|On this page…|
The most important requirement of a practical drivetrain is the ability to transfer rotational motion and torque among spinning components at different speeds and gear ratios. A single set of gears is usually not sufficient to accomplish this. Clutches are the critical components that allow the drivetrain to selectively transfer motion and torque at different gear ratios under manual or automatic control.
This section explains how to model and use clutches in driveline models without and with frictional losses and braking.
A common problem in drivetrain design is transferring motion and torque at different fixed gear ratios. Drivetrains are typically designed to switch among a set of discrete gear ratios. Implementing the switch from one gear ratio to another requires gradually disengaging one set of driveline couplings and engaging another set. Clutches allow you to gradually engage and disengage driveline shafts from one another.
The Controllable Friction Clutch block represents a standard surface friction-based clutch that models this behavior and requires no more than modest preparation to use. This section uses this block. You also can model clutches in greater detail using the Fundamental Friction Clutch block, which requires you to specify the static and kinetic clutch friction more completely. See Model Friction Clutches at a Fundamental Level following.
Tip You can model continuous motion-torque transfer with the Torque Converter block, which simulates fluid viscosity instead of surface friction and which never locks.
A clutch makes two shafts spinning at different rates spin at a single rate by applying forces that tend to accelerate one shaft and decelerate the other. The most common way for a clutch to accomplish this is with surface friction. Such a clutch can operate in one of three modes of motion:
Disengaged: the clutch applies no friction at all.
Engaged but unlocked: the clutch applies kinetic friction, and the two shafts spin at different rates.
Engaged and locked: the clutch applies static friction, and the two shafts spin together.
A clutch consists of mated frictional surfaces overlapping one another and connected on either side to a shaft. If the clutch is disengaged, the frictional surfaces have no contact and the shafts spin independently. To engage the clutch, a moderate amount of contact between two surfaces is induced by applying clutch pressure (a force normal to the surfaces). The two surfaces in contact and moving relative to one another experience kinetic friction, which causes them to narrow their relative velocity. The faster surface tends to slow down (unless an external torque is acting) and the slower one to speed up. At some critical combination of reduced relative speed and pressure (normal force), the clutch locks, so that the two shafts are spinning at the same rate. The locking of the shafts is controlled by static friction, which holds the shafts together as long as sufficient normal force is applied and no relative torque is large enough to overcome the locking. If the clutch unlocks but is still engaged, it again applies kinetic rather than static friction.
Note: The transition between unlocked and locked states is discontinuous. Modeling a clutch locking or unlocking requires searching for the correct combination of pressure and torque acting on the clutch.
The locking and unlocking are determined during simulation by accurate zero-crossing detection and repeated mode iteration. In the default case, this mode iteration induces algebraic loops in Simulink®, non-time-based simulation steps that trigger warnings at the MATLAB command line. You can change this default behavior through your driveline's Driveline Environment block.
Here you construct a simple model that simulates a gear being engaged, then disengaged, by a clutch. Torque and motion are transferred from one shaft to another over a finite time interval. Start with the simple gear model of the last section or with the drive_sgeardrive_sgear example. The completed clutch model is the drive_sclutchdrive_sclutch example.
From the SimDriveline™ block library, you need a Controllable Friction Clutch block. Also copy a Signal Builder and a Constant block from the Simulink block library.
Remove the Torque Sensor blocks, insert the Clutch between Inertia1 and Simple Gear, then reconnect the connection lines.
In the Clutch dialog, select the Output clutch mode check box, but leave the other defaults. Rearrange and connect the blocks as shown here.
Simple Clutch Model with Programmed Clutch Pressure
Use the Constant block as the input torque signal in place of the sinusoidal signal. Reconfigure Mux and the first Scope blocks to accept three signals, the two angular velocities and the clutch pressure. Connect the second Scope to display the Clutch mode signal.
Open Signal Builder and construct the following signal. Signal Builder specifies the clutch pressure signal, which is normalized between 0 and 1. (The Peak normal force field in Clutch determines the maximum clutch pressure.)
|Time Range (Seconds)||Signal Value|
|0 – 2||0|
|2 – 4||0 – 0.8 with constant slope|
|4 – 6||0.8|
|6 – 7||0.8 – 0 with constant slope|
|7 – 10||0|
Open the Scopes and start the simulation.
The normalized clutch pressure signal follows the profile you created in Signal Builder and determines the model's behavior.
From 0 to 2 seconds, the velocity of Inertia1 increases linearly because it is subject to a constant torque.
At 2 seconds, the clutch begins to engage, and Inertia2 begins to spin. The velocity of Inertia1 continues to rise, although at a slower rate, because the two inertias now share the external torque.
At 4 seconds, the pressure reaches its maximum, and the clutch locks. The driveshafts connected by the clutch spin together. Inertia1 and Inertia2 continue to speed up at constant accelerations.
At 6 seconds, the clutch begins to disengage as the pressure drops. Inertia1 and Inertia2 continue to accelerate with the applied torque.
The clutch unlocks at 6.77 seconds and fully disengages at 7 seconds. (The clutch unlocks a little before completely disengaging because the pressure, even before vanishing, becomes too small to maintain the lock.) Inertia1 is still accelerating. But Inertia2 now free of the drive shaft and its torque, no longer accelerates and instead spins at a constant rate without frictional loss.
While the two shafts are locked, between 4 and 6.77 seconds, Inertia1 and Inertia2 spin in a fixed 2:1 ratio. The Simple Gear, with a gear ratio of 2 between follower and base, transforms Inertia2's velocity to half that of Inertia1.
How the Clutch Mode Indicates Locking and Unlocking. The Clutch mode signal indicates the relative motion of its two connected shafts. From 0 to 4 seconds, the two shafts are moving relative to one another. The follower (driven) shaft is slower than the base (drive) shaft, so the mode signal is -1. Once the two shafts lock, their relative velocity is 0, and the mode signal switches to 0. At 6.77 seconds, they unlock, and the drive (base) shaft starts accelerating faster than the driven (follower) shaft. The mode signal switches back to -1.
To see the two Inertias of the preceding model locked and spinning at the same rate,
Remove Simple Gear and connect Inertia1 directly to the Clutch. Change the Peak normal force value in Clutch to 2.5 (newtons).
Restart the simulation. Inertia1 and Inertia2 now spin at the same rate while the clutch is locked between 4 and 6.77 seconds.
Simple Clutch Model with No Gear
To make your clutch system model more realistic, you should add frictional damping to the spinning shafts of drive_sclutchdrive_sclutch. Here you add a kinetic friction torque proportional to the angular velocity to both sides of the clutch. A simple way to do this is to create a friction subsystem that applies such a torque to any driveline axis it is connected to. Then you can copy the subsystem and modify your existing clutch model by connecting the two copies on either side of the clutch.
Tip The velocity used for this damping is the absolute velocity of a single shaft relative to rest (as defined by a Housing block, for example). If you had two driveline shafts and wanted to exert a relative damping between them as a function of their relative velocities, you could use the Torsional Spring-Damper block. In general, this block applies a mixture of spring-like and damping torques between the two connected axes. But you can apply a pure damping torque by simply setting the spring constant to zero.
The frictional torque is τfric = -μω, where μ is the frictional proportionality constant. To apply the frictional torque proportional to the velocity, you need to
Measure the angular velocity of the driveline axis
Multiply it by -μ, because the frictional torque opposes the motion
Apply the resulting torque back to the driveline axis
To implement kinetic damping torque:
Copy Motion Sensor and Torque Actuator blocks and, from the Simulink library, a Gain block, into your model window.
Connect the angular velocity port Vel to the inport of the Gain block and the outport of the Gain block to the torque inport of the Torque Actuator block. Enter -0.3 for the Gain value in the Gain dialog, leaving the other defaults.
With your cursor, select the connected Sensor-Gain-Actuator block set, and create a subsystem. Call it Damper. When you create the subsystem, the port appearing on its block is a driveline connector port , not a Simulink port >.
Now create a second copy of Damper.
Rotational Kinetic Damping Subsystem
Complete and run the model.
Connect the two Damper subsystems to the driveline of your previous clutch model as shown.
Damped Simple Clutch Model
Change the simulation time to 20 seconds. Then open the Scope blocks and click Start.
Readjust the horizontal axes of the Scopes with Autoscale to see the full plots. The clutch pressure and external torques are applied as before. But the shaft rotations are different now because of the damping.
Inertia1, as before, begins to spin when the clutch starts to engage at 2 seconds. After the clutch locks at 4 seconds, the body continues to accelerate, but at a slower rate than it did without damping. At 6 seconds, the clutch begins to disengage and completely disengages at 7 seconds. Unlike the friction-free case, Inertia1, subject to friction, now starts to slow down. Its angular velocity drops exponentially with time once the external torque is removed.
The behavior of Inertia is more complex. It begins to spin up, but at a lower rate than before, because of the damping. Between 2 and 7 seconds, Inertia has to share the external torque with Inertia1 via the Clutch and the Simple Gear. After seven seconds, the external torque applies to Inertia alone. It continues to accelerate, but at an ever-slowing rate, because of the damping. If you let the simulation run without stopping, Inertia will approach its terminal angular velocity, a state where the frictional torque exactly balances the externally applied torque. The terminal velocity is ωterm= τext/μ or 1/0.3 = 3.3333 radians/second in this case. The Scope plot shows this terminal value.
A special case of transferring motion occurs when you want to brake the spinning of a driveline component, slowing it down until it stops. The common way to brake the motion is to couple the spinning component to a fixed housing, which effectively has infinite inertia and is represented by a SimDriveline Housing block. Because the housing cannot move, a driveline axis locked to a housing also cannot move. You can implement the gradual engagement or disengagement of a driveline component with a housing using a clutch, just as you use a clutch to gradually couple or uncouple two spinning shafts.
The drive_clutch_engagedrive_clutch_engage example model is an elaboration on the preceding models of this chapter and features two clutches, one of which acts as a brake. The model also includes frictional damping for greater realism. The simulation time is set to inf (infinity).
Simple Clutch Model with Brake Clutch
This model again uses the basic structure of inertia-clutch-gear-inertia. The first body, Inertia, is still driven by an external torque, and the initial velocities are still 0. There is, however, another clutch for the second body, Inertia1, that can couple Inertia1 to the Housing and bring it to a stop. Another new feature, compared to the preceding models, is the switching assembly made of the Clutch Switch and Flipper blocks. You can flip this switch to apply a constant clutch pressure signal to either Gear Clutch or Brake Clutch. The two Damper subsystems are identical to those you constructed in Model Realistic Clutch Systems with Loss, except that the frictional constants, the Gain values of the Gain blocks, are set to -0.1.
Start the model with the Clutch Switch set to 1. The clutch pressure is then applied to Gear Clutch, which engages and locks the driver and driven shafts and causes Inertia and Inertia1 to rotate together.
The angular velocity of Inertia1 (2.5 radians/second) is half that of Inertia (5 radians/second) because the gear ratio of the Simple Gear block is 2, follower to base. In this switch mode, no clutch pressure is applied to Brake Clutch, which remains unengaged. The mode of Brake Clutch is then -1, because Brake Clutch's follower, the Housing block, is at rest, while the base, Inertia1, is spinning. The mode of Gear Clutch is 0, because its base and follower, the driver and driven shafts, are locked together.
After an initial transient, the system settles into a steady state of motion where the external torque is exactly balanced by the frictional losses. The effective frictional constant, with two dampers, is 0.2. With an external torque of 1 newton-meter, the terminal angular velocity of Inertia is then ω = 1/0.2 = 5 radians/second.
With the simulation running, now change the Clutch Switch to 0 to disengage Gear Clutch and engage Brake Clutch. The system undergoes another transient while Gear Clutch disengages and Brake Clutch engages.
The angular velocity of Inertia and the driver shaft settles down to a new steady state of 10 radians/second, twice its old speed. The mode of Gear Clutch is now -1, because the driven shaft (follower) is not moving, while the driver shaft (base) continues to spin.
Because Gear Clutch is now disengaged, Inertia is no longer subject to the second frictional damping block, Damper1. The effective frictional constant drops in half, to 0.1, and the terminal velocity doubles. At the same time, Inertia1 is no longer receiving torque through Gear Clutch. But Brake Clutch is engaged and couples Inertia1 to the immobile Housing. Once engaged, the kinetic friction of Brake Clutch and Damper1 bring the driven shaft and Inertia1 to a stop. Because it locks, Brake Clutch's mode becomes 0.
To see the transient behavior at simulation start and when you switch the clutches,
Start the simulation and let it run for a short time. Then switch Clutch Switch to the other mode.
After another short time, stop the simulation. Use the Autoscale feature of the Scopes to capture the entire simulation sequence. The transients from the starting behavior and the switching transition will be visible.
For example, in these plots, the model was started with Clutch Switch set to 1 (Gear Clutch locked, Brake Clutch disengaged, no braking). The velocities quickly climbed to their steady-state values. Then Clutch Switch was changed at about 682 seconds of simulation time. Gear Clutch disengaged and Brake Clutch engaged, braking the driven shaft. The driver shaft's angular velocity rose from 5 to 10 radians/second. The driven shaft's angular velocity dropped to 0.
The Controllable Friction Clutch block is easy to use, requiring only a single normalized pressure signal to modulate the kinetic friction. You fix all its other characteristics before starting simulation.
Modeling a friction clutch at a fundamental level requires direct control over the kinetic and static friction torques. The Fundamental Friction Clutch block gives you that greater control. With this block, you must specify, by either external signals or internal sensor-actuator feedback, the clutch's kinetic friction and static friction limits (positive and negative) as functions of time.