1.1.1 Inertia

• Rotational inertia is a function of unbalanced torques, and the polar moment of inertia. This gives rise to another form of d’Alambert’s equation.

• The rotational inertia is about the center of mass and will be oriented in the negative rotational direction. (Therefore it is on the opposite side of the sum of torques equation)

• Consider the example below,

1.1.2 Springs

• Rotational springs are commonly found in products like measuring tapes and children’s toys.

• Essentially springs behave like members under torsion.

• We can use Hooke’s law to deal with an engineered torsional spring.

• Note: the angles used should be in radians.

• As long as the spring is deforming elastically the results will be valid. When it deforms plastically the spring constant value will probably increase, and the undeformed position will change.

• Practice problem,

1.1.3 Viscous Friction

• As before we can assume that some viscous liquid exists between two rotating shafts. The torque this viscous barrier will exert will become higher as the velocity increases.

• If the viscous boundary is between two moving surfaces, then the difference between the two will determine the torque.

• These calculations should be done using radians as the units.

• Practice problem,

1.1.4 Levers

• A lever is a simple device used to balance moments in a system.

• These are very common in engineered systems.

• These are well behaved when the displacements are small.

• Practice problem,

1.1.5 Gears and Belts

• Gears act as solid connections between mechanical components.

• Gears come in a variety of shapes, but all of these transmit forces at a fixed radius.

Spur Gears - Round gears with teeth parallel to the rotational axis

Rack - A straight gear (used with a small round gear called a pinion

Helical Gear - The teeth follow a helix around the rotational axis.

etc.

• The contact forces between the gears are always equal and opposite. The forces are also tangential to the pitch radius of the gear.

• Gears will typically have different radii to create a mechanical advantage. This also results in a ratio between angular motion. The number of teeth on a gear is proportional to the diameter.

• Quite often we will have a sequence of gears to give a significant gear ratio in a compact space. We will also have gears that can be engaged different ways to get multiple gear ratios, such as standard transmissions in cars.

• If we have a rack and pinion gear set (a straight rack gear) then we convert rotational motion to translational motion.

• When dealing with belts in system we use the same principles as with gears (except there are no teeth).

- A belt wound around a drum will act like a rack and pinion gear pair.

- A belt around two or more pulleys will act like gears.

• Practice problem,