Sensors and Actuators A 324 (2021) 112687 Contents lists available at ScienceDirect Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna Design and modeling of an improved bridge-type compliant mechanism with its application for hydraulic piezo-valves Mingxiang Ling a,∗ , Jiulong Wang a , Mengxiang Wu a , Lei Cao a , Bo Fu b a b Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, 621900, China School of Mechanical Engineering, Sichuan University, Chengdu, 610065, China a r t i c l e i n f o Article history: Received February 2021 Received in revised form 27 February 2021 Accepted 10 March 2021 Available online 15 March 2021 Keywords: Compliant mechanisms Piezoelectric actuator Mechanical displacement amplifier Flow control valve Dynamic stiffness matrix Smart materials a b s t r a c t Bridge-type compliant mechanisms have been frequently utilized as the micro displacement amplifier for a variety of precision manipulation applications However, the inertial movement of internal actuators (e.g piezo-stacks) limits the system’s dynamic bandwidth in some traditional design This paper presents an improved bridge-type compliant mechanism with double output ports that can generate homodromous bi-motions actuated by only one group of piezo-stacks The inertial motion of piezo-stacks is avoided and the dynamic bandwidth is enhanced The two-port dynamic stiffness model is established to straightforwardly capture its kinetostatic and dynamic characteristics from the perspective of input and output ports The displacement amplification ratio, input stiffness, fundamental frequency and dynamic response spectrum of the improved bridge-type compliant mechanism are curved against key geometric parameters, then the optimal performance can be confirmed Of particular interest for the mechanism application is applying it to develop a new type of piezoelectric two-stage flow control valve with relatively fast dynamic response and large flow rate The inner leakage and oil contamination is effectively overcome in contrast to traditional nozzle-flapper servovalves The presented piezoelectric flow control valve is fabricated and experimentally measured with the step response time of 8.5 ms, frequency bandwidth of 120 Hz, and stroke of ±0.8 mm (corresponding to the flow rate of 180 L/min at the supply pressure of 210 bar) © 2021 Elsevier B.V All rights reserved Introduction Advances in smart materials have resulted in a renewed interest in developing actuators with high control precision, compact structure and low cost for use in micro/nano- manipulations, ultraprecision machining and many other engineering fields [1–3] The main goal is multi-fold to obtain high-frequency response, sufficient stroke, multiple degrees of freedom, fine resolution, and high reliability With this in mind, many researchers have made significant strides in developing all kinds of actuators, such as shape memory alloys, piezoelectric actuators, magnetostrictive transducers, voice coil motors, and so forth [4,5] Among these types, piezoelectrically actuated compliant mechanisms have been preferred for some precision applications due to the following advantages: (a) High energy density and simplicity in structure of piezoelectric actuators with large driving force (Piezo- ∗ Corresponding author E-mail address: ling mx@163.com (M Ling) https://doi.org/10.1016/j.sna.2021.112687 0924-4247/© 2021 Elsevier B.V All rights reserved electric stacks), nano-scale resolution and fast dynamic response (b) The benefits of monolithic structure of flexure-based compliant mechanisms without wear, friction, backlash and reduced requirement of assembly process [6] Strokes of commercially available piezo-stacks are very small with the deformation of about 0.1 % to 0.15 % of their length Therefore, flexure-based amplifying mechanisms are often designed for the application with strokes of several hundreds of microns and even millimeter ranges [7] Except for the functionalities of mechanical amplifying and motion guiding, another purpose of using flexure-based compliant mechanisms for piezoelectric applications is the fact that piezo-stacks can support little tensile or shear force loads, and compliant mechanisms can provide preload and restoring force through elastic deformation, rendering piezo-stacks suitable for use in dynamic applications At present, leveraged [8–11], bridge-type [12–15] and ScottRussell [16,17] compliant mechanisms are frequently used as mechanical displacement amplifiers Some derivative or newly developed amplifying mechanisms have also been proposed [18–23] Although leveraged compliant mechanisms can provide M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 a desirable displacement amplification ratio, they require large spaces Bridge-type compliant mechanisms were originated from the early Moonie-type, Cymbal-type and Rainbow-type amplifying mechanisms [24] and are widely used in precision engineering with a satisfying displacement amplification ratio in a compact size Moreover, the structural symmetry in the bridge-type amplifying mechanisms prevents shearing loads being applied to piezo-stacks One disadvantage of some previous bridge-type compliant mechanisms, however, lies in the inertial motion of internal actuators (e.g piezo-stacks), limiting the system’s dynamic bandwidth [12–15] To this end, we introduce an improved bridge-type compliant mechanism with two output ports in the current study, which can generate homodromous bi-motions in the same direction with only a single group of piezo-stacks With this improved design, the inertial motion of piezo-stacks is avoided, and the frequency bandwidth is consequently enhanced Based on the unique structure of the proposed bridge-type compliant mechanism with double output ports generating bi-motions, a new type of piezoelectrically actuated two-stage four-way hydraulic servovalve is designed, fabricated and tested In comparison to previous four-way servovalves actuated by two or four pairs of piezo-stacks [25,26], only one group of piezo-stacks is required in the current design The controller complexity and cost are reduced to a certain extent In addition, the issue of oil contamination and inner leakage in traditional nozzle-flapper hydraulic servovalves is effectively overcome, allowing lower oil cleaning level with higher reliability This paper progresses as follows: Section describes the configuration of the improved bridge-type compliant mechanism The kinetostatic and dynamic analyses by using a two-port dynamic stiffness model are implemented in Section 3, followed by design and experimental testing for a new type of piezoelectric two-stage servovalve in Section Finally, concluding remarks are provided in Section on the triangle and flextensional principles The leaf-spring flexure hinge is shown in Fig as an example but other flexure hinges, such as circular, corner-filleted, elliptic and hyperbolic profiles [27,28], are also commonly applied for designing bridge-type compliant mechanisms In the traditional bridge-type compliant mechanisms shown in Fig 1(a), one output port is fixed to block while another port is served as the output displacement port In such a traditional configuration, except for the longitudinal expansion deformation, piezo-stacks have an extra transverse inertial movement with a half magnitude of output displacement This inertial movement limits the system’s dynamic bandwidth The bridge-type compliant mechanism shown in Fig 1(b) was presented by Tian, et al [29] for use in designing a micro-gripper by further combining a leveraged amplifying mechanism, which is not specified here One input port of this type of mechanism is fixed to block with two output ports free to generate bi-motions in an opposite direction As can be seen from the deformation nephogram in Fig 1(b), the inertial movement of piezo-stacks is eliminated but the output ports still have parasitic motions along the longitudinal direction An improved bridge-type compliant mechanism is proposed in the current paper shown in Fig 1(c) The proposed improved bridge-type compliant mechanism consists of leaf-spring hinges, rigid links as well as guiding flexible beams fixed to the block Micro-displacement of piezoelectric stacks is mechanically amplified without inertial movement nor parasitic motion by adding four pairs of guiding flexible beams In addition, homodromous bimotions in the same direction can be generated by re-organizing the relative position of flexure hinges in comparison to the configuration in Fig 1(a) and Fig 1(b), which will be utilized to drive a new type of flow control valve in the current study Since the thickness of guiding flexible beams is much smaller than that of other parts, the output displacement of the improved bridge-type compliant mechanism will not be heavily attenuated Design of the improved compliant mechanism Kinetostatic and dynamic analyses Fig compares the schematic configuration of three types of piezoelectrically actuated bridge-type compliant mechanisms These bridge-type compliant mechanisms consist of flexure hinges and rigid links Micro motion of piezo-stacks is amplified based 3.1 Two-port dynamic stiffness modeling In order to size the geometric parameters, the two-port dynamic stiffness model of the improved bridge-type compliant mecha- Fig Comparison of bridge-type compliant amplifying mechanisms (a) Traditional bridge-type compliant mechanism with one output port (b) Derivative bridge-type compliant mechanism with two output ports generating reversed bi-motions for use in designing a micro-gripper in Ref [29] (c) Improved bridge-type compliant mechanism with double output ports generating homodromous bi-motions for designing a flow control piezo-servovalve in the current paper M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Fig Condensed configuration of the improved bridge-type compliant mechanism is the circular frequency Di (ω) = R Ti · De (ω) · R i , here De (ω) is the dynamic stiffness matrix of flexible beams in their local coordinate frame o-xi yi [30]: De (ω) = K − ω2 M − ω4 M − ω6 M where K0 , M1 , M2 , M3 are the static stiffness matrix and the first three-order mass matrices of flexible beam elements, respectively, and the detailed expressions are listed in Appendix Coordinate transformation matrix Ri is determined by the orientation  i of the ith flexible beam with respect to the reference frame o-xy, as shown in Fig 3: Fig Geometric parameters and discretization of the presented bridge-type compliant mechanism ⎡ cos  sin  i i ⎢ − sin  cos  i i ⎢ ⎢ ⎢ 0 Ri = ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 nism is formulated, and the geometric parameters are defined in Fig The mechanism is discretized into flexure hinges, rigid links and lumped mass (output ports) with only kinetic energy Flexure hinges and rigid links are regarded as flexible beams and are denoted serially from elements (1) to (20) connecting with nodes from number to number 16 All the clamped nodes are numbered as The input force actuated by piezo-stacks is denoted as fp As shown in Fig 3, each flexible beam has six degrees of freedom: axial displacements uj and uk , transverse deflections wj and wk , rotations ϕj and ϕk at the two nodes j and k The frequencydependent nodal forces and nodal displacements of the ith flexible beam can be correlated by their dynamic stiffness matrix Di (ω) in the reference frame o-xy [30,31]: F i,k (ω) = Di (ω) · xi,j xi,k = ki,1 ki,2 ki,3 ki,4 · xi,j ⎤ 0 0 0 0⎥ ⎥ 0 cos Âi sin Âi − sin Âi cos Âi 0 ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎦ 0⎥ (3) The value of each orientation  i for the flexible beam elements in Fig is summarized in Table Since the two nodes of the rigid links are not aligned at the central axis, they are equivalent to be flexible beams with an orientation Â=actan(H/l4 ), as shown in Fig On the other hand, the transfer matrix of the ith flexible beam element, which also describes the relationship between nodal force and nodal displacement of flexible beams, can be easily derived based on Eq (1) as: Fig Definition of nodal forces and nodal displacements of the ith flexible beam element with respect to the reference coordinate frame F i,j (ω) (2) xi,k F i,k = = Ti · −k−1 i,2 xi,j F i,j = · ki,1 ki,3 − ki,4 · k−1 i,2 · ki,1 t i,1 t i,2 t i,3 t i,4 xi,j · k−1 i,2 ki,4 · k−1 i,2 F i,j · xi,j (4) F i,j Based on the improved transfer matrix method in [31], the total transfer matrices of four branch limbs from the input ports to the output ports in Fig can be directly derived as: ⎧ T = T · (ST ) · (ST ) · (SQ T ) ⎪ ⎪ ⎪ ⎪ ⎨ T = T 10 · (ST ) · (ST ) · (SQ T ) (1) xi,k ⎪ T = T 15 · (ST 14 ) · (ST 13 ) · (SQ 12 T 11 ) ⎪ ⎪ ⎪ ⎩ where Fi,j (ω)=[Nj ; Qj ; Mj ], Fi,k (ω)=[Nk ; Qk ; Mk ] and xi,j =[uj ; wj ; ϕj ], xi,k =[uk ; wk ; ϕk ] are the two nodal forces and nodal displacements of the ith flexible beam in the reference coordinate frame ω T = T 20 · (ST 19 ) · (ST 18 ) · (SQ 17 T 16 ) (5) M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Table The value of each orientation  i for the flexible beam elements (Unit: Degree) Parameters Values Parameters Values Parameters Values Â1 Â2 Â3 Â4 Â5 Â6 Â7 90 −90 −actan(H/l4 ) 90 −90 Â8 Â9  10  11  12  13  14 180 180+actan(H/l4 ) 180 −90 90 180 180+actan(H/l4 )  15  16  17  18  19  20 180 −90 90 −actan(H/l4 ) In Eq (5), the specific values of Qi (i = 2, 7, 12, 17) can be expressed as [31]: ⎧ I3 ⎪ ⎪ ⎪ ⎪ ⎨Q2 = k 2,4 I3 ⎪ ⎪ ⎪ ⎪ ⎩Q7 = k 7,4 O3 I3 O3 I3 ,Q 12 = ,Q 17 = I3 O3 k12,4 I3 I3 O3 k17,4 I3 where i = 1, 2, 3, is the number of four condensed limbs ti,1 , ti,2 , ti,3 and ti,4 are the block sub-matrices of the condensed transfer matrix Ti in Eq (5) Taking the four port nodes in Fig as the study objects, forces imposed on each node are the summation of the inverse nodal force of its connected equivalent limbs, the inertial force of the node itself if it is a lumped mass and external forces according to d’Alembert’s principle Therefore, the following force equilibrium equation sets can be established: (6) ⎧ f i1 = F 1,j + F 4,j ⎪ ⎪ ⎪ ⎪ ⎨ f i2 = F 2,j + F 3,j where ki,4 (i = 2, 7, 12, 17) are the last three rows and three columns in Di (ω) I3 and O3 are respectively the unit and zero matrices with the dimension of × Based on the above condensation, the improved bridge-type compliant mechanism in Fig can be further equivalent as the two-port mechanical model, as shown in Fig The equivalent relationship between the nodal force and nodal displacement of the condensed limbs can be obtained again based on Eq (5): F i,j F i,k = i = D (ω) · −t −1 i,2 xi,j xi,k · t i,1 t i,3 − t i,4 · t −1 · t i,1 i,2 = ki,1 ki,2 ki,3 ki,4 t −1 i,2 t i,4 · t −1 i,2 · · xi,j f o2 = F 3,k + F 4,k + M · xo2 in which, fo1 and fo2 are dummy forces assumed at the output ports The dynamic stiffness matrix M of the output ports can be calculated as follows [30,31]: ⎡ xi,j xi,k (8) ⎪ f o1 = F 1,k + F 2,k + M · xo1 ⎪ ⎪ ⎪ ⎩ ⎢ M(ω) = ⎣ (7) −mω2 0 −mω2 0 −0 · ω2 xi,k ⎤ ⎥ ⎦ (9) where m is the mass of the output ports Fig Comparison of the static and dynamic performances with the presented two-port dynamic stiffness model and finite elemental results (Four sets of FEM with H = 0.2 mm, 0.5 mm, mm and mm) M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Table Key geometric parameters of the improved bridge-type compliant mechanism Parameters Values Parameters Values h1 (mm) h2 (mm) h3 (mm) h4 (mm) m (g) 6.0 0.3 0.5 4.0 1.88 l1 (mm) l2 (mm) l3 (mm) l4 (mm) d (mm) 25 10 4.0 10 15 By substituting Eq (7) into Eq (8), there has: ⎧ f i1 = (k1,1 xi1 + k1,2 xo1 ) + (k4,1 xi1 + k4,2 xo2 ) ⎪ ⎪ ⎪ ⎪ ⎨ f i2 = (k2,1 xi2 + k2,2 xo1 ) + (k3,1 xi2 + k3,2 xo2 ) ⎪ f o1 = (k1,3 xi1 + k1,4 xo1 ) + (k2,3 xi2 + k2,4 xo1 ) + M · xo1 ⎪ ⎪ ⎪ ⎩ (10) f o2 = (k4,3 xi1 + k4,4 xo2 ) + (k3,3 xi2 + k3,4 xo2 ) + M · xo2 Fig Theoretical prediction of the displacement amplification ratio of the two output ports by offsetting the node position of the input ports Rewriting Eq (10) as the form of matrix, the two-port dynamic stiffness model of the improved bridge-type compliant mechanism can be regularly expressed as: ⎧ f i1 ⎪ ⎪ ⎪ ⎪ ⎨ f i2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ f o1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ f o2 ⎡ k1,1 + k4,1 k1,2 k4,2 k2,1 + k3,1 k2,2 k3,2 ⎣ k1,3 k2,3 k1,4 + k2,4 + M k4,3 k3,3 k4,4 + k3,4 + M ⎢ ⎢0 =⎢ ⎢ · ⎤ ⎧ xi1 ⎪ ⎪ ⎪ ⎪ ⎨ xi2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ xo1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎥ ⎥ ⎥ ⎥ ⎦ (11) xo2 Fig Picture of the improved bridge-type compliant mechanism actuated by piezo-stacks where fi2 =-fi1 =[fp ; 0; 0] and fo1 =fo2 =0 are the input and output forces fp is the input force actuated by piezo-stacks xi1 , xi2 , xo1 and xo2 are the input and output displacements quency response of output displacements can be calculated and one is curved in Fig 5(c) versus frequency f with geometric parameter H = 0.5 mm Similarly, the fundamental frequency against different geometric parameter H can be easily obtained by checking the peaks of the frequency response curve, as shown in Fig 5(d) One can see that the theoretical model well describes the change trend of the static and dynamic performances with respect to the finite elemental results The optimal geometric parameter H = 0.5 mm can be straightforwardly confirmed from these curves to fabricate the prototype In addition, it can be seen from Fig 5(a) that the displacements of the two output ports become inconsistent with the increase of H due to the transverse asymmetry of the improved bridge-type compliant mechanism This discordance can be relieved by adjusting the position of piezo-stacks in actual applications In order to support this viewpoint, the displacements of the two output ports were theoretically calculated with the same process as that in Fig 5(a) but by offsetting the node position of the input ports During calculation, the node position of input ports can be regarded as the force points of piezo-stacks From the results in Fig 6, the displacements of the two output ports become equal in the domain of geometric parameter H = 0.5 mm when the offset of the node position is 0.2 mm 3.2 Parameter influence analysis Actuating forces with the magnitude of 100 N and the frequency range from Hz to 1000 Hz were exerted on the input ports in the x-direction to calculate the static and dynamic performances The material was selected as spring steel with Young’s modulus of 200 GPa and density of 7850 kg/m3 The geometric parameters are listed in Table Four sets of finite element simulation with the commercial software package ANSYS Workbench15.0 were employed to verify the two-port dynamic stiffness model The Solid186 element was chosen to build the model and the advanced size function of proximity and curvature was adopted to refine the element The results were proven to be convergent and accurate enough The static analysis and harmonic analysis in ANSYS Workbench15.0 were carried out to evaluate the static and dynamic performances and then compared with the theoretical results When setting frequency ω = rad/s, the displacement amplification ratio R=xo /xi and input stiffness Kin =fi /xi can be calculated by solving the linear equation sets of Eq (11) Those two static indexes were plotted versus geometric parameter H shown in Fig 5(a) and (b) Moreover, starting from the initial value of frequency f = Hz (ω = 2f) and incrementing step by step with f = Hz, the fre5 M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Fig Experimental results of the static and dynamic performances of the improved bridge-type compliant mechanism (a) Output displacement under frequency of Hz (b) Frequency response curve Fig Schematic diagram and three-dimensional view of the presented piezoelectric two-stage servovalve based on the improved bridge-type compliant amplifying mechanism Piezo-stacks (model NAC5023-H40) with the axial stiffness of 150 N/m was adopted as the micro motion generator The mechanism was fabricated with the Invar steel and by the wire cut electrical discharge machining technique Fig provides the prototype with the assembled piezo-stacks, whose dimension is 70 mm × 64 mm × 15 mm From the experimental results in Fig 8, the output displacement of 0.24 mm for one output port can be obtained, and the first-order resonance frequency of the improved bridge-type compliant mechanism was measured as 880 Hz (including piezostacks) In addition, hysteresis error of piezoelectric material can be observed in Fig 8(a), and this type of error can be further compensated by some control strategies [32] but is not the emphasis of the current study On the other hand, the theoretical prediction is in good agreement with the finite element simulation with small errors as shown in Fig.5, while the larger discrepancy between the theoretical results and experimental testing comes from the displacement attenuation effect of piezo-stacks under a spring load and prototyping errors, such as mismatching of structural parameters, machining errors, misalignment and assembling errors Two output ports of the improved bridge-type compliant mechanism are directly connected to the two pilot sliding spools The input voltage generates linear deformation of piezo-stacks and the amplified homodromous motion moves the two identical pilot spools sliding back and forth, opening one orifice (port a or port b) to oil supply port P while simultaneously connecting another orifice to oil return port T This generates a pressure difference and force imbalance across the main spool that causes it to move back and forth The sliding movement of the main spool opens or overlays the load control orifice windows A and B on sleeve and thus modulates the fluid direction and flow rate It should be noted that the elastic deformation of compliant mechanism provides a restoring force for dynamic motion of pilot spool and no extra spring is needed in the current design The design with sliding spools also promises to be less oil contamination and low leakage In comparison to previous four-way servovalves actuated by two or four pairs of piezo-stacks [25,26], only a group of piezo-stacks is required avoiding the complexity of controller thanks to the bi-motions of the improved bridge-type compliant mechanism Application for designing a piezo-valve 4.2 Prototype and experimental setup 4.1 Conceptual design The spool and sleeve of the proposed piezo-valve are mode of Cr12MoV steel, and the prototype is shown in Fig 10 The piezoelectric servovalve was fixed to the valve test rig to evaluate its static and dynamic performances, and the experimental setup is shown in Fig 11 Anti-wear hydraulic oil (Mode 46) was used as the working medium The flow characteristics were recorded and analyzed with a data recorder (Yiheng Inc., China) A power amplifier with the maximum output voltage of 200 V developed in our lab was A two-stage four-way proportional flow control valve with the pilot stage being two sliding spools driven by the improved piezoelectric bridge-type compliant mechanism is designed, as schematically illustrated in Fig The second stage employs the hydraulic amplifier for a large flow rate, which has been widely adopted in traditional nozzle-flapper, jet-deflector or direct-drive hydraulic servovalves [33,34] M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Fig 12 Measured results of step response of the main spool Fig 10 Prototype of the presented piezoelectric servovalve Fig 13 Measured results of the dynamic bandwidth of the main spool time reaching 60 % and 100 % of the maximum opening orifice (corresponding to a maximum spool position of ±0.8 mm) is respectively 5.9 ms and 8.5 ms The fast response time in a matter of microseconds is comparable with typical smart-material based nozzle-flapper and direct-drive flow control valves in literature [25,26,33,34], but with the advantage of having larger strokes (larger flow rate) A second experiment was performed to show the dynamic behavior of the designed two-stage piezo-valve in frequency domain In this case, sine sweep voltages ranging from Hz to 150 Hz were applied to the piezo-stacks As shown in Fig 13, the dynamic bandwidth with the supply pressure of 30 bar can be measured as about 120 Hz under the amplitude attenuation of -3 dB Conclusions This paper describes the detailed design, modeling and analysis of an improved bridge-type compliant mechanism to amplify the micro stroke of piezoelectric stacks with enhanced dynamic bandwidth The kinetostatics and dynamics of the presented bridge-type compliant mechanism are formulated and analyzed by using the two-port dynamic stiffness model The improved bridge-type compliant mechanism is further applied to design a piezo-actuated two-stage four-way hydraulic servovalve With the unique structure of the improved bridge-type compliant mechanism with double output ports, only one group of piezo-stacks is required avoiding the complexity of controller and reducing the cost Measuring results show the step response time of 5.9 ms and 8.5 ms (60 % and 100 % of the maximum stroke), as well as the frequency bandwidth of 120 Hz (-3 dB) at the supply pressure of 30 bar Fig 11 Experimental setup for evaluating the static and dynamic performances utilized to drive the piezo-stacks Displacements of the main spool was measured by a LVDT displacement sensor (UM-375, Univo Sensor Inc.) and then feedback to a controller for tracking an opening orifice of ±0.8 mm All tests were intended to be performed under no-load conditions 4.3 Measured results Fig 12 shows the step response of the main spool at a supply pressure of 30 bar It is shown that the mechanical response M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 Author statement of high-order mass matrices, possible readers are recommended to refer to our previous study [30] ⎡ Mingxiang Ling: Conceptualization, Modeling, Original draft preparation, revision Jiulong Wang: CAD-Drawing, Design of the compliant mechanism Mengxiang Wu: Design of the piezo-valve, Experimental measuring Lei Cao: Software and numerical validation Bo Fu: Reviewing and Editing All authors read and contributed to the manuscript ⎢ ⎢ ⎢ E ⎢ K0 = ⎢ l ⎢ ⎢ ⎢ ⎣ Al2 0 −Al2 0 12I 6Il −12I 6Il −6Il 2Il2 Al2 4Il2 Sym 12I ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −6Il ⎦ 4Il2 ⎡ ⎢ ⎢ ⎢ Al ⎢ M1 = ⎢ 420 ⎢ ⎢ ⎣ Declaration of competing interest No conflict of interest exits in the submission of this manuscript, and the manuscript is approved by all authors for publication I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part 140 0 70 156 22l 54 13l 4l2 Sym ⎤ −13l ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 140 0 ⎥ 156 −22l ⎦ −3l2 4l2 ⎡ EA˛4 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ M2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Acknowledgment The authors acknowledge the research projects funded by the National Natural Science Foundation of China [grant number 52075179], the Applied Basic Research Program of Science and Technology Department of Sichuan Province of China [grant number 20YYJC0312], and the Presidential Foundation of China 45l ⎤ 0 7EA˛4 360l 0 59EIˇ8 161700l3 223EIˇ8 2910600l2 1279EIˇ8 3880800l3 − 1681EIˇ8 23284800l2 71EIˇ8 4365900l 1681EIˇ8 23284800l2 − 1097EIˇ8 69854400l EA˛4 45l 0 59EIˇ8 161700l3 − Sym 223EIˇ8 2910600l2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 71EIˇ8 4365900l ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ M3 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2EA˛6 945l ⎤ 31EA˛6 15120l 0 3547EIˇ12 23837814000l2 5801EIˇ12 8475667200l3 112631EIˇ12 − 76810048000l2 127EIˇ12 3972969000l 112631EIˇ12 76810048000l2 2EA˛6 945l 0 551EIˇ12 794593800l3 551EIˇ12 794593800l3 Sym ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 899EIˇ12 ⎥ − 28252224000l ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 12 3547EIˇ ⎥ − ⎥ 23837814000l2 ⎥ ⎦ 12 127EIˇ 3972969000l Appendix B Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.sna.2021 112687 Academy of Engineering Physics [grant number YZJJLX2019008], for the financial support in carrying out this work They are also thankful to all members of Micro Actuator and Sensor Group for helpful discussion and support References [1] N.S Kumar, R.P Suvarna, K.C.B Naidu, et al., Piezoelectric actuators and their applications Actuators: Fundamentals, Principles, Mater Appl (2020) 1–16 [2] M.X Ling, L.L Howell, J.Y Cao, et al., Kinetostatic and dynamic modeling of flexure-based compliant mechanisms: a survey, Appl Mech Rev 72 (3) (2020) [3] P Mercorelli, N Werner, An adaptive resonance regulator design for motion control of intake valves in camless engine systems, Ieee Trans Ind Electron 64 (4) (2017) 3413–3422 [4] J Tang, H Fan, J Liu, et al., Suppressing the backward motion of a stick–slip piezoelectric actuator by means of the sequential control method (SCM), Mech Syst Signal Process 143 (2020), 106855 Appendix A For flexible straight beams, the expressions of static stiffness matrix and the first three-order mass matrices in Eq (2) are listed as follows, where E is the Young’s modulus, is the density, l is the length, h and d are in-plane and out-of-plane thickness, A=h·d, and I= h3 ·d/12 are the area and moment of inertia about the neutral axis of the cross-section, ˛2 = l2 /E and ˇ4 = l4 A/EI For more details M Ling, J Wang, M Wu et al Sensors and Actuators A 324 (2021) 112687 [5] P Mercorelli, N Werner, Integrating a piezoelectric actuator with mechanical and hydraulic devices to control camless engines, Mech Syst Signal Process 78 (2016) 55–70 [6] S Iqbal, A Malik, A review on MEMS based micro displacement amplification mechanisms, Sens Actuators A Phys 300 (2019), 111666 [7] J Hricko, Sˇ Havlík, Compliant mechanisms for motion/force amplifiers for robotics[C] International Conference on Robotics in Alpe-Adria Danube Region, Springer, Cham, 2019, pp 26–33 [8] W Chen, J Qu, W Chen, et al., A compliant dual-axis gripper with integrated position and force sensing, 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Mechanical Engineering in Xi’an Jiaotong University and received the PhD degree in 2019 From 2017 to 2018, He was a visiting scholar with the Compliant Mechanisms Lab at the Department of Mechanical Engineering, Brigham Young University, USA He is now an associate research fellow at China Academy of Engineering Physics His main research interests include compliant mechanisms, mechanical dynamics, piezoelectric actuator and sensor (acoustic sensing) Jiulong Wang received the Bachelor degree from Sichuan University, China, in 2015 and the Master degree in mechanical engineering from Xi’an Jiaotong University, China, in 2018 He is now an engineer fellow at China Academy of Engineering Physics His research interests include dynamic test and evaluation, nondestructive testing and evaluation of mechanical structure Mengxiang Wu received the Bachelor degree in the Department of Mechanical Engineering, Three Gorges University, China, in 2015, and the Master degree in the Department of Mechanical Engineering from Shantou University, China, in 2019 He is now an assistant engineer at China Academy of Engineering Physics His main research interests include piezoelectric actuator, mechanical optimization design Lei Cao received his Bachelor and Master degrees in the Department of New Energy, from China University of Petroleum(East China), in 2016 and 2020, respectively He is now an assistant Engineer at China Academy of Engineering Physics His main research interests include precision control for piezoelectric actuator, mechanical dynamics Bo Fu received the Bachelor degree in 1991 and the Master degree in 1994 from Sichuan University, Chengdu, China, and the PhD degree in 2005 from Paderborn University, Germany, all in mechanical engineering In 1994, he joined Sichuan University, where he is currently a Professor with the School of Mechanical Engineering His current research interests include fluid power transmission and control, piezoelectric systems and ultrasonic technologies, and multi-objective optimization ... improved bridge- type compliant mecha- Fig Comparison of bridge- type compliant amplifying mechanisms (a) Traditional bridge- type compliant mechanism with one output port (b) Derivative bridge- type compliant. .. smaller than that of other parts, the output displacement of the improved bridge- type compliant mechanism will not be heavily attenuated Design of the improved compliant mechanism Kinetostatic and. .. controller thanks to the bi-motions of the improved bridge- type compliant mechanism Application for designing a piezo- valve 4.2 Prototype and experimental setup 4.1 Conceptual design The spool and sleeve