Tire (Magic Formula)

In this block, a tire is considered as a rigid combination of a wheel and a tire that is in contact with the road and is prone to slipping. When the torque drives the wheel axle, the tire transmits the longitudinal force. , on the road. The tire transmits the resulting reaction in the form of force back to the wheel. This action rotates the wheel, creating a longitudinal motion. If we simulate the pliability of a tire, it will also deform elastically under load. If for the parameter Effective rolling radius model the value is set Load and velocity dependent (Magic Formula), then the radius of the tire will also change depending on the load and rotation speed.

The figure shows the forces acting on the tire.

Variables that determine the bus model:

You can simulate rolling, sliding and deformation.

Pumping and sliding_

The equation of translational motion of a non-sliding, non-deformable tire looks like this: . The sliding of the tires leads to a change in the longitudinal force .

The sliding speed of the contact spot is . For a tire without deformation .

The non-smoothed value of the slip coefficient of the contact spot is:

The block defines the denominator of the slip coefficient as:

where — parameter value Lower boundary of slip denominator, VXLOW.

Meaning smoothly changes to in transition areas:

,

.

The unit determines the sliding coefficient according to:

where

Meaning it changes smoothly in transition areas:

,

.

The block defines the smoothing threshold of the sliding coefficient as:

For this equation, the locked, sliding wheel has . For perfect rolling .

The formation

If the check box is selected Compliance, then the block considers the tire as elastic. When the tire is deformed, the point of contact of the tire with the road rotates with angular velocity slightly different from the speed of the wheel , which causes the contact patch to slip. The block defines a deformable tire as a translational spring damper with rigidity and damping .

If the checkbox is not checked Compliance Then , and there is no longitudinal deformation of the tire at any time in the simulation, then .

This block consists of several sub-components. The equivalent block diagram is shown below.

The block simulates the transient and stationary modes, as well as the start and stop. Blocks Translational Spring and Translational Damper are equivalent to rigidity and damping tires, respectively.

Block Tire-Road Interaction (Magic Formula) represents the longitudinal force on the bus as a function and using the empirical formula X .Guys, where — an independent sliding variable, and — input signal on port N.

Block Wheel and Axle — this is the rolling radius of the tire . The inertia value is the effective inertia. . Characteristic function of the bus determines the longitudinal force . Together with the torque of the drive shaft applied to the wheel axis, determines the angular and longitudinal movement of the wheel.

If the tire deformation is not modeled, then the block does not take into account the sub-components. Translational Spring and Translational Damper in an equivalent circuit, and the contact variables are returned to the wheel variables. In this case, the tire actually has infinite stiffness, and the P port of the block Wheel and Axle connects directly to the C port of the unit Force Sensor.

If in the block Rolling Resistance specified other than Pressure and velocity dependent (Magic Formula) rolling resistance model, then the equivalent circuit will not use the block Force Sensor and the T block port Rolling Resistance. If the checkbox is not checked Compliance, then the P port of the block Wheel and Axle connects directly to the C port of the unit Force Sensor or the T port of the unit Tire-Road Interaction (Magic Formula).

This block determines the effective rolling radius, taking into account its increase or decrease due to centrifugal forces, so that

where