How can the wobble of a rotating cantilever shaft be reduced?

Question

I have a cantilever shaft 6inches in length supported on one end using a ball bearing. I am having problems of shaft wobbling in the bearing. I cannot have support on the other end.

How can I help this situation?

The shaft is made of aluminum, rotating at around 120 rpm.

Answer 1

Rotating shafts are subject to deflections, even without applied (transverse) loads. To reduce excessive shaft deflection:

• Avoid operating speeds near the critical shaft speed, where the shaft vibrates at its natural frequencies.
• Ensure the shaft is concentric, where its center of mass rotates about its geometric center.

Assuming the shaft is concentric and does not operate at the critical shaft speed, deflections are caused by applied loads but depend on geometry and material. From your diagram, construct a Free Body Diagram and system of equations to solve for the reaction moment at the bearing(s). $$\sum F: \qquad 0 = R_{bearing} - (F_{pulley}+F_{distrib})$$ $$\sum M_{@bearing}: \qquad 0 = M_{bearing} + F_{pulley}d_{pulley} - F_{distrib}d_{distrib}$$ ^{Note that $F_{distrib}$ is the resultant force of the distributed load.}

From the shear and bending moment diagrams, the bearing(s) must resist the reaction moment, $M_{bearing}$, required by static equilbrium (or the shaft is free to rotate about its transverse axes). Therefore:

• Decrease force by reducing shaft speed

• Select bearing(s) rated for the reaction moment, reaction forces, and operating speeds.

  Modify bearing type or configuration:

  A single ball bearing is not designed to resist bending moments. Instead, consider using wide needle bearings, double bearings, or (preferably) two separated bearings. Multiple bearings resolve the reaction moment to reaction forces applied at a distance ($M = Fd$), where greater distance between bearings reduce $R_{bearing}$. These design options are illustrated below.

From the deflection equations, the following design parameters further reduce shaft deflections (note that these parameters are only applicable if the bearing(s) resist the bending moments):

• Increase shaft diameter
• Select a stiffer material

Used in combination, these parameters are used to design shafts according to design requirements (geometry, material, loads, allowable deflections).

Answer 2

Your shaft is a beam. You must have something like this:

X========X========

Where the X's are bearings.

Things you can do to improve the situation:

• Slower rate of rotation
• Use stiffer material for the shaft
• Use bearings that control the orientation of the shaft, not just it's
  location (e.g. don't use bearings embedded in a sphere)
• Increase the diameter of the shaft
• Make the stick out distance shorter
• make the portion of the shaft between the bearings larger in diameter
• Change the distance between the bearings. Exactly 'to what' depends on more information than you've given.

The phenomenon you're dealing with is 'shaft whirl' - you can read about it in a Millwright's book.

Answer 3

If the dimensions are fixed, a careful selection of alloy or hardness that has a higher inherent damping would also reduce deflection while the shaft is rotating.