It is proved in this paper that the hodograph of a frequency term (for example the kth frequency term) of the shaking moment of spatial mechanisms is an ellipse. Furthermore, expressions are provided for the lengths and attitudes of the semi-axes of this ellipse in terms of Fourier coefficients of the shaking moment. Accordingly, two equivalent systems of the kth frequency term of the shaking moment can be found; consequently two types of their balancers are proposed. A 7-R linkage is used as a numerical example for demonstrating the effects of the balancers. Additionally, it is also shown that the result of a previous work by other researchers, concerning the optimum balancing of a frequency term of pitching and yawing components of the shaking moment of high-speed machinery, is a special case of this study; furthermore, additional to the balancer they proposed, another is provided.
LowenG. G.BerkofR. S.Survey of investigations into the balancing of linkage. J. Mechanisms, 1968, 3, 221–231.
2.
BerkofR. S.LowenG. G.TepperF. R.Balancing of linkages. Shock and Vibr. Dig., 1977, 9(6), 3–10.
3.
LowenG. G.TepperF. R.BerkofR. S.Balancing of linkages-an update. Mechanism and Mach. Theory, 1983, 18(3), 213–220.
4.
AdlerU. (Editor-in-Chief) Automotive Handbook, 3rd edition (GirlingP., Editor-in-Chief in English translation), 1993, pp. 384–385 (Robert Bosch, GmbH).
5.
LowenG. G.TepperF. R.BerkofR. S.The quantitative influence of complete force balancing on the forces and moments of certain families of four-bar linkages. Mechanism and Mach. Theory, 1974, 9, 299–323.
6.
MewesE.Unbalanced inertia force in slider-crank mechanisms of large eccentricity. Trans. ASME, J. Appl. Mechanics, 1958, 80, 225–232.
7.
HanC.-Y.Balancing of high speed machinery. Trans. ASME, J. Engng for Industry, 1967, 89, 111–118.
8.
GappoevT. T.Balancing of inertia forces and their moment in a slider-crank mechanism (in Russian). Izv. Vyssh. Ucheb. Zaved-Mashinostroenie, 1967, 7, 25–29.
9.
DaviesT. H.The kinematics and design of linkages, balancing mechanisms and machines. Mach. Des. Engng, 1968, 40–51.
10.
Shchepetil'nikovV. A.The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing. J. Mechanisms, 1968, 3, 367–389.
11.
BagciC.Shaking force and shaking moment balancing of the plane slider-crank mechanism. Proceedings of the Fourth Oklahoma State University Applied Mechanisms Conference, 1975, paper 25.
12.
Shchepetil'nikovV. A.Balancing of pinned linkages with links of nonsymmetrical shape (in Russian). In Balancing of Rotors and Mechanisms (Ed. Shchepetil'nikovV. A.), 1978, pp. 224–230 (Mashinostroenie, Moscow).
13.
UrbaA. L.The position of corresponding mass for the best balancing of plane mechanisms (in Russian). In Balancing of Machines and Instruments (Ed. Shchepetil'nikovV. A.), 1979, pp. 226–230 (Mashinostroenie, Moscow).
14.
DaviesT. H.A versatile computer package for mechanism analysis; part 2: Dynamics and balancing. Proceedings of 1984 Mechanisms Conference, 1984 (Cranfield Institute of Technology and IMechE).
15.
KochevI. S.Contribution to the theory of torque, shaking force and shaking moment balancing of planar linkages. Mechanism and Mach. Theory, 1991, 26(3), 275–284.
16.
ChiouS.-T.DaviesT. H.The ideal locations for the contra-rotating shafts of generalised Lanchester balancers. Proc. Instn Mech. Engrs, Part C, 1994, 208(C1), 29–37.
17.
DaviesT. H.NiuG. H.On the retrospective balancing of installed planar mechanisms. Proc. Instn Mech. Engrs, Part C, 1994, 208(C1), 39–45.
18.
GappoevT. T.Peculiarities of balancing spatial mechanisms (in Russian). In Balancing of Machines and Instruments (Ed. Shchepetil'nikovV. A.), 1979, pp. 243–250 (Mashinostroenie, Moscow).
19.
GappoevT. T.TabuevD. B.Peculiarities of the dynamic balancing of spatial four-bar mechanisms (in Russian). Proceedings of Fifth World Congress on The Theory of Machines and Mechanisms, Montreal, Canada, 1979, pp. 1420–1423 (American Society of Mechanical Engineers, New York).
20.
FreudensteinF.MaceyJ. P.MakiE. R.Optimum balancing of combined pitching and yawing moments in high-speed machinery. Trans. ASME, J. Mech. Des., 1981, 103, 571–577.
21.
SemenovM. V.Balancing of spatial mechanisms. J. Mechanisms, 1968, 3, 355–365.
22.
ChiouS.-T.TsaiR.-J.The ideal shaft locations of three-rotating-mass balancers for spatial mechanisms. Mechanism and Mach. Theory, 1995, 30(3), 405–416.
23.
StevensenE. N.Jr.Balancing of machines. Trans, ASME, J. Engng for Industry, 1973, 95, 650–656.
24.
ChiouS.-T.ShiehM.-G.TsaiR.-J.The two-rotating-mass balancers for partial balancing of spatial mechanisms. Mechanism and Mach. Theory, 1996 (accepted).
25.
TepperF. R.LowenG. G.On the distribution of the r.m.s. shaking moment of unbalanced planar mechanisms: Theory of isomomental ellipses. Trans. ASME, J. Engng for Industry, 1977, 665–671.
26.
ChenF. Y.Isomomental ellipsoids and the distribution of shaking moments in spatial mechanisms. ASME paper 74-DET-28, 1974, pp. 5–9.
27.
ChiouS.-T.TsaiR.-J.The distribution of the r.m.s. shaking moment of spatial mechanisms. J. Chin. Soc. Mech. Engrs, 1994, 15(2), 187–197.
28.
DavydovYa. S. Balancing of mechanisms (in Russian). In Rechnoi Transport, 1963, 24 pp. (Izdat, Moscow).
29.
BaranovG. G.A Course in the Theory of Mechanisms and Machines (in Russian), 4th edition, 1967 (Mashinostroenie, Moscow).
30.
TsaiL.-W.Oldham-coupling second-harmonic balancer. Trans. ASME, J. Mechanisms, Transm., and Automn in Des., 1984, 106, 285–290.
31.
TsaiL.-W.MakiE. R.Planetary-gear-type second-harmonic balancers. Trans. ASME, J. Mechanisms, Transm., and Automn in Des., 1989, 111, 530–536.
32.
ChiouS.-T.ShiehM.-G.The explicit formulae for designing the two-contra-rotating-mass balancers of spatial mechanisms. Mechanism and Mach. Theory, 1996, (accepted).
33.
ChiouS.-T.TzouJ.-C.The hodograph and the balancer of any frequency term of the shaking force of spatial mechanisms. Proc. Instn Mech. Engrs, Part C, 1996, 210(C2), 135–141.
34.
SandorG. N.ErdmanA. G.Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, 1984, pp. 366–367 (Prentice-Hall, Englewood Cliffs, New Jersey).
35.
LeeH. Y.LiangC. G.Displacement analysis of the general spatial 7-link 7R mechanism. Mechanism and Mach. Theory, 1988, 23(3), 219–226.
36.
DenavitJ.HartenbergR. S.A kinematic notation for low-pair mechanisms based on matrices. Trans. ASME, J. Appl. Mechanics, 1955, 22, 215–221.
