Gearbox intermediate shaft

An intermediate shaft carries a large gear, driven by the input stage, and a small pinion that drives the output stage. Their tooth forces act in both bending planes, and the torque passes from the gear to the pinion through the shaft. This tutorial rebuilds the Gearbox intermediate shaft example from a blank project and then fixes its failing deflection check.

The numbers are those of this version of Axomek. To compare as you go: open the Gearbox intermediate shaft example. It starts a new project from the example in place of yours, so save yours first (the app asks) and come back to it with Switch project (Ctrl+P).

The model

The tooth forces follow from the torque. The gear's tangential force is $2\cdot T/d=2\times 150000/200=1500\mathrm{N}$, the pinion's $2\times 150000/60=5000\mathrm{N}$. The radial forces are the tangential forces times tan 20°. Here the tangential forces act along z and the radial forces along y.

Step 1: the sections

1. Start a Blank project and choose Add a first shaft section.
2. Select S1 and set Length 25 (keep Ø30).
3. Choose + Section, then set S2: Outer diameter 38, Length 95.
4. + Section, S3: 45, 10.
5. + Section, S4: 38, 105.
6. + Section, S5: 30, 25.

Each new section is added at the right end with the previous section's size, so you only change what differs. The shaft is now 260 mm long.

Step 2: supports and bearings

1. Supports tab, + Support. B1 appears at x = 26 mm, locating. Set Name Input-side bearing, Designation 6306, Position 12.5.
2. + Support. B2 appears at x = 234 mm, floating. Set Name Output-side bearing, Designation NU 206, Position 247.5.
3. Select B1. In Bearing choose 6306 deep groove ball bearing under "New bearing: fits the Ø30 mm seat".
4. Select B2. In Bearing choose NU 206 cylindrical roller bearing, under "New bearing: fits the Ø30 mm seat" too.

The floating side takes the NU 206: its outer ring has no ribs, so the shaft can slide axially as it warms. An NU bearing cannot locate a shaft. Both bearings have a 30 mm bore, the diameter of the end sections they sit on; a bearing whose bore does not match the shaft under it fails the shaft seat check.

Step 3: the gear and the pinion

1. Loads tab, + Load. Set Name Gear, Position 80, Force y -546, Force z 1500, Torque 150000 (N·mm).
2. Under The part (mass, unbalance, inertia), set Part mass 6 kg.
3. + Load again. Set Name Pinion, Position 180, Force y 1819.9, Force z 5000, Torque -150000, Part mass 0.8.

The two torques sum to zero, as they must. The masses do not add weight here (gravity is off); they count in the critical speeds.

Step 4: limits and bearing requirements

1. Material tab, Limits for the checks: Max deflection 0.05 mm, Max tilt at bearings 4 ′, Max von Mises stress 200 MPa. A gear mesh needs a stiffer shaft than a plain support, hence the tight limits.
2. Open each bearing in the bearing module (Open BR1 in the bearing module → under the support's Bearing field). For BR1 and BR2 set Speed 950 rpm, and under Requirements Minimum life 20000 h and Minimum static safety s₀ 2.

Step 5: read the results

The shaft's banner reads 1 of 3 checks fails.

Switch between xy plane and xz plane above the drawing. The larger forces act in z, so the xz plane shows most of the bending. The deflection the check uses is the resultant of both planes.

The bearings pass all their checks:

The system life of the two is 138 509 h, limited by BR1. Note that BR1's basic life alone would fail 20 000 h; the modified life includes the lubrication and cleanliness.

Step 6: stiffen the shaft

The largest deflection sits between the gear and the pinion, where the shaft is Ø38. Make both Ø38 sections thicker.

1. Select S2 and set Outer diameter 42.
2. Select S4 and set Outer diameter 42.

The banner reads All 3 checks pass:

The deflection check passes at 90 % of its limit, so its row shows the warning mark. That is a passing design with little margin. Ø44 brings it to 47.9 µm on S4 alone; you can explore this quickly from the command line:

axm new box.axm.json --template gearbox-intermediate
axm sweep box.axm.json sections.S4.outerDiameter --from 38 --to 46 --steps 5

The bearing loads do not change with the diameters: the shaft is statically determinate on two rigid supports. They would change with Couple linked bearings switched on, because the bearings then act as springs.

Going further

• Add a shaft strength check under Material to assess the collar and the bearing seats after DIN 743 (Shaft strength).
• Replace the two loads by real gear pairs mounted on two more shafts, as in the single-stage gearbox tutorial.

Related

• Shaft
• Bearings
• Units and conventions
• Deflection, critical speeds and whirl
• Tutorial: a simple shaft on two bearings