An New Elliptic Integral Solution To The Multiple Inflection Points Of Large Deflection Beams In Compliant Mechanisms

The compliant mechanism is a new type of mechanism to transform force,movement and energy through the elastic deformation of its member. Compared withtraditional rigid mechanisms, compliant mechanisms have many advantages such asreduced noise and friction of kinematic pairs, improved precision, reducedmanufacturing and operating cost, reduced maintenance. Therefore, compliantmechanisms have attracted widespread interest from scholars and became the newfield of research worldwide.Due to the large deflection with geometric nonlinearity in the beam, the analysisof deflection beam is usually complex. The research of end angle in the range of[-πφ+,φ](φ denotes the direction of the end force) becomes relatively mature atpresent. For the end angle over the interval [-πφ+,φ](φ of beam which has aninflection point and multiple inflection points, current methods such as the finiteelement method and Adomian decomposition method are time-consuming and lowefficiency, the elliptic integral method need to determine the load type and thenchoose equations corresponding, the calculation process is complicated.This paper puts forward a method of elliptic integrals to solve the multipleinflection points of large deflection problems. The method can be applied to arbitraryend angle and end load, and it can determine the position of inflection points in thelarge deflection beam. Through the application of the Bernoulli Euler equations, largedeflection equations of beams that contain inflection points or no inflection point areobtained. By splitting into two parts the large deflection beam equation with oneinflection point and multiple inflection points, the large deflection beam equation doesnot contain the inflection points, it can greatly simplify the large deflection equationsof beams. Through the application of elliptic integrals to rewrite large deflectionequation of beams, deflection equations of large deflection beam contain elliptic integrals. Through solving the rewritten deflection equation of large deflection beam,the large deflection beam in the compliant mechanism could solve. By solvingequations of large deflection beams that contain single inflection point and multipleinflection points, the new elliptic integrals method of multiple inflection points beamfor solving large deflection has been verified. By solving partially compliant four-barmechanisms, circular-guided compliant mechanisms and slider-crank compliantmechanisms with this new method and comparing with the finite element method, theproposed method can be proved. The method shows more efficient than the finiteelement method to calculate when the end angle of beams is arbitrary and the beamcontains multiple inflection points.