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Power transmission element with frictional belt wrapped around pulley circumference

**Library:**Simscape / Driveline / Couplings & Drives

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.

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:

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

where:

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

,*β*= 1. Otherwise,*β*= -1.*V*is the relative velocity between the belt and pulley periphery._{rel}*V*= 0 for the ideal pulley case._{rel}*V*is the branch A linear velocity._{A}*V*is the branch B linear velocity._{B}*V*is the pulley linear velocity at its center. If the pulley is not translating, this variable is fixed to 0._{C}*ω*is the pulley angular velocity._{S}*R*is the pulley radius.*F*is the belt centrifugal force._{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}*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.*V*is the friction velocity threshold._{thr}

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:

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

where:

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

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

*T*is the pulley torque._{S}*b*is the bearing viscous damping.*F*is the force threshold._{thr}

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

*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.