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Analyze Degrees of Freedom

About Driveline Degrees of Freedom and Constraints

Identifying rotational degrees of freedom is important for building and analyzing a driveline, particularly a complex system with many constraints and external actuations. Simulink® represents driveline DoFs as states, among all states of a model, including the pure Simulink states. For more about driveline actuation, see Actuate Drivelines with Torques and Motions. See Find and Use Driveline States following for more about states and how they are related to DoFs.

This section discusses how to identify driveline DoFs, take constraints into account, and extract the true or independent DoFs from a complete driveline diagram. It includes these basic steps:

  • The basic elements of a driveline diagram:

    • Connection lines

    • Dynamic elements and internal torques

    • Constraints

  • Sensors and actuators

    • Importing and exporting information into and from your driveline

    • Terminating DoFs

With these pieces, you can enumerate all the DoFs of any driveline. The section closes with an example of how to do this.

Identify Degrees of Freedom

In a Simscape™ Driveline™ model, all mechanical motions are rotational. Because absolute angles are not used in Simscape Driveline software, it is simplest to identify a driveline degree of freedom (DoF) with an angular velocity. (Some blocks use the relative angle between two driveline shafts to determine the torques generated by internal driveline dynamic elements.) A DoF represents a single, distinct angular velocity. Each DoF responds to the torques acting on the inertias making up the driveline. Integrating Newton's equations of rotational motion determines the angular motions. In fundamental terms, mechanical DoFs are properties of rotating inertias. It is consistent and simpler to identify a single Simscape Driveline DoF as a driveline axis (idealized driveshaft) with its connected inertias, because the inertias are rigidly attached to their idealized shaft.

Thus, to identify and count DoFs in a driveline, you need to look at a Simscape Driveline diagram starting with its Physical Modeling driveline connection lines first, before considering its blocks. Driveline blocks modify the DoFs represented by connection lines by

  • Imposing torques that act relatively between driveline axes

  • Adding constraints among the driveline axes

  • Imposing externally actuated torques and motions

Fundamental Degrees of Freedom

The basic unit of driveline motion is the degree of freedom (DoF) represented by an unbroken driveline connection line. Such lines represent idealized massless and perfectly rigid driveshafts. Rotating bodies with rotational inertias, represented by Inertia blocks, are rigidly attached to these lines and rotate with the axes.

Driveline Axes as Fundamental Degrees of Freedom

A driveline connection line anchored by driveline connector ports represents an idealized driveline axis. The connection line enforces the constraint that the two connected driveline components rotate at the same angular velocity.

You measure the angular velocity of an axis with a Motion Sensor block. For the Simscape Driveline analysis of a single axis, only angular velocity is important. The absolute angle of an axis is internally undefined.

Measuring Driveline Axis Motion with a Motion Sensor

Defining Relative and Absolute Angles.  Relative angle is sometimes necessary to compute internally generated torques between pairs of axes (see Connected Degrees of Freedom following). To determine a relative angle, the block integrates the relative angular velocity of the pair of axes and adds the result to the initial relative angle that you specify in the cases where it is needed.

You can define an absolute angle of rotation for a single axis only when you measure its motion with a Motion Sensor block. The sensor defines the absolute angle by integrating the angular velocity of the axis and adding an arbitrary absolute reference angle that you provide in the Motion Sensor dialog.

Rigidly Rotating Inertias Attached to Driveline Axes

By itself, a driveline connection line represents a single DoF. You cannot subject this DoF to any torques, because it lacks rotational inertia. The other basic element needed to construct a functioning driveline model is one or more Inertia blocks. In a real mechanical system, the spinning bodies carry both inertia and DoFs. Simscape Driveline spinning bodies are rigidly attached to a driveline axis. It is simpler to take the equivalent point of view that the driveline axis is the fundamental DoF and the bodies carry only inertia.

You attach Inertias to driveline connection lines by branching the lines. The attached inertias are subject to whatever torque is transmitted by the connection line, which imposes the constraint that everything attached to a single line must be spinning at the same rate.

Driveline Axis Branching Rules and Constraints

You can branch driveline connection lines. You can only connect the end of any branch of a driveline connection line to a driveline connector port . All driveline components connected to the ends of a set of branched lines rotate at the same angular velocity. A set of unbroken, branched connection lines represents a single DoF.

Branched Connection Lines and Angular Velocity Constraints

Connected Degrees of Freedom

You can connect two independent driveline axes, representing two independent degrees of freedom (DoFs), by an internal dynamic element. A dynamic element generates a relative torque from the relative angle and/or motion of the two axes. This relative torque acts between the two axes, which remain independent DoFs and which transmit the relative torque (with equal magnitude and opposite sign on the base and follower axes) to their respective attached inertias.

Dynamic Elements and Internal Torque Generation

The Dynamics Elements library contains blocks representing driveline elements that generate internal torques. A single relative torque is applied with positive sign to the follower (F) axis and negative sign to the base (B) axis.

  • Hard Stop and Torsional Spring-Damper generate spring-like and damping torques acting on the driveline axes connected to them. These torques are a function of the relative angle and angular velocity of the two axes.

  • Torque Converter generates a viscous torque acting on the driveline axes connected to it. This torque is a function of the relative angular velocity of the two axes. In normal operation (forward power flow), the impeller (I) is equivalent to the base (B), and the turbine (T) to the follower (F).

Clutches and Conditional Connections

A clutch is a conditional or dynamic constraint.

A clutch, if unlocked, also connects two driveline axes and can impose a relative torque between them, leaving the two axes independent. The unlocked clutch is either completely unengaged, imposing no torque at all, or engaged, imposing kinetic friction as a function of the relative velocity of the two connected axes. The Controllable Friction Clutch block models such a clutch.

If a clutch locks, applying only static friction between the two connected axes, the two axes are no longer independent. Instead, they act as a single axis, spinning at the same rate. See Constrained Degrees of Freedom following.

Constrained Degrees of Freedom

Certain driveline elements couple driveline axes in such a way as to eliminate their freedom to move independently. Such elements impose constraints on the motions of the connected axes. A constrained axis is no longer independent of other axes and does not count toward the total net or independent motions of the driveline. Such constraints remove independent degrees of freedom (DoFs) from the system.

Not all constraints are independent. Closing branched connection lines into loops makes some of the constraints contained within the loops redundant. The number of effective or independent constraints is the number of constraints arising from blocks minus the number of independent closed driveline connection line loops.

Except for clutches, driveline constraints are unconditional or static constraints, that is, unchanging over the simulation. Clutches impose conditional or dynamic constraints.

Locking a Driveline Axis

Connecting a driveline connection line to a Housing block freezes the motion of the corresponding driveline axis. It cannot move, and its angular velocity is constrained to be zero during a simulation. Such an axis has no associated independent DoF.

Locking Two Driveline Axes Together with a Clutch

A locked clutch, as long as the conditions for locking are valid, constrains the two connected driveline axes to spin together. The two axes remain distinct, but only one represents an independent DoF. The other is dependent. See Controllable Friction Clutch for more details.

An unlocked clutch, even if it continues to apply a relative kinetic friction torque between the axes, no longer imposes a constraint. Instead, it acts as a dynamic element. See Connected Degrees of Freedom preceding.

Coupling Driveline Axes with Gears

A gear coupling between two or more driveline axes reduces the independent DoFs of the driveline by imposing constraints. The nature of those constraints depends on the gear being used. Gear blocks with two connected axes impose one such constraint and reduce the two axes to a single independent DoF.

Multiaxis gears impose more than one constraint. For example, a planetary gear imposes two constraints on three axes, reducing the axes to one independent DoF. (This count does not include the fourth, internal DoF, the planetary wheel, which is not connected to an axis.)

Closed Loops, Effective Constraints, and Constraint Consistency

The actual constraint count used to determine the number of DoFs is the number of effective or independent constraints. You must take special care in counting constraints in a driveline diagram when connection lines form closed loops. The presence of closed loops in a diagram reduces the effective constraint count by rendering some of the constraints redundant:

Number of independent constraints = Number of constraints from blocks – Number of independent loops

You can reliably count the number of independent loops by counting the fundamental loops. Fundamental loops have no subloops. You can trace a fundamental loop with only one path. By counting only fundamental loops, you avoid overcounting loops that overlap.

For example, this diagram clearly has two independent loops.

In this diagram, you can draw three loops: two inner loops, left and right, and the outer loop. The outer loop encompasses both inner loops.

There are two independent loops in this diagram, because only two are fundamental. The outer loop is not fundamental.

Consistency of Constraints.  A closed loop renders redundant one of the constraints contained within it as long as all the angular velocities constrained by line branchings are equal over the whole loop. (See Driveline Axis Branching Rules and Constraints preceding.) The angular velocities not directly connected by lines must also be consistent if, for example, they are transferred through gears.

If the angular velocities along a closed loop cannot be made consistent, the driveline is overconstrained and cannot move.

Actuate, Sense, and Terminate Degrees of Freedom

You can use Simscape Driveline blocks with only one driveline connector port to originate and/or terminate degrees of freedom (DoFs) because they can end a driveline connection line. Such blocks include:

These blocks do not have to end a connection line.

They can instead be branched like this:

Terminating a connection line does not actually create or destroy a DoF, of course, but it does limit the DoF. If the termination is an actuator, the termination can modify the DoFs of the driveline. On the other hand, sensors have no effect on driveline DoFs.

Directionality, Actuating, and Sensing

Driveline connection lines have no inherent directionality. The direction of motion and torque flow is determined by the driveline dynamics once you simulate. You should contrast this with the inherent directionality of Simulink ports > and signal lines.

Although driveline connection lines are nondirectional, directionality is implicitly introduced into a driveline model when you attach actuator blocks to the diagram (see Sensors & Actuators), because these blocks interface pure Simscape Driveline blocks with the rest of Simulink. The actuator's effect on the driveline is determined by the (signed) Simulink input signal entering on one side.

The motion that results from the driveline simulation in turn determines the sign of the Simulink output signals that emerge from sensor blocks.

The Effect of Torque Actuation on Degrees of Freedom

Connecting a Torque Actuator to a driveline applies the torque specified by a Simulink input signal to the driveline. Such an actuation has no effect on the number of system DoFs. The driveline axes transmit the torque to their connected Inertias, and the driveline is free to respond to the imposed torques. The motion is simulated by integrating the driveline accelerations (a result of the imposed torques) to obtain the driveline velocities.

The Effect of Motion Actuation on Degrees of Freedom

Connecting a Motion Actuator to a driveline axis removes the freedom of that axis to respond to torques and instead specifies the axis motion during the simulation from the actuator's Simulink input signal. Motion actuation, unlike torque actuation, removes an independent DoF from the system.

Count Independent Degrees of Freedom

To determine the number of independent degrees of freedom (DoFs) in your driveline,

  1. Count all the continuous, unbroken driveline connection lines (lumping together connected sets of branched lines) in the Simscape Driveline portion of your model diagram. Call the total of such lines NCL.

    These lines connect two driveline connector ports or terminate on one driveline connector port , as discussed in Fundamental Degrees of Freedom and Actuate, Sense, and Terminate Degrees of Freedom preceding.

  2. Count all the constraints arising from blocks that impose constraints on their connected driveline axes. Call the total of such constraints Nbconstr.

    In most cases, each such block imposes one constraint, but complex gears impose more than one. See Constrained Degrees of Freedom preceding for details.

  3. Count the number of independent loops Nloop. The effective number of constraints is Nconstr = NbconstrNloop. Refer to Closed Loops, Effective Constraints, and Constraint Consistency preceding for more information.

  4. Count all the motion actuations in your driveline, by counting each Motion Actuator block. Consult the preceding section, Actuate, Sense, and Terminate Degrees of Freedom, for further discussion. Call the total of such motion actuations Nmact.

The number NDoF of independent DoFs in your driveline is

NDoF = NCLNconstrNmact = NCL – [NbconstrNloop] – Nmact

A necessary (although not sufficient) condition for driveline motion and successful driveline simulation is that NDoF be positive.

Conditional Degrees of Freedom with Clutches

Unlike other driveline components, clutches can undergo a discrete mode change during the course of a simulation. The number of independent DoFs of a driveline is not, in general, constant during its motion. Each mode change of one or more clutches changes the independent DoF count. Different collective states of a driveline's clutches, taken as a whole, can have different total net DoFs. To understand a driveline completely, you must examine each possible collective state of its clutch modes to identify its independent DoFs and possibly invalid configurations.

See Troubleshooting Simulation Errors preceding and Combine Clutches and Gears into Transmissions.

Count Degrees of Freedom in Simple Driveline with Clutch

Consider the simple transmission model drive_strans_ideal.

Simple Transmission

This system has five apparent DoFs, represented by these driveline axes:

  • Branched axis with Inertia1

  • Branched axis with Inertia2

  • Axis connecting the Hi Gear Clutch to Simple Gear 2:1

  • Axis connecting the Lo Gear Clutch to Simple Gear 5:1

  • Axis connecting the Brake Clutch to the Housing

There is an apparent closed loop formed by the Gears and Gear Clutches. This loop is real only if both Gear Clutches are locked.

The actual number of independent DoFs depends on the state of the clutches. The model has no Motion Actuators, so we need consider only Gears and Clutches as constraints.

  • The two Gears are always acting, thus yielding two ever-present constraints.

  • The fifth axis is always connected to the Housing. These three constraints reduce five DoFs to two.

Now consider the clutches.

  • Consider first the case where the Brake Clutch is disabled (free).

    • If both Hi Gear Clutch and Lo Gear Clutch are unlocked, the system has two independent DoFs, essentially one on the left of the Gear Clutches and the other between the Gear Clutches and the Brake Clutch.

    • If one of these Gear Clutches is locked, the additional constraint reduces the system to one independent DoF, essentially everything to the left of the Brake Clutch. (The clutch control schedule is set up to prevent both of these clutches from being locked at the same time.)

  • If the Brake Clutch is enabled, then the clutch control schedule keeps the two Gear Clutches disabled.

    • If the Brake Clutch is unlocked, the driveline has two independent DoFs, the same two as above: essentially, to the left of the Gear Clutches and between the Gear Clutches and the Brake Clutch.

    • If the Brake Clutch is locked, the system is reduced to one DoF, essentially to the left of the Gear Clutches. Everything to the right of the Gear Clutches is locked to the Housing in this case.

This table and abstract diagram summarize the possibilities available in this model.

Brake EnablingClutch LockingIndependent DoFs
Brake disabledBoth Gear Clutches unlockedTwo: On the left and on the right of the Gear Clutches
One Gear Clutch lockedOne: On the left of the Brake Clutch
Brake enabledBrake Clutch unlockedTwo: On the left and on the right of the Gear Clutches
Brake Clutch lockedOne: On the left of the Gear Clutches

Degrees of Freedom in the Simple Transmission

Possible But Nonphysical Configurations

It is also worth considering possibilities not available in the model because the clutch schedule design, implemented in the Clutch Control subsystem, excludes them.

Both Gear Clutches Locked, Brake Clutch Unlocked.  This configuration creates a conflict of DoFs and reduces the independent DoFs to one. The driveline axis to the right of the Gear Clutches tries to spin at two different rates, as required by two different gear ratios. Two locked clutches enforce two additional constraints on the two remaining DoFs, but form a closed loop, nominally leaving one freedom in the mechanism. Because of the DoF conflict, attempting to simulate such a configuration leads to a Simscape Driveline error.

If the two Gears had identical gear ratios, the DoFs would not conflict, and the simulation would run without error.

One Gear Clutch Locked, Brake Clutch Locked.  This configuration also creates a conflict of DoFs and yields zero DoFs. The two locked clutches enforce two additional constraints on the two remaining DoFs and leave no freedom in the mechanism. In addition, the driveline axis between the Gear Clutches, driven by the driveline axis to the left, tries to spin but finds itself locked to the Housing. Attempting to simulate such a configuration leads to a Simscape Driveline error.

Both Gear Clutches Locked, Brake Clutch Locked.  This configuration is also overconstrained. Three locked clutches enforce two effective constraints on the remaining two DoFs (after taking into account the closed loop) and yield NDoF = 0. In addition, the driveline axis to the right of the Gear Clutches tries to spin at two different nonzero rates, while at the same time remaining locked to the Housing, creating two distinct DoF conflicts.

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