7. Rotation

• Forces and torques (moments) are complementary in mechanical systems. When either are out of balance, we get some sort of acceleration.


7.1 Mathematical Elements

• There are a number of rotational elements that will create torques,



viscous friction

gears and belts


• We can also consider the energy and power in a rotational system


Problem 7.1 Given the initial state of a rotating mass, find the state 5 seconds later.

7.1.1 Inertia

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• 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,


7.1.2 Springs

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• 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.

Problem 7.2 If we have a 1/2" 1020 steel rod that is 1 yard long, what is the torsional spring coefficient?

7.1.3 Viscous Friction

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• 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.

Problem 7.3 If a wheel (J=5kg m**2) is turning at 150 rpm and the damping coefficient is 1Nms/rad, what is the deceleration?

7.1.4 Levers

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• 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.

Problem 7.4 Given a lever set to lift a 1000 Kg rock: if the lever is 2m long and the distance from the fulcrum to the rock is 10cm, how much force is required to lift it?

7.1.5 Gears and Belts

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• 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.


• 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.

Problem 7.5 We have a gear train that has an input gear with 20 teeth, a center gear that has 100 teeth, and an output gear that has 40 teeth. If the input shaft is rotating at 5 rad/sec what is the rotation speed of the output shaft? What if the center gear is removed?

7.2 System Examples

• These problems are set up solve much like the translation problems.

• Consider the example below,


Problem 7.6 (Problem 4.24 pg. 133, Close and Frederick) Write the equations to relate the force ‘F’ and displacement ‘x’.


7.3 Problems

Problem 7.7 Draw the FBDs and write the differential equations for the mechanism below. The right most shaft is fixed in a wall.

Answer 7.7

Problem 7.8 Draw the FBDs and write the differential equations for the mechanism below. Assume the pulleys are massless.

Answer 7.8

Problem 7.9 Find the polar moments of inertia of area and mass for a round cross section with known radius and mass per unit area. How are they related?

Answer 7.9


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