Vibration Reduction Optimization Of Rotating Flexible Structure

This paper studies the structural optimization design method to reduce the residual vibration of rotating flexible structures.Rotating flexible structure is widely used in practical industrial equipment.The vibration reduction problem caused by its lightweight demand have attracted much attention.The elastic deformation and the rigid rotating motion of rotating flexible structures are coupled with each other.Compared with the usual non-rotating flexible structure,the dynamic analysis and optimization of rotating flexible structure have more challenges in theory and method.In this paper,the residual vibration of rotating flexible structures is studied as a rigid-flexible coupling dynamic problem.Two commonly used methods of rigid-flexible coupling dynamics,the absolute nodal coordinate formulation(ANCF)and the floating frame coordinate formulation(FFR),are applied to the finite element modeling for dynamic analysis of rotating flexible structures.Combined with the method of structural optimization design under impact load,an efficient vibration reduction optimization design method of rotating flexibale structure under impact load and subject to the given material volume constraint is proposed.The main research contents are as follows:1.The ANCF proposed by Shabana describes the large rotation by introducing slope coordinates as nodal degrees of freedom.For composite structures composed of beams,plates,shells or blocks,the discontinuities of the slope coordinates of the element nodes at the structural joints where several elements meet bring the difficulties in the modeling and analysis of composite structures.In this paper the intermediate element coordinate method of dealing with the problem of slope discontinuity in the literature of ANCF was discussed.The slope discontinuity problems in the classical example of folded beam and in the straight beam with stepped section were investigated in detail.The intermediate coordinate method in the existing literature of ANCF and the constrained function method,which correctly models the slope discontinuity are applied to calculate the axial strains of the beam elements in the two examples.The numerical results show that the intermediate coordinate method overlooks the difference between the slope degree of freedom and the rotation degree of freedom,and thus the correct axial strains cannot be achieved at the joint node with slope discontinuity.Since that,the floating frame coordinate formulation is applied in this thesis for the finite element modeling of rotating flexible structures.2.An efficient method based on quadratic performance index and Lyapunov’s second methods is developed for the vibration reduction optimization of frame structure rotating at a constant angular velocity.Four optimization formulations and algorithms are studied for bo th low and high angular velocity.These formulations consider both linear vs nonlinear dynamic models as well as direct integration and Lypunov method to evaluate performance index.Three numerical examples are presented to compare the effectiveness of the four algorithms.Numerical results show that all the four optimization methods can reduce vibration at low rotating speed and find very similar optimal designs.However,at high rotating speed,dynamic model of rotating structure must consider nonlinear axial strain in order to obtain correct dynamic response.Since the Lyapunov’s second method is not applicable for the nonlinear model and the NFTI(Nonlinear FFR dynamic model with nonlinear axial strain and Time Integration)method is very inefficient,the present study develops a perturbation method,linearizes the nonlinear dynamic model and establishes the LNFLA(Linearized Nonlinear FFR dynamic model with nonlinear axial strain,the Lyapunov’s second method and Adjoint sensitivity algorithm).Numerical examples show the optimization method LNFLA provides an efficient numerical method for vibration reduction of rotating frame structure with both low and high rotating speed.3.The topology optimization of rotating plate around the fixed axis at a constant angularvelocity is studied to reduce the vibration under impact load.A quadratic performance index inintegral form is used as the objective and the artificial density of element is used as designvariables.The plate rotates at its own plane and the in-plane vibration is studied.Consideringthe assumption of small strain,the differential equations of motion are established by usingmulti-body dynamics theory and finite element method.Lyapunov’s second method is appliedto simplify the objective.Adjoint variable method for sensitivity analysis and modesuperposition method are used to improve computational efficiency.Numerical results showthe proposed topology optimization method can reduce the vibration of rotating plate and theoptimized topology configuration is different from that of maximum frequency.Thetopological configuration and fundamental frequency of optimized designs change with theincrease of rotational speed.The effect of boundary conditions,location of impact orobservation point and plate aspect ratio on optimized designs is discussed.