Shafts and bearings: one load, two choices

Strong, stiff and long-lived

• The shaft must be strong and stiff enough, and its bearings must last as long as the pump. Both choices come from the same loads.
• Draw the bending-moment diagram, find the worst section (the gear seat, where moment and notch meet) and step the shaft down where the moment is small.

The DE-Goodman shaft equation

• Solving Goodman for the diameter gives $d = [\tfrac{16n}{\pi}(2K_fM_a/S_e + \sqrt3K_{fs}T_m/S_{ut})]^{1/3}$.
• With the shoulder data and $n = 2$: $d = \mathbf{29.0}$ mm, so choose the standard 30 mm, which gives $n = 2.21$. The peak moment is $1.017 \times 70 = 71.2$ N·m.

Why bearing life has a cube

• For rolling bearings, the catalogue gives the basic dynamic load rating $C$. The L10 life, the revolutions that 90% survive, is $L_{10} = (C/P)^3 \times 10^6$ for ball bearings.
• Double the load and the life falls to an eighth. A required life fixes $C_{req} = P(L/10^6)^{1/3}$.

Choosing the bearing

Lives beyond 90%

• $L_{10}$ is the life 90% of bearings reach. For a pump that must not stop, ask for 99%: the ISO 281 factor $a_1 = 0.25$ scales the life.
• The 6206's 27 000 h becomes $0.25 \times 27\,000 = \mathbf{6750}$ h at 99%, short of 20 000. Holding 99% needs $C \ge 2.94 \times (216/0.25)^{1/3} = \mathbf{28.0}$ kN: a size up.

Worked example: is the shaft stiff enough?

• Treat the 30 mm shaft as simply supported on a 200 mm span with 1565 N at 70 mm: $I = 3.98 \times 10^{-8}$ m⁴.
• The slope at the gear is $Pab(b - a)/3EIL = \mathbf{0.00018}$ rad, inside the 0.0005 limit for good tooth contact. At the near bearing it is 0.00049 rad, within what a ball bearing tolerates.

Three shaft slips

Recap