Variability of Blade Vibration in Mistuned Bladed Discs
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VARIABILITY OF BLADE VIBRATION IN MISTUNED BLADED DISCS A thesis submitted to the University of London for the degree of Doctor of Philosophy By Yum Ji Chan Department of Mechanical Engineering Imperial College London October 2009 Abstract The huge variation in forced vibration response levels of mistuned bladed discs, which are bladed discs with slightly different blades, is investigated in this thesis. The issue is known as the blade mistuning problem. A mis- tuning management strategy is proposed and assembled to manage the high vibration response levels of mistuned bladed discs. This strategy involves (i) evaluation of the range of the response level, (ii) achieving a better bladed disc design, and (iii) monitoring the status of the actual hardware. The severity of the vibration problem of a mistuned bladed disc is usually quantified by the “amplification factor", which is clearly defined in this thesis to remove inconsistency among previous definitions. A new procedure is proposed to estimate the small probabilities of high amplification factors more accurately and more efficiently than is possible with typical Monte Carlo simulations. By casting the blade mistuning problem as a robust design problem, parameter design and tolerance design concepts are used to find methods of improving robustness of bladed discs. The design parameters of the bladed disc design are changed in parameter design while, in contrast, the mistuning parameter distribution on a bladed disc is controlled in tolerance design. If the blades and the disc form a single component (also known as a blisk), the uncertainties of joint properties are removed and the responses of the bladed disc can be predicted. A response-prediction procedure proposed previously is validated experimentally, and the quality of experimental data required is evaluated. The investigation in this thesis has shown that a mistuning management strategy is viable. A new procedure is developed to fulfil (i), design guide- lines are formed to facilitate (ii), and the monitoring in (iii) is practical in the foreseeable future. i Acknowledgments I am grateful to Prof. D. J. Ewins for his commitment after his end of full-time service at Imperial College London. His guidance demonstrated good research practices, and his advice in concise technical writing can be seen everywhere in this thesis. The feedback from Prof. M. Imregun on my research is acknowledged. I enjoy the professional and social discussions with my colleagues in the Dynamics Section, including Dr. E. P. Petrov, Dr. C. Zang, Dr. C. Schwingshackl, Dr. M. Nikolic, Dr. Sen Huang, Dr. D. Di Maio, Mr. A. Gondhalekar and the ERASMUS students. Special acknowledgment is given to Dr. D. Di Maio for his help in setting up the experimental rig, on which the results in Chapter 7 of this thesis is based. The help of Ms. N. Hancock on the administrative work is appreciated. My thanks is also due to several staff members from the Department of Mathematics. The discussion with Dr. G. Moore was constructive, and the content in Appendix E is based on the advice of Dr. R. Jacobs. The pastoral care from Dr. G. Stephenson is exceptional, and his support went as far as reading my thesis voluntarily. My studies at Imperial College would be much less enjoyable, or even impossible, without the support of several persons outside the College. Dr. M. G. Sainsbury stimulated my interest in structural dynamics and rec- ommended me reading a doctoral degree at his alma mater. My parents, Fu-Chai Chan and Yuen-Yee Lai, have provided the finances and encour- aged my intellectual explorations, and my sisters, Wancy and Mancy, have advised on my well-being with their expertise. Last but not least, I wish to express my gratitude to Leonie Lee, who has been understanding, supportive and caring throughout my academic adventures in Europe. ii Contents Nomenclature vii Abbreviations xi List of Figures xii List of Tables xvi 1 Introduction 1 1.1 Overview . .1 1.2 A Mistuning Management Strategy . .3 1.3 Objectives of the current research . .4 1.4 An overview of the thesis . .4 2 Literature review 6 2.1 Introduction . .6 2.2 Current developments in the blade mistuning problem . .7 2.2.1 Our ability of analysing mistuned bladed discs . .8 2.2.2 Range of response levels in mistuned bladed discs . 11 2.2.3 Sensitivity studies . 17 2.2.4 Intentional mistuning approach . 18 2.2.5 Blisks and the need of mistuning identification . 19 2.3 Questions to be answered in this thesis . 20 2.4 A brief review of robust design concept . 21 2.5 Summary . 23 3 Evaluation of the range of the amplification factor 24 3.1 Introduction . 25 iii CONTENTS 3.2 Evaluation of the worst scenario: the quest to find a realistic maximum amplification factor . 29 3.2.1 Upper bound of the amplification factor . 29 3.2.2 Numerical search of the maximum amplification factor 33 3.3 Current approach of amplification factor distribution prediction 36 3.4 A new procedure to predict the probabilities of occurrence of extreme amplification factors . 39 3.4.1 Introduction to the importance sampling method . 40 3.4.2 Importance sampling method in the blade mistuning problem . 43 3.4.3 Demonstration . 44 3.5 Summary . 48 4 Application of robust design concepts to the blade mistuning problem 49 4.1 Introduction . 50 4.2 Terminology . 51 4.3 Introduction to major robust design methods . 53 4.3.1 The Taguchi method . 53 4.3.2 The Robust Optimisation method . 57 4.4 Casting the blade mistuning problem as a robust design problem 59 4.4.1 Input parameters to the amplification factor function . 59 4.4.2 Robustness functions to be investigated . 61 4.4.3 Outline and approach of investigation . 64 4.5 Summary . 66 5 Improving robustness of bladed discs by parameter design 67 5.1 Introduction . 68 5.2 Significance of the blade mistuning problem in high cycle fatigue 68 5.3 Dependence of robustness on bladed disc design parameters . 70 5.3.1 Models to be used in the analyses . 71 5.3.2 Maximum adjusted amplification factor of a 6-DOF model . 74 5.3.3 Maximum adjusted amplification factors of four 64- sector models . 76 5.3.4 Maximum amplification factors of six 24-sector blisks 80 iv CONTENTS 5.4 Robustness of bladed disc designs in previous research . 82 5.5 Amplification factor distribution in special situations . 87 5.5.1 Amplification factor distribution in bladed discs with damping mistuning . 87 5.5.2 Amplification factor distribution with excitation of modes in veering regions . 88 5.5.3 Amplification factor distribution in apparently-tuned bladed discs . 91 5.6 Summary . 92 6 Improving robustness of bladed discs by tolerance design 94 6.1 Introduction . 95 6.1.1 Issues related to the \large mistuning concept" . 96 6.2 Maximum amplification factor sensitivity to mass mistune . 97 6.2.1 Derivation . 97 6.2.2 Analysis . 99 6.2.3 Demonstration . 100 6.3 A new definition of interblade coupling ratio . 102 6.4 Managing blade responses using a small mistuning approach . 104 6.4.1 Dependence of robustness on level of input variability 106 6.4.2 Excitation of mode shapes in the veering region . 110 6.5 Feasibility of the intentional mistuning approach . 111 6.5.1 Determination of the consequences of intentional mis- tuning using the importance sampling method . 112 6.5.2 Evaluation of a linear intentional mistuning pattern . 113 6.6 Summary . 117 7 Predicting vibration response levels of integral bladed discs (blisks) 119 7.1 Introduction . 119 7.2 Outline of a response level prediction procedure . 121 7.3 Experimental demonstration of the proposed procedure . 123 7.3.1 Mistuning pattern 1 . 126 7.3.2 Mistuning pattern 2 . 128 7.4 Robustness of the proposed procedure . 132 7.4.1 Major sources of error in modal testing . 133 v CONTENTS 7.4.2 Robustness of the predicted response . 135 7.4.3 Identification of the damping mistuning pattern . 138 7.5 Recommendations for future experimental work . 140 7.6 Summary . 141 8 Conclusions 143 8.1 Conclusions . 143 8.1.1 Efficient estimation of small probabilities . 144 8.1.2 Reduction of the variability of responses in blades . 145 8.1.3 Validation of a response-prediction procedure . 148 8.2 Major contributions to knowledge . 149 8.3 Recommendations for future research . 150 A Modal properties of bladed discs 151 B Implementation of the cross entropy method 154 B.1 Outline of the algorithm . 156 C First-order derivatives of natural frequencies and mode shapes158 C.1 Derivatives of non-repeated natural frequencies . 158 C.2 Derivative of a mode shape . 159 C.3 Natural frequency and mode shape derivatives of double modes160 C.4 Summary . 161 D Sensitivity of the maximum adjusted amplification factor 162 D.1 Simplifying the vibration responses derivatives vector . 162 D.2 Analysing the adjusted amplification factor sensitivity . 165 E Pdf of the sum of random numbers of different distributions167 F The Fundamental Mistuning Model (FMM) and FMM-ID algorithms 169 F.1 The Fundamental Mistuning Model (FMM) algorithm . 169 F.2 The FMM-ID algorithm . 173 G Parameters of an intentionally-mistuned blisk test piece 177 References 180 vi Nomenclature [Φ] Multiple mode shapes of a vibration system, where double modes are arbitrarily assigned [Ψ] Multiple mode shapes of a vibration system, where double modes are specified Ω Forced harmonic excitation frequency (in rad·s-1 unless otherwise specified) Ωr Rotation speed of bladed disc (in rev/min) α Scaling coefficient fβg Modal weighting factors vector γ The adjustment factor in the conjugate gradient optimisation method γ The pass ratio in the cross entropy method δ Degree of intentional