# 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. When you set Belt type to `Ideal - No slip`, the belt does not slip relative to the pulley surfaces.

The block accounts for friction between the flexible belt and the pulley periphery. If the friction force is not sufficient to drive the load, the block allows slip. The relationship between the tensions in the driving and driven branches conforms to the capstan equation, also known as the Euler-Eytelwein equation. The block 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 simulate 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 functional block diagram for the Belt Drive block.

### Equations

The diagrams show the open and crossed belt drive configurations. When you set Drive type to `Open belt`, both pulleys tend to rotate in the same direction and the larger pulley has a larger belt wrap angle. When you set Drive type to ```Crossed belt```, the pulleys tend to rotate in opposite directions and have the same wrap angle.

Belt Drive Diagrams

The figures and equations refer to these quantities:

• 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.

Open Belt

When you set Drive type to ```Open belt```, the block calculates the wrap angle of the belt around each pulley as:

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

The diagram shows the wrap angles and parameters.

Crossed Belt

When you set Drive type to ```Crossed belt```, the two wrap angles are equal and the wrap angle of the belt around each pulley:

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

The diagram shows the 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. To parameterize separate friction interactions, use two Belt Pulley blocks.

## Ports

### Conserving

expand all

Mechanical rotational conserving port associated with the shaft of pulley A.

Mechanical rotational conserving port associated with the shaft of pulley B.

## Parameters

expand all

### Belt

Belt type selection. The belt type affects slip conditions

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

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

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

Option to simulate compliance.

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

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

#### Dependencies

To enable this parameter, set Belt type to `Ideal - No slip`.

Sheave angle of the V-belt.

#### Dependencies

To enable this parameter, set Belt type to `V-belt`.

Number of V-belts.

The block rounds noninteger values to the nearest integer. Increasing the number of belts increases the friction force, effective mass per unit length, and maximum allowable tension.

#### Dependencies

To enable this parameter, set Belt type to `V-belt`.

Option to include the effects of centrifugal force. If you set this parameter to `Model centrifugal force`, centrifugal force saturates to approximately 90% of the value of the force on each belt end.

#### Dependencies

To enable this parameter, set Belt type to `Flat belt` or `V-belt`.

Centrifugal force contribution in terms of linear density.

#### Dependencies

To enable this parameter, set Belt type to `Flat belt` or `V-belt` and Centrifugal force to ```Model centrifugal force```.

Effective stiffness of the belt.

#### Dependencies

To enable this parameter, set Belt type to `Ideal - No slip` and Compliance to ```Specify stiffness and damping``` or set Belt type to `Flat belt` or `V-belt`.

Effective damping of the belt.

To enable this parameter, either set

• Belt type to ```Ideal - No slip``` and Compliance to `Specify stiffness and damping`, or

• Belt type to ```Flat belt``` or `V-belt`

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

To enable this parameter, either set

• Belt type to ```Ideal - No slip``` and Compliance to `Specify stiffness and damping`, or

• Belt type to ```Flat belt``` or `V-belt`

Option to specify a maximum tension. If you set this parameter to `Specify maximum tension` and the belt tension on either end of the belt meets or exceeds the value that you specify for the Belt maximum tension parameter, the simulation stops and generates an assertion error.

#### Dependencies

To enable this parameter, either set

• Belt type to ```Ideal - No slip``` and Compliance to ```Specify stiffness and damping```, or

• Belt type to ```Flat belt``` or `V-belt`

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.

To enable this parameter, either set

• Belt type to ```Ideal - No slip```, Compliance to ```Specify stiffness and damping```, and Maximum tension to ```Specify maximum tension```, or

• Belt type to ```Flat belt``` or `V-belt` and Maximum tension to ```Specify maximum tension```

### Pulley A

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

Option to parameterize rotational inertia with an initial velocity.

Rotational inertia of the pulley.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

Initial rotational velocity of the pulley.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

### Pulley B

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

Option to parameterize rotational inertia with an initial velocity.

Rotational inertia of the pulley.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

Initial rotational velocity of the pulley.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

### Contact

To enable these settings, set Belt type to ```Flat belt``` or `V-belt`.

Option to parameterize continuous or modal friction.

Option to initialize the simulation with pulley A locked or unlocked.

#### Dependencies

To enable this parameter, set Friction model to `Modal`.

Option to initialize the simulation with pulley B locked or unlocked.

#### Dependencies

To enable this parameter, set Friction model to `Modal`.

Belt friction while maintaining static contact.

#### Dependencies

To enable this parameter, set Friction model to `Modal`.

Belt friction while slipping.

#### Dependencies

To enable this parameter, set Friction model to `Modal`.

Coulomb friction coefficient between the belt and the pulley surface.

#### Dependencies

To enable this parameter, set Friction model to `Continuous`.

Option to calculate the wrap angle by the pulley center separation or to specify those values directly.

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

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

Distance between the centers of the pulleys.

#### Dependencies

To enable this parameter, set Wrap angle calculation to ```Specify pulley center separation```.

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

#### Dependencies

To enable this parameter, set Wrap angle calculation to ```Specify wrap angles```.

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

#### Dependencies

To enable this parameter, set Wrap angle calculation to ```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.

Relative force required for peak kinetic friction in the contact.

#### Dependencies

To enable this parameter, set Belt type to `Flat belt` or `V-belt`.

expand all

## References

[1] Johnson, Kenneth L. Contact Mechanics. Cambridge: Cambridge Univ. Press, 2003.

## Version History

Introduced in R2012a