Belt Drive

Power transmission system with taut belt connecting two pulleys

Libraries:
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.

To learn more about the construction of the block, see About Composite Components and the reference pages for the individual component blocks.

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:

θ_A is the wrap angle of pulley A.
θ_B is the wrap angle of pulley B.
R_A is the effective radius of pulley A.
R_B 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:

$\begin{array}{l} θ_{A} = π + 2 * \sin^{- 1} \frac{R_{A} - R_{B}}{C}, \\ θ_{B} = π - 2 * \sin^{- 1} \frac{R_{A} - R_{B}}{C} \end{array}$

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:

$θ_{A} = θ_{B} = π + 2 * \sin^{- 1} \frac{R_{A} + R_{B}}{C} .$

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

A — Pulley A shaft
mechanical rotational

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

B — Pulley B shaft
mechanical rotational

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

Parameters

expand all

Belt

Drive type — Belt drive configuration
`Open belt` (default) | `Crossed belt`

Belt drive configuration. For more information, see Belt Drive Diagrams.

Belt type — Belt parameterization
`Ideal - No slip` (default) | `Flat belt` | `V-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.

Compliance — Compliance option
`No compliance - Suitable for HIL simulation` (default) | `Specify stiffness and damping`

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.

V-belt sheave angle — Sheave angle
`30` `deg` (default) | nonnegative scalar

Sheave angle of the V-belt.

Dependencies

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

Number of V-belts — Number of belts
`1` (default) | positive scalar integer

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.

Centrifugal force — Centrifugal force option
`Do not model centrifugal force - Suitable for HIL simulation` (default) | `Model centrifugal force`

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.

Belt mass per unit length — Mass per unit length
`0.6` `kg/m` (default) | positive scalar

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.

Longitudinal stiffness — Belt effective stiffness
`1e4` `N/m` (default) | nonnegative scalar

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.

Longitudinal damping — Belt effective damping
`1e3` `N/(m/s)` (default) | nonnegative scalar

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

Pre-tension — Resting belt tension
`100` `N` (default) | positive scalar

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

Maximum tension — Tension threshold option
`No maximum tension` (default) | `Specify maximum tension`

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

Belt maximum tension — Maximal tension threshold
`1e5` `N` (default) | positive scalar

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

Pulley radius — Pulley radius
`.15` `m` (default) | positive scalar

Radius of the pulley.

Bearing viscous friction coefficient — Bearing viscous friction
`0` `N*m/(rad/s)` (default) | scalar

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

Inertia — Rotational inertia
`No inertia` (default) | `Specify inertia and initial velocity`

Option to parameterize rotational inertia with an initial velocity.

Pulley inertia — Pulley inertia
`0.01` `kg*m^2` (default) | positive scalar

Rotational inertia of the pulley.

Dependencies

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

Pulley initial velocity — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

Initial rotational velocity of the pulley.

Dependencies

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

Pulley B

Pulley radius — Pulley radius
`.30` `m` (default) | positive scalar

Radius of the pulley.

Bearing viscous friction coefficient — Bearing viscous friction
`0.001` `N*m/(rad/s)` (default) | scalar

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

Inertia — Rotational inertia
`No inertia` (default) | `Specify inertia and initial velocity`

Option to parameterize rotational inertia with an initial velocity.

Pulley inertia — Pulley inertia
`0.03` `kg*m^2` (default) | positive scalar

Rotational inertia of the pulley.

Dependencies

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

Pulley initial velocity — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

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.

Friction model — Friction method selection
`Modal` (default) | `Continuous`

Option to parameterize continuous or modal friction.

Pulley A initial state — Initial state of pulley A
`Unlocked` (default) | `Locked`

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

Dependencies

To enable this parameter, set Friction model to Modal.

Pulley B initial state — Initial state of pulley B
`Unlocked` (default) | `Locked`

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

Dependencies

To enable this parameter, set Friction model to Modal.

Static friction coefficient — Static friction coefficient
`0.5` (default) | positive scalar

Belt friction while maintaining static contact.

Dependencies

To enable this parameter, set Friction model to Modal.

Kinetic friction coefficient — Kinetic friction coefficient
`0.42` (default) | positive scalar

Belt friction while slipping.

Dependencies

To enable this parameter, set Friction model to Modal.

Contact friction coefficient — Contact friction coefficient
`0.5` (default) | positive scalar

Coulomb friction coefficient between the belt and the pulley surface.

Dependencies

To enable this parameter, set Friction model to Continuous.

Wrap angle calculation — Angle calculation type
`Specify pulley center separation` (default) | `Specify wrap angles`

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.

Pulley center separation — Pulley center separation distance
`0.5` `m` (default) | positive scalar

Distance between the centers of the pulleys.

Dependencies

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

Pulley A wrap angle — Pulley wrap angle
`145` `deg` (default) | scalar

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.

Pulley B wrap angle — Pulley wrap angle
`215` `deg` (default) | scalar

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.

Velocity threshold — Contact threshold
`0.001` `m/s` (default) | positive scalar

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.

Force threshold — Force threshold
`0.01` `N` (default) | positive scalar

Relative force required for peak kinetic friction in the contact.

Dependencies

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

More About

expand all

Hardware-in-the-Loop Simulation

For optimal simulation performance, set the Belt > Centrifugal force parameter to Do not model centrifugal force - Suitable for HIL simulation.

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2012a

Belt Drive

Description

Equations

Assumptions and Limitations

Ports

Conserving

A — Pulley A shaft mechanical rotational

B — Pulley B shaft mechanical rotational

Parameters

Belt

Drive type — Belt drive configuration Open belt (default) | Crossed belt

Belt type — Belt parameterization Ideal - No slip (default) | Flat belt | V-belt

Compliance — Compliance option No compliance - Suitable for HIL simulation (default) | Specify stiffness and damping

Dependencies

V-belt sheave angle — Sheave angle 30 deg (default) | nonnegative scalar

Dependencies

Number of V-belts — Number of belts 1 (default) | positive scalar integer

Dependencies

Centrifugal force — Centrifugal force option Do not model centrifugal force - Suitable for HIL simulation (default) | Model centrifugal force

Dependencies

Belt mass per unit length — Mass per unit length 0.6 kg/m (default) | positive scalar

Dependencies

Longitudinal stiffness — Belt effective stiffness 1e4 N/m (default) | nonnegative scalar

Dependencies

Longitudinal damping — Belt effective damping 1e3 N/(m/s) (default) | nonnegative scalar

Pre-tension — Resting belt tension 100 N (default) | positive scalar

Maximum tension — Tension threshold option No maximum tension (default) | Specify maximum tension

Dependencies

Belt maximum tension — Maximal tension threshold 1e5 N (default) | positive scalar

Pulley A

Pulley radius — Pulley radius .15 m (default) | positive scalar

Bearing viscous friction coefficient — Bearing viscous friction 0 N*m/(rad/s) (default) | scalar

Inertia — Rotational inertia No inertia (default) | Specify inertia and initial velocity

Pulley inertia — Pulley inertia 0.01 kg*m^2 (default) | positive scalar

Dependencies

Pulley initial velocity — Initial pulley rotational velocity 0 rad/s (default) | scalar

Dependencies

Pulley B

Pulley radius — Pulley radius .30 m (default) | positive scalar

Bearing viscous friction coefficient — Bearing viscous friction 0.001 N*m/(rad/s) (default) | scalar

Inertia — Rotational inertia No inertia (default) | Specify inertia and initial velocity

Pulley inertia — Pulley inertia 0.03 kg*m^2 (default) | positive scalar

Dependencies

Pulley initial velocity — Initial pulley rotational velocity 0 rad/s (default) | scalar

Dependencies

Contact

Friction model — Friction method selection Modal (default) | Continuous

Pulley A initial state — Initial state of pulley A Unlocked (default) | Locked

Dependencies

Pulley B initial state — Initial state of pulley B Unlocked (default) | Locked

Dependencies

Static friction coefficient — Static friction coefficient 0.5 (default) | positive scalar

Dependencies

Kinetic friction coefficient — Kinetic friction coefficient 0.42 (default) | positive scalar

Dependencies

Contact friction coefficient — Contact friction coefficient 0.5 (default) | positive scalar

Dependencies

Wrap angle calculation — Angle calculation type Specify pulley center separation (default) | Specify wrap angles

Pulley center separation — Pulley center separation distance 0.5 m (default) | positive scalar

Dependencies

Pulley A wrap angle — Pulley wrap angle 145 deg (default) | scalar

Dependencies

Pulley B wrap angle — Pulley wrap angle 215 deg (default) | scalar

Dependencies

Velocity threshold — Contact threshold 0.001 m/s (default) | positive scalar

Force threshold — Force threshold 0.01 N (default) | positive scalar

Dependencies

More About

Hardware-in-the-Loop Simulation

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

Topics

A — Pulley A shaft
mechanical rotational

B — Pulley B shaft
mechanical rotational

Drive type — Belt drive configuration
`Open belt` (default) | `Crossed belt`

Belt type — Belt parameterization
`Ideal - No slip` (default) | `Flat belt` | `V-belt`

Compliance — Compliance option
`No compliance - Suitable for HIL simulation` (default) | `Specify stiffness and damping`

V-belt sheave angle — Sheave angle
`30` `deg` (default) | nonnegative scalar

Number of V-belts — Number of belts
`1` (default) | positive scalar integer

Centrifugal force — Centrifugal force option
`Do not model centrifugal force - Suitable for HIL simulation` (default) | `Model centrifugal force`

Belt mass per unit length — Mass per unit length
`0.6` `kg/m` (default) | positive scalar

Longitudinal stiffness — Belt effective stiffness
`1e4` `N/m` (default) | nonnegative scalar

Longitudinal damping — Belt effective damping
`1e3` `N/(m/s)` (default) | nonnegative scalar

Pre-tension — Resting belt tension
`100` `N` (default) | positive scalar

Maximum tension — Tension threshold option
`No maximum tension` (default) | `Specify maximum tension`

Belt maximum tension — Maximal tension threshold
`1e5` `N` (default) | positive scalar

Pulley radius — Pulley radius
`.15` `m` (default) | positive scalar

Bearing viscous friction coefficient — Bearing viscous friction
`0` `N*m/(rad/s)` (default) | scalar

Inertia — Rotational inertia
`No inertia` (default) | `Specify inertia and initial velocity`

Pulley inertia — Pulley inertia
`0.01` `kg*m^2` (default) | positive scalar

Pulley initial velocity — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

Pulley radius — Pulley radius
`.30` `m` (default) | positive scalar

Bearing viscous friction coefficient — Bearing viscous friction
`0.001` `N*m/(rad/s)` (default) | scalar

Inertia — Rotational inertia
`No inertia` (default) | `Specify inertia and initial velocity`

Pulley inertia — Pulley inertia
`0.03` `kg*m^2` (default) | positive scalar

Pulley initial velocity — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

Friction model — Friction method selection
`Modal` (default) | `Continuous`

Pulley A initial state — Initial state of pulley A
`Unlocked` (default) | `Locked`

Pulley B initial state — Initial state of pulley B
`Unlocked` (default) | `Locked`

Static friction coefficient — Static friction coefficient
`0.5` (default) | positive scalar

Kinetic friction coefficient — Kinetic friction coefficient
`0.42` (default) | positive scalar

Contact friction coefficient — Contact friction coefficient
`0.5` (default) | positive scalar

Wrap angle calculation — Angle calculation type
`Specify pulley center separation` (default) | `Specify wrap angles`

Pulley center separation — Pulley center separation distance
`0.5` `m` (default) | positive scalar

Pulley A wrap angle — Pulley wrap angle
`145` `deg` (default) | scalar

Pulley B wrap angle — Pulley wrap angle
`215` `deg` (default) | scalar

Velocity threshold — Contact threshold
`0.001` `m/s` (default) | positive scalar

Force threshold — Force threshold
`0.01` `N` (default) | positive scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.