# 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. When you set **Belt type** to
`Ideal - No slip`

, the belt does not slip relative to the
pulley surfaces. You can optionally enable pulley linear translation.

The block accounts for friction between the flexible belt and the pulley surface. The tensioner slips when the load exceeds the friction force. The block accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.

The belt ends can either move in the same direction or in the opposite direction.

The block equations relate power transmission between the belt branches or to or from the pulley. The driving and driven 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 by adding mass to
at least one of the belt ends or by adding a tensioner like the Rope
block. 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.

You can set **Friction model** to `Modal`

to use a
modal parameterization for the pulley. Choose the modal parameterization for greater
numerical robustness. To enable this parameter, set **Pulley
translation** to `Off`

.

The Belt Pulley block uses a composite implementation of the Fundamental Friction Clutch block to produce the conditions for the modal parameterization.

### Equations

For either setting of the **Friction model** parameter, the block
equations refer to these quantities:

*β*is the belt direction sign. When you set**Belt direction**to`Ends move in same direction`

,*β*= 1. Otherwise,*β*= -1.*V*is the relative velocity between the belt and pulley periphery._{rel}*V*= 0 when_{rel}**Belt type**is`Ideal - No slip`

.*V*is the linear velocity of branch A._{A}*V*is the linear velocity of branch B._{B}*V*is the linear velocity of the pulley at its center. If_{C}**Pulley translation**is`Off`

, the block constrains this to 0.*ω*is the angular velocity of the pulley contact surface._{S}*R*is the radius of the pulley.*F*is centrifugal force of the belt._{centrifugal}*F*is the force acting through the pulley centroid. When you specify a value for_{C}**Inertia**,*F*includes force due to the pulley mass acceleration._{C}*ρ*is the belt linear density.*F*is the friction force between the pulley and the belt._{fr}*F*is the force acting along branch A._{A}*F*is the force acting along branch B._{B}*θ*is the contact wrap angle.*τ*is the pulley torque._{S}

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 you set **Pulley translation** to
`On`

, the force balancing equation is:

$${F}_{C}=(\beta {F}_{A}-{F}_{B}-{F}_{centrifugal,smooth,sat})\cdot \mathrm{sin}\left(\frac{\theta}{2}\right).$$

To calculate
*F _{centrifugal,smooth,sat}*, the block

Smooths

*F*using_{centrifugal}*F*. You can increase smoothing by raising_{thr}*F*, and you can decrease smoothing by lowering_{thr}*F*._{thr}Saturates values above

*βF*._{A}-F_{B}

The sign convention is such that, when **Belt direction** is
`Ends move in opposite direction`

, a positive rotation
in port **S** results in a negative translation for port
**A** and a positive translation for port
**B**.

To enable the **Friction model** parameter, set **Belt
type** to `Flat belt`

or
`V-belt`

and **Pulley translation** to
`Off`

.

**Continuous Friction**

When you set **Friction model** to `Continuous`

,
the block equations refer to these quantities:

*μ*is the**Contact friction coefficient**parameter.*μ*is the instantaneous value of the friction coefficient._{smoothed}*V*is the_{thr}**Velocity threshold**parameter.*b*is the viscous damping of the bearing.*F*is the_{thr}**Force threshold**parameter.

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

$${\mu}_{smoothed}={\mu}_{sheave}\mathrm{tanh}\left(4\frac{{V}_{rel}}{{V}_{thr}}\right),$$

where the hyperbolic tangent function maintains numerical robustness by ensuring a smooth
and continuous output for *V _{rel}*
zero-crossings.

For a V-belt, the block derives the contact friction value using the sheave angle:

$${\mu}_{sheave}=\frac{\mu}{\mathrm{sin}\left(\frac{\varphi}{2}\right)},$$

where:

*μ*is the effective friction coefficient._{sheave}*Φ*is the sheave angle.

For a flat belt, *μ _{sheave}* =

*μ*.

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:

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

which follows the form of the capstan equation, also known as the Euler-Eytelwein equation. The torque acting on the pulley is

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

where *σ* = 1 when you set **Belt type** to
`Ideal - No slip`

. Otherwise,

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

**Modal Friction**

For a V-belt, the block derives the static and kinetic friction values by using the sheave angle:

$$\begin{array}{l}{\mu}_{Static,sheave}=\frac{{\mu}_{Static}}{\mathrm{sin}\left(\frac{\varphi}{2}\right)}\\ {\mu}_{Kinetic,sheave}=\frac{{\mu}_{Kinetic}}{\mathrm{sin}\left(\frac{\varphi}{2}\right)}\end{array}$$

where:

*μ*is the_{Static}**Static friction coefficient**parameter.*μ*is effective static friction coefficient._{Static,sheave}*μ*is the_{Kinetic}**Kinetic friction coefficient**parameter.*μ*is effective kinetic friction coefficient._{Kinetic,sheave}*Φ*is the sheave angle.

For a flat belt, *μ _{Static,sheave}* =

*μ*, and

_{Static}*μ*=

_{Kinetic,sheave}*μ*.

_{Kinetic}When you set **Friction model** to
`Modal`

, the block calculates the maximum static
friction force before slipping as

$${F}_{Friction,Static,Max}=\left({F}_{Drive}-{F}_{Centrifugal}\right)\cdot \left(1-{e}^{-{\mu}_{Static,sheave}\theta}\right),$$

where *F _{Friction,Static,Max}* is the maximum force
magnitude due to static friction, and

*F*is the force due to tension felt at the end of the pulley with the greatest force magnitude. As with

_{Drive}*F*, the block uses

_{centrifugal}*F*to smooth

_{thr}*F*. To increase or decrease the smoothing on

_{Drive}*F*, raise of lower the value of the

_{Drive}**Force threshold**parameter.

The block smoothly saturates
*F _{Friction,Static,Max}* to be
greater than or equal to 0 as

$${F}_{Friction,Static,Max,Smooth}=0.5\left({F}_{Friction,Static,Max}+\sqrt{{F}_{Friction,Static,Max}^{2}+{\left(R{F}_{thr}\right)}^{2}}\right).$$

During slip,

$${F}_{Friction,Slip}=\frac{{\mu}_{Kinetic,sheave}}{{\mu}_{Static,sheave}}{F}_{Friction,Static,Max},$$

where
*μ _{Kinetic,sheave}* is the kinetic
friction coefficient. The block resolves the torque balance using

$${\tau}_{S}={\omega}_{S}b+R{F}_{Friction},$$

and

$$R\beta {F}_{A}+R{F}_{B}=-R{F}_{Friction},$$

where Simscape™ logs *RF _{Friction}* as

`fundamental_clutch.torque`

.## Assumptions and Limitations

The block assumes noncompliance along the length of the belt.

The block assumes 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

*V*and force_{C}*F*only account for the component along this line of motion._{C}The Eytelwein equation for belt friction neglects the effect of pulley translation on friction.

## Ports

### Conserving

## Parameters

## Model Examples

## More About

## References

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

## Extended Capabilities

## Version History

**Introduced in R2012a**