Frictional brake with two pivoted shoes diametrically positioned about a rotating drum

**Library:**Simscape / Driveline / Brakes & Detents / Rotational

The Double-Shoe Brake 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 the 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. The equations for the friction torque that the leading and trailing shoes develop are:

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

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

$$c={r}_{a}+{r}_{p}\mathrm{cos}{\theta}_{p},$$

where for $$0\le {\theta}_{s}\le \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$,

$${\theta}_{a}={\theta}_{s},$$

and for $${\theta}_{s}\ge \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$,

$${\theta}_{a}=\frac{\pi}{2}.$$

Where:

*T*is the brake torque the leading shoe develops._{LS}*T*is the brake torque the trailing shoe develops._{TS}*μ*is the effective contact friction coefficient.*p*is the maximum linear pressure in the leading shoe-drum contact._{a}*p*is the maximum linear pressure in the trailing shoe-drum contact._{b}*r*is the drum radius._{D}*θ*is the shoe beginning angle._{sb}*θ*is the shoe span angle._{s}*θ*is the angle from hinge pin to maximum pressure point._{a}*c*is the arm length of the cylinder force with respect to the hinge pin.*r*is the pin location radius._{p}*θ*is the hinge pin location angle._{p}*r*is the actuator location radius._{a}

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 hyperbolic function

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

where:

*μ*is the effective contact friction coefficient.*μ*is the contact friction coefficient._{Coulomb}*ω*is the shaft velocity._{shaft}*ω*is the angular velocity threshold._{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 equations for determining the balance of moments for the leading shoe are

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

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

and

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

where:

*F*is the actuation force.*M*is the moment acting on the leading shoe due to normal force._{N}*M*is the moment acting on the leading shoe due to friction force._{F}*c*is the arm length of the cylinder force with respect to the hinge pin.*p*is the maximum linear pressure at the shoe-drum contact surface._{a}*r*is the pin location radius._{p}*θ*is the hinge pin location angle._{p}*r*is the actuator location radius._{a}

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,
that is, when *M _{F}* >

The balance of moments for the trailing shoe is

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

The net braking torque is

$$T={T}_{LS}+{T}_{TS}+{\mu}_{visc}*{\omega}_{shaft},$$

where *μ _{visc}* is the viscous friction
coefficient.

You can model the effects of heat flow and temperature change by exposing the optional thermal
port. To expose the port, in the **Friction** settings, set the
**Thermal Port** parameter to `Model`

.
Exposing the port also exposes or changes the default value for these related settings,
parameters, and variables:

**Friction**>**Temperature****Friction**>**Static friction coefficient vector****Friction**>**Coulomb friction coefficient vector****Friction**>**Contact friction coefficient vector****Thermal Port**>**Thermal mass****Variables**>**Temperature**

Use the **Variables** settings to set the priority and initial target values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables (Simscape).

Variable settings are visible only when, in the **Friction** settings, the
**Thermal port** parameter is set to
`Model`

.

Contact angles smaller than 45° produce less accurate results.

The brake uses the long-shoe approximation.

The brake geometry does not self-lock.

The model does not account for actuator flow consumption.