Documentation |
Frictional brake with two pivoted shoes diametrically positioned about rotating drum
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:
Parameter | Description |
---|---|
T_{LS} | Brake torque the leading shoe develops |
T_{TS} | Brake torque the trailing shoe develops |
μ | Effective contact friction coefficient |
p_{a} | Maximum linear pressure in the leading shoe-drum contact |
p_{b} | Maximum linear pressure in the trailing shoe-drum contact |
r_{D} | Drum radius |
θ_{sb} | Shoe beginning angle |
θ_{s} | Shoe span angle |
θ_{a} | Angle from hinge pin to maximum pressure point $${\theta}_{a}=\{\begin{array}{cc}{\theta}_{s}& if\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {\theta}_{s}\le \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.& if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta}_{s}\ge \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$$ |
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:
Parameter | Description |
---|---|
μ | Effective contact friction coefficient |
μ_{Coulomb} | Contact friction coefficient |
ω_{shaft} | Shaft velocity |
ω_{threshold} | 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 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:
Parameter | Description |
---|---|
F | Actuation Force |
M_{N} | Moment acting on the leading shoe due to normal force |
M_{F} | Moment acting on the leading shoe due to friction force |
c | Arm length of the cylinder force with respect to the hinge pin |
The following equations give M_{N}, M_{F}, 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:
Parameter | Description |
---|---|
p_{a} | Maximum linear pressure at the shoe-drum contact surface |
r_{p} | Pin location radius |
θ_{p} | Hinge pin location angle |
r_{a} | 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:
M_{F}>M_{N}
The following formula provides the balance of moments for the trailing shoe.
$$F=\frac{{M}_{N}+{M}_{F}}{c}$$
Formulas and parameters for M_{N}, M_{F}, 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 μ_{visc} is the viscous friction coefficient.
The brake uses the long-shoe approximation
The brake geometry does not self-lock
The model does not account for actuator flow consumption
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.
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.
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 ≤ θ_{sb} ≤ (π-pin location angle). The default value is 5 deg.
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). The default value is 130 deg.
Value of the viscous friction coefficient at the contact surface. The parameter must be greater than or equal to zero. The default value is 0.01 n*m/(rad/s).
Friction coefficient at the shoe-drum contact surface. The parameter must be greater than zero. The default value is 0.3.
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.