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structures can be successfully carried out using the complete and partial similarities. In the design of the scaled structures under complete similarity condition, all the derived scaling laws must be satisfied between the model and prototype which yields the perfect similarity between the two scales. However, the design of a scaled-down structure which is perfectly similar to its prototype has the practical limitation, especially for laminated structures. Relaxing some of the scaling laws may eliminate the limitation of the design under complete similarity condition and yields the scaled models that are partially similar to their prototype. However, the design of the scaled structures under the partial similarity condition must follow a deliberate methodology to ensure the accuracy of the scaled structure in predicting the structural response of the prototype. Scaled models can be designed to replicate the dynamic characteristic (e.g. frequencies, mode shapes and damping ratios) of their full-scale counterparts. However, appropriate response scaling laws need to be derived to predict the dynamic response of the full-scale prototype from the experimental data of the scaled model.
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1059:{\displaystyle {\begin{aligned}&R_{e}=\left({\frac {\rho VL}{\mu }}\right)&\longrightarrow &V_{\text{model}}=V_{\text{application}}\times \left({\frac {\rho _{a}}{\rho _{m}}}\right)\times \left({\frac {L_{a}}{L_{m}}}\right)\times \left({\frac {\mu _{m}}{\mu _{a}}}\right)\\&C_{p}=\left({\frac {2\Delta p}{\rho V^{2}}}\right),F=\Delta pL^{2}&\longrightarrow &F_{\text{application}}=F_{\text{model}}\times \left({\frac {\rho _{a}}{\rho _{m}}}\right)\times \left({\frac {V_{a}}{V_{m}}}\right)^{2}\times \left({\frac {L_{a}}{L_{m}}}\right)^{2}.\end{aligned}}}
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especially by wave motions at the interface between the water and the air. The scaling requirements for each of these phenomena differ, so models cannot replicate what happens to a full sized vessel nearly so well as can be done for an aircraft or submarine—each of which operates entirely within one medium.
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Similitude is a term used widely in fracture mechanics relating to the strain life approach. Under given loading conditions the fatigue damage in an un-notched specimen is comparable to that of a notched specimen. Similitude suggests that the component fatigue life of the two objects will also be
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Similitude analysis is a powerful engineering tool to design the scaled-down structures. Although both dimensional analysis and direct use of the governing equations may be used to derive the scaling laws, the latter results in more specific scaling laws. The design of the scaled-down composite
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It is often impossible to achieve strict similitude during a model test. The greater the departure from the application's operating conditions, the more difficult achieving similitude is. In these cases some aspects of similitude may be neglected, focusing on only the most important parameters.
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The design of marine vessels remains more of an art than a science in large part because dynamic similitude is especially difficult to attain for a vessel that is partially submerged: a ship is affected by wind forces in the air above it, by hydrodynamic forces within the water under it, and
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Similitude has been well documented for a large number of engineering problems and is the basis of many textbook formulas and dimensionless quantities. These formulas and quantities are easy to use without having to repeat the laborious task of dimensional analysis and formula derivation.
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modeled at 1/40th scale. The application operates in sea water at 0.5 °C, moving at 5 m/s. The model will be tested in fresh water at 20 °C. Find the power required for the submarine to operate at the stated speed.
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Similitude can be used to predict the performance of a new design based on data from an existing, similar design. In this case, the model is the existing design. Another use of similitude and models is in validation of
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Engineering models are used to study complex fluid dynamics problems where calculations and computer simulations aren't reliable. Models are usually smaller than the final design, but not always.
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Construction of a scale model, however, must be accompanied by an analysis to determine what conditions it is tested under. While the geometry may be simply scaled, other parameters, such as
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Note that even though the model is scaled smaller, the water velocity needs to be increased for testing. This remarkable result shows how similitude in nature is often counterintuitive.
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Another application of similitude is to replace the operating fluid with a different test fluid. Wind tunnels, for example, have trouble with air liquefying in certain conditions so
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Simplification of the formulas (by neglecting some aspects of similitude) is common, and needs to be reviewed by the engineer for each application.
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is sometimes used. Other applications may operate in dangerous or expensive fluids so the testing is carried out in a more convenient substitute.
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may need to be altered. Similitude is achieved when testing conditions are created such that the test results are applicable to the real design.
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The values of the dimensionless parameters are held to be the same for both the scale model and application. This can be done because they are
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Rezaeepazhand, J.; Simitses, G.J.; Starnes, J.H. (1996). "Scale models for laminated cylindrical shells subjected to axial compression".
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Scaled composite laminated I-beams with different scales and lamination schemes designed based on structural similitude analysis.
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Schematic of scaled composite laminated I-beams: prototype (top) and models with different scales and layups (bottom)
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and will ensure dynamic similitude between the model and the application. The resulting equations are used to derive
1893:. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 6. Springer. pp. 115–123.
1823:. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 4. Springer. pp. 115–126.
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allow testing of a design prior to building, and in many cases are a critical step in the development process.
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is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.
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1817:"Similitude Analysis of Composite I-Beams with Application to Subcomponent Testing of Wind Turbine Blades"
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is constructed and the relevant relationships of force and velocity are formulated using techniques from
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shows that the system can be described with two dimensionless numbers and one independent variable.
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De Rosa, S.; Franco, F. (2015). "Analytical similitudes applied to thin cylindrical shells".
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A model test is then conducted at that velocity and the force that is measured in the model (
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with the ultimate goal of eliminating the need for physical models altogether.
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Some common applications of similitude and associated dimensionless numbers;
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MIT open courseware lecture notes on
Similitude for marine engineering
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Eydani Asl, M.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2015).
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1355:{\displaystyle F_{\text{application}}=F_{\text{model}}\times 3.44}
1206:{\displaystyle V_{\text{model}}=V_{\text{application}}\times 21.9}
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Scale models in engineering : fundamentals and applications
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Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2017).
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Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2016).
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To satisfy the above conditions the application is analyzed;
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5 variables - 3 fundamental units ⇒ 2 dimensionless numbers.
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The following criteria are required to achieve similitude;
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models. It is also the primary theory behind many textbook
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Concept applicable to the testing of engineering models
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This example has five independent variables and three
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2152:Emori, Richard I.; Schuring, Dieterich J. (2016).
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2133:Advances in Aircraft and Spacecraft Science
559:is used to rearrange the units to form the
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161:is a concept applicable to the testing of
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1704:This article includes a list of general
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1650:Solid mechanics: structural similitude
1891:Special Topics in Structural Dynamics
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1769:can be expressed in terms of kg·m/s.
201:Similitude's main application is in
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2029:Similitude and Approximation Theory
150:test. The test is designed to have
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2096:Journal of Hydraulic Research
2055:Environmental Fluid Mechanics
1951:and help improve the section.
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1504:List of dimensionless numbers
534:. The fundamental units are:
2116:10.1080/00221686.2011.578914
1899:10.1007/978-3-319-15048-2_11
1829:10.1007/978-3-319-22449-7_14
1802:10.1016/0263-8223(95)00154-9
1981:Binder, Raymond C. (1973).
32:Similitude (disambiguation)
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2156:(2nd ed.). Elsevier.
2027:Kline, Stephen J. (1986).
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168:. A model is said to have
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2145:10.12989/aas.2015.2.4.403
2075:10.1007/s10652-008-9078-3
2008:Howarth, L., ed. (1953).
1871:10.1016/j.tws.2017.01.020
1681:Similitude of ship models
1249:{\displaystyle F_{model}}
1625:Boundary layer thickness
319:dimensionless parameters
2022:– via HathiTrust.
1725:more precise citations.
1601:Centrifugal compressors
1157:{\displaystyle F/L^{2}}
1858:Thin-Walled Structures
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61:improve this article
30:For other uses, see
2108:2011JHydR..49..293H
2090:Heller, V. (2011).
2067:2009EFM.....9..125C
2014:. Clarendon Press.
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886:
883:
880:
877:
873:
865:
861:
857:
852:
849:
846:
840:
836:
831:
827:
823:
821:
817:
810:
806:
800:
796:
790:
786:
782:
775:
771:
765:
761:
755:
751:
747:
740:
736:
730:
726:
720:
716:
707:
703:
694:
690:
688:
685:
682:
677:
673:
670:
667:
661:
657:
652:
648:
644:
642:
608:
604:
577:
573:
526:
525:
519:
517:to be measured
514:
509:
501:
500:
494:
491:
488:
472:
461:
460:
457:
454:
451:
435:
424:
423:
420:
415:
412:
404:
403:
400:
397:
394:
393:of submarine)
386:
385:
382:
379:
376:
350:
347:
334:
333:
322:
312:
301:
300:
292:
284:
237:
234:
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2210:
2199:
2196:
2194:
2191:
2190:
2188:
2179:
2176:
2175:
2171:
2165:
2159:
2155:
2150:
2146:
2142:
2138:
2134:
2129:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2038:0-387-16518-5
2034:
2030:
2025:
2021:
2017:
2013:
2012:
2006:
2002:
1998:
1994:
1988:
1984:
1979:
1978:
1971:
1968:
1960:
1950:
1949:editing guide
1944:
1940:
1935:
1926:
1925:
1919:
1910:
1904:
1900:
1896:
1892:
1888:
1881:
1878:
1872:
1867:
1863:
1859:
1855:
1848:
1845:
1840:
1834:
1830:
1826:
1822:
1818:
1811:
1808:
1803:
1799:
1795:
1791:
1784:
1781:
1775:
1772:
1768:
1764:
1758:
1755:
1745:
1742:
1734:
1724:
1720:
1714:
1713:
1707:
1702:
1693:
1692:
1686:
1682:
1679:
1678:
1674:
1672:
1664:
1656:
1649:
1643:
1642:
1639:
1635:
1631:
1628:
1626:
1623:
1622:
1619:
1615:
1611:
1607:
1604:
1602:
1599:
1598:
1595:
1592:
1590:
1587:
1586:
1583:
1579:
1575:
1571:
1568:
1566:
1563:
1562:
1558:
1554:
1553:Froude number
1550:
1546:
1543:
1540:
1537:
1536:
1533:
1530:
1528:
1523:
1521:
1515:
1508:
1505:
1497:
1495:
1473:
1462:
1459:
1439:
1435:
1426:
1422:
1413:
1409:
1395:
1388:
1387:
1386:
1372:
1349:
1346:
1337:
1333:
1324:
1316:
1315:
1314:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1264:
1241:
1238:
1235:
1232:
1229:
1225:
1200:
1197:
1188:
1184:
1175:
1167:
1166:
1165:
1149:
1145:
1140:
1136:
1116:
1096:
1076:
1049:
1044:
1039:
1032:
1028:
1022:
1018:
1012:
1007:
1002:
997:
990:
986:
980:
976:
970:
965:
961:
954:
950:
944:
940:
934:
930:
921:
917:
908:
895:
891:
887:
881:
878:
875:
871:
863:
859:
855:
850:
844:
838:
834:
829:
825:
815:
808:
804:
798:
794:
788:
784:
780:
773:
769:
763:
759:
753:
749:
745:
738:
734:
728:
724:
718:
714:
705:
701:
692:
680:
675:
671:
668:
665:
659:
655:
650:
646:
633:
632:
631:
630:
629:Scaling laws:
626:
624:
606:
602:
593:
575:
571:
562:
558:
554:
552:
549:Invoking the
547:
545:
541:
537:
533:
523:
520:
518:
515:
513:
510:
507:
503:
502:
498:
495:
492:
489:
486:
470:
463:
462:
458:
455:
452:
449:
433:
426:
425:
421:
419:
416:
413:
410:
406:
405:
401:
398:
395:
392:
388:
387:
383:
381:Scaled model
380:
377:
374:
373:
370:
368:
364:
359:
356:
348:
346:
342:
338:
331:
327:
326:dimensionless
323:
320:
316:
313:
310:
306:
305:
304:
298:
297:
293:
290:
289:
285:
282:
281:
277:
276:
275:
268:
264:
262:
258:
254:
250:
245:
243:
235:
233:
231:
226:
224:
220:
216:
212:
208:
204:
199:
197:
196:
191:
187:
183:
179:
175:
171:
167:
164:
160:
153:
149:
146:
143:A full scale
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
2153:
2136:
2132:
2099:
2095:
2058:
2054:
2031:. Springer.
2028:
2010:
1982:
1963:
1954:
1942:
1890:
1880:
1861:
1857:
1847:
1820:
1810:
1796:(4): 371–9.
1793:
1789:
1783:
1774:
1757:
1737:
1728:
1709:
1669:
1624:
1600:
1588:
1564:
1557:Weber number
1538:
1531:
1524:
1516:
1512:
1493:
1364:
1216:
1068:
628:
627:
622:
555:
548:
529:
516:
511:
417:
378:Application
360:
352:
343:
339:
335:
330:scaling laws
329:
325:
321:as possible.
302:
294:
286:
278:
273:
259:and type of
246:
242:Scale models
239:
227:
200:
194:
193:
189:
185:
184:similarity.
176:similarity,
169:
158:
157:
151:
123:
114:
104:
97:
90:
83:
76:"Similitude"
71:
59:Please help
54:verification
51:
1864:: 151–161.
1723:introducing
1610:Mach number
1574:Mach number
1431:application
1418:application
1329:application
1193:application
913:application
710:application
499:·s (N s/m)
353:Consider a
253:temperature
163:engineering
148:wind tunnel
2187:Categories
1941:" section
1706:references
1687:References
1502:See also:
1365:The power
349:An example
211:fluid flow
190:similitude
186:Similarity
170:similitude
159:Similitude
87:newspapers
2124:121563448
2083:121960118
2020:572735435
1957:July 2023
1460:×
1423:×
1347:×
1198:×
1008:×
966:×
951:ρ
941:ρ
931:×
903:⟶
885:Δ
856:ρ
848:Δ
805:μ
795:μ
785:×
750:×
735:ρ
725:ρ
715:×
687:⟶
676:μ
666:ρ
512:calculate
471:μ
434:ρ
418:calculate
375:Variable
355:submarine
345:similar.
203:hydraulic
178:kinematic
174:geometric
2049:(2009).
1731:May 2009
1675:See also
540:kilogram
493:1.00x10
490:1.88x10
391:diameter
257:velocity
249:pressure
236:Overview
219:formulas
209:to test
117:May 2009
2104:Bibcode
2063:Bibcode
1767:newtons
1761:In the
1719:improve
459:(kg/m)
448:density
255:or the
182:dynamic
101:scholar
2160:
2122:
2081:
2035:
2018:
2001:393400
1999:
1989:
1937:This "
1905:
1835:
1708:, but
1527:helium
1466:
590:) and
544:second
422:(m/s)
384:Units
215:scaled
166:models
103:
96:
89:
82:
74:
2120:S2CID
2079:S2CID
1444:model
1342:model
1180:model
926:model
697:model
536:meter
506:force
453:1028
409:speed
399:1/40
261:fluid
108:JSTOR
94:books
2158:ISBN
2033:ISBN
2016:OCLC
1997:OCLC
1987:ISBN
1903:ISBN
1833:ISBN
1555:and
1463:17.2
1350:3.44
1201:21.9
456:998
402:(m)
205:and
188:and
145:X-43
80:news
2141:doi
2112:doi
2071:doi
1895:doi
1866:doi
1862:113
1825:doi
1798:doi
1551:, (
1313:):
504:F (
407:V (
389:L (
221:in
63:by
2189::
2135:.
2118:.
2110:.
2100:49
2098:.
2094:.
2077:.
2069:.
2057:.
2053:.
1995:.
1901:.
1889:.
1860:.
1856:.
1831:.
1819:.
1794:34
1792:.
1763:SI
1636:,
1632:,
1616:,
1612:,
1608:,
1580:,
1576:,
1572:,
1547:,
546:.
542:,
538:,
508:)
497:Pa
487:)
450:)
414:5
411:)
396:1
361:A
251:,
232:.
225:.
2166:.
2147:.
2143::
2137:2
2126:.
2114::
2106::
2085:.
2073::
2065::
2059:9
2041:.
2003:.
1970:)
1964:(
1959:)
1955:(
1945:.
1911:.
1897::
1874:.
1868::
1841:.
1827::
1804:.
1800::
1744:)
1738:(
1733:)
1729:(
1715:.
1478:s
1474:/
1470:m
1457:]
1453:N
1449:[
1440:F
1436:=
1427:V
1414:F
1410:=
1407:]
1403:W
1399:[
1396:P
1373:P
1338:F
1334:=
1325:F
1299:n
1296:o
1293:i
1290:t
1287:a
1284:c
1281:i
1278:l
1275:p
1272:p
1269:a
1265:F
1242:l
1239:e
1236:d
1233:o
1230:m
1226:F
1213:.
1189:V
1185:=
1176:V
1150:2
1146:L
1141:/
1137:F
1117:p
1097:F
1077:p
1050:.
1045:2
1040:)
1033:m
1029:L
1023:a
1019:L
1013:(
1003:2
998:)
991:m
987:V
981:a
977:V
971:(
962:)
955:m
945:a
935:(
922:F
918:=
909:F
896:2
892:L
888:p
882:=
879:F
876:,
872:)
864:2
860:V
851:p
845:2
839:(
835:=
830:p
826:C
816:)
809:a
799:m
789:(
781:)
774:m
770:L
764:a
760:L
754:(
746:)
739:m
729:a
719:(
706:V
702:=
693:V
681:)
672:L
669:V
660:(
656:=
651:e
647:R
623:F
607:p
603:C
594:(
576:e
572:R
563:(
522:N
483:(
446:(
311:.
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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