Multi-body systems

Gears

• Larger gear - > slower spin
• Accelerations - > tangential are equal but centripetal are not
• Opposite direction

Standard sign convention

1. \( \omega_{1z} \) and \( \omega_{2z} \) have opposite signs (the gears rotate in opposite direction)
2. Smaller gears rotate faster

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Example: Rotating gears. #rkc-gsc

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Example: Rotating gears with chain. #rkc-csc

Acceleration in contact (no slip)

Gear kinetics

When two gears in equilibrium mesh,

• There is a "contact force" between the gears—this comes from Newton's third law.
• This contact force is perpendicular to the radius of the gear and produces a moment on each gear.
• For each gear,
  $$ I \alpha = M + F_c \ r $$
  where \( M \) is an external moment and \( F_c \ r \) is the moment created by the contact force.

Four-bar linkages

Image from Wikimedia. CC BY 2.0

A four-bar linkage is a common mechanism that is the simplest closed-chain movable linkage. It consists of four members, connected in a loop by four joints. One member is typically fixed to the ground by pin supports.

Here is is something to keep in mind when solving problems involving four-bar linkages:

• The tangential velocity of the points at the pins is zero.

Slider cranks

Image from Wikimedia. CC BY 2.0

A slider crank mechanism is a common mechanical system that converts rotational motion into linear motion. It consists of a crank, a connecting rod, and a slider. The crank rotates about a fixed axis, the connecting rod connects the crank to the slider, and the slider moves linearly along a guide or track.

Here are some things to keep in mind when solving problems involving slider-crank mechanisms:

• The velocity of the slider is purely tangential to the path of the slider.
• The acceleration of the slider is purely tangential to the path of the slider.
• The slider has a reaction force normal to the path of the slider.