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## Represent and Transfer Driveline Motion and Torque

### About Inertia, Motion, and Gears

The purpose of a gear set is to transfer rotational motion and torque at a known ratio from one driveline axis to another. This section introduces you to modeling gears and using them to couple bodies rotating on driveline axes.

### Couple Rotational Motion with Gears

A gear set consists of two or more meshed gears corotating at some specified gear ratios. The ratios might or might not be constant. The gear ratios determine how angular velocity and torque are transferred from one driveline component to another.

#### Gear Coupling Rules

Ideal gears mesh and corotate at a point of contact without frictional loss or slippage.

The simplest gear coupling consists of two circular gear wheels of radii r1 and r2, spinning with angular velocities ω1 and ω2, respectively, and lying in the same plane. Their connected shafts are parallel and carry torques τ1 and τ2. The gear ratio of gear 2 to gear 1 is the ratio of their respective radii: g21 = r2/r1. The power transferred along either shaft is ω·τ.

The gear coupling is often specified in terms of the number of gear teeth on each gear, N1 and N2. The gear ratio of gear 2 to gear 1 is then g21 = N2/N1 = r2/r1.

The fundamental conditions on the simple gear coupling of rotational motion are ω21 = ±1/g21 and τ21 = ±g21. That is, the ratio of angular velocities is the reciprocal of the ratio of radii, while the ratio of torques is the ratio of radii. The transferred power, being the product of angular velocity and torque, is the same on either shaft.

The choice of signs indicates that the gears can spin in the same or in opposite directions. If the gears are external to one another (corotating on their respective outside surfaces), they rotate in opposite directions. If the gears are internal to one another (corotating with the outside of the smaller gear meshing with inside of the larger gear), they rotate in the same direction.

 Warning   Gear ratios should always be strictly positive. If a gear ratio vanishes or becomes negative, the SimDriveline™ simulation stops with an error.

#### Generalized Gear Coupling Rules

You need the general ideal gear coupling conditions if you are coupling gears that are not constant in radii, not lying in the same plane, or not circular.

The general velocity constraint requires that the linear velocities of the gears at the point of contact be the same. This is a vector condition on the angular velocities ω1 and ω2 and the radius vectors r1 and r2: ω1 x r1 = ω2 x r2. The alternative form in terms of the number of gear teeth is equivalent to this linear velocity constraint. For the gear teeth to mesh, the number of teeth per unit length of gear circumference must be the same on the two gears.

The general torque condition arises from the force equilibrium at the point of contact. If there is no linear motion of the whole gear assembly, the forces at contact F must be equal and opposite. The ratio of torques is then:

|τ2|/|τ1| = |r2 x F|/|r1 x F|

The power transferred along either shaft is conserved across ideal gear couplings:

ω2·(r2 x F) = ω1·(r1 x F)

### Couple Two Spinning Inertias with Simple Gear

In this example, you couple two spinning inertias, first, along a single shaft (driveline axis), so that they spin with the same angular velocity; then spinning along two shafts and coupled by a gear so that they spin at different velocities; and finally, coupled by a gear and actuated by an external torque, spinning at different rates and experiencing different torques. You use the most basic SimDriveline blocks, such as Inertia, Simple Gear, and Driveline Environment.

#### Modeling Two Spinning Inertias

Here you create the first version of the simplest driveline model, two inertias spinning together along the same axis. Open the SimDriveline and Simulink® block libraries and a new Simulink model window.

1. Drag and drop two Inertia, two Motion Sensor, and one Initial Condition blocks into the model window.

2. Every topologically distinct driveline block diagram requires exactly one Driveline Environment block, found in the SimDriveline Solver & Inertias library. Copy one such block into your model.

3. From the Simulink library, drag and drop a Scope and a Mux block. Then connect the blocks as shown.

Two Spinning Inertias

4. Open the Initial Condition block. In the Initial angular velocity field, replace its default 0 entry with pi radians/second (rad/s). Click OK.

If you do not connect an Initial Condition block to a driveline axis, the axis by default starts the simulation with zero angular velocity. You must ensure that the initial angular velocities of your coupled driveline axes are consistent with one another. If they are not, the simulation stops with an error.

5. Open the Scope block and start the simulation. The two angular velocities are constant at 3.14 radians/second.

#### Coupling Two Spinning Inertias with a Simple Gear

Now you modify the model you just created by coupling the two spinning inertias with a simple, ideal gear with a fixed gear ratio.

1. From the SimDriveline block library, drag and drop a Simple Gear block into your model. Open the block. Leave the default follower-base gear ratio value at 2. Clear the Follower and base rotate in opposite directions check box and click OK. The simple gear then represents two gear wheels corotating in the same direction, with the smaller wheel inside the larger. Reconnect the blocks as shown.

Two Spinning Inertias Coupled by a Gear

Leave the initial angular velocities at pi in the Initial Condition block. SimDriveline software automatically sets the correct initial angular velocity for Inertia.

2. Open the Scope and start the simulation. The two angular velocities are constant at 3.14 and 6.28 radians/second for Inertia1 and Inertia, respectively. The follower-base gear ratio is 2, and the angular velocity of Inertia is twice that of Inertia1, with the same sign, because the two bodies are spinning in the same direction.

3. Now select the Follower and base rotate in opposite directions check box. The simple gear then becomes two wheels corotating in opposite directions, with the two wheels meshed on their respective outer surfaces.

4. Restart the simulation. The two angular velocities are 3.14 and -6.28 radians for Inertia1 and Inertia, respectively. The second angular velocity is twice the first and with opposite sign, because the two bodies are spinning in opposite directions.

5. Finally, again clear the Follower and base rotate in opposite directions check box.

#### Torque-Actuating Two Coupled, Spinning Inertias

In the final version of the simple gear model, you actuate the inertias with an external torque instead of starting them with fixed initial angular velocities. The external torque varies sinusoidally. You can find a completed version of this model in the example drive_sgeardrive_sgear.

1. From the SimDriveline block library, copy a Torque Actuator and two Torque Sensor blocks. From the Simulink block library, drag and drop a second Scope block, a second Mux block, and a Sine Wave block.

2. Disconnect the Inertia blocks from the Simple Gear and insert the Torque Sensors. Disconnect and delete the Initial Condition block. The two axes will now default back to zero angular velocities.

Connect the other blocks as shown.

Two Spinning Inertias Coupled by a Gear and Actuated with Torque

3. Open both Scope blocks and start the simulation.

The measured torques and angular velocities vary sinusoidally. The angular velocity of Inertia1 is half that of Inertia, as you saw in the previous models. But the torque in the second shaft is twice that in the first, as required by the laws of gear coupling.

If you select the Follower and base rotate in opposite directions check box in Simple Gear and restart the simulation, the same angular velocities and torques result, except that the values associated with Inertia1 and the second shaft are negative, because the second body and second shaft are spinning in opposite directions.

#### Sensing and Actuating Motion and Torque

The Sensor and Actuator blocks you use in the preceding models illustrate their dual nature: they act as driveline components themselves, but also let you connect driveline blocks with the rest of Simulink.

• Sensor & Actuator blocks have both driveline connector ports and normal Simulink ports >. You can extract sensor signal information with a block's Simulink outports. You can actuate motion or apply external torques by feeding in actuation signals with a block's Simulink inports.

Many other SimDriveline blocks feature Simulink ports for inserting and measuring signals.

• You connect a Torque Sensor along a driveline axis, by placing it in series with other driveline components.

• You connect the other Sensor and Actuator blocks across a driveline axis, by branching the driveline connection line off to one side and connecting this secondary line to the block; or by connecting the block to the end of a driveline axis.

### Couple Two Spinning Inertias with Variable Gear

You can modify the simple gear model further by replacing the fixed-ratio gear with a gear whose gear ratio varies in time. You specify the gear ratio variation with a Simulink signal. Start with the simple gear model you built in the preceding section or by opening and editing the drive_sgeardrive_sgear example.

1. From the SimDriveline block library, drag and drop a Variable Ratio Gear block and replace the Simple Gear block with it. Open Variable Ratio Gear and ensure that the Follower and base rotate in opposite directions check box is selected (the default). The two shafts will spin in opposite directions.

2. The Variable Ratio Gear block accepts the continuously varying gear ratio as a Simulink signal through the extra inport labeled r. For this example, create a Simulink signal for the gear ratio with a Signal Builder block from the Simulink block library. Build a signal that rises with constant slope from 1 to 2 over 10 seconds. Then connect the Signal Builder block to the r port.

Simple Variable Ratio Gear Model

3. Leave the other, original settings of the simple gear model unchanged. Open both Scopes and start the simulation.

The two shafts' angular velocities and torques have opposite signs. Apart from this sign difference, the ratios of angular velocities and torques start at 1, because the initial gear ratio is 1. But as the gear ratio increases toward 2, the angular velocity of Inertia1 becomes smaller than that of Inertia, while the associated torque in the second shaft (apart from the opposite sign) becomes larger than that in the first shaft. Because of the changing gear ratio, the motion and the torques are no longer strictly sinusoidal, even though the actuating external torque is.

The drive_vgeardrive_vgear example is a full model of this type. To learn more about how to use variable gears, including the Coriolis acceleration, consult the Variable Ratio Gear block reference page.

### Couple Three Spinning Inertias with Planetary Gear

You can further modify the simple gear model and use it as a starting point for studying more complex gear sets. One of the most important is the planetary gear, which has three wheels, the ring, the sun, and the planet, all held in place by a common carrier body. The planetary gear is interesting in its own right, but also important because it is a common component in complex, realistic transmissions.

1. Replace the Simple Gear in your model with a Planetary Gear from the SimDriveline block library. A planetary gear splits input angular motion from the carrier between the ring and sun wheels, each connected to their respective bodies.

2. Copy another Inertia and another Motion Sensor as well. Connect the blocks to form the new diagram as shown.

Simple Planetary Gear Model

3. Enter 2 for the Ring/Sun gear ratio in Planetary Gear. Open the Scope and start the simulation to observe the angular velocities of the ring, carrier, and sun, from largest to smallest. The ratio of the ring to sun gear velocities is always two.

4. To see the ring and sun wheels spinning alone, you must lock the carrier. In this case, you switch the torque actuation to the ring wheel. Copy a Housing block from the SimDriveline block library. Disconnect and delete Inertia, replacing it on the carrier driveline axis with Housing, and reconnect the Driveline Environment block to this connection line.

5. Insert a Torque Actuator and move the Sine Wave block next to it. Connect it to the inport.

Simple Planetary Gear Model with Locked Carrier

6. Open the Scope and start your model. Observe the angular velocities of the ring, carrier, and sun.

The carrier, connected to Housing, does not move. The ring is driven with a sinusoidal torque, and the sun responds by spinning in the opposite direction (ring and sun gear wheels are external to one another) at twice the rate. The ring wheel has twice the radius (or twice the number of teeth) as the sun, so it spins half as fast.