micromachines
Article Modeling and Experimental Study of Double-Row Bow-Type Micro-Displacement Amplifier for Direct-Drive Servo Valves
Guoping Liu 1,*, Zhongbo He 1,*, Guo Bai 2, Jiawei Zheng 3, Jingtao Zhou 1 and Ming Chang 4 1 Army Engineering University Shijiazhuang Campus, Shijiazhuang 050003, China; [email protected] 2 Unit 63926 of PLA, Beijing 100192, China; [email protected] 3 Unit 32184 of PLA, Beijing 100072, China; [email protected] 4 Department of Mechanized Infantry, Shijiazhuang Branch, Army Infantry Academy, Shijiazhuang 050083, China; [email protected] * Correspondence: [email protected] (G.L.); [email protected] (Z.H.)
Received: 22 February 2020; Accepted: 13 March 2020; Published: 16 March 2020
Abstract: Giant magnetostrictive actuators (GMA) are widely used in the field of servo valves, but the displacement of GMA is limited, which renders meeting the requirements of large flow direct-drive electro-hydraulic servo valves (DDV) difficult. In order to solve these problems, this study proposes a double-row bow-type micro-displacement amplifier (DBMA), used to increase output displacement of GMA to meet the requirements. This study, by static analysis, analyzes the force of a flexure hinge based on theoretical mechanics and material mechanics, derives the stiffness matrix of the flexure hinge by the influence coefficient method, establishes the pseudo-rigid model, and derives the amplification ratio of a DBMA. Also, by kinetic analysis, using Castigliano’s second theorem, a formula of equivalent stiffness and natural frequency of DBMA were derived and the influences of different parameters on them were analyzed, respectively. After that, we analyzed the amplifier using finite element method (FEM) simulation software and verified the model by manufacturing a prototype and building a test system. Theoretical calculations and experimental results showed that the amplification ratio of the DBMA fluctuated between 15.43 and 16.25. The natural frequency was about 305 Hz to 314 Hz and the response bandwidth was up to 300 Hz. The error among the theoretical, simulated, and experimental values was within 8%, supporting the accuracy of the model.
Keywords: micro-displacement amplifier; giant magnetostrictive actuator (GMA); pseudo-rigid model; servo valve
1. Introduction Giant magnetostrictive material (GMM) possesses excellent mechanical properties and frequency response characteristics [1,2]. Giant magnetostrictive actuators (GMA), which are designed with GMM as their core, have the faster response and larger output displacement than actuators made of similar materials (such as piezoelectric actuator and shape memory alloy actuator), etc. [3] With great load capacity, GMA exhibit broad application prospects in the field of electro-hydraulic servo systems. Based on GMA, the new Direct Drive Servo Valve (GMM-DDV) can improve response speed and reliability under the premise of ensuring a certain flow rate [4–10]. However, in respect to output requirements, the output displacement of GMA is limited, usually micrometers, and it is difficult to directly drive a mechanism that requires a large stroke. Therefore, an appropriate micro-displacement amplifier must be designed [11–13]. At present, micro-displacement amplifiers used in the field of precision machinery can be roughly divided into mechanical type and hydraulic type. Among them, the hydraulic amplifier is mainly based
Micromachines 2020, 11, 312; doi:10.3390/mi11030312 www.mdpi.com/journal/micromachines Micromachines 2020, 11, 312 2 of 15 on Pascal’s theorem. Hwan-Sik Yoon et al. developed a hydraulic amplifier for piezoelectric actuators and verified it experimentally [14]. Yu et al. improved the hydraulic amplifier with a diaphragm structure and applied it to direct drive servo valves; the displacement amplification ratio was 9, and the governing error of valve spool was less than 1% [15]. Mechanical amplifiers usually feature a flexure hinge structure, which has the advantages of no back stroke, no friction, smooth motion, and high resolution. Paros and Weisbord derived a design formula for flexure hinges [16,17]. Sujit designed and manufactured a micro-displacement amplification mechanism based on the triangle principle, established a pseudo-rigid body model, and performed FEM simulations and experiments. The actual magnification was about 1.5 times [18]. Huang designed a diamond-shaped micro-displacement amplifier with bidirectional active output that features a symmetrical structure design and uses 4 actuators [19]. Huang et al. designed a skew-symmetric amplifier that converts a linear input into an angular output [20]. Lobontiu et al. used Castigliano’s second theorem to derive a closed-loop analytical formula of the stiffness of the bow-type amplifier and studied the influence of the relevant geometric parameters on its stiffness [21]. Wan used the kinetic energy theorem to consider the stiffness of the overall moving pair and the rotational stiffness of the flexible hinge, and derived the parameters affecting amplifier displacement and amplification ratios [22]. Muraoka used a honeycomb design and proposed an array-type mechanical amplifier with a more compact structure [23]. Wei used a hybrid flexible hinge to design a bow-and-pin magnification mechanism with multi-stage magnification [24]. Ling et al. designed a two-stage bow-type amplifier with a large range and high frequency and established a displacement amplification ratio formula by defining the impedance factor [25,26]. Zhang et al. designed a bow-type amplifier and established its simplified model [27]. Ye et al. used the matrix method to establish the flexibility matrix of the bow-type amplifier and derived its output displacement formula and amplification formula [28,29]. The theory of micro-displacement amplifiers for valves has developed from simple studies of stress-strain relationships to the comprehensive utilization of the law of energy, Castigliano’s second theorem, vibration theory, and matrix theory, which have much improved theoretical research and made models more precise and accurate. Nevertheless, the amplifier requires too much space, as compared with the actuator, and the space utilization rate is too low [30]. Therefore, the structural design needs to be optimized to improve the space utilization rate.
2. Static Analysis of Double-Row Bow-type Micro-Displacement Amplifier (DBMA) According to the principle of energy conservation, when the output energy W of the GMA is determined, the amplification ratio R provided by the micro-displacement amplifier is larger, and the force Fout at the output end of the amplifier is smaller, as in Equation (1): W = xin Fin = xout Fout + Wl × × xout = xin R (1) × Fout F /R ≤ in where xin and Fin are respectively the displacement and force provided by the GMA, xout is the output displacement of the amplifier, and Wl is the loss during amplification. Therefore, the amplification ratio of the micro-displacement amplifier cannot be unrestricted, and the optimum amplification ratio should be reasonably selected in consideration of practical applications. The basic principle of the bow-type micro-displacement amplifier is the Pythagorean theorem, as shown in Figure1. The three sides of the right triangle are a, b, and c, and the length of the hypotenuse c is constant. When the leg a is elongated by ∆a, the leg b is shortened by ∆b. θ is the angle between the shortest leg and the hypotenuse in the triangle. Micromachines 2019, 10, x 3 of 16
Δb Micromachines 2020, 11, 312 3 of 15 Micromachines 2019, 10, x 3 of 16
Δb c b
θc b Δ a a Figure 1. Amplification method based on the Pythagorean theorem. θ Δ The three sides of the right triangle are a, b, anda c, anda the length of the hypotenuse c is constant. When the leg a is elongated by Δa, the leg b is shortened by Δb. θ is the angle between the shortest FigureFigure 1. AmplificationAmplification method method based based on on the the Pythagorean Pythagorean theorem. leg and the hypotenuse in the triangle. According to Pythagorean theorem TheAccording three sides to Pythagorean of the right theoremtriangle are a, b, and c, and the length of the hypotenuse c is constant. When the leg a is elongated by Δa, (the leg22+= b is shortened 2 by Δb. θ is the angle between the shortest a2ab+ b2 = c c2 (2) leg and the hypotenuse in the triangle. −Δ2 22 + +Δ2 = 2 2 (2) (a()()aa∆a) + (b bb+ ∆b) = cc According to Pythagorean theorem − Ignoring the second orderorder infinitesimal,infinitesimal,22+= thethe amplificationamplification 2 raterate isis ab c (2) ()()−ΔΔab22 + +Δ = 2 aaλθ=≈=∆a b bbtan c λ = Δ = tan θ, ,(3) (3) ∆bba≈ a Ignoring the second order infinitesimal, the amplification rate is When b > aa, ,λλ >> 1.1. Therefore, Therefore, in inthe the amplifier amplifier design design process, process, when when the thehypotenuse hypotenuse is unchanged, is unchanged, the Δab thelong long leg should leg should be elongated, be elongated, and and the theshortλθ short=≈= leg legshould shouldtan be shortened. be shortened. Δ , (3) The bow-type amplifieramplifier isis shownshown inin FigureFigure2ba .2. Four Four oblique oblique sides sidesa a, ,b b, ,c c, ,d d,, and and hypotenuses hypotenusese eand andf f form four right triangles. The length of the hypotenuse a is constant; when the input end elongates, Whenform four b > a right, λ > 1. triangles. Therefore, The in lengththe amplifier of the hypotenusedesign process,a is constant;when the whenhypotenuse the input is unchanged, end elongates, the the leg g of the triangle elongates as well, and the leg h shortens as a result. This means that the output longthe leg legg shouldof the triangle be elongated, elongates and as the well, short and leg the should leg h shortens be shortened. as a result. This means that the output end outputs displacement to both sides, and the ratio of displacement amplification depends on the end outputsThe bow-type displacement amplifier to is both shown sides, in Figure and the 2. ratio Four of oblique displacement sides a, amplificationb, c, d, and hypotenuses depends on e and the angle between the sides a and h. fangle form betweenfour right the triangles. sides a and Theh length. of the hypotenuse a is constant; when the input end elongates, the leg g of the triangle elongates as well, and the leg h shortens as a result. This means that the output end outputs displacement to bothRight sides, Triangle and the ratio of displacement amplification depends on the angle between the sides a and h. h a Output b g Right Triangle f eh a InputOutput b g f Output d e c Input
Figure 2. SimplifiedSimplified diagram of a bow-type amplifier. amplifier. c Output d DBMA is shown in Figure3. In order to increase the amplifier’s amplification ratio and meet the DBMA is shown in Figure 3. In order to increase the amplifier’s amplification ratio and meet the output force requirements simultaneously,simultaneously, thethe twotwo bow-typebow-type amplifiersamplifiersare are used used in in series. series. Figure 2. Simplified diagram of a bow-type amplifier.
DBMA is shown in Figure 3. In order to increase the amplifier’s amplification ratio and meet the output force requirements simultaneously, the two bow-type amplifiers are used in series.
Micromachines 2019, 10, x 4 of 16 Micromachines 2020, 11, 312 4 of 15 Micromachines 2019, 10, x 4 of 16 Micromachines 2019, 10, x l ls 4 of 16 l m10 ml11s m12 l ls me 10 m11 m12c m10 m11 m12 l m e c m13 l 9 e c ll m9 m13 ll m9 m13 m8 m14 m4 lm mm3 8 mm14 m8 m145 m4 lm m4 lm armm3 m5 m3 m5 m6 arm m2 arm flexure hinge m6 m6 m2 m2 flexure hinge flexurem1 hinge m7
m1 m7 m1 m7 Figure 3. Structure of the double-row bow-type micro-displacement amplifier (DBMA).
FigureFigure 3.3. Structure ofof thethe double-rowdouble-row bow-typebow-type micro-displacementmicro-displacement amplifieramplifier (DBMA).(DBMA). In DBMA,Figure a 3.large Structure dimensional of the double-row difference bow-type exists between micro-displacement the flexure amplifier hinge and (DBMA). the arm, and the deformationInIn DBMA,DBMA, concentrates aa largelarge dimensionaldimensional on the flexure didifferencefference hinge. existsexists Therefore, betweenbetween the the thearm flexureflexure can be hingehinge regarded andand thetheas a arm,arm, rigid andand body, thethe In DBMA, a large dimensional difference exists between the flexure hinge and the arm, and the anddeformationdeformation the flexure concentratesconcentrates hinge can be on regarded the flexureflexure as a hinge. hinge.flexible Ther Therefore, rod,efore, or a thin the platearm can fixed be at regarded both ends. as The a rigid working body, deformation concentrates on the flexure hinge. Therefore, the arm can be regarded as a rigid body, principleandand thethe flexureflexure of the hinge flexurehinge cancan hinge bebe regardedregarded is shown asas in a aFigure flexibleflexible 4. rod, rod, oror aa thinthin plateplate fixedfixed atat bothboth ends.ends. TheThe workingworking and the flexure hinge can be regarded as a flexible rod, or a thin plate fixed at both ends. The working principleprinciple ofof thethe flexureflexure hingehinge isis shownshown inin FigureFigure4 .4. principle of the flexure hinge is shown in Figure 4. Δx
Δx Δx Δy Δθ
Δy Δθ M ΔFy Δθ M F M F
Figure 4. The working principle of the flexure hinge.
FigureFigure 4.4. TheThe workingworking principleprinciple ofof thethe flexureflexure hinge.hinge. It shows that, whenFigure the flexure 4. The workinghinge isprinciple subjected of the to flexure the horizontal hinge. force F, a horizontal deformationItIt showsshows Δ that, xthat, will when occur;when the when flexurethe flexure su hingebjected hinge is subjectedto the is momentsubjected to the horizontalM to, a verticalthe horizontal force deformationF, a horizontal force ΔFy, and deformationa horizontal an angle It shows that, when the flexure hinge is subjected to the horizontal force F, a horizontal Δθ∆deformationx will occur;occur. Δ whenx will subjected occur; when to the su momentbjected toM the, a verticalmoment deformation M, a vertical∆ deformationy and an angle Δy∆ andθ will an occur.angle deformation Δx will occur; when subjected to the moment M, a vertical deformation Δy and an angle Δθ willThe occur. force of the arm is shown in Figure5 5.. Δθ will occur. The force of the arm is shown in Figure 5. The force of the arm is shown in Figure 5. FA A
FA A MA FA A h MA MB FB MA h MB FB h MB B FB
Figure 5. The force of the arm. B Figure 5. The force of the arm. B
The two sides of the arm are respectivelyFigure 5. The subjected force of the to arm. the horizontal forces FA and FB, which The two sides of the arm are respectivelyFigure 5. The subjectedforce of the to arm. the horizontal forces FA and FB, which generate the moments MA and MB at the flexure hinges A and B. It is easy to draw conclusions from generate the moments MA and MB at the flexure hinges A and B. It is easy to draw conclusions from the staticThe balancetwo sides theory: of the arm are respectively subjected to the horizontal forces FA and FB, which the staticThe twobalance sides theory: of the arm are respectively( subjected to the horizontal forces FA and FB, which generate the moments MA and MB at the FflexureA = FB hinges A and B. It is easy to draw conclusions from generate the moments MA and MB at the flexure hinges A and B. It is easy to draw conclusions from(4) = the static balance theory: 2FFMABA = 2MB = FA h the static balance theory: · 22M ==⋅MFh (4) FFABA= Since DBMA is a symmetric structure, itsAB= 1/8 model is chosen, as shown in Figure6. FFAB 22M ==⋅MFh (4) ABA==⋅ (4) Since DBMA is a symmetric structure, 22M itsABA 1/8 MFhmodel is chosen, as shown in Figure 6. Since DBMA is a symmetric structure, its 1/8 model is chosen, as shown in Figure 6. Since DBMA is a symmetric structure, its 1/8 model is chosen, as shown in Figure 6.
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Micromachines 2020, 11, 312 5 of 15 Micromachines 2019, 10, x a e 5 of 16 l1 b l2 a me 10 m11 c l2 l1 l1 b l2 m10 m11 c m9 l2 l l1
m9 l d
d Figure 6. DBMA’s 1/8 model.
Since the output displacement ofFigure the GMA6. DBMA’s is smal 1/8 1/8l model. compared to the size of the amplifier, the bending angle Δθ of the flexure hinge is also small, so the chord length causing by the rotation of the Since the output displacement of the GMA is small compared to the size of the amplifier, arm isSince approximately the output displacementequivalent to theof the arc GMAlength. is Thsmalerefore,l compared it can beto seenthe size that of under the amplifier, the action the of the bending angle ∆θ of the flexure hinge is also small, so the chord length causing by the rotation of bendingthe input angle force, Δθ the of horizontal the flexure displacement hinge is also dsmall,x and verticalso the chord displacement length causing dy in 1/8 by DBMA the rotation model of are: the the arm is approximately equivalent to the arc length. Therefore, it can be seen that under the action of arm is approximately equivalent to the arc length. Therefore, it can be seen that under the action of the input force, the horizontal displacementdyhdx=Δ+⋅Δand2 verticalθ displacement dy in 1/8 DBMA model are: the input force, the horizontal displacement dx x and vertical displacement dy in 1/8 DBMA model are: =Δ+⋅Δθ (5) ( dxey 2 ==Δ+⋅Δ+ θ ddyhxx 22∆y h ∆θ · (5)(5) F F Mdy ==Δ+⋅Δ2∆x + e ∆θ Where, Δx= , Δy= and Δθ = dxey are2 respectively· the x-axis and y-axis deformation and Klx K ly Kθ Where, ∆x = F , ∆y = F and ∆θ = M are respectively the x-axis and y-axis deformation and FKlx FKly MKθ =− the rotationWhere, deformationΔx= , Δy of= flexure and hinge, Δθ = hl are12 lrespectively is the height the difference x-axis and between y-axis deformation the two flexure and the rotation deformationK ofK flexure hinge, Kh = l l is the height difference between the two flexure Klx ly θ 1 − 2 hinges, KKlxlx and andK ly arely are respectively respectively the the stiff stiffnessness=− of flexure of flexure hinge hinge on the on xthe-axis x-axis and yand-axis, y-axis, and Kandθ is theKθ the rotation deformation of flexure hinge, hl12 l is the height difference between the two flexure angularis the angular stiffness stiffness of the of supporting the supporting arm. arm. K K K hinges,According lx and to the ly arepseudo-rigid-body respectively the model, stiffness the of deformation flexure hinge of on the the flexure flexure x-axis hinge and can y-axis, be regarded and θ asis the a stretch-bend angular stiffness combinationcombination of the supporting deformation,deformation, arm. which which can can simplify simplify the the flexure flexure hinge hinge as as a a combination combination of of a tensiona tensionAccording spring spring and to and the a torsiona pseudo-rigid-body torsion spring. spring. The The pseudo-rigid-bodymodel, pseudo-rigid-body the deformation model model of ofthe of DBMA flexureDBMA is hingeis shown shown can in bein Figure Figureregarded7. 7. as a stretch-bend combination deformation, which can simplify the flexure hinge as a combination of a tension spring and a torsion spring.a The pseudo-rigid-body modelb of DBMA is shown in Figure 7.
a b
e f d c e a b f d c a b
e f d c e Figure 7. Pseudo-rigid body model of DBMA. f d c Using the influenceinfluence coecoefficientffiFigurecient method, 7.method, Pseudo-rigid as as shown shown body in inmodel Figure Figure of8, DBMA. the8, the sti ffstiffnessness coe coefficientfficient matrix matrix of the of hingesthe hinges at both at both ends ends of the of armthe arm can can be obtained. be obtained. Using the influence coefficient method, as shown in Figure 8, the stiffness coefficient matrix of the hinges at both ends of the arm can be obtained.
Micromachines 2020, 11, 312 6 of 15 Micromachines 2019, 10, x 6 of 16
e l
(a) ω=0，θ=1
e l
(b) ω=1，θ=0
Figure 8. The influence coefficient method. (a) ω = 0, θ = 1 (b) ω = 1, θ = 0. Figure 8. The influence coefficient method. (a) ω = 0, θ = 1 (b) ω = 1, θ = 0. In Figure8a, ω = 0, θ = 1; in Figure8b, ω = 1, θ = 0. So the stiffness coefficient matrix is In Figure 8a, ω = 0, θ = 1; in Figure 8b, ω = 1, θ = 0. So the stiffness coefficient matrix is 12EIz 0 12EIl3z K = 2(2l+0e)EI (6) l3 0 z K = l2 (6) 2(2leEI+ ) 0 z where EIz is the bending stiffness of the flexure hinge, lland2 e are the length of the flexure hinge and db3 the arm respectively, E is the elastic modulus of the material, Iz = 12 is the moment of inertia of the flexurewhere hinge, EIz andis theb bendingis the thickness stiffness of of the the flexure flexure hinge, hinge,d lis and the ethickness are the length of the of amplifier the flexure (also hinge the 3 width of the flexure hinge). = db and the arm respectively, E is the elastic modulus of the material, Iz is the moment of inertia Therefore, the amplification ratio of DBMA is 12 of the flexure hinge, and b is the thickness of the flexure hinge, d is the thickness of the amplifier (also 2 2 the width of the flexure hinge). 4dy 4l(3h + 8l + 4el) R = = (7) Therefore, the amplification ratio of dDBMAx 4 isb2 e + 8b2l + 3ehl
The partial derivatives of l, h, e, b to 4Rd, respectively,43lh()22++84 l is el R ==y (7) 22++ 16(8b2e2l+3db2xeh2+48332beb2el2+32 blb2l3+3 ehle2hl2+12ehl3) ∂R = > 0 ∂l (4b2e+8b2l+3ehl)2 2 2 2 2 2 3 The partial derivatives ∂ Rof l, h12, le(,8 b ehto+ R16,b respectively,hl 4e l +3eh l 8isel ) = − 2 − < 0 ∂h (4b2e+8b2l+3ehl) ()22++ 22 22 3 2 2 +++ 23 2 2 3 (8) ∂R 16∂R 8bel12hl 3( beh4b h+3h l 32+8 bell ) 32 bl 3 ehl 12 ehl => = 2 < 0 ∂e (4b2e+8b2l+3ehl) 2 0 ∂l − 22 32bl(e+2l)(()3483hbe2+8l++2+4 blel) ehl ∂R = < 0 ∂b 2 2 2 222223− (4b e+8b l+3ehl) ∂R 12lbehbhlel() 8+−+− 16 4 3 ehlel 8 =<0 It can be seen from ∂h Equation (8) that22 in++ the positive2 range, R increases as l increases, and decreases ()483be bl ehl as h, e and b increase. The relationship between R and the parameters h, l, e and b, respectively, is shown(8) 223++ in Figure9. ∂R 12hl()438 b h h l l =− <0 ∂ 22 2 e ()483be++ bl ehl 22 ∂R 32bl() e+++ 2 l() 3 h 8 l4 el =− < 0 ∂b ()22++ 2 483be bl ehl
It can be seen from Equation (8) that in the positive range, R increases as l increases, and decreases as h, e and b increase. The relationship between R and the parameters h, l, e and b, respectively, is shown in Figure 9.
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FigureFigure 9. 9.Relationship Relationship between betweenR Rand andh ,hl,, le,, eb., b.
3.3. Kinetic Kinetic Analysis Analysis of of DBMA DBMA SinceSince the the DBMA DBMA is is used used to to drive drive the the servo servo valve valve spool spool to move,to move, it is it necessary is necessary to pay to pay attention attention to theto outputthe output force force change change while while considering considering the amplificationthe amplification ratio. ratio. Known Known as as
F =F =klkl (9)(9) When the output displacement l is a constant, the larger the stiffness k, the larger the output When the output displacement l is a constant, the larger the stiffness k, the larger the output force force F. In order to meet the demand for the output force of the DBMA, the equivalent stiffness k F. In order to meet the demand for the output force of the DBMA, the equivalent stiffness k should be should be increased as much as possible under the demand of the output displacement. increased as much as possible under the demand of the output displacement. The DBMA could be regarded as a mass-spring system, and its vibration equation can be The DBMA could be regarded as a mass-spring system, and its vibration equation can be expressed as expressed as .. MuMu+ +=Ku Ku= 00 (10)(10) ConsiderConsider the the DBMA DBMA as as a a single-degree-of-freedom single-degree-of-freedom system, system, using using its its simplified simplified model, model, as as shown shown inin Figure Figure6. 6. According According to to the the vibration vibration theory, theory, its its natural natural frequency frequency is is
1r K f = e n 1 π Ke (11) fn = 2 me (11) 2π me where Ke and me are respectively the equivalent stiffness and the equivalent mass of the DBMA. where K and m are respectively the equivalent stiffness and the equivalent mass of the DBMA. Accordinge e to the pseudo-rigid body theory, the flexure hinge can be regarded as an ideal According to the pseudo-rigid body theory, the flexure hinge can be regarded as an ideal rotating rotating joint. As shown in Figure7, the overall elastic potential energy of the system is joint. As shown in Figure7, the overall elastic potential energy of the system is 111222 =Δ+Δ+Δ() () ()θ UKxKyK161 lx1 ly 1 θ (12) U = 16 222K (∆x)2 + K (∆y)2 + K (∆θ)2 (12) 2 lx 2 ly 2 θ According to the Castigliano’s second theorem According to the Castigliano’s second theorem uF K ! = ! (13) u θ FM K = (13) θ M ∂U u = where ∂ is the displacement of the arm in the x-axis direction, and θ is the arm rotation = ∂U F where u ∂F is the displacement of the arm in the x-axis direction, and θ is the arm rotation angle. angle.According to Equation (13), the flexure hinge coupling stiffness is According to Equation (13), the flexure hinge coupling stiffness is 3 EbEbd3( de ()+ e+2l2) l Ke = = (14) K e 2 2 2 2 2 4 3 (14) 3232b lbl22+ 16++++ 16eb ebll 2+ 6h 6 hll 22+ 32 32l l 4+ 16 16el el 3
The partial derivatives of K e with respect to l, h, e, b, d, respectively, are
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The partial derivatives of Ke with respect to l, h, e, b, d, respectively, are 3 2 2 2 22 22 2 3 22 4 ∂K Eb d()4161612348348 b e+++++++ b el b l e l eh l el h l l e =− <0 ∂ 22 22 3 22 l3 2 2 2 lblebhllel()162 2++++ 82 32 162 8 3 2 2 4 ∂Ke Eb d(4b e +16b el+16b l +12e l +3eh l+48el +3h l +48l ) 3 = ∂K 32Eb dh() e+ l 2 < 0 ∂l e =−l2(16b2l+8eb2+3h2
FigureFigure 10. 10.Relationship Relationship between betweenKe Kande andh, hl,, el, eb,, bd., d.
TheThe kinetic kinetic energy energy of of the the DBMA DBMA consists consists of of the the vibration vibration along along the thex -axisx-axis and andy -axisy-axis and and the the rotationalrotational kinetic kinetic energy energy around around the the z-axis z-axis of of each each arm. arm. X111414X14 14 =++1 . 2 22. 2 1 ω2 2 T =TmuuJm kxu()+ u y+ J ω kk (16)(16) k xk kkyk k k 2 22==2 k=kk1 11k=1 where m is the mass of each arm (where mmmmmm======m m, where mk is thek mass of each arm (where m1 = m3 = m5 13578101214= m7 = m8 = m10 = m12 = m14, m2 ====m6 = m9 = m13ud), =u x2k = 2dxk is the displacement of each arm in the horizontal direction, mmmm26913), xkkx is the displacement of each arm in the horizontal direction, and and uyk = 2dyk is the displacement of each arm in the vertical direction. ud= 2 is the displacement of each arm in the vertical direction. ykkThe momenty of inertia Jk(k = 1, 3, 5, 7, 8, 10, 12, 14) and the angular velocity of rotation ω of the arm canThe be moment expressed of as inertia Jk(= 1,3,5,7,8,10,12,14) and the angular velocity of rotation ω of the k h i = 1 ( )2 + ( + )2 arm can be expressed as Jk 12 mk l2 l1 l e . − uy (17) ω = q 1 2 2 22 J 4=−++(l2 mlll1) +(()()l+e) le kk− 21 12 Substituting Equation (17) into Equation (16), theu following equation can be obtained. (17) ω = y −++22 1 9 4(ll212 )5 ( le )m4 . 2 T = + m + + m2 + + 2m u (18) R2 4 1 R2 4 2 11 y Substituting Equation (17) into Equation (16), the following equation can be obtained. Therefore, the equivalent mass of the DBMA is 1 925 m Tmmmu=+++ + 4 +2 2 1 229 122 5 m4 11 y (18) me = RR+ 42m1 + + 4 m2 + + 2m11 (19) R2 4 R2 4 2 Therefore, the equivalent mass of the DBMA is
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=+1 925 +++ m4 + mmmme 122 11 (19) RR224 42
MicromachinesAccording2020, to11 ,Equation 312 (14) and (19), the natural frequency of the DBMA is 9 of 15
Ebde3 ()+ 2 l According to Equations (14) and (19), the22 natural++++ frequency 2 22 of the 4 DBMA 3 is = 11Ke 32b l 16 eb l 6 h l 32 l 16 el fn = (20) 2ππm 2 uv19 25 m e u +++mmmEb3d(e+2l) + 4 +2 r t 2 1 2 2 11 1 K 1 R 4 2 2+ R2 + 42 2+ 4+2 3 f = e = 32b l 16eb l 6h l 32l 16el n m (20) 2π me 2π 1 9 2 5 4 2 + 4 m1 + 2 + 4 m2 + 2 + 2m11 The partial derivatives of K e and R toRf n , respectively,R is
The partial derivatives of K and R to f , respectively,3 is ∂f e ()mmEdbeln++2(2) n =>12 0 3222224333 R ∂2fnπ R()32 bl(++++m 161+2 eblm2) √Edb 6 hl(e+2l 32) l 16 elm = > 0 e R 3 2 2 2 2 2 4 3 3 (21) 2πR √(32b l +16eb l+6h l +32l +16el )me 22q (21) ∂ ∂ fn ++2 + 2 + fn 483be4b ble 8b l eh3lehl =>K = 2 2 >0 0 e K22e++l(3h +8l +4el) KeeKl(84)3hl el It can be seen from Equation (21) that the natural frequency fn of the DBMA is positively correlated It can be seen from Equation (21) that the natural frequency fn of the DBMA is positively with the amplification ratio R and the equivalent stiffness Ke, that is, increasing either R or Ke, the natural frequencycorrelated canwith be the increased. amplification ratio R and the equivalent stiffness Ke, that is, increasing either R or Ke, the natural frequency can be increased. 4. Simulation Research on DBMA 4. Simulation Research on DBMA The DBMA was analyzed by finite element method (FEM) software. The values of various parametersThe DBMA are shown was analyzed in Table1. by finite element method (FEM) software. The values of various parameters are shown in Table 1. Table 1. Size of DBMA for simulation. Table 1. Size of DBMA for simulation. Parameters Description Value Parameters Description Value c Width of arms 1,3,5,7,8,10,12,14 10 mm c Width of arms 1,3,5,7,8,10,12,14 10 mm a Width of arms 2,6,9,13 10 mm d a Thickness of the amplifierWidth of (also arms the 2,6,9,13 width of the flexure hinge) 10 mm 10 mm l d Thickness of the amplifierLength of(also the th flexuree width hinge of the flexure hinge) 10 mm 4.41 mm e l LengthLength of of the the flexure arms 2,6,9,11 hinge 4.41 19.90mm mm ls e LengthWidth of the of arms arm 2,6,9,11 11 19.90 mm 9 mm ll ls LengthWidth of of arms arm 2,6,9,13 11 9 mm 45.47 mm E ll ElasticLength modulus of arms of 2,6,9,13 the material 45.47 215mm Mpa E Elastic modulus of the material 215 Mpa
The meshing resultresult isis shownshown in in Figure Figure 11 11.. And And static static deformation deformation simulation simulation results results are are shown shown in Figurein Figure 12 .12.
Figure 11. Meshing result in finite finite element method (FEM) software.
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(a(a) ) (b(b) )
FigureFigure 12. 12. Static Static deformation deformation simulation simulation simulation results results results in in in FEM FEM FEM software. software. software. ( a(a) )(The aThe) The deformation deformation deformation of of DBMA, ofDBMA, DBMA, ( b(b) ) (ThebThe) The stress stress stress distribution distribution distribution of of DBMA. ofDBMA. DBMA.
WeWe selected selectedselected di differentdifferentfferent b and bb andandh for hh for 30for FEM 3030 FEMFEM simulations; simulations;simulations; the theoretical thethe theoreticaltheoretical values valuesvalues and simulation andand simulationsimulation results wereresultsresults compared, were were compared, compared, as shown as as inshown shown Figure in in 13Figure Figure. 13. 13.
(a(a) ) (b(b) )
FigureFigure 13. 13. The TheThe effect eeffectffect of of b b and and h h hon onon R R Rrespectively. respectively.respectively. ( a(a) ( )aTheoretical )Theoretical Theoretical and and and simulation simulation simulation values values values of of R ofR and andR and b b, ,( b(b), ) (TheoreticalbTheoretical) Theoretical and and and simulation simulation simulation values values values of of R ofR and andR and h h. . h.
ItIt can cancan be bebe seen seenseen from fromfrom Figure FigureFigure 13 that 1313 that whenthat whenwhen the thickness thethe thicknessthicknessb of the b flexureb ofof thethe hinge flexureflexure changed, hingehinge the changed,changed, theoretical thethe valuetheoreticaltheoretical of the value amplificationvalue of of the the amplification amplification ratio R fit the simulatedratio ratio R R fit fit valuethe the simulated simulated better, and value value the errorbetter, better, was and and within the the error 7%.error With was was respectwithin within to7%.7%. the With With relationship respect respect to to between the the relationship relationshiph and R, between whenbetweenh > h h 1.2and and mm, R R, ,when when the theoretical h h > > 1.2 1.2 mm, mm, results the the theoretical theoretical were very results results closeto were were the simulationveryvery close close to to results. the the simulation simulation When h results.< results.1.2 mm, When When the h theoreticalh < < 1.2 1.2 mm, mm, valuethe the theoretical theoretical of R differed value value greatly of of R R differed differed from the greatly greatly simulated from from value.thethe simulatedsimulated The simulated value.value. value TheThe simulated reachedsimulated the vava maximumluelue reachedreached point thethe when maximummaximumh = 1.2 mm. pointpoint In when thewhen theoretical hh == 1.21.2 mm. calculation,mm. InIn thethe Rtheoreticaltheoreticalwas negatively calculation, calculation, correlated R R was was with negatively negativelyh. This correlated correlated was because with with when h h. .This This the was wash was because because small, when when the height the the h h was diwasff erencesmall, small, betweenthethe height height the difference difference two flexure between between hinges the wasthe two two also flexure flexure small, andhing hing theeses was deformationwas also also small, small, mainly and and manifestedthe the deformation deformation as the mainly tensilemainly deformationmanifestedmanifested as as of the the the tensile tensile flexure deformatio deformatio hinge. Thenn of of the propertiesthe flexure flexure hinge. ofhinge. the The materialThe properties properties were of takenof the the material intomaterial account were were intaken taken the simulationintointo account account calculation, in in the the simulation simulation but were calculation, calculation, not included but but in were were the theoreticalnot not included included calculation, in in the the theoretical theoretical thus causing calculation, calculation, an error. thus thus causingcausingWe an selectedan error. error.h = 2.6 mm, b = 0.6 mm, re-established the model, applied a uniform force of 0–800 N, and recordedWeWe selected selected the h inputh = = 2.6 2.6 mm, displacementmm, b b = = 0.6 0.6 mm, mm, and re-established re-established output displacement the the model, model, respectively, applied applied a a uniform uniform as shown force force in of of Figure 0–800N, 0–800N, 14. Itandand can recorded recorded be seen fromthe the input input the analysis displacement displacement that the and and input output output displacement displacement displacement change respectively, respectively, of the DBMA as as shown shown was approximately in in Figure Figure 14. 14. linearItIt can can be withbe seen seen the from from output the the displacementanalysis analysis that that the change.the input input Therefore, displacement displacement under change change the uniform of of the the DBMA DBMA force of was was 0–800N, approximately approximately the static deformationlinearlinear with with the the of output output the DBMA displacement displacement remained change. change. nearly Therefor Therefor unchanged,e,e, under under and the the uniform uniform amplification force force of of ratio 0–800N, 0–800N, was the stablethe static static at arounddeformationdeformation 15.3. Comparedofof thethe DBMADBMA with remainedremained the theoretical nearlynearly calculation, unchanged,unchanged, the andand relative thethe amplificationamplification error was 6.06%. ratioratio waswas stablestable atat aroundaround 15.3. 15.3. Compared Compared with with the the theoretical theoretical calculation, calculation, the the relative relative error error was was 6.06%. 6.06%.
MicromachinesMicromachinesMicromachines 20202019 2019,, 11,10 10,, 312,x x 111111 ofof of 1516 16
FigureFigureFigure 14.14. 14. Relationship Relationship between between input input and and output. output.
UsingUsingUsing the the modal modal simulation simulation functionfunction function inin in thethe the FEM FEM software, software, thethe the firstfirst first sixsix six modes modes of of the the DBMA DBMA were were obtainedobtainedobtained asas as shownshown shown inin in FigureFigure Figure 1515. 15..
(a(a) ) (b(b) )
(c(c) ) (d(d) )
(e(e) ) (f(f) )
FigureFigureFigure 15.15. 15. First First six six modemode mode shapes.shapes. shapes. ( a(a) ) firstfirst first order order modemode mode shape,shape, shape, (( b(b) ) secondsecond second orderorder order modemode mode shape,shape, shape, (( c()c) ) thirdthird third orderorderorder modemode mode shape, shape, shape, (d ( )d(d fourth) )fourth fourth order order order mode mode mode shape, shape, shape, (e) fifth( e(e) )fifth orderfifth order order mode mode mode shape, shape, shape, (f) sixth ( f()f order)sixth sixth modeorder order shape.mode mode shape.shape.
Micromachines 2019, 10, x 12 of 16
It can be seen from Figure 15 that the fourth-order mode shape of the DBMA was the same as that of the single-degree-of-freedom model, and other mode shapes caused displacement in other directions. In the FEM simulation results, the frequency of the fourth mode was 314.13 Hz, and the natural frequency is 293.9 Hz in the theoretical calculation. The difference between them was 6.44%. The theoretical calculation was in good agreement with the simulation results. Micromachines 2020, 11, 312 12 of 15 5. Experimental Verification of DBMA According to the theoretical calculation and simulation research, a DBMA-GMA prototype was It can be seen from Figure 15 that the fourth-order mode shape of the DBMA was the same as made and a corresponding test system was built, as shown in Figure 16b. that of the single-degree-of-freedom model, and other mode shapes caused displacement in other As is shown in Figure 16, a Rigol-DG1022U signal generator (Shenzhen Keruijie Technology Co., directions. In the FEM simulation results, the frequency of the fourth mode was 314.13 Hz, and the Ltd., Shenzhen, China) was used to generate the excitation signal, which was amplified by the natural frequency is 293.9 Hz in the theoretical calculation. The difference between them was 6.44%. GF800W power amplifier (Army Engineering University, Shijiazhuang, China) to drive the DBMA- The theoretical calculation was in good agreement with the simulation results. GMA. Displacement was measured using the MicrotrakTM 3-LTS-025-02 laser displacement sensor (MTI5. Experimental Instruments, Verification Inc., Washington, of DBMA USA), which was supplied by the IT6932A programmable voltage source. The input current value was monitored by the PICO-TA189 current clamp and the According to the theoretical calculation and simulation research, a DBMA-GMA prototype was test data were acquired by the PICO scope digital oscilloscope (PicoTechnology Inc, Cambridgeshire, made and a corresponding test system was built, as shown in Figure 16b. UK).
(a) (b)
FigureFigure 16. TestTest System. ( (aa)) Monitoring Monitoring part part of of the the test test system, system, ( (bb)) Measuring Measuring part part of of the the test test system. system.
AAs sinusoidal is shown insignal Figure was 16 ,used a Rigol-DG1022U to excite the signalDBMA-GMA, generator with (Shenzhen the signal Keruijie frequency Technology changing Co., betweenLtd., Shenzhen, 30 Hz and China) 70 Hz was with used 10 to Hz generate as the thegradient excitation and signal,the input which current was amplifiedchanged between by the GF800W 1 A–5 Apower with 1 amplifier A as the gradient, (Army Engineering and the axial University, and transverse Shijiazhuang, displacement China) of the to DBMA-GMA drive the DBMA-GMA. is measured TM byDisplacement the laser displacement was measured sensor. using During the Microtrakthe experiment,3-LTS-025-02 10 measurements laser displacement were made for sensor each point. (MTI TheInstruments, maximum Inc., and Washington,minimum value USA), in the which data waswere supplied removed by during the IT6932A processing, programmable and the mean voltage value ofsource. the remaining The input 8 current data was value taken. was Finally, monitored the bydisplacement the PICO-TA189 response current curve clamp of the and DBMA-GMA the test data underwere acquired the excitation by the of PICO the sinusoidal scope digital signal oscilloscope could be (PicoTechnologyobtained as Figure Inc, 17 Cambridgeshire,shows. UK). AsA sinusoidalcan be seen signal from wasFigure used 17, to both excite the the inpu DBMA-GMA,t and output withdisplacements the signal were frequency approximately changing proportionalbetween 30 Hz to andthe current, 70 Hz with with 10 g Hzood as linearity. the gradient It shows and thethat input in the current range changedof 30–70 Hz, between the dynamic 1 A–5 A performancewith 1 A as the of gradient,the DBMA and was the relati axialvely and stable, transverse and the displacement amplification of ratio the DBMA-GMAfluctuates between is measured 15.43– 16.25.by the The laser error displacement between the sensor. experimental During the results experiment, and the 10 theoretical measurements calculation were made was between for each point.1.6% andThe 3.56% maximum and andwas minimumbetween 2.81% value and in the 8.17% data from were the removed simulation during calculation. processing, and the mean value of theSinusoidal remaining frequency 8 data was sweep taken. signal Finally, of the0–400 displacement Hz is applied response to the curve DBMA-GMA, of the DBMA-GMA and the current under wasthe excitationset as 4A. of After the sinusoidal the measured signal data could were be obtainedsorted out, as Figurethe relationship 17 shows. between frequency and displacementAs can be was seen obtained from Figure as shown 17, bothin Figure the input 18. and output displacements were approximately proportional to the current, with good linearity. It shows that in the range of 30–70 Hz, the dynamic performance of the DBMA was relatively stable, and the amplification ratio fluctuates between 15.43–16.25. The error between the experimental results and the theoretical calculation was between 1.6% and 3.56% and was between 2.81% and 8.17% from the simulation calculation. Sinusoidal frequency sweep signal of 0–400 Hz is applied to the DBMA-GMA, and the current was set as 4A. After the measured data were sorted out, the relationship between frequency and displacement was obtained as shown in Figure 18.
Micromachines 2019, 10, x 13 of 16 Micromachines 2019, 10, x 13 of 16 Micromachines 2020, 11, 312 13 of 15
(a) (b) (a) (b)
(c) (c) Figure 17. Response of sinusoidal excitation. (a) Input displacement & current, (b) Output Figure 17. Response of sinusoidal excitation. (a) Input displacement & current, (b) Output displacement Figuredisplacement 17. Response & current, of ( csinusoidal) Amplification excitation. ratio & (current.a) Input displacement & current, (b) Output displacement& current, (c) & Amplification current, (c) Amplification ratio & current. ratio & current.
Figure 18. DisplacementDisplacement & & Frequency. Frequency. Figure 18. Displacement & Frequency. It can be seen from Figure 18 that the measured natural frequency frequency of of the the DBMA DBMA was was 305.01 305.01 Hz, Hz, withIt an can error be seen of 3.67% from from Figure the 18 theoreticaltheoretical that the measured calculationcalculation natural andand 2.87%2.87% frequency fromfromthe theof the simulationsimulation DBMA was result.result. 305.01 Hz, with an error of 3.67% from the theoretical calculation and 2.87% from the simulation result. 6. Discussion Discussion 6. Discussion In thisthis study, study, we we designed designed a DBMA a DBMA and performed and performed mathematical mathematical modeling, modeling, simulation simulation modeling, modeling,andIn experimental this and study, experimental verification. we designed verification. In thea DBMA experimental In andthe expe verification,perimentalrformed mathematical verification, we chose a larger wemodeling, chose size prototypea simulationlarger size for modeling,prototypeanalysis. There andfor analysis. experimental were two There reasons verification. were for two this reasons choice:In the Oneforexperimental this was choice: to avoid verification, One the was influence to we avoid chose of machining the a influence larger errors size of prototypemachiningon the experimental for errors analysis. on accuracy; the There experimental thewere other two was reasonsaccuracy; that suchfor the this a largeother choice: amplifier was One that was was such requiredto avoida large duethe amplifier influence to the size was of of machining errors on the experimental accuracy; the other was that such a large amplifier was
Micromachines 2020, 11, 312 14 of 15 the giant magnetostrictive actuator. However, we believe that the size of the prototype had no effect on the accuracy of the model for the following reasons:
(a) In mathematical modeling, we did not discuss the size of the model, so mathematical models were established at the micron and millimeter levels. However, it should be noted that the condition for the assumption in Figure4 was that the thickness of the arm needed to be much larger than the flexure hinge, in order to treat the arm as a rigid body, and the flexure hinge as a thin plate with fixed ends to simplify the model. (b) In the simulation analysis, we actually established more than one size model. The simulation results show that when all the size parameters of the model were enlarged or reduced in proportion, there was no effect on the amplification ratio. When only a few parameters were changed, the results changed. These results are shown in Figures9, 10 and 13. (c) Since the size of the actuator of the giant magnetostrictive servo valve reached the millimeter level and an amplifier was used with the actuator, we did not make a smaller prototype for experiments. However, from the results of mathematical modeling and simulation, we believe that such structures and modeling methods would be correct at larger or smaller sizes.
7. Conclusions a. This paper establishes a pseudo-rigid model of DBMA, analyzes the stress form of flexure hinge, and derives the stiffness matrix of flexure hinge by the influence coefficient method. b. A static and kinetic analysis of DBMA was carried out, and a formula of displacement magnification was derived. Using Castigliano’s second theorem, a formula of equivalent stiffness and natural frequency of DBMA were derived. The effects of different parameters on the magnification, equivalent stiffness, and natural frequency were analyzed, respectively. c. A simulation of DBMA was carried out using FEM simulation software. A prototype of DBMA was fabricated, a corresponding test system was built, and experimental research was conducted. Errors of theoretical calculation, simulation results, and experimental data were all within 8%, which supported the correctness of the model. d. Theoretical research and experimental results show that the displacement amplification ratio of DBMA was stable at about 15.5, the natural frequency is about 305Hz to 314Hz, and the response bandwidth was up to 300 Hz, which satisfy the servo valve output requirements.
Author Contributions: Conceptualization, Z.H.; methodology, J.Z. (Jiawei Zheng) and G.L.; software, G.B. and J.Z. (Jingtao Zhou); formal analysis, G.L.; data curation, G.L., G.B. and M.C.; writing—original draft preparation, G.L.; project administration, Z.H. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by the National Natural Science Foundation of China (No. 51275525). Conflicts of Interest: The authors declare no conflict of interest.
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