# Double-Shoe Brake

Frictional brake with two pivoted shoes diametrically positioned about rotating drum

## Library

Brakes & Detents/Rotational

## Description

The block represents a frictional brake with two pivoted rigid shoes that press against a rotating drum to produce a braking action. The rigid shoes sit inside or outside the rotating drum in a diametrically opposed configuration. A positive actuating force causes the rigid shoes to press against the rotating drum. Viscous and contact friction between the drum and the rigid shoe surfaces cause the rotating drum to decelerate. Double-shoe brakes provide high braking torque with small actuator deflections in applications that include motor vehicles and some heavy machinery. The model employs a simple parameterization with readily accessible brake geometry and friction parameters.

In this schematic, a) represents an internal double-shoe brake, and b) represents an external double-shoe brake. In both configurations, a positive actuation force F brings the shoe and drum friction surfaces into contact. The result is a friction torque that causes deceleration of the rotating drum. Zero and negative forces do not bring the shoe and drum friction surfaces into contact and produce zero braking torque.

The model uses the long-shoe approximation. Contact angles smaller than 45° produce less accurate results. The following formulas provide the friction torque the leading and trailing shoes develop, respectively:

`${T}_{LS}=\frac{\mu \cdot {p}_{a}\cdot {r}_{D}{}^{2}\left(\mathrm{cos}{\theta }_{sb}-\mathrm{cos}{\theta }_{s}\right)}{\mathrm{sin}{\theta }_{a}}$`
`${T}_{TS}=\frac{\mu \cdot {p}_{b}\cdot {r}_{D}{}^{2}\left(\mathrm{cos}{\theta }_{sb}-\mathrm{cos}{\theta }_{s}\right)}{\mathrm{sin}{\theta }_{a}}$`

In the formulas, the parameters have the following meaning:

ParameterDescription
TLSBrake torque the leading shoe develops
TTSBrake torque the trailing shoe develops
μEffective contact friction coefficient
paMaximum linear pressure in the leading shoe-drum contact
pbMaximum linear pressure in the trailing shoe-drum contact
θsbShoe beginning angle
θsShoe span angle
θaAngle from hinge pin to maximum pressure point
`${\theta }_{a}=\left\{\begin{array}{cc}{\theta }_{s}& if\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {\theta }_{s}\le \pi }{2}\\ \pi }{2}& if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta }_{s}\ge \pi }{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$`

The model assumes that only Coulomb friction acts at the shoe-drum surface contact. Zero relative velocity between the drum and the shoes produces zero Coulomb friction. To avoid discontinuity at zero relative velocity, the friction coefficient formula employs the following hyperbolic function:

`$\mu ={\mu }_{Coulomb}\cdot \mathrm{tanh}\left(\frac{4{\omega }_{shaft}}{{\omega }_{threshold}}\right)$`

In the formula, the parameters have the following meaning:

ParameterDescription
μEffective contact friction coefficient
μCoulombContact friction coefficient
ωshaftShaft velocity
ωthresholdAngular velocity threshold

Balancing the moments that act on each shoe with respect to the pin yields the pressure acting at the shoe-drum surface contact. The following formula provides the balance of moments for the leading shoe.

`$F=\frac{{M}_{N}-{M}_{F}}{c}$`

In the formula, the parameters have the following meaning:

ParameterDescription
FActuation Force
MNMoment acting on the leading shoe due to normal force
MFMoment acting on the leading shoe due to friction force
cArm length of the cylinder force with respect to the hinge pin

The following equations give MN, MF, and c, respectively.

`${M}_{N}=\frac{{p}_{a}{r}_{p}{r}_{D}}{\mathrm{sin}{\theta }_{a}}\left(\frac{1}{2}\left[{\theta }_{s}-{\theta }_{sb}\right]-\frac{1}{4}\left[\mathrm{sin}2{\theta }_{s}-\mathrm{sin}2{\theta }_{sb}\right]\right)$`
`${M}_{F}=\frac{\mu {p}_{a}{r}_{D}}{\mathrm{sin}{\theta }_{a}}\left({r}_{D}\left[\mathrm{cos}{\theta }_{sb}-\mathrm{cos}{\theta }_{s}\right]+\frac{{r}_{p}}{4}\left[\mathrm{cos}2{\theta }_{s}-\mathrm{cos}2{\theta }_{sb}\right]\right)$`
`$c={r}_{a}+{r}_{p}\mathrm{cos}{\theta }_{p}$`

The parameters have the following meaning:

ParameterDescription
paMaximum linear pressure at the shoe-drum contact surface
θpHinge pin location angle

The model does not simulate self-locking brakes. If brake geometry and friction parameters cause a self-locking condition, the model produces a simulation error. A brake self-locks if the friction moment exceeds the moment due to normal forces:

MF>MN

The following formula provides the balance of moments for the trailing shoe.

`$F=\frac{{M}_{N}+{M}_{F}}{c}$`

Formulas and parameters for MN, MF, and c share the definition in the leading shoe section.

The net braking torque has the formula:

`$T={T}_{LS}+{T}_{TS}+{\mu }_{visc}$`

In the formula, parameter μvisc is the viscous friction coefficient.

## 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 brake uses the long-shoe approximation

• The brake geometry does not self-lock

• The model does not account for actuator flow consumption

## Ports

`F`

Physical signal port that represents the brake actuating 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`.

Distance between the drum center and the force line of action. The parameter must be greater than zero. The default value is `100` `mm`.

Distance between the hinge pin and drum centers. The parameter must be greater than zero. The default value is `125` `mm`.

Pin location angle

Angular coordinate of the hinge pin location from the brake symmetry axis. The parameter must be greater than or equal to zero. The default value is `15` `deg`.

Shoe beginning angle

Angle between the hinge pin and the beginning of the friction material linen of the shoe. The value of the parameter must be in the range 0 ≤ θsb ≤ (π-pin location angle). The default value is `5` `deg`.

Shoe span angle

Angle between the beginning and the end of the friction material linen on the shoe. The value of the parameter must be in the range 0 < θsb ≤ (π -pin location angle - shoe beginning angle). The default value is `120` `deg`.

### Friction

Viscous friction coefficient

Value of the viscous friction coefficient at the 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 contact friction coefficient 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`.