Shaft Deflection
Single Point Shaft Deflection In applications where a support rail is not used, shaft deflection can become critical in the function of the bearing. If deflection is greater than the misalignment capabilities of a standard pillow block, binding can occur. One solution would be to increase shaft and bearing size (to lessen the amount of deflection) or to use an open bearing configuration with a support rail. Follow the formulas below to check shaft deflection and sag.
Total Deflection ø1 in. Shaft
24 in. Length (L) 250 lb. load (W) Deflection =
W • L 3 D (from table)
= 250 lb. • (24 in.) 3 6.83 • 10 7 = 3,456,000 in 3 lb. 68,300,000 in 2 lb.
Deflection at Center
1/2 L
F
Deflection =
0.0506 in.
Displacement Angle
L 4 S (from table)
SAG
=
Deflection plus Sag
Formula for Inch and Metric Shafting Deflection Total shaft deflection in horizontal applications: Total Deflection ( δ ) = Def + Sag = FL 3 + 55L 4 48EI 384EI Deflection = w x L 3 / D Sag = L 4 / S Deflection = Pure deflection due to load at center of shaft* Sag = Deflection of shaft due to its own weight* L = Shaft unsupported length*
(24) 4 4.92 • 10 8
=
331,776 in 4 492,000,000 in 3
SAG
=
SAG .000674 in. Total Deflection ( δ ) = Deflection + SAG =
F = load being applied at center of shaft** D = Deflection coefficient (D = 48 • E x I) S = Sag coefficient (S = E • I • 384 / (5 • sw)) a = Displacement angle
= 0.0506 in. + .000674 in.
Total Deflection ( δ ) = 0.0513 in.
* inch or mm ** lb or N Notes: I = π • diam 4 / 64
sw = π • diam 2 / 4 • density E = Modulus of Elasticity (Young’s modulus)
pbclinear.com • Engineering Your Linear Motion Solutions 145
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