The stress in a spinning ring is its density multiplied by the square of its rim speed, σ = ρ × v², so mild steel of 250 MPa yield and 7,800 kg/m³ density reaches yield at a rim speed of 179 m/s whatever the rotor’s size. A 100 mm radius rotor therefore yields at 17,100 RPM, and with a safety factor of 3 on stress it should never exceed 9,900 RPM, because stress grows with the square of speed and a rotor 10% over speed carries 21% more stress.
Part of Centrifuge Building
A speed limit for a rotor is the highest rim speed at which its material, loaded by its own rotation and its contents, stays well below its yield strength.
The Derivation: A Ring Holds Itself Together
Cut a thin spinning ring in half. Each element of mass dm at radius r is pulled outward with a force dm × ω² × r. Add the pulls over a half ring and the outward force on the cut is 2 × ρ × A × ω² × r², where A is the ring’s cross-section. It is held by the tension σ × A at each of the two cuts, so 2σA = 2ρAω²r² and
σ = ρ × ω² × r² = ρ × v²
The size of the ring and its wall thickness drop out; only the rim speed and the density matter. Because stress goes with v², a rotor run 10% over speed has a stress of 1.1² = 1.21 times the original, 21% more. This holds for a thin ring. A solid disc is less highly stressed, with a peak at its center of (3 + ν) ÷ 8 × ρ × v², about 0.41 times the ring value for steel with Poisson’s ratio ν = 0.3 (a formula from general knowledge, not from this wiki). A bar-shaped rotor arm has a stress at its center of ρ × v² ÷ 2 from its own mass, plus the pull of anything hung on its ends.
Worked Example 1: The Limit For Mild Steel
Use the values the carrier uses: density 7,800 kg/m³, yield 250 MPa.
- Rim speed at yield: v = √(250 × 10⁶ ÷ 7,800) = √32,051 = 179.0 m/s.
- Stress safety factor 3 on the stress: the working stress is 250 ÷ 3 = 83.3 MPa, so v = √(83.3 × 10⁶ ÷ 7,800) = 103.4 m/s. That is 179.0 ÷ √3, since stress goes with v².
- RPM from rim speed: N = 60 × v ÷ (2π × r) = 9.549 × v ÷ r (r in meters).
| Radius | RPM at yield (179 m/s) | RPM at safety factor 3 (103.4 m/s) | RCF at that RPM |
|---|---|---|---|
| 50 mm | 34,190 | 19,740 | 21,800 g |
| 75 mm | 22,800 | 13,160 | 14,500 g |
| 100 mm | 17,100 | 9,870 | 10,900 g |
| 150 mm | 11,400 | 6,580 | 7,260 g |
| 300 mm | 5,700 | 3,290 | 3,630 g |
Spot check, 100 mm: 9.549 × 103.4 ÷ 0.1 = 9,874 RPM, and v² ÷ (g × r) = 10,692 ÷ 0.981 = 10,900 g.
A correction to the carrier article. Centrifuge Building gave a maximum “roughly 200 m/s” which would be 19,000 RPM for a 100 mm rotor, and then took one third of the RPM, 6,300. The first figure is above the 179 m/s the carrier’s own material numbers allow, and dividing the speed by 3 divides the stress by 9, not by 3: 6,300 RPM at 100 mm is a rim speed of 66 m/s and a stress of 7,800 × 66² = 34 MPa, a margin of about 7 on yield. The figures above (17,100 at yield, 9,900 with a factor of 3 on stress) follow from the same inputs. The carrier’s 6,300 RPM is conservative, not unsafe.
Worked Example 2: The Separator Bowl Corners
The bowl in Bowl Construction carries the ring stress plus the pressure of the milk on its wall (σ₂ = p × R ÷ t, with p = ½ρ_milk ω² (R² − r_i²), r_i = 20 mm assumed, wall 3.5 mm). Four combinations from the carrier’s ranges (diameter 150-200 mm, speed 6,000-10,000 RPM):
| Bowl | RPM | Ring stress | Milk-pressure stress | Total | Margin on 250 MPa |
|---|---|---|---|---|---|
| 150 mm | 8,000 | 30.8 | 40.5 | 71 MPa | 3.5 |
| 150 mm | 10,000 | 48.1 | 63.2 | 111 MPa | 2.2 |
| 200 mm | 6,000 | 30.8 | 55.8 | 87 MPa | 2.9 |
| 200 mm | 10,000 | 85.5 | 154.9 | 240 MPa | 1.04 |
So the top corner of the carrier’s range, a 200 mm bowl at 10,000 RPM, has essentially no margin in a 3.5 mm mild-steel wall. The ranges cannot be combined freely; check the pair you choose. The lower combinations are comfortable, and the reference 150 mm at 8,000 RPM, which gives 5,366 g, keeps a margin of about 3.5.
Worked Example 3: A Tube-Rotor Arm
The tube centrifuge in Tube Centrifuge at 5,180 RPM (3,000 g at 100 mm): ω² = 294,200 and rim speed 54.2 m/s. Take an arm 20 mm × 6 mm (A = 1.2 × 10⁻⁴ m²) with a 10 mm bore for the shaft at its center, and 40 g of bucket, tube and sample at each end (assumptions).
- Pull of one bucket: 0.040 × 294,200 × 0.1 = 1,177 N.
- Pull of one half of the arm’s own mass: ρ × A × ω² × L² ÷ 2 = 7,800 × 1.2 × 10⁻⁴ × 294,200 × 0.01 ÷ 2 = 1,377 N.
- Tension across the center: 1,177 + 1,377 = 2,554 N. Net area at the bore: (20 − 10) × 6 = 60 mm² = 6 × 10⁻⁵ m². Stress = 2,554 ÷ 6 × 10⁻⁵ = 42.6 MPa.
- A hole concentrates stress locally by a factor often taken as 2-3 for a small hole in a plate (general knowledge); at 2.5, the peak is about 106 MPa, a margin of 250 ÷ 106 = 2.4.
So the arm is under the factor of 3, even though the rim speed of 54 m/s is half the ring limit. In a bar rotor the holes decide, not the ring formula. Thicken the arm at the bore or lower the speed.
Why Overspeed Is Brutal
- 10% over speed: 1.1² = 1.21 times the stress.
- A crank turned at 90 RPM instead of the designed 60: 1.5² = 2.25 times the stress. A rotor designed to a factor of 3 is left with 3 ÷ 2.25 = 1.33.
- Hotter or corroded metal, welds and fatigue from repeated starts all lower the real strength below the table.
Put a limit in the mechanism: a fixed gear ratio and a belt that slips are limits; a person’s arm is not.
What You Need
- The material’s density and yield strength; the carrier’s mild-steel figures are used above, and anything else (salvaged steel, aluminum, cast iron) needs its own figures and a test
- A measure of the rotor’s radius to the rim and of any holes and welds
- A margin; see Safety Factors
Method: Setting The Speed Limit
Step 1: compute the rim speed at the intended RPM: v = 2π × r × N ÷ 60. Step 2: compute the ring stress ρ × v² and add the contents’ load (liquid pressure or cake). Step 3: divide by the fraction of metal left after holes and multiply by 2-3 where the geometry concentrates stress. Step 4: check the result is no more than a third of yield. Step 5: build in the limit, and mark the maximum RPM on the machine.
Common Mistakes
- Dividing the speed by the safety factor. The stress goes as the square: divide the stress, and take the square root for the speed.
- Taking the largest figure in each range. See Example 2.
- Ignoring holes. See Example 3.
- Using a safe speed for the rotor and forgetting the hand that can crank faster.
Safety
A rotor does not fail gradually
Past its limit a spinning ring fails at one point and the rest follows within a turn. The fragments leave at the rim speed, 54-63 m/s in the examples here. Never raise the speed of a rotor you have not rechecked, never run one with a crack, a deep scratch or a poor weld, and keep everyone out of the plane of the rotor. See Containment Shield.
What’s Next
- Learn next: Containment Shield — the shield for the failure the limit does not prevent
- Combine with: Centrifugal Force Calculations — the g that each limit allows