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Gear Constraints

You can represent gear constraints in a multibody model. To do this, SimMechanics™ provides a Gears and Couplings library. This library contains gear blocks that you can use to constrain the motion of two rigid body frames. The figure shows the gear blocks that the library provides.

Gear Types

The Gears and Couplings > Gears library provides blocks for modeling gears. The table summarizes the gears you can model with these blocks.

BlockDescription
Common Gear ConstraintTransfer rotational motion between two frames spinning about parallel axes
Rack and Pinion ConstraintTransfer rotational motion at a pinion into translational motion at a rack and vice-versa.
Bevel Gear ConstraintTransfer rotational motion between two frames spinning about arbitrarily aligned axes.

Featured Examples

SimMechanics provides two featured examples that highlight the use of gear blocks. The table lists these examples. To open an example model, at the MATLAB® command line, enter the model name, e.g., sm_cardan_gear.

Featured ExampleModel NameGear Blocks Used
Cardan gearsm_cardan_gearCommon Gear Constraint
Windshield wipersm_windshield_wiperRack and Pinion Constraint
Robotic wrist sm_robotic_wristBevel Gear Constraint

Open the models and examine the blocks for examples of how to connect the gear blocks and specify their parameters.

Inertia, Geometry, and Efficiency

Each gear block represents a kinematic constraint between two rigid body frames. This constraint does not account for the effects of inertia or power transmission losses. It also does not provide gear visualization. If necessary, consider modeling these effects using other SimMechanics and Simscape™ blocks. To represent gear inertia and geometry, use the Solid block.

Using Gear Blocks

To apply a gear constraint between two rigid bodies, connect the base and follower frames of the gear block to the rigid body frames that you want to constrain. Then, open the gear block dialog box and specify the gear parameters. Parameters can include gear dimensions and ratio.

Featured example sm_cardan_gear illustrates an application of the Common Gear block. In this model, two Common Gear blocks connect three gear rigid bodies. Subsystems Planet Gear A, Planet B and Link, and Sun Gear represent these rigid bodies. One Common Gear block constrains the motion of subsystem Planet Gear A with respect to subsystem Sun Gear. The other Common Gear block constrains the motion of subsystem Planet B and Link with respect to subsystem Planet Gear A. The figure shows the block diagram of this model.

So that the three gear subsystems can rotate with respect to each other, the model includes three Revolute Joint blocks. Each Revolute Joint block provides one rotational degree of freedom between one gear subsystem and the gear carrier—a rigid body that holds the three rotating gears. The figure shows the Mechanics Explorer display of this model.

Assembling Rigid Bodies with Gear Constraints

To assemble successfully, a model must satisfy the constraints that a gear block imposes. These include distance and orientation constraints that are specific to each block. The table summarize these constraints.

Gear ConstraintDescription
Frame DistanceThe model must maintain a fixed distance between the base and follower gear frames. The value of this distance depends on the gear block that you use.
Frame OrientationThe model must orient the base and follower gear frames according to rules that are specific to each block.

The rigid body frames that the gear block connects must have the proper number and type of degrees of freedom. For a Common Gear block, the frames must have two rotational degrees of freedom with respect to each other. For a Rack and Pinion block, the frames must have one translational and one rotational degree of freedom with respect to each other. You provide these degrees of freedom using joint blocks.

  • Use joint blocks with revolute primitives to provide the rotational degrees of freedom.

  • Use joint blocks with prismatic primitives to provide the translational degrees of freedom.

Common Gear Assembly and Simulation

During assembly, the Common Gear block requires that the base and follower frame Z axes align. These are the rotation axes of the two gear frames. Failure to align the Z axes of the two gear frames results in assembly failure during model update. The figure illustrates the common gear rigid bodies, frames, and distance constraints.

Connect the gear rigid bodies to joints possessing one (or more) revolute joint primitives. The rotational axis of the revolute primitive must align with the Z axis of the gear frame that it connects to. This ensures that the gear frames possess a rotational degree of freedom about the correct axis (Z).

Common Gear Types

With the Common Gear block, you can represent internal and external gear constraints. If the gear constraint is internal, the gear frames rotate in the same direction. If it is external, the gear frames rotate in opposite directions. The figure illustrates the two common gear types that you can represent and their relative rotation senses.

Gear Dimensions

In the block dialog box, you specify the gear dimensions. Depending on the specification method that you choose, you can specify the center-to-center distance between gears or the pitch circle radii. During model assembly, the Common Gear block imposes this distance constraint between the two gear frames. This ensures that the gear assembles properly or, if issues arise, that you can correct any assembly issues early on.

You specify the gear relative sizes in the block dialog box. If you select the Center Distance and Ratio specification method, the gear ratio specifies which of the two gears is the larger one. If the gear ratio is greater than one, the follower gear is the larger gear. If the gear ratio is smaller than one, the base gear is the larger gear.

If you specify an internal gear type, the larger gear is the ring gear. A gear ratio greater than unity makes the follower gear the ring gear. A gear ratio smaller than unity makes the base gear the ring gear.

Gear Pitch Circles

The pitch circle of a gear is an imaginary circle that passes through the contact point between gears. The pitch radius of a gear is the radius of this imaginary circle. The figure illustrates the pitch circles of two meshing gears and their pitch radii. These are the gear radii that you enter in the block dialog box when you select the Pitch Circle Radii specification method.

Simulation

During simulation, the Common Gear block requires that the model maintain the proper distance between gear frames. This distance must equal either the center-to-center distance or the sum of base and follower gear pitch radii that you specify in the block dialog box. The structure of the model must be such that the gears maintain this distance between them. Failure to maintain this distance results in an error during simulation.

In the Cardan Gear example, the Carrier rigid body fixes the distances between the three gears. As long as these distances match the gear dimensions that you specify in the block dialog box, the model should simulate without an issue.

Rack and Pinion Assembly and Simulation

The base frame of the Rack and Pinion block represents the pinion. It can rotate about its Z axis. The follower frame of the same block represents the rack. It can translate along its Z axis. During assembly, the Rack and Pinion block requires that the base and follower frame Z axes be mutually orthogonal.

When the gear is in its zero configuration—a configuration in which the angle and displacement between base and follower frames are taken as zero—the follower frame Z axis is also parallel to the base frame X axis, and base and follower frame Y axes are parallel to each other. The follower frame origin lies along the base frame -Y axis, at a distance equal to the base gear pitch radius. The figure illustrates these constraints.

To ensure the rack and pinion can move with respect to each other, you must connect the rack and pinion rigid bodies to joints blocks. The joint block on the rack side must have one (or more) prismatic primitives. At least one primitive axis must align with the Z axis of the follower gear frame. The joint block on the pinion side must have one (or more) revolute primitives. At least one revolute axis must align with the Z axis of the base gear frame.

Gear Pitch Circles

The pitch circle of a rack and pinion gear is the imaginary circle that passes through the contact point between the pinion and the rack. The pitch radius is the radius of this imaginary circle. The figure illustrates the pitch circle for a rack and pinion. This is the circle whose radius you enter in the block dialog box.

Simulation

During simulation, the Rack and Pinion block requires that the model maintain the proper distance between gear frames. The distance between the base frame origin (pinion) and the follower frame Z axis must equal the pinion radius. Failure to maintain this distance between gear frames results in a simulation error.

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