micromachines Article
Vibration-Assisted Roll-Type Polishing System Based on Compliant Micro-Motion Stage Yan Gu 1, * , Xiuyuan Chen 1 , Jieqiong Lin 1, *, Mingming Lu 1 , Faxiang Lu 1 , Zheming Zhang 1 and Hao Yang 2 1
2
*
School of Mechatronic Engineering, Changchun University of Technology, Changchun 130012, China;
[email protected] (X.C.);
[email protected] (M.L.);
[email protected] (F.L.);
[email protected] (Z.Z.) Changchun Equipment and Technology Research Institute, Norinco Group, Changchun 130012, China;
[email protected] Correspondence:
[email protected] (Y.G.);
[email protected] (J.L.); Tel.: +86-431-8571-6288 (Y.G. & J.L.)
Received: 3 September 2018; Accepted: 27 September 2018; Published: 29 September 2018
Abstract: This paper aims to create a high-quality surface based on the linear contact material removal mechanism. For this paper, a piezo-driven, flexure-based micro-motion stage was developed for the vibration-assisted roll-type precision polishing system. Meanwhile, the compliance matrix method was employed to establish the amplification ratio and compliance model of the flexure mechanism. The dimensions of the mechanism were optimized using the grey wolves optimization (GWO) algorithm, aiming to maximize the natural frequencies. Using the optimal parameters, the established models for the mechanical performance evaluation of the flexure stage were verified with the finite-element method. Through closed-loop test, it was proven that the proposed micro-motion stage performs well in positioning micro motions. Finally, high quality surface using silicon carbide (SiC) ceramic with 36 nm Sa was generated by the independently developed vibration-assisted roll-type polishing machine to validate the performance of the established polishing system. Keywords: vibration-assisted; roll-type polishing system; micro-motion stage; grey wolves optimization (GWO) algorithm; silicon carbide (SiC) ceramic
1. Introduction Advanced ceramic materials have been found to be used in a new generation of space-to-ground optical information collection systems, so as to make the optical systems constantly have thermal stability and a high stiffness to weight ratio. The growth of optical camera key components that are made of a high-quality optical reflector is also encouraged [1]. Silicon carbide (SiC) ceramic is a competitive representative material for establishing the ideal space for reflecting mirrors by the advantages of chemical inertness and corrosion resistance. However, there also remains difficulties with SiC linked with the inherent properties of high brittleness and low fracture toughness [2,3]. Among the various machining methods for SiC ceramic materials, polishing stands out as the most suitable way to produce ultra-precision surfaces. In particular, polishing is key in aeronautics for better mechanical performance [4,5]. Traditionally, the computer numerical control (CNC) polishing method is widely adopted for processing high-cost, aspheric optics. There is a positive correlation between the accuracy and duration of polishing. To remove contour errors, Becker et al. designed a CNC polisher with a coordinate measuring machine (CMM), which feeds error data draw back to the polisher and determines a proper polishing duration [6]. Inspired by uncertainty processes, Jones et al. developed a large CNC aspheric Micromachines 2018, 9, 499; doi:10.3390/mi9100499
www.mdpi.com/journal/micromachines
Micromachines 2018, 9, 499
2 of 25
optics polishing machine that polished the target optical profile alternatively by interferometric inspection method [7]. Mori et al. extended the theory on an atomic scale fracture to CNC shaping and finishing of semiconductors, creating a polishing tool that applied a constant load from the normal direction on the substrate surface; in the meantime, loose abrasives were supplied. Unlike the conventional CNC polishing method, this tool wrapped up the rotating sphere with a relatively soft polyurethane material [8,9]. Though diamond polishing is still the most widely used finishing process, some subsurface damages and cracks may also be generated by the great force and high temperature. In particular, the rapid blunting and wear of the polishing tool will shorten the service life, suppress the productivity, and push up the cost. To solve these defects, many studies have been done to apprehend the polishing mechanism. For example, vibration assistance has been introduced to enhance the surface integrity and machining efficiency of hard and brittle materials through the alteration of process kinematics. Suzuki et al. applied ultrasonic vibration transversely to the workpiece surface under constant pressure, using loose abrasive and a micro-spherical polyurethane tool. In the subsequent study, an ultrasonic two-axis vibration-assisted polishing machine with hybrid piezo-electric actuators was developed for finishing micro-aspheric optics molds with ultra-precise high numerical aperture (NA) [10,11]. Both non-rotating tools were applicable to small aspherical targets, which were based on the measured material removal depth and the surface morphologies. Zhao et al. conducted ultrasonic vibration-assisted polishing (UVAP) of cylindrical groove arrays on silicon carbide (SiC), revealing that the two-body abrasion mechanism could be transferred to the three-body abrasion mechanism at low relatively speed and polishing force [12]. To improve the machining efficiency, elliptical ultrasonic-assisted grinding (EUAG) has been proposed by Liang et al. and successfully applied to the machining of mono-crystal sapphire, and carried out the brittle-ductile transition characteristics using single diamond abrasive grain [13]. Through 2D ultrasonic assisted polishing, Yu et al. enhanced the material removal rate (MRR) and reduced the surface roughness of Inconel 718 nickel-based alloy [14]. Despite the aforementioned efforts, most vibration-assisted polishing devices are still resonant types that convert electrical energy to mechanical energy using the piezo-electric transducers. Some of them even resort to ultrasonic frequencies (>20 kHz). The resonant types are more energy efficient, but they can only work at certain discrete frequencies. Comparatively speaking, non-resonant types typically use the piezo-driven flexure mechanism and can operate at a large range of continuous frequencies. Considering the complexity of the polishing process, it is highly necessary to design non-resonant, multi-dimensional vibration-assisted polishing device. So many researchers have reported numerous research results on non-resonant, multi-dimensional devices in recent years [15,16]. Below is a brief review of related studies. Suzuki et al. proposed a low contact force polishing system using 2D low frequency vibrations (2DLFV) based on the piezo-driven flexure mechanism, which was capable of deterministic surface error correction of the complicated shapes on micro molds, as a result of its flexibility in view of applicable polishing tools [17]. Micromilling is in direct competition with laser and other mechanical surface treatments [18]. In order to generate specific surface textures and design hydrophobicity surface property, Chen et al. developed a method of the surface texture formation using non-resonant vibration assisted micro milling [19]. Gu et al. developed a non-resonant vibration-assisted polishing device (VAPD) with parallel-driven piezo-electric (PZT) actuators aiming to solve the processing limitations of hard and brittle materials, and the maximum working bandwidth could reach up to 1879 Hz [20]. Nevertheless, the new electronics manufacturing process is suitable for continuous production lines, but not compatible with the conventional batch-type polishing process. To overcome the limitations of the traditional polishing system, the roll-type linear chemical mechanical polishing (roll-CMP) system came into being, which relied on a low-friction removal material system for linear contact to reduce the flexural deformation of ultrathin substrates or brittle fracture of large rectangular glass substrates [21]. Targeting the material removal system, Lee et al. investigated the influences of the process parameters on the mean MRR (MRRavg) and non-uniformity (NU) in
Micromachines 2018, 9, 499
3 of 25
a roll-CMP system and proposed a mathematical model for linear roll-CMP to disclose the effects of polishing pads on the MRR of Cu, which laid a solid theoretical basis for the roll-CMP process [22,23]. Micromachines 2018, 9, x 3 of 26 Motivated by the above-mentioned problems, we target the development of an elliptical and proposed a mathematical model for linear roll-CMP to disclose theflexural effects of polishing pads on vibration-assisted roll-type polishing (EARP) system considering the mechanism to extend the the MRR of Cu, which laid a solid theoretical basis for the roll-CMP processhas [22,23]. capabilities of existing mechanisms for precision polishing. This system the following advantages: Motivated by the above-mentioned problems, we target the development of an elliptical it works in the non-resonant mode and can deliver large working stroke; benefiting from the principle vibration-assisted roll-type polishing (EARP) system considering the flexural mechanism to extend of linearthecontact, ultrathin substrates without fracture could realized; thefollowing flexible system capabilities of existing mechanisms for brittle precision polishing. Thisbe system has the supports frequencyitadjustment suits the continuous production lineworking in newstroke; electronics production; advantages: works in theand non-resonant mode and can deliver large benefiting high polishing becontact, realized through the integration of feed andcould machining modules. from the accuracy principle ofcan linear ultrathin substrates without brittle fracture be realized; the flexible system supports frequency adjustment and suits the continuous production line in new electronics production; high polishing accuracy can be realized through the integration of feed and 2. Structural Design and Prototype Development machining modules.
2.1. Assembly of the EARP System 2. Structural Design and Prototype Development
The EARP system consists of a ball-screw feed drive system, a polishing roller processing system, of the EARP System and an 2.1. XYAssembly micro-motion stage with a large workspace. In particular, the ball-screw feed drive system should be noted, which includes servo motors, ball-screws, supporting bearings, The EARP system consists of a ball-screw feed drive system, acouplings, polishing roller processing system, and an XY micro-motion stage with a large workspace. In particular, the ball-screw feed linear guides, and other machine structures. Another part is the polishing roller processing system, drive system should be noted, which includes servo motors, ball-screws, supporting which involves an aluminum roller, a hard polyurethane polishing padcouplings, (to ensure the flatness of bearings, linear guides, and other machine structures. Another part is the polishing roller processing substrate surface), and a DC motor. The polishing roller processing system could achieve its own system, which involves an aluminum roller, a hard polyurethane polishing pad (to ensure the rotationflatness and translation in the XZ plane. The resolution of motion of the ball-screw feed drive system of substrate surface), and a DC motor. The polishing roller processing system could achieve is 1 µm its in own the Zrotation direction. The valueinofthe theXZ roller scanning and rotation speed calculated based on and translation plane. The resolution of motion of theisball-screw feed the requirement of the workpiece surface quality and the NC program is generated by the personal drive system is 1 μm in the Z direction. The value of the roller scanning and rotation speed is calculated on the requirement of the workpiece surface quality(PZT1 and the NCPZT2), program is to its computer. The XYbased micro-motion stage is driven by two PZT actuators and owing generated by the personal computer. The holes XY micro-motion stage is driven by twoa PZT actuators high input stiffness. Considering that fixing can be easily machined with tolerance of ±5 µm, (PZT1 and PZT2), owing to its high input stiffness. Considering that fixing holes can be easily precise assembly of the components is not impossible nowadays. machined with a tolerance of ±5 µ m, precise assembly of the components is not impossible The computer-aided design assembly processes of the EARP system are shown graphically in nowadays. Figure 1. The of design the stage can be assembled through two graphically steps (Figure Thecomponents computer-aided assembly processes of the together EARP system are shown in 1a,b). The fixing bolts in Figure 1a are the stage fixedcan holes constrained in all directions, and(Figure the support Figure 1. The components of the be assembled together through two steps 1a,b). plate The1b fixing bolts limited in Figure by 1a are fixed holes all directions, support plate in shown in Figure is only thethe contact areaconstrained in the X in direction. In and the the additional steps Figure 1bthe is only limited by the contact in the X direction. In the additional shown in roller in Figure 1c–e, micro-motion stage, thearea ball-screw feed drive system, andsteps the polishing Figure 1c–e, the micro-motion stage, the ball-screw feed drive system, and the polishing roller processing system are assembled together and fixed at the base. The front, side, and top views of the processing system are assembled together and fixed at the base. The front, side, and top views of the assembled modelmodel are shown in Figure 1f–h. assembled are shown in Figure 1f–h.
Figure 1. Elliptical vibration-assisted roll-type polishing (EARP) assembly procedure. (a–c) components assembly; (d,e) base mounting; (f–h) front, side, and top views. PZT—piezo-electric.
Micromachines 2018, 9, x
4 of 26
Micromachines 2018, 9, 499 Figure 1. Elliptical vibration-assisted roll-type polishing (EARP) assembly procedure. (a–c) components assembly; (d,e) base mounting; (f–h) front, side, and top views. PZT—piezo-electric.
4 of 25
2.2. Vibration Trajectory of the Micro-Motion 2.2. Vibration Trajectory of XY the XY Micro-MotionStage Stage As reviewed in the introduction, some parameters can becan arbitrarily alteredaltered by theby non-resonant As reviewed in the introduction, some parameters be arbitrarily the non-resonant vibration It is simple and directly generates with two orthogonal vibration device. It is simple device. and directly generates the ellipse with the twoellipse orthogonal actuators, compared actuators, compared with those devices with two parallel actuators. The micro-motion stage aims with those devices with two parallel actuators. The micro-motion stage aims to vibrate attoa certain vibrate at a certain frequency and time about the center point O in the plane XOY. According to the frequency and time about the center point O in the plane XOY. According to the basic vibration basic vibration principles, the vibration process of the 2D micro-motion stage with variable principles, the vibration process parameters can be expressed as of the 2D micro-motion stage with variable parameters can be expressed as x Ax sin 2 f1t x = A x sin (2π f 1 t) (1) (1) y (1 ) A sin 2 f 2t y = (1 − η 0 ) · Ay y sin (2π f 2 t + ϕ) where Ax and Ay are the signal amplitudes in the x and y directions, respectively; f1 and f2 are the
where Ax and Ay are the signal amplitudes in the x and y directions, respectively; f 1 and f 2 are the signals frequencies in the directions of x and y, respectively; is the phase difference between the signals frequencies in the directions of x and y, respectively; ϕ is the phase difference between the initial phases in x and y directions; and is the displacement loss rate induced by hinges 7, 8, 9, initial phases in x and y directions; and η 0 is the displacement loss rate induced by hinges 7, 8, 9, and 10 and 10 in the y direction. Then, the stage trajectories were analyzed with different vibration in the yparameters. direction. Then, the stage trajectories were analyzed with different vibration parameters. The trajectory of the stage in in the exhibited a Lissajous with The trajectory of the stage theXOY XOY plane plane exhibited as as a Lissajous curvecurve when when signals signals with different frequencies applied directions of xx and parameters of f1 of = ff2,1 = f are 2 different frequencies werewere applied totodirections andy.y.The The parameters , ϕ = π/2 are 2 provided in Equation (1). Figure 2 shows the effects 2D vibration amplitudes closed provided in Equation (1). Figure 2 shows the effects of 2Dofvibration amplitudes on on thethe closed trajectory motion of the stage. motiontrajectory of the stage.
The composition trajectoryfor fordifferent different amplitudes. (a) (a) same amplitudes: Ax = AyA=x 5=μm; Figure Figure 2. The2.composition trajectory amplitudes. same amplitudes: Ay = 5 µm; (b) different amplitudes: Ax = 6 μm, Ay = 5 μm; (c) different amplitudes: Ax = 5 μm, Ay = 6 μm; (d) (b) different amplitudes: A x = 6 µm, Ay = 5 µm; (c) different amplitudes: A x = 5 µm, Ay = 6 µm; same amplitudes: Ax = Ay = 6 μm. (d) same amplitudes: A x = Ay = 6 µm.
As shown in Figure 2, the vibration amplitudes in the x and y directions formed a square
Astrajectory shown in Figure thesame vibration amplitudes in the of x and y directions square trajectory area under2,the conditions. The magnitude amplitudes affectsformed only the asize for the area under the same conditions. The magnitude of amplitudes affects only the size for the area, without area, without affecting the uniformity of the Lissajous figures. affecting the uniformity of the Lissajous figures. 2.3. Design of the EARP System
2.3. Design The of theEARP EARP System system is designed with a XY micro-motion stage, a roller, and a pair of PZT actuators, as shown in Figure 3. The substrate should be placed on the stage and attached to the The EARP system is designed with a XY micro-motion stage, a roller, and a pair of PZT actuators, rubber backing layer. The polishing pad is tightly coiled around the high-stiffness roller, which can as shown in Figure 3. The substrate should be placed on the stage and attached to the rubber backing dampen the vibrations at a high rolling speed. layer. The As polishing pad is tightly around high-stiffness which can dampen the the conventional roll-typecoiled polishing systemthe performs polishing roller, with line-contact material vibrations at a mechanism, high rollingthespeed. removal polishing contact area is much smaller than that of the rotary CMP system. As the conventional roll-type polishing system performs polishing with line-contact material removal mechanism, the polishing contact area is much smaller than that of the rotary CMP system. Accordingly, the abrasives directly involved in the polishing are insufficient, which impacts the surface roughness. In order to raise the amount of abrasives involved in polishing, an elliptical vibration is introduced in the processing, as shown in Figure 4. L is the roller length, D is the roller diameter, and A is the apparent area of contact between the pad and the substrate.
Micromachines Micromachines 2018, 2018, 99,, xx
55 of of 26 26
Micromachines 2018, 9, xabrasives Accordingly, the Accordingly, the abrasives
5 ofthe 26 directly directly involved involved in in the the polishing polishing are are insufficient, insufficient, which which impacts impacts the surface surface roughness. roughness. In In order order to to raise raise the the amount amount of of abrasives abrasives involved involved in in polishing, polishing, an an elliptical elliptical Accordingly, the abrasivesthe directly involved in the polishing are insufficient, which impacts the Micromachines 2018, 9,introduced 499 vibration vibration is is introduced in in the processing, processing, as as shown shown in in Figure Figure 4. 4. LL is is the the roller roller length, length, D D is is the the roller roller surface roughness. In order to raise the amount of abrasives involved in polishing, an elliptical diameter, diameter, and and A A is is the the apparent apparent area area of of contact contact between between the the pad pad and and the the substrate. substrate. vibration is introduced in the processing, as shown in Figure 4. L is the roller length, D is the roller diameter, and A is the apparent area of contact between the pad and the substrate.
5 of 25
The diagram elliptical vibration-assisted polishing (EARP) system. Figure The schematic schematic diagram of ellipticalvibration-assisted vibration-assisted roll-type roll-type polishing (EARP) system.system. Figure Figure 3. The3.3.schematic diagram ofof elliptical roll-type polishing (EARP) Figure 3. The schematic diagram of elliptical vibration-assisted roll-type polishing (EARP) system.
Figure 4. of contact the EARP system. Figure 4. Schematic Schematicof ofcontact contact area area inin the EARP system. Figure 4. Schematic areain the EARP system. Figure 4. Schematic of contact area in the EARP system.
3. of Ratio and Compliance Modeling 3. Determination Determination of Amplification Amplification Ratio andCompliance Compliance Matrix Matrix Modeling 3. Determination of Amplification Ratio and Matrix Modeling 3.3.1. Determination Ratio Compliance Matrix Modeling Determinationof of Amplification Amplification Ratio andand Compliance Matrix Modelling
3.1. Determination of Amplification Ratio and Compliance Matrix Modelling 3.1. Determination of Amplification Ratio and Compliance Matrix Modelling
Modelling be way improve use the Modelling may may be the the suitable suitable way toCompliance improve the the use of of the EARP EARP on on several several applications applications [24]. [24]. 3.1. Determination of Amplification Ratio andto Matrix Modelling
Modelling be theanalysis, suitableititway to improve the usethe ofbridge-type the EARP mechanism on severalas applications [24]. Before kinematics is to an Before the themay kinematics analysis, is necessary necessary to simplify simplify the bridge-type mechanism as an ideal ideal Modelling mayanalysis, be the suitable to improve the usein of the on several applications [24]. Before multi-rigid the kinematics it is way necessary simplify theEARP bridge-type as body ideal pivots, as 5. length multi-rigid body mechanism mechanism with with ideal pivots, to as shown shown in Figure Figure 5. Let Let llaa be be the themechanism length of of the the arm arman ideal Before thethe kinematics analysis, it is necessary to simplify theand bridge-type mechanism an ideal and be angle the line pivots the multi-rigid with ideal pivots, asarm shown inand Figure 5. Let la line. be theas length of the arm and ααbody be themechanism angle between between the connecting connecting line of of arm pivots the horizontal horizontal line. multi-rigid body mechanism with ideal pivots, as shown in Figure 5. Let la be the length of the arm and α be the angle between the connecting line of arm pivots and the horizontal line. and α be the angle between the connecting line of arm pivots and the horizontal line.
Figure Figure 5. 5. Ideal Ideal model model of of bridge-type bridge-type mechanism. mechanism. Figure Ideal model model ofofbridge-type mechanism. Figure 5.5.Ideal bridge-type mechanism.
Considering the huge topological differences, each arm in the bridge-type mechanism is simulated as a link between two flexure hinges, and each flexure hinge is modelled as an elastic beam. As the structure is symmetrical, only one bridge arm of the mechanical model needs to be analyzed. Through kinematic analysis, the static mechanics model of the flexure hinge AB can be simplified as a quarter mechanic model, as shown in Figure 6.
Micromachines 2018, 9, x
6 of 26
Micromachines 2018, 9, x
6 of 26
Considering the huge topological differences, each arm in the bridge-type mechanism is simulated as a link between two flexure hinges, and each flexure hinge is modelled as an elastic Considering the huge topological differences, each arm in the bridge-type mechanism is beam. As the structure is symmetrical, only one bridge arm of the mechanical model needs to be simulated as a link between two flexure hinges, and each flexure hinge is modelled as an elastic Micromachines 2018, 9, 499 analyzed. Through kinematic analysis, the static mechanics model of the flexure hinge AB can be beam. As the structure is symmetrical, only one bridge arm of the mechanical model needs to be simplified as a quarter mechanic model, as shown in Figure 6. analyzed. Through kinematic analysis, the static mechanics model of the flexure hinge AB can be simplified as a quarter mechanic model, as shown in Figure 6.
6 of 25
6. Quarter mechanicmodel model of flexure hinge.hinge. FigureFigure 6. Quarter mechanic ofbridge-type bridge-type flexure Figure 6. Quarter mechanic model of bridge-type flexure hinge.
As shown in Figure 7, there are only two symmetric, horizontally balanced forces at point B. As shown in Figure 7, there are only two symmetric, horizontally balanced forces at point B. Meanwhile, the arm AB has two moments, and the force at point A is provided by point B. From the As shown in has Figure 7, there are only twothe symmetric,at horizontally forces atpoint point B. Meanwhile, the arm AB two moments, isbalanced provided B. From the force equilibrium theory, the equationsand showing force that A point can be FB F Aand 2M A 2M Bby Meanwhile, the arm AB has two moments, and the force atFpoint A is provided by pointB.MFrom the force equilibrium theory,theory, the equations showing that F F = FB = F and 2M A = 2MB = M can be easily equilibrium derived. force the equations showing that A A FB F and 2M A 2M B M can be easily derived. easily derived.
Figure 7. The mechanics of bridge-type flexure hinge. Figure 7. The mechanics of bridge-type flexure hinge.
Figure 7. The mechanics bridge-type flexure hinge.7) [25]. The equation would appear when the forceof F is applied at point A (Figure 2 K Fh The equation would appear when theMforce F isapplied at point A (Figure 7) [25]. (2) The equation would appear when the force F is applied at point A (Figure 7) [25].
Fl F Fh cos 2 KKl l M Fl FFh cos K2K l lα ∆α
M = = Kl ∆ll Fh Fll l F Mcos α F cos At point A on arm AB, F x F At point A on arm AB,
(2)
(2) (3)
F x often Fl has lM F cos l deformation, Fh (3) The an extremely small that is, the is very At point A onbridge-type arm AB,mechanism
small. Therefore, the chord lengthoften produced byextremely the rigid body is approximately the istovery The bridge-type mechanism has an smallrotation deformation, that is, the equal arc length corresponding to [26]. Then, y can be expressed as follows: F∆x length = Fl ·produced ∆l + M by · ∆α F cos α ·rotation ∆l + Fh · ∆α small. Therefore, the chord the= rigid body is approximately equal to the arc length corresponding to [26]. Then, as follows: y yl can cos be expressed , (4)
(3)
∆y = la cos α · ∆α,
(4)
a
The bridge-type mechanism often has an small deformation, that is, the ∆α is very y extremely la matrix cos According to beam theory, the compliance of, the right-angle hinge can be obtained(4) by small. Therefore, the chord length produced by the rigid body rotation is approximately equal to the the following [27]: According to beam theory, the compliance matrix of the right-angle hinge can be obtained by arc lengththe corresponding to ∆α [26]. Then, ∆y can be expressed as follows: following [27]:
According to beam theory, the compliance matrix of the right-angle hinge can be obtained by the following [27]: l 0 0 Ebw 4l 3 l 6l 2 + Gbw Ch = 0 (5) 3Ebw3 Ebw3 0
6l 2 Ebw3
12l Ebw3
where E is the elastic modulus and G is the shear modulus of the hinge material. The coordinate system of the right-angle hinge is shown in Figure 8.
Ebw Ch 0 0
3
4l l 3Ebw3 Gbw 6l 2
6l Ebw3 12l Ebw3 2
(5)
Ebw3 Micromachines 2018, 9, 499 where E is the elastic modulus and G is the shear modulus of the hinge material. The coordinate system of the right-angle hinge is shown in Figure 8.
7 of 25
8. Coordinate system of right-angle flexure hinge. FigureFigure 8. Coordinate system of right-angle flexure hinge.
According to Equation (5), K l and K can be derived as follows:
According to Equation (5), Kl and Kα can be derived as follows: Ebw3 Ebw , K 12 l 3 l Ebw Ebw
Kl
Kl =
(6)
, Kα =
12l hinge, the theoretic displacement Considering the elastic stiffness of lthe right-angle amplification ratio can be calculated as
(6)
Considering the elastic stiffness of the right-angle hinge, the theoretic displacement amplification l cos y Ramp1 = a h ratio can be calculated as x cos2 w2 (7) ∆y la cos α 6h Ramp1 = = cos + h (7) 2 α · w2 ∆x 6h model is The ideal displacement amplification ratio of the lever
The ideal displacement amplification ratio of the Llever model is Ramp2
m
(8)
L
where L is the total length of the lever and Ramp m is input 2 =length. m Thus, the actual amplification ratio of the displacement amplifier can be obtained as
(8)
where L is the total length of the lever and m islainput cos Llength. x direction 2 cosdisplacement 2 Thus, the actual amplification ratio of the amplifier can be obtained as ( h) m Ramp
6h Ramp Ramp1 Ramp2 l cos α· L L 1— cos2a lα2a·ωcos y direction 2 ( cos6h +2h)m = Ramp1 · Ramp2 = ( h)lma cos α· L 0 (1 − 6ηh ) · cos2 α·ω2 ( 6h +h)m
x direction
(9)
y direction
(9)
3.2. Compliance Matrix Modelling of XY Micro-Motion Stage
3.2. Compliance Matrix Modelling of XY Micro-Motion Stage 3.2.1. Compliance Matrix Method
3.2.1. Compliance Matrix Method
According to the simplified model of Wu & Zhou [28], the compliance matrix of the
right-circular in its local coordinate system can be defined follows: According to thehinge simplified model of Wu & Zhou [28], the as compliance matrix of the right-circular hinge in its local coordinate system can be defined as follows:
c11 f C = 0 0
0 c22 c32
0 c23 c33
(10)
As the links have much higher stiffness than flexure hinges, the motions of micro-motion stage can be viewed as the elastic deformation of the flexure hinges [29]. According to Figures 8 and 9, h iT when a load vector F = f x f y mz is applied on point Oi of the flexure in/around certain axes, h iT the displacement D = d x dy θz of that point in its local coordinate system Oi − xy can be described as D = COi F (11) where COi is the compliance matrix of the flexure hinge in its local coordinate system Oi − xy [30]. Note that the compliance factors in the matrices of the right-angle hinge and the right-circular hinge can be derived from Equations (5) and (10), respectively.
Micromachines 2018, 9, x
8 of 26
0 c11 0 C 0 c22 c23 0 c 32 c33
Micromachines 2018, 9, 499
8 of 25
f
(10)
The coordinate system of the right-circular hinge is shown in Figure 9. The coordinate system of the right-circular hinge is shown in Figure 9.
Figure Coordinate system system ofofright-circular hinge. Figure 9. 9. Coordinate right-circular hinge.
As the links have much higher stiffness than flexure hinges, the motions of micro-motion stage
For a flexure hinge, the compliance matrix can be transferred from the local coordinate system can be viewed as the elastic deformation of the flexure hinges [29]. According to Figures 8 and 9, Oi − xy to another coordinate system OT j − xy by the following equation when a load vector F = f x f y mz is applied on point Oi of the flexure in/around certain axes, T T j j Ti COiin Titsi local coordinate system Oi xy can be (12) the displacement D d x d y z ofCOj that=point described as j
where the transformation matrix Ti can be described, D COi F
(11)
where COi is the compliance matrix of the flexure hinge j j in its local coordinate system Oi xy [30]. j
T = R ·P
i right-angle i i of the Note that the compliance factors in the matrices hinge and the right-circular hinge can be derived from Equations (5) and (10), respectively. j The matrix Pi can hinge, be obtained by For a flexure the compliance matrix can be transferred from the local coordinate system to another coordinate system O j xy by the following Oi xy equation
1 0 ry j T j C T j T C Pi =Oj 0i Oi 1 i−r x
0
0
1
where the transformation matrix Ti j can be described, j
Ti i Pi it can be written as If the matrix Ri rotates around the x, y and zR axes, j
j
j
(12)
(13)
(14)
(13)
j
by The matrix Pi can be obtained 1 0 0 cos β 0 − sin β cos ε j j j 1 0 ry R x = 0 cos α , R = sin α , Ry = j0 1 0 − sin ε z Pi 0 1 rx 0 − sin α cos α sin β 0 cos β 0 0 0
1
sin ε 0 cos ε 0 (14) 0 1
(15)
where α, β, and ε are the jrotation angles of x, y, and z axes, respectively. If the matrix R i rotates around the x, y and z axes, it can be written as 3.2.2. Output Compliance 0 0 1 Modelling cos 0 sin cos
sin 0 0 cos sin 0 R j sin cos 0 j j R R 1 0 , , (15) y z property, a quarter of the bridge-type Because of thex double symmetric mechanism is modelled 0 sin cos sin 0 cos 0 0 1
(Figure 10). In the coordinate system E-xy, the compliance of flexure hinge d can be expressed as follows: where α, β, and ε are the rotation angles of x, y, andz axes, Trespectively. T
CdE = C1E + C2E = T1E C1 T1E
+ T2E C2 T2E
(16)
where CiE is defined as the compliance of flexure hinge i with respect to the point E in the coordinate system E-xy, Ci is the compliance of flexure hinge i in the local coordinate system, and TiE is the transformation matrix from the coordinate system E-xy to the local coordinate system.
Because of the double symmetric property, a quarter of the bridge-type mechanism is modelled Thanks to the symmetric property, half of the lever mechanism is simulated (Figure 11). Then, (Figure 10). In the coordinate system E-xy, the compliance of flexure hinge d can be expressed as the compliance of the left limb M in the coordinate system C-xy can be described as follows: 1
1 T T 1 T T E C E C E C T C CLeft TEC CLeft TEC C E T3CCCE3 T3C EC T4C CT4E T4C T 2 TT2E5 C5 T5 T C d 1 2 1 1 1 2 Micromachines 2018,E9, 499
T
T
T6C C6 T6C
(16) (18)
where Ci is defined as the compliance of flexure hinge i with respect to the point E in the coordinate system E-xy, Ci is the compliance of flexure hinge i in the local coordinate system, and
9 of 25
Ti E is the transformation matrix from the coordinate system E-xy to the local coordinate system.
The compliance of the left section can be written as
E CLeft CdE RxE CdE RxE
T
(17)
Thanks to the symmetric property, half of the lever mechanism is simulated (Figure 11). Then, the compliance of the left limb M in the coordinate system C-xy can be described as C E CLeft TEC CLeft TEC
T
T 1
T3C C3 T3C
1 T 1
T4C C4 T4C
C C T5 C5 T5
T
T6C C6 T6C
T
(18)
Figure Quartermodel model of mechanism. Figure 10.10. Quarter of bridge-type bridge-type mechanism.
The compliance of the left section can be written as T E E E E E CLe f t = Cd + R x ( π )Cd R x ( π )
(17)
Thanks to the symmetric property, half of the lever mechanism is simulated (Figure 11). Then, the compliance of the left limb M in the coordinate system C-xy can be described as C CLe ft =
h
C E TEC CLe f t TE
T
+ T3C C3 T3C
T i −1
h T i −1 −1 T T + T4C C4 T4C + T5C C5 T5C + T6C C6 T6C
(18)
Figure 10. Quarter model of bridge-type mechanism. Figure 11. Half model of lever mechanism.
Hence, the compliance of limb M in the coordinate system O-xy can be obtained as C O CM TCO CLeft
1
C RyC CLeft RyC
T
1 1
T
O T C
(19)
Similarly, the compliance matrix of flexible limb N can be expressed as
Figure 11.11.Half oflever levermechanism. mechanism. Figure Halfmodel model of
Hence,Hence, the compliance of limb MM ininthe system O-xy be obtained as the compliance of limb thecoordinate coordinate system O-xy cancan be obtained as CO M
=
( C −1 CO TCO C Left OM C TC CLef t
1 1
+ RRC (π ) CC C R RC(π )T−1T 1
C y
y
C Left
Le f t
T
C y
y
T1 O − C
)
TCO
T
(19)
(19)
Similarly, the compliance matrix of flexible limb N can be expressed as
Similarly, the compliance matrix of flexible limb N can be expressed as T N CN = TON CO M TO
(20)
Therefore, the output compliance of the micro-motion stage in the coordinate system O-xy can be derived as h i −1 h O −1 −1 −1 i −1 −1 OC O T O −1 (21) KO = (CO )−1 = TN + C7 + C8O + C9O + C10 + CO N TN M 3.2.3. Input Compliance Modelling Excluding the bridge-type amplifier in limb M, the compliance model of the micro-motion stage is simulated as the stiffness model in Figure 12. The bridge-type amplifier will tolerate the force applied
1 B 1 8
C
C78 C7B
1
(22)
The two hinges are connected to the limb N of chains 9 and 10; thus, we have
Micromachines 2018, 9, 499
C910 N =TNB CN TNB
T
1 B 1 10
C 1
C9B
(23)
10 of 25
Therefore, the compliance of point D can be calculated as 1 1compliance C D 1integral by the rest of the stage at the interface D. of chains 7 and 8(24) with respect to D D CD =Then, 78 C910 N +CC point B can be derived as Dwith respect Let CDA denote the compliance of point to−input −1 1 −1end A can be derived by C78 = C7B + CT8B (22)
CDA TDACD TDA
(25)
Figure 12. Stiffness the micro-motion stage thethe y-direction with one limb actuated. Figure 12. Stiffness model model of theofmicro-motion stageinin y-direction with one limb actuated.
Figure 13 shows the free body diagram of the quarter model of bridge-type mechanism.
The twoAssuming hinges are connected tofixed the [31], limbtheNfollowing of chains 9 andcan 10;bethus, weaccording have to the that point D remains equations obtained force-deflection relationship at the input end A T −1 −1 −1 c F B B u Ay cB22 FDy c23 M B C910N = TN CN TN + C9Az 11+in C10 Az c32 FDy c33 M Az 0
(26)
(23)
(27)
Therefore, the compliance of point D can be calculated as CD =
D C78
−1
+
D C910N
−1 −1
+ CCD
Micromachines 2018, 9, x
(24) 11 of 26
A denote the compliance of point D with respect to input end A can be derived Let CD where cij (i, j = 1, 2 and 3) is the compliance factors in the i-th row and j-th column of the matrix CDA by ; and FDy defines the force applied by the micro-motion T stage excepting the bridge-type Ais actuated A by a force mechanism along the y direction, which CD = TD CD TDA Fin (input force of PZT). Accordingly, the output displacement u Ay can be obtained as a function of input displacement
(25)
uin shows the free body diagram of the quarter model of bridge-type mechanism. Figure 13 Ramp1 uin Assuming that point D remains fixed [31], uthe equations can be obtained Ay following (28) according to the force-deflection relationship at the input end A where Ramp1 is the amplification ratio of the bridge-type model. Then, the relationship between the load and deflection along the y-direction can be obtained as
u Ay = c22 FDy u + cd23FM Az + c11 Fin Ay
22 Dy
(29)
where d 22 is a compliance factor of matrix D. + c θ Az = c32 FCDy 33 M Az = 0 According to Equations (26)–(29), the input compliance for the micro-motion stage in the can3) beis derived as follows jY-direction = 1, 2 and the compliance factors in the i-th row and j-th column of the
(26) (27)
A; where cij (i, matrix CD Ramp and FDy defines the force appliedk by micro-motion excepting the bridge-type mechanism c cstage c Fthe in 1 1 22 23 32 (30) in in 11 d 22 F c33 d 22 force c along the y direction, which is actuated uby aforce (input of PZT). in
13. The free body diagram of quarter model of bridge-type mechanism. Figure 13.Figure The free body diagram of quarter model of bridge-type mechanism.
The stiffness model of the micro-motion stage with one limb actuated is established to calculate the input stiffness in the x-direction, as shown in Figure 14. The compliance of the point D can be calculated by 1 D 1 78
C
D CD = CM
1
D C910 C
The compliance of point D with respect to input end A is
(31)
Micromachines 2018, 9, 499
11 of 25
Accordingly, the output displacement u Ay can be obtained as a function of input displacement uin u Ay = Ramp1 · uin
(28)
where Ramp1 is the amplification ratio of the bridge-type model. Then, the relationship between the load and deflection along the y-direction can be obtained as u Ay = −d22 FDy
(29)
where d22 is a compliance factor of matrix CD . According to Equations (26)–(29), the input compliance for the micro-motion stage in the Y-direction can be derived as follows Fin c22 c23 · c32 Ramp1 (30) k in = = 1+ − · uin d22 c33 · d22 c11 The stiffness model of the micro-motion stage with one limb actuated is established to calculate the input stiffness in the x-direction, as shown in Figure 14. The compliance of the point D 0 can be calculated by −1 0 −1 −1 D0 D D0 CD 0 = C M + C78 + C910C (31) 0 Micromachines 2018, 9, x
12 of 26
14. Stiffness model of the micro-motion stage stage in withwith one limb Figure Figure 14. Stiffness model of the micro-motion inthe thex-direction x-direction one actuated. limb actuated.
4. Dynamics Model
The compliance of point D 0 with respect to input end A0 is Compared with vector equations, energy methods can easily derive the energy equations of 0 T compliance mechanisms [33]. Thus, the of the 2D micro-motion stage are A0 frequencies A A0 natural (32) CD 0 = TD 0 CD 0 TD 0 determined by energy equilibrium-based Lagrange equation, aiming to give a panorama of the free vibrations in the 2D micro-motion stage. Then, the compliance forcompliant the micro-motion in the cancoordinates, be derivedthat as follows: Theinput kinetic energies of the system arestage expressed byx-direction the generalized is, the input-displacement variable DI dix diy . It is assumed that the kinetic energies come from
c
0
c
0
·c
0
Ramp
32 1 0 the rigid links between thek in flexure in ·Figure 15, translational motions are = 1hinges + 22[34]. − As23shown (33) 0 0·d 0 0 d c c 22 33 22 11 generated by links e, j, and k in limb M; rotational motions are generated by links g and f in that limb; and both translational and rotational motions are generated by links a, b, c, d, h, and i. Moreover, the 0 A , d0 where c0central 1, 2 and the compliance factors the i-th row and j-th matrix CD 0 22 ij (i, j =point O of 3) theismotion stage moves along in x and y axes without anycolumn rotation.of The kinetic is a compliance factor matrix of CDthe Ramp1mechanism is the amplification ofcan thebebridge-type 0 , and energy of limb M,of consisting bridge-type and the leverratio model, expressed as model.
Machine stiffness could also be obtained in the d experimental testing [32]. TA
4. Dynamics Model
1 1 1 1 1 1 1 1 1 iy mt ( diy ) 2 ( )mt l12 ( ) 2 mt dix 2 me diy 2 + mv dix 2 m p ( d ix ) 2 2 2 2 12 l1 2 2 2 2 2
diy d 1 1 1 1 1 m p diy 2 ( )m p l2 2 ( ix ) 2 ( )mq l32 ( ) 2 vector energy methods can 2 equations, 2 12 l2 2 12 l3 easily
(34)
Compared with derive the energy equations of compliance mechanisms [33]. Thus, the natural frequencies of the 2D micro-motion stage are where determined by energy equilibrium-based Lagrange equation, aiming to give a panorama of the mt ma mb mc md (35) free vibrations in the 2D micro-motion stage. mv m j mk
(36)
m p mh mi
(37)
mq mg m f
(38)
Because of the symmetric property, the kinetic energies of limbs N in the x direction can be considered as
Micromachines 2018, 9, x
13 of 26
Micromachines 2018, 9, 499
12 of 25
1 mo diy 2 (41) 2 The kinetic energies of the compliant system are expressed by the generalized coordinates, that is, TO
h i where mo is the mass of the motion the input-displacement variable D I = stage. dix diy . It is assumed that the kinetic energies come from Then, the kinetic energy of the whole mechanism can be obtained as follows the rigid links between the flexure hinges [34]. As shown in Figure 15, translational motions are 2 1 2 1 1 1 2 1 2 generated by links e, j, Tand ( kmin mv M;mrotational mq + mWare )dix 2generated ( mt mby m p g and f in that limb; t limb p me motions v links 3 2 3 2 24 2 3 2 3 and both translational and 1rotational motions are generated by links a, b, c, d, h, and i. (42) Moreover, 1 1 mo me mq )diy 2 the central point O of the motion 2 2stage24moves along x and y axes without any rotation. The kinetic energy of limb M, consisting of the bridge-type mechanism and the lever model, can be expressed as Next, the kinetic energy should be substituted into the Lagrange’s equation below, TA =
. 2d T . .T . . . 2 2(43) Q1j 1 1 1 1 1 1 1 2 ( diy ) + 1 m d 2 2 2 m ( d ) + ( ) m l dt DI + 2 me diy + 2 mv dix + 2 m p ( 2 dix ) D t t t 1 iy ix 2 2 2 12 2I l1 . 2 . 2 . dthe dix 1 1 of undamped 1 vibration 1 1 2 2 2 The dynamics equation free of + 2 m p diy + 2 ( 12 )m p l2 ( l2 ) + 2 ( 12 )mq l3 ( liy3 ) compliant system can be derived as
MD KD 0
where
(34)
(44)
According to the characteristic equation, we have
mt = m a + mb + mc + md
(35)
K M i 2 0
(45)
mv = m j + mk The natural frequency can be calculated by
m p = m h + mi
(36) (37)
1 K f i ( ) mq =2m g 2+ mM f
(46)
(38)
Figure 15. 15. TheThe 2-dimensional micro-motion stage. Figure 2-dimensional micro-motion stage. 5. Workspace Material Analysis Because of the and symmetric property, the kinetic energies of limbs N in the x direction can be considered as The relationship between the actual and nominal maximum output displacement of the PZT in the steady state can be expressed as
TB =
. 2 . . . 2 1 1 1 1 1 2 dix 2 + 1m d 2 d t e D f iy ix 2 mt ( 2 dix ) + 2 ( 12 ) mt l1 ( l1 ) + 2 km 2 p pl . 2 D= . 2 . k k d s 1 1 p + 12 m p dix 2 + 12 ( 12 )m p l2 2 ( liy2 ) + 21 ( 12 )mq l3 2 ( dlix3 )
.
.
+ 21 mv diy 2 + 21 m p ( 12 diy )
2
(47)
(39)
where k p is the stiffness of the PZT, k s is the stiffness of spring load on the PZT, f pl is the
As preload limb Wonundergoes motions contains limb load, M, the platform, ks kinmobile PZT, D istranslational the nominal stroke of theand PZT without external andcentre when the and the micro-motion decoupler ofstage ideal hasprismatic no externaljoints, load. the kinetic energies of limbs W in the x direction can be derived as . 1 TW = mW · dix 2 (40) 2 The kinetic energy TO of the centre mobile platform can be calculated by TO = where mo is the mass of the motion stage.
. 1 mo · diy 2 2
(41)
Micromachines 2018, 9, 499
13 of 25
Then, the kinetic energy of the whole mechanism can be obtained as follows T=
( 23 mt + 12 mv + 23 m p + 12 me + . 1 mq )diy 2 + 12 mo + 12 me + 24
1 24 mq
.
+ 21 mW )dix 2 + ( 23 mt + 21 mv + 23 m p
(42)
Next, the kinetic energy should be substituted into the Lagrange’s equation below, d ∂T ∂T = Qj . − dt ∂ D I ∂D I
(43)
The dynamics equation of undamped free vibration of the compliant system can be derived as ..
M D + KD = 0
(44)
According to the characteristic equation, we have
|K − Mλi 2 | = 0
(45)
The natural frequency can be calculated by 1 λ f = i =( ) 2π 2π
r
K M
(46)
5. Workspace and Material Analysis The relationship between the actual and nominal maximum output displacement of the PZT in the steady state can be expressed as k p ∆D − f pl D= (47) k p + ks where k p is the stiffness of the PZT, k s is the stiffness of spring load on the PZT, f pl is the preload on PZT, ∆D is the nominal stroke of the PZT without external load, and k s = k in when the micro-motion stage has no external load. 5.1. Maximum Stress Subject to Rotation If a flexure hinge bears a bending moment around its rotation axis, the maximum angular displacement θ max occurs when the maximum stress σmax reaches the yield stress σy . The maximum stress usually appears at each outer surface of the thinnest part of the hinge. Therefore, the relationship between θ max and σmax can be expressed as σmax =
E(1 + η )9/20 max θ η 2 f (η )
(48)
where η1 = w/l and η2 = t/2r are dimensionless geometry factors and f (η ) is a dimensionless compliance factor. The latter can be defined as s " # 1 3 + 4η + 2η 2 6(1 + η ) 2+η −1 f (η ) = + tan η 2η + η 2 (1 + η )(2η + η 2 ) (2η + η 2 )3/2
(49)
According to the geometry of the micro-motion stage, the maximum angular deflection may occur on the hinge, which belongs to either bridge-type mechanism or level model. If the stage is actuated with a full stroke of the PZT, it could arrive at the maximum values [35]. Under such a case, we have d1 = Ramp1 D and d2 = RampD.
Micromachines 2018, 9, 499
14 of 25
Concerning the bridge-type mechanism, the flexure hinges rotation can be derived by θ1 m =
d1 la
(50)
For the lever model, the deformation of the flexure hinges can be expressed as θ2 m =
d2 lb
(51)
According to Equations (48) and (51), the maximum stresses subject to the rotation of the flexures are obtained by E(1 + η1 )9/20 Ramp1 D E(1 + η2 )9/20 RampD m σ1 m = , σ = (52) 2 η1 2 f ( η1 ) l a η2 2 f ( η2 ) l b The following relationship must be satisfied to prevent material failure: S f σ1 m ≤ σy , S f σ2 m ≤ σy
(53)
where S f ∈ (1, +∞) is a specified safety factor. Substituting Equation (52) into (53) allows the derivation of the relationships, la ≥
E(1 + η1 )9/20 Ramp1 DS f η1 2 f (η1 )σy
, lb ≥
E(1 + η2 )9/20 RampDS f η2 2 f (η2 )σy
(54)
5.2. Maximum Tensile Stress Calculation Under the axial load, the maximum tensile stress may occur on the thinnest portions of the flexure hinges constructing the bridge-type mechanism, which can be determined by σ3 m =
Fin K D = in Smin 2bw
(55)
It should be noted that bw is multiplied by 2 because flexures a and d simultaneously bear the axial load. The maximum tensile stress should be limited to the allowable stress σy , 2σy Kin D ≤ bw Sf
(56)
In the other links of the stage, the maximum tensile stress in the thinnest part of flexure hinges can be expressed as u Ay /d22 F σ4m = 1 = (57) Smin dt where Smin is the minimum cross-sectional area of the hinge. Then, it will lead to another constraint relationship for this design, u Ay σy ≤ (58) dt · d22 Sf This is another guideline to avoid the risk of plastic failure in the size design of the stage. 6. Dimension Optimization This chapter optimizes the dimensions of the 2D XY micro-motion stage to maximize the natural frequency, the determinant of the dynamic performance of the stage. Thus, the dimensions r, t, and w should be optimized. The analytical models on compliance and natural frequency show that the frequency has a nonlinear positive correlation with t and w, and a negative one with
Micromachines 2018, 9, 499
15 of 25
r. The increase of the frequency can significantly enhance the input stiffness and slightly reduce the quality. The parameters ta1 , ta2 of the lever model and la of the bridge-type mechanism are optimized to obtain the maximum amplification ratio. Further, parameter lb is optimized to achieve a compact structure. However, the optimal values of these parameters must be determined by computational method. In this paper, grey wolves optimization (GWO) is introduced for dimensional optimization. Inspired by the leadership hierarchy and hunting mechanism of grey wolves. The applications of solving classical engineering design problems and the real problem in optical engineering have proven that the algorithm is suitable for challenging problems with unknown search spaces [36]. Compared with well-know heuristics genetic algorithms (GA), GWO is able highly competitive. GWO has a very high level of local optimal avoidance, which enhances the probability of finding proper approximations of the optimal weight and biases. Moreover, the accuracy of the optimal values for weights and biases is very high because of the high exploitation of GWO [37]. The optimization conditions are as follows: (1) (2) (3)
Objective: maximize the natural frequency (f ) Related parameters: r, t, w, ta1 , ta2 , la , and lb Constraints: parameters of right-circular flexure hinge: 0.05 ≤ t/r ≤ 0.65; amplification ratio:Am ≥ 6; constraint Equations (54), (56), and (58); ranges of parameters: 0.3 mm ≤ t ≤ 0.9 mm, 0.3 mm ≤ w ≤ 0.9 mm, 17 mm ≤ la ≤ 22 mm, 0.3 mm ≤ ta1 ≤ 0.9 mm, 0.3 mm ≤ ta1 ≤ 0.9 mm, 5 mm ≤ lb ≤ 10 mm, and 2 mm ≤ r ≤ 4 mm.
The constraints are determined by the following factors: The value of t/r is limited to ensure the accuracy of the selected compliance factors Ci . To prevent plastic failure of the material, the constraint equations of the stress analysis must be satisfied, and the safety factor Sf is set to 1.5. In addition, the thinnest part of the flexible hinge should not be smaller than 0.3 mm; otherwise, the wire electro-discharge machining (WEDM) technique of the micro-motion stage cannot ensure the tolerance of 0.01 mm. Under lever bending and hinge stretching, the displacement loss may produce a sub-optimal lever amplification ratio in the optimization of lever dimension. The maximum number of iterations is set to 552. The convergence process of the GWO is presented Micromachines 2018, 9, x 26 in Figure 16. The optimal results are as follows: t = w = ta1 = 0.6 mm, ta2 = 0.4 mm, la 16 = of20.88 mm, lb = 8 mm, r = 3 mm (t , t are the thickness of the right-angle hinge and the right-circular hinge, a1 a2 la = 20.88 mm, lb = 8 mm, r = 3 mm (ta1, ta2 are the thickness of the right-angle hinge and the respectively, in the lever model). Based in onthe these optimal dimensions, the natural is computed right-circular hinge, respectively, lever model). Based on these optimalfrequency dimensions, the natural as 234.885 Hz. frequency is computed as 234.885 Hz.
16. Convergence processof ofthe the grey grey wolves optimization (GWO). FigureFigure 16. Convergence process wolves optimization (GWO).
7. Performance Evaluation Finite-ElementAnalysis Analysis 7. Performance Evaluation withwith Finite-Element The output stiffness, input stiffness, amplification ratio, and natural frequency of the XY The output stiffness, input stiffness, amplification ratio, and natural frequency of the XY micro-motion stage were validated without damping on the finite element software micro-motion stage were validated without damping on the finite element software ABAQUS/Explicit ABAQUS/Explicit (Abaqus Inc., Palo Alto, CA, USA). The material used in the simulations was 6061 (AbaqusAl,Inc., Palo Alto, CA, USA). The material the simulations was 6061 Al, was for which for which the modulus of elasticity was 69,000used Mpa,in possion’s ratio was 0.3, and density 3 3 the modulus of elasticity 69,000 Mpa, grid possion’s ratioin was density was 2700 2700 kg/m . Further, awas 3.9 mm tetrahedral was chosen mesh.0.3, Theand boundary conditions werekg/m . up mm aftertetrahedral the geometric model and the input ends of the proposedwere micro-motion Further, set a 3.9 grid wasmeshing chosenstep in mesh. The boundary conditions set up after the stage weremeshing applied tostep the load. Theinput parameters were micro-motion described in Table 1. Sensitivity geometric model and the ends of ofthe themodel proposed stage were applied to and convergence tests were performed at points B and C, respectively, as shown in Figure 15. During the EARP process, the hinges suffer from reciprocating tensile deformation, resulting in internal stress. Accordingly, internal stress on the compliance structure was tested in this paper. Specifically, the input load and output motion of the XY micro-motion stage were obtained as a given displacement on the input end of bridge-type mechanism; then, the input stiffness and amplification ratio of the flexure-based stage were determined based on the input load and output
Micromachines 2018, 9, 499
16 of 25 Figure 16. Convergence process of the grey wolves optimization (GWO).
the load. The parameters of thewith model were described in Table 1. Sensitivity and convergence tests 7. Performance Evaluation Finite-Element Analysis were performed at points B and C, respectively, as shown in Figure 15. During the EARP process, The output stiffness, input stiffness, amplification ratio, and natural frequency of the XY the hinges suffer from reciprocating tensile deformation, resulting in internal stress. Accordingly, micro-motion stage were validated without damping on the finite element software internalABAQUS/Explicit stress on the compliance testedThe in material this paper. the was input load and (Abaqus Inc.,structure Palo Alto, was CA, USA). used Specifically, in the simulations 6061 output motion of the XY micro-motion stage were obtained as a given displacement on the input end Al, for which the modulus of elasticity was 69,000 Mpa, possion’s ratio was 0.3, and density was 3 2700 kg/m . Further, a 3.9 mmthe tetrahedral grid wasand chosen in mesh. Theratio boundary conditions were stage of bridge-type mechanism; then, input stiffness amplification of the flexure-based set up after based the geometric step and themotion input ends of the17). proposed micro-motion were determined on themodel inputmeshing load and output (Figure Furthermore, the output stage were applied to the load. The parameters of the model were described in Table 1. Sensitivity stiffness can be evaluated by exerting an external force onto the stage. and convergence tests were performed at points B and C, respectively, as shown in Figure 15. During the EARP process, the hinges suffer from reciprocating tensile deformation, resulting in Table 1. Main parameters of the flexure-based stage. internal stress. Accordingly, internal stress on the compliance structure was tested in this paper. Specifically, the input load and output motion of the XY micro-motion stage were obtained as a r b t w ta1 ta2 given displacement on the input end of bridge-type mechanism; then, the input stiffness and 3 mm 15 mm 0.6 mm 0.6 mm 0.6 mm 0.4 mm amplification ratio of the flexure-based stage were determined based on the input load and output la lb E σ µ ρ motion (Figure 17). Furthermore, the output stiffness can be evaluated by exerting an external 3force 20.88 mm 8 mm 69,000 MPa 228 MPa 0.3 2700 kg/m onto the stage.
Figure Deformationmodel model XY stage. Figure 17. 17. Deformation XYmicro-motion micro-motion stage. Table 1. Main parameters of the flexure-based stage. The force-displacement curves of XY micro-motion stage are presented in Figure 18. It can tin a1 the x direction ta2 t w be seen from Figure 18a rthat the bdriving point displacement increased linearly 15 mmwhen 0.6the mmstage0.6 mmdriven 0.6 mm 0.4 mm with the corresponding3 mm input load, was in x and y directions, respectively. la E σ ρ The finite-element analysis showslbthat the input stiffness was μ30.57 N/µm, 6.9% smaller than the 20.88 mm 8 mm 69,000 MPa 228 MPa 0.3 2700 kg/m3 theoretical value obtained by compliance matrix method. As shown in Figure 18c, the stiffness (31.2 N/µm) obtained by finite-element analysis was 7.4% smaller than that the theoretical value (33.71 N/µm) acquired by compliance matrix method. The output stiffness was tested by applying an external force on the flexure-based stage, and the resulting force-displacement curves are shown in Figure 18b,d. In the x direction, the output stiffness was 9.02 N/µm, 18.3% smaller than the theoretical value 10.67 N/µm. In the y direction, the output stiffness was 11.31 N/µm, 18.6% smaller than the theoretical value 13.41 N/µm. The deviation mainly comes from the accuracy of the compliance factors equations and the hypothesized rigidity of these links in the matrix model. Next, the finite-element analysis software was employed to measure the output displacements of the points on the sensor brackets. Similarly, the input displacements can be obtained by measuring the displacement of the input end in the bridge-type mechanism. The input-output displacement relationship of the flexure-based stage is displayed in Figure 19. It can be seen that the displacement amplification ratio was around 6.78 along the x axis, 11% smaller than the theoretical value of 7.55; that along the y axis was around 5.53, 6.5% greater than the theoretical value of 5.89. The differences are attributable to the fact that the deflections of the lever model reduced the displacement of the flexure-based stage. The deviation about displacement ratio between the x and y directions resulted from the displacement loss of the flexure hinges 7, 8, 9, and 10.
the corresponding input load, when the stage was driven in x and y directions, respectively. The finite-element analysis shows that the input stiffness was 30.57 N μm , 6.9% smaller than the theoretical value obtained by compliance matrix method. As shown in Figure 18c, the stiffness (31.2 N μm ) obtained by finite-element analysis was 7.4% smaller than that the theoretical value (33.71 N μm ) acquired by compliance matrix method. The output stiffness was tested by applying an
external Micromachines 2018,force 9, 499on the flexure-based stage, and the resulting force‐displacement curves are shown in 17 of 25 Figure 18b,d.
Figure The force‐displacement curvesof of XY XY micro-motion stage. (a) Input displacement in the y in the Figure 18. The18.force-displacement curves micro-motion stage. (a) Input displacement direction; (b) Output displacement in the y direction; (c) Input displacement in the x direction; (d) y direction; (b) Output displacement in the y direction; (c) Input displacement in the x direction; Output displacement in the x direction. Micromachines 2018, 9, x 18 of 26 (d) Output displacement in the x direction.
In the x direction, the output stiffness was 9.02 N μm , 18.3% smaller than the theoretical value 10.67 N μm . In the y direction, the output stiffness was 11.31 N μm , 18.6% smaller than the theoretical value 13.41 N μm . The deviation mainly comes from the accuracy of the compliance factors equations and the hypothesized rigidity of these links in the matrix model. Next, the finite-element analysis software was employed to measure the output displacements of the points on the sensor brackets. Similarly, the input displacements can be obtained by measuring the displacement of the input end in the bridge-type mechanism. The input‐output displacement relationship of the flexure-based stage is displayed in Figure 19. It can be seen that the displacement amplification ratio was around 6.78 along the x axis, 11% smaller than the theoretical value of 7.55; that along the y axis was around 5.53, 6.5% greater than the theoretical value of 5.89. The differences are attributable to the fact that the deflections of the lever model reduced the displacement of the flexure-based stage. The deviation about displacement ratio between the x and y directions resulted from the displacement loss of the flexure hinges 7, 8, 9, and 10.
19. Input‐output displacementrelationship relationship ofof the flexure-based stage.stage. FigureFigure 19. Input-output displacement the flexure-based
The maximum stress of the thinnest hinge must be lower than the yield stress of the material
The maximum stress of the thinnest hinge must be lower than the yield stress of the material (152 MPa). This is to prevent the hinges from failure in the production and application processes. To (152 MPa). This is to prevent the hinges from failure in the production and application processes. T test the stress on the flexure-based stage, an input force F 1010,1010 was applied at the input To test the stress on the flexure-based stage, an input force FII = [1010, 1010] T was applied at the input end so that the analysis results indicated the developed XY micro-motion stage possesses the end so that the analysis results indicated the developed XY micro-motion stage possesses the potential potential to achieve the maximum workspace around 112 μm 89 μm , while the maximum von Mises to achieve the maximum workspace around 112 µm × 89 µm, while the maximum von Mises stress are stress are 62.09 MPa and 102.98 MPa, occurring in the right-angle hinge and right-circular hinge, 62.09 MPa and 102.98 MPa,material occurring inasthe right-angle and right-circular hinge, respectively, respectively, without failure shown in Figure hinge 20. without material failure as shown in Figure 20.
The maximum stress of the thinnest hinge must be lower than the yield stress of the material (152 MPa). This is to prevent the hinges from failure in the production and application processes. To test the stress on the flexure-based stage, an input force FI 1010,1010
T
was applied at the input
end so that the analysis results indicated the developed XY micro-motion stage possesses the potential to achieve the maximum workspace around 112 μm 89 μm , while the maximum von Mises Micromachines 499 MPa and 102.98 MPa, occurring in the right-angle hinge and right-circular hinge, 18 of 25 stress2018, are 9,62.09
respectively, without material failure as shown in Figure 20.
Figure Stresstest testof of the the flexure-based stage. Figure 20.20. Stress flexure-based stage.
The firstmode four mode shapes of the flexure-based stage 21)21) show that that the two The first four shapes of the flexure-based stage(Figure (Figure show thetranslational two translational vibrations occurred at the first and the second modes with the frequencies 220.43 Hz and 233.67 Hz, vibrations occurred at the first and the second modes with the frequencies 220.43 Hz and 233.67 Hz, respectively. Because of the structure symmetry, the first two modes had almost the same respectively. Because of the structure symmetry, the first two modes had almost the same frequencies. frequencies. The corresponding theoretical values obtained from Lagrange’s equation were both The corresponding theoretical values obtained from Lagrange’s equation were both 239.7 8.7% and 239.7 Hz, 8.7% and 2.6% higher than the results of the finite-element analysis, respectively. TheHz, third 2.6% higher than the results of the finite-element analysis, respectively. The third and fourth and fourth modes had frequencies of 513.20 Hz and 628.36 Hz, leading to rotation and twisting, modes respectively. These two modes were Hz, neglected in analytical modelling, owing respectively. to the simulation had frequencies of 513.20 Hz and 628.36 leading to rotation and twisting, These two difficulties. modes were neglected in analytical modelling, owing to the simulation difficulties. Micromachines 2018, 9, x
19 of 26
Figure First fourmode mode shapes shapes ofofthe flexure-based stage. Figure 21. 21. First four the flexure-based stage.
8. Testing Experiments 8. Testing Experiments Several offline tests were carried out to verify the performance of the XY micro-motion stage.
Several offline tests were carried out to verify the performance of the XY micro-motion stage. The experimental setup is schematically shown in Figure 22. Two PZT actuators (40 vs. 12, Core The experimental setup isCo., schematically shown in Figure Two (40 vs. 12, Core Tomorrow Science Ltd., Harbin, China), whose 22. sizes arePZT 12actuators mm 51.5 mm and Tomorrow Science Co., Ltd., Harbin, China), whose sizes are ϕ12 mm × 51.5 mm and K pzt = 35 N/µm, K pzt 35 N μm , respectively, were embedded into the drive structures to achieve impact structure respectively, were embedded into the drive structures to achieve impact structure sizes. A power sizes. A power amplifier (PI, E-500) was utilized to amplify the excitation signals, which were amplifier (PI, E-500) waspower utilized to amplify theA excitation signals, which were generated generated by the PMAC controller. four-channel capacitive displacement sensor by the power PMAC controller. A four-channel displacement sensor (Micro-sense DE 5300-013) (Micro-sense DE 5300-013) was chosencapacitive for dynamic position measurements. To reduce external disturbances on the sensing and measurement system, a vibration-isolated air-floating platform was sensing was chosen for dynamic position measurements. To reduce external disturbances on the used to mount the XY micro-motion stage.
8. Testing Experiments Several offline tests were carried out to verify the performance of the XY micro-motion stage. The experimental setup is schematically shown in Figure 22. Two PZT actuators (40 vs. 12, Core Tomorrow Science Co., Ltd., Harbin, China), whose sizes are 12 mm 51.5 mm and
K pzt 35 N μm , respectively, were embedded into the drive structures to achieve impact structure
Micromachines 2018, 9, 499
19 of 25
sizes. A power amplifier (PI, E-500) was utilized to amplify the excitation signals, which were generated by the power PMAC controller. A four-channel capacitive displacement sensor (Micro-sense DE 5300-013) was chosen for dynamic position measurements. To reduce external and measurement system, a vibration-isolated air-floating platform was used to mount the XY disturbances on the sensing and measurement system, a vibration-isolated air-floating platform was micro-motion stage. the XY micro-motion stage. used to mount
Figure Schematicconfiguration configuration ofofthe system. Figure 22. 22. Schematic theexperimental experimental system.
Sine Sweep Response 8.1. Sine8.1. Sweep Response The swept excitation method is chosen to study the dynamic characteristic as it is an expedient
The swept excitation method is chosen to study the dynamic characteristic as it is an expedient method to use. A variable frequency sine sweep signal was applied to the PZT actuators for each methodaxis, to use. A displacement variable frequency sinerecorded sweep and signal was applied to the PZT actuators and the response was analyzed by fast Fourier transform (FFT). Thefor each axis, andMicromachines the displacement response was recorded and analyzed by fast Fourier transform 2018, 9, x 20 of 26 (FFT). The measured result along y and x axes are shown in Figure 23a,b. It can be obtained from Figure 23a measured result along y and x axes are shown in Figure 23a,b. It can be obtained from Figure 23a that the first three natural frequencies were 237.51 Hz, 361.43 Hz, and 522.75 Hz, respectively. that the first three natural frequencies were 237.51 Hz, 361.43 Hz, and 522.75 Hz, respectively. Figure 23b shows the first three natural frequencies were 205.39 Hz, 294.80 Hz, and 502.17 Hz, Figure 23b shows the first three natural frequencies were 205.39 Hz, 294.80 Hz, and 502.17 Hz, which which were withthethe first three natural frequencies measured x axis were coincident coincident with first three natural frequencies measured along thealong x axis the as well. Theas well. The marginal difference bymanufacturing manufacturing errors additional mass. According to marginal differencemay maybe be caused caused by errors and and additional mass. According to a increase in equivalent the equivalent massescompared compared with in stiffness, the frequencies a higherhigher increase in the masses withthe theincrease increase in stiffness, the frequencies are relatively smaller. Theworking working bandwidth of the system system is limitedisbylimited the obtainedobtained are relatively smaller. The bandwidth of machining the machining by lowest first natural frequency. If the contacts between PZT actuators and input ends perfectly, a the lowest first natural frequency. If the contacts between PZT actuators and input ends perfectly, micro-motion in a high frequency range can be achieved [38]. However, the resonance frequencies a micro-motion inrequirements a high frequency range can beatachieved [38]. satisfy the of precision polishing medium or low However, roll speed. the resonance frequencies satisfy the requirements of precision polishing at medium or low roll speed.
Figure 23. Dynamic responsesalong along (a) (b)(b) x axis. Figure 23. Dynamic responses (a)yyaxis axisand and x axis.
8.2.and Stroke and Resolution 8.2. Stroke Resolution Test Test The surface roughness, a key precision polishing, is positively correlated with thewith stroke of stroke The surface roughness, a key to toprecision polishing, is positively correlated the XY micro-motion stage. In light of this, a stair control voltage with maximum displacement of 72 µ m of XY micro-motion stage. In light of this, a stair control voltage with maximum displacement was given to the x direction actuator. Similarly, the maximum displacement of 84 µ m was adopted of 72 µmaswas givensignal to the x direction Similarly, the maximum displacement of 84 µm the driving in the y direction.actuator. Figure 24 records the displacements in the x and y directions. was adopted as theCMD driving signal in the y direction. Figure 24 records the displacements In the figure, means command displacement, while ACT refers to the actual displacement. It in the be seen that 71 µ m and 83command µ m along displacement, x and y axes, respectively. Moreover, x and y can directions. In the the stroke figure,was CMD means while ACT refers to the Figure 24b shows that the compliant mechanism cannot track the maximum input command immediately, which demonstrates the tested compliant mechanism is working in the maximum limitation of displacement output. There is a huge difference between simulated and actual displacements along the x axis. A possible reason lies in the friction between the flexure-based stage and fixed base plate, which is generated by manufacturing accuracy and assembly errors. If the contact surface is sufficiently smooth, the actual displacement could approximate the simulated
Micromachines 2018, 9, 499
20 of 25
actual displacement. It can be seen that the stroke was 71 µm and 83 µm along x and y axes, respectively. Moreover, Figure 24b shows that the compliant mechanism cannot track the maximum input command immediately, which demonstrates the tested compliant mechanism is working in the maximum limitation of displacement output. There is a huge difference between simulated and actual displacements along the x axis. A possible reason lies in the friction between the flexure-based stage and fixed base plate, is generated by manufacturing accuracy and assembly errors. If26the contact Micromachines 2018, 9,which x 21 of surface is sufficiently smooth, the actual displacement could approximate the simulated results. Micromachines 2018, 9, x
21 of 26
Figure 24. 24. Motion stroke axisand and(b)(b) y axis. CMD—command displacement; ACT—actual Figure Motion strokeof of (a) (a) x x axis y axis. CMD—command displacement; ACT—actual displacement. displacement. Figure 24. Motion stroke of (a) x axis and (b) y axis. CMD—command displacement; ACT—actual displacement.
The resolution is a major criterion forflexure-based the flexure-based In theory, the The resolution is a major designdesign criterion for the stage.stage. In theory, the piezo-electricpiezo-electric-driven flexure-based stage can achieve a high resolution. To identify the resolution of driven flexure-based stage can a high criterion resolution. the resolution the flexure-based The resolution is aachieve major design for To theidentify flexure-based stage. Inof theory, the the flexure-based stage, a stair-step command signal was generated from a digital computer, which piezo-electric-driven flexure-based stage can achieve a high resolution. To identify the resolution of stage,was a stair-step command signal was generated from a digital computer, which was provided to provided to the PEAs to drive flexure stage. In each step, chattering was observed mainly the flexure-based stage, a stair-step command signal was generated from a digital computer, which the PEAs to drive flexure stage. In each step, chattering was observed mainly because of the inherent because of the inherent noise effect of the capacitance sensor. The responses in Figure 25a,b show was provided to the PEAs to drive flexure stage. In each step, chattering was observed mainly noise that effect the capacitance in70 Figure 25a,b show that flexure-based stage theofflexure-based stage sensor. achievedThe the responses resolution of nm and 56 nm along thethe x and y axes, because of the inherent noise effect of the capacitance sensor. The responses in Figure 25a,b show respectively. If the performance of the capacitance sensor could be improved and external achievedthat thethe resolution of 70 nm and 56 nm along the x and y axes, respectively. If the performance flexure-based stage achieved the resolution of 70 nm and 56 nm along the x and y axes, is reduced, a higherbe output resolution (~50 nm) can disturbance be obtained [39]. of thedisturbance capacitance sensor improved and external is reduced, respectively. If the could performance of the capacitance sensor could be improved anda higher external output resolution (~50 nm) can be obtained [39]. disturbance is reduced, a higher output resolution (~50 nm) can be obtained [39].
Figure 25. Resolution along the (a) x axis and (b) y axis. Figure 25. Resolutionalong along the the (a) (b)(b) y axis. Figure 25. Resolution (a)xxaxis axisand and y axis. 8.3. Step Response and Sine Response Test
The andSine sine responses along Stepstep Response and Response Sine Response Test the two directions were examined to inspect the XY 8.3. Step8.3. Response and Test micro-motion stage performance. A typical proportional-integral-derivative (PID) controller was
The step and sine responses along the two directions were examined to inspect the XY The step and sine responses alongstage. the According two directions werein examined inspect implemented to position the flexure-based to the results Figure 26, thetorise times the XY micro-motion stage performance. A typical proportional-integral-derivative (PID) controller was of the x axis and yperformance. axis motions were approximately 29 ms and 24 ms, respectively. Moreover, very micro-motion stage A typical proportional-integral-derivative (PID) controller was implemented to position the flexure-based stage. According to the results in Figure 26, the rise times small and steady errors and overshoots were observed thanks to the feedback control during the implemented position the flexure-based stage. According the results inMoreover, Figure 26, of the xto axis and y axis motions were approximately 29 ms and 24toms, respectively. verythe rise positioning process. times ofsmall the xand axissteady and yerrors axis and motions were were approximately 29 ms andfeedback 24 ms, respectively. overshoots observed thanks to the control during Moreover, the positioning process. very small and steady errors and overshoots were observed thanks to the feedback control during the positioning process.
Micromachines 9, x Micromachines 2018,2018, 9, 499
22 of 26
21 of 25
Figure 26. 26. Step responses theapparatus apparatus in (a) x direction and the y direction. Figure Step responses of of the in (a) the the x direction and (b) the (b) y direction. To up, sumall up, theresults test results thatproposed the proposed EARP system follow external To sum thealltest showshow that the EARP system cancan follow external commands commands with fast response speed and high accuracy, and thus can be applied to precision with fast response speed and high accuracy, and thus can be applied to precision polishing. polishing.
9. Polishing Experiment
9. Polishing Experiment
9.1. Experiment Setup 9.1. Experiment Setup The polishing experiments performed independently developed vibration-assisted The polishing experimentswere were performed onon thethe independently developed vibration-assisted roll-type polishing machine. Figure Figure 27 of the The polishing roll is roll-type polishing machine. 27 shows showsphotographs photographs of machine. the machine. The polishing roll is mounted on XZ an XZ table linear guides. guides. The micro-motion stagestage is mounted on a B-axis mounted on an table bybylinear The micro-motion is mounted on atilting B-axis tilting table and the workpiece is mounted on the flexure stage with bonding wax. The experiment setup table and the workpiece is mounted on the flexure stage with bonding wax. The experiment setup consists of the machines, the micro-motion stage, the charge amplifer (Type 5018, Kistler, consists of the machines, the micro-motion stage, the charge amplifer (Type 5018, Kistler, Winterthur, Winterthur, Switzerland), the force sensor (Type 9211B, Kistler), the PEAs (40 vs. 12, Core Tomorrow Switzerland), theLtd., force sensor (Typesignal 9211B, Kistler), the PEAs vs. 12, Corepower Tomorrow Science Co., Science Co., Harbin, China), generator (DG4162, Rigol,(40 Beijing, China), amplifier Ltd., Harbin, China), (DG4162, Rigol, Beijing, China), were power amplifier (PI, E-500), (PI, E-500), and so signal on. Thegenerator control signals provided by the signal generator amplified using amplifier andsignals were then sent to drive thesignal piezo-electric stacks. To achieve accurate polishing, and sopower on. The control provided by the generator were amplified using power amplifier the force applied to piezo-electric detect abrasive-workpiece and the charge amplifier and were thensensor sent was to drive the stacks. Toengagement achieve accurate polishing, thewas force sensor used to measure the polishing forces. was applied to detect abrasive-workpiece engagement and the charge amplifier was used to measure the polishing forces. Micromachines 2018, 9, x
23 of 26
Figure 27. Polishing experimental setup. Figure 27. Polishing experimental setup.
The specific experimental parameters were as follows: roll speed: 240 r/min; frequency (f1, f2): 120 Hz, 120 Hz; amplitude (A1, A2): 5 μm, 6 μm; input voltage (V1, V2): 4 V, 8 V; phase difference: 90 . The workpiece material was ground prior to polishing of SiC ceramic, as shown in Figure 28. The white-light interferometer (ZygoNewview, Middlefield, CT, USA) was employed to capture the topography features.
Micromachines 2018, 9, 499
22 of 25 Figure 27. Polishing experimental setup.
The specific experimental parameters were as follows: roll speed: 240 r/min; frequency (f 1 , f 2 ): 120 Hz, 120The Hz;specific amplitude (A1 , Aparameters µm;asinput voltage (V1 , 240 V2 ):r/min; 4 V, frequency 8 V; phase experimental follows: roll speed: (f1,difference: f2): 2 ): 5 µm, 6were ◦ 120 Hz, 120 Hz; amplitude (A 1 , A 2 ): 5 μm, 6 μm; input voltage (V 1 , V 2 ): 4 V, 8 V; phase difference: 90 . The workpiece material was ground prior to polishing of SiC ceramic, as shown in Figure 28. material was ground priorMiddlefield, to polishing ofCT, SiC USA) ceramic, as shown in Figure 28. 90 . The workpiece The white-light interferometer (ZygoNewview, was employed to capture the The white-light interferometer (ZygoNewview, Middlefield, CT, USA) was employed to capture the topography features. topography features.
Figure Siliconcarbide carbide ceramic workpiece. Figure 28.28. Silicon ceramic workpiece.
9.2. Results and Discussions 9.2. Results and Discussions The workpiece surface morphology with non-vibration and vibration polishing are given in
The workpiece surface morphology with non-vibration and vibration polishing are given in Figure 29. Because of the unstable control of polishing slurry, some marks can be observed at the Figure 29. Because themethods. unstableFigure control polishing slurry, some was marks can befrom observed workpiece withofboth 29a of shows the surface roughness improved 95 nm at the workpiece with both methods. Figure 29a shows the surface roughness was improved from 95 nm Sa Sa to 80 nm Sa and 504 nm Sz reduced with form deviation by non-vibration polishing. The polishing apparentlywith be observed in the area. shown in Figure 29b, the scratches to 80 nm Sa andscratches 504 nm can Sz reduced form deviation byAs non-vibration polishing. The polishing were compared with thein former as a result of moreinabrasives involved in polishingwere (44 nm scratches canincreased apparently be observed the area. As shown Figure 29b, the scratches increased Sa), and 856 nm Sz smaller than non-vibration by applying vibration. The performance of the compared with the former as a result of more abrasives involved in polishing (44 nm Sa), and 856 nm Sz 2018, 9, x 24 of 26 smaller Micromachines than non-vibration by applying vibration. The performance of the vibration assisted roll-type polishing was evaluated by experiment, indicating that the developed method could effectively vibration assisted roll-type polishing was evaluated by experiment, indicating that the developed improve the surface quality. method could effectively improve the surface quality.
Figure 29. Workpiece surface morphology(a) (a)non-vibration non-vibration polishing andand (b) vibration polishing. Figure 29. Workpiece surface morphology polishing (b) vibration polishing.
10. Conclusions 10. Conclusions In this paper, the aim was the generation of a high-quality surface based on the linear contact
In this paper, the aim was the generation of a high-quality surface based on the linear contact material removal mechanism. The theoretical model for the micro-motion stage’s statics, materialcompliance, removal mechanism. The theoretical model the micro-motion stage’s statics, compliance, and dynamic properties were given andfor validated by the GWO, followed by prototype and dynamic properties were given andpolishing validated by the GWO, followed by prototype fabrication. fabrication. A closed-loop test and experiment were determined to meet the critical requirements for precision finishing. Compared with existing mechanisms, dedicated to polishing, the advantages of the proposed EARP system are listed as follows: (1) It is capable to deliver large 2D vibration amplitudes, while maintaining favourable resolution. Experimental tests performed show that the vibration strokes in the x- and y-directions can reach 71 μm and 83 μm, respectively, while the resolution at points B and C of the micro-motion stage are 70 nm and 56 nm, respctively.
Micromachines 2018, 9, 499
23 of 25
A closed-loop test and polishing experiment were determined to meet the critical requirements for precision finishing. Compared with existing mechanisms, dedicated to polishing, the advantages of the proposed EARP system are listed as follows: (1)
(2)
(3)
It is capable to deliver large 2D vibration amplitudes, while maintaining favourable resolution. Experimental tests performed show that the vibration strokes in the x- and y-directions can reach 71 µm and 83 µm, respectively, while the resolution at points B and C of the micro-motion stage are 70 nm and 56 nm, respctively. Because of the high stiffness of the proposed flexure-based stage, natural frequencies that meet the requirements are also achieved, which were examined to be 205.39 Hz and 237.51 Hz in the xand y-directions, respectively. Compared with the non-vibration roll-type polishing system, 44 nm Sa and 856 nm Sz are improved by proposed EARP system used independently developed polishing machine. Accordingly, the micro-motion stage could increase the number of abrasive particles involved in polishing.
Author Contributions: Y.G. and X.C. designed the experiments, analyzed the results and wrote the paper; F.L. and Z.Z. performed the simulation work; J.L., M.L. and H.Y. contributed reagents/ materials/analysis tools. Funding: This work was supported by Research on Key Technology of Manufacturing of Information-enabled for complex light machine parts Complex Optical Machine Functional Parts, National Ministry of Science and Technology International Cooperation Project, number: 2016YFE0105100. This work was also supported by the Micro-Nano and Ultra-Precision Key Laboratory of Jilin Province (grant No. 20140622008JC), Science and Technology Development Projects of Jilin Province (grant Nos. 20180623034TC, 20180201052GX), and Jilin Provincial Education Department Scientific Research Planning Project (grant No. JJKH20181036KJ). Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12.
Zhang, Y.; Zhang, J.; Han, J.; He, X.; Yao, W. Large-scale fabrication of lightweight Si/SiC ceramic composite optical mirror. Mater. Lett. 2004, 58, 1204–1208. [CrossRef] Zhang, Z.; Yang, X.; Zheng, L.; Xue, D. High-performance grinding of a 2-m scale silicon carbide mirror blank for the space-based telescope. Int. J. Adv. Manuf. Technol. 2017, 89, 463–473. [CrossRef] Agarwal, S.; Rao, P.V. Grinding characteristics, material removal and damage formation mechanisms in high removal rate grinding of silicon carbide. Int. J. Mach. Tools Manuf. 2010, 50, 1077–1087. [CrossRef] Gu, Y.; Zhu, W.; Lin, J.; Lu, M.; Kang, M. Subsurface Damage in Polishing Process of Silicon Carbide Ceramic. Materials 2018, 11, 506. [CrossRef] [PubMed] López de Lacalle, L.N.; Rodriguez, A.; Lamikiz, A.; Celaya, A.; Alberdi, R. Five-axis machining and burnishing of complex parts for the improvement of surface roughness. Mater. Manuf. Process. 2011, 26, 997–1003. [CrossRef] Becker, K. M400 and P400-A pair of machines for computer-controlled fine correction of optical surfaces. In Proceedings of the International Congress for Ultraprecision Technology, Aachen, Germany, 19 May 1988; p. 212. Jones, R.A. Computer-controlled optical surfacing with orbital tool motion. Opt. Eng. 1986, 25, 256–785. [CrossRef] Mori, Y.; Ikawa, N.; Okuda, T.; Yamagata, K. Numerically controlled elastic emission machining. Technol. Rep. Osaka Univ. 1976, 26, 283–294. Mori, Y.; Yamauchi, K.; Endo, K. Elastic emission machining. Precis. Eng. 1987, 9, 123–128. [CrossRef] Suzuki, H.; Moriwaki, T.; Okino, T.; Ando, Y. Development of ultrasonic vibration assisted polishing machine for micro aspheric die and mold. CIRP. Ann.-Manuf. Technol. 2006, 55, 385–388. [CrossRef] Suzuki, H.; Hamada, S.; Okino, T.; Kondo, M.; Yamagata, Y. Ultraprecision finishing of micro-aspheric surface by ultrasonic two-axis vibration assisted polishing. CIRP Ann. 2010, 59, 347–350. [CrossRef] Zhao, Q.; Sun, Z.; Guo, B. Material removal mechanism in ultrasonic vibration assisted polishing of micro cylindrical surface on SiC. Int. J. Mach. Tools Manuf. 2016, 103, 28–39. [CrossRef]
Micromachines 2018, 9, 499
13.
14. 15. 16. 17.
18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
24 of 25
Liang, Z.; Wang, X.; Wu, Y.; Xie, L.; Jiao, L.; Zhao, W. Experimental study on brittle–ductile transition in elliptical ultrasonic assisted grinding (EUAG) of monocrystal sapphire using single diamond abrasive grain. Int. J. Mach. Tools Manuf. 2013, 71, 41–51. [CrossRef] Yu, T.; Yang, X.; An, J.; Yu, X.; Zhao, J. Material removal mechanism of two-dimensional ultrasonic vibration assisted polishing Inconel718 nickel-based alloy. Int. J. Adv. Manuf. Techol. 2018, 96, 657–667. [CrossRef] Chern, G.L.; Chang, Y.C. Using two-dimensional vibration cutting for micro-milling. Int. J. Mach. Tools Manuf. 2006, 46, 659–666. [CrossRef] Chen, W.; Huo, D.; Shi, Y.; Hale, J.M. State-of-the-art review on vibration-assisted milling: Principle, system design, and application. Int. J. Adv. Manuf. Technol. 2018, 97, 2033–2049. [CrossRef] Chee, S.K.; Suzuki, H.; Uehara, J.; Yano, T.; Higuchi, T.; Lin, W. A Low Contact Force Polishing System for Micro Molds that Utilizes 2-Dimensional Low Frequency Vibrations (2DLFV) with Piezoelectric Actuators (PZT) and a Mechanical Transformer Mechanism. IJAT 2013, 7, 71–82. [CrossRef] Martínez, S.; Lamikiz, A.; Ukar, E.; Calleja, A.; Arrizubieta, J.A.; de Lacalle, L.L. Analysis of the regimes in the scanner-based laser hardening process. Opt. Laser Eng. 2017, 90, 72–80. [CrossRef] Chen, W.; Zheng, L.; Huo, D.; Chen, Y. Surface texture formation by non-resonant vibration assisted micro milling. J. Micromech. Microeng. 2018, 28, 025006. [CrossRef] Gu, Y.; Zhou, Y.; Lin, J.; Lu, M.; Zhang, C.; Chen, X. Design, analysis, and testing of a flexure-based vibration-assisted polishing device. AIP Adv. 2018, 8, 055113. [CrossRef] Wang, H.; Lee, H.; Jeong, H. Statistical Analysis on Process Variables in Linear Roll-CMP. J. Korean Soc. Tribol. Lubr. Eng. 2014, 30, 139–145. Lee, H.; Wang, H.; Park, J.; Jeong, H. Experimental investigation of process parameters for roll-type linear chemical mechanical polishing (Roll-CMP) system. Precis. Eng. 2014, 38, 928–934. [CrossRef] Lee, H. Mathematical modelling of material removal rate in roll-type linear CMP (roll-CMP) process: Effect of polishing pad. Int. J. Precis. Eng. Manuf. 2016, 17, 495–501. [CrossRef] Tabernero, I.; Lamikiz, A.; Martinez, S.; Ukar, E.; De Lacalle, L.L. Modelling of energy attenuation due to powder flow-laser beam interaction during laser cladding process. J. Mater. Process. Technol. 2012, 212, 516–522. [CrossRef] Ma, H.W.; Yao, S.M.; Wang, L.Q.; Zhong, Z. Analysis of the displacement amplification ratio of bridge-type flexure hinge. Sens. Actuator A Phys. 2006, 132, 730–736. [CrossRef] Qi, K.Q.; Xiang, Y.; Fang, C.; Zhang, Y.; Yu, C.S. Analysis of the displacement amplification ratio of bridge-type mechanism. Mech. Mach. Theory 2015, 87, 45–56. [CrossRef] Kim, J.H.; Kim, S.H.; Kwak, Y.K. Development and optimization of 3-D bridge-type hinge mechanisms. Sens. Actuator A Phys. 2004, 116, 530–538. [CrossRef] Wu, Y.; Zhou, Z. Design calculations for flexure hinges. Rev. Sci. Instrum. 2002, 73, 3101–3106. [CrossRef] Pham, H.H.; Chen, I.M. Stiffness modelling of flexure parallel mechanism. Precis. Eng. 2005, 29, 467–478. [CrossRef] Li, Y.; Huang, J.; Tang, H. A compliant parallel XY micromotion stage with complete kinematic decoupling. IEEE. Trans. Autom. Sci. Eng. 2012, 9, 538–553. [CrossRef] Li, Y.; Xu, Q. Design and analysis of a totally decoupled flexure-based XY parallel micromanipulator. IEEE Trans. Robot. 2009, 25, 645–657. Olvera, D.; De Lacalle, L.L.; Compeán, F.I.; Fz-Valdivielso, A.; Lamikiz, A.; Campa, F.J. Analysis of the tool tip radial stiffness of turn-milling centers. Int. J. Adv. Manuf. Technol. 2012, 60, 883–891. [CrossRef] Xiao, S.; Li, Y. Optimal Design, Fabrication, and Control of an XY Micropositioning Stage Driven by Electromagnetic Actuators. IEEE. Trans. Ind. Electron. 2013, 60, 4613–4626. [CrossRef] Li, Y.; Xu, Q. A totally decoupled piezo-driven XYZ flexure parallel micropositioning stage for micro/nanomanipulation. IEEE. Trans. Autom. Sci. Eng. 2011, 8, 265–279. [CrossRef] Li, Y.; Xu, Q. A novel piezoactuated XY stage with parallel, decoupled, and stacked flexure structure for micro-/nanopositioning. IEEE. Trans. Ind. Electron. 2011, 58, 3601–3615. [CrossRef] Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 2014, 69, 46–61. [CrossRef] Mirjalili, S. How effective is the Grey Wolf optimizer in training multi-layer perceptrons. Appl. Intell. 2015, 43, 150–161. [CrossRef]
Micromachines 2018, 9, 499
38. 39.
25 of 25
Jia, S.; Yuan, Q.; Cai, W.; Li, M.; Li, Z. Energy modeling method of machine-operator system for sustainable machining. Energy Convers. Manag. 2018, 172, 265–276. [CrossRef] Cai, K.; Tian, Y.; Wang, F.; Zhang, D.; Shirinzadeh, B. Development of a piezo-driven 3-DOF stage with T-shape flexible hinge mechanism. Robot. Comput.-Intergr. Manuf. 2016, 37, 125–138. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
