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# Belt Pulley

Power transmission element with frictional belt wrapped around pulley circumference

• Library:
• Simscape / Driveline / Couplings & Drives

## Description

The Belt Pulley block represents a pulley wrapped in a flexible ideal, flat, or V-shaped belt. The ideal belt does not slip relative to the pulley surface. The pulley can optionally translate through port C, as is the case in a block and tackle system.

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 block allows you to select the relative belt direction of motion. The two belt ends can move in equal or opposite directions. The block model assumes noncompliance in the belt and no resistance to motion due to wrapping around the pulley.

The block equations model power transmission between the belt branches or to/from the pulley. The tight and loose branches use the same calculation. Without sufficient tension, the frictional force is not enough to transmit power between the pulley and belt.

The model is valid when both ends of the belt are in tension. An optional warning can display in the Simulink® Diagnostic Viewer when the leading belt end loses tension. When assembling a model, ensure that tension is maintained throughout the simulation. This can be done by adding mass to at least one of the belt ends or by adding a tensioner into your model. Use the Variable Viewer to ensure that any springs attached the belt are in tension. For more details on building a tensioner, see Best Practices for Modeling Pulley Networks.

### Equations

If the relative velocity between the belt and pulley is positive or zero, that is ${V}_{rel}\ge 0$, the Belt Pulley block calculates friction force as

`${F}_{fr}={F}_{B}-{F}_{centrifugal}=\left({F}_{A}-{F}_{centrifugal}\right)*\mathrm{exp}\left(f*\theta \right).$`

If the relative velocity is negative, that is ${V}_{rel}<0$, the friction force is calculated as

`${F}_{fr}={F}_{A}-{F}_{centrifugal}=\left({F}_{B}-{F}_{centrifugal}\right)\ast \mathrm{exp}\left(f\ast \theta \right).$`

The relative velocity is:

`${V}_{rel}={V}_{A}-{\omega }_{S}\ast R-{V}_{C}$`

`${V}_{rel}=-{V}_{B}+{\omega }_{S}\ast R+{V}_{C}$`

If Belt type is set to either `V-belt` or `Flat belt` and Centrifugal force is set to `Model centrifugal force`, the centrifugal force is:

`${F}_{centrifugal}=\rho \ast {\left({V}_{B}-{V}_{C}\right)}^{2}$`

where:

• Vrel is the relative velocity between the belt and pulley periphery.

• VA is the branch A linear velocity.

• VB is the branch B linear velocity.

• VC is the pulley linear velocity at its center. If the pulley is not translating, this value is 0.

• ωS is the pulley angular velocity.

• R is the pulley radius.

• Fcentrifugal is the belt centrifugal force.

• ρ is the belt linear density.

• Ffr is the friction force between the pulley and the belt.

• FA is the force acting along branch A.

• FB is the force acting along branch B.

• f is the friction coefficient.

• θ is the contact wrap angle.

For a flat belt, specify the value of f directly in the block parameters dialog box. For a V-belt, the model calculates the value as

`$f\text{'}=\frac{f}{\mathrm{sin}\left(\varphi }{2}\right)},$`

where:

• f' is the effective friction coefficient for a V-belt.

• Φ is the V-belt sheave angle.

The idealization of the discontinuity at Vrel = 0 is both difficult for the solver to resolve and not physically accurate. To alleviate this issue, the friction coefficient is assumed to change its value as a function of the relative velocity such that

`$\mu =-f\ast \mathrm{tanh}\left(4\ast {V}_{rel}}{{V}_{thr}}\right),$`

where

• μ is the instantaneous value of the friction coefficient.

• f is the steady-state value of the friction coefficient.

• Vthr is the friction velocity threshold.

The friction velocity threshold controls the width of the region within which the friction coefficient changes its value from zero to a steady-state maximum. The friction velocity threshold specifies the velocity at which the hyperbolic tangent equals 0.999. The smaller the value, the steeper is the change of μ.

This friction force is calculated as

`${F}_{fr}={F}_{A}-{F}_{centrifugal}=\left({F}_{B}-{F}_{centrifugal}\right)\ast \mathrm{exp}\left(\mu \ast \theta \right).$`

The resulting torque delivered by the pulley is given as

`${T}_{S}=\left({F}_{A}+{F}_{B}\right)\ast R\ast \text{tanh}\left(\text{4}\frac{{V}_{\text{rel}}}{{V}_{\text{thr}}}\right)\ast \text{tanh}\left(\frac{{F}_{B}}{{F}_{\text{thr}}}\right)-{\omega }_{S}\ast b.$`

where:

• TS is the pulley torque.

• b is the bearing viscous damping.

• Fthr is the force threshold.

The resulting force exerted by the pulley center is:

`${F}_{C}=\left({F}_{A}+{F}_{B}\right)\ast \mathrm{sin}\left(\frac{\varphi }{2}\right).$`

## Assumptions and Limitations

• The model does not account for compliance along the length of the belt.

• Both belt ends maintain adequate tension throughout the simulation.

• The translation of the pulley center is assumed to be planar and travels along the bisect of the pulley wrap angle. The center velocity VC and force FC only account for the component along this line of motion.

## Ports

The sign convention is such that, when Belt direction is set to `Ends move in opposite direction`, a positive rotation in port S tends to give a negative translation for port A and a positive translation for port B.

### Conserving

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Rotational conserving port associated with the pulley shaft.

Translational conserving port associated with belt end A.

Translational conserving port associated with belt end B.

Translational conserving port associated with pulley translational velocity. The pulley moves within the plane and along the bisect of the pulley wrap angle. When the relative velocity is positive and pulley translation is enabled, the pulley center moves.

#### Dependencies

To expose this port, set Pulley translation to `On`.

## 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.

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`.

Option to include the effects of centrifugal force. 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 when 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.

Relative direction of translational motion of one belt end with respect to the other.

#### Dependencies

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

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

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.

The Belt maximum tension parameter is visible only when the Maximum tension parameter is set to `Specify maximum tension`.

Whether the block generates a warning when the tension at either end of the belt falls below zero.

### Pulley

Whether to model pulley linear motion. Setting this parameter to `On` exposes port C.

Radius of the pulley.

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 mass for inertia calculation.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter when Pulley translation is set to `On` exposes this parameter.

Initial translational velocity of the pulley.

#### Dependencies

Selecting ```Specify inertia and initial velocity``` for the Inertia parameter when Pulley translation is set to `On` exposes this parameter.

### Contact

Contact settings are only visible if the Belt type parameter in the Belt settings is set to `Flat belt` or `V-belt`

Coulomb friction coefficient between the belt and the pulley surface.

Radial contact angle between the belt and the pulley.

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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## See Also

### Topics

Introduced in R2012a

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