# Fundamental Friction Clutch

Represent friction clutch controlled by kinetic and upper and lower static friction signals

Dynamic Elements

## Description

A friction clutch transfers torque between two driveline axes by coupling them with friction. The Fundamental Friction Clutch block models a standard friction clutch with kinetic friction and static (locking) friction acting on the two axes. The motion is measured as the slip ω of follower (F) axis relative to base (B) axis, ω = ωF – ωB.

The Fundamental Friction Clutch requires three input signals:

• Kinetic friction torque τK ≥ 0 (port τk)

• Static friction upper limit torque τS+ (port τ+)

• Static friction lower limit torque τS (port τ)

### Clutch Directionality

The friction clutch has two possible directionalities:

• Bidirectional (ω ≤ 0 or ω ≥ 0), allowing the follower to rotate relative to the base in either direction

• Unidirectional (ω ≥ 0), allowing the follower to rotate relative to the base in the forward direction only

#### Modeling Unidirectional Clutches

A unidirectional clutch is equivalent to a friction clutch connected in parallel to a one-way clutch that disengages when the relative motion reverses.

You can model a pure one-way clutch using a unidirectional clutch block with zero kinetic and static friction inputs. In that case, forward relative motion is friction-free, and reverse relative motion is forbidden.

If you want a unidirectional clutch that allows the follower to rotate relative to the base in the reverse direction only, connect the Fundamental Friction block in your driveline with reversed orientation, follower (F) to base (B).

### Clutch States and Transitions

A friction clutch can be in one of three states:

• Unengaged (ω ≠ 0 and τK = 0), when the clutch applies no friction at all. The frictional surfaces are not in contact. The follower and base are independent, and no torque is transferred between them. No power is dissipated by the clutch in this state.

• Engaged (but not locked, ω ≠ 0 and τK > 0), when the clutch applies kinetic friction as the frictional surfaces touch and slip past one another. The follower and base remain independent, but some torque is transferred between them.

The clutch dissipates power only in this state. The power dissipated is |ω·τK|.

• Locked (ω = 0 and τK > 0), when the clutch applies static friction. The frictional surfaces are locked together and do not slip. The follower and base effectively form a single axis. This state transfers the maximum torque possible. Because static constraints do no work, no power is dissipated by the clutch in this state.

There is also a fourth, virtual state called the wait state (see Friction Clutch Theory and Implementation following).

#### Clutch Locking and Unlocking

Locking requires that the:

• Relative speed |ω| be smaller than a velocity threshold ωTol.

• Kinetic friction torque τK be positive.

The static friction torque controls the unlocking of a friction clutch. (You can optionally lock the clutch at the start of the simulation as well.) When the clutch is locked, it remains locked unless the torque transferred across the clutch exceeds the static friction torque limits.

#### Clutches, Constraints, and Degrees of Freedom

If it locks, a Fundamental Friction Clutch block imposes a constraint on your driveline by requiring that two otherwise independent angular velocities be equal. A locked clutch thus reduces the number of independent degrees of freedom by one.

On the other hand, a clutch unlocking restores an independent degree of freedom to a driveline.

A locking clutch imposes a dynamic constraint because its constraint can appear and disappear during the simulation.

### Clutch Mode Iteration

 Mode Iteration and Algebraic Loops   A clutch locks or unlocks after a set of locking conditions are tested (see Friction Clutch Theory and Implementation following). This testing is called mode iteration.In the default case, mode iteration requires non-time-increment simulation steps (algebraic loops) that trigger warnings at the MATLAB® command line.

#### Enabling and Disabling Mode Iteration

You can turn off mode iteration for a whole driveline from that driveline's Driveline Environment block. In that case, the friction clutches of your model are tested without mode iteration. Instead, the locking and unlocking tests are applied over multiple time steps, improving your simulation performance, but possibly decreasing its accuracy.

#### Mode Iteration and Zero-Crossing Detection

The Fundamental Friction Clutch uses a specialized type of zero-crossing detection (ZCD) to solve the locking and unlocking conditions. To avoid infinite loops and zero-crossing conflicts, disable any other ZCD conditions applied to a clutch-connected driveshaft by normal Simulink® blocks connected directly or indirectly to its driveline connection line.

#### Mode Iteration and Code Generation

In the default mode, clutch mode changes are simulated with mode iteration turned on. However, in the generated code versions of SimDriveline™ models, mode iteration is turned off automatically. Clutch locking and unlocking are determined over multiple time steps.

### Setting the Clutch Velocity Tolerance

 Clutch Velocity Tolerance and Solvers   Variable-step solvers allow you to use either automatically computed or explicitly specified values for ωTol. Fixed-step solvers require you to specify a value for ωTol, either in each clutch or in the connected Driveline Environment block.

You can set the clutch velocity tolerance or threshold ωTol for a clutch in a number of ways, using a value specified in the clutch itself or in the Driveline Environment block connected to its driveline, and depending on whether you are using a variable- or fixed-step solver.

• You can allow the clutch to use the driveline-wide default velocity tolerance settings specified in the Driveline Environment block.

This is the default configuration of the clutch.

• You can override the driveline-wide default velocity tolerance settings by allowing a particular clutch to automatically compute a velocity tolerance from solver settings.

This option is available only if you use a variable-step solver.

• You can override the driveline-wide default velocity tolerance settings by specifying an individual velocity tolerance value for a particular clutch.

### Using Dynamic Element Blocks

Use the blocks of the Dynamic Elements library as a starting point for vehicle modeling. To see how a Dynamic Element block models a driveline component, look under the block mask. The blocks of this library serve as suggestions for developing variant or entirely new models to simulate the same components. Break the block's library link before modifying it and creating your own version.

## Dialog Box and Parameters

Directionality

Select `Bidirectional` or `Unidirectional` to determine how the follower axis can turn relative to the base, in both directions or only in the forward direction, respectively. The default is `Bidirectional`.

Use default clutch velocity tolerance from the Driveline Environment block

Select to require this clutch to use the driveline-wide velocity tolerance ωTol specified in the Driveline Environment block connected to the driveline. The default is selected.

If you unselect this check box, you enable the Use automatic clutch velocity tolerance for variable-step solvers check box and the Clutch velocity tolerance field (see following).

Clutch velocity tolerance

Sets the minimum angular velocity ωTol above which the clutch cannot lock. Below this velocity, the clutch can lock. (See the diagram, Clutch States and Transitions following.) The units are radians/second. The default is `1e-3`.

For a fixed-step solver, this clutch always uses this value, if you do not specify a default velocity tolerance through the Driveline Environment block.

This field is enabled only if the Use default clutch velocity tolerance from the Driveline Environment block check box is unselected.

Use automatic clutch velocity tolerance for variable-step solvers

Select to require this clutch to compute the velocity tolerance ωTol automatically from solver settings. The default is selected.

This check box is enabled only if the Use default clutch velocity tolerance from the Driveline Environment block check box is unselected.

Start simulation with clutch in locked mode

Select to start the simulation with the clutch already locked. The default is unselected.

Output clutch slip (S)

Select to make available the Simulink outport for the clutch slippage signal. The default is unselected.

The clutch slippage is the relative angular velocity ω of the two coupled driveline axes. The signal measures the clutch slippage in radians/second.

Output clutch mode (M)

Select to make available the Simulink port for the discrete clutch mode signal. The default is unselected.

The signal value is a function of the clutch state. See the table, Clutch States and Modes, following.

## Friction Clutch Theory and Implementation

The Fundamental Friction Clutch block can apply two kinds of friction to driveline motion, kinetic and static.

• The clutch applies kinetic friction torque, specified as a positive input signal, only when one driveline axis is spinning relative to the other driveline axis; that is, when the clutch is unlocked and the slip nonzero.

• The clutch applies static friction torque when the two driveline axes are locked and spin together, without slip.

You specify static friction limits as input signals. These upper and lower limits define a locked range of static friction. If the friction across the clutch remains within this range, the clutch remains locked.

The block iterates through multistep testing to decide when to lock and unlock the clutch.

### Clutch State, Transition, and Variable Summary

The first chart summarizes the possible states and transitions of a bidirectional clutch. The states and transitions of a unidirectional clutch consist of just one side, left or right, of the chart. The second diagram summarizes the physical differences between the locked and unlocked states. The final table summarizes the clutch variables.

Clutch States and Transitions

Clutch Slip vs. Friction Torque

Clutch Variables

SymbolDefinitionInportSignificance
ωRelative angular velocity (slip) ωF – ω B
αRelative angular accelerationdω/dt
ωTolRelative angular velocity tolerance for clutch lockingFirst locking condition: |ω| ≤ ωTol
τKKinetic friction torqueτkSecond locking condition: τK > 0
τS±Static friction torque limitsτ±Defines locked range
τTotal torque transferred across clutch Clutch remains locked if τS < τ < τS+

Clutch States and Modes

StateMode Signal
Forward or Wait Forward+1
Locked0
Reverse or Wait Reverse-1
Default Initial State0

### Unlocked State: Kinetic Friction

The kinetic friction torque τK applied between the base and follower driveshafts is specified by the incoming signal at the τk inport. This signal should be positive or zero.

The Fundamental Friction Clutch applies this torque as long as the clutch remains unlocked.

### Locked State: Static Friction

Once the friction clutch locks, it remains locked as long as the total torque τ transferred across the clutch remains within the range defined by the static friction torque limits:

τS < τ < τS+

You specify the static friction torque limits τS± by the incoming signals at the τ+ and τ inports.

In general, τS+ and τS are independent, as long as

τS < 0 < τS+

### How the Friction Clutch Locks and Unlocks

The locking and unlocking of a friction clutch proceed through an intermediate Wait state.

#### Wait State

The Wait state is a virtual state that continues the motion of the clutch's previous state but tests for locking or unlocking.

• If the clutch moved to Wait from Locked, it remains locked while in Wait.

• If the clutch moved to Wait from Unlocked, it remains unlocked while in Wait.

#### Clutch Locking

The friction clutch locks the two connected driveline axes together when both these conditions hold:

• τK> 0

• Either of these conditions:

• |ω| ≤ ωTol

• ω changes sign while the clutch is unlocked

If the ω changes sign while the clutch is unlocked, but τK = 0, the clutch enters the Wait state. While the clutch is in the Wait state, the driveshafts continue to slip relative to one another, subject to τK. If while in the Wait state, the clutch locking conditions become true, the clutch moves to Locked.

 Note:   You can also lock a clutch before the simulation starts with the Start simulation with clutch in locked mode option in the dialog. (See Manually Locking a Clutch at Simulation Start following.)

#### Clutch Unlocking

If the total torque across the two driveline axes moves outside the static friction limit range, the clutch enters the Wait state. While the clutch is in the Wait state, it remains locked but tests for unlocking.

The unlocking of a friction clutch is a conditional, multistep process implemented internally.

• If you turn off mode iteration for your driveline model (in the Driveline Environment block dialog), the clutch unlocks over multiple simulation time steps.

• If you leave it on, the simulation suspends the time steps and starts mode iteration to determine whether to unlock the clutch.

The Wait state encompasses the steps that test the entire driveline for unlocking.

1. The block first checks the relative acceleration α = dω/dt of the two driveline axes, given the torques present when the clutch enters the Wait state.

The clutch returns from the Wait state to the Locked state if

• The whole driveline requires the axes to turn in the relative forward direction, but α is negative.

• The whole driveline requires the axes to turn in the relative reverse direction, but α is positive.

2. If the clutch remains in the Wait state instead of returning to Locked, the relative acceleration is integrated in time to obtain the absolute value of the virtual angular speed and checks this result against angular velocity tolerance ωTol. If the result is less than ωTol, the clutch returns to the start of the Wait state and the relative acceleration check. If the result exceeds ωTol, the clutch unlocks.

3. In the Unlocked state, the clutch begins applying kinetic friction again.

### Default Initial State

When your driveline simulation starts, the physical state of the clutch is undetermined, unless you require the clutch to be locked beforehand. (See Manually Locking a Clutch at Simulation Start following.)

In the model, the clutch starts in a temporary Default Initial state. When the simulation gets underway, the clutch immediately tests its condition to see if it should be locked or unlocked and moves itself to the appropriate state.

### Manually Locking a Clutch at Simulation Start

For any particular clutch, you can override the initial clutch mode iteration at the simulation start (see Default Initial State preceding) by selecting the Start simulation with clutch in locked mode check box in that clutch's dialog. In this case, the simulation starts with that clutch already in the Locked state, with no initial tests of clutch conditions.

## Examples

The Controllable Friction Clutch is a subsystem built from the Fundamental Friction Clutch. Consult its block reference page for further details.

The Controllable Friction Clutch and models using it are SimDriveline examples of friction clutches.

## Reference

[1] Moler, C. B., Numerical Computing with MATLAB, Philadelphia, Society for Industrial and Applied Mathematics, 2004, Chapter 7.

## See Also

In the Simulink documentation, see the Signal Routing Library and the Sources Library, and Zero-Crossing Detection.

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