Double-Shoe Brake

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

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Libraries:
Simscape / Driveline / Brakes & Detents / Rotational

Description

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.

Equations

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:

$$T_{LS}=\frac{c\mu p_a{r}_D{}^2\left(\cos {\theta }_{sb}-\cos {\theta }_s\right)}{\sin {\theta }_a},$$

$$T_{TS}=\frac{c\mu p_b{r}_D{}^2\left(\cos {\theta }_{sb}-\cos {\theta }_s\right)}{\sin {\theta }_a},$$

$$c=r_a+r_p\cos {\theta }_p,$$

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_LS is the brake torque the leading shoe develops.

• T_TS is the brake torque the trailing shoe develops.

• μ is the effective contact friction coefficient.

• p_a is the maximum linear pressure in the leading shoe-drum contact.

• p_b is the maximum linear pressure in the trailing shoe-drum contact.

• r_D is the drum radius.

• θ_sb is the shoe beginning angle.

• θ_s is the shoe span angle.

• θ_a is the angle from hinge pin to maximum pressure point.

• c is the arm length of the cylinder force with respect to the hinge pin.

• r_p is the pin location radius.

• θ_p is the hinge pin location angle.

• r_a is the actuator location radius.

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

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

where:

• μ is the effective contact friction coefficient.

• μ_Coulomb is the contact friction coefficient.

• ω_shaft is the shaft velocity.

• ω_threshold is the angular 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 equations for determining the balance of moments for the leading shoe are

$$F=\frac{M_N-M_F}{c},$$

$$M_N=\frac{p_a{r}_p{r}_D}{\sin {\theta }_a}\left(\frac{1}{2}\left[{\theta }_s-{\theta }_{sb}\right]-\frac{1}{4}\left[\sin 2{\theta }_s-\sin 2{\theta }_{sb}\right]\right),$$

and

$$M_F=\frac{\mu p_a{r}_D}{\sin {\theta }_a}\left(r_D\left[\cos {\theta }_{sb}-\cos {\theta }_s\right]+\frac{r_p}{4}\left[\cos 2{\theta }_s-\cos 2{\theta }_{sb}\right]\right),$$

where:

• F is the actuation force.

• M_N is the moment acting on the leading shoe due to normal force.

• M_F is the moment acting on the leading shoe due to friction force.

• c is the arm length of the cylinder force with respect to the hinge pin.

• p_a is the maximum linear pressure at the shoe-drum contact surface.

• r_p is the pin location radius.

• θ_p is the hinge pin location angle.

• r_a is the actuator location radius.

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 > M_N.

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.

Faults

To model a fault in the Double-Shoe Brake block, in the Faults section, click the Add fault hyperlink next to the fault that you want to model. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

When you trigger a fault, the block applies the value of the Belt force when faulted parameter instead of the value at port F for the remainder of the simulation. When the value is 0, no braking occurs. When the value is relatively large, the brake is stuck.

Thermal Model

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

Limitations and Assumptions

• 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.

Ports

Input

F — Actuating force

physical signal

Physical signal input port associated with the applied actuating force.

Conserving

S — Drum shaft rotation

rotational mechanical

Rotational conserving port associated with the rotating drum shaft.

H — Heat flow

thermal

Thermal conserving port associated with heat flow.

Dependencies

This port is visible only when, in the Friction settings, the Thermal Port parameter is set to Model.

Exposing this port makes related settings visible.

Parameters

Geometry

Drum radius — Contact surface radius

150 mm (default) | positive scalar

Radius of the drum contact surface. The value must be greater than zero.

Actuator location radius — Center-to-center distance between drum and force line of action

100mm (default) | positive scalar

Distance between the drum center and the force line of action. The value must be greater than zero.

Pin location radius — Center-to-center distance between hinge pin and drum

125mm (default) | positive scalar

Distance between the hinge pin and drum centers. The parameter must be greater than zero.

Pin location angle — Angular distance from hinge pin to brake symmetry axis

15deg (default) | nonnegative scalar

Angular coordinate of the hinge pin location from the brake symmetry axis. The value must be greater than or equal to zero.

Shoe beginning angle — Angle between hings pin and friction material

5deg (default) | positive scalar

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 range0 ≤ θ_sb ≤ (π-pin location angle).

Shoe span angle — Friction material angle

120deg (default) | positive scalar

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 < θ_sb ≤ (π -pin location angle - shoe beginning angle).

Friction

Viscous friction coefficient — Viscous friction

.01 n*m/(rad/s) (default) | nonnegatvie scalar

Viscous friction coefficient at the contact surface. The value must be greater than or equal to zero.

Thermal port — Thermal model

Omit (default) | Model

Model for heat flow and temperature change:

• Omit — Neglect thermal dynamics.

• Model — Include thermal dynamics.

Dependencies

When this parameter is set to Model, thermal port H and related settings are visible.

Temperature — Temperature

[280, 300, 320] K (default) | increasing vector

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase left to right.

Dependencies

This parameter is only visible when the Thermal Port parameter is set to Model.

Contact friction coefficient — Coulomb friction

0.3 (default) | [0.1, 0.05, 0.03] (thermal model default) | positive scalar or vector

Coulomb friction coefficient at the belt-drum contact surface. The value must be greater than zero. For the thermal model:

• The number of elements in the vector must be the same as the number of elements in the specified vector for the Temperature parameter

• The values increase left to right.

• Each value must be greater than zero.

Dependencies

This parameter is specified as a:

• Scalar when the Thermal Port parameter is set to Omit.

• Vector when the Thermal Port parameter is set to Model.

Angular velocity threshold — Rotational velocity required for near steady-state contact friction

0.01 rad/s (default) | positive scalar

Angular velocity at which the contact friction coefficient practically reaches its steady-state value. The value must be greater than zero.

Faults

To modify the faults, create a fault and, in the block dialog, click Open fault properties. In the Property Inspector, click the Fault behavior link to open the faults.

Belt tension fault — Option to model belt tension fault

Add fault

Whether to model a belt tension fault in the block. To add a fault, click the Add fault hyperlink.

Belt force when faulted — Faulted belt force

0 N (default) | nonnegative scalar

Belt force when faulted. When you trigger a fault, the block applies the value of the Belt force when faulted parameter instead of the value at port F for the remainder of the simulation. When the value is 0, no braking occurs. When the value is relatively large, the brake is stuck.

Dependencies

To enable this parameter, first enable faults for the block by clicking the Add fault hyperlink.

Thermal Port

Thermal mass — Resistance to temperature change

50 kJ/K (default) | scalar

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.

Dependencies

To enable this parameter, set Thermal Port to Model.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2012b

R2024a: Model faults with the Simscape faults interface

The Double-Shoe Brake block now supports the Simscape faults interface. To learn more, see Introduction to Simscape Faults.