Coulomb friction acts along the plane of contact between two solid surfaces, in opposition to their actual or potential relative motion, and in proportion to the normal force pushing the surfaces together. It encompasses both kinetic friction, applied when the surfaces are in relative motion, and static friction, applied when they are locked together. Coulomb friction is the basis for clutches and clutch-like elements that rely on normal forces to keep surfaces in contact. When the relative speed of the surfaces becomes small enough, these elements lock and move together.
Realistic friction models often include viscous friction. This type of frictional force or torque is proportional to the relative translational or rotational velocity of the two surfaces in contact.
The Clutches library contains models of standard clutches that couple two rotating driveline axes. The Brakes & Drives and Couplings & Drives libraries provide blocks to construct clutch-like elements of your own, including detents, brakes, and specialized clutches. These clutch-like elements apply Coulomb friction forces or torques between pairs of translating or rotating axes in loaded contact. They also allow inclusion of viscous friction. Once engaged, clutches and brakes act to decelerate the relative motion of surfaces in contact and can lock the surfaces together under certain conditions.
Clutches and clutch-like elements have a dual role in a driveline model. When engaged but not locked, they act as dynamic elements, generating torques and forces between driveline axes in relative motion. When locked, they act as conditional or dynamic constraints, locking driveline axes to move together. Such constraints are conditional, because they can unlock, unlike gears.
The main loss in a clutch system coupling two driveshafts comes from viscous friction at the two shaft bearings. Consider the sdl_simple_clutch model, presented in Engage and Disengage Gears with Clutches. Here you add a kinetic friction torque proportional to the angular velocity on both sides of the clutch (viscous friction). The Simscape™ Foundation library provides a Rotational Damper block that represents such a damper. The angular motion of the driveshafts is relative to another component. Here the angular velocities of the shafts are measured relative to rotational ground, represented by Mechanical Rotational Reference. You can make a friction subsystem that applies such a torque to any driveline axis connected to it. You can copy the subsystem and modify the existing clutch model by connecting the two copies on either side of the clutch.
Note: The velocity used in this damping is the absolute velocity of a single shaft relative to rest. If you had two rotating driveline shafts and wanted to exert a relative damping between them as a function of their relative velocities, use the same Rotational Damper block connected between the two axes.
The viscous friction torque is τfric = –μω, where μ is the viscous friction coefficient. To implement this torque:
From the Simscape library, copy Mechanical Rotational Reference, Rotational Damper, and Connection Port into your model window.
Connect the Mechanical Rotational Reference to the case (C) port of the Rotational Damper and the rod (R) port of the Rotational Damper to the Connection Port.
Open the Rotational Damper. For Damping
0.3. Leave the
default units. Close the dialog box.
Select the whole connected three-block set, and create
a subsystem. Call it
Create a second copy of
Rotational 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. Open the Scope blocks and click Start.
Readjust the horizontal axes of the Scope with Autoscale to see the full plots. The clutch pressure and external torques are applied as before. The shaft rotations are now different because of the damping.
As before, Inertia2 begins to spin when the clutch starts to engage at 2 seconds. After the clutch locks at 4 seconds, the body continues to accelerate, at a slower rate than it did without damping. At about 6.7 seconds, the clutch begins to disengage and completely disengages at 7 seconds. Subject to friction, Inertia2 now starts to slow down, unlike in the friction-free case. Once the external torque is removed, its angular velocity drops exponentially with time.
The behavior of Inertia1 is more complex. It begins to spin up, at a lower rate than before, because of the damping. Between 2 and 7 seconds, Inertia1 shares the external torque with Inertia2 via the Clutch and the Simple Gear. After 7 seconds, the external torque applies to Inertia1 alone. It continues to accelerate, 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. This terminal velocity is ωterm = τext/μ or 1/0.3 = 3.3333 radians/second. The third Scope plot approaches this terminal value.
The most critical addition that you can make to clutch models for greater realism is to change the clutch pressure signals from step functions (0 to 1, or 1 to 0) to signals with a smooth rise and fall. This greater realism results in a more complex model. At any simulation time, it is critical for your model to determine transmission motion by locking exactly the correct number of clutches. (If all clutches are unlocked, the transmission is in neutral.) Changing a transmission's gear settings while maintaining this requirement is one of the central problems of transmission design.
Such transmission and vehicle models as sdl_crcr and sdl_vehicle switch gear settings without placing their transmissions in neutral. Controlling an actual manual transmission requires moving the transmission out of gear and into neutral, picking a new gear setting, and then putting the transmission into the new gear.
In the sdl_crcr example, with manual transmission control, you can mimic these steps by turning on the Neutral Switch, changing the gear setting, then turning off slipping the Neutral Switch.
In a programmed transmission control model, you can filter clutch pressures with Transfer Fcn blocks, shaping the pressure signals from sharp steps to smooth rises or falls.
The sdl_dual_clutch example model illustrates important points about both physical and control design modeling using the Simscape and Simscape Driveline™ environment and libraries.
The model represents an automatic transmission with two clutches. In an automatic transmission, the decision of when to change gear ratio and what the next gear ratio will be is made by an engine management subsystem. The pedal deflection imposed by a vehicle driver is converted into a demanded engine torque. The torque demanded and the current forward vehicle speed together determine which gear ratio the transmission will switch to before it actually switches (gear preselection). By gradually lowering the clutch pressure, the transmission control system smoothly unlocks and disengages the clutch configuration for the current gear ratio. At the same time, the control system gradually raises the clutch pressures to achieve the new gear ratio by engaging and locking the new clutch configuration.
The physical components of the transmission, from the engine, gears, and clutches, to the vehicle body and tire, are modeled using Simscape Driveline blocks, with physical ports and connections.
The algorithmic control of the transmission, including the gear-switching and transmission control, is modeled using normal Simulink® blocks, with signal ports and lines, as well as enabled subsystems.
Dual-Clutch Model with Gear Preselection
The model is also set up to switch between two common simulation configurations.
If you double-click the subsystem icon for desktop simulation, you configure the model to simulate with a variable-step global solver, without fixed-cost or fixed-step local solver options.
If you double-click the subsystem icon for real-time simulation, you configure the model to simulate with a discrete-time, fixed-step global solver, as well as a fixed-cost, fixed-step local solver.
You can directly adjust all the solver options by opening the model Configuration Parameters and the network Solver Configuration dialog boxes.