# Band Brake

Frictional brake with flexible band wrapped around rotating drum

## Library

Brakes & Detents/Rotational

## Description

The block represents a frictional brake with a flexible band that wraps around the periphery of a rotating drum to produce a braking action. A positive actuating force causes the band to tighten around the rotating drum and it places the friction surfaces in contact. Viscous and contact friction between the drum and the flexible band surfaces cause the rotating drum to decelerate. Band brakes provide high braking torque at the cost of reduced braking precision in applications that include winch drums, chainsaws, go-karts and mini-bikes. The model employs a simple parameterization with readily accessible brake geometry and friction parameters.

The following formula provides the braking torque as a function of the external brake actuation force that tightens the belt.

`$T=\left({F}_{TB}-{F}_{A}\right)\cdot {r}_{D}+{\mu }_{visc}\cdot \omega ,$`

In the formula, the parameters have the following meaning:

ParameterDescription
TBraking torque
FTBForce acting on the tense branch of the band
FAExternal brake actuation force
μviscViscous friction coefficient
μContact friction coefficient
ϕWrap angle

Forces FTB and FA satisfy the relationship:

`$\frac{{F}_{TB}}{{F}_{A}}={e}^{\mu \varphi },$`

Replacing the relationship in the braking torque formula eliminates force FTB:

`$T={F}_{A}\left({e}^{\mu \varphi }-1\right)\cdot {r}_{D}+{\mu }_{visc}\cdot \omega ,$`

To avoid discontinuity at zero relative velocity, the model defines the actuation force FS as a hyperbolic function:

`${F}_{A}={F}_{in}\cdot \mathrm{tanh}\left(\frac{4\omega }{{\omega }_{threshold}}\right),$`

In the previous formula, the parameters represent the following quantities:

ParameterDescription
FinForce input signal
ωthresholdAngular velocity threshold

Connection F is a physical signal port that represents the external tensioning force applied on the belt. Connection S is a conserving rotational port that represents the rotating drum shaft.

## Modeling Thermal Effects

You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, right-click the block in your model and, from the context menu, select Simscape > Block choices. Choose a variant that includes a thermal port. Specify the associated thermal parameters for the component.

## Assumptions and Limitations

• The model does not account for actuator flow consumption

## Ports

`F`

Physical signal port that represents the belt tensioning force

`S`

Rotational conserving port that represents the rotating drum shaft

`H`

Thermal conserving port. The thermal port is optional and is hidden by default. To expose the port, select a variant that includes a thermal port.

## Parameters

### Geometry

Radius of the drum contact surface. The parameter must be greater than zero. The default value is `150` `mm`.

Wrap angle

Contact angle between the flexible belt and the rotating drum. The parameter must be greater than zero. The default value is `270` `deg`.

### Friction

Viscous friction coefficient

Value of the viscous friction coefficient at the belt-drum contact surface. The parameter must be greater than or equal to zero. The default value is `.01` `n*m/(rad/s)`.

Temperature

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase left to right. The default value is `[280.0, 300.0, 320.0]` `K`.

This parameter is visible only when you select a block variant that includes a thermal port.

Contact friction coefficient

Value of the Coulomb friction coefficient at the belt-drum contact surface. The value is greater than zero. Unless you select a block variant that includes a thermal port, the default value is `0.3`.

If you select a block variant that includes a thermal port, you specify this parameter as array. The array is the same size as the array for the Temperature parameter and the values increase left to right. The default value for the thermal variant is `[0.1, 0.05, 0.03]`.

Angular velocity threshold

Angular velocity at which the friction coefficient at the belt-drum contact surface practically reaches its steady-state value. The parameter must be greater than zero. The default value is `0.01` `rad/s`.

### Thermal Port

These thermal parameters are visible only when you select a block variant that includes a thermal port.

Thermal mass

Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change. The default value is `50` `kJ/K`.

Initial temperature

Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses. The default value is `300` `K`.