Belt Pulley

Power transmission element with frictional belt wrapped around pulley circumference

Library:
Simscape / Driveline / Couplings & Drives

Description

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 pulley can optionally translate through port C, as is the case in a block and tackle system.

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 the leading belt end loses tension. When assembling a model, ensure that tension is maintained throughout the simulation. This can be done by adding mass to at least one of the belt ends or by adding a tensioner into your model. Use the Variable Viewer to ensure that any springs attached the belt are in tension. For more details on building a tensioner, see Best Practices for Modeling Pulley Networks.

Equations

If the relative velocity between the belt and pulley is positive or zero, that is $V_{r e l} \geq 0$ , the Belt Pulley block calculates friction force as

$F_{f r} = F_{B} - F_{c e n t r i f u g a l} = (F_{A} - F_{c e n t r i f u g a l}) * \exp (f * θ) .$

If the relative velocity is negative, that is $V_{r e l} < 0$ , the friction force is calculated as

$F_{f r} = F_{A} - F_{c e n t r i f u g a l} = (F_{B} - F_{c e n t r i f u g a l}) * \exp (f * θ) .$

The relative velocity is:

$V_{r e l} = V_{A} - ω_{S} * R - V_{C}$

$V_{r e l} = - V_{B} + ω_{S} * R + V_{C}$

If Belt type is set to either V-belt or Flat belt and Centrifugal force is set to Model centrifugal force, the centrifugal force is:

$F_{c e n t r i f u g a l} = ρ * {(V_{B} - V_{C})}^{2}$

where:

V_rel is the relative velocity between the belt and pulley periphery.
V_A is the branch A linear velocity.
V_B is the branch B linear velocity.
V_C is the pulley linear velocity at its center. If the pulley is not translating, this value is 0.
ω_S is the pulley angular velocity.
R is the pulley radius.
F_centrifugal is the belt centrifugal force.
ρ is the belt linear density.
F_fr is the friction force between the pulley and the belt.
F_A is the force acting along branch A.
F_B is the force acting along branch 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

$f' = \frac{f}{\sin (\frac{ϕ}{2})},$

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

$μ = - f * \tanh (4 * \frac{V_{r e l}}{V_{t h r}}),$

where

μ is the instantaneous value of the friction coefficient.
f is the steady-state value of the friction coefficient.
V_thr 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 friction force is calculated as

$F_{f r} = F_{A} - F_{c e n t r i f u g a l} = (F_{B} - F_{c e n t r i f u g a l}) * \exp (μ * θ) .$

The resulting torque delivered by the pulley is given as

$T_{S} = (F_{A} + F_{B}) * R * tanh (4 \frac{V_{rel}}{V_{thr}}) * tanh (\frac{F_{B}}{F_{thr}}) - ω_{S} * b .$

where:

T_S is the pulley torque.
b is the bearing viscous damping.
F_thr is the force threshold.

The resulting force exerted by the pulley center is:

$F_{C} = (F_{A} + F_{B}) * \sin (\frac{ϕ}{2}) .$

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.
The translation of the pulley center is assumed to be planar and travels along the bisect of the pulley wrap angle. The center velocity V_C and force F_C only account for the component along this line of motion.

Ports

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.

Conserving

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`S` — Pulley shaft angular velocity
mechanical rotational

Rotational conserving port associated with the pulley shaft.

`A` — Belt end A linear velocity
mechanical translational

Translational conserving port associated with belt end A.

`B` — Belt end B linear velocity
mechanical translational

Translational conserving port associated with belt end B.

`C` — Pulley center linear velocity
mechanical translational

Translational conserving port associated with pulley translational velocity. The pulley moves within the plane and along the bisect of the pulley wrap angle. When the relative velocity is positive and pulley translation is enabled, the pulley center moves.

Dependencies

To expose this port, set Pulley translation to On.

Parameters

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Belt

`Belt type` — Belt model
`Ideal - No slip` (default) | `Flat belt` | `V-belt`

Belt model:

Ideal - No slip — Model an ideal belt, which does not slip relative to the pulley.
Flat belt — Model a belt with a rectangular cross-section.
V-belt— Model a belt with a V-shaped cross-section.

Dependencies

This parameter affects the visibility of related belt parameters and the Contact settings.

`V-belt sheave angle` — Sheave angle
`30` `deg` (default) | positive scalar

Sheave angle of the V-belt.

Dependencies

This parameter is visible only when Belt type is set to V-belt.

`Number of V-belts` — Number of belts
`1` (default) | positive scalar integer

Number of V-belts.

Noninteger 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.

Dependencies

This parameter is visible only when Belt type is set to V-belt.

`Centrifugal force` — Centrifugal force model
`Do not model centrifugal force - Suitable for HIL simulation` (default) | `Model centrifugal force`

Option to include the effects of centrifugal force. If included, centrifugal force saturates to approximately 90 percent of the value of the force on each belt end.

Dependencies

This parameter is visible only when Belt type is set to Flat belt or V-belt.

If this parameter is set to Model centrifugal force, the Belt mass per unit length parameter is exposed.

`Belt mass per unit length` — Mass per unit length
`0.6` `kg/m` (default) | positive scalar

Centrifugal force contribution in terms of linear density expressed as mass per unit length.

Dependencies

Selecting Model centrifugal force for the Centrifugal force parameter exposes this parameter.

`Belt direction` — Initial belt end relative motion direction
`Ends move in opposite direction` (default) | `Ends move in same direction`

Relative direction of translational motion of one belt end with respect to the other.

Dependencies

This parameter is visible only when Belt type is set to Flat belt or V-belt.

`Maximum tension` — Tension threshold model
`No maximum tension` (default) | `Specify maximum tension`

Tension threshold model. If Specify maximum tension is selected and the belt tension on either end of the belt meets or exceeds the value that you specify for Belt maximum tension, the simulation stops and generates an assertion error.

Dependencies

Selecting Specify maximum tension exposes the Belt maximum tension parameter.

`Belt maximum tension` — Maximal tension threshold
`1e5` `N` (default) | positive scalar

Maximum allowable tension for each belt. When the tension on either end of the belt meets or exceeds this value, the simulation stops and generates an assertion error.

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

`Tension warning` — Slack threshold reporting
`Do not check tension` (default) | `Warn if leading end loses tension`

Whether the block generates a warning when the tension at either end of the belt falls below zero.

Pulley

`Pulley translation` — Whether to model pulley linear motion
`Off` (default) | `On`

Whether to model pulley linear motion. Setting this parameter to On exposes port C.

`Pulley radius` — Pulley radius
`.15` `m` (default) | positive scalar

Radius of the pulley.

`Bearing viscous friction coefficient` — Bearing viscous friction
`0` `N*m/(rad/s)` (default) | scalar

Viscous friction associated with the bearings that hold the axis of the pulley.

`Inertia` — Rotational inertia model
`No inertia` (default) | `Specify inertia and initial velocity`

Rotational inertia model.

Dependencies

Selecting Specify inertia and initial velocity exposes the Pulley inertia and Pulley initial velocity parameters.

`Pulley inertia` — Pulley inertia
`0.01` `kg*m^2` (default) | positive scalar

Rotational inertia of the pulley.

Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter exposes this parameter.

`Pulley initial rotational velocity` — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

Initial rotational velocity of the pulley.

Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter exposes this parameter.

`Pulley mass` — Pulley mass
`0.01` `kg` (default) | positive scalar

Pulley mass for inertia calculation.

Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter when Pulley translation is set to On exposes this parameter.

`Pulley initial translational velocity` — Initial pulley rotational velocity
`0` `m/s` (default) | scalar

Initial translational velocity of the pulley.

Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter when Pulley translation is set to On exposes this parameter.

Contact

Contact settings are only visible if the Belt type parameter in the Belt settings is set to Flat belt or V-belt

`Contact friction coefficient` — Friction coefficient
`0.5` (default) | positive scalar

Coulomb friction coefficient between the belt and the pulley surface.

`Wrap angle` — Belt-to-pulley contact angle
`180` `deg` (default) | positive scalar

Radial contact angle between the belt and the pulley.

`Velocity threshold` — Velocity threshold
`0.001` `n/s` (default) | positive scalar

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.

Model Examples

Power Window System

A motor-driven power window system. A DC motor drives the power window mechanism via a self-locking worm gear with the ratio 1 : 50. The power window mechanism consists of a cable drum and four pulleys all connected by a cable. The window is attached to the cable at two points by the lift plate. This ensures both sides of the window move at the same speed and in the same direction, keeping the window level. The model also includes viscous friction in the guide rails.

More About

expand all

Hardware-in-the-Loop Simulation

For optimal simulation performance, set the Belt > Centrifugal force parameter to Do not model centrifugal force - Suitable for HIL simulation.

Documentation

Belt Pulley

Description

Equations

Assumptions and Limitations

Ports

Conserving

S — Pulley shaft angular velocity mechanical rotational

A — Belt end A linear velocity mechanical translational

B — Belt end B linear velocity mechanical translational

C — Pulley center linear velocity mechanical translational

Dependencies

Parameters

Belt

Belt type — Belt model Ideal - No slip (default) | Flat belt | V-belt

Dependencies

V-belt sheave angle — Sheave angle 30 deg (default) | positive scalar

Dependencies

Number of V-belts — Number of belts 1 (default) | positive scalar integer

Dependencies

Centrifugal force — Centrifugal force model Do not model centrifugal force - Suitable for HIL simulation (default) | Model centrifugal force

Dependencies

Belt mass per unit length — Mass per unit length 0.6 kg/m (default) | positive scalar

Dependencies

Belt direction — Initial belt end relative motion direction Ends move in opposite direction (default) | Ends move in same direction

Dependencies

Maximum tension — Tension threshold model No maximum tension (default) | Specify maximum tension

Dependencies

Belt maximum tension — Maximal tension threshold 1e5 N (default) | positive scalar

Tension warning — Slack threshold reporting Do not check tension (default) | Warn if leading end loses tension

Pulley

Pulley translation — Whether to model pulley linear motion Off (default) | On

Pulley radius — Pulley radius .15 m (default) | positive scalar

Bearing viscous friction coefficient — Bearing viscous friction 0 N*m/(rad/s) (default) | scalar

Inertia — Rotational inertia model No inertia (default) | Specify inertia and initial velocity

Dependencies

Pulley inertia — Pulley inertia 0.01 kg*m^2 (default) | positive scalar

Dependencies

Pulley initial rotational velocity — Initial pulley rotational velocity 0 rad/s (default) | scalar

Dependencies

Pulley mass — Pulley mass 0.01 kg (default) | positive scalar

Dependencies

Pulley initial translational velocity — Initial pulley rotational velocity 0 m/s (default) | scalar

Dependencies

Contact

Contact friction coefficient — Friction coefficient 0.5 (default) | positive scalar

Wrap angle — Belt-to-pulley contact angle 180 deg (default) | positive scalar

Velocity threshold — Velocity threshold 0.001 n/s (default) | positive scalar

Model Examples

Power Window System

More About

Hardware-in-the-Loop Simulation

Extended Capabilities

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

See Also

Topics

Simscape Driveline Documentation

Support

`S` — Pulley shaft angular velocity
mechanical rotational

`A` — Belt end A linear velocity
mechanical translational

`B` — Belt end B linear velocity
mechanical translational

`C` — Pulley center linear velocity
mechanical translational

`Belt type` — Belt model
`Ideal - No slip` (default) | `Flat belt` | `V-belt`

`V-belt sheave angle` — Sheave angle
`30` `deg` (default) | positive scalar

`Number of V-belts` — Number of belts
`1` (default) | positive scalar integer

`Centrifugal force` — Centrifugal force model
`Do not model centrifugal force - Suitable for HIL simulation` (default) | `Model centrifugal force`

`Belt mass per unit length` — Mass per unit length
`0.6` `kg/m` (default) | positive scalar

`Belt direction` — Initial belt end relative motion direction
`Ends move in opposite direction` (default) | `Ends move in same direction`

`Maximum tension` — Tension threshold model
`No maximum tension` (default) | `Specify maximum tension`

`Belt maximum tension` — Maximal tension threshold
`1e5` `N` (default) | positive scalar

`Tension warning` — Slack threshold reporting
`Do not check tension` (default) | `Warn if leading end loses tension`

`Pulley translation` — Whether to model pulley linear motion
`Off` (default) | `On`

`Pulley radius` — Pulley radius
`.15` `m` (default) | positive scalar

`Bearing viscous friction coefficient` — Bearing viscous friction
`0` `N*m/(rad/s)` (default) | scalar

`Inertia` — Rotational inertia model
`No inertia` (default) | `Specify inertia and initial velocity`

`Pulley inertia` — Pulley inertia
`0.01` `kg*m^2` (default) | positive scalar

`Pulley initial rotational velocity` — Initial pulley rotational velocity
`0` `rad/s` (default) | scalar

`Pulley mass` — Pulley mass
`0.01` `kg` (default) | positive scalar

`Pulley initial translational velocity` — Initial pulley rotational velocity
`0` `m/s` (default) | scalar

`Contact friction coefficient` — Friction coefficient
`0.5` (default) | positive scalar

`Wrap angle` — Belt-to-pulley contact angle
`180` `deg` (default) | positive scalar

`Velocity threshold` — Velocity threshold
`0.001` `n/s` (default) | positive scalar

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