Study on Vibratory Feeders: Calculation of Natural Freq uency

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 be the angle between OA and OH. OH is perpendicular to the center line of the spring. If the bowl is rotated by an angle 6, the upper end of the leaf Nomenclature b = width of leaf spring E = Young’s modulus /nl = natural frequency of fixed type vibratory feeder /—2 = natural frequency of floating and semi-floating type vibratory feeder G = shear modulus h = thickness of leaf spring /;. = geometrical moment of inertia J = inertia moment about vertical axis of bowl ke = equivalent spring constant ki = numerical factor K.E. = kinetic energy of bowl / = length of leaf spring M = mass of bowl Me = equivalent inertia mass Ms = bending moment at end of spring (in width direction) M, = torsional moment n = number of leaf springs Rs = shearing force at end of spring (in width direction) r r0 u2 x A» = 8 = K 4> radius of base circle radius of setting circle on bowl strain energy in width direction strain energy in thickness direction strain energy in torsion distance from cramping part on leaf spring slope of deflection at upper end of spring (in width direction) angular displacement at upper end of spring ratio of equivalent inertia mass of bowl to that of base inclination of leaf spring vibration direction angle deflection at upper end of leaf spring width direction component of deflection at upper end of leaf spring rotation of bowl torsional displacement of leaf spring offset factor offset angle 250/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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spring is moved from A to C Denoting the displacement AC’ by 5, and the angle The following cos( + 0) e2 = rŠ +rdtan a = 6 cos 7 relations also exist: PSP MSP 3EIZ psp + 2EIZ MA = «. Where 2EIZ EIZ Ps : shearing force at the end of spring, Ms : bending moment at the end of spring, E : Young’s modulus, , h2h\ Iz : geometrical moment of inertia ( = ŠŠ J, /: length of leaf spring, 12 b : width of leaf spring, h : thickness of leaf spring. Hence P.= Eb3b i) M = (f-l) EVh /cd_ P \ 3 2 (6) (7) Substituting equations (1-3) in equations (6) and (7), and neglecting small terms because of small 8, we obtain EWh ( l \ -ocos (3sin27IKŠ – I P* = -2rP Ms = ŠŠ b cos 2rl P^2y(\-±) (8) (9) (10) where K is the offset factor, defined by rtan /cos 7 Then the strain energy in the width direction £/, is represented by 2EIZ Eb2h 24lr ji (Psx+Ms)2dx b2 cos213 sin2(27) (3/c2 -3K+ 1) (ID Next, consider the strain energy for the thickness direction. The deformation in the thickness direction can be ap proximated as that of parallel leaf springs and the strain energy U2 can be represented by the following equation: Ehhi U2=1j-S2cos2p (13) The strain energy for torsion U3 is calculated by 1 £/, 2k, btf G !> dx (13) where k, is a numerical factor depending on the ratio b/h, G is the shear modulus and M, is the torsional moment of the spring. The angle of torsion at the upper end of the spring is geometrically given by 6′ = 8 sin 7, hence 6 cos /3 sin2 7 d’= (H) and the torsional moment M, is expressed as: £,M3G—, M,=-l (15) Substituting equations (14) and (15) in equation (13) gives kxbh^Gh2 cos2(3sin47 £/,=-2lr2 (16) The equivalent spring constant of bowl-type feeder ke can be calculated from y<52 =«(£/,+£/2 + £/3) where n is a number of leaf springs. Then ke is written as: nEbh3 (17) K = P t1 + A(l)2 (T)2 sin2(2^)(3K2-3K+1) + k(Š\(Š Y sin47Jcos2/3 (18) Generally, the third term is much smaller than the first and second terms and Eq. (18) becomes approximately , nEbh1 r 1 / b \2 ( I \2 . ,,— , ^=^^11 + T2(T) (T) sin(27) C(3K2-3K+1)J cos2 (3 If it is assumed that the angle 8 is small, then tan S3 =? tan sin 7 and cos2/3 in equation (19) becomes 1 cos2 (3 = 1 + [^sin(27))2 (19) (20) (21) Substituting equation (21) in equation (19), the approximate spring constant can be calculated numerically for any given condition of the leaf spring. 5. Calculation of Equivalent Inertia Term Since the bowl has an angular vibration about its vertical axis together with a vertical vibration, the inertia term is related to both the mass of bowl M and the inertia moment J. The kinematic energy of the bowl is represented by K.E. = -Mb2 cos2/3cos27+ X-J82 Łit ., J sin2 7 2V V Š Mcos27 + cos2|3’52 Hence the equivalent inertia Me is given by J sin2 7′ M— / , 7sin27\ , — IMcos27^ jŠ 1 cos2/3 (22) (23) Journal of Mechanical Design JANUARY 1981, Vol. 103 / 251 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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40 en o iŠ u 30 ix. z o iŠ < or tan 7′ =tan7. Ji + («7 COS7)2 (27) (28) Consider the case when the setting positions of the springs are varied as seen in Fig. 5(a) while the radius of the base circle r is held constant. The relation between 7′ and 7 is shown in Fig. 6 for l/r = 1. Note that the vibration direction angle 7′ coincides with 7 at K = 0. It is also seen from this diagram that 7′ is slightly different from 7 when K is nearly unity. Similarly, Fig. 7 shows the case when the setting positions of the springs are varied while the radius of the setting circle on bowl r0 (corresponds to the distance OA) is held constant, as seen in Fig. 5(b). As is shown in this diagram, 7′ is considerably different from 7 when r is small and K is nearly unity. Referring to these results, it is convenient to set the offset factor at K = 0 for selecting any vibration direction. 9. Experiment In Fig. 8, a photographic view of the experimental apparatus and its main is shown. A vibrating table (T), on which a bowl should be fixed, is supported on three sets of inclined leaf springs (2). The displacement of the table is detected through a differential transformer (3). By controlling the screw (4). a desirable static load is applied to the table. The applied load is detected through strain gauges which are mounted on the load detecter ring © Ł The displacement of the table and the applied load are recorded simultaneously with an X-Y recorder @. Fig. 9 shows an example of a load-displacement diagram for various cramping torques of leaf springs. As seen in this diagram, the equivalent spring characteristic of this system exhibits a hysteresis loop. At the same time, the softening tendency of the spring stiffness is large when the cramping torque is small. From these results, it may be concluded that the mirlo-slip occurs at the cramping parts when the displacement becomes large and, therefore, a large resultant moment is applied. If the displacement range is small and the cramping torque is large, the spring characteristic can be considered linear. The experimental spring constant in this report is obtained in this linear range. Figs. 10 and 11 show the experimental results of the spring constant compared with the theoretical values. It is seen from these results that the theoretical values are in good agreement with experimental results when the width of the spring is relatively small. If width becomes large, however, the ex perimental values are smaller than the theoretical values because of micro-slip and insufficient rigidity of the cramping parts. Journal of Mechanical Design JANUARY 1981, Vol. 103/253 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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(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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