Ball Screw Calculator
Size a ball screw drive: lead and feed rate give the rotational speed and drive torque, while diameter, unsupported length and end bearings give the critical speed, the DN value and the buckling load. The calculator checks the operating speed against the 80 percent limit and rates DN value and buckling stability with a traffic light - live with every input.
Calculation
Model: standard ball-screw sizing formulas (kinematics, drive torque, critical speed, DN value, Euler buckling load, L10 service life) with catalogue reference values. Preliminary sizing tool - final selection follows the manufacturer's catalogue data.
Results
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Formulas and fundamentals
Kinematics and drive torque
The kinematics link the lead P (travel per revolution) to the speed (v in m/s, P in m, n in rpm):
The drive torque combines the static feed term and the acceleration term:
Where F is the axial process force, η the drive efficiency and J·α the acceleration torque. The moving linear mass m reduced to the screw is:
The angular acceleration follows from the linear acceleration as:
Critical speed and DN value
The bending-critical speed limits the rotational speed of the slender screw (d0 and L in mm):
The end-bearing factor f depends on the mounting (fixed-fixed 25.5; fixed-supported 17.7; supported-supported 11.5; fixed-free 3.9). The operating limit is:
In addition the DN value is checked against the manufacturer limit (typically 70,000 to 150,000 mm/min):
Buckling load and service life
The axial compression stability is assessed with Euler, using the second moment of area of the root section (d3 = root diameter):
Only half the buckling load is admissible:
The nominal service life of the rolling contact follows the bearing relation with the dynamic axial load rating Ca, in operating hours:
Worked example
A screw with nominal diameter d0 = 25 mm and lead P = 10 mm moves an axis at v = 0.5 m/s. The speed is n = v·60/P = 0.5·60/0.01 = 3000 rpm. With an axial process force F = 5000 N and efficiency η = 0.9 the static drive torque is M_drive = F·P/(2·π·η) = 5000·0.01/(2·π·0.9) = 8.84 Nm.
With fixed-supported bearings and an unsupported length L = 800 mm the critical speed is n_crit = 17.7·25/800²·10⁷ = 6914 rpm. The operating speed of 3000 rpm sits at 43 percent, well below the 80 percent limit (5531 rpm) - the speed check passes.
The DN value DN = d0·n = 25·3000 = 75,000 mm/min is within the usual range. With root diameter d3 = 21 mm and E = 210,000 N/mm² the buckling load is uncritical. The example shows the typical case: not the strength but the critical speed of the slender, long-supported screw governs at high traverse speeds.
Frequently asked questions
Why does the critical speed limit the screw?
A slender rotating screw goes into resonance at its bending-critical speed and whips. The operating speed is therefore limited to 80 percent of the critical speed. Critical speed rises with diameter and falls with the square of the unsupported length - a shorter support span or a stiffer bearing arrangement help the most.
What is the DN value?
The DN value is the product of nominal diameter d0 in mm and speed n in rpm. It characterises the circulation speed of the balls in the nut and is limited by the manufacturer (typically 70,000 to 150,000 mm/min, external recirculation lower, internal higher). It acts in addition to the critical speed and often governs for small, fast-turning screws.
Which mounting is best?
Fixed-fixed gives the highest critical speed (factor 25.5) and the highest buckling load (Euler factor 4), but needs axial fixed bearings at both ends and thermal preload control. Fixed-supported (17.7) is the usual compromise. Fixed-free (3.9) suits only short, slow axes.
When does buckling govern?
For long, thin screws under high compressive force. The Euler buckling load grows with the fourth power of the root diameter and falls with the square of the length. Only half the buckling load is admissible (safety factor 2). Under tension the buckling check is dropped; tensile strength then governs.
How is service life calculated?
Via the bearing relation L10 = (Ca/F)³·10⁶ revolutions with the dynamic axial load rating Ca from the manufacturer catalogue and the acting axial force F. The cube makes life very sensitive to load: doubling the force cuts life to one eighth. In operating hours L10h = L10/(60·n).
Does this replace manufacturer selection?
No. It is a preliminary sizing tool with the accepted standard formulas and catalogue reference values. The final choice of nut type, preload, load rating and admissible speed is made from the concrete catalogue data of the selected manufacturer (e.g. Steinmeyer, THK, NTN-SNR).
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