Dynamic Topology Optimization For The Beam Structures Considering Rotating Effects

Helicopter rotor blades,wind turbine blades,and airplane propellers are some typical rotating beam structures in engineering.Developing the optimization method to improve the dynamic performance of these rotating beams has gained extensive attention from researchers and engineers.The essential features of rotating systems are the existence of rotating effects,including gyroscopic effects and centrifugal stiffening effects.In addition,most of the rotating beams in engineering are anisotropic and inhomogeneous thin-walled structures,which exhibit significant warping deformation.This will heavily influence the beam cross-sectional properties.Therefore,it is meaningful to establish a reasonable structural design method considering rotating effects and warping deformation to improve the dynamic performance of rotating beams.In this dissertation,the cross-sectional topology,the location of the concentrated mass,the axial shape,and the fiber orientation of composite material are taken as design variables to investigate the dynamic optimization method for the rotating beam structures.A series of work is carried out as follows.Based on the relationship between the cross-sectional topology and properties considering the warping of sections and coupling among deformations,the dynamic equation of the rotating beam structure is derived.Then a topology optimization model of the rotating beam cross-section for the maximum fundamental eigenfrequency or eigenfrequency gap is established.The concurrent optimization of the cross-sectional configuration and axial characteristics(including the location of the concentrated mass and the axial shape)is investigated to improve the dynamic performance of rotating beams.A topology optimization method of the rotating composite beam is proposed,and a concurrent optimization model of the cross-sectional topology and the fiber orientation is established.Thus,rotating beams with more excellent dynamic performance can be obtained by simultaneous designing the structural configuration and material.The main contents and results are obtained as follows:1.A new topology optimization design method of the beam cross-section considering rotating effects.Based on the Giavotto beam theory,the mapping relationship between the cross-sectional topology and cross-sectional properties considering warping deformation is established.Then,the dynamic governing equation of the rotating beam considering centrifugal stiffening effects and gyroscopic effects is derived.On this basis,the topology optimization model of the rotating beam cross-section is formulated,and the objectives are to maximize the fundamental eigenfrequency and the gap between two consecutive eigenfrequencies.The eigensolution of the rotating beam has complex eigenvalues due to the gyroscopic terms in the governing equations,and this work mainly focuses on with the steady rotating state,in which the real parts of all complex eigenvalues equal to zero.The sensitivities of eigenfrequencies with respect to design variables are derived in the complex space.In numerical examples optimized rotating beam cross-sections with different angular velocities are obtained.results show that the angular velocity has a significant influence on the optimal cross-sect and validate the effectiveness and necessity of the proposed method.2.Concurrent optimization design for the location of the concentrated mass and the cr sectional topology of rotating beams.The structural stiffness and mass are two main fac that significantly influence the structural dynamic performance.Simultaneously optimizing structural stiffness and mass can effectively improve the dynamic performance of rotat(?)beams.A Dirac's delta function is employed to express the concentrated mass at an arbitrta location of the beam,and a parameterization method for the location of the concentrated ma is proposed.Then a concurrent optimization model for the beam cross-section and the locat(?)of the concentrated mass is formulated.Rotating beams considering the concentrated mass w(?)fixed location and varying magnitude is studied.The results show that rotating beams wi different magnitude of the concentrated mass have different optimal cross-sectional topologie Thus,the magnitude of the concentrated mass could significantly influence the optimal cros sectional topology of the rotating beam.Then,rotating beams considering the concentrat mass with fixed magnitude and the varying location is studied.The results demonstrate that(?)dynamic performance of rotating beams can be greatly improved by simultaneously optimizi the location of the concentrated mass and the cross-sectional topology.3.Concurrent optimization design for the axial shape and the cross-sectional topology rotating nonuniform beams.In a(?)dition to the design of the beam cross-section,the structu dynamic performance can be fur(?)er improved by optimizing the axial beam shape.Howev(?)multaneously optimizing the cr(?)s-section and the axial shape needs to compute the propert(?)all cross-sections,which sig(?)ficantly increases the computational cost.For solving(?)(?)blem,a mapping function is(?)eveloped to calculate the sectional properties of any cro(?)(?)tions from one reference cr(?)section.Two types of design variables are introduced.T(?)t set of variables are the ma(?)ng parameters that is used to define the shape of the bear(?)ile another set of variables(?)e the general density variables that is use to describe t(?)(?)pology of the reference cross(?)action.A concurrent optimization model for the axial sha?nd the cross-section of rotatii(?)non-uniform beams is established,and a quickly solvi(?)method for the sensitivities of(?)jective functions with respect to those two types of des(?)variables is presented.Numer(?)l examples verify the effectiveness and efficiency of t(?)proposed method.4.Concurrent optimization esign for the cross-sectional topology and the fiber orientati(?)of rotating composite beams(?)e dynamic topology optimization method of rotating bea cross-section is applied to rotating composite beam structures,and the integrated optimization design for the cross-sectional topology and the fiber orientation is realized.Compared with the single material optimization problem,composite material optimization faces the problems of complicated material properties and numerous design variables.Therefore,the Discrete Material Optimization method is introduced to formulate a simultaneous optimization model of rotating composite beams.Numerical examples show that the rotating composite beams at different angular velocities have different optimal cross-sectional topologies and fiber orientations.The dynamic performance of rotating composite beams can be improved effectively by simultaneously optimizing the cross-sectional topology and the fiber orientation compared with the single material optimization problem.