Power transmission element with frictional belt wrapped around pulley circumference

Simscape / Driveline / Couplings & Drives

This block represents a pulley wrapped in a flexible flat or V-shaped belt. 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 dialog box enables you to select the relative belt direction of motion. Depending on the chosen setting, 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 at the MATLAB^{®} command line when either belt end loses tension. When assembling a model,
ensure that tension is maintained throughout the simulation. Consider this requirement
when you interpret the simulation results.

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

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

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

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

In both cases:

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

$${F}_{C}=\rho \ast {V}_{B}^{2}$$

$${V}_{A}=-{V}_{B}$$

Where:

*V*is the relative velocity between the belt and pulley periphery._{rel}*V*is the Branch A linear velocity._{A}*V*is the Branch B linear velocity._{B}*ω*is the pulley angular velocity._{S}*R*is the pulley radius.*F*is the belt centrifugal force._{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.*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

$$f\text{'}=\frac{f}{\mathrm{sin}\left(\raisebox{1ex}{$\varphi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)},$$

where:

*f*is the effective friction coefficient for a V-belt.^{'}*Φ*is the V-belt sheave angle.

The idealization of the discontinuity at
*V _{rel}* = 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

$$\mu =-f\ast \mathrm{tanh}\left(4\ast \raisebox{1ex}{${V}_{rel}$}\!\left/ \!\raisebox{-1ex}{${V}_{thr}$}\right.\right),$$

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 is the change of
*μ*.

This friction force is calculated as

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

The resulting torque on 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:

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

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

Both belt ends maintain adequate tension throughout the simulation.

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**.

**Belt type**Type of flexible element modeled.

`Flat belt`

— (Default)`V-belt`

— Exposes the**V-belt sheave angle**and**Number of V-belts**parameters.

**V-belt sheave angle**Sheave angle of the V-belt. Default is

`30 deg`

.The

**V-belt sheave angle**parameter is visible only when the**Belt type**parameter is`V-belt`

.**Number of V-belts**Number of parallel V-belts. The default value is

`1`

.Non-integer 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.

The

**Number of V-belts**parameter is visible only when the**Belt type**parameter is`V-belt`

.**Centrifugal force**Select whether 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. Options are:

`Do not model centrifugal force - Suitable for HIL simulation`

— Default option which does not include centrifugal force contributions.`Model centrifugal force`

— Exposes the**Belt mass per unit length**parameter for configuring the centrifugal force contribution.

**Belt mass per unit length**Linear density of each belt. Default is

`0.6 kg/m`

.The model uses this parameter to calculate the centrifugal loading. This parameter is enabled only if you select

`Model centrifugal force`

for the**Centrifugal force**parameter.**Belt direction**.Relative direction of translational motion of one belt end with respect to the other. Options include:

`Ends move in opposite direction`

`Ends move in same direction`

**Maximum tension**Specifies whether the block throws an assertion when the belt tension is too large.

`No maximum tension`

— (Default)`Specify maximum tension`

— Throws an assertion when the belt tension is too high. Exposes the**Belt maximum tension**parameter.

**Belt maximum tension**Maximum allowable tension for each belt. Default is

`1e+5 N`

.When the tension on either end of the belt meets or exceeds this value, the simulation ends with an error.

The

**Belt maximum tension**parameter is visible only when the**Maximum tension**parameter is`Specify maximum tension`

.**Tension warning**Specifies whether the block writes a warning at the MATLAB command line when the tension at either end of the belt falls below zero.

`Do not check tension`

— (Default)`Warn if either end loses tension`

Select the

**Pulley**tab to specify the pulley characteristics.

**Pulley radius**Radius of the pulley. Default is

`0.15 m`

.**Bearing viscous friction coefficient**Viscous friction associated with the bearings that hold the axis of the pulley. Default is

`0.001 N*m/(rad/s)`

.**Inertia**Specifies whether the block models rotational inertia of the pulley.

`No inertia`

— (Default)`Specify inertia and initial velocity`

— Models rotational inertia. Exposes the**Pulley inertia**and**Pulley initial velocity**parameters.

**Pulley inertia**Rotational inertia of the pulley. Default is

`0.01 kg*m^2`

.The

**Pulley inertia**parameter is visible only when the**Inertia**parameter is`Specify inertia and initial velocity`

.**Pulley initial velocity**Initial rotational velocity of the pulley. Default is

`0 rad/s`

.The

**Pulley initial velocity**parameter is visible only when the**Inertia**parameter is`Specify inertia and initial velocity`

.to specify contact characteristics, use the

**Contact**pane.

**Contact friction coefficient**Coulomb friction coefficient between the belt and the pulley surface. Default is

`0.5`

.**Wrap angle**Angle of contact between the belt and pulley. Default is

`180 deg`

.**Friction velocity threshold**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. Default is

`0.001 m/s`

.

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

.