**by V Feeders · 1979 · Cited by 38 — This paper treats a method of calculating natural frequency of vibratory feeders. In a bowl-type feeder, the deformation of the spring is complicated and **

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Study on Vibratory S.Okabe Assoc. Professor. Y. Yokoyama Feeders: Calculation of Natural Freq uency of Bowl-Type Vibratory Feeders Professor. Faculty of Engineering. Kanazawa University. Kanazawa. Japan This paper treats a method of calculating natural frequency of vibratory feeders. In a bowl-type feeder, the deformation of the spring is complicated and the exact calculation of its constant is difficult. Therefore an approximate calculation is presented under some assumptions. The relations between spring constant and spring setting condition are clarified and shown in various diagrams. The equations of natural frequency for the fixed type and the semi-floating type feeder are represented briefly. The vibration direction of bowl-type feeder is also discussed. The theoretical results are confirmed by experimental studies. 1. Introduction Vibratory feeders are a very useful method for conveying or feeding various parts and materials in automatic asssembly system [I]. Much theoretical and experimental researchers related to the conveying mechanism of feeders has been reported [2-7]. As seen in previous studies, the feeding or conveying velocity of parts is influenced by the vibration amplitude of the trough or bowl. Generally, most vibratory feeders are used at the resonant or near-resonant frequency of the mechanical system to improve feeding efficiency. Therefore, it is very desirable to predict the natural frequency of the feeder, although until now feeders have been designed perimentally. This paper treats the method of calculating natural frequency of vibratory feeders. In a linear type vibratory feeder, the spring constant and inertia term can be calculated easily. But in bowl-type feeder, the deformation of the spring is complicated and the exact calculation of spring constant is difficult. In this paper, an approximate calculation is made and some relations between natural frequency and the setting condition of the spring are shown. These results will be useful for design, development and practical use of bowl-type feeders. 2. Equivalent Model of Bowl-Type Vibratory Feeder The bowl-type vibratory feeder is made up of four main parts, that is, bowl, springs, base and exciter. The bowl is usually supported on three or four sets of incIinded leaf springs fixed to the base, and is vibrated by an tromagnetic exciter mounted on the base. Contributed by the Design Engineering Division for publication in the JOURNAL OF MECHANICAL-DESIGN. Manuscript received at ASME headquarters Nov. 1979. Journal of Mechanical Design (a) TYPE VIBRATORY FEEDER (c) EQUIVALENT MOOEL OF TYPE FEEDER (b) EQUIVALENT MODEL IF NG TYPE FEEDER (d) EQUIVALENT tv()DEL OF FIXED TYPE FEEDER Fig. 1 Equivalent model of bowl·type vibratory feeder JANUARY 1981, Vol. 1031249 Downloaded 07 Oct 2010 to 128.187.97.3. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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CENTER LINE OF LEAF SPRING r^^yjy/^//////^ B ‘B’ 5: 6r 0 AC’ B’C ^C’AB’ CENTER LINE OF LEAF SPRING (a) (b) Fig. 2 Deflection of leaf spring in bowl-type vibratory feeder The vibratory feeder is often mounted on vibration isolators, for example rubber feet, as shown schematically in Fig. 1(a), (b), to minimize the force transmission to the foundation. The relations between mounting conditions and dynamic characteristics of the feeder have already been clarified [8]. According to this previous study, if the stiffness of the vibration isolator is less than about one-fifth of that of the lead spring, the vibratory characteristics can be ap proximated by those of a floating type feeder which is sup ported at the nodal point of spring, as shown in Fig. 1(c). If the feeder is mounted on a foundation without a vibration isolator, the equivalent model of the feeder is presented in Fig. 1(d). 3. Deformation of Leaf Spring and Some Assumptions for Analyses In a bowl-type feeder, three or four sets of inclined leaf springs are arranged along a circumference. Then the movement of the bowl has an angular vibration about its vertical axis together with a vertical vibration. In this case, the deformation of each spring is very complicated. Therefore, in order to simplify the discussion, the following assumptions are presented: The deformation of the leaf spring is influenced by the deformations in (i) thickness direction, (ii) width direction and (Hi) torsion. These deformations are independent of one another without any geometrical constraint, so that the total deformation can be calculated by means of vector addition of each deformation. 4. Calculation of Spring Constant Consider a leaf spring inclinded at an angle y to the horizontal and fixed to a base at point D and to a bowl at point A, as shown in Fig. 2. Let O be the center of the circle (named base circle) which is inscribed tangent to the center lines of leaf springs, as shown in Fig. 2(a). LeU^be the radius of this base circle and

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spring is moved from A to C Denoting the displacement AC’ by 5, and the angle

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40 en o iŠ u 30 ix. z o iŠ <

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(a)VIE\4OFAPPARATUS(b) 1N PARTFig.8ExperimentalapparatusandItsmainpart10.Conclusions1.Equationsfortheequivalentspringconstantandnaturalfrequencyofbowl-typefeederscanbederivedanalytically.2.Inanordinarybowl-typefeeder,thedeformationofthespringiscomplicatedandtheequivalentspringconstantisabout2- 40timesaslargeasthatinthecasewhereparallelleafspringsareused.3.Theequivalentspringconstantisinfluencedbytheoffsetfactorwhichisrelatedtothesettingpositionoftheleafspring.WhentheoffsetfactorKisvariedfromzerotounitywhiletherestoftheparametersareheldconstant,theequivalentspringconstanttakesaminimumvalueatK=0.5.4.Theequivalentspringconstantisinfluencedbytheinclinationoftheleafspring.Whentheinclinationoftheleafspringisvaried,thespringconstanttakesamaximumat’Y=45°.254/Vol.103,JANUARY19815.Thevibrationdirectionangleisnotequaltotheclinationofleafspring,exceptinthecasewhenK=O.6.Thenaturalfrequencyofthebowl-typefeederisproximatelyproportionaltothesquarerootofthethicknessoftheleafspringandtothetwothirdspowerofitswidth,whileinthelinear-typefeederitisproportionaltothetwothirdspowerofthicknessandthesquarerootofwidth.7.Theinertiatermisrelatedtothemassofthebowlandtheinertiamomentaboutitsverticalaxis.8.Ifthecrampingtorqueofspringissmall,theequivalentspringcharacteristicexhibitsahysteresisloopforlargeamplitudeofvibration.9.Ithasbeenconfirmedthatthetheoreticalresultsagreewellwiththeexperimentalresultsintherangeofpracticaluse.TransactionsoftheASMEDownloaded 07 Oct 2010 to 128.187.97.3. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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Acknowledgment The authors gratefully thank Professor Y. Jimbo, University of Tokyo, for his generous suggestions throughout this work, and also wish to thank Mr. H. NOMURA and Mr. H. IWASAKI for their kind assistance in experiments. References 1 Boothroyd.G., and Redford, A.H., Mechanized Assembly, McGraw-Hill, New York, 1968. 2 Booth, J.H., and McCallion, H., “On Predicting the Mean Conveying Velocity of a Vibratory Conveyor,” Proc. Insln. Mech. Engrs. Pt. 1, Vol. 178, No. 20, 1964, pp. 521-538. 3 Redford, A.H., and Boothroyd, G., “Vibratory Feeding,” Proc. Instn. Mech, Engrs., pt. 1, Vol. 182, No. 6, pp. 135-152. 4 Sakaguchi, K., and Taniguchi, O., “Studies on Vibratory Feeders,” Trans. JSME, Vol. 35, No. 279, 1969, pp. 2183-2189. 5 Morcos, W.A., “On Design of Oscillating Conveyors,” ASME, Journal of Engineering for Industry, Vol. 92, No. 1,1970, pp. 53-61. 6 Jimbo, Y., Yokoyama, Y., and Okabe, S., “Vibratory Conveying,” Bull. Japan Soc. ofPrec. Engg., Vol. 4, No. 3,1970, pp. 59-64. 7 Mansour, W.A., “Analog and Digital Analysis and Synthesis of Oscillatory Track,” ASME, Journal of Engineering for Industry, Vol. 94, No. 2, 1972, pp. 488-494. 8 Yokoyama, Y., Okabe, S., Nomura, H., and Iwasaki, H., “Setting Method and Dynamics of Vibratory Feeders,” Memoirs of the Faculty of Technology Kanazawa Univ., Vol. 11, No. 1,1977, pp. 59-69. 256/ Vol. 103, JANUARY 1981 Transactions of the ASME Downloaded 07 Oct 2010 to 128.187.97.3. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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