flange

driveshaft

flangex

-

Axle longitudinal flange

flange_x

fz

-

Real input port for axle vertical force

f_z

 

Parameters

Symbol

Condition

Default

Units

Description

ID

formulation

Linear

Defines the tire formulation type:

• 

Linear: linear formulation with saturation

• 

Magic: Magic formula

tiremode

ϵv

0.1

ms

A small constant added to the denominator of the slip ratio to prevent singularity

epsilon_v

Fz0

formulation &equals; Linear

1500

N

Nominal vertical load

Fz0

Fx0

formulation &equals; Linear

2000

N

Traction force at nominal vertical load

Fx0

k0

formulation &equals; Linear

0.15

Slip ratio at peak traction force

sr0

J

2

kg·m2

Inertia of the two wheels and the axle

J

r

0.35

m

Wheel radius

r

B&comma;C&comma;D&comma;E

formulation &equals; Magic

10&comma; 2&comma; 1&comma; 1

Coefficients used in the Magic formula.

BCDE

K

formulation &equals; Magic

1

Extra scaling coefficient for the Magic formula.

K

Sv

formulation &equals; Magic

0

N

Residual tire force at zero slip (Use Sv0 for models with fixed-step integration)

Sv

 

 

Initial Conditions

Symbol

Condition

Default

Units

Description

ID

&omega;0

rads

Initial tire angular velocity

w

 

See Also

MapleSim Driveline Library Overview

MapleSim Library Overview

1-D Mechanical Overview

Longitudinal Vehicle Dynamics

Tire Component Library

Vehicle and Tire

Vehicle

 

 

References

Pacejka, Hans B. Tire and Vehicle Dynamics. Second Edition. SAE International and Elsevier, 2005.