Deflection, slope and critical speed

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

Strength is often not what sizes a shaft. Gears that do not stay parallel wear out; bearings that are misaligned lose most of their life; a shaft that runs near a critical speed tears itself apart. All three are stiffness problems, and stiffness scales with d⁴ rather than d³ — so the deflection check moves faster than the stress check when you change a diameter.

Slope at the bearings

Every bearing type has a misalignment it will tolerate. Exceed it and the load concentrates on one edge of the rolling elements.

Bearing	typical slope limit [rad]
tapered roller	0.0005 – 0.0012
cylindrical roller	0.0008 – 0.0012
deep-groove ball	0.001 – 0.003
spherical / self-aligning	0.026 – 0.052
uncrowned spur gear (at the mesh)	< 0.0005

Those are guideline ranges, not a code — the bearing manufacturer’s catalogue always wins. ShaftLab shades the allowable band on the slope chart for the bearing type you pick.

Sizing for deflection

If a deflection or slope is too large, you do not have to iterate blindly. Since everything scales with d⁴:

d_new = d_old · ( n_d · y_oldy_allowable )^1/4

Work out that ratio for the worst constraint and multiply all the diameters by it. The tight constraint becomes exactly satisfied and the rest get looser.

Twist

θ = Σ T_i l_iG J_i

Torsional stiffness rarely limits a power-transmission shaft, but it matters a lot for anything positioning-related: a machine-tool feed shaft, a robot joint, an encoder drive. A common rule of thumb is to keep the twist under about 0.25° per metre; for a servo axis you will want an order of magnitude less.

Critical speed

A shaft with mass is a spring–mass system, and it has a speed at which running it excites its own bending mode. For a uniform simply supported shaft:

ω₁ = (πl)² √( E Im )

and for a real shaft with lumped gears and pulleys, Rayleigh’s method uses the static deflection under the weights themselves:

ω₁ = √( g · Σ w_i y_iΣ w_i y_i² )

Rayleigh always over-estimates the first critical speed slightly, so treat it as an upper bound. The design rule is simple: keep the operating speed below half the first critical speed. ShaftLab reports the ratio.

💡 Adding mass at mid-span lowers the critical speed fast; adding it near a bearing barely changes it. If you are close to a critical speed, moving the heavy component towards a bearing is usually cheaper than making the shaft thicker.