Research On Mechanical Properties Of Castigliano’s Second Theorem And Elastic Beam Theory Based 2-DOF Compliant Positioning Stage

As the basis of many advanced manufacturing industries,precision positioning technology is widely used in biological cell manipulation,micro/nano assembly,optical fibers alignment,nano-imprint lithography,scanning probe microscope,and fast tool servo.The precision positioning stages based on the compliant mechanisms and piezoelectric ceramic actuators have gradually become the focus of precision engineering because it can provide nanometer-scale step displacement and fast frequency response.According to the requirements of high precision,high frequency response and large stroke in two-dimensional precision positioning field,two kinds of decoupled 2-DOF(degrees-of-freedom)compliant positioning stages based on bridge-type amplification mechanism and piezoelectric ceramic actuators were designed.Based on the structural design,the statics models of the compliant positioning stages were established by the Castigliano’s second theorem and the Elastic beam theory,and the dynamics model was performed utilizing the Lagrangian equation.Theoretical model was verified by FEA(finite element analysis)and prototype experiments,and the results show that the proposed kinematic amplification modeling method had higher accuracy and was better than most typical modeling methods at present.The main research was as follows.Firstly,by analyzing the stiffness characteristics of the flexible hinges of different notch types and the coupling characteristics of the parallelogram mechanisms,the 4-PP configuration(P represents the prismatic joint),bridge-type amplification mechanisms,guiding mechanism composed of leaf-type flexible beams,and piezoelectric ceramic actuators were selected to complete the design of two kinds of compliant positioning stages according to the design requirements and 2-DOFs rigid mechanisms.Secondly,the statics model of the bridge-type amplification mechanism and the stiffness model of the guiding mechanism were established by the Castigliano’s second theorem and the Elastic beam theory.The two models were combined according to Newton’s third law,and the statics model of the entire stages was finally obtained.To analyze the natural frequency of the positioning stages,dynamics model was performed based on the Lagrangian equation.Thirdly,the FEA of the bridge-type amplification mechanism was conducted by ANSYS.The FEM(finite element model),the proposed model and the typical models in the existing literatures were compared and the comparison results demonstrated that the kinematic amplification model proposed in this paper features the smallest error.When the interval w of the flexible hinges on both sides of the connecting beams was taken as the independent variable,the error of the proposed model at the peak of amplification ratio with respect to the FEM is 6.76%,which is lower than the most presented typical models.Fourthly,the FEA was performed on the static and dynamic characteristics of the two kinds of compliant positioning stages such as displacement amplification ratio,input/output stiffness(compliance),maximum stress,buckling failure,decoupling performance and natural frequency.The FEA results were compared with the theoretical model established and the results show that the two kinds of designed stages can work stably in the range of elastic deformation corresponding to theoretical analysis.Based on the results of FEA,the applicable occasions of the two stages were discussed,and the concept of impedance factor was introduced to reveal the essence that the guiding mechanism has a loading effect on the amplification ratio of the bridge-type amplification mechanism.Finally,the experimental testing system was set up to conduct experiments on decoupled 2-DOF compliant positioning stages.The prototypes were processed by EDM technology,and their displacement amplification ratios,cross-coupling errors,and workspaces were tested experimentally.In addition,the experimental results,the FEA results and the theoretical models were compared,which effectively verified the correction of the theoretical models with the causes of errors analysis.