The POD Method Based On Transient Response And Applications For Dimension Reduction Of Rotor Systems

Most of the actual engineering dynamics systems are high-dimensional and complex nonlinear systems,such as the rotor system of aero-engine,gas turbine,etc.The qualitative analysis of this type of system is difficult and the computational expense is very expensive.So the order reduction for the original system should be studied,then the reduced model will be used to represent the original(high-dimensional and complex)model.The design uncertainties should be considered in the actual engineering systems,for example,the mass,stiffness or other parameters should be stochastic in the ragne of stated tolerance.The order reduction for uncertain systems plays a significant role in the area of nonlinear dynamics.The order reduction method in deterministic systems and dimensionality reduction method in stochastic systems will be discussed in this thesis.Firstly,the order reduction methods of high-dimensional deterministic systems and the corresponding applications in the rotor systems are studied in this thesis.Meanwhile,the dynamical characteristics of the rotor system with looseness fault and the bifurcation behaviors of the reduced models are discussed.Finally,the order reduction methods of the rotor system with uncertainties and its dynamical characteristics are studied.The main contents of this thesis can be divided into several aspects as follows:The traditional proper orthogonal decomposition(POD)method is modified based on the intertial manifold theory and the transient POD method is improved.Two numerical examples of rotor-bearing systems with loosenss are provided: first,the transient POD method is applied to the 23 degrees of freedom(DOF)rotor system with pedestal looseness at left end,the 2-DOF reduced model reserves the bifurcation and amplitude-frequency characteristics of the original model,the efficiency of the transient POD method is verified via comparing with the traditional POD method;second,the transient POD method is applied to 7-DOF rotor system with ball bearings at both ends and loosenss at one end,the comparion of the dynamical characteristics between reduced and original systems verify the efficiency of the transient POD method once more.The POM energy method is proposed in this thesis,and the physical significance is definite.This thesis studies the dynamical characteristics of different models with loosenss fault.First,the 15-DOF rotor system with looseness at one end and 16-DOF rotor system at both ends are established.The dynamical characteristics of the 16-DOF system are more complex than the 15-DOF one,more fractional and inteteral frequencies occur.The reduced model obtained by the transient POD method can reserve the dynamical behaviors of the original model very well,the optimal order reduction condition of dynamical system is proposed based on the proper orthogonal mode(POM)energy method.Meanwhile,the initial values of displacement and velocity are studied,the perturbation of the initial values will change the frequency components of the system,but the order reduction efficiency will not be affected.The transient POD method is compared with the structure order reduction method(SOR),the results show that both the two methods can be applied to the rotor system and the efficiency and accuracy of the transient POD method can also be verified.Singularity theory of the 6-DOF rotor system supported by cubic nonlinearity stiffness is analyzed in this thesis.The transient POD method is applied to reduce the original model to a single DOF one,the co-dimension of the reduced model is analyzed and the whole bifurcation behaviors of the rotor system are provided.Then a method to seek main bifurcation parameters of a class of nonlinear dynamics system is proposed based on the effects of parametric variation of the dynamical system on the eigenvalues of the Frechet matrix.The equivalence of physical and unfolding parameters is verified,the comparision of bifurcation behaviors between engineering and unversial unfolding demonstrates that engineering unfolding can reserve the main bifurcation behaviors of the unversial unfolding,which can satify the requirements of confirming system parameters.The PDD method is generalized to dynamical system models in the thesis.The PDD method is applied to 2-DOF spring system with uncertain stiffness,damping,mass,and the first two order moments of the dynamical system are analyzed.The PDD method can reserve the amplitude-frequency characteristics of the exact solution(MCS).This thesis studies the order of the PDD method,it can approximate to the accurate solution better when the order increases.The PDD method is applied to the linear rotor system and the 6-DOF rotor system supported by cubic nonlinearity stiffness respectively.This thesis studies the multi uncertainties of the stochastic system and discusses the multi-varibale PDD method.The PDD method can approximate to the amplitude-frequency characteristics of accurate solution,which verifies the accuracy and efficiency of the PDD method.The preliminary application of the PDD method in the nonlinear rotor system provides the theory guidance to work on the more complex rotor systems in the future.