An elastica-type analytical solution to the problem of large deflections of slightly curved spring-strips, fixed at one end and loaded at the other with couples and with forces of various directions, is obtained in this work. The main methods of calculation of elliptic integrals are studied, and the limits of their applicability are established as functions of the required degrees of accuracy and of the loading conditions of the spring-strips.
The results obtained with the proposed method are then compared to particular cases already developed in the literature with different approaches. It is shown that in all the cases considered the method permits accurate results to be obtained.
JonesR. V.Some uses of elasticity in instrument design. J. Sci. Instrum., 1962, 39, 193–203.
2.
WeinsteinW. D.Flexure-pivot bearings—part 1. Mack Des., 10 June 1965, 150–157.
3.
WeinsteinW. D.Flexure-pivot bearings—part 2. Mack Des., 8 July 1965, 136–145.
4.
TrilynskiW.Fine Mechanisms and Precision Instruments—Principles of Design, 1971 (Pergamon Press, Oxford).
5.
AndersonE. H.MooreD. M.FansonJ. L.EaleyM. A.Development of an active truss element for control of precision structures. Opt. Engng, 1990, 29, 1333–1341.
6.
ReissR. S.Opto-mechanical instrument design. Proc. SPIE, 1987, 817, 154–170.
7.
SlocumA.Precision Machine Design, 1992 (Prentice-Hall, Englewood Cliffs, New Jersey).
8.
FangW.WickertJ. A.Post buckling of micro-machined beams. J. Micromeck, 1994, 4, 116–122.
9.
LiuCTsaoT.TaiY. CLeuT. S.HoC. M.TangW. L.MiuD.Out of plane permalloy actuators for delta-wing control. Proceedings of IEEE Conference on Micro Electro Mechanical Systems, Amsterdam, 1995, pp. 7–12.
10.
HaringxJ. A.The cross-spring pivot as a constructional element. Appl. Sci. Res., Part A, 1949, 1, 313–332.
11.
PlainevauxJ. E.Guidage par suspension sur lames elastiques. Nuovo Cimento, 1953, X, 1451–1460.
12.
TheocharisP. S.PanayotounakosD. E.Exact solution of the non-linear differential equation concerning the elastic line of straight rod due to terminal loading. Int. J. Nonlinear Mechanics, 1982, 17(5/6), 395–402.
13.
TimoshcnkoS. T.History of Strength of Materials, 1953, (McGraw-Hill, New York).
14.
TimoshenkoS. T.GereJ. M.Theory of Elastic Stability, 2nd edition, 1961 (McGraw-Hill, New York).
15.
BartenH. J.On the deflection of a cantilever beam. Q. Appl. Math., 1945, II, 168–171; III, 275–276.
16.
BisshoppK. E.DruckerD. C.Large deflection of cantilever beams. Q. Appl. Math., 1945, III, 272–275.
17.
ConwayH. D.The large deflection of simply supported beams. Phil. Mag., 1947, 38, 905–911.
18.
ScottE. J.CarverD. R.On the nonlinear differential equation for beam deflection. Trans. ASME. J. Appl. Mech., 1955, 22(77), 245–248.
19.
GospodneticD.Deflection curve of a simply supported beam. Trans. ASME, J. Appl. Mech, 1956, 23(78), 675–676.
20.
Frisch-FayR.A new approach to the analysis of the deflection of thin cantilevers. Trans. ASME, J. Appl. Mech., 1961, 28(83), 87–90.
21.
ManuelF. S.LeeS. L.Flexible bars subjected to arbitrary discrete loads and boundary conditions. J. Franklin Inst., 1968, 285, 452–474.
22.
MattiassonK.Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals. Int. J. Numer. Meth. Engng, 1981, 16, 145–153.
23.
NaveeS.EllingR. E.Equilibrium configurations of cantilever beams subjected to inclined end loads. Trans. ASME, J. Appl. Mech, 1992, 59, 572–579.
24.
HymansF.Flat springs with large deflections. Trans. ASME, J. Appl. Mech., 1946, 13(68), A-223–A-230.
25.
SeamesA. E.ConwayH. D.A numerical procedure for calculating the large deflections of straight and curved beams. Trans. ASME, J. Appl. Mech., 1957, 79, 289–294.
26.
van WijngaardenA.Large distortions of circular rings and straight rods-parts I and II. Proc. Kon. Ned. Akad v. Wetensch. (Proc. R. Acad.), 1949, 49, 648–664.
27.
ConwayH. D.The nonlinear bending of thin circular rods. Trans. ASME, J. Appl. Mech., 1956, 23(78), 7–10.
28.
MitchellT. P.The nonlinear bending of thin rods. Trans. ASME, J. Appl. Mech., 1959, 26(81), 40–43.
29.
DornW. S.McCrackenD. D.Numerical Methods with Fortran IV Case studies, 1972 (John Wiley, New York).
30.
AbramowitzM.StegunI. A.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1970 (Dover Publications, New York).
31.
KingL. V.On the Direct Numerical Calculation of Elliptic Functions and Integrals, 1924 (Cambridge University Press, Cambridge).
32.
GradshteynI. S.RyzhikI. M.Table of Integrals, Series and Products, 1980 (Academic Press, San Diego).
33.
PressW. H.FlanneryB. P.TenkolskyS. A.VetterlingW. T.Numerical Recipes—The Art of Scientific Computing, 1986 (Cambridge University Press, Cambridge).
34.
JenkinsJ. A.SeitzT. B.PrzemienieckiJ. S.Large deflections of diamond-shaped frames. Int. J. Solids Structs, 1966, 2, 591–603.
35.
KerrC. N.Large deflections of a square frame. Q. J. Mech. and Appl. Math., 1964, XVII(I), 23–38.
36.
DysonA.EvansH. P.SnidleR. W.A simple, accurate method for calculation of stresses and deformations in elliptical Hertzian contacts. Proc. Instn Mech. Engrs, 1992, 206, 139–141.
