38:
47:
31:
2636:
2307:
119:
A holding capstan is a ratchet device that can turn only in one direction; once a load is pulled into place in that direction, it can be held with a much smaller force. A powered capstan, also called a winch, rotates so that the applied tension is multiplied by the friction between rope and capstan.
994:
supercarriers (97,000 tons each, but for the baby it would be only a little more than 1 kg). The large number of turns around the capstan combined with such a high friction coefficient mean that very little additional force is necessary to hold such heavy weight in place. The cables necessary
1500:
1313:
55:
535:
2423:
2144:
2135:
41:
An example of when knowledge of the capstan equation might have been useful. The bent white tube contains a cord to raise and lower a blind. The tube is bent 40 degrees in two places. The blue line indicates a more efficient
1590:
1691:
1344:
1168:
1196:
213:
598:
It is important that the line is not rigid, in which case significant force would be lost in the bending of the line tightly around the cylinder. (The equation must be modified for this case.) For instance a
987:
For instance, the factor "153,552,935" (5 turns around a capstan with a coefficient of friction of 0.6) means, in theory, that a newborn baby would be capable of holding (not moving) the weight of two
1976:
409:
124:
a holding capstan and a powered capstan are used in tandem so that a small force can be used to raise a heavy sail and then the rope can be easily removed from the powered capstan and tied off.
2711:
Metzger, Andreas; Konyukhov, Alexander; Schweizerhof, Karl (2011). "Finite
Element implementation for the EULER–EYTELWEIN problem and further use in FEM-simulation of common nautical knots".
2631:{\displaystyle T_{0}e^{-\mu _{\tau }ks\,{\sqrt {\cos ^{2}\alpha -\sin ^{2}\alpha /\mu _{g}^{2}}}}\leq T\leq T_{0}e^{\mu _{\tau }ks\,{\sqrt {\cos ^{2}\alpha -\sin ^{2}\alpha /\mu _{g}^{2}}}}.}
1048:
2418:
654:
344:
569:
401:
374:
270:
243:
1837:
2387:
2302:{\displaystyle \omega =\mu _{\tau }{\sqrt {k_{n}^{2}-{\frac {k_{g}^{2}}{\mu _{g}^{2}}}}}=\mu _{\tau }k{\sqrt {\cos ^{2}\alpha -{\frac {\sin ^{2}\alpha }{\mu _{g}^{2}}}}},}
1740:
1336:
1191:
961:(a rope to the loose side of a sail) wound more than three turns around a winch. The force gain would be extreme besides being counter-productive since there is risk of a
1097:
1896:
593:
314:
1916:
1714:
108:
Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small
2336:
2030:
1864:
290:
2360:
2003:
1790:
1068:
976:
It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (
980:
or sail sheet) from sliding down. The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is
2037:
1050:
is a function of the total angle subtended by the rope on the capstan. On the verge of slipping, this is also the frictional force, which is by definition
1495:{\displaystyle {\frac {dT_{\mathrm {load} }(\varphi )}{d\varphi }}=\mu T_{\mathrm {load} }(\varphi ),\qquad T_{\mathrm {load} }(0)=T_{\mathrm {hold} },}
1511:
1615:
2747:
1308:{\textstyle \delta R(\varphi )\approx T_{\mathrm {load} }(\varphi )\sin(\delta \varphi )\approx T_{\mathrm {load} }(\varphi )\delta \varphi }
1102:
1716:
is the angle (in radians) between the two flat sides of the pulley that the v-belt presses against. A flat belt has an effective angle of
156:
995:
to support this weight, as well as the capstan's ability to withstand the crushing force of those cables, are separate considerations.
2821:
1923:
530:{\displaystyle F=T_{\text{load}}-T_{\text{hold}}=(e^{\mu \varphi }-1)~T_{\text{hold}}=(1-e^{-\mu \varphi })~T_{\text{load}}}
2938:
2687:
2923:
2657:
2896:
2773:
349:
For dynamic applications such as belt drives or brakes the quantity of interest is the force difference between
37:
2395:
1006:
618:
with the coefficient of friction, the number of turns around the cylinder, and the angle of contact. Note that
293:
79:
1756:
For the same power transmission, a V-belt requires less tension than a flat belt, increasing bearing life.
316:
is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle
1753:
tends to wedge into the mating groove in a pulley as the load increases, improving torque transmission.
628:
319:
2851:
1765:
607:
547:
379:
352:
248:
221:
1797:
2751:
2365:
1719:
2877:
2728:
2339:
1318:
1173:
1073:
615:
571:
is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller
91:
2869:
1874:
578:
299:
1901:
1699:
82:), relates the hold-force to the load-force if a flexible line is wound around a cylinder (a
17:
2943:
2859:
2720:
2662:
981:
46:
2314:
2008:
1842:
1760:
Generalization of the capstan equation for a rope lying on an arbitrary orthotropic surface
966:
2803:
1750:
958:
275:
30:
2855:
116:
force on the other side; this is the principle by which a capstan-type device operates.
58:
An example of holding capstans and a powered capstan used to raise sails on a tall ship.
54:
2345:
1988:
1775:
75:
2130:{\displaystyle T_{0}e^{-\int _{s}\omega \,ds}\leq T\leq T_{0}e^{\int _{s}\omega \,ds}}
1053:
27:
Relates the hold-force to the load-force if a flexible line is wound around a cylinder
2932:
2881:
2732:
2652:
144:
128:
600:
1585:{\displaystyle T_{\mathrm {load} }(\varphi )=T_{\mathrm {hold} }e^{\mu \varphi }}
962:
603:
is to some extent rigid and doesn't obey the principles of the capstan equation.
1686:{\displaystyle T_{\text{load}}=T_{\text{hold}}e^{\mu \varphi /\sin(\alpha /2)}}
988:
136:
102:
98:
2873:
140:
121:
2864:
2839:
2724:
965:, result being that the sheet will foul, form a knot and not run out when
2913:
Arne
Kihlberg, Kompendium i Mekanik för E1, del II, Göteborg 1980, 60–62.
2897:"Introduction to Computational Contact Mechanics: A Geometrical Approach"
1163:{\textstyle \delta R(\varphi )=R(\varphi +\delta \varphi )-R(\varphi )}
83:
1764:
If a rope is lying in equilibrium under tangential forces on a rough
1746:
1606:
977:
208:{\displaystyle T_{\text{load}}=T_{\text{hold}}\ e^{\mu \varphi }~,}
1918:
are satisfying the following criteria for all points of the curve
132:
87:
53:
45:
36:
29:
272:
is the resulting force exerted at the other side of the capstan,
970:
540:
Several assumptions must be true for the equations to be valid:
620:
the radius of the cylinder has no influence on the force gain
1768:
surface then all three following conditions are satisfied:
2389:
is a coefficient of friction in the tangential direction.
656:
based on the number of turns and coefficient of friction
1315:. Combining these and considering infinitesimally small
97:
It also applies for fractions of one turn as occur with
984:(loose end is pulled clear), by hand or a self-tailer.
1971:{\displaystyle -\mu _{g}<\tan \alpha <+\mu _{g}}
1321:
1199:
1176:
1105:
1056:
1009:
2426:
2398:
2368:
2348:
2317:
2147:
2040:
2011:
1991:
1926:
1904:
1877:
1845:
1800:
1778:
1722:
1702:
1618:
1514:
1347:
1076:
631:
581:
550:
412:
382:
355:
322:
302:
278:
251:
224:
159:
2643:
This generalization has been obtained by
Konyukhov.
1601:
Generalization of the capstan equation for a V-belt
957:From the table it is evident why one seldom sees a
2802:Moradmand, Jamshid; Marcks, Russell; Looker, Tom.
2630:
2412:
2381:
2354:
2330:
2301:
2129:
2024:
1997:
1970:
1910:
1890:
1858:
1831:
1784:
1734:
1708:
1685:
1584:
1494:
1330:
1307:
1185:
1162:
1099:. By simple geometry, the additional normal force
1091:
1062:
1042:
648:
587:
563:
529:
395:
368:
338:
308:
284:
264:
237:
207:
112:force exerted on one side can carry a much larger
2840:"Contact of ropes and orthotropic rough surfaces"
614:It can be observed that the force gain increases
544:The rope is on the verge of full sliding, i.e.
1792:is positive for all points of the rope curve:
131:this effect allows a lighter person to hold (
8:
2844:Journal of Applied Mathematics and Mechanics
296:between the rope and capstan materials, and
50:Schematic of quantities for capstan equation
2797:
2795:
2665:, a device that exploits the capstan effect
1170:when increasing the angle by a small angle
625:The table below lists values of the factor
1043:{\textstyle T_{\mathrm {load} }(\varphi )}
2863:
2681:
2679:
2615:
2610:
2601:
2589:
2570:
2564:
2563:
2551:
2546:
2536:
2513:
2508:
2499:
2487:
2468:
2462:
2461:
2449:
2441:
2431:
2425:
2413:{\displaystyle \omega ={\text{constant}}}
2405:
2397:
2373:
2367:
2347:
2322:
2316:
2286:
2281:
2264:
2257:
2242:
2236:
2227:
2210:
2205:
2195:
2190:
2184:
2175:
2170:
2164:
2158:
2146:
2118:
2109:
2104:
2094:
2072:
2063:
2055:
2045:
2039:
2016:
2010:
1990:
1962:
1934:
1925:
1903:
1882:
1876:
1850:
1844:
1814:
1799:
1777:
1721:
1701:
1670:
1653:
1646:
1636:
1623:
1617:
1573:
1553:
1552:
1520:
1519:
1513:
1473:
1472:
1440:
1439:
1406:
1405:
1359:
1358:
1348:
1346:
1320:
1274:
1273:
1223:
1222:
1198:
1175:
1104:
1075:
1055:
1015:
1014:
1008:
645:
636:
630:
580:
555:
549:
521:
499:
477:
452:
436:
423:
411:
387:
381:
360:
354:
335:
321:
301:
277:
256:
250:
229:
223:
190:
177:
164:
158:
2689:The Mechanics of Friction in Rope Rescue
2032:are satisfying the following inequality
1866:is a normal curvature of the rope curve.
662:
2675:
1982:Limit values of the tangential forces:
2692:. International Tech Rescue Symposium
7:
1985:The forces at both ends of the rope
245:is the applied tension on the line,
2838:Konyukhov, Alexander (2015-04-01).
2686:Attaway, Stephen W. (1999-11-01).
1563:
1560:
1557:
1554:
1530:
1527:
1524:
1521:
1483:
1480:
1477:
1474:
1450:
1447:
1444:
1441:
1416:
1413:
1410:
1407:
1369:
1366:
1363:
1360:
1284:
1281:
1278:
1275:
1233:
1230:
1227:
1224:
1025:
1022:
1019:
1016:
649:{\displaystyle e^{\mu \varphi }\,}
25:
1871:Dragging coefficient of friction
1605:The belt friction equation for a
1338:yields the differential equation
2362:is a curvature of a rope curve,
1772:No separation – normal reaction
339:{\displaystyle \varphi =2\pi \,}
1434:
564:{\displaystyle T_{\text{load}}}
396:{\displaystyle T_{\text{hold}}}
369:{\displaystyle T_{\text{load}}}
265:{\displaystyle T_{\text{hold}}}
238:{\displaystyle T_{\text{load}}}
1832:{\displaystyle N=-k_{n}T>0}
1678:
1664:
1542:
1536:
1462:
1456:
1428:
1422:
1381:
1375:
1296:
1290:
1263:
1254:
1245:
1239:
1212:
1206:
1157:
1151:
1142:
1127:
1118:
1112:
1086:
1080:
1037:
1031:
511:
486:
467:
445:
1:
18:Proof of the capstan equation
2658:Frictional contact mechanics
2382:{\displaystyle \mu _{\tau }}
1735:{\displaystyle \alpha =\pi }
1331:{\textstyle \delta \varphi }
1186:{\textstyle \delta \varphi }
2924:Capstan equation calculator
2746:Mann, Herman (5 May 2005).
2713:PAMM Proc. Appl. Math. Mech
1092:{\displaystyle R(\varphi )}
2960:
2820:Slocum, Alexander (2008).
403:. The formula for this is
969:(by slacking grip on the
892:
835:
789:
753:
727:
701:
696:
693:
690:
687:
684:
681:
678:
670:
665:
2822:"FUNdaMENTALS of Design"
2804:"Belt and Wrap Friction"
1891:{\displaystyle \mu _{g}}
1193:is well approximated by
671:Coefficient of friction
588:{\displaystyle \varphi }
309:{\displaystyle \varphi }
135:) a heavier person when
2895:Konyukhov, A.; Izi, R.
2772:Johnson, K. L. (1985).
1911:{\displaystyle \alpha }
1709:{\displaystyle \alpha }
1070:times the normal force
294:coefficient of friction
80:Johann Albert Eytelwein
72:Euler–Eytelwein formula
2865:10.1002/zamm.201300129
2725:10.1002/pamm.201110116
2632:
2414:
2383:
2356:
2332:
2303:
2131:
2026:
1999:
1972:
1912:
1892:
1860:
1833:
1786:
1736:
1710:
1687:
1586:
1496:
1332:
1309:
1187:
1164:
1093:
1064:
1044:
650:
589:
565:
531:
397:
370:
340:
310:
286:
266:
239:
209:
68:belt friction equation
59:
51:
43:
34:
2633:
2415:
2384:
2357:
2333:
2331:{\displaystyle k_{g}}
2304:
2132:
2027:
2025:{\displaystyle T_{0}}
2000:
1973:
1913:
1893:
1861:
1859:{\displaystyle k_{n}}
1834:
1787:
1737:
1711:
1688:
1587:
1497:
1333:
1310:
1188:
1165:
1094:
1065:
1045:
651:
590:
566:
532:
398:
371:
341:
311:
287:
267:
240:
210:
57:
49:
40:
33:
2939:Equations of physics
2424:
2396:
2366:
2346:
2315:
2145:
2038:
2009:
1989:
1924:
1902:
1875:
1843:
1798:
1776:
1720:
1700:
1616:
1512:
1345:
1319:
1197:
1174:
1103:
1074:
1054:
1007:
1003:The applied tension
629:
579:
548:
410:
380:
353:
320:
300:
285:{\displaystyle \mu }
276:
249:
222:
157:
139:, and also produces
2856:2015ZaMM...95..406K
2620:
2518:
2342:of the rope curve,
2291:
2215:
2200:
2180:
2628:
2606:
2504:
2410:
2379:
2352:
2340:geodesic curvature
2328:
2299:
2277:
2201:
2186:
2166:
2127:
2022:
1995:
1968:
1908:
1888:
1856:
1829:
1782:
1745:The material of a
1732:
1706:
1683:
1582:
1505:whose solution is
1492:
1328:
1305:
1183:
1160:
1089:
1060:
1040:
646:
585:
561:
527:
393:
366:
336:
306:
282:
262:
235:
205:
60:
52:
44:
35:
2775:Contact Mechanics
2621:
2519:
2408:
2355:{\displaystyle k}
2294:
2292:
2218:
2216:
1998:{\displaystyle T}
1785:{\displaystyle N}
1639:
1626:
1393:
1063:{\textstyle \mu }
955:
954:
558:
524:
516:
480:
472:
439:
426:
390:
363:
259:
232:
201:
185:
180:
167:
16:(Redirected from
2951:
2901:
2900:
2892:
2886:
2885:
2867:
2835:
2829:
2828:
2826:
2817:
2811:
2810:
2808:
2799:
2790:
2789:
2787:
2785:
2780:
2769:
2763:
2762:
2760:
2759:
2750:. Archived from
2743:
2737:
2736:
2708:
2702:
2701:
2699:
2697:
2683:
2663:Torque amplifier
2637:
2635:
2634:
2629:
2624:
2623:
2622:
2619:
2614:
2605:
2594:
2593:
2575:
2574:
2565:
2556:
2555:
2541:
2540:
2522:
2521:
2520:
2517:
2512:
2503:
2492:
2491:
2473:
2472:
2463:
2454:
2453:
2436:
2435:
2419:
2417:
2416:
2411:
2409:
2406:
2388:
2386:
2385:
2380:
2378:
2377:
2361:
2359:
2358:
2353:
2337:
2335:
2334:
2329:
2327:
2326:
2308:
2306:
2305:
2300:
2295:
2293:
2290:
2285:
2276:
2269:
2268:
2258:
2247:
2246:
2237:
2232:
2231:
2219:
2217:
2214:
2209:
2199:
2194:
2185:
2179:
2174:
2165:
2163:
2162:
2136:
2134:
2133:
2128:
2126:
2125:
2114:
2113:
2099:
2098:
2080:
2079:
2068:
2067:
2050:
2049:
2031:
2029:
2028:
2023:
2021:
2020:
2004:
2002:
2001:
1996:
1977:
1975:
1974:
1969:
1967:
1966:
1939:
1938:
1917:
1915:
1914:
1909:
1897:
1895:
1894:
1889:
1887:
1886:
1865:
1863:
1862:
1857:
1855:
1854:
1838:
1836:
1835:
1830:
1819:
1818:
1791:
1789:
1788:
1783:
1741:
1739:
1738:
1733:
1715:
1713:
1712:
1707:
1692:
1690:
1689:
1684:
1682:
1681:
1674:
1657:
1641:
1640:
1637:
1628:
1627:
1624:
1591:
1589:
1588:
1583:
1581:
1580:
1568:
1567:
1566:
1535:
1534:
1533:
1501:
1499:
1498:
1493:
1488:
1487:
1486:
1455:
1454:
1453:
1421:
1420:
1419:
1394:
1392:
1384:
1374:
1373:
1372:
1349:
1337:
1335:
1334:
1329:
1314:
1312:
1311:
1306:
1289:
1288:
1287:
1238:
1237:
1236:
1192:
1190:
1189:
1184:
1169:
1167:
1166:
1161:
1098:
1096:
1095:
1090:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1030:
1029:
1028:
951:
950:
947:
944:
937:
936:
933:
926:
925:
922:
915:
914:
907:
906:
888:
887:
884:
877:
876:
873:
866:
865:
858:
857:
850:
849:
831:
830:
823:
822:
815:
814:
807:
806:
785:
784:
777:
776:
663:
655:
653:
652:
647:
644:
643:
606:The line is non-
594:
592:
591:
586:
570:
568:
567:
562:
560:
559:
556:
536:
534:
533:
528:
526:
525:
522:
514:
510:
509:
482:
481:
478:
470:
460:
459:
441:
440:
437:
428:
427:
424:
402:
400:
399:
394:
392:
391:
388:
375:
373:
372:
367:
365:
364:
361:
345:
343:
342:
337:
315:
313:
312:
307:
291:
289:
288:
283:
271:
269:
268:
263:
261:
260:
257:
244:
242:
241:
236:
234:
233:
230:
214:
212:
211:
206:
199:
198:
197:
183:
182:
181:
178:
169:
168:
165:
70:, also known as
64:capstan equation
21:
2959:
2958:
2954:
2953:
2952:
2950:
2949:
2948:
2929:
2928:
2920:
2910:
2908:Further reading
2905:
2904:
2894:
2893:
2889:
2837:
2836:
2832:
2824:
2819:
2818:
2814:
2806:
2801:
2800:
2793:
2783:
2781:
2778:
2771:
2770:
2766:
2757:
2755:
2748:"Belt Friction"
2745:
2744:
2740:
2710:
2709:
2705:
2695:
2693:
2685:
2684:
2677:
2672:
2649:
2585:
2566:
2547:
2542:
2532:
2483:
2464:
2445:
2437:
2427:
2422:
2421:
2394:
2393:
2369:
2364:
2363:
2344:
2343:
2318:
2313:
2312:
2260:
2259:
2238:
2223:
2154:
2143:
2142:
2105:
2100:
2090:
2059:
2051:
2041:
2036:
2035:
2012:
2007:
2006:
1987:
1986:
1958:
1930:
1922:
1921:
1900:
1899:
1878:
1873:
1872:
1846:
1841:
1840:
1810:
1796:
1795:
1774:
1773:
1762:
1751:serpentine belt
1718:
1717:
1698:
1697:
1642:
1632:
1619:
1614:
1613:
1603:
1598:
1596:Generalizations
1569:
1548:
1515:
1510:
1509:
1468:
1435:
1401:
1385:
1354:
1350:
1343:
1342:
1317:
1316:
1269:
1218:
1195:
1194:
1172:
1171:
1101:
1100:
1072:
1071:
1052:
1051:
1010:
1005:
1004:
1001:
948:
945:
942:
940:
934:
931:
929:
923:
920:
918:
912:
910:
904:
902:
885:
882:
880:
874:
871:
869:
863:
861:
855:
853:
847:
845:
828:
826:
820:
818:
812:
810:
804:
802:
782:
780:
774:
772:
667:
632:
627:
626:
577:
576:
551:
546:
545:
517:
495:
473:
448:
432:
419:
408:
407:
383:
378:
377:
356:
351:
350:
318:
317:
298:
297:
274:
273:
252:
247:
246:
225:
220:
219:
186:
173:
160:
155:
154:
150:The formula is
28:
23:
22:
15:
12:
11:
5:
2957:
2955:
2947:
2946:
2941:
2931:
2930:
2927:
2926:
2919:
2918:External links
2916:
2915:
2914:
2909:
2906:
2903:
2902:
2887:
2850:(4): 406–423.
2830:
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2019:
2015:
1994:
1980:
1979:
1978:
1965:
1961:
1957:
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1951:
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1945:
1942:
1937:
1933:
1929:
1907:
1885:
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1324:
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1244:
1241:
1235:
1232:
1229:
1226:
1221:
1217:
1214:
1211:
1208:
1205:
1202:
1182:
1179:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1088:
1085:
1082:
1079:
1059:
1039:
1036:
1033:
1027:
1024:
1021:
1018:
1013:
1000:
997:
953:
952:
938:
927:
916:
908:
900:
897:
894:
890:
889:
878:
867:
859:
851:
843:
840:
837:
833:
832:
824:
816:
808:
800:
797:
794:
791:
787:
786:
778:
770:
767:
764:
761:
758:
755:
751:
750:
747:
744:
741:
738:
735:
732:
729:
725:
724:
721:
718:
715:
712:
709:
706:
703:
699:
698:
695:
692:
689:
686:
683:
680:
676:
675:
669:
642:
639:
635:
612:
611:
604:
596:
584:
575:contact angle
554:
538:
537:
520:
513:
508:
505:
502:
498:
494:
491:
488:
485:
476:
469:
466:
463:
458:
455:
451:
447:
444:
435:
431:
422:
418:
415:
386:
359:
334:
331:
328:
325:
305:
281:
255:
228:
216:
215:
204:
196:
193:
189:
176:
172:
163:
76:Leonhard Euler
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2956:
2945:
2942:
2940:
2937:
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2934:
2925:
2922:
2921:
2917:
2912:
2911:
2907:
2898:
2891:
2888:
2883:
2879:
2875:
2871:
2866:
2861:
2857:
2853:
2849:
2845:
2841:
2834:
2831:
2823:
2816:
2813:
2805:
2798:
2796:
2792:
2777:
2776:
2768:
2765:
2754:on 2007-08-02
2753:
2749:
2742:
2739:
2734:
2730:
2726:
2722:
2718:
2714:
2707:
2704:
2691:
2690:
2682:
2680:
2676:
2669:
2664:
2661:
2659:
2656:
2654:
2653:Belt friction
2651:
2650:
2646:
2644:
2625:
2616:
2611:
2607:
2602:
2598:
2595:
2590:
2586:
2582:
2579:
2576:
2571:
2567:
2560:
2557:
2552:
2548:
2543:
2537:
2533:
2529:
2526:
2523:
2514:
2509:
2505:
2500:
2496:
2493:
2488:
2484:
2480:
2477:
2474:
2469:
2465:
2458:
2455:
2450:
2446:
2442:
2438:
2432:
2428:
2402:
2399:
2391:
2374:
2370:
2349:
2341:
2323:
2319:
2310:
2296:
2287:
2282:
2278:
2273:
2270:
2265:
2261:
2254:
2251:
2248:
2243:
2239:
2233:
2228:
2224:
2220:
2211:
2206:
2202:
2196:
2191:
2187:
2181:
2176:
2171:
2167:
2159:
2155:
2151:
2148:
2140:
2122:
2119:
2115:
2110:
2106:
2101:
2095:
2091:
2087:
2084:
2081:
2076:
2073:
2069:
2064:
2060:
2056:
2052:
2046:
2042:
2034:
2033:
2017:
2013:
1992:
1984:
1983:
1981:
1963:
1959:
1955:
1952:
1949:
1946:
1943:
1940:
1935:
1931:
1927:
1920:
1919:
1905:
1883:
1879:
1870:
1851:
1847:
1826:
1823:
1820:
1815:
1811:
1807:
1804:
1801:
1794:
1793:
1779:
1771:
1770:
1769:
1767:
1759:
1757:
1754:
1752:
1748:
1743:
1729:
1726:
1723:
1703:
1675:
1671:
1667:
1661:
1658:
1654:
1650:
1647:
1643:
1633:
1629:
1620:
1612:
1611:
1610:
1608:
1600:
1595:
1577:
1574:
1570:
1549:
1545:
1539:
1516:
1508:
1507:
1506:
1489:
1469:
1465:
1459:
1436:
1431:
1425:
1402:
1398:
1395:
1389:
1386:
1378:
1355:
1351:
1341:
1340:
1339:
1325:
1322:
1302:
1299:
1293:
1270:
1266:
1260:
1257:
1251:
1248:
1242:
1219:
1215:
1209:
1203:
1200:
1180:
1177:
1154:
1148:
1145:
1139:
1136:
1133:
1130:
1124:
1121:
1115:
1109:
1106:
1083:
1077:
1057:
1034:
1011:
998:
996:
993:
992:
985:
983:
979:
974:
973:(free end)).
972:
968:
964:
960:
939:
928:
917:
909:
901:
898:
895:
891:
879:
868:
860:
852:
844:
841:
838:
834:
825:
817:
809:
801:
798:
795:
792:
788:
779:
771:
768:
765:
762:
759:
756:
752:
748:
745:
742:
739:
736:
733:
730:
726:
722:
719:
716:
713:
710:
707:
704:
700:
677:
674:
664:
661:
659:
640:
637:
633:
623:
621:
617:
616:exponentially
609:
605:
602:
597:
582:
574:
552:
543:
542:
541:
518:
506:
503:
500:
496:
492:
489:
483:
474:
464:
461:
456:
453:
449:
442:
433:
429:
420:
416:
413:
406:
405:
404:
384:
357:
347:
332:
329:
326:
323:
303:
295:
279:
253:
226:
202:
194:
191:
187:
174:
170:
161:
153:
152:
151:
148:
146:
145:lead climbing
142:
138:
134:
130:
129:rock climbing
125:
123:
117:
115:
111:
106:
104:
100:
95:
93:
89:
85:
81:
77:
73:
69:
65:
56:
48:
39:
32:
19:
2890:
2847:
2843:
2833:
2827:. pagea 5–9.
2815:
2784:February 14,
2782:. Retrieved
2774:
2767:
2756:. Retrieved
2752:the original
2741:
2716:
2712:
2706:
2694:. Retrieved
2688:
2642:
1763:
1755:
1744:
1695:
1604:
1504:
1002:
990:
986:
975:
956:
672:
657:
624:
619:
613:
601:Bowden cable
572:
539:
348:
217:
149:
126:
118:
113:
109:
107:
96:
71:
67:
63:
61:
2719:: 249–250.
1766:orthotropic
1749:or multi-V
978:anchor warp
963:riding turn
103:band brakes
99:rope drives
2933:Categories
2758:2013-02-23
2670:References
1898:and angle
999:Derivation
137:top-roping
2882:122410452
2874:1521-4001
2733:119597604
2608:μ
2599:α
2596:
2583:−
2580:α
2577:
2553:τ
2549:μ
2530:≤
2524:≤
2506:μ
2497:α
2494:
2481:−
2478:α
2475:
2451:τ
2447:μ
2443:−
2400:ω
2375:τ
2371:μ
2279:μ
2274:α
2271:
2255:−
2252:α
2249:
2229:τ
2225:μ
2203:μ
2182:−
2160:τ
2156:μ
2149:ω
2116:ω
2107:∫
2088:≤
2082:≤
2070:ω
2061:∫
2057:−
1960:μ
1950:α
1947:
1932:μ
1928:−
1906:α
1880:μ
1808:−
1730:π
1724:α
1704:α
1668:α
1662:
1651:φ
1648:μ
1578:φ
1575:μ
1540:φ
1426:φ
1399:μ
1390:φ
1379:φ
1326:φ
1323:δ
1303:φ
1300:δ
1294:φ
1267:≈
1261:φ
1258:δ
1252:
1243:φ
1216:≈
1210:φ
1201:δ
1181:φ
1178:δ
1155:φ
1146:−
1140:φ
1137:δ
1131:φ
1116:φ
1107:δ
1084:φ
1058:μ
1035:φ
989:USS
641:φ
638:μ
583:φ
573:effective
507:φ
504:μ
501:−
493:−
462:−
457:φ
454:μ
430:−
333:π
324:φ
304:φ
280:μ
195:φ
192:μ
141:rope drag
122:tall ship
2899:. Wiley.
2647:See also
2407:constant
1839:, where
668:of turns
2944:Winches
2852:Bibcode
608:elastic
292:is the
143:during
114:loading
110:holding
92:capstan
84:bollard
74:(after
42:design.
2880:
2872:
2731:
2696:23 Nov
2311:where
1747:V-belt
1696:where
1607:v-belt
991:Nimitz
982:tailed
666:Number
515:
471:
218:where
200:
184:
120:On a
2878:S2CID
2825:(PDF)
2807:(PDF)
2779:(PDF)
2729:S2CID
2420:then
2338:is a
2141:with
967:eased
959:sheet
133:belay
90:or a
88:winch
2870:ISSN
2786:2011
2698:2022
2005:and
1953:<
1941:<
1824:>
1638:hold
1625:load
1609:is:
971:tail
702:0.5
697:0.7
694:0.6
691:0.5
688:0.4
685:0.3
682:0.2
679:0.1
557:load
523:load
479:hold
438:hold
425:load
389:hold
376:and
362:load
258:hold
231:load
179:hold
166:load
86:, a
78:and
62:The
2860:doi
2721:doi
2587:sin
2568:cos
2485:sin
2466:cos
2392:If
2262:sin
2240:cos
1944:tan
1659:sin
1249:sin
949:281
946:321
943:553
935:935
932:552
930:153
924:624
921:635
913:751
911:286
905:392
899:535
886:631
883:702
875:026
872:540
864:751
862:286
856:228
848:881
842:152
829:503
827:537
821:612
813:392
805:881
799:286
793:6.6
783:661
775:881
769:535
766:152
757:3.5
749:81
737:6.6
734:3.5
731:1.9
720:6.6
717:4.8
714:3.5
711:2.6
708:1.9
705:1.4
346:).
127:In
101:or
94:).
66:or
2935::
2876:.
2868:.
2858:.
2848:95
2846:.
2842:.
2794:^
2727:.
2717:11
2715:.
2678:^
1742:.
903:12
896:23
893:5
881:43
854:23
839:12
836:4
819:81
811:12
796:43
790:3
763:43
760:12
754:2
746:43
743:23
740:12
728:1
723:9
660:.
622:.
147:.
105:.
2884:.
2862::
2854::
2809:.
2788:.
2761:.
2735:.
2723::
2700:.
2626:.
2617:2
2612:g
2603:/
2591:2
2572:2
2561:s
2558:k
2544:e
2538:0
2534:T
2527:T
2515:2
2510:g
2501:/
2489:2
2470:2
2459:s
2456:k
2439:e
2433:0
2429:T
2403:=
2350:k
2324:g
2320:k
2297:,
2288:2
2283:g
2266:2
2244:2
2234:k
2221:=
2212:2
2207:g
2197:2
2192:g
2188:k
2177:2
2172:n
2168:k
2152:=
2123:s
2120:d
2111:s
2102:e
2096:0
2092:T
2085:T
2077:s
2074:d
2065:s
2053:e
2047:0
2043:T
2018:0
2014:T
1993:T
1964:g
1956:+
1936:g
1884:g
1852:n
1848:k
1827:0
1821:T
1816:n
1812:k
1805:=
1802:N
1780:N
1727:=
1679:)
1676:2
1672:/
1665:(
1655:/
1644:e
1634:T
1630:=
1621:T
1571:e
1564:d
1561:l
1558:o
1555:h
1550:T
1546:=
1543:)
1537:(
1531:d
1528:a
1525:o
1522:l
1517:T
1490:,
1484:d
1481:l
1478:o
1475:h
1470:T
1466:=
1463:)
1460:0
1457:(
1451:d
1448:a
1445:o
1442:l
1437:T
1432:,
1429:)
1423:(
1417:d
1414:a
1411:o
1408:l
1403:T
1396:=
1387:d
1382:)
1376:(
1370:d
1367:a
1364:o
1361:l
1356:T
1352:d
1297:)
1291:(
1285:d
1282:a
1279:o
1276:l
1271:T
1264:)
1255:(
1246:)
1240:(
1234:d
1231:a
1228:o
1225:l
1220:T
1213:)
1207:(
1204:R
1158:)
1152:(
1149:R
1143:)
1134:+
1128:(
1125:R
1122:=
1119:)
1113:(
1110:R
1087:)
1081:(
1078:R
1038:)
1032:(
1026:d
1023:a
1020:o
1017:l
1012:T
941:3
919:6
870:3
846:1
803:1
781:6
773:1
673:μ
658:μ
634:e
610:.
595:.
553:T
519:T
512:)
497:e
490:1
487:(
484:=
475:T
468:)
465:1
450:e
446:(
443:=
434:T
421:T
417:=
414:F
385:T
358:T
330:2
327:=
254:T
227:T
203:,
188:e
175:T
171:=
162:T
20:)
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