Chain Drives

Chain Drives

Chain Drives: Method and Sources

A roller chain is a row of links one pitch long that wraps the sprockets as a polygon. Everything about the layout of a drive follows from that picture, and the tool works it as geometry and statics. The power a chain may carry is a different matter: it is set by fatigue of the link plates and by impact fatigue of the rollers and bushings, and it comes from the rating equations of the American roller-chain standard. The standard (ASME B29.1, with ISO 606 as its international counterpart) and the makers' rating tables are cited, not reproduced; every rating constant is an input of the form, and the page says below how each default was obtained and what was checked against a printed document. The default drive (5 hp at 1,750 rev/min on No. 40 chain, 17 and 51 teeth, 20 in centres) is illustrative, not a published example.

Geometry

Pitch diameter, ratio, speed. The roller centres of a sprocket with N teeth lie on a regular polygon with sides of one pitch p, so the circle through them has the diameter D = p / sin(180°/N). The speed ratio is N₂/N₁. The chain advances one pitch per tooth, so its mean speed is V = N₁ p n₁.

Chain length. With lengths counted in pitches and C the centre distance, the usual approximation is L/p = 2C/p + (N₁ + N₂)/2 + (N₂ − N₁)² / (4π² C/p). The tool also evaluates the tangent-line length: the two strands are tangent to the pitch circles and inclined to the centre line by α, sin α = (D₂ − D₁)/(2C), the small sprocket is wrapped over 180° − 2α and the large one over 180° + 2α, and each sprocket holds teeth in proportion to its wrap, so L/p = 2(C/p) cos α + (N₁ + N₂)/2 + (N₂ − N₁) α/π. The length is rounded up to an even number of pitches (an odd number needs an offset link, which the ratings do not cover), or the entered length is used.

Centre distance for that length. Inverting the approximation gives the closed form C/p = ¼ [L′ + √(L′² − 2(N₂ − N₁)²/π²)] with L′ = L/p − (N₁ + N₂)/2. The tangent-line length has no closed inverse and is solved by Newton iteration safeguarded by bisection; the residual printed with the result is below 10⁻¹⁰ pitch. On 40 random drives the two centre distances differ by less than 0.02 pitch when the tooth difference is under 60 and by up to 0.12 pitch for an 11-tooth pinion driving 88 teeth at 21 pitches, so the root is the one to machine to. Both treat the wrap as a circle; the true polygon adds a variation of the same order as the chordal rise as the sprockets turn, which the slack strand absorbs.

Layout guides. The text of the American Chain Association ratings (adapted from the standard) asks for at least 120° of wrap on the small sprocket, calls 30 to 50 pitches of centre distance most desirable and says large sprockets should normally not exceed 120 teeth; the tool notes a departure from each.

Chordal action and sprocket dimensions

The strand leaves a sprocket from a link whose distance from the centre swings between D/2 (roller at the tangent point) and (D/2) cos(180°/N) (link square to the radius). At a constant sprocket speed the chain speed therefore falls from its peak by ΔV/V = 1 − cos(180°/N) once per tooth, the strand rises and falls by (D/2)(1 − cos(180°/N)), and the mean speed is sin(x)/x of the peak with x = π/N. The variation is 13.4 % for 6 teeth, 1.7 % for 17 and 0.5 % for 32, which is the reason for the common advice of at least 17 teeth on a fast pinion.

Sprocket diameters follow the tooth-form proportions of the standard, written as mathematics: outside diameter of a turned sprocket p (0.6 + cot(180°/N)), bottom diameter D − D_r with D_r the roller diameter, and for an odd tooth count the caliper diameter D cos(90°/N) − D_r, measured over nearly opposite tooth gaps. These three forms were confirmed only from secondary descriptions of the standard (Machinery's Handbook, cited); the roller diameter is an input, and when it is left blank the tool takes 5/8 of the pitch, the proportion of the standard series as the builder recalls it.

Power rating

The two equations. With the power in hp, the pitch in inches and the speed of the small sprocket in rev/min, the single-strand rating is the lower of the link-plate fatigue limit H₁ = K_s k_lp N₁^a n₁^b p^{(3 − 0.07p)}, which rises with speed, and the roller and bushing impact limit H₂ = 1000 K_r N₁^1.5 p^0.8 / n₁^1.5, which falls. Plotted against speed on logarithmic axes the two lines make the tent of the rating chart; they cross at n = [1000 K_r N₁^{1.5 − a} p^{0.07p − 2.2} / (K_s k_lp)]^{1/(b + 1.5)}. Above a third limit, the speed at which the joints gall whatever the lubrication, the rating is zero.

Where the default constants come from. The American Chain Association published its ratings of December 2001 as a free document ("local reproduction authorized"): an introduction adapted from ASME B29.1 and one table of horsepower against tooth count and speed for each chain number. The document prints no equation. The builder parsed its tables and fitted the form above to the link-plate region; the fit is K_s = 0.0044, a = 1.00, b = 0.96 with the pitch exponent 3 − 0.07p, and the roller side is reproduced by K_r = 17 for No. 40 to 240, 29 for No. 25 and 35 and 3.4 for No. 41, with k_lp = 0.55 for the light No. 41. With these constants the tool's equations reproduce 5,223 of the 5,224 standard-series cells at or below 85 % of the galling speed to within the printed rounding (0.005 hp plus 0.4 %); the one exception is No. 41 at 35 teeth and 1,200 rev/min (9.80 printed, 9.73 computed). The tables themselves are not shipped with the tool.

Earlier ratings. Machine-design texts print the constants of the previous standard, K_s = 0.004, a = 1.08, b = 0.9, with the same roller equation; the select offers them. That form is the builder's recollection of the texts and was not checked against a printed copy; against the 2001 tables it is 10 to 30 % lower at moderate and high speeds, which is what the 2001 document says of its own improvement.

Galling speed. The earlier standard's galling expression, as the builder recalls it, does not reproduce the 2001 tables, so it is not used. When the field is blank the tool estimates the speed at which the 2001 tables fall to zero by its own fit, n_g = 31,020 / (N₁^0.797 p^1.013) rev/min: 2 % rms and 6 % at worst on the 64 cells that lie on the falling edge, and inside the last-nonzero to first-zero bracket widened by 6 % in all 227 rows. The tables do not drop at once: between about 85 % and 100 % of this speed they fall below the two equations, and the tool says so when the drive runs there.

Factors and conditions. Design power = service factor × transmitted power; the rating of a multiple-strand chain = strand factor × single-strand rating. The service factors (1.0 to 1.7 by type of load and of driver) and the strand factors 1.7, 2.5 and 3.3 are Tables 1 and 2 of the 2001 document. Its ratings assume the recommended lubrication, about 100 pitches of chain, aligned sprockets on parallel horizontal shafts and press-fit connecting links, and correspond to about 15,000 hours at full load. The margin printed is rating / design power − 1.

Lubrication. The 2001 tables mark their speed columns Type A (manual or drip), Type B (bath or disc) or Type C (oil stream); the boundaries are speeds of the small sprocket for each chain number, not chain speeds. The two defaults filled by the chain select are the first Type B and the first Type C column of that number's table, read by a script from the page layout (No. 40 checked by eye: 180 and 1,600 rev/min). They are inputs.

Heavy series and the No. 160 page. The heavy-series tables follow the same form with a link-plate factor of about 1.05 to 1.15; the tool does not ship those factors (enter one). In the document as distributed, the page headed No. 160 repeats the No. 160H table cell for cell, so No. 160 could not be checked and its lubrication speeds are those of that page.

Loads

The tight strand carries the pull F_t = P/V. Both strands carry the centrifugal tension (w/g)V², which is balanced inside the chain and does not load the shafts, and the tension that holds the slack strand at its sag y over a level span S, w S²/(8y) + w y (parabola). The shaft load is the vector sum of the two strand pulls without the centrifugal part, √(A² + B² + 2AB cos 2α) with A = F_t + F_sag and B = F_sag. The static factor is the tensile strength divided by the tight-strand tension; it is information, not the design check, because a running chain is governed by the fatigue rating, far below its tensile strength.

The chain weight and the tensile strength are the maker's data and are not shipped. Left blank, the tool estimates the weight as 1.7 p² lb/ft per strand and the minimum tensile strength as 12,500 p² lb per strand (p in inches). Both are the builder's recollection (catalogue weights within about 15 %; the strength rule gives the 3,125 lb usually quoted for No. 40) and are labelled as estimates on the results.

Selection table, wear

The selection table repeats the rating at the entered tooth count, speed and design power for every standard number from 25 to 240, with the entered link-plate constants, each number's roller constant, the estimated galling speed and the number of strands needed by the factors 1, 1.7, 2.5, 3.3. A chain is usually replaced at 3 % elongation (less with many teeth or fixed centres); the figure is common practice recalled by the builder, and the tool only shows how far the worn chain rides out on the large sprocket.

Conveyor chain pull

For a conveyor of length L inclined at θ, with the conveyed weight W on the loaded run, chain weight w per unit length (all strands) and friction coefficient f, the loaded run resists with (W + wL)(f cos θ + sin θ) and the return run with wL(f cos θ − sin θ). Walking round the loop from the point where the chain leaves the drive sprocket, with the tension nowhere negative, gives the largest tension at the drive and the effective pull F_e = W(f cos θ + sin θ) + 2wLf cos θ; the drive power is F_eV. When tan θ exceeds f the return run slides back under its own weight and the tool reports the tension it keeps at the drive. The working load each strand must have is service factor × largest tension / strands, to be compared with the maker's rated working load. This is plain statics; sprocket friction, skirt drag and unsupported catenaries are left to the service factor.

Lifting chain slings

The tension in each of n legs at θ from the vertical is W/(n cos θ); a four-leg sling is counted as three. 29 CFR 1910.184 printed Table N-184-1, rated capacities of alloy steel chain slings from 1/4 to 1 3/4 in, until 2011; since then the rule requires the maker's permanently affixed rating and the table number carries the minimum allowable link size instead. The tool reproduces the 2004 printing as a sizing guide and picks the smallest size whose single-leg rating covers the leg tension. The multi-leg columns of the table are legs × rating × cos θ rounded (all within 2.5 %), except two misprints in the 45° double-sling column (5,800 lb for the 1 in chain and 61,000 lb for the 1 1/4 in chain, where the statics give about 54,800 and 81,300 lb). The Navy's rigging rule of thumb for ordinary open-link chain, safe working load = 8 D² tons with D in inches, is shown for comparison. The rating marked on the sling governs.

Not covered, not verified

Not covered: inverted-tooth (silent) chain, double-pitch and engineering-steel chains (named only), idlers and drives with more than two sprockets, inclined or vertical drive spans in the sag tension, wear life, and the heavy-series factors. Verified against a printed document: the service and strand factors, the layout guides, the rating constants (against the 2001 tables), the sling table. From the builder's memory and labelled so: the earlier rating constants, the sprocket tooth-form expressions (confirmed only through secondary descriptions), the 5/8 roller proportion, the weight and strength estimates, the 3 % elongation limit. The numerical work is cross-checked by an independent numpy and scipy script.