Documentation |
Frictional brake with flexible band wrapped around rotating drum
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:
Parameter | Description |
---|---|
T | Braking torque |
F_{TB} | Force acting on the tense branch of the band |
F_{A} | External brake actuation force |
r_{D} | Drum radius |
μ_{visc} | Viscous friction coefficient |
μ | Contact friction coefficient |
ϕ | Wrap angle |
Forces F_{TB} and F_{A} satisfy the relationship:
$$\frac{{F}_{TB}}{{F}_{A}}={e}^{\mu \varphi},$$
Replacing the relationship in the braking torque formula eliminates force F_{TB}:
$$T={F}_{A}\left({e}^{\mu \varphi}-1\right)\cdot {r}_{D}+{\mu}_{visc}\cdot \omega ,$$
To avoid discontinuity at zero relative velocity, the model defines the actuation force F_{S} 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:
Parameter | Description |
---|---|
F_{in} | Force input signal |
ω_{threshold} | Angular 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.
Radius of the drum contact surface. The parameter must be greater than zero. The default value is 150 mm.
Contact angle between the flexible belt and the rotating drum. The parameter must be greater than zero. The default value is 130 mm.
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).
Value of the Coulomb friction coefficient at the belt-drum contact surface. The parameter must be greater than zero. The default value is 0.3.
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.