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## Model Planetary Gear Train

### Model Overview

Planetary gear trains are common in industrial, automotive, and aerospace systems. A typical application is the automatic transmission system of car. From a kinematic point of view, what sets this mechanism apart is the kinematic constraint set between gear pairs. These constraints fix the angular velocity ratios of the gear pairs, causing the gears in each pair to move in sync.

In SimMechanics™, you represent the kinematic constraint between meshed gears using blocks from the Gears sublibrary. This tutorial shows you how to use these blocks to model a planetary gear train. The gear train contains four rigid bodies:

• Sun gear

• Planet gear

• Ring gear

• Planet carrier

Each rigid body, including the planet carrier, can spin about its central axis. In addition, each planet gear can revolve about the sun gear. Joint blocks provide the required degrees of freedom, while gear constraint blocks ensure the gears move as if they were meshed.

### Model Sun-Planet Gear Set

Model the gear rigid bodies and connect them with the proper degrees of freedom. In a later step, you add gear constraints to this model.

1. Drag these blocks to a new model.

LibraryBlockQuantity
Body ElementsSolid2
JointsRevolute Joint1
JointsPlanar Joint1
Frames and TransformsRigid Transform1
Frames and TransformsWorld Frame1
UtilitiesMechanism Configuration1
Simscape > UtilitiesSolver Configuration1

2. Connect and name the blocks as shown.

3. In the Sun Gear block dialog box, specify these parameters.

ParameterSetting
Geometry > ShapeSelect General Extrusion.
Geometry > Cross-Section

Enter simmechanics.demohelpers.gear_profile(2*Sun.R,Sun.N,A). Select units of cm.

Geometry > LengthEnter T. Select units of cm.
Inertia > DensityEnter Rho.
Graphic > Visual Properties > ColorEnter Sun.RGB.

The function simmechanics.demohelpers.gear_profile produces a rough approximation of an involute gear profile.

4. In the Planet Gear block dialog box, specify these parameters.

ParameterSetting
Geometry > ShapeSelect General Extrusion.
Geometry > Cross-SectionEnter simmechanics.demohelpers.gear_profile(2*Planet.R,Planet.N,A). Select units of cm.
Geometry > LengthEnter T. Select units of cm.
Inertia > DensityEnter Rho.
Graphic > Visual Properties > ColorEnter Planet.RGB.

5. In the Rigid Transform block dialog box, specify these parameters.

ParameterSetting
Translation > MethodSelect Standard Axis.
Translation > AxisSelect +Y.
Translation > OffsetEnter Sun.R + Planet.R. Select units of cm.

6. In the model workspace, define the block parameters using MATLAB® code:

```% Common Parameters
Rho = 2700;
T = 3;
A = 0.8; % Gear Addendum

% Sun Gear Parameters
Sun.RGB = [0.75 0.75 0.75];
Sun.R = 15;
Sun.N = 40;

% Planet Gear Parameters
Planet.RGB = [0.65 0.65 0.65];
Planet.R = 7.5;
Planet.N = Planet.R/Sun.R*Sun.N;```
7. Simulate the model. To induce motion, try adjusting the velocity state targets in the joint block dialog boxes. Notice that the sun and planet gears move independently of each other. To constrain gear motion, you must add a gear constraint block between the gear solid blocks.

You can open a copy of the resulting model. At the MATLAB command line, enter smdoc_planetary_gear_a.

### Constrain Sun-Planet Gear Motion

Specify the kinematic constraints acting between the sun and planet gears. These constraints ensure that the gears move in a meshed fashion.

1. Drag these blocks to the sun-planet gear model.

LibraryBlock
ConstraintsDistance Constraint
Gears and Couplings > GearsCommon Gear Constraint

2. Connect the blocks as shown. The new blocks are highlighted.

3. In the Common Gear Constraint block dialog box, specify these parameters.

ParameterSetting
Specification MethodSelect Pitch Circle Radii.
Specification Method > Base Gear RadiusEnter Sun.R. Select units of cm.
Specification Method > Follower Gear RadiusEnter Planet.R. Select units of cm.

4. In the Distance Constraint block dialog box, specify this parameter:

• Distance — Enter Sun.R + Planet.R. Select units of cm.

5. Simulate the model. To induce motion, try adjusting the velocity state targets in the joint block dialog boxes. Notice that the sun and planet gears now move in sync.

You can open a copy of the resulting model. At the MATLAB command line, enter smdoc_planetary_gear_b.

### Add Ring Gear

Model the ring gear rigid body, connect it with the proper degrees of freedom, and constrain its motion with respect to the planet gear.

1. Add these blocks to the sun-planet gear model.

LibraryBlock
Body ElementsSolid
JointsRevolute Joint
Gears and Couplings > GearsCommon Gear Constraint

2. Connect and name the blocks as shown. The new blocks are highlighted.

3. In the Ring Gear block dialog box, specify these parameters.

ParameterSetting
Geometry > ShapeSelect General Extrusion.
Geometry > Cross-SectionEnter Ring.CS. Select units of cm.
Geometry > LengthEnter T.
Inertia > DensityEnter Rho.
Graphic > Visual Properties > ColorEnter Ring.RGB.

4. In the Common Gear Constraint1 block dialog box, specify these parameters.

ParameterSetting
TypeSelect Internal.
Specification MethodSelect Pitch Circle Radii.
Specification Method > Base Gear RadiusEnter Planet.R. Select units of cm.
Specification Method > Follower Gear RadiusEnter Ring.R. Select units of cm.

5. In the model workspace, define the Ring Gear block parameters using MATLAB code:

```% Ring Gear Parameters
Ring.RGB = [0.85 0.45 0];
Ring.R = Sun.R + 2*Planet.R;
Ring.N = Ring.R/Planet.R*Planet.N;

Ring.Theta = linspace(-pi/Ring.N,2*pi-pi/Ring.N,100)';
Ring.RO = 1.1*Ring.R;
Ring.CSO = [Ring.RO*cos(Ring.Theta) Ring.RO*sin(Ring.Theta)];
Ring.CSI = simmechanics.demohelpers.gear_profile(2*Ring.R,Ring.N,A);
Ring.CSI = [Ring.CSI; Ring.CSI(1,:)];
Ring.CS = [Ring.CSO; flipud(Ring.CSI)];```
6. Simulate the model. To induce motion, try adjusting the velocity state targets in the joint block dialog boxes. Notice that the sun, planet, and ring gears move in a meshed fashion.

You can open a copy of the resulting model. At the MATLAB command line, enter smdoc_planetary_gear_c.

### Add Gear Carrier

Up to now, you have kept the sun and planet gears at a fixed distance using a Distance Constraint block. In an actual planetary gear, a gear carrier enforces this constraint. Model the gear carrier and connect it between the sun and planet gears.

1. Remove these blocks from the planetary gear model:

• Planar Joint

• Rigid Transform

• Distance Constraint

2. Add these blocks to the planetary gear model.

LibraryBlockQuantity
Body ElementsSolid1
JointsRevolute Joint2
Frames and TransformsRigid Transform2

3. Connect and name the blocks as shown.

Pay close attention to the Rigid Transform block orientation: the B frame ports should face the Solid block. The new blocks are highlighted.

4. In the Carrier block dialog box, specify these parameters.

ParameterSetting
Geometry > ShapeSelect General Extrusion.
Geometry > Cross-SectionEnter Carrier.CS. Select units of cm.
Geometry > LengthEnter Carrier.T.
Inertia > DensityEnter Rho.
Graphic > Visual Properties > ColorEnter Carrier.RGB.

5. In the Rigid Transform block dialog box, specify these parameters.

ParameterSetting
Translation > MethodSelect Cartesian.
Translation > OffsetEnter [Carrier.L/2 0 -(Carrier.T+T)/2]. Select units of cm.

6. In the Rigid Transform1 block dialog box, specify these parameters.

ParameterSetting
Translation > MethodSelect Cartesian.
Translation > OffsetEnter [-Carrier.L/2 0 -(Carrier.T+T)/2]. Select units of cm.

7. In the model workspace, define the Carrier block parameters using MATLAB code:

```% Gear Carrier Parameters
Carrier.RGB = [0.25 0.4 0.7];
Carrier.L = Sun.R + Planet.R;
Carrier.W = 2*T;
Carrier.T = T/2;

Theta = (90:1:270)'*pi/180;
Beta = (-90:1:90)'*pi/180;

Carrier.CS = [-Carrier.L/2 + Carrier.W/2*cos(Theta) ...
Carrier.W/2*sin(Theta); Carrier.L/2 + Carrier.W/2*cos(Beta), ...
Carrier.W/2*sin(Beta)];```
8. Simulate the model. To induce motion, try adjusting the velocity state targets in the joint block dialog boxes. Notice that the gear carrier now performs the task of the Distance Constraint block.

You can open a copy of the resulting model. At the MATLAB command line, enter smdoc_planetary_gear_d.

### Add More Planet Gears

Experiment with the model by adding more planet gears. Remember that you must change the Carrier rigid body to accommodate any additional planet gears. To see an example with four planet gears, at the MATLAB command line enter smdoc_planetary_gear_e.

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