Dynamic modeling and experimental verification of a piezoelectric
part feeder in a structure with parallel bimorph beams
Paul C.-P. Chao
a,*, Chien-Yu Shen
baDepartment of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC bDepartment of Mechanical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan, ROC
Received 6 September 2006; received in revised form 28 January 2007; accepted 4 February 2007 Available online 12 February 2007
Abstract
The study is aimed to perform dynamic modeling of a part feeder powered by piezoelectric actuation. This part feeder consists mainly of a horizontal platform vibrated by a pair of parallel piezoelectric bimorph beams. Owing to intermittent impacts with the platform, the transported part on the platform is able to march forward from one end to another. Dynamic modeling of the feeder is accomplished by essentially using the Rayleigh–Ritz decomposition method. The process of modeling first incorporates material properties and constitu-tive equations of the piezoelectric materials, and then captures the complex dynamics of the parallel-beam piezo-feeder by three low-order assumed-modes in the transverse direction of the vibrating beams. Applying Lagrange’s equations on the kinetic and strain energies formulated in terms of generalized coordinates associated with the first three modes, the system dynamics is then represented by three coupled discrete equations of motion. Based on these equations, motions of the platform can be obtained. With platform motion in hand, the intermittent impacts between the parts and the platform are modeled, rendering the marching speed of the part. Numerical simulations are conducted along with the experiments. The closeness found between the theoretical predicted transporting speed of the part and the experimental counterparts verify the effectiveness of the models established.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Piezoelectric bimorph beam; Part feeder; Rayleigh–Ritz method
1. Introduction
Piezoelectric part feeders in varied structures are often used nowadays to transport small parts such as screws, nuts and IC components along a manufacture line in many automatic factories, especially those in semi-con-ductor industry. These piezoelectric feeders own merits of low cost, simple structure and free of electromagnetic effects. This study is dedicated to establish a dynamic model and perform dynamic analysis on the piezoelectric feeder in a simple form as shown inFig. 1—a rectangular platform supported by a pair of parallel bimorph piezo-electric bimorph beams tilted by some angle. Applying harmonic voltages on the two piezoelectric beams in a
synchronous fashion, the platform would be in pure translational motions in the direction perpendicularly to the lengths of the piezo-beams. The parts to be trans-ported on the platform are then moved forward to another end of the platform by intermittent contacts/col-lision with the platform.
The modeling of the bimorph piezoelectric beam has been intensively investigated in the recent years. Some research works focused on incorporating nonlinear factors into the lumped model of the beam dynamics[1–3], while others further developed various control schemes to achieve precision positioning or reduce structure vibrations via piezoelectric structures [4–12]. Within these works, Choi et al.[6–9]proposed control designs for piezoelectric beams to perform position tracking. The designs were based on the discrete beam model derived from assumed-modes. Gosavi and Kelkar [10] constructed the model of
0041-624X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2007.02.002
*
Corresponding author. Tel.: +886 3 5131377; fax: +886 3 5752469. E-mail address:[email protected](P.C.-P. Chao).
a piezo-actuated flexible beam by Lagrangian formulation for robust control design. Han et al.[11]developed an ana-lytical model of the laminated composite beam by classical laminated beam theory and Ritz method. Bailey et al.[12]
designed an active vibration damper for a cantilever beam by a piezoelectric polymer and distributed-parameter con-trol theory. Recently, efforts were paid to utilize the tech-nique of finite elements to model the piezo-structures. Piefort and Preumont[13]modeled coupled piezoelectrical systems by finite element formulation. Cappelleri et al.[14]
constructed finite element simulations of the PZT bimorph actuators with aim for computational inexpensiveness. Fung et al. [15] and Fung and Chao [16] modeled the dynamic behavior of the piezoelectric beams by Hamilton’s principle and the finite elements.
Besides the past studies on the dynamics of piezo-beams in varied forms, few research works devote effort on piezoelectric part feeder. Choi and Lee [17] conducted modal analysis and control of a bowl parts piezo-feeder. Jiang et al. [18]developed a simulation software for parts feeding. Doi et al.[19]achieved feedback control of a par-allel beam parts feeder; however, the main focus was on the controller development and thus the associated mod-eling derived only lumped model and the collision between parts and feeder were not explored. In this study, the method of Rayleigh–Ritz decomposition featuring assumed modes[20]is employed herein to predict dynam-ics of parallel piezoelectric beams, then using basic colli-sion theory [21] to estimate transporting speed of the part on the platform. The transporting velocity of the objective on the feeder platform depends on various designs of the feeder such as sizes/mass/material proper-ties of piezoelectric beams and platform, even that the
tilt-ing angle of the two parallel piezo-beams has profound effects on the transporting velocity. The final goal of the current study is to establish reliable theoretical models for designers to predict well about the feeder performance— the transporting velocity, prior to mass production. Experiments are conducted to verify the theoretically-pre-dicted transportation speeds of the part. General closeness between simulation data and experiment counterparts are found, which verify the effectiveness of the models developed.
2. Dynamic modeling
The physical structure of the piezo-feeder as well as deno-tations of components is shown inFig. 1. The main compo-nents include (1) a horizontal platform carrying transported parts, (2) two parallel bimorph piezoelectric beams, each of which has leaf spring bolted with a steel plate sandwiched by piezoelectric patches. With the beams tilted by an angle a from vertical, the parallel beams would be flexurally deformed reciprocally with harmonic voltages applied on piezoelectric patches. The platform is consequently in pure translational motions perpendicular to the lengths of the piezo-beams. The assumption of synchronism in planar movements of two beams regardless of manufacturing toler-ance is set to initialize the ensuing dynamic modeling. This assumption keeps the platform in constant horizontal while vibrating. The dynamics of the beams are analyzed first in the following subsection to predict platform motion, which is followed by experimental model verification in Section2.2. Subsequently, the impact dynamics of the part to the platform is analyzed to predict transporting speed in Section2.3.
2.1. Bimorph piezoelectric beams
A simplified model of a piezoelectric bimorph beam is shown in Fig. 2, one end of which is fixed to baseplate (ground) while another is in oscillatory vibrations with a constant angle a to the horizontal.Fig. 2a denotes impor-tant dimensions and coordinates, where L is the total length of the piezo-beam; L1 is the length of portion of the steel plate sandwiched by piezo-patch; x is the axis along the beam while y captures the transverse deflection of the beam at given instant; i.e., y = y(x, t).Fig. 2b shows all forces acting on the beam. In bothFig. 2a,b, the plat-form and mounting bolts are simplified as point masses
and located at appropriate positions. The two piezoelectric patches on the steel plate produce a harmonic bending moment Mv. The moment results in harmonic motion of the entire beam, rendering the horizontal platform vibrat-ing in pure harmonic translational motions with a constant angle a to the horizontal and then moving the part on the platform forward by the impacts between the part and plat-form. The objective of the analysis in this section is to fetch the harmonic motion of the platform by performing dynamic modeling of the piezoelectric bimorph beam to obtain the motion of the beam tip before analyzing the impact dynamic of the transported part to the platform in the following section.
The dynamic modeling of the piezoelectric bimorph beams is carried out via the method of the assumed-mode accompanied with the technique of Rayleigh–Ritz decom-position [20] and linear constitutive physical laws of the piezoelectric materials. It is first assumed the piezo-beams are primarily in flexural motion, with negligible axial motion. Based on the method of the assumed-mode, the flexural motions of the beam y(x, t) as defined in Fig. 2
can be approximated by
yðx; tÞ ¼X n
i¼1
/iðxÞqiðtÞ; ð1Þ
where /i(x) is the assumed-mode shape function, qi(t) is the generalized temporal modal coordinate, and n is the num-ber of modes. The associated boundary conditions of the beam are
yðx; tÞ ¼ 0; x ¼ 0 and y0ðx; tÞ ¼ 0; x ¼ 0; ð2aÞ to reflect the fixed end at the bottom, while
y0ðx; tÞ ¼ 0; x ¼ L and y000ðx; tÞ ¼ 0; x ¼ L ð2bÞ to realize the condition of the fixed tilt angle a and free shear force at the other end, respectively. This non-zero fixed angle of a in fact makes possible horizontal, pure translational motions of the platform. It should be noted at this point that since the piezoelectric beam of feeder is not only fixed to ground but also connected to the horizon-tal platform with a fixed tilt angle, the ensuing modeling process is unique and different from those for piezoelectric cantilever beams, which were well done in many past works
[6–10]. Satisfying the four boundary conditions in Eq.(2), the trail functions for applying the assumed-mode method are set as the mode shapes, which are
/iðxÞ ¼ ðsin bix sinh bixÞ þ aiðcos bix cosh bixÞ;
i¼ 1; 2; . . . ; ð3Þ
where
ai¼
ðcos biL cosh biLÞ ðsin biLþ sinh biLÞ
and the values of bi’s can be easily calculated by tan biL + tanh biL = 0. Note that one can follow a standard proce-dure of solving vibration of a beam[20]to obtain the mode
1 L 2 x 1 x 2 ( ) 2 M m + g 1 m g v M 1 cos m g 1 sin m g 2 ( ) cos 2 M m + g 2 ( ) sin 2 M m + g Vibrating Ground (Baseplate) L Ground (Baseplate) Platform mass Mounting bolt Leaf spring Steel plates Piezo-patches Fixed Fixed
Fig. 2. (a) The configuration and each component of the simplified model and (b) acting forces on the piezo-beam.
shapes in Eq.(3). With calculated bi’s the first three mode shapes of /i(x) are depicted inFig. 3, which would be uti-lized later in this study. The kinetic energy of the system can then be expressed as
TðtÞ ¼1 2 Z L1 0 mbðxÞ _y2ðx; tÞ dx þ 1 2 Z x2 x1 mpðxÞ_y2ðx; tÞ dx þ1 2m1_y 2ðL 1; tÞ þ 1 2 Z L L1 msðxÞ_y2ðx; tÞ dx þ1 2 m2þ M 2 _y2ðL; tÞ; ð4Þ
where mb(x), mp(x) and ms(x), respectively, denote the masses of steel beam, piezoceramic and leaf spring, while m1and m2are the mass of mounting bolts and M is the mass of the horizontal platform. It can be seen from the term of M/2 in Eq.(4) that the total mass of the platform is evenly distributed at two beam tips. Incorporating Eq.
(1) into Eq.(4) yields
TðtÞ ¼1 2 Xn i¼1 Xn j¼1 mij_qiðtÞ _qjðtÞ; ð5Þ where mij¼ mji¼ Z L1 0 mbðxÞ/iðxÞ/jðxÞdx þ Z x2 x1 mpðxÞ/iðxÞ/jðxÞdx þ m1/iðL1Þ/jðL1Þ þ Z L L1 msðxÞ/iðxÞ/jðxÞdx þ m2þ M 2 /iðLÞ/jðLÞ; i; j¼ 1; . . . ; n: ð6Þ
Assuming small deflections of the piezo-beams and the constant cross section along the length, the bending mo-ment induced is
Momðx; tÞ ¼EbIbðxÞo 2y
ðx; tÞ
ox2 ; ð7Þ
where Eband Ibare Young’s modulus and moment of iner-tia of the steel beam, respectively. The strain energy of steel beam is VbðtÞ ¼ 1 2 Z V rbebdV ; ð8Þ
where rband eb are stress and strain of the beam, respec-tively, and they are functions of axial position x only. The stress can be expressed in terms of the bending mo-ment as
rbðx; tÞ ¼
Mom yðx; tÞ
IðxÞ : ð9Þ
Incorporating Eqs. (7) and (9) into Eq. (8), and using Hooke’s law (rb= Ebeb) yield
VbðtÞ ¼ 1 2 Z L1 0 EbIbðxÞ o2yðx; tÞ ox2 2 dx; ð10Þ where Ib¼ bbh3b
12 with bb and hb being width and thickness the steel beam, respectively. Consider the linear constitu-tive equations for piezoelectric materials in the form of
r11¼ cDd11 h31D; ð11Þ
E3¼ h31d11þ b33D; ð12Þ
where r11 and d11 are the stress and strain in the axial direction of piezoelectric elements; E3 is the applied elec-tric field and D is the elecelec-tric displacement which is posi-tive when the electric field and the polarization are in the same direction. cD, h31and b33 are the elastic stiffness for constant electric displacement, the piezoelectric strain constant and dielectric constant, respectively. The strain energy of the two piezoelectric patches in the bimorph structure is then VpðtÞ ¼ 1 2 Z VL ðr11d11þ DE3Þ dV þ 1 2 Z VR ðr11d11þ DE3Þ dV ; ð13Þ where VLand VRare individual volumes of the two piezo-patches at right and left sides of the steel plate, respectively. Substituting Eqs.(11) and (12)into Eq.(13)and also incor-porating Eqs.(7) and (9), Vp(t) arrive at
VpðtÞ ¼ Z x2 x1 E2pIp cD o2yðx; tÞ ox2 2 dx Ap Z x2 x1 h231D2 cD dx þ Ap Z x2 x1 b33D2dx; ð14Þ where Ip¼ bph3p 12 þ Apd 2
bp with Ep, bp, hp and Ip being the Young’s modulus, width, thickness and the moment of inertia of the piezo-patch, respectively, and finally dbp is the distance between the neutral axes of the steel and the piezo-patches. On the other hand, the strain energy of the leaf spring is VsðtÞ ¼ 1 2 Z L L1 EsIsðxÞ o2yðx; tÞ ox2 2 dx: ð15Þ
The total strain energy of the entire beam system including the masses of platform and mounting bolts can be obtained as VðtÞ ¼1 2 Z L1 0 EbIbðxÞ o2yðx; tÞ ox2 2 dxþ Z x2 x1 E2pIp cD o2yðx; tÞ ox2 2 dx Ap Z x2 x1 h231D2 cD dxþ Ap Z x2 x1 b33D 2dx þ1 2 Z L L1 EsIsðxÞ o2yðx; tÞ ox2 2 dxþ m1gcos a L1 ðy0ðL 1; tÞÞ 2 2 ! þ m2þ M 2 gcos a L ðy 0ðL; tÞÞ2 2 ! : ð16Þ
Substituting Eq. (1) into Eq. (16), the total strain energy V(t) can further be expressed as
VðtÞ ¼1 2 Xn i¼1 Xn j¼1 kijqiðtÞqjðtÞ Ap Z x2 x1 h2 31D 2 cD dxþ Ap Z x2 x1 b33D 2 dx þX n i¼1 Xn j¼1 m1gcos a L1 /0iðL1Þ/0jðL1Þ 2 ! qiðtÞqjðtÞ þX n i¼1 Xn j¼1 m2þ M 2 gcos a L / 0 iðLÞ/ 0 jðLÞ 2 ! qiðtÞqjðtÞ; ð17Þ where kij¼ kji ¼ Z L1 EbIb o2/iðxÞ ox2 o2 /jðxÞ ox2 ! " # dx þ 2 Z x2x1 E2pIp cD o2/iðxÞ ox2 o2 /jðxÞ ox2 ! " # dx þ Z LL1 EsIs o2/iðxÞ ox2 o2 /jðxÞ ox2 ! " # dx; i; j¼ 1; . . . ; n: ð18Þ
With mass and stiffness matrices derived as in Eqs.(6) and (18), respectively. The external work done by the applied voltage and gravity is derived next, which is started with expressing the work as
W ¼ MvðtÞ ½y0ðx2; tÞ y0ðx1; tÞ þ m1gsin a yðL1; tÞ
þ m2þ M
2
gsin a yðL2; tÞ; ð19Þ
where Mv(t) is the bending moment generated by the piez-oceramic bimorph patches due to the application of har-monic voltage ~VðtÞ. In fact,
MvðtÞ ¼ 2epEphpbp
hpþ hb 2
¼ c ~VðtÞ; ð20Þ
where epis the induced strain in the piezoceramic patches, which is arisen from application of the input voltage. The proportionality of epto applied voltage ~VðtÞ in fact makes possible the expression of MvðtÞ ¼ c eVðtÞ in the second line of Eq. (20), where c is a constant and can be seen as
the generated bending moment per voltage. The work in Eq. (19) can further be expressed in terms of generalized coordinates of the assumed modes in Eq. (1)as
W ¼X n i¼1 MvðtÞ ½/0iðx2Þ /0iðx1ÞqiðtÞ þ X3 i¼1 m1gsin a /iðL1ÞqiðtÞ þX n i¼1 m2þ M 2 gsin a /iðLÞqiðtÞ: ð21Þ
With the kinetic/potential energies and work obtained, the Lagrangian can be formulated by summing Eqs. (4), (17) and (21), which is prepared for further derivation of system equations of motion, yielding
^ L¼ T V þ W ¼1 2 Xn i¼1 Xn j¼1 mij_qiðtÞ _qjðtÞ 1 2 Xn i¼1 Xn j¼1 kijqiðtÞqjðtÞ þ Ap Z x2 x1 h231D2 cD dx Ap Z x2 x1 b33D 2 dx X n i¼1 Xn j¼1 m1gcos a L1 /0iðL1Þ/0jðL1Þ 2 ! qiðtÞqjðtÞ X n i¼1 Xn j¼1 ðm2þ M 2Þg cos a L /0iðLÞ/0jðLÞ 2 ! qiðtÞqjðtÞ þX n i¼1 MvðtÞ ½/0iðx2Þ /0iðx1ÞqiðtÞ þ Xn i¼1 m1gsin a /iðL1ÞqiðtÞ þX n i¼1 m2þ M 2 gsin a /iðLÞqiðtÞ: ð22Þ
Using Lagrange’s equations d dt o^L o _qiðtÞ o^L oqiðtÞ¼ 0; i¼ 1; . . . ; n; ð23Þ one can derive the discrete equations of motion for the sys-tem as Xn j¼1 mij€qjðtÞ þ Xn j¼1 kijqjðtÞ þ Xn j¼1 m1gcosa L1/0iðL1Þ/0jðL1ÞqjðtÞ þX 3 j¼1 m2þ M 2 gcosa L/0iðLÞ/ 0 jðLÞqjðtÞ ¼ MvðtÞ ½/0iðx2Þ /0iðx1Þ þ m1gsina /iðL1Þ þ m2þ M 2 gsina /iðLÞ: ð24Þ
Note that an additional Lagrange’s equation governing the electric displacement D other than those in Eq. (23)is not considered herein since it leads to no mechanical dynamic equation due to the fact that the piezoelectric effects are al-ready incorporated in mechanical work in Eq.(21)through the moment Mv(t). To re-arrange Eq. (24) into standard forms of dynamic equations, define
Mij¼ mij; and ð25Þ Kij ¼ kijþ m1gcos a L1/0iðL1Þ/0jðL1Þ þ m2þ M 2 gcos a L/0iðLÞ/ 0 jðLÞ; i; j¼ 1; . . . ; n; ð26Þ
as symmetric mass coefficient and stiffness coefficient matri-ces, respectively. Assuming further the material damping of the system as the Rayleigh damping, the damping matrix can be expressed as a linear combination of the mass and stiffness matrices of the form
Cij¼ ^aMijþ ^bKij; i; j¼ 1; . . . ; n: ð27Þ where ^a and ^b are to be determined for different material and/or experiments. With the assumed MvðtÞ ¼ c eVðtÞ in Eq. (20), one can cast the external force, RHS of Eq.
(28), into the form
Fi eVðtÞ ¼ MvðtÞ ½/0iðx2Þ / 0 iðx1Þ þ m1gsin a /iðL1Þ þ m2þ M 2 gsin a /iðLÞ ð28Þ
with the coefficient Fias
Fi¼ c ½/0iðx2Þ /0iðx1Þ
þm1gsin a /iðL1Þ þ mð 2þ M=2Þg sin a /iðLÞ
VoltðtÞ : ð29Þ
Finally, the decomposed equations of motion for analysis can be expressed as
Mij€qjðtÞ þ Cij_qjðtÞ þ KijqjðtÞ ¼ Fi eVðtÞ; i; j¼ 1; . . . ; n: ð30Þ where {Mij, Cij, Kij} are given by Eqs.(6),(18),(25),(27) and
(26)and Fiis given by Eq.(29). With the discrete model in Eq.(30)via the Rayleigh–Ritz decomposition method, the solutions of generalized coordinates qj(t)’s can be easily ob-tained. One can subsequently utilize the original assumed expression of y(x, t) in Eq.(1) with the assumed-modes in Eq.(3)to derive the flexural motion of piezoelectric bimo-rph beam, further rendering the translational motions of the platform—the same as that of the beam tip.
2.2. Experimental verification of the predicted beam dynamics
Experiments are conducted to obtain practical fre-quency response of the piezoelectric feeder. The results are compared with those theoreticals predicted by the dynamic model established in the last subsection. The crit-ical dimensions and material properties of the feeder are listed in Table 1. In addition, the masses of the platform M, and top/lower mounting bolt assemblies, m1 and m2 (as shown in Fig. 1) are calibrated, resulting in M = 0.1057 kg, m1= 0.0046 kg and m2= 0.0038 kg, respec-tively. The experiment system is set up as shown in
Fig. 4a, where the inter-relation between the feeder and
equipments is illustrated. The dynamic signal analyzer pro-vides swept-sine input voltages to the piezoelectric patches, ranging from 10 Hz to 1 kHz—the upper operation limit. The input voltage is powered by an amplifier before it is sent to the piezo-patches. With the top platform of the fee-der in harmonic translational motions and the part unfee-der
transporting, a laser displacement sensor is utilized to mea-sure the horizontal displacement of the feeder. Meamea-sure- Measure-ments are taken and feedbacked to the dynamic signal analyzer. Calculated by the analyzer, the gain and phase of the frequency response from the applied voltage to the horizontal displacement of the platform can be obtained, which are shown in Fig. 4b, along with the theoretical response. As to the numerical simulated responses, they are obtained based on the derived discrete version of the mathematical model in Eq.(30)with the practical parame-ters inTable 1employed. Considering the first three modes; i.e., n = 1, . . ., 3 and assuming ^a¼ 0:05 and ^b¼ 6 105 for the Rayleigh damping Cij’s defined in Eq.(27), the gain and phase of frequency responses are calculated and also shown inFig. 4b. Note that due to the non-zero tilting angle of the piezo-beam to the vertical, the transverse flexural direc-tion of the piezo-beam is not in horizontal. Therefore, the theoretically-calculated gain shown inFig. 4b is multiplied by the cosine of the tilting angle of the piezo-beam to reflect actual beam tip flexural motion before plotted inFig. 4b.
It is seen from Fig. 4b that a resonance occurs around 330 Hz, and most importantly, the closeness between the theoretical predicted response and experimental counter-part is clearly present, verifying the effectiveness of the dynamic model established in the previous subsection. Note that the closeness validates that even though the hys-teresis effects of the piezo-actuation are not considered by the theoretical model(30), the vibration of the piezo-beam in amplitude can be predicted accurately since the hystere-sis induces primarily only the phase shift (time delay) of the piezo-beam response. Therefore, the transported part speed could consequently be accurately estimated. It is also observed fromFig. 4b that the feeder owns fairly flat gain response before the resonance of 330 Hz. This dynamic characteristic allows the piezo-feeder to be usually designed to be operated before or close to the resonance frequency to have fast part-transporting speeds. Finally,Fig. 4c pre-sents the corresponding time-domain experimental and the-oretically displacements of the platform in the horizontal direction with input voltage of 150 V and frequency of Table 1
Material properties and dimensions of the piezoelectric feeder
Piezo-patches Steel plate Leaf spring
Total length l (mm) 19 38.9 26 Width b (mm) 13 16 16 Thickness h (mm) 1 2.3 1 Young’s modulus E (1010N/m2) 6.0 20.7 9.56 Density q (kg/m3) 7500 7800 2250 Tilt Angle a (°) 15
Elastic stiffness cD(1010N/m2) 4.22 N/A N/A
Piezoelectric strain constant d31(1010m/V)
2.7 N/A N/A
Piezoelectric stress constant h31(108N/Coulomb) 4.39 N/A N/A Dielectric impermeability b33(10 6 Vm/Coulomb) 6.393 N/A N/A
250 Hz. It is seen from this figure that satisfactory correla-tion between experimentals and theoreticals is clearly pres-ent, showing the effectiveness of the established dynamic model in Eq.(30)for predicting dynamics of the platform.
2.3. The impact dynamics of the parts
The interactive dynamics between the transported part and the vibratory platform is explored in this section to find the transporting speed of the part subjected to some harmonic voltage applied. Consider first a horizontal plat-form supported by a pair of tilted parallel bimorph
piezo-electric beams as shown in Fig. 5, where the platform is excited to undergo oscillatory pure translational motions along Y-direction—the motions between solid-lined and dashed-lined positions. These oscillations of the platform in Y-direction first bring the transported part upward along the Y-direction as the platform in uprising, and then allow the part in free falling as the platform is descending. While the platform changing its oscillatory direction from descending to ascending, the transported part has intermit-tent impacts with the platform until the platform brings the part upward again. In summary, the interactive dynamics between the transported part and the platform can be Fig. 4. Experimental verification of platform dynamics; (a) the experiment system for system identification; (b) frequency response of the theoretical and realistic models in horizontal displacement of the vibrating platform; and (c) time-domain horizontal displacement of the platform by experiment and numerical simulation. The input voltage is 150 V in frequency of 250 Hz.
divided into three stages: (1) part arising with the platform; (2) part free falling; and (3) part impacting the platform. This series of interactive three-stage dynamics is repeated between platform oscillation cycles, moving the part from one end to another end of the platform. To predict the transporting speed of the part, the following dynamic anal-ysis is conducted. Assuming first the input voltage to the bimorph piezo-beam is in a harmonic form VeðtÞ ¼
Vsinð2pf tÞ, where f is the oscillation frequency in Hz and V is the magnitude of the input voltage. The motion of the beam tip in the Y-direction is then also in a harmonic form of
yðL2; tÞ ¼ A sinð2pf tÞ; ð31Þ
where A is the amplitude of the beam tip response, which can be derived by solving Eqs.(29) and (30). Note that A is also amplitude of platform oscillation. With A solved, the horizontal and vertical motions of the platform, equal to that of the beam tip, can be expressed by
dpl;hðtÞ ¼ yðL2; tÞ cos a ¼ A sinð2pftÞ cos a; ð32Þ dpl;vðtÞ ¼ yðL2; tÞ sin a ¼ A cosð2pftÞ sin a; ð33Þ
where dpl,h(t) and dpl,v(t) denote the horizontal/vertical dis-placements of the platform, respectively; and a is the tilt angle of the piezo-beams. The horizontal and vertical velocities of the platform are then
vpl;hðtÞ ¼ _yðL2; tÞ cos a ¼ 2pfA cosð2pf tÞ cos a; ð34Þ vpl;vðtÞ ¼ _yðL2; tÞ sin a ¼ 2pfA cosð2pf tÞ sin a; ð35Þ where vpl,h(t) and vpl,v(t) denote horizontal/vertical veloci-ties of the platform, respectively. With the motion of plat-form in hand, the three stages of impact dynamics for the transported part are exploited next.
2.3.1. Part arising with platform
In this stage, the transported part sticks to the platform; therefore, the displacement/velocity of the part are same as the platform. The displacement/speed of the part can be captured by motion of the platform as follows:
dpa;hðtiÞ ¼ dpl;hðtiÞ; ð36Þ
dpa;vðtiÞ ¼ dpl;vðtiÞ; ð37Þ
vpa;hðtiÞ ¼ vpl;hðtiÞ; ð38Þ
vpa;vðtiÞ ¼ vpl;vðtiÞ; ð39Þ
where ti denotes the arbitrary time instant when the part arises with the platform; dpa,h(ti), dpa,v(ti), vpa,h (ti) and vpa,v(ti) denote the horizontal/vertical displacements and speeds of the transported part, respectively.
2.3.2. Part freely falling (separated from platform) As the part is lifted by the platform toward the highest horizontal position, the vertical motion of the platform decelerates to zero. The vertical acceleration of the platform in this period can be easily be derived from Eq.(33)or(35)
as
apl;vðtÞ ¼ ð2pf Þ 2
A sinð2pf tÞ sin a: ð40Þ It stands a chance that before the platform reaches the highest horizontal position, the acceleration of the plat-form falls below the gravity; that is,
apl;vðtÞ < g: ð41Þ
The above condition gives the moment when the transported part is separated from the oscillating platform and then expe-riencing free-falling motion. If the condition are not satisfied, the part sticks to the platform all the time, rendering no part transportation. The vertical position and speed of the part after separation are, respectively, as follows:
dpa;vðtiÞ ¼ dpa;vðti1Þ þ vpa;vðti1Þ ðti ti1Þ
1
2g ðti ti1Þ 2
; ð42Þ
vpa;vðtiÞ ¼ vpa;vðti1Þ gðti ti1Þ; ð43Þ where {ti, ti1} are two arbitrary time instants during part free-falling. As to the horizontal motion of the part, it is a constant motion after the separation. In the other words, the horizontal speed of the part maintains the last speed Fig. 5. Parts in transportation on the platform.
as separated from the platform; therefore, the horizontal position and speed of the part after separation are, respec-tively, as follows:
dpa;hðtiÞ ¼ dpa;hðti1Þ þ vpa;hðti1Þ ðti ti1Þ; ð44Þ
vpa;hðtiÞ ¼ vpa;hðti1Þ: ð45Þ
2.3.3. Part impacting the platform
As the platform decelerates its vertical motion to com-plete almost an oscillation cycle, the free-falling part with a constant downward acceleration of gravity is bound to impact the platform. Assuming negligible horizontal fric-tion between the part and platform, it can be deducted that with the platform kept in horizontal, the impact only affects the vertical motions of the part before and after the impact; i.e., the horizontal part-transporting speed are the same around the impact instant. Based on the aforementioned, the basic theory of one-dimensional colli-sion[21]with coefficient of restitution e being employed to characterize the impact dynamics of the transported part, which asserts the principle
ðv0 1 v
0
2Þ ¼ e ðv1 v2Þ; ð46Þ
where v0
1and v02are the velocities of the colliding objectives after impacting, and v1and v2are the velocities of the ob-jects before impacting. Applying principle(48)and conser-vation of momentum, the after-impact vertical velocities of the part and the platform can be derived, respectively, as
v0pa;vðtiÞ ¼
ðmpa eMÞ vpa;vðtiÞ þ Mð1 þ eÞ vpl;vðtiÞ ðmpaþ MÞ ; ð47Þ v0pl;vðtiÞ ¼ mpað1 þ eÞ vpa;vðtiÞ þ ðM empaÞ vpl;vðtiÞ ðmpaþ MÞ ; ð48Þ
where mpaand M denote the masses of the parts and the platform; vpa,v(ti) and vpl,v(ti) are the velocities of the part and the platform before impact, respectively; v0
pa;vðtiÞ and v0pl;vðtiÞ represent the ones after impact. Assuming mass of the platform is much larger than that of the transported part; i.e., M mpa, the after-impact velocities of the plat-form and transported part in Eqs.(47) and (48)can be well approximated by
v0pa;vðtiÞ ¼ e vpa;vðtiÞ þ ð1 þ eÞ vpl;vðtiÞ; ð49Þ
v0pl;vðtiÞ ¼ vpl;vðtiÞ: ð50Þ
Eq.(50)indicates that the motion of the platform is unaf-fected by the part impact. This justifies the dynamic mod-eling of the piezo-beam in Section 2.1 to leave out the dynamic influence from part impacting. On the other hand, as the friction is neglected in horizontal, the horizontal velocity of the part is the same before and after the impact, that is,
vpa;hðtiÞ ¼ vpa;hðti1Þ: ð51Þ
With the dynamic modeling accomplished for the piezo-beams and transported part in Section 2, simulations are conducted in the following steps to predict the
part-trans-portation speed. Suppose first the part and the platform are both initially still. In the first stage the input voltage of eV ¼ V sinð2pf tÞ is applied to the piezo-patches of the platform to raise the part up until the downward accel-eration of the platform exceeds gravity. After this happens, in the second stage, the part is separated from the platform and in free falling. As the downward speed of the platform further decelerates, the parts will impact intermittently with the platform in the third stage until the part sticks with the platform again. Then the simulation repeats itself from the first stage until the designated end of simulation period. In this way of simulation, the motions of the part and plat-form are solved, and finally the average transport velocity of the part on the piezo-feeder can be computed by
vt¼
RTþDT
T vpa;hðtÞdt
DT ; ð52Þ
where T and DT are initial time and period of simulation, respectively.
3. Numerical simulation and experiment verification
Experiments are conducted to verify the effectiveness of the previously-established mathematical models for pre-dicting the transporting velocity of the part. The practical part feeder and the experiment system for measuring part-transporting speed are shown in Fig. 6. The signal generator is used to generate sinusoidal input signals, which are amplified by a power amplifier to vibrate the piezo-feeder. A laser displacement sensor (MT250) is used to measure the horizontal/vertical motions of the vibratory platform for dynamic verification. Due to the fact that the part in transportation is in high-frequency up-and-down motions and its size is usually small, it is highly difficult to keep pointing the laser beam of the non-contact dis-placement sensor right at the transported part for measur-ing part displacement. To overcome the difficulty, a CCD camera is instead utilized to record the part-transportation motion for the purpose of estimating transporting speed. The dimensions and material properties of the piezo-feeder are shown as inTable 1. The input frequency range is usu-ally 60–300 Hz to be below and close to the resonance of 330 Hz, as found in Fig. 4b. The feeder is operated under the resonance frequency of 330 Hz to avoid structural break-down. The test part to be transported is a small screw weighted 1 g; i.e., mpa= 0.001 kg. The coefficient of restitution for collision is identified as e 0.3 based on a simple drop test.
Setting the input voltage as 230 V and 150 V in the same frequency of 250 Hz, theoretical responses of the platform and part are computed based on the discrete dynamic Eq.
(30) featuring the first three modes. Figs. 7 and 8 show the resulted time responses, respectively, for cases of 230 V and 150 V, respectively. Fig. 7a presents the simu-lated displacements of the platform and part over two oscillating cycles of the platform. In this figure, the critical
times denotes when the part is separated from the platform and when collisions occur. It is seen from this figure that the part is separated from the oscillating platform around 0.4 ms before the platform reaches its highest position. The separation timing is in fact right at the moment the gravity is equal to the vertical acceleration of the platform, as evident from Fig. 7b. It is also seen from Fig. 7a that after the part is separated from the platform, the part falls freely until it collides with the platform around 3.1 ms. With several collisions, the part sticks with the platform again after approximately 3.7 ms, then starting another round of stiction with the platform, separation and colli-sions. As to the horizontal displacement of the transported part—the transportation distance of the part, it can be deducted that the part moves horizontally with the plat-form in the period of stiction, while it undergoes horizontal motion in a contact velocity in periods of free-falling and collisions. Therefore, the horizontal displacement of the transported part can be easily obtained and depicted in
Fig. 7d, where between the separation and the last collision the part displacement conforms to a constant velocity hor-izontal motion, while between the last collision and separa-tion the horizontal displacement of the part is the same as that of the platform, as shown inFig. 7e. Finally, the trans-ported velocity of the part is obtained by differentiating the previously-computed horizontal displacement of the part in
Fig. 7d. The results are shown inFig. 7e. Making use of the results in Fig. 7e, where it is seen that it can be computed that the averaged transporting speed is 34 mm/s. For another case with less input voltage of 150 V, the previous analysis/computation are conducted again to observe the effects of level of input voltage on the transportation speed. The same computations as those for input voltage of 230 V are carried out to present the results inFig. 8a–e. A scrupu-lous comparison between Figs. 7 and 8 reveals that the
smaller input voltage of 150 V leads to a longer period of part stiction to the platform (as shown in Fig. 8a), and results in slower averaged transportation speed of 6.85 mm/s (as shown inFig. 8e). Therefore, a general rule of maximizing the part traveling speed is to increase the input voltage for a shorter period of part stiction to the platform.
In order to verify the effectiveness of the theoretical models established previously for platform vibration and part transportation, theoretical transportation motions of the part over the full length of the platform are next com-puted, along with one single run of experimental displace-ments of the transported part measured by a CCD camera. The results are shown inFig. 9a, where the solid curve rep-resents theoretical predictions, while the asterisks denote the measured traveling distances of the transported part by a CCD camera. Also shown in this figure are the traveling times of the part over the total length of the plat-form—5.01 and 4.96 s from experiments and theoretical predictions, respectively. Note that the solid curve is in fact combination of thousand of cycles of horizontal part dis-placements shown inFig. 8d that lasts 0.008 s. It can be generally seen fromFig. 9a that the closeness between the theoreticals and experimentals are clearly present, showing the effectiveness of the theoretical models established in this study.
It should be noted at this point that the measured trav-eling motion of the part in fact varies from run to run, pos-sibly due to the facts that the transported screw is not a point mass as assumed and the existence of friction in hor-izontal direction that was not considered. For further con-firmation on prediction accuracy in part-transporting speed, one hundred runs of part traveling for the total plat-form length of 34 mm are conducted with the CCD camera to record the total traveling time. Fig. 9b depicts the Fig. 6. Experiment system for measuring motions of the platform and the transported part.
Fig. 7. Simulated responses of the platform and part subjected to input voltage of 230 V in frequency of 250 Hz: (a) vertical displacement of the platform and part; (b) vertical acceleration of the platform; (c) horizontal displacement of the platform; (d) horizontal displacement of the transported part; and (e) transportation speed of the part.
Fig. 8. Simulated responses of the platform and part subjected to input voltage of 150 V in frequency of 250 Hz: (a) vertical displacement of the platform and part; (b) vertical acceleration of the platform; (c) horizontal displacement of the platform; (d) horizontal displacement of the transported part; and (e) transportation speed of the part.
distribution of the recorded traveling times for 100 runs in a bar chart. It can be seen from this chart that the distribu-tion of the measured traveling time centers around the the-oretically-predicted traveling time, 4.96 s, as shown in
Fig. 9a, within an acceptable range, demonstrating the
effectiveness of the theoretical model. Furthermore, the theoretically-predicted traveling speed of 6.85 mm/s as shown inFig. 9a is also close to 7.0 mm/s, the experimental averaged traveling speed herein inFig. 9b over 100 runs of transportation.
4. Conclusions
A thorough modeling of the piezoelectric part feeder with the aim on predicting part-transporting speed is accomplished in this study. It starts with an establishment
of the dynamic equations of motion of the piezo-beams in the piezo-feeder via the Rayleigh–Ritz method and is then followed by the modeling on the impact dynamics between the platform and transported part, which is accomplished via basic collision theory. The validity of the established models is further ensured by experiments. The closeness shown between the theoretically-predicted dynamics of the feeder/part and those experimental counterparts finally confirm the success of the modelings. Based on the analyt-ical and experimental results obtained, the following con-clusions can be drawn.
(1) The first three assumed-modes proposed in Eq. (3)
are capable of predicting the realistic dynamics of a flexurally-vibrated piezoelectric beam, as evident from the experimental results inFig. 4b and c. Fig. 9. Traveling history of the transported part subjected to input voltage of 150 V in frequency of 250 Hz: (a) time history of part traveling distance for a particular run—horizontal displacement over the length of the platform, 34 mm; (b) distribution of experimental traveling times over the total length of the platform, 34 mm.
(2) The basic collision theory is pertinent to be employed to predict the impact dynamics between the platform and transported part, leading to well-predicted part-transporting speeds as evident fromFig. 9a and b. (3) It is theoretically predicted and experimentally
con-firmed that the transported part experiences three stages of different types of motions: (1) part arising with the platform; (2) part free falling; and (3) part impacting the platform. This series of interactive three-stage dynamics is repeated between platform oscillation cycles, moving the part from one end to another of the platform. The transported part moves horizontally with the platform in the period of arising with the platform, while it undergoes horizontal motion in a contact velocity in periods of free-falling and collisions.
(4) A few general rules for control strategy have been distilled:
i. Based on experimental operation, the feeder should be operated under the resonance frequency of 330 Hz to avoid structural break-down.
ii. A basic rule of maximizing the part traveling speed is to increase the input voltage for a shorter period of part stiction to the platform.
Although the part-traveling speeds theoretically pre-dicted in this study anticipate well about the realistics, a broader distribution range of experimental part-traveling speeds as compared to the theoreticals is present, as shown inFig. 9b. This might be due to the facts that (1) the trans-ported part is not a point mass and (2) the ignorance on the impact friction between the part and platform. To further improve modeling, the part needs to be assumed as a rigid body and one needs to investigate the dynamic effects of the impact friction on the part-transporting speed in the future.
Acknowledgements
The authors are greatly indebted to the National Science Council of ROC for the support of the research through Contract Nos. 95-2221-E-009-367 and 95-2745-E-033-004-URD.
References
[1] P. Ge, M. Jouaneh, Modeling hysteresis in piezoceramic actuators, Precision Engineering 17 (3) (1995) 211–221.
[2] M. Goldfarb, N. Celanovic, Modeling piezoelectric stack actuators for control of micromanipulation, IEEE Transactions on Control Systems (1997) 69–79.
[3] D. Croft, G. Shed, S. Devasia, Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy applica-tion, ASME Journal of Dynamic Systems, Measurement 123 (2001) 35–43.
[4] P. Ge, M. Jouaneh, Tracking control of piezoceramic actuator, IEEE Transactions on Control Systems Technology 4 (3) (1996) 209–216. [5] S.B. Jung, S.W. Kim, Improvement of scanning accuracy of PZT
piezoelectric actuators by feed-forward model-reference control, Precision Engineering 16 (1) (1994) 49–55.
[6] S.B. Choi, S.S. Cho, Y.P. Park, Vibration and position tracking control of piezoceramic-based smart structures via QFT, ASME Journal of Dynamic Systems, Measurement, and Control 121 (1999) 27–32.
[7] S.B. Choi, H.K. Kim, S.C. Lim, Y.P. Park, Position tracking control of an optical pick-up device using piezoceramic actuator, Mecha-tronics 11 (2001) 691–705.
[8] S.B. Choi, H.C. Shin, A hybrid actuator scheme for robust position control of a flexible single-link manipulator, Journal of Robotic Systems 13 (6) (1996) 359–370.
[9] S.B. Choi, C.C. Cheong, B.S. Thompson, M.V. Gandhi, Vibration control of flexible linkage mechanisms using piezoelectric films, Mechanism and Machine Theory 29 (4) (1994) 535–546.
[10] S.V. Gosavi, G. Kelkar, Modeling identification, and passivity-based robust control of piezo-actuated flexible beam, ASME Journal of Vibration and Acoustics 126 (2) (2004) 260–271.
[11] J.H. Han, K.H. Rew, I. Lee, An experimental study of active vibration control of composite structures with a piezo-ceramic actuator and a piezo-film sensor, Smart Materials and Structure 6 (1997) 549–558.
[12] T. Bailey, J.E. Hubbard Jr., Distributed piezoelectric-polymer active vibration control of a Cantilever beam, AIAA Journal of Guidance and Control 8 (5) (1985) 605–611.
[13] V. Piefort, A. Preumont, Finite element modeling of piezoelectric structures, Samtech User’s Conference, January 2001.
[14] D.J. Cappelleri, M.I. Frecker, T.W. Simpson, Optimal design of a PZT bimorph actuator for minimally invasive surgery, in: Proceed-ings of SPIE, vol. 3984, 2000, pp. 321–335.
[15] R.F. Fung, S.C. Chao, Y.S. Kung, Piezothermoelastic analysis of an optical beam deflector, Sensors and Actuators A: Physical 87 (2001) 179–187.
[16] R.F. Fung, S.C. Chao, Dynamic analysis of an optical beam deflector, Sensors and Actuators A: Physical 84 (2000) 1–6. [17] S.B. Choi, D.H. Lee, Modal analysis and control of a bowl parts
feeder activated by piezoceramic actuators, Journal of Sound and Vibration 275 (2004) 452–458.
[18] W.H. Jiang, P.S.K. Chua, F.L. Tan, Simulation software for parts feeding in a vibratory bowl feeder, International Journal of Produc-tion Research 41 (9) (2003) 2037–2055.
[19] T. Doi, K. Yoshida, Y. Tamai, K. Kono, K. Naito, T. Ono, Feedback control for electromagnetic vibration feeder, JSME International Journal, Series C 44 (1) (2001) 44–52.
[20] L. Meirovitch, Fundamentals of Vibrations, McGraw Hill Int., New York, USA, 2001.
[21] A.P. Boresi, R.J. Schmidt, Engineering Mechanics: Dynamics, Brooks/Cole, CA, USA, 2000.
