Model a Compound 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 Simscape™ Multibody™, 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 bodies:

• Sun gear

• Planet gear

• Ring gear

• Planet carrier

Each 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 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.

   Library	Block	Quantity
   Body Elements	Extruded Solid	2
   Joints	Revolute Joint	1
   Joints	Planar Joint	1
   Frames and Transforms	Rigid Transform	1
   Frames and Transforms	World Frame	1
   Utilities	Mechanism Configuration	1
   Simscape > Utilities	Solver Configuration	1

2. Connect and name the blocks as shown.

   

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

   Parameter	Setting
   Geometry > Cross-Section	Enter simmechanics.demohelpers.gear_profile(2*Sun.R,Sun.N,A). Select units of cm.
   Geometry > Length	Enter T. Select units of cm.
   Inertia > Density	Enter Rho.
   Graphic > Visual Properties > Color	Enter Sun.RGB.

   The simmechanics.demohelpers.gear_profile function generates the cross-section matrix for an external gear with an involute tooth profile. The cross-section is approximate. Use the function as an example only.

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

   Parameter	Setting
   Geometry > Cross-Section	Enter simmechanics.demohelpers.gear_profile(2*Planet.R,Planet.N,A). Select units of cm.
   Geometry > Length	Enter T. Select units of cm.
   Inertia > Density	Enter Rho.
   Graphic > Visual Properties > Color	Enter Planet.RGB.

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

   Parameter	Setting
   Translation > Method	Select Standard Axis.
   Translation > Axis	Select +Y.
   Translation > Offset	Enter 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.

   Library	Block
   Constraints	Distance Constraint
   Gears and Couplings > Gears	Common 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.

   Parameter	Setting
   Specification Method	Select Pitch Circle Radii.
   Specification Method > Base Gear Radius	Enter Sun.R. Select units of cm.
   Specification Method > Follower Gear Radius	Enter 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 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.

   Library	Block
   Body Elements	Extruded Solid
   Joints	Revolute Joint
   Gears and Couplings > Gears	Common 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.

   Parameter	Setting
   Geometry > Cross-Section	Enter Ring.CS. Select units of cm.
   Geometry > Length	Enter T.
   Inertia > Density	Enter Rho.
   Graphic > Visual Properties > Color	Enter Ring.RGB.

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

   Parameter	Setting
   Type	Select Internal.
   Specification Method	Select Pitch Circle Radii.
   Specification Method > Base Gear Radius	Enter Planet.R. Select units of cm.
   Specification Method > Follower Gear Radius	Enter 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.

   Library	Block	Quantity
   Body Elements	Extruded Solid	1
   Joints	Revolute Joint	2
   Frames and Transforms	Rigid Transform	2

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.

   Parameter	Setting
   Geometry > Cross-Section	Enter Carrier.CS. Select units of cm.
   Geometry > Length	Enter Carrier.T.
   Inertia > Density	Enter Rho.
   Graphic > Visual Properties > Color	Enter Carrier.RGB.

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

   Parameter	Setting
   Translation > Method	Select Cartesian.
   Translation > Offset	Enter [Carrier.L/2 0 -(Carrier.T+T)/2]. Select units of cm.

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

   Parameter	Setting
   Translation > Method	Select Cartesian.
   Translation > Offset	Enter [-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 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.