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Simulation of Motion Interactions of a 2-DOF Linear Piezoelectric Impact Drive Mechanism with a Single Friction Interface

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Simulation of Motion Interactions of a 2-DOF Linear Piezoelectric Impact Drive Mechanism with a Single Friction Interface applied sciences Article Simulation of Motion Interactions of a 2-DOF Linear Piezoelectric Impact Drive Mechanism with a Single Friction Interface Haojie Xia †, Liling Han *,†, Chengliang Pan, Huakun Jia and Liandong Yu * School of Instrument Science and Opto-electronics Engineering, Hefei University of Technology, Hefei 230009, China; [email protected] (H.X.); [email protected] (C.P.); [email protected] (H.J.) * Correspondence: [email protected] (L.H.); [email protected] (L.Y.); Tel.: +86-551-6290-1780 (L.H.) † Haojie Xia and Liling Han contributed equally to this work. Received: 20 July 2018; Accepted: 16 August 2018; Published: 19 August 2018 Featured Application: This study provides theoretical and technical references for the practical development of a parallel piezoelectric impact drive mechanism. A controllable complex curve motion can be realized by the method of using coupled motions. Abstract: A two-degrees-of-freedom (2-DOF) linear piezoelectric impact drive mechanism (PIDM) is actuated by two independent piezoelectric actuators (PAs). The coupled motion interactions of a two orthogonal DOF linear PIDM with a single friction interface are introduced and analyzed. A complete dynamic model of the 2-DOF PIDM is established with the Karnopp friction model considering the distribution of friction in the x-axis and y-axis. The output displacements of the 2-DOF PIDM and two corresponding independent 1-DOF PIDMs are investigated numerically. When the two input exciting signals of a 2-DOF PIDM have the same driving voltage of 100 V with a duty ratio of 98% at 10 Hz and two 1-DOF PIDMs are driving under the same conditions, the step displacements in the two axes of 2-DOF PIDM are improved compared to the corresponding 1-DOF PIDM. When the two input exciting signals of a 2-DOF PIDM have the same driving voltages of 100 V with a duty ratio of 98% but the driving frequency is 10 Hz in the x-axis and 20 Hz in the y-axis, the results show that the displacement of high frequency achieves a slight decrease and displacement of low frequency shows a large increase compared to the two corresponding 1-DOF PIDMs. Keywords: piezoelectric; coupled motion; simulation; motion interaction 1. Introduction Because piezoelectric impact drive mechanisms (PIDMs) have a compact structure, high positioning accuracy, and a long stroke, they have been widely used in the fields of precision positioning [1,2]. Multiple-degrees-of-freedom (DOF) motions of stages and manipulators with PIDMs can achieve flexible and wide control in actuators. Multi-DOF motions of PIDM can be designed in two structures. The first kind of structure is the complex structure in series [3–5], for which the motions are independent and their corresponding controls are relatively simple. The second kind of structure is the compact structure in parallel [6–9], for which the motions often share a common friction interface [6,8] that leads to a coupled motion when the motions are operated at the same time [8,10]. Recent studies on parallel PIDMs have attained multi-DOF motions by using different modes of vibration under ultrasonic driving [8,11–15], and studies on series PIDMs have achieved multi-DOF motions by avoiding coupled motions [4,5,16]. However, there are few studies on the coupled motion characteristics of the parallel impact PIDM [17,18]. Appl. Sci. 2018, 8, 1400; doi:10.3390/app8081400 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 10 achieved multi-DOF motions by avoiding coupled motions [4,5,16]. However, there are few studies Appl.on the Sci. coupled2018, 8, 1400 motion characteristics of the parallel impact PIDM [17,18]. 2 of 10 Establishing the dynamic model of parallel multi-DOF PIDMs is foundational to understanding coupled motion interactions and estimating the influences on motion performances. PIDMs are usuallyEstablishing combined the by dynamicthree basic model elements: of parallel piezoelectric multi-DOF actuator PIDMs (PAs), is foundational a stage, and to understanding a slider. The workingcoupled motionprocess interactions is a stick-slip and motion estimating with fricti the influenceson actuation on motionunder a performances. sawtooth driving PIDMs voltage are usually signal [19].combined In the by dynamic three basic analysis, elements: the PA piezoelectric and stage actuatorare simplified (PAs), as a stage,a spring–mass–damper and a slider. The workinglumped processmodel, and is a the stick-slip slider is motion considered with as friction a mass. actuation The motion under of athe sawtooth slider relies driving on the voltage friction signal between [19]. theIn the stage dynamic and slider analysis, [20]. The the PAfriction and force stage is are nonlinear, simplified so as an a accurate spring–mass–damper friction model lumpedis important model, to studyand the coupled slider is motion considered interactions. as a mass. Many The motionfriction of models, the slider such relies as onthe theCoulomb, friction betweenKarnopp, the LuGre, stage Leuven,and slider and [20 elastoplastic]. The friction friction force models, is nonlinear, have sobeen an widely accurate used friction in the model friction is importantanalysis of to PIDMs study [21–25].coupled The motion Coulomb interactions. model Manyis a basic friction static models, friction such model as the that Coulomb, cannot describe Karnopp, friction LuGre, when Leuven, the relativeand elastoplastic speed is zero. friction The models,Karnopp have model been is an widely improved used static in the friction friction model analysis that ofprovides PIDMs a [simple21–25]. Themanner Coulomb to avoid model the detection is a basic of static zero frictionrelative modelvelocity that by cannotdefining describe a zero velocity friction interval. when the Dynamic relative frictionspeed is models, zero. The such Karnopp as such model as theis LuGre, an improved Leuven static, and elastoplastic friction model models, that provides have high a simple accuracy manner and canto avoid explain the detectionthe presliding, of zero Stribeck, relative and velocity hysteretic by defining effects, a zeroalthough velocity the interval. definitions Dynamic of the frictionmodel parametersmodels, such are as difficult such as [26–30]. the LuGre, The Leuven,Karnopp and friction elastoplastic model is models,used for haveanalysis high when accuracy the precision and can isexplain low. the presliding, Stribeck, and hysteretic effects, although the definitions of the model parameters are difficultIn this [paper,26–30]. a The2-DOF Karnopp parallel friction linear model PIDM is is used considered for analysis as the when research the precision object. A is simplified low. modelIn for this the paper, orthogonal a 2-DOF motions parallel of linear the PIDMPIDM isis considered established as and the a research two-dimensional object. A simplifiedextended Karnoppmodel for thefriction orthogonal model motionsis introduced. of the PIDM Interactio is establishedns of the and orthogonal a two-dimensional motions extended are investigated Karnopp numericallyfriction model by is si introduced.mulations [31]. Interactions of the orthogonal motions are investigated numerically by simulations [31]. 2. Structure Design, Working Process, and Dynamic Model Analysis 2. Structure Design, Working Process, and Dynamic Model Analysis The proposed 2-DOF linear PIDM is shown in Figure 1a and is made up of two orthogonal PAs, The proposed 2-DOF linear PIDM is shown in Figure1a and is made up of two orthogonal PAs, a stage, and a slider. According to present studies, the 2-DOF linear PIDM can be regarded as two a stage, and a slider. According to present studies, the 2-DOF linear PIDM can be regarded as two spring–mass–damper systems. The simplified system of the PIDM is shown in Figure 1b, where mpx spring–mass–damper systems. The simplified system of the PIDM is shown in Figure1b, where mpx is is the equivalent mass of the PA and stage in the x-axis, xp is the distance of the PA in the x-axis, xs is the equivalent mass of the PA and stage in the x-axis, xp is the distance of the PA in the x-axis, xs is the the distance of the slider in the x-axis, kx is the stiffness of the PA and stage in the x-axis, cx is the distance of the slider in the x-axis, kx is the stiffness of the PA and stage in the x-axis, cx is the damper damper of the PA and stage in the x-axis, Fpx(t) is the equivalent driving force of the PA in the x-axis, of the PA and stage in the x-axis, Fpx(t) is the equivalent driving force of the PA in the x-axis, fx is the fx is the friction force in the x-axis, msx is the mass of the slider in the x-axis, mpy is the equivalent mass friction force in the x-axis, msx is the mass of the slider in the x-axis, mpy is the equivalent mass of the of the PA and stage in the y-axis, yp is the distance of the PA in the y-axis, ys is the distance of the PA and stage in the y-axis, yp is the distance of the PA in the y-axis, ys is the distance of the slider in the slider in the y-axis ky is the stiffness of the PA and stage in the y-axis, cy is the damper of the PA and y-axis ky is the stiffness of the PA and stage in the y-axis, cy is the damper of the PA and stage in the stage in the y-axis, Fpy(t) is the equivalent driving force of the PA in the y-axis, fy is the friction force y-axis, Fpy(t) is the equivalent driving force of the PA in the y-axis, fy is the friction force in the y-axis, in the y-axis, and msy is the mass of the slider in the y-axis [32,33].
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