Belt Pulley

Power transmission element with frictional belt wrapped around pulley circumference

Libraries:
Simscape / Driveline / Couplings & Drives

Description

The Belt Pulley block represents a pulley wrapped in a flexible ideal, flat, or V-shaped belt. When you set Belt type to Ideal - No slip, the belt does not slip relative to the pulley surfaces. You can optionally enable pulley linear translation.

The block accounts for friction between the flexible belt and the pulley surface. The tensioner slips when the load exceeds the friction force. The block accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.

The belt ends can either move in the same direction or in the opposite direction.

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

Your model is valid when both ends of the belt are in tension. You can choose to display a warning in the Simulink^® Diagnostic Viewer when the leading belt end loses tension. When assembling a model, ensure that tension is maintained throughout the simulation by adding mass to at least one of the belt ends or by adding a tensioner like the Cable block. Use the Variable Viewer to ensure that any springs attached to the belt are in tension. For more details on building a tensioner, see Best Practices for Modeling Pulley Networks.

You can set Friction model to Modal to use a modal parameterization for the pulley. Choose the modal parameterization for greater numerical robustness. To enable this parameter, set Pulley translation to Off.

The Belt Pulley block uses a composite implementation of the Fundamental Friction Clutch block to produce the conditions for the modal parameterization.

Equations

For either setting of the Friction model parameter, the block equations refer to these quantities:

β is the belt direction sign. When you set Belt direction to Ends move in same direction, β = 1. Otherwise, β = -1.
V_rel is the relative velocity between the belt and pulley periphery. V_rel = 0 when Belt type is Ideal - No slip.
V_A is the linear velocity of branch A.
V_B is the linear velocity of branch B.
V_C is the linear velocity of the pulley at its center. If Pulley translation is Off, the block constrains this to 0.
ω_S is the angular velocity of the pulley contact surface.
R is the radius of the pulley.
F_centrifugal is centrifugal force of the belt.
F_C is the force acting through the pulley centroid. When you specify a value for Inertia, F_C includes force due to the pulley mass acceleration.
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.
θ is the contact wrap angle.
τ_S is the pulley torque.

The kinematic constraints between the pulley and belt are:

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

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

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

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

When you set Pulley translation to On, the force balancing equation is:

$F_{C} = (β F_{A} - F_{B} - F_{c e n t r i f u g a l, s m o o t h, s a t}) \cdot \sin (\frac{θ}{2}) .$

To calculate F_{centrifugal,smooth,sat}, the block

Smooths F_centrifugal using F_thr. You can increase smoothing by raising F_thr, and you can decrease smoothing by lowering F_thr.
Saturates values above βF_A-F_B.

The sign convention is such that, when Belt direction is Ends move in opposite direction, a positive rotation in port S results in a negative translation for port A and a positive translation for port B.

To enable the Friction model parameter, set Belt type to Flat belt or V-belt and Pulley translation to Off.

Continuous Friction

When you set Friction model to Continuous, the block equations refer to these quantities:

μ is the Contact friction coefficient parameter.
μ_smoothed is the instantaneous value of the friction coefficient.
V_thr is the Velocity threshold parameter.
b is the viscous damping of the bearing.
F_thr is the Force threshold parameter.

The instantaneous friction coefficient is a function of the relative velocity such that

$μ_{s m o o t h e d} = μ_{s h e a v e} \tanh (4 \frac{V_{r e l}}{V_{t h r}}),$

where the hyperbolic tangent function maintains numerical robustness by ensuring a smooth and continuous output for V_rel zero-crossings.

For a V-belt, the block derives the contact friction value using the sheave angle:

$μ_{s h e a v e} = \frac{μ}{\sin (\frac{ϕ}{2})},$

where:

μ_sheave is the effective friction coefficient.
Φ is the sheave angle.

For a flat belt, μ_sheave = μ.

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 the change of μ.

The block determines the effect of friction on the force at the belt ends as:

$- β F_{A} - F_{c e n t r i f u g a l} = (F_{B} - F_{c e n t r i f u g a l}) e^{- μ_{s m o o t h e d} θ},$

which follows the form of the capstan equation, also known as the Euler-Eytelwein equation. The torque acting on the pulley is

$τ_{S} = (- β F_{A} - F_{B}) R σ + ω_{S} b,$

where σ = 1 when you set Belt type to Ideal - No slip. Otherwise,

$σ = \tanh (4 \frac{V_{r e l}}{V_{t h r}}) \tanh (\frac{F_{B}}{F_{t h r}}) .$

Modal Friction

For a V-belt, the block derives the static and kinetic friction values by using the sheave angle:

$\begin{array}{l} μ_{S t a t i c, s h e a v e} = \frac{μ_{S t a t i c}}{\sin (\frac{ϕ}{2})} \\ μ_{K i n e t i c, s h e a v e} = \frac{μ_{K i n e t i c}}{\sin (\frac{ϕ}{2})} \end{array}$

where:

μ_Static is the Static friction coefficient parameter.
μ_{Static,sheave} is effective static friction coefficient.
μ_Kinetic is the Kinetic friction coefficient parameter.
μ_{Kinetic,sheave} is effective kinetic friction coefficient.
Φ is the sheave angle.

For a flat belt, μ_{Static,sheave} = μ_Static, and μ_{Kinetic,sheave} = μ_Kinetic.

When you set Friction model to Modal, the block calculates the maximum static friction force before slipping as

$F_{F r i c t i o n, S t a t i c, M a x} = (F_{D r i v e} - F_{C e n t r i f u g a l}) \cdot (1 - e^{- μ_{S t a t i c, s h e a v e} θ}),$

where F_{Friction,Static,Max} is the maximum force magnitude due to static friction, and F_Drive is the force due to tension felt at the end of the pulley with the greatest force magnitude. As with F_centrifugal, the block uses F_thr to smooth F_Drive. To increase or decrease the smoothing on F_Drive, raise of lower the value of the Force threshold parameter.

The block smoothly saturates F_{Friction,Static,Max} to be greater than or equal to 0 as

$F_{F r i c t i o n, S t a t i c, M a x, S m o o t h} = 0.5 (F_{F r i c t i o n, S t a t i c, M a x} + \sqrt{F_{F r i c t i o n, S t a t i c, M a x}^{2} + {(R F_{t h r})}^{2}}) .$

During slip,

$F_{F r i c t i o n, S l i p} = \frac{μ_{K i n e t i c, s h e a v e}}{μ_{S t a t i c, s h e a v e}} F_{F r i c t i o n, S t a t i c, M a x},$

where μ_{Kinetic,sheave} is the kinetic friction coefficient. The block resolves the torque balance using

$τ_{S} = ω_{S} b + R F_{F r i c t i o n},$

and

$R β F_{A} + R F_{B} = - R F_{F r i c t i o n},$

where Simscape™ logs RF_Friction as fundamental_clutch.torque.

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.

Open Model

Compound Pulley

A block and tackle system that is represented by the Belt Pulley and Cable blocks from the Simscape™ Driveline™ library. This compound pulley is rigged as a threefold purchase with two triple blocks. The system has a mechanical advantage of 6.

Open Model

Assumptions and Limitations

The block assumes noncompliance along the length of the belt.
The block assumes both belt ends maintain adequate tension throughout the simulation.
The block treats the translation of the pulley center as planar where the pulley 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.
The Eytelwein equation for belt friction neglects the effect of pulley translation on friction.

Ports

Conserving

expand all

S — Pulley shaft angular velocity
mechanical rotational

Mechanical rotational conserving port associated with the angular velocity of the pulley shaft.

A — Belt end A linear velocity
mechanical translational

Mechanical translational conserving port associated with the linear velocity of belt end A.

B — Belt end B linear velocity
mechanical translational

Mechanical translational conserving port associated with the linear velocity of belt end B.

C — Pulley center linear velocity
mechanical translational

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

Dependencies

To enable this port, set Pulley translation to On.

Parameters

expand all

Belt

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

Belt type selection. The belt type affects slip conditions.

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

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

Sheave angle of the V-belt.

Dependencies

To enable this parameter, set Belt type to V-belt.

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

Number of V-belts.

The block rounds noninteger values to the nearest integer. Increasing the number of belts increases the effective mass per unit length and maximum allowable tension.

Dependencies

To enable this parameter, set Belt type to V-belt.

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

Option to include the effects of centrifugal force. If you set this parameter to Model centrifugal force, centrifugal force saturates to approximately 90% of the value of the force on each belt end.

Dependencies

To enable this parameter, set Belt type to Flat belt or V-belt.

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

To enable this parameter, set Belt type to Flat belt or V-belt and Centrifugal force to Model centrifugal force.

Belt direction — Initial relative motion direction of the belt end
`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.

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

Option to specify a maximum tension. If you select Specify maximum tension and the belt tension on either end of the belt meets or exceeds the value that you specify for the Belt maximum tension parameter, the simulation stops and generates an assertion error.

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.

Dependencies

To enable this parameter, set Maximum tension 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.

Note

When combining Belt Pulley blocks with Rope blocks to form loops, set the Belt Pulley block Tension warning parameter to Do not check tension and set the Rope block Tension warning parameter to Warn if rope loses tension.

Pulley

Pulley translation — Option to allow linear motion
`Off` (default) | `On`

Option to allow the pulley to translate. Setting this parameter to On enables port C.

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

Radius of the pulley.

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

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

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

Option to parameterize the rotational inertia with an initial velocity.

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

Rotational inertia of the pulley.

Dependencies

To enable this parameter, set Inertia to Specify inertia and initial velocity.

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

Initial rotational velocity of the pulley.

Dependencies

To enable this parameter, set Inertia to Specify inertia and initial velocity.

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

Pulley mass for the inertia calculation.

Dependencies

To enable this parameter, set

Pulley translation to On
Inertia to Specify inertia and initial velocity

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

Initial translational velocity of the pulley.

Dependencies

To enable this parameter, set

Pulley translation to On
Inertia to Specify inertia and initial velocity

Contact

To enable these settings, set Belt type to Flat belt or V-belt.

Friction model — Friction method selection
`Modal` (default) | `Continuous`

Option to parameterize continuous or modal friction.

Dependencies

To enable this parameter, set Pulley translation to Off.

Initial state — Initial state of pulley
`Unlocked` (default) | `Locked`

Option to initialize the simulation with the pulley locked or unlocked.

Dependencies

To enable this parameter, set

Pulley translation to Off
Friction model to Modal

Static friction coefficient — Static friction coefficient
`0.5` (default) | positive scalar

Belt friction while maintaining static contact.

Dependencies

To enable this parameter, set Friction model to Modal.

Kinetic friction coefficient — Kinetic friction coefficient
`0.42` (default) | positive scalar

Belt friction while slipping.

Dependencies

To enable this parameter, set Friction model to Modal.

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

Coulomb friction coefficient between the belt and the pulley surface.

Dependencies

To enable this parameter, set Friction model to Continuous.

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` `m/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.

Force threshold — Force threshold
`0.01` `N` (default) | positive scalar

Relative force required for peak kinetic friction in the contact.

Dependencies

To enable this parameter, set Belt type to Flat belt or V-belt.

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.

References

[1] Johnson, Kenneth L. Contact Mechanics. Cambridge: Cambridge Univ. Press, 2003.

Extended Capabilities

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

Version History

Introduced in R2012a

Belt Pulley

Description

Equations

Examples

Power Window System

Compound Pulley

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 parameterization Ideal - No slip (default) | Flat belt | V-belt

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 option 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 relative motion direction of the belt end Ends move in opposite direction (default) | Ends move in same direction

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

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

Dependencies

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

Pulley

Pulley translation — Option to allow linear motion Off (default) | On

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

Bearing viscous friction coefficient — Bearing viscous friction 0.001 N*m/(rad/s) (default) | scalar

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

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

Friction model — Friction method selection Modal (default) | Continuous

Dependencies

Initial state — Initial state of pulley Unlocked (default) | Locked

Dependencies

Static friction coefficient — Static friction coefficient 0.5 (default) | positive scalar

Dependencies

Kinetic friction coefficient — Kinetic friction coefficient 0.42 (default) | positive scalar

Dependencies

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

Dependencies

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

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

Force threshold — Force threshold 0.01 N (default) | positive scalar

Dependencies

More About

Hardware-in-the-Loop Simulation

References

Extended Capabilities

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

Version History

See Also

Topics

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 parameterization
`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 option
`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 relative motion direction of the belt end
`Ends move in opposite direction` (default) | `Ends move in same direction`

Maximum tension — Tension threshold option
`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 — Option to allow linear motion
`Off` (default) | `On`

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

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

Inertia — Rotational inertia
`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

Friction model — Friction method selection
`Modal` (default) | `Continuous`

Initial state — Initial state of pulley
`Unlocked` (default) | `Locked`

Static friction coefficient — Static friction coefficient
`0.5` (default) | positive scalar

Kinetic friction coefficient — Kinetic friction coefficient
`0.42` (default) | positive scalar

Contact friction coefficient — Contact 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` `m/s` (default) | positive scalar

Force threshold — Force threshold
`0.01` `N` (default) | positive scalar

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