Power Screws and Ball Screws: Method

Power Screws and Ball Screws: Method and Sources

A screw converts torque into thrust. In a power screw the nut slides on the thread flanks, so friction takes a large part of the work and, in return, usually holds the load when the drive stops. In a ball screw the load rolls on recirculating balls: nine tenths of the work reaches the load, the screw does not hold by itself, and the design questions become rating life, whirling speed and stiffness. The page treats both. Thread proportions come from NBS Handbook H28 (1957) Part III, a U.S. government publication; the standards now in force (ASME B1.5 and B1.8, ISO 2901 to 2904, ISO 3408) and makers' ratings are cited, and their numbers are entered by the user.

Thread geometry

The tool takes the major diameter d, the pitch p and the number of starts, and builds the thread from the proportions of the form chosen. In every form the thread is p/2 thick at the pitch line, which lies half the basic height h inside the major diameter:

d_m = d − h,   d_r = d − 2h − allowance,   lead l = starts × p,   tan λ = l/(π d_m).

With flank angles α₁ (loaded flank) and α₂ measured in an axial section, the crest flat is p[0.5 − (h/2p)(tan α₁ + tan α₂)] and the thickness at the root, which the mating thread leaves, is p minus that flat. The forms offered are the square thread (flanks at 0°, h = 0.5p); the Acme thread of H28 Section XII (29° included angle, h = 0.5p, flat 0.3707p, pitch diameter D − h, minor diameter D − 2h); the stub Acme of Section XIII (h = 0.3p, flat 0.4224p); the buttress thread of Section XIV (7° pressure flank, 45° clearance flank, h = 0.6p, crest flat 0.16316p); and the 30° metric trapezoidal profile (h = 0.5p, flat 0.366p), whose proportions are those of ISO 2901 stated as mathematics. The test script reproduces the flats and heights of H28 Tables XII.1, XIII.1 and XIV.1 and the pitch diameters, minor diameters and lead angles of all 23 sizes of the Acme series in Table XII.3. H28 sets the largest minor diameter of a general-purpose Acme screw 0.020 in below basic for 10 threads per inch and coarser and 0.010 in for finer pitches; that is the default allowance. Dimension tables of the standards are not shipped.

Torque, efficiency and self-locking

Unwrap one turn of the thread at the pitch diameter: the nut is a block on an incline of angle λ, pushed along the incline by the tangential force and pressed on it by the load F. The flank is tilted, so the normal force leans away from the axis; seen in the section normal to the helix the tilt is α_n, with tan α_n = tan α cos λ. Equilibrium of the block with friction μ opposing the motion gives

T_R = (F d_m/2) (tan λ + μ sec α_n)/(1 − μ sec α_n tan λ)   to move against the load,

T_L = (F d_m/2) (μ sec α_n − tan λ)/(1 + μ sec α_n tan λ)   to move with it.

Textbooks usually print sec α, the axial-section angle, in place of sec α_n; at ordinary lead angles the two differ by hundredths of a percent, and the output shows both. A thrust collar of mean diameter d_c and friction μ_c adds μ_c F d_c/2 to each. Without friction the torque would be F l/(2π), so the efficiency is F l/(2π T). The U.S. Army handbook AMCP 706-346 (Elevating Mechanisms, 1964) prints this efficiency for a screw-type drive as its equations 16 and 17, (cos β − μ tan λ)/(cos β + μ cot λ) and the same with the thrust-bearing term, and works an example in paragraph 48: pressure angle 14½°, tan λ = 0.0765, μ = 0.10 gives 0.422, and with a thrust bearing of 1.5 in on a 2.6 in pitch diameter at μ_b = 0.01, 0.408. The library returns 0.4221 and 0.4091 (the handbook rounded its denominator to 1.04 + 1.31), and an independent numerical solution of the three-dimensional free body agrees with the closed form to rounding error for forty random screws.

The thread is self-locking when the lowering torque is positive, μ sec α_n ≥ tan λ, that is when μ is at least tan λ cos α_n. The same handbook states the condition as tan λ less than μ. A self-locking thread has an efficiency below one half, which is the dotted boundary on the efficiency chart. When the thread does not lock, the load drives the screw with the torque −T_L, less the collar friction. Friction coefficients are entered: the handbook assumes 0.10 for a lubricated, protected drive; the range in use for oiled steel on bronze is roughly 0.10 to 0.16 and vibration lowers the value that can be relied on for holding.

Stresses

The body carries F on the root area, σ_a = 4F/(π d_r²), and the raising torque in torsion, τ = 16T/(π d_r³). Each thread is a short cantilever: the load on it acts at mid-height, h/2 from the root, on a section π d_r long and b thick, so the bending stress is 3 F₁ h/(π d_r b²) and the transverse shear 3 F₁/(2π d_r b). For the square thread with b = h = p/2 these are the familiar 6F/(π d_r n p) and 3F/(π d_r n p). The nut thread has the same thickness at the major diameter d. Threads do not share the load evenly; the first engaged thread carries the most, and the share 0.38 in common textbook use is the default of an entered factor (the tool never uses less than the even share p/L_n). At the root surface of that thread the bending stress, the body axial stress across it and the torsional shear combine into the von Mises stress σ′ = √(½[(σ_b − σ_y)² + σ_y² + σ_b²] + 3τ²) and the maximum shear stress (half the spread of the principal stresses); a screw in compression has σ_y opposite in sign to the bending stress, the worse case. Margins are allowable/stress − 1 with S_y/n for von Mises and bending, 0.5 S_y/n for maximum shear and 0.577 S_y/n for thread shear, n being the entered design factor.

The bearing pressure is F over the projected flank area π d_m h of the engaged threads. It governs wear, and its allowable is an entered value (about 2,500 psi is commonly allowed for steel on bronze at low speed; the figure is a textbook one and is not from a public document). The required nut length is the largest of the lengths that meet the bearing allowable and the thread-shear allowables of nut and screw. H28 gives the shear area of an Acme thread per inch of engagement as π K_n[0.5 + n tan 14½° (E_s − K_n)]; at basic dimensions this is π d_r (b/p), the section used here.

Column buckling

A screw in compression is a round column of the root diameter, radius of gyration d_r/4. Euler's load C π² E A/(L/k)² applies above the slenderness (L/k)₁ = √(2π² C E/S_y), the Johnson parabola A[S_y − (S_y L/(2πk))²/(C E)] below it. The fixity coefficients are the theoretical ones, 0.25, 1, 2.046 and 4; the third is computed from tan x = x. A finite-element column confirms all four.

Wear and power

At n revolutions per unit time the nut travels n l and the flanks rub at π d_m n/cos λ. The product of bearing pressure and rubbing speed is reported and, when the nut maker's limit is entered, compared with it. Input power is torque times angular speed; useful power is F times travel speed.

Ball screw torque and preload

With efficiency η the drive torque is T = F l/(2π η); when the load drives the screw the shaft receives F l η′/(2π). A NASA symposium paper (Chiba and others, 1991) measured η = 0.95 on a solid-lubricated 25 mm screw in vacuum and calls it nearly that of a grease-lubricated screw; 0.90 and 0.80 are the defaults. A preloaded nut is two nut halves pressed against each other through the screw with the force F_pr. Turning the screw drives one half against that force and is helped by the other, so the net drag is F_pr l/(2π) × (1/η − η′). Makers publish their own coefficients, which can be entered instead.

Rating life

Ball screws are rated like ball bearings. The life reached by 90 % of a population is L₁₀ = (C_a/F)³ million revolutions, C_a being the load for one million revolutions; the exponent 3 for ball contacts and the linear accumulation of damage are set out in NASA/TP-2013-215305 (Zaretsky). Over a duty cycle of loads F_i at speeds n_i for shares q_i of the time, linear damage makes the cycle equivalent to the cubic-mean load F_m = (Σ F_i³ n_i q_i/Σ n_i q_i)^1/3 at the mean speed. The tool applies an entered load factor f_w to F_m, gives the life in revolutions, hours and travel, and the static factor C₀/F_max. The ratings themselves are the maker's (ISO 3408-5 defines them); the defaults are illustrative. The preload is not added to the working load, and the life of the support bearings is a separate check.

Critical speed, d-n value and buckling

The shaft whirls at its first bending natural frequency, n_cr = λ²/(2π L²) √(E I/(ρ A)), where √(I/A) = d_r/4 for the root section and λ is the first root of the frequency equation of a uniform beam: 1.8751 fixed-free (cos λ cosh λ = −1), π supported-supported, 3.9266 fixed-supported (tan λ = tanh λ), 4.7300 fixed-fixed (cos λ cosh λ = 1). The library solves these equations; a finite-element beam reproduces the roots. Service speed is kept to a fraction of n_cr, 0.8 by custom. The d-n value, nominal diameter in millimetres times revolutions per minute, limits the ball speed in the return path; the limit is the maker's and is entered. The buckling load is Euler's with the same four fixities over the loaded length, divided by the design factor. The two charts are the selection charts of the makers' catalogues, computed for the entered root diameter.

Stiffness, resonance, heat and accuracy

Between the nut and the bearing that holds it axially the shaft is a bar: k = E A/x with one locating bearing, softest when the nut is farthest; k = E A (1/x + 1/(L − x)) with both ends fixed, softest at mid-span where it is 4 E A/L. Nut and bearing stiffnesses add in series. The moved mass on this spring resonates at (1/2π)√(k/m). A temperature rise ΔT lengthens the shaft by α L ΔT; a shaft fixed at both ends is pretensioned by α ΔT E A so that it does not go slack. The positioning error estimate is the plain sum of the lead error (the permitted variation in 300 mm of the accuracy grade, entered, scaled with the travel), the elastic deflection at the largest load and the uncompensated thermal growth over the travel. It is an upper estimate for comparison of options, not an acceptance figure: the standards give the tolerance on long travels in their own tables.

Limits

Thread friction is a single entered number; starting friction, lubrication regime and wear-in are not modelled. The first-thread share and the bearing allowable are entered factors. Stress concentration at the thread root and fatigue are not treated. The column check takes the root diameter over the whole length and theoretical end conditions. For the ball screw the tool does not compute ratings from the ball and groove geometry, the preload's effect on life and stiffness, the nut's Hertzian deflection, acceleration torque or the support-bearing life.