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# Belt Pulley

Flexible belt wrapped around a pulley

## Library

Couplings & Drives

## Description

The block represents a pulley wrapped with a flexible flat band or V-shaped belt. 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 sign convention for the model is as shown.

Pulley rotation in the positive direction causes branch B to move in the positive direction and branch A to move in the negative direction. The model assumes that the belt is noncompliant along its length, and that the belt exhibits no resistance to motion when flexed around the pulley.

The Belt Pulley block uses the following equations to calculate friction force:

• Vrel is the relative velocity between the belt and pulley periphery.

• VA is the Branch A linear velocity.

• VB is the Branch B linear velocity.

• ωS is the pulley angular velocity.

• R is the pulley radius.

• FC is the belt centrifugal force.

• ρ is the belt linear density.

• Ffr is the friction force between the pulley and the belt.

• FA is the force acting along branch A.

• FB is the force acting along branch B.

• f is the friction coefficient.

• θ is the contact wrap angle.

For a flat band, specify the value of f directly in the block parameters dialog box. For a V-belt, the model calculates the value using this equation:

• f' is the effective friction coefficient for a V-belt.

• Φ is the V-belt sheave angle.

The idealization of the discontinuity at Vrel = 0 is both difficult for the solver to resolve and not completely physically accurate. To alleviate this issue, the friction coefficient is assumed to change its value as a function of the relative velocity, with the following relationship:

• μ is the instantaneous value of the friction coefficient.

• f is the steady-state value of the friction coefficient.

• Vthr is the friction velocity threshold.

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 parameter uses the following equation to calculate the friction force:

The resulting torque on the pulley is given by the following equation:

• TS is the pulley torque.

• b is the bearing viscous damping.

The equations that this model uses are effective for transmitting power to either branch of the belt or to and from the pulley. The tight and loose branch 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 at the MATLAB® command line when either belt end loses tension. When assembling a model, ensure that tension is maintained throughout the simulation. Interpret the results with this in mind.

Connection S is a conserving rotational terminal associated with the pulley shaft. Connections A and B are the conserving mechanical translational ports associated with the belt ends. The sign convention is such that a positive rotation in port S tends to give a negative translation for port A and a positive translation for port B.

## Assumptions and Limitations

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

• Both belt ends maintain adequate tension throughout the simulation.

## Dialog Box and Parameters

The dialog box includes three tabs. Use the Belt pane to specify belt characteristics.

### Belt type

Type of flexible element modeled.

• Flat band — (Default)

• V-belt— Exposes the V-belt sheave angle and Number of V-belts parameters.

### V-belt sheave angle

Sheave angle of the V-belt. Default is 30 deg.

The V-belt sheave angle parameter is visible only when the Belt type parameter is V-belt.

### Number of V-belts

Number of parallel V-belts. Default is 1.

Non-integer values are rounded to the nearest integer. Increasing the number of belts increases the friction force, effective mass per unit length, and maximum allowable tension. Non-integer values are rounded to the nearest integer.

The Number of V-belts parameter is visible only when the Belt type parameter is V-belt.

### Belt mass per unit length

Linear density of each belt. Default is 0.6 kg/m.

### Maximum tension

Specifies whether the block throws an assertion when the belt tension is too large.

• No maximum tension — (Default)

• Specify maximum tension — Throws an assertion when the belt tension is too high. Exposes the Belt maximum tension parameter.

### Belt maximum tension

Maximum allowable tension for each belt. Default is 1e+5 N.

When the tension on either end of the belt meets or exceeds this value, the simulation ends with an error.

The Belt maximum tension parameter is visible only when the Maximum tension parameter is Specify maximum tension.

### Tension Warning

Specifies whether the block writes a warning at the MATLAB command line when the tension at either end of the belt falls below zero.

• Do not check belt tension — (Default)

• Warn if either belt end loses tension

Select the Pulley tab to specify the pulley characteristics.

Radius of the pulley. Default is 0.15 m.

### Bearing viscous friction coefficient

Viscous friction associated with the bearings that hold the axis of the pulley. Default is 0.001 N*m/(rad/s).

### Inertia

Specifies whether the block models rotational inertia of the pulley.

• No inertia — (Default)

• Specify inertia and initial velocity — Models rotational inertia. Exposes the Pulley inertia and Pulley initial velocity parameters.

### Pulley inertia

Rotational inertia of the pulley. Default is 0.01 kg*m^2.

The Pulley inertia parameter is visible only when the Inertia parameter is Specify inertia and initial velocity.

### Pulley initial velocity

Initial rotational velocity of the pulley. Default is 0 rad/s.

The Pulley initial velocity parameter is visible only when the Inertia parameter is Specify inertia and initial velocity.

Use the Contact pane to specify contact characteristics.

### Contact friction coefficient

Coulomb friction coefficient between the belt and the pulley surface. Default is 0.5.

### Wrap angle

Angle of contact between the belt and pulley. Default is 180 deg.

### Friction velocity threshold

Relative velocity required for peak kinetic friction in the contact. The friction velocity threshold improves the numerical stability of the simulation by ensuring that the force is continuous when the direction of the velocity changes. Default is 0.001 m/s.

## Ports

PortDescription

S

Conserving rotational port associated with the pulley shaft

A

Conserving translational port associated with belt end A

B

Conserving translational port associated with belt end B

## Example

The sdl_power_windowsdl_power_window SimDriveline™ example model shows the use of the Belt Pulley block.