# 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, which is necessary in a block and tackle system.

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 tight and loose branches conforms to the Euler equation. The block accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.

You can select the relative belt direction of motion. The two belt ends can move in equal or opposite directions. The block assumes noncompliance in the belt and neglects losses due to wrapping the belt around the pulley.

The block equations model power transmission between the belt branches or to or 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.

Your model is valid when both ends of the belt are in tension. You can choose to display a warning 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

The kinematic constraints between the pulley and belt are:

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

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

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

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

When the pulley can translate, the force balancing equation is:

`$0={F}_{C}+\left(-\beta {F}_{A}+{F}_{B}\right)\cdot \mathrm{sin}\left(\frac{\theta }{2}\right),$`

where:

• β is the belt direction sign. When you set Belt direction to ```Ends move in same direction```, β = 1. Otherwise, β = -1.

• Vrel is the relative velocity between the belt and pulley periphery. Vrel = 0 for the ideal pulley case.

• 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 variable is fixed to 0.

• ωS is the pulley angular velocity.

• R is the pulley radius.

• Fcentrifugal is the belt centrifugal force.

• FC is the force acting through the pulley centroid. When you specify a value for Inertia, FC includes force due to the pulley mass acceleration.

• 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. This is equivalent to the Contact friction coefficient parameter.

• θ is the contact wrap angle.

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.

For a flat belt, specify the value of f as the Contact friction coefficient parameter. For a V-belt, the block calculates the value as

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

where:

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

• Φ is the V-belt sheave angle.

The friction coefficient is a function of the relative velocity such that

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

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 the change of μ.

The block determines the effect of friction on the force at the belt ends as:

`$-\beta {F}_{A}-{F}_{centrifugal}=\left({F}_{B}-{F}_{centrifugal}\right){e}^{\mu \theta },$`

which follows the form of Eytelwein's formula for belt friction. The torque acting on the pulley is:

`${T}_{S}=\left(-\beta {F}_{A}-{F}_{B}\right)R\sigma +{\omega }_{S}b,$`

where:

• σ = 1 when you set Belt type to `Ideal - No slip`. Otherwise,

`$\sigma =\mathrm{tanh}\left(4\frac{{V}_{rel}}{{V}_{thr}}\right)\mathrm{tanh}\left(\frac{{F}_{B}}{{F}_{thr}}\right).$`

• TS is the pulley torque.

• b is the bearing viscous damping.

• Fthr is the force threshold.

## Assumptions and Limitations

• The block neglects compliance along the length of the belt.

• Both belt ends maintain adequate tension throughout the simulation.

• The block treats the translation of the pulley center as planar where the pulley 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.

• The Eytelwein equation for belt friction neglects the effect of pulley translation on friction.

## Ports

### Conserving

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Mechanical rotational conserving port associated with the angular veocity of the pulley shaft.

Mechanical translational conserving port associated with the linear velocity of belt end A.

Mechanical translational conserving port associated with the linear velocity of belt end B.

Mechanical 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, 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 that 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.

Sheave angle of the V-belt.

#### Dependencies

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

Number of V-belts.

Noninteger values are rounded to the nearest integer. Increasing the number of belts increases the 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 included, 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 expressed as mass per unit length.

#### Dependencies

To enable this parameter, set Centrifugal force to ```Model centrifugal force```.

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

Tension threshold limit. If the belt tension reaches the value of the Belt maximum tension parameter, the simulation stops and generates an assertion error.

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

To enable this parameter, set Maximum tension to `Specify maximum tension`.

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

Note

When combining Belt Pulley blocks with Rope blocks to form loops, set the Belt Pulley block Tension warning parameter to `Do not check tension`, and set the Rope block Tension warning parameter to ```Warn if rope loses tension```.

### Pulley

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

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

Option to model rotational inertia.

#### Dependencies

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

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 mass for inertia calculation.

#### Dependencies

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

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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