Methods Study On Stochastic Reliability Analysis Of High-speed Dynamic Balancing Machine

With the spread of the concept of reliability in engineering, reliability assessment of performance of mechanical equipment is becoming a necessary performance evaluation criterion during phase of the design, manufacture and use of equipment. Therefore, the study of mechanical performance of typical equipment and its corresponding reliability and life assessment methods will be very important in practice. This dissertation originates from an 863 project(2009AA04Z419) named Re-evaluation and Growth Technologies on Reliability and Lifetime of Typical Major Equipment in Service. In this dissertation, a high-speed balancing machine was taken as object of study, then a thorough and systematic study on the assessment methods of mechanical performance and stochastic reliability of its dynamic balancing rotor system was performed, and we hope this work will provide corresponding theoretical and practical experiences to similar rotating equipment.The mechanical system of high-speed dynamic balancing machine belongs to rotor dynamics. And its mechanical study is classified into two kinds, one is only to study its bearing support, another is to study the interaction system of support-bearing-rotor. A deep understanding to the static and dynamic performance of these two systems is of importance for the balancing precision and operation safety of high-speed dynamic balancing machine. However, due to uncertainties in high-speed dynamic balancing machine, its performances are also uncertain. As a result, the deterministic performance may not reflect the true operation status of equipment. Stochastic uncertainty analysis and reliability assessment are the major ways of dealing with uncertainties. Performance estimation from the perspective of stochastic reliability can remarkably improve the safety marginal of operation of high-speed dynamic balancing machine.In fact, high-speed dynamic balancing machine belongs to small samples and high reliability equipment, and often has the characteristics of complex model, large calculation cost, high non-linear, high dimension etc.. With these characteristics in mind, we first analyzed the static and dynamic performance of high-speed dynamic balancing machine. And then, the methods of stochastic reliability analysis were studied, including the stochastic uncertainty analysis methods of key responses and the reliability calculation methods of key failure modes for high-speed dynamic balancing machine. Details are as follows:(1) Static bearing capacity, static/dynamic stiffness characteristics, and fatigue stress-life of bearing support of high-speed dynamic balancing machine were studied.Static finite element model of bearing support was first established, and then the static bearing capacity and the variation law of radial static stiffness of bearing support were respectively discussed based on this finite element model. The experimental dynamic stiffness of bearing support was measured by hammering method. And then the dynamic stiffness behaviors of bearing support were discussed according to the tested dynamic stiffness data. A modal identification strategy of bearing support was then proposed when only limit experimental dynamic stiffness data was available. This strategy is convenient for engineering application. At last, the fatigue stress-life of bearing support for several typical rotor weights was estimated based on the estimated unbalance force amplification coefficient and first critical speed after obtaining the stress state of main rod of bearing support under the action of pseudo-unbalance force.(2) The beam finite element modeling theory and direct differentiation method of response sensitivity analysis were studied for high-speed dynamic balancing machine. And the equivalent stiffness and damping coefficients formulas of support-bearing system were also given. From these equivalent coefficients, the formulas of the first critical speed and unbalance force amplification coefficient were also derived.The finite element equation of high-speed dynamic balancing machine was derived from beam finite element theory of rotor system. After the definition parameters of model were totally listed, the direct differentiation formulas of sensitive analysis of modal frequency and critical speed with respect to these definition parameters were presented. And also the level relationships of direct differentiation of coefficient matrix with respect to these definition parameters were also presented. The equivalent stiffness and damping coefficient formulas of support-bearing system were derived by ignoring the cross coefficient terms. And then the estimation formulas of the first order critical speed and the unbalance force amplification coefficient were also derived from the simplified support-bearing-rotor system. These formulas are very useful for quickly estimating the fatigue stress-life of bearing support. At last, the dynamic behaviors of a 50 MW steam turbine rotor dynamic balancing system and its first order eigenvalue sensitivities were analyzed.(3) The Rackwitz-Fiessler method of the random space transformation for the random vector of high-speed dynamic balancing machine was studied. The forward and backward processes and variation relationship of correlation coefficients in Rackwitz-Fiessler(R-F) method were discussed in details. And the enhanced R-F method was proposed subsequently.From two views(one is according to the proposed equivalent R-F conditions, another is mathematical derivation), the transformation process of R-F method from physical X space to standard normal Y space was confirmed to obey isoprobabilistic marginal mapping transformation of Nataf-Pearson(N-P) method. According to this conclusion, the correlation coefficient variation formula in equivalent normal process of R-F method should be the same to Der Kiureghian and Liu’s formula. Then the enhanced R-F method was proposed if such a formula was used in R-F method. At last, comparisons of calculation cost and efficiency between enhanced R-F method and N-P method were discussed.(4) Then improved R-F methods and generalized R-F methods were proposed based on copula theory.Due to some deficiencies of linear correlation coefficient, improved R-F methods based on Spearman and Kendall correlation coefficient were respectively proposed. And in further, as there is Gaussian copula assumption in traditional R-F method, so the generalized R-F methods were proposed based on the elliptical copula family. Meanwhile, in order to make the traditional FORM match with generalized R-F methods, the generalized first order reliability index and generalized FORM were proposed based on the same concept of MPP in the uncorrelated standard spherical U space. Finally, the reliability of infinite fatigue life of main rod of bearing support was analyzed.(5) An adaptive MPP searching algorithm, short for aHLRF, was proposed. The global convergence was proved and several key issues for improving algorithm efficiency were also discussed.On the basis of existing HLRF-based MPP search methods, an adaptive MPP search algorithm a HLRF was presented, in which the algorithm parameters were estimated automatically by using the existing iteration information. aHLRF was proved to be global convergence and its convergence rate was at least linear. Several key issues affecting the efficiency of the aHLRF were discussed, and eleven examples were selected to validate aHLRF. Finally, the reliability of finite fatigue stress-life of main rod of bearing support, and the reliability of stability margin of rotor system under operating speed were analyzed.(6) The stochastic uncertainty analysis on critical speed and unbalance response of high-speed dynamic balancing machine were studied based on spectral stochastic finite element method with non- intrusive and intrusive way, respectively.Approximation error of Gaussian random field were discussed when K-L expansion with wavelet-Galerkin scheme was applied to deal with random field. Then the Gaussian random field approximations of both Young modulus of the shaft and the rotor distribution unbalance were done with this K-L expansion scheme. Further, based on Sudret’s Box-Ball filling strategy, the detailed programming steps were listed for automatically generating polynomial chaos basis functions with arbitrary number of dimensions and order. In order to automatically rewrite the right hand side terms of the spectral stochastic finite element equation in term of polynomial chaos basis, we proposed a recursive algorithm. This algorithm is limited to the case only when the right hand side terms were expressed in term of polynomial. Finally, based on non-invasive spectral stochastic finite element method, uncertainty analysis of critical speed of a 50 MW steam turbine rotor balancing system was done; based on invasive spectral stochastic finite element method, uncertain analysis of unbalance response of a 50 MW steam turbine rotor balancing system under the action of random point unbalance and distributed unbalance was also done.In conclusion, this dissertation accomplished the study on the static and dynamic performances analysis and the methods of stochastic reliability analysis of high-speed dynamic balancing machine. And the characteristics of high-speed dynamic balancing machine had been given adequate consideration, including complex model, large calculation cost, high non-linear, high dimension etc., and also belonging to small samples and high reliability equipment. The methods used for high-speed dynamic balancing machine, such as Rackwitz-Fiessler methods, aHLRF, and uncertainty analysis based on spectral methods, are also suitable to analyze the stochastic reliability of other similar rotating equipment.