Research On Model Order Reduction Based On POD Method And Application For Complex Rotor-bearing Systems

High-dimensional complex systems can be widely found in many fields,such as aviation,spaceflight,shipping,etc.Model order reduction(MOR)of the complex nonlinear system has become one of the central issues in mechanical engineering technology.Moreover,aero-engines and gas turbines are currently the major engineering problems in our country that need to be captured.The rotor-bearing system is the core component of an aero-engine and gas turbine,which has the features of complicated structure,high degrees of freedom(DOFs),coexisting of multi-nonlinearities and coupling of multi-physics.Consequently,there are many thorny problems,for example,it is difficult to fast analyze the dynamic characteristics of system,control and optimize the structural parameters,and evaluate the reliability,etc.Therefore,it is essential to develop the fast and accurate MOR methods to solve the high-dimensional complex systems,based on which its dynamic characteristics can be investigated in detail by a reduced-order model(ROM)with a low degrees of freedom.Correspondingly,the structural parameters can be optimized efficiently,and the development period can be shortened.Thus,this method study has a great application value for the design and development of rotor structure of our country's aero-engine and gas turbine.In this thesis,the complex rotor-bearing systems of aero-engine and gas turbine are taken as the study objects,upon which the MOR approaches are investigated.Based on the Proper orthogonal decomposition(POD)method,the new MOR methods which solve the basic problems and limitations of the current method are proposed and applied on the complex rotor-bearing systems of aero-engine and gas turbine.The experiment studies are carried out as well.The main contents and achievements obtained in this thesis are listed as following.A new adaptive POD method is proposed to address the weakness of local property of the interpolation tangent-space of Grassmann manifold(ITGM)method based on the relationships of Grassmann manifold,geodesic line,and so on.Then,this method is applied to the MOR of a high-dimensional nonlinear rotor system with a pair of liquid-film bearings and a pedestal looseness fault.The numerical results demonstrate that the method has a good robustness for a large interpolating span,the smaller interpolating spans of parameter,the lower mode numbers and the higher accuracy of ROMs.Furthermore,this method and the ITGM method are proved to be equivalent in a local-parametric region,but the latter fails to reduce order and lose the parameter adaptation for a large interpolating span.Therefore,the new method can be applied to the MOR of high-dimensional complex systems in a wider parametric region.A parametric domain transient POD method with the minimum error of bifurcation is proposed to obtain an invariance ROM in parameter space to overcome the ever-changing ROMs derive from the adaptive POD method or the non-unique ROMs under the different sampling conditions,and the MOR condition of a parametric domain is defined as well.If one of the proper orthogonal modes(POMs)of the different sampling conditions satisfies the small total average least-square truncation error,an invariance ROM can be obtained by this POM in the parametric domain,and the dynamic characteristics of the ROM are almost the same with the original systems.In addition,the ROMs are equivalent under different sampling conditions when the total average least-square truncation errors are small enough.Meanwhile,there exists an optimal sampling length among all of the sampling lengths for the parametric domain reduction when the other parameters are determined,so an invariance ROM can be obtained by the POM for the transient signal of this length,and the ROM has the minimum error of bifurcation,which is an essential condition for the parametric domain reduction.Finally,the parametric domain transient POD method is applied to a high-dimensional nonlinear rotor-bearings system with a pedestal looseness fault.The effects of the rotational speeds,initial conditions,sampling lengths and mode numbers on the ROMs are analyzed,the existence of the optimal sampling length is verified,and the ROMs with 2 and 3 DOFs are obtained by the optimal sampling length in a wide range of rotation speed.The initial condition has a great effect on the ROM,and it will lead to the failure of the parametric domain reduction when the initial value is too small to excite the intrinsic mode of the structure.MOR of the complex rotor systems supported by rolling element bearings are investigated,but the first study is to reveal the nonlinear resonance characteristics of varying compliance(VC)vibration of rolling element bearing.The numerical and experimental results indicate that the VC contact resonance in the vertical direction presents the soft spring characteristic,while the soft and hard spring characteristics coexist for the horizontal resonance.Later,the parametric domain POD method is utilized in the MOR of the two complex experimental rotor systems with the misalignment of coupling: the multi-discs-single rotor-bearing system and the dual rotor-bearing system.The numerical and experimental results indicate that the vibration features of the complex rotor-bearing systems can be calculated exactly and efficiently by this method in the low-order primary resonance rotation speed region.Meanwhile,the reduction of experimental data of the original system is discussed by using POD method.It is found that the traditional sampling viewpoint of “the higher the POMs ratio,the higher the precision” is not always correct,and the accuracy of ROM is related to the numbers(lengths)of sampling;the high accuracy of ROM of the original data system needs a certain number of sampling;for different POMs,the corresponding sampling lengths are different.These results are consistent with the theoretical conclusions of the parametric domain POD method.Furthermore,the vibration response feature of the two experimental rotor-bearing systems are measured and analyzed,it may provide a criterion for the misalignment of the rotor-bearing system.The dynamic model of the dual rotor-bearing system of aero-engine combining the cylindrical shell-conical shell-disc is established based on the finite element method.The POD with mode expansion second-MOR method is proposed by combining the component mode synthesis(CMS)method and POD method,and it is utilized in the MOR of the structural dynamic system above.The mode expansion second-MOR method has a higher accuracy than the directly second-MOR by using POD method,because the method involves the higher-order modes of the complex systems,and establishes a mapping between the second ROM and the original system.Besides,the vibration response characteristics of the dual rotor-bearing system of aero-engine are studied by this second-MOR method in the case of a large clearance of bearing.The results indicate that the dynamic characteristics of the dual rotor-bearing system are complicated,and show an obvious misalignment or decentraction.Furthermore,the signals of VC vibrations are extremely weak because of the large number of rolling elements of aero-engine bearings,the complex nonlinear phenomena of the system are induced by Hertzian contact,clearance of bearing and multi-excitation sources,thus,controlling and optimization of the clearance of bearing is important to decrease the vibrations of the aero-engine rotors.