Two stability limits instead of a single moment analysis
A long, high-speed ball screw rarely fails due to drive torque. It fails due to stability: Under compressive load, it can buckle, and at high speeds, it enters bending resonance. Both limits depend on the same parameter—the unsupported length L between the bearings—and the relationship is quadratic: Doubling L reduces the buckling load by a factor of four.
Euler buckling load
The theoretical buckling load follows the Euler method; F_k/S is permissible. For ball screws, S ≥ 2 is the standard rule of thumb listed in catalogs; higher values apply where there is a risk to personnel—the manufacturer’s specifications are binding. The moment of inertia I depends on the fourth power of the core diameter d_r —a diameter of 16 mm instead of 14 mm already results in approximately 70% higher buckling load:
I = π · d_r⁴ / 64
F_k = n_L · π² · E · I / L²
F_k = Buckling load (N) · E = 210,000 N/mm² (steel) · d_r = Core diameter (mm) · L = unsupported length between bearings (mm) · n_L = Bearing coefficient. Catalogs sometimes approximate d_r as the nominal diameter minus the ball diameter.
Validity: Euler’s formula applies only to slender spindles. The slenderness ratio is defined as λ = L / (√n_L · d_r/4); Euler’s formula applies only above the critical slenderness ratio (empirical value for steel is approximately 90–105; use the higher value when in doubt). For short, stocky spindles, the manufacturer’s buckling diagram applies. Second limit: Regardless of the buckling load, the allowable compressive stress of the spindle as well as the load ratings of the nut and fixed bearing limit the axial force—the smallest value is decisive.
Bearing Types and Their Coefficients
The bearing arrangement is fully reflected in the coefficient n_L —ranging from 0.25 (fixed–free) to 4 (fixed–fixed), representing a factor of 16 between the worst and best arrangements. The decisive factor here is how strongly a bearing clamps the spindle end against tilting—these are bending boundary conditions, not the question of whether the bearing holds the spindle axially in place. The same system provides the speed coefficient f_n for the critical speed:
- Fixed–fixed (both ends supported, e.g., a pair of angular contact ball bearings): n_L = 4 · f_n = 27.6
- Fixed–pinned (one fixed bearing, one floating bearing): n_L = 2 · f_n = 19.0
- Articulated–Articulated (both ends mounted with tilting motion): n_L = 1 · f_n = 12.2
- Fixed–Free (one end fixed, the other free): n_L = 0.25 · f_n = 4.3
Critical speed
At high rotational speeds, the spindle enters bending resonance. The critical speed n_krit is the speed at which this resonance begins; 80% of this value is considered permissible:
n_krit = f_n · (d_r / L²) · 10⁷
n_zul = 0.8 · n_krit
n_krit, n_zul in min⁻¹ · d_r, L in mm (numerical-value equation, coefficients for steel: E = 210,000 N/mm², ρ = 7.85 kg/dm³). In addition, the speed parameter of the nut (d₀ · n) limits the speed – the smaller value applies. Caution when comparing with catalogs: some already include the factor 0.8 in the coefficient (then about 9.7 / 15.1 / 21.9 / 3.4) and give n_zul directly – do not multiply by 0.8 again.
Ball screw catalogs specify a maximum axial force of 50% of the buckling load (safety factor S = 2)—a standard rule of thumb for ball screws; this value is higher when there is a risk to personnel, and the manufacturer’s specifications are binding. The ball screw calculator is preset with the stricter value S = 2.5. Screw jack manufacturers often calculate a buckling safety factor of 3 for the lead screw; here, too, the manufacturer’s specifications are binding.
Important Note
Worked example
Given
Core diameter d_r = 15 mm, bearing spacing L = 800 mm, fixed–pinned bearing arrangement (n_L = 2, f_n = 19.0).
Calculation
- I = π · 15⁴ / 64 ≈ 2,485 mm⁴
- F_k = 2 · π² · 210,000 · 2,485 / 800² ≈ 16,095 N → acceptable (S = 2): F_zul ≈ 8,048 N
- Slenderness λ = 800 / (√2 · 15/4) ≈ 151 – well above the critical slenderness ratio; Euler’s formula applies
- n_krit = 19.0 · (15 / 800²) · 10⁷ ≈ 4,453 min⁻¹ → n_zul = 0.8 · 4,453 ≈ 3,562 min⁻¹
For verification: Ball screw calculator (opens in a new tab)
- Marked point 1: Example: 16,095 N
- Marked point 1: Example: 16,095 N
Description and values of the figure
All four curves apply to a core diameter d_r = 15 mm and a bearing spacing L ranging from 800 to 1,600 mm, and they vary with the square of L. In this range, the slenderness ratio of all points is 107 or higher—the spindle is slender, and Euler’s formula applies. Fixed-Fixed (bearing coefficient 4) withstands the highest load, while Fixed-Free (bearing coefficient 0.25) withstands the lowest. The highlighted Fixed–pinned curve corresponds to the bearing configuration in the worked example for this unit (L = 800 mm, 16,095 N). The dashed line shows the allowable load F_zul=F_k/2 with a safety factor S=2 (standard rule of thumb in the catalog; the linked online calculator uses S=2.5 by default—this is not a contradiction, just a different standard value). The dashed line coincides with the “Hinge-Hinge” curve (2/2 = 1).
| Bearing spacing L (mm) | Fest–Fest (n_L=4) (N) | Fest–Gelenkig (n_L=2) (N) | Gelenkig–Gelenkig (n_L=1) (N) | Fest–Frei (n_L=0,25) (N) | F_zul = F_k/2 (S=2, bezogen auf Fest-Gelenkig) (N) |
|---|---|---|---|---|---|
| 800 | 32,191 | 16,095 | 8,048 | 2,012 | 8,048 |
| 1,000 | 20,602 | 10,301 | 5,151 | 1,288 | 5,151 |
| 1,200 | 14,307 | 7,154 | 3,577 | 894 | 3,577 |
| 1,400 | 10,511 | 5,256 | 2,628 | 657 | 2,628 |
| 1,600 | 8,048 | 4,024 | 2,012 | 503 | 2,012 |
Knowledge check
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