Shaft Design Considerations

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Shaft Design Considerations

• Design Considerations
• to minimize both shaft deflections and stresses,
  shafts should be as short as possible
• cantilevers are not recommended unless required
  for serviceability (i.e. access to v-belts, etc.)
• avoid stress concentrations at regions of high
  bending moments minimize their effect through
  generous reliefs

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Shaft Design Constraints

• Design Constraints
• deflections at gears should not exceed 0.1 mm
• relative slopes between mating gear axes should
  be held to less than 0.0005 rad
• shaft slope at bearings should be kept less than
• cylindrical roller bearings 0.0001 rad
• tapered roller bearings 0.0005 rad
• deep-groove roller bearings 0.004 rad
• spherical ball bearings 0.0087 rad

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Multiaxial Loading

• What happens if there is loading in more than one
  plane (i.e. vertical and horizontal planes)?
• the simplest method is to create separate bending
  moment diagrams for each plane and then determine
  the resultant moment at sites of interest using
  Pythagorems theorem

• the same procedure should be used for determining
  shaft slopes and deflections

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Design for Fluctuating Bending and Torsion

• The von Mises stress amplitude component sa and
  mean component sm are given by

where A and B are the radicals in the above
equations. The Gerber fatigue failure criterion
is defined by
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Design for Fluctuating Bending and Torsion

• solving for the shaft diameter d

or, solving for the factor of safety, n
where
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Determining Shaft Deflections

• Various method exist to determine the deflections
  of beams due to bending.
• The complicating factor for the design of shafts
  is typically the presence of step changes in
  shaft diameter along its length (shoulders,
  etc.). Thus, one commonly used method is the
  Integration Method with aid of Singularity
  Functions.

distributed load function
shear force function
moment function
slope function
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Determining Shaft Deflections

• Integrating

The integration constants C1 and C2 are the
boundary conditions on the shear and moment
function, which are simply the reaction forces
imposed on the beam. Thus, if the reaction
forces are used in the analysis (which is a very
good idea)
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Singularity Functions

• Singularity functions are use to represent
  discrete entities (loads, moments, etc.) applied
  in a discontinuous fashion over the beam length.
• Denoted by the binominal function in angled
  brackets

where x is the variable of interest a is a
user defined parameter to denote where in x
the singularity acts n is the power of
the function
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Singularity Functions

• Commonly used functions are

unit parabolic function
unit ramp function
unit step function
unit impulse function
Integration of Singularity Functions
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Torsional Deflection

• Angular deflection of a shaft from torsional
  loads is

where T is the torque l is shaft length G is
the shear modulus J is the polar moment
If the shaft is stepped or has multiple torques
applied to it, the angular deflection can be
determined from the sum of the deflections of
each shaft segment
where, i is the shaft segment
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Example