# Belt Drive

Power transmission system with taut belt connecting two pulleys

• Library:
• Simscape / Driveline / Couplings & Drives

## Description

The Belt Drive block represents a pair of pulleys connected with a flexible ideal, flat, or V-shaped belt. The ideal belt does not slip relative to the pulley surfaces.

The model accounts for friction between the flexible belt and the pulley periphery. If the friction force is not sufficient to drive the load, the model allows slip. The relationship between the tensions in the tight and loose branches conforms to the Euler equation. The model accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.

The Belt Drive block is a structural component based on the Simscape™ Driveline™ Belt Pulley block and Simscape Translational Spring and Translational Damper blocks. The Translational Spring and Translational Damper blocks model the compliance of the belt. For the equations governing the contact dynamics between the belt and the pulley, see the Belt Pulley block. The figure shows the block diagram for the Belt Drive block.

### Equations

The diagrams show the open and crossed belt drive configurations. In the open belt configuration, both pulleys tend to rotate in the same direction and the larger pulley has a larger belt wrap angle. In the crossed belt configuration, the pulleys tend to rotate in opposite directions and have the same wrap angle.

Belt Drive Diagrams

In the open configuration, the wrap angle of the belt around each pulley is given by the expressions:

`$\begin{array}{l}{\theta }_{A}=\pi +2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}-{R}_{B}}{C},\\ {\theta }_{B}=\pi -2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}-{R}_{B}}{C}\end{array}$`

where:

• θA is the wrap angle of pulley A.

• θB is the wrap angle of pulley B.

• RA is the effective radius of pulley A.

• RB is the effective radius of pulley B.

• C is the distance between the centers of pulleys A and B.

The diagram shows the open-configuration wrap angles and parameters.

In the crossed configuration, the two wrap angles are equal and the wrap angle of the belt around each pulley is

`${\theta }_{A}={\theta }_{B}=\pi +2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}+{R}_{B}}{C}$`

The diagram shows the closed-configuration wrap angles and parameters.

## Assumptions and Limitations

• The pulleys do not translate.

• The friction coefficient and friction velocity threshold between the belt and each of the pulleys is the same.

## Ports

### Conserving

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Rotational conserving port associated with the shaft of pulley A.

Rotational conserving port associated with the shaft of pulley B.

## Parameters

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### Belt

Belt model:

• `Ideal - No slip` — Model an ideal belt, which does not slip relative to the pulley.

• `Flat belt` — Model a belt with a rectangular cross-section.

• `V-belt`— Model a belt with a V-shaped cross-section.

#### Dependencies

This parameter affects the visibility of related belt parameters and the Contact settings.

Compliance model.

• ```No compliance - Suitable for HIL simulation``` — Model a noncompliant belt, which does not yield elastically when subjected to a force. To prioritize performance, select this model.

• `Specify stiffness and damping` — Model a compliant belt, which does yield elastically when subjected to a force. To prioritize fidelity, select this model.

#### Dependencies

This parameter is visible only if Belt type is set to `Ideal - No slip`. This parameter affects the visibility of related belt parameters.

Sheave angle of the V-belt.

#### Dependencies

This parameter is visible only when Belt type is set to `V-belt`.

Number of V-belts.

Noninteger values are rounded to the nearest integer. Increasing the number of belts increases the friction force, effective mass per unit length, and maximum allowable tension.

#### Dependencies

This parameter is visible only when Belt type is set to `V-belt`.

Centrifugal force model. If included, centrifugal force saturates to approximately 90 percent of the value of the force on each belt end.

#### Dependencies

This parameter is visible only if Belt type is set to `Flat belt` or `V-belt`. If this parameter is set to `Model centrifugal force`, the Belt mass per unit length parameter is exposed.

Centrifugal force contribution in terms of linear density expressed as mass per unit length.

#### Dependencies

Selecting `Model centrifugal force` for the Centrifugal force parameter exposes this parameter.

Effective stiffness of the belt.

#### Dependencies

This parameter not visible if Belt type is set to `Ideal - No slip` and Compliance is set to ```No compliance - Suitable for HIL simulation```.

Effective damping of the belt.

#### Dependencies

This parameter not visible if Belt type is set to `Ideal - No slip` and Compliance is set to ```No compliance - Suitable for HIL simulation```.

Tension in the belt when the belt and pulleys are at rest. The value must be positive.

#### Dependencies

This parameter not visible if Belt type is set to `Ideal - No slip` and Compliance is set to ```No compliance - Suitable for HIL simulation```.

Tension threshold model. If ```Specify maximum tension``` is selected and the belt tension on either end of the belt meets or exceeds the value that you specify for Belt maximum tension, the simulation stops and generates an assertion error.

#### Dependencies

This parameter not visible if Belt type is set to `Ideal - No slip` and Compliance is set to ```No compliance - Suitable for HIL simulation```.

For all three belt types, selecting ```Specify maximum tension``` exposes the Belt maximum tension parameter.

Maximum allowable tension for each belt. When the tension on either end of the belt meets or exceeds this value, the simulation stops and generates an assertion error.

#### Dependencies

This parameter not visible if Belt type is set to `Ideal - No slip` and Compliance is set to ```No compliance - Suitable for HIL simulation```.

For all three belt types, the Belt maximum tension parameter is visible only when the Maximum tension parameter is set to `Specify maximum tension`.

### Pulley A

Viscous friction associated with the bearings that hold the axis of the pulley.

Rotational inertia model.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` exposes the Pulley inertia and Pulley initial velocity parameters.

Rotational inertia of the pulley.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter exposes this parameter.

Initial rotational velocity of the pulley.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter exposes this parameter.

### Pulley B

Viscous friction associated with the bearings that hold the axis of the pulley.

```Specify inertia and initial velocity```— To models rotational inertia. Exposes the Pulley inertia and Pulley initial velocity parameters.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` exposes the Pulley inertia and Pulley initial velocity parameters.

Rotational inertia of the pulley.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter exposes this parameter.

Initial rotational velocity of the pulley.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter exposes this parameter.

### Contact

Coulomb friction coefficient between the belt and the pulley surface. Default is `0.5`.

Whether the block calculates the wrap angle of the belt on the pulleys using the pulley radii, center separation, and drive type or if the wrap angles are specified directly. Options are:

• ```Specify pulley center separation``` — The block calculates the wrap angle of the belt on the pulleys using the pulley radii, center separation, and drive type.

Exposes the Pulley center separation parameter.

• `Specify wrap angles` — The block uses the wrap angles that you specify.

#### Dependencies

Selecting ```Specify pulley center separation``` exposes the Pulley center separation parameter.

Selecting `Specify wrap angles` exposes the Pulley A wrap angle and Pulley B wrap angle parameters.

Distance between the centers of the pulleys.

#### Dependencies

The Pulley center separation parameter is visible only when the Wrap angle calculation parameter is ```Specify pulley center separation```.

Angle of contact between the belt and pulley attached to port A.

#### Dependencies

The Pulley A wrap angle parameter is visible only when the Wrap angle calculation parameter is `Specify wrap angles`.

Angle of contact between the belt and pulley attached to port B.

The Pulley B wrap angle parameter is visible only when the Wrap angle calculation parameter is `Specify wrap angles`.

Relative velocity required for peak kinetic friction in the contact. The friction velocity threshold improves the numerical stability of the simulation by ensuring that the force is continuous when the direction of the velocity changes.

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