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# Magic Formula Tire Force and Torque

Apply steady-state tire force and torque by using Magic Formula tire equations

Since R2021b

Libraries:
Simscape / Multibody / Forces and Torques

## Description

The Magic Formula Tire Force and Torque block implements the combined slip steady-state magic formula model and can optionally include turn slip effects [1]. The block applies the force and torque to the follower frame.

The block calculates only the tire force and torque. To model the geometry and inertia properties of the tire, you must use a solid block, such as the Cylindrical Solid block. The block supports only passenger car tires. The magic formula tire model assumes that passenger car tires are disks, as shown in the diagram.

The contact frame is located at the contact point between the tire and ground plane. The follower frame is located at the center of the tire and rotates with the tire. To correctly orient the tires on a vehicle, you must align the z-axes of the follower frames with the blue arrows, as shown in the diagram.

To specify the properties of a tire model, generate a scalar structure array by using the `simscape.multibody.tirread` function and enter the array in the Tire Parameters parameter. Note that the block observes the ISO sign convention. To indicate which side of the vehicle the tire is mounted to, use the Tire Side parameter.

The B port, which represents the surface that the tire contacts, must be connected to an Infinite Plane or Grid Surface block. The geometry of the contact surface can move or be fixed relative to the world frame. The F port represents the follower frame. Note that if the follower frame penetrates the ground connected to the B port, the magic formula tire force and torque equations become degenerate and the block does not apply force and torque. This case happens when the follower frame of the block is below the geometry connected to the B port.

The yaw, camber, and spin angles correspond to a y-x-z sequence rotation about the follower frame of a tire. The image shows the contact and follower frames of the tire at zero configuration.

## Ports

### Geometry

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Base geometry that represents the ground pane that the tire contacts. You must connect this port to the Infinite Plane or Grid Surface block.

### Frame

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Follower frame that represents the tire. The frame origin is located at the center of the tire.

### Input

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Physical signal port that accepts the scaling factor of the longitudinal friction coefficient of the tire.

#### Dependencies

To enable this port, under Scaling Coefficients, set LMUX to `Provided by Input`.

Physical signal port that accepts the scaling factor of the lateral friction coefficient of the tire.

#### Dependencies

To enable this port, under Scaling Coefficients, set LMUY to `Provided by Input`.

### Output

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Force/Torque

Physical signal output port that provides the magic formula tire force that is applied to the follower frame of the block. The output contains three parts:

• Fx is the longitudinal force tangential to the ground surface at the contact point.

• Fy is the lateral force which is orthogonal to the plane defined by Fx and Fz.

• Fz is the normal force that is normal to the ground surface at the contact point.

#### Dependencies

To enable this port, under Sensing > Force/Torque, select Tire Force.

Physical signal output port that provides the magic formula tire torque that is applied to the follower frame of the block.

#### Dependencies

To enable this port, under Sensing > Force/Torque, select Tire Torque.

Physical signal output port that provides the distance from the contact point to the point of the resultant lateral force. The value has a unit of length.

You can use the pneumatic trail to compute the aligning torque, Mz.

#### Dependencies

To enable this port, under Sensing > Force/Torque, select Pneumatic Trail.

Slip

Physical signal output port that provides the ratio of the longitudinal slip velocity to the longitudinal speed of the tire. The value is dimensionless.

#### Dependencies

To enable this port, under Sensing > Slip, select Longitudinal Slip.

Physical signal output port that provides the longitudinal slip saturated to always be within the limits defined by the KPUMIN and KPUMAX parameters.

#### Dependencies

To enable this port, under Sensing > Slip, select Saturated Longitudinal Slip.

Physical signal output port that provides the angle of the right triangle made by the lateral slip velocity, Vsy and the longitudinal speed, Vx. The value has a unit of angle.

#### Dependencies

To enable this port, under Sensing > Slip, select Slip Angle.

Physical signal output port that provides the slip angle saturated to always be within the limits defined by the ALPMIN and ALPMAX parameters. The value has a unit of angle.

#### Dependencies

To enable this port, under Sensing > Slip, select Saturated Slip Angle.

Physical signal output port that provides the ratio of the tire yaw velocity to the magnitude of the tire velocity in the xy-plane of the contact frame. The value has a unit of angle/length.

Turn slip is useful when modeling low speed cornering, such as parking maneuvers.

#### Dependencies

To enable this port, under Sensing > Slip, select Turn Slip.

Linear Velocity

Physical signal output port that provides the component of the relative velocity between the tire frame and the contact point on the geometry along the x-direction of the contact frame. The value has a unit of length/time.

#### Dependencies

To enable this port, under Sensing > Linear Velocity, select Relative Longitudinal Velocity.

Physical signal output port that provides the component of the relative velocity between the tire frame and the contact point on the geometry along the y-direction of the contact frame. The unit of the meters/sec.

#### Dependencies

To enable this port, under Sensing > Linear Velocity, select Relative Lateral Velocity.

Physical signal output port that provides the component of the relative velocity between the slip point on the tire and the coincident point on the geometry along the x-direction of the contact frame. The value has a unit of length/time.

#### Dependencies

To enable this port, under Sensing > Linear Velocity, select Longitudinal Slip Velocity.

Physical signal output port that provides the component of the relative velocity between the slip point on the tire and the coincident point on the geometry along the y-direction of the contact frame. The value has a unit of length/time.

#### Dependencies

To enable this port, under Sensing > Linear Velocity, select Lateral Slip Velocity.

Yaw

Physical signal output port that provides the first derivative of the yaw angle. The value has a unit of angle/time.

#### Dependencies

To enable this port, under Sensing > Yaw, select Velocity.

Camber

Physical signal output port that provides the camber angle of the tire. The value has a unit of angle.

#### Dependencies

To enable this port, under Sensing > Camber, select Angle.

Physical signal output port that provides the camber angle of the tire saturated to always be within the limits defined by the CAMMIN and CAMMAX parameters. The value has a unit of angle.

#### Dependencies

To enable this port, under Sensing > Camber, select Angle.

Physical signal output port that provides the first derivative of the camber angle. The value has a unit of angle/time.

#### Dependencies

To enable this port, under Sensing > Camber, select Velocity.

Spin

Physical signal output port that provides the first derivative of the spin angle. The value has a unit of angle/time.

#### Dependencies

To enable this port, under Sensing > Spin, select Velocity.

Tire Radius

Physical signal output port that provides the free radius of the tire. The radius increases as the tire rotates faster. The value has a unit of meter.

#### Dependencies

To enable this port, under Sensing > Tire Radius, select Free Radius.

Physical signal output port that provides the distance from the tire frame to the contact point. The value has a unit of meter.

#### Dependencies

To enable this port, under Sensing > Tire Radius, select Loaded Radius.

Physical signal output port that provides the distance from the tire frame to the slip point. The value has a unit of meter.

#### Dependencies

To enable this port, under Sensing > Tire Radius, select Effective Rolling Radius.

Friction

Physical signal output port that provides the longitudinal friction coefficient of the tire computed by the magic formula equations.

#### Dependencies

To enable this port, under Sensing > Friction, select Longitudinal Friction Coefficient.

Physical signal output port that provides the lateral friction coefficient of the tire computed by the magic formula equations.

#### Dependencies

To enable this port, under Sensing > Friction, select Lateral Friction Coefficient.

Contact Frame

Physical signal port that outputs a 3-by-3 rotation matrix that maps the vectors in the contact frame to vectors in the reference frame of the base geometry. The output signal is resolved in the reference frame associated with the base geometry.

#### Dependencies

To enable this port, in the Sensing > Contact Frame section, select Base Rotation.

Physical signal port that outputs at 3-by-1 vector that contains the coordinates of the origin of the contact frame resolved in the reference frame of the base geometry.

#### Dependencies

To enable this port, in the Sensing > Contact Frame section, select Base Translation.

Physical signal port that outputs a 3-by-3 rotation matrix that maps vectors in the contact frame to vectors in the reference frame of the follower geometry. The output signal is resolved in the reference frame associated with the follower geometry.

#### Dependencies

To enable this port, in the Sensing > Contact Frame section, select Follower Rotation.

Physical signal port that outputs at 3-by-1 vector that contains the coordinates of the origin of the contact frame resolved in the reference frame of the follower geometry.

#### Dependencies

To enable this port, in the Sensing > Contact Frame section, select Follower Translation.

## Parameters

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Tire position during modeling, specified as either `Left` or `Right`. Set the parameter to the side of the vehicle to which the tire is mounted.

Tire parameters, specified as a scalar structure array. Use the `simscape.multibody.tirread` function to generate the structure array from a TIR file.

Slip mode, specified as either `Combined` or `Combined + Turn`.

To model combined slip, select `Combined`. To model combined slip with turn slip effects, select ```Combined + Turn```.

### Scaling Coefficients

Scaling factor for the longitudinal friction coefficient. Select ```From Tire Parameters``` to use the constant scaling factor provided by the TIR file, or select `Provided by Input` to use input signals as scaling factors.

Scaling factor for the lateral friction coefficient. Select ```From Tire Parameters``` to use the constant scaling factor provided by the TIR file, or select `Provided by Input` to use input signals as scaling factors.

### Sensing

Force/Torque

Frame used to resolve the calculated tire force and torque, specified as either `Contact` or `Follower`.

## References

[1] Pacejka, Hans B., and Igo Besselink. Tire and Vehicle Dynamics. 3rd. Engineering Automotive Engineering. Amsterdam: Elsevier/Butterworth-Heinemann, 2012.

[2] Besselink, I. J.M., A. J.C. Schmeitz, and H. B. Pacejka. “An Improved Magic Formula/Swift Tyre Model That Can Handle Inflation Pressure Changes.” Vehicle System Dynamics 48, no. sup1 (December 2010): 337–52. https://doi.org/10.1080/00423111003748088.

[3] van der Hofstad, R. H. M. T. “Study on improving the MF-Swift tyre model.” (2010).

## Version History

Introduced in R2021b

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