Power transmission element with frictional belt wrapped around pulley circumference

**Library:**Simscape / Driveline / Couplings & Drives

The Belt Pulley block represents a pulley wrapped in a flexible ideal, flat, or V-shaped belt. The ideal belt does not slip relative to the pulley surface.

The model accounts for friction between the flexible belt and the pulley periphery. If the friction force is not sufficient to drive the load, the model allows slip. The relationship between the tensions in the tight and loose branches conforms to the Euler equation. The model accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.

The block allows you to select the relative belt direction of motion. The two belt ends can move in equal or opposite directions. The block model assumes noncompliance in the belt and no resistance to motion due to wrapping around the pulley.

The block equations model power transmission between the belt branches or to/from the pulley. The tight and loose branches use the same calculation. Without sufficient tension, the frictional force is not enough to transmit power between the pulley and belt.

The model is valid when both ends of the belt are in tension. An optional warning can
display in the Simulink^{®} Diagnostic Viewer when either belt end loses tension. When assembling a
model, ensure that tension is maintained throughout the simulation. Consider this
requirement when you interpret the simulation results.

If the relative velocity between the belt and pulley is positive or zero, that is $${V}_{rel}\ge 0$$, the Belt Pulley block calculates friction force as

$${F}_{fr}={F}_{B}-{F}_{C}=\left({F}_{A}-{F}_{C}\right)*\mathrm{exp}\left(f*\theta \right).$$

If the relative velocity is negative, that is $${V}_{rel}<0$$, the friction force is calculated as

$${F}_{fr}={F}_{A}-{F}_{C}=\left({F}_{B}-{F}_{C}\right)\ast \mathrm{exp}\left(f\ast \theta \right).$$

In both cases:

$${V}_{rel}={V}_{A}-{\omega}_{S}\ast R$$

$${F}_{C}=\rho \ast {V}_{B}^{2}$$

$${V}_{A}=-{V}_{B}$$

where:

*V*is the relative velocity between the belt and pulley periphery._{rel}*V*is the branch A linear velocity._{A}*V*is the branch B linear velocity._{B}*ω*is the pulley angular velocity._{S}*R*is the pulley radius.*F*is the belt centrifugal force._{C}*ρ*is the belt linear density.*F*is the friction force between the pulley and the belt._{fr}*F*is the force acting along branch A._{A}*F*is the force acting along branch B._{B}*f*is the friction coefficient.*θ*is the contact wrap angle.

For a flat belt, specify the value of *f* directly in the block
parameters dialog box. For a V-belt, the model calculates the value as

$$f\text{'}=\frac{f}{\mathrm{sin}\left(\raisebox{1ex}{$\varphi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)},$$

where:

*f*is the effective friction coefficient for a V-belt.^{'}*Φ*is the V-belt sheave angle.

The idealization of the discontinuity at
*V _{rel}* = 0 is both difficult for the
solver to resolve and not physically accurate. To alleviate this issue, the friction
coefficient is assumed to change its value as a function of the relative velocity
such that

$$\mu =-f\ast \mathrm{tanh}\left(4\ast \raisebox{1ex}{${V}_{rel}$}\!\left/ \!\raisebox{-1ex}{${V}_{thr}$}\right.\right),$$

where

*μ*is the instantaneous value of the friction coefficient.*f*is the steady-state value of the friction coefficient.*V*is the friction velocity threshold._{thr}

The friction velocity threshold controls the width of the region within which the
friction coefficient changes its value from zero to a steady-state maximum. The
friction velocity threshold specifies the velocity at which the hyperbolic tangent
equals 0.999. The smaller the value, the steeper is the change of
*μ*.

This friction force is calculated as

$${F}_{fr}={F}_{A}-{F}_{C}=\left({F}_{B}-{F}_{C}\right)\ast \mathrm{exp}\left(\mu \ast \theta \right).$$

The resulting torque on the pulley is given as

$${T}_{S}=\left({F}_{A}+{F}_{B}\right)\ast R\ast \text{tanh}\left(\text{4}\frac{{V}_{\text{rel}}}{{V}_{\text{thr}}}\right)\ast \text{tanh}\left(\frac{{F}_{B}}{{F}_{\text{thr}}}\right)-{\omega}_{S}\ast b.$$

where:

*T*is the pulley torque._{S}*b*is the bearing viscous damping.*F*is the force threshold._{thr}

The model does not account for compliance along the length of the belt.

Both belt ends maintain adequate tension throughout the simulation.

The sign convention is such that, when **Belt direction** is set to
`Ends move in opposite direction`

, a positive rotation in
port **S** tends to give a negative translation for port
**A** and a positive translation for port
**B**.