DEVELOPMENT OF MICROSLIP FRICTION MODELS AND FORCED RESPONSE PREDICTION METHODS FOR FRICTIONALLY CONSTRAINED TURBINE BLADES

Transcription

1 DEVELOPMENT OF MICROSLIP FRICTION MODELS AND FORCED RESPONSE PREDICTION METHODS FOR FRICTIONALLY CONSTRAINED TURBINE BLADES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ender Cigeroglu, BS, MS ***** The Ohio State University 2007 Dissertation Committee: Professor Chi-Hsiang Menq, Adviser Professor Ahmet Kahraman Professor Daniel A. Mendelsohn Approved by Adviser Graduate Program in Mechanical Engineering

2 ABSTRACT High cycle fatigue failure of turbine and compressor blades due to resonance in the operating frequency range is one of the main problems in the design of gas turbine engines. In order to suppress excessive vibration of blades and prevent high cycle fatigue, dry friction dampers are used by the engine manufacturers. However, due to the nonlinear nature of dry friction, analysis of such systems becomes complicated. Because of its mathematical simplicity, macroslip models are widely used and they perform well if the contact pressure acting on the friction interface is low. However, in case of high contact pressure and/or large contact areas, partial slip in the friction interface occurs, which cannot be modeled by the macroslip model. In order to obtain accurate forced response predictions for frictionally constrained bladed disks, microslip modeling is necessary. However, due its mathematical complexity, microslip friction models in literature are developed for simple cases, which leave the realistic bladed disk models unaddressed. In this research, it is aimed to develop microslip friction models and forced response prediction methods, which can be used in realistic bladed disk models, usually modeled by finite elements, and can handle complex contact kinematics resulting in friction damper applications. This leads to the development of the following microslip ii

3 friction models: one-dimensional dynamic microslip friction model, distributed microslip friction model for one-dimensional motion with normal load variation induced by normal motion, microslip friction model for two-dimensional periodic motion with normal load variation, three-dimensional microslip friction model with normal load variation. The application of these models in the forced response prediction of frictionally constrained structures is as well presented. In case of non-planar friction contact, it is possible to have variation of the friction interface resulting in variation of the contact stiffnesses associated with the friction contact. The microslip friction models developed do not consider the variation of contact stiffness; therefore, transition criteria for two-dimensional motion with stiffness and normal load variation and three-dimensional motion with stiffness and normal load variation are developed. In the vibration suppression of turbine blades, one of the most commonly used damper type is wedge damper, which is also referred as underplatform damper. A wedge damper model and forced prediction method for frictionally constrained blades is developed. The model developed considers the damper as a solid body with rigid body modes and elastic modes, where dampers and blades are modeled by finite elements. Damper moves freely in between adjacent blades; therefore, damper motion is not constrained artificially. Forced response prediction method is as well presented and predictions for a test case are compared with test data. iii

4 The microslip friction models developed are associated with contact stiffness through out the contact interface. Unlike other model parameters, contact stiffnesses are not available and they have to be determined. In order to estimate these contact stiffnesses, a method is developed which is based on the difference between the modal model and finite element model. The method is applied on the test case with wedge dampers in order to obtain contact stiffnesses and good agreement between the forced response predictions and test data is observed. iv

5 To My Parents Nigar and Ramazan Ciğeroğlu v

6 To My Fiancée Demet Denizlioğlu vi

7 ACKNOWLEDGMENTS I am most grateful to my adviser Prof. Dr. Chia-Hsiang Menq for his continuous suggestions, interest and expert guidance during the preparation of this dissertation. I thank Dr. Ning An for valuable discussions and his continuous support in the development of computer codes for this research. I wish to thank Demet Denizlioğlu, for her continuous support and encouragement during the preparation of this dissertation. I would like to thank to my family for their lifetime help, support and understanding they have shown to me. This research is supported GUIde Consortium of the Carnegie-Mellon University, which is sponsored by the Air Force Research Laboratory. The financial assistance is gratefully acknowledged. vii

8 VITA May 21, June 1999 August Present Born, Manisa, Turkey BS Department of Mechanical Engineering Middle East Technical University Ankara, Turkey MS Department of Mechanical Engineering Middle East Technical University Graduate Teaching and Research Assistant Middle East Technical University Graduate Research Assistant The Ohio State University Columbus OH PUBLICATIONS Cigeroglu E, An N, Menq CH, Wedge Damper Modeling and Forced Response Prediction of Frictionally Constrained Blades, ASME Turbo Expo 2007, May, Montréal, Canada. Cigeroglu E, An N, Menq CH, A Microslip Friction Model with Normal Load Variation Induced by Normal Motion, Nonlinear Dynamics, (accepted for publication) Ciğeroğlu E, Özgüven HN, Non-linear Vibration Analysis of Bladed Disks with Dry Friction Dampers, Journal of Sound and Vibration, 295 (2006) Cigeroglu E, Lu W, Menq CH, One-dimensional Dynamic Microslip Friction Model, Journal of Sound and Vibration, 292 (2006) viii

9 FIELDS OF STUDY Major Field: Mechanical Engineering Minor Field: Structural Dynamics, Vibrations, Friction Modeling, Nonlinear Vibrations, Numerical Methods ix

10 TABLE OF CONTENTS ABSTRACT. DEDICATION. ACKNOWLEDGMENTS... VITA LIST OF TABLES... LIST OF FIGURES. ii v vii viii xiii xiv CHAPTER PAGE 1. INTRODUCTION Background and Motivation Research Objective and Scope Literature Review Friction Models Forced Response Analysis Dissertation Overview ONE-DIMENSIONAL MICROSLIP FRICTION MODEL Introduction Determination of Stick-Slip Transitions Determination of Contact Elastic Force Two-Region Friction Interface Three-Region Friction Interface results Stick-Slip Transitions Equivalent Spring Force and Damping Force Effect of Normal Load Distribution Comparison with Point Contact Model Comparison with Numerical Solution conclusions A MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION INDUCED BY NORMAL MOTION Introduction Two-Dimensional Microslip Friction Model with Normal Load Variation Stick, Slip and Separation Transition Transition Angles x

11 Stick-slip-separation Map Distribution of Fourier Coefficients Forced response Multi-Mode Solution Method Examples D Bar Model Blade to Ground Damper Conclusion MODELING MICROSLIP FOR TWO-DIMENSIONAL PERIODIC MOTION AND ITS EFFECTS ON BLADE VIBRATION Introduction Microslip friction model for Two-dimensional Periodic Motion with normal load variation Transition Angles Distribution of Fourier Coefficients Formulation Multi-Harmonic Solution Method Friction Interface Map numerical results Effect of Normal Load Distribution Effect of Multiple Harmonics Friction Interface Maps Conclusion THREE-DIMENSIONAL MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION INDUCED BY NORMAL MOTION Introduction three-dimensional Microslip friction model with normal load variation Stick, Slip and Separation Transition Criteria Transition Angles Distribution of Fourier Coefficients Multi-Mode Solution Method Friction Interface Map numerical results Friction Interface for Specified Motion Blade to Ground Damper Conclusion FRICTION MODELS FOR STIFFNESS VARIATION Introduction Two-Dimensional Motion with Stiffness and Normal Load Variation Stick, Slip and Separation Transition Three-Dimensional Motion with Stiffness and Normal Load Variation Numerical Example Two-dimensional motion with stiffness and normal load variation Three-dimensional motion with stiffness and normal load variation xi

12 6.5. Conclusion WEDGE DAMPER MODELING AND FORCED RESPONSE PREDICTION OF FRICTIONALLY CONSTRAINED BLADES Introduction Wedge Damper Model Blade and Wedge Damper Contact Model Friction Contact Model and Forced Response Prediction Initial Preload on Contact Surfaces Forced Response Method Numerical Results Tuned Bladed Disk System Comparison with the Test Case Data Conclusions ESTIMATION OF CONTACT STIFFNESS Introduction Calculation Method for Contact Stiffness Normal Contact Stiffness Tangential Contact Stiffness Block-to-Block Example Contact Stiffnesses for Bladed Disk System with Wedge Dampers Conclusion CONCLUSIONS AND recommended FUTURE WORK Conclusions Future Work Unknown Coefficients for Three-Region Friction Interface TRANSITION CRITERIA FOR 3D MOTION Bibliography xii

13 LIST OF TABLES Table Page 5.1 State-by-state simulation with and without convergence acceleration xiii

14 LIST OF FIGURES Figure Page 1.1 Blade-to-ground damper Blade-to-blade damper Shroud fan stages Wedge damper D Microslip model Effect of excitation frequency on contact elastic force Build up of contact elastic force and its comparison with various normal load distributions Two-region friction interface due to convex normal load distribution Three-region friction interface due to concave normal load distribution Temporal change of two-region friction interface Temporal change of three-region friction interface Effect of excitation frequency on hysteresis: two-region friction interface Effect of excitation frequency on Fourier coefficient F s : two-region friction interface Effect of excitation frequency on Fourier coefficient F c : two-region friction interface Effect of excitation frequency on hysteresis: three-region friction interface Effect of excitation frequency on Fourier coefficient F s : three-region friction interface Effect of excitation frequency on Fourier coefficient F c : three-region friction interface Comparison of F c for both friction interfaces (ω=0.3ω c ) Comparison of F s for both friction interfaces (ω=0.3ω c ) Equivalent point contact model Comparison of F s for microslip and point contact models Comparison of F c for microslip and point contact models Comparison of numerical and analytical solutions Planar contact of two bodies Distributed contact model for 2D motion D bar model xiv

15 3.4 Build-up friction force for constant normal load distribution: (a) model, F( t) for (b)<(c)<(d) Build-up friction force for quadratic normal load distribution: (a) model, F( t) for (b)<(c)<(d) Effect of number of modes at 1000Hz a) constant b) quadratic normal load distributions: 1 mode, 3 modes, 10 modes Effect of number of modes at 3100Hz a) constant b) quadratic normal load distributions: 1 mode, 3 modes, 10 modes Phase difference along slip-to-stick transition boundary at 1000Hz a) constant b) quadratic normal load distributions Blade to ground damper model Finite element model for the blade (Maybe modified) Forced response for excitation force in x direction: free, -5e5, 1000, 7500, 1e4, 5e4, 1e5, 2e5, stuck Forced response for excitation force in y direction: free, -2e4, 80, 350, 500, 1000, 1e4, 2e4, stuck Stable and unstable solutions for n 0 = 1000: stable solution, unstable solution Friction interface for n 0 = 1000 before jump (a) 0, (b) 90, (c) 180, (d) 270 ; after jump (e) 0, (f) 90, (g) 180, (h) Friction interface for n 0 = 1000 and θ = D periodic motion Friction force in one cycle Hysteresis curve Harmonic coefficients for the friction force Hysteresis curve Friction force in one cylcle Amplitude of harmonic coefficients Hysteresis curve Friction force in one cycle Normal load distributions Effect of normal load distribution stuck, constant, distribution 1, distribution 2, distribution 3, distribution 4, distribution Forced response for preload case: distribution 3 stuck, single-harmonic: 1.0e6, 5.0e5, 2.5e5, 1.0e5, 5.0e4, 1.0e4, multi-harmonic: 1.0e6, 5.0e5, 2.5e5, 1.0e5, 5.0e4, 1.0e xv

16 4.13 Forced response for constant initial gap: free, singleharmonic, multi-harmonic Friction interface plots for constant normal load distribution Friction interface plots for 3 rd normal load distribution Friction interface plots for 5 th normal load distribution Planar contact of two bodies Distributed point contact model for 3D motion Friction interface maps for prescribed motion: a) θ=0, b) θ=45, c) θ=90, d) θ=135, e) θ=180, f) θ=225, g) θ=270, h) θ= Finite element model for the blade Tracking plots for blade to ground damper system: free, -1x10 6, 2x10 5, 5x10 5, 1.5x10 6, 2x10 6, stuck Propagation of stick-slip boundary: ( ) θ=27.0, ( ) θ=27.1, ( ) θ=27.2, ( ) θ=27.3 (stick on the right, slip on the left) Propagation of stick-slip boundary: ( ) θ=211.0, ( ) θ=212.0, ( ) θ=212.5, ( ) θ=213.0 (stick on the left, slip on the right) Distributed contact model for 2D motion with stiffness variation Distributed contact model for 3D motion with stiffness variation Hemi-sphere to flat surface contact Hysteresis curves for no separation case Hysteresis curves for no separation case Normalized hysteresis curves for different k Friction force trajectory for no separation case Friction force trajectory for separation case Contact model for wedge damper Blade and damper coordinate systems Wedge damper contact planes and coordinate systems Finite element models for a) blade b) wedge damper Tracking plot for first mode Tracking plot for second mode Optimal and frequency shift curves: 1 st mode Optimal and frequency shift curves: 2 nd mode Tracking plots a) 3 rd b) 7 th mode Optimum and frequency shift curves a) 3 rd b) 7 th mode Effect of normal load on rigid body motion of damper (dc component) Effect of normal load on rigid body motion of damper (ac component) Effect of rotational modes for normal load a) 100 b) 200 c) 1000 d) translational and rotational modes, translational modes Effect of number of contact points for normal load a)100 b) 200 c) 1000 d) Contact status of sample points on a) left b) right contact planes: stick, slip, separation) Schematic view of test case Finite element model for the test case xvi

17 7.18 Frequency shift curve Predicted normalized optimal curve and test data Block-to-block example Deformed shape and contour plot of total displacement Normal stiffness distribution a) small block b) large block Total normal contact stiffness Contact pressure Effect of number of modes used Tangential contact stiffness a) small block b) large block Total tangential contact stiffness Percent difference in tangential stiffness calculated for positive and negative traction Percent difference in normal contact stiffness a) small block b) large block Percent difference in tangential contact stiffness a) small block b) large block Parallel planes to apply normal and tangential forces for wedge damper Schematic view for bounded configuration Normalized contact stiffness in t-direction Normalized contact stiffness in v-direction Normalized contact stiffness in w-direction xvii

18 CHAPTER 1 1. INTRODUCTION 1.1. BACKGROUND AND MOTIVATION Forced vibration analysis of bladed disk assemblies is an important part of the design of gas turbine engines. One of the main problems of the gas turbine engines is the high cycle fatigue failure of turbine and compressor blades due to blade vibration resonance in the operating range. Preventing turbomachinery blade failures is an important issue for the gas turbine engine developers and therefore, avoiding excessive vibration in turbomachinery blading is crucial. Dry friction dampers dissipate energy in the form of heat due to the rubbing motion of the contacting surfaces resulting from relative motion. They are easy to manufacture, install, and they can withstand high temperatures; therefore, dry friction dampers are widely used by the gas turbine engine developers to suppress excessive vibration amplitudes. 1

19 There are several types of friction dampers used by the gas turbine engine manufacturers, which can be categorized in the following groups: blade-to-ground dampers, blade-to-blade dampers, shrouds and underplatform dampers. Figure 1.1 shows blade-to-ground damper, where damper is in contact with the coverplate or the disk, which is stationary with respect to the blade, and energy is dissipated through the friction contact due to the relative motion between the blade and the so-called ground. On the other hand, in blade-to-blade damper, damper is pushed against the adjacent blades as shown in Figure 1.2, and energy is dissipated due to the relative motion between the adjacent blades. Shroud as shown in Figure 1.3 is a metal piece, which sticks out from the sides of the blade. During engine operation, shrouds come into contact and energy is dissipated due to the relative motion at the contact interface. Underplatform dampers can as be considered as a special group of blade-to-blade dampers where the damper is placed under the blade platforms and comes into contact with the blade due to the centrifugal forces. Wedge damper, which is given in Figure 1.4, is a commonly used underplatform damper type where the damper has two inclined surfaces in contact with the adjacent blades. It is also referred to as a cottage-roof damper. Due to centrifugal forces created by the rotation of the engine, the damper contacts with the adjacent blade platforms and energy is dissipated at the two friction surfaces due the relative motion between the blades and the damper. However, due to the nonlinear nature of dry friction, dynamic analysis of the system becomes complicated. 2

20 Damper Blade 1 Damper Blade 2 Rivetted to Coverplate Disk Figure 1.1 Blade-to-ground damper Blade 1 Blade 2 Damper Disk Figure 1.2 Blade-to-blade damper 3

21 Shrouds Friction contact Interface with the adjacent blade Figure 1.3 Shroud fan stages Blade 1 Blade 2 Damper Disk Figure 1.4 Wedge damper 4

22 In the dynamic response analysis of structures having friction contact, two types of approaches are used mainly: the macroslip and the microslip methods. Due to its mathematical simplicity, the macroslip approach is widely used, in which the friction interface is modeled as a rigid body, and is entirely in slip stick or separation states. In the simplest macroslip model, the normal load acting on the surface is assumed to be constant and the contact point experiences stick or slip states. This simple friction model was applied to blade-to-ground dampers [Griffin, 1980; Dowell and Schwartz, 1983; Sirinivasan and Cutts, 1983; Menq and Griffin, 1985; Cameron et al., 1990; Wang and Chen, 1993] in order to predict forced responses and it is observed that, this simple friction model was accurate enough for the analysis of blade-to-ground dampers. However, as the contact kinematics gets complicated, more comprehensive friction models are necessary. In blade-to-blade dampers, shroud contacts or underplatform dampers, the relative motion at the contact interface can be elliptical (2D) instead of linear and in order to model these cases two-dimensional friction models were developed by Menq et al. [1991], Sanliturk and Ewins [1996] and Menq and Yang [1998]. When the relative motion stays on the contact plane, the contact normal load remains constant during the course of motion and the interface experiences stick-slip friction induced by the tangential motion. This type of contact kinematics arises from either the specific design of friction contact [Griffin, 1980; Dowell and Schwartz, 1983] or from the simplification of the analysis [Cameron et al., 1990; Ferri, 1996]. More generally, if the relative motion has normal component perpendicular to the contact plane, the normal motion will cause normal load variation and possible intermittent separation of the two 5

23 contacting surfaces. It can occur in various systems, such as shroud contact interfaces of fan blades and wedge damper contact of turbine blades in turbine jet engines [Menq et al., 1986a; Yang and Menq, 1998a, 1998b]. Friction models including normal load variation was developed by Menq et al. [1986b] for 1D tangential motion with in phase normal motion, by Yang et al. [1998] for out-of-phase normal motion, by Yang and Menq [1998c] for 3D contact motion. The effect of normal load variation was also addressed by Yang and Menq [1998c] for three-dimensional harmonic motion and by Chen et al. [2001] for three-dimensional periodic motion. Later, utilizing the similar criteria developed in [Yang and Menq, 1998c], Petrov and Ewins [2003] published their work for one-dimensional tangential motion with normal load variation and described an algorithm to determine transition angles numerically for periodic motion, similar to that reported in [Chen et al., 2001]. It should be noted that, all the above-mentioned friction models are for point contact, in which the entire interface is either in stick, slip, or separation states and partial slip is not possible. This so-called macroslip approach is widely used and works well if the normal load is small. However, microslip, or partial slip of the friction interface, becomes important and needs to be taken into account when the friction contact pressure is large. An interesting aspect of the microslip approach [Menq et al., 1986c, 1986d; Csaba, 1998; Quinn and Segalman, 2005] is the assumption that the friction force is transmitted across a contact area rather than through a point of contact and that a distributed version of Coulomb's law of friction determines which part of the contact 6

24 surface slips. The effects of microslip on the vibration of frictionally constrained structures and its significance were experimentally verified [Menq et al., 1986d; Filippi et al., 2004; Koh et al., 2005]. It is important to note that, due to its mathematical complexity, most of the models developed for microslip friction are for simple structures and for simple contact kinematics, which leaves real contact problems unaddressed. Microslip friction model, due to its mathematical complexity, is studied by limited number of researchers unlike the macroslip model. A continuous microslip friction model was developed by Menq et al. [1986c] where the damper is modeled as an elastic beam with a shear layer. Csaba [1998] developed a similar microslip friction model based on the model in [Menq et al., 1986c] where quadratic normal load distribution instead of was used and shear layer was removed for simplicity. The microslip friction models presented above can be applied to simple contact kinematics with one-dimensional motion and they do not consider the damper inertia. However, the slip motion in blade-to-blade and wedge damper is two-dimensional and hence it is necessary to develop microslip friction models that can model twodimensional slip motion. Moreover, the normal load distribution acting on the friction interface is kept constant which is not the case for shroud contacts, or wedge dampers [Menq et al., 1986a; Yang and Menq, 1998a]. Therefore, in order to accurately predict forced response of systems with complex contact kinematics, more sophisticated microslip friction models are necessary. 7

25 Finite element models are often used in the forced response analysis of complex structures, which result in many degrees of freedom (DOFs). Due to the friction contact, this results in large systems of nonlinear equations which need to be solved iteratively. This is a computationally expensive and also an inefficient process. Menq and Griffin [1985] developed a nonlinear forced response analysis method for steady state response of frictionally damped structures using finite element models. This method where the unknowns are reduced to only the nonlinear DOFs is used by several researchers [Chen and Menq, 1998; Yang and Menq, 1998b; Chen and Menq, 2001; Petrov and Ewins 2003; Petrov, 2004]. In addition to this structural modification techniques are as well applied to the analysis of bladed disk systems by Sanliturk et al. [2001] and Ciğeroğlu and Özgüven [2006], where the nonlinear dynamic stiffness matrix is obtained by structural modification techniques RESEARCH OBJECTIVE AND SCOPE The objectives of this research are to develop microslip friction models and a forced response analysis method for the analysis of bladed disk systems with friction dampers. In the development of the microslip friction models and the forced response method, the following issues are considered: 8

26 Damper inertia, Non-uniform normal load distribution, Variation of normal load distribution, 3D contact motion (2D tangential motion and 1D normal motion), Variation of contact stiffness (due to variation of contact area), Mode shape superposition approach. Most of the friction models available in literature are point contact (macroslip) models and majority of the microslip friction models does not consider the damper inertia. Therefore, in the modeling of microslip friction the following aims are identified. Firstly, the effect of damper inertia is to be included into the models developed. Different from macroslip where the contact of two bodies is a point, in microslip, the contact interface is a line or an area/surface; therefore, distributed friction models are needed to analyze microslip behavior. Secondly, for 1D, 2D and 3D motion stick and slip transition criteria will be developed for the distributed friction model. It should be noted that, the motion of contacting bodies can have a component along the normal direction of the friction interface which results in variation of the normal load acting on the contact interface. As a result of this, the transitions of stick and slip will be affected and it is even possible to have separation of contact interfaces; therefore, these effects should be included into the development of microslip friction modeling. Thirdly, since the contact interface is a line or an area/surface, it is possible to have contact parameters (preload, friction coefficient etc.) as functions of spatial variables, and the variation of contact parameters with respect 9

27 to spatial variables is to be included into the models to be developed. Finally, depending on the contact geometries, it is possible to have variation of the contact interface due to forces acting on the structures. This variation of contact interface can result in temporal variations of the contact parameters which will affect the stick, slip and separation states of contact interface. Consequently, microslip models which can model these effects will be developed. All of these will result in the development of the following friction models: 1D dynamic microslip friction model, 1D microslip friction model with normal load variation induced by normal motion, 3D microslip friction model with normal load variation 1D microslip friction model with stiffness and normal load variation, 3D microslip friction model with stiffness and normal load variation. Forced response of structures with nonlinearities is analyzed by applying receptance methods, where the DOFs are divided in to linear and nonlinear DOFs, and solving only the nonlinear equations obtained from the nonlinear DOFs. In these receptance methods, the total DOFs is reduced to the number of nonlinear DOFs; however, in order to accurately model microslip phenomenon, high numbers of nonlinear elements are needed. In addition to this, cyclic symmetry of bladed disk structures is damaged due to manufacturing tolerances; as a result of this the system has to be analyzed without using cyclic symmetry simplifications which increases the number of 10

28 nonlinear DOFs; consequently, the number of nonlinear equations to be solved. Therefore, it is aimed to develop a forced response analysis method in order to analyze frictionally constrained turbine blades efficiently. In addition to this, the method to be developed should be applicable to single and multiple harmonic forced response analysis LITERATURE REVIEW Friction Models In order to model complex contact kinematics simple stick slip friction was not sophisticated enough; therefore, more complicated friction models were developed. In order to model shroud fan stages Menq et al. [1986b] developed a friction model where the normal load acting on the contact interface varies in phase with tangential motion; in addition to this, authors derived analytical formulas for the resulting friction force. A more comprehensive friction model that considers normal load variation was developed by Yang et al. [1998]. Authors developed transition criteria for stick, slip and separation states for the case of general time varying normal load and for simple harmonic relative motion analytical formulas for the transition angles are derived. 11

29 Friction models explained above assumes tangential motion only in one direction. However, as in the case of shroud contact, blade-to-blade damper and underplatform damper, complex contact kinematics may result in two-dimensional tangential or slip motion. Menq et al. [1991] developed a two-dimensional friction model where the friction force is estimated by using an interpolation method. Later Menq and Yang [1998c], developed analytical transition criteria for stick, slip transitions. For circular motion, which is special case of two-dimensional motion, analytical expressions for transition angles and friction forces were derived. In this friction model flexible element at the contact interface is characterized by a 2x2 stiffness matrix. On the other hand, Sanliturk and Ewins [1996] extended the two-dimensional friction model developed by Menq et al. [1991]. In this model, flexibility of the friction contact is expressed by a single stiffness element. Yang and Menq [1998c] developed a three-dimensional friction model where the motion is decomposed into a two-dimensional in-plane tangential motion and a normal motion perpendicular to the tangential plane. The flexibility of the friction contact is expressed by a 3x3 stiffness matrix. Authors developed analytical transition criteria for stick, slip and separation states for general motion. Using a state by state simulation hysteresis curves for different circumstances are generated which are used in forced response calculations. 12

30 Menq et al. [1986c] developed a continuous microslip friction model, in which the friction damper was modeled as an elastic bar in contact with rigid ground and connected to a spring at the left end. A shear layer was included between the bar and the ground; therefore, it is possible to have linear deformation relative to the support before the beginning of slip [Bowden and Tabor, 1950]. Under the effect of uniform normal load distribution, partial-slip and gross-slip of the bar were studied. The developed microslip friction damper was analyzed with a single degree of freedom oscillator using Harmonic Balance Method (HBM), and the authors assessed that this microslip model showed approximately fifty percent reductions in the resonant response for high normal load cases. Furthermore, this developed friction model was used to explain turbine blade friction damping data and shroud damping data in [Menq et al., 1986d]. Csaba [1998] proposed a microslip friction model with a quadratic normal load distribution based on the model developed by Menq et al. [1986c] in which the shear layer was removed for simplicity. A single blade with a friction damper attached to the ground was analyzed in frequency domain and the author evaluated that macroslip predicted the response amplitudes much higher than the microslip model used. The beam model in [Menq et al., 1986c] was also addressed by Quinn and Segalman [2005]; where in order to investigate joint dynamics, authors obtain analytical expressions for quasistatic case excluding the inertial term, and developed a discrete model in order to numerically solve the same problem with the inertial effects included. From the 13

31 numerical analysis, authors deduced that, for low frequency excitations quasi-static model approximates the system response closely. Filippi et al. [2004] described a measurement method in order to determine the friction characteristics between two surfaces. The authors estimated the possible measurement errors and tried to eliminate or avoid them in the measurement process. Specimens were selected in order to have negligible inertial effects and hysteresis curves for different displacement amplitudes were given which showed the microslip behavior. Friction coefficient between the surfaces and contact stiffness in the direction of motion was also determined through the experiment. Koh et al. [2005] performed experiments on a spherical damper, where authors used a quasi-static setup, and a dynamic setup given in [Filippi et al., 2004]. Hysteresis curves for the spherical damper, which show the microslip phenomena, were obtained and equivalent stiffness and equivalent damping vs. displacement amplitude curves were calculated. These results were compared with the equivalent stiffness and damping curves determined from Mindlin s model. For quasistatic tests (low frequency) good agreement between the experimental results and Mindlin s model was observed. Equivalent stiffness calculated from dynamic test result was similar to the one obtained from Mindlin s model; however, the equivalent damping curve was overestimated by the model. Song et al. [2004] added a parallel spring to the parallel-series Iwan model and used this model to estimate the friction in the joints. The model parameters estimating the microslip behavior were determined from experimental results by applying a neural network algorithm. 14

32 Forced Response Analysis The equation of motion of a frictionally constrained structure can be given in matrix form as follows. ( t) (,, ) Mx + Cx + Kx = f + f x x t, (1.1) exc n where MCK,, are the mass matrix, damping matrix and stiffness matrix, and f, f are the excitation and nonlinear forcing vectors, respectively. The nonlinear nature of the exc n problem is due to f which is a function of the motion x ( t). In order to solve Equation n (1.1), three different approaches are used in literature: analytical approach, time domain simulation and frequency domain simulation by using Harmonic Balance Method (HBM). Analytical approach is proposed by Den Hartog [1931] where the nonlinear equation of motion is divided into two linear parts using appropriate boundary conditions and closed form solutions for a single degree of freedom oscillator. This method was extended to a two degrees of freedom oscillator by Yeh [1964]. Some other analytical works on friction nonlinearity were developed by Kato [1974], Marui and Kato [1984]. All these analytical works were developed for simple structures and they become impractical, for systems with large DOFs. 15

33 In time domain approach, the nonlinear equations of motion are integrated numerically [Menq and Griffin, 1985; Anderson and Ferri, 1990; Sanliturk et al., 1997; Yang et al., 1998; Yang and Menq, 1998c; Csaba, 1998; Chen and Menq, 2001, Ciğeroğlu and Özgüven, 2006] using a numerical integration technique, such as finite difference method or well known Runge-Kutta method. In time domain integration, the equations of motion together with the excitation and nonlinear forces are discritized in time domain and the resulting linear system of equations are solved to obtain the solution step by step. Time domain integration method can be implemented easily; however, steady state solutions can be obtained through transient solutions which increase the computational time considerably. The computation time increases even more due to stiff problems where small time steps are necessary to avoid numerical instabilities. Due to long computation times, time domain integration method is rarely used for practical problems; however, the time domain solution is valuable in comparing the accuracy of other numerical methods. In some cases, in order to decrease the time spent in transient solutions steady state solution estimates, which are obtained by other solution methods, are used as initial conditions [Csaba, 1998]. In Harmonic Balance Method (HBM), also called as Describing Function Method, the nonlinear forces are represented by Fourier series approximation and it is a widely used method in the forced response analysis of frictionally damped structures. Approximating the nonlinear forces by Fourier series, differential equations of motion are converted to a set of nonlinear algebraic equations, which are then solved for the 16

34 unknown harmonics. In general, friction forces are represented by single harmonic Fourier series [Menq and Griffin, 1985; Ostachowicz, 1990, Ferri and Bindemann, 1992; Whiteman and Ferri, 1996; Yang and Menq, 1998b; Yang et al., 1998; Sanliturk et al., 1997, 2001]; whereas, multi harmonic representations can be found in [Pierre et al., 1985; Ferri and Dowell, 1988; Wang and Chen, 1993; Kuran and Özgüven, 1996; Chen et al., 2000; Petrov and Ewins, 2003]. Analytical equations of motion are not available for complex structures; therefore, FE models are commonly employed. For structures having many DOFs this results in large systems of nonlinear equations, and it should be noted that, for multi-harmonic HBM the number of unknowns will increase as a multiple of number of harmonics included into the analysis. In order to decrease computation time, reduction methods with receptance approach was developed by Menq and Griffin [1985]. In this method, DOFs of the structure is divided into linear and nonlinear DOFs and iterative solution is only performed on the nonlinear DOFs. Since in frictionally damped structures, the nonlinear DOFs are only present at the friction contacts, and this, results in a substantial decrease in the number of nonlinear equations to be solved. This method is used by several researchers in the forced response analysis of frictionally damped bladed disks [Chen and Menq, 1998; Yang and Menq, 1998b; Chen and Menq, 2001; Petrov and Ewins 2003; Petrov, 2004]. 17

35 Structural modification techniques are as well employed in order to decrese computational efforts. Kuran and Özgüven [1996] proposed a modal supposition method for steady state harmonic analysis of nonlinear structures in order to decrease the number of unknowns by using the linear modes of the system. Tanrıkulu et al. [1993], Sanliturk et al. [2001] and Ciğeroğlu and Özgüven [2006] applied structural modification methods to determine the inverse of dynamic stiffness matrix, where nonlinear DOFs are considered as modifications to the linear system. Tuned or cyclically symmetric bladed disks have rotationally periodic sections, also called as sectors, which can be used to form the whole structure by simply rotating this section. If the excitation forces acting on each sector are identical through a phase difference, which is the case for turbomachinery applications, the response of whole bladed disk system can be obtained by only analyzing one sector. Therefore, cyclic symmetry reduces total number of nonlinear equations to the number of nonlinear DOFs associated with a sector. Bladed disks with cyclic symmetry are studied by Chen and Menq [1998, 2001], Sextro et al. [2001], Petrov [2004], Panning et al. [2004]. In the analysis microslip friction, in order to capture friction contact accurately, the number of contact points at the contact interface has to be large, which increases the number of nonlinear DOFs even for a tuned system. Moreover, manufacturing tolerances and techniques result in mistuning in the system which destroys the cyclic symmetry and therefore, complete bladed disk system has to be analyzed, which increases the number of 18

36 nonlinear equations and the methods described above become impractical, even though not impossible DISSERTATION OVERVIEW In the first chapter of this dissertation background information, and objective and scope of this research are provided. Chapter 2 describes the development of the analytical one-dimensional dynamic microslip friction model. In Chapter 3, 4 and 5, distributed microslip friction models are introduced, and development of 1D microslip friction model with normal load variation induced by normal motion, microslip friction model for 2D periodic motion and 3D microslip friction model with normal load variation induced by normal motion are presented. Developed models are applied to a blade-to-ground damper example. Chapter 6 describes distributed friction models for stiffness variation and transition crieteria for 2D and 3D microslip friction models with stiffness variation are presented. In Chapter 7 a wedge damper model and forced response prediction method for frictionally constrained blades are presented. Chapter 8 presents a method in order to estimate contact stiffnesses used in the friction models and finally, conclusion and future work is given in Chapter 9. 19

37 CHAPTER 2 2. ONE-DIMENSIONAL MICROSLIP FRICTION MODEL 2.1. INTRODUCTION In the dynamic response analysis of structures having friction contact, two types of approaches are used, the macroslip and the microslip methods. Due to its mathematical simplicity, the macroslip approach is widely used, in which the friction interface is modeled as a rigid body, and is entirely in slip or stick states. This method is acceptable if gross-slip occurs at the friction interface, which is possible if the normal load acting on the interface is small. On the other hand, the microslip method is mathematically complicated; however, since the friction interface is modeled as an elastic body, it is capable of modeling partial slip, which occurs if the normal load acting on the interface is high. For those cases, macroslip model results in stuck interface and estimates no energy dissipation through friction contact. 20

38 Menq et al. [1986c] developed a microslip model, in which the friction damper was modeled as an elastic bar in contact with a rigid ground and connected to a spring at the left end. A shear layer was included between the bar and the ground; therefore, it is possible to have linear deformation relative to the support before the beginning of slip [Bowden and Tabor, 1950]. Under the effect of uniform normal load distribution, partialslip and gross-slip of the bar were studied. A single degree of freedom oscillator with a friction damper was analyzed by applying Harmonic Balance Method (HBM), and the authors assessed that for high normal load cases this microslip model showed approximately fifty percent reductions in the resonant response. In addition to this, Menq et al. [1986d] explained experimental friction damping data by using this microslip friction model. Csaba [1998] proposed a microslip friction model with a quadratic normal load distribution based on the model developed by Menq et al. [1986c] in which the shear layer was removed for simplicity. A single blade with a friction damper attached to the ground was analyzed in frequency domain and the author evaluated that macroslip predicted the response amplitudes much higher than the microslip model used. Filippi et al. [2004] described a measurement method in order to determine the friction characteristics between two surfaces. The authors estimated the possible measurement errors and tried to eliminate or avoid them in the measurement process. Specimens were selected in order to have negligible inertial effects and hysteresis curves for different displacement amplitudes were given which showed the microslip behavior. Friction 21

39 coefficient between the surfaces and contact stiffness in the direction of motion was also determined through the experiment. Song et al. [2004] added a parallel spring to the parallel-series Iwan model and used this model to estimate the friction in the joints. The model parameters estimating the microslip behavior were determined from experimental results by applying a neural network algorithm. The objective of this chapter is to develop a dynamic one-dimensional microslip friction model by including the inertia of the damper. For this purpose, a one-dimensional beam model that is similar to the one introduced by Menq et al. [1986c], but includes the inertia of the beam and has a non-uniform normal load distribution, is proposed and is shown in Figure 2.1. The beam is connected to the ground from the left end with a spring in order to include strain hardening effects, and a harmonic excitation is applied to the right end of the beam. A shear layer, which permits elastic deformation of the beam before the occurrence of slip, is inserted between the beam and the ground. The shear layer can be visualized as distributed springs connected to the beam and in contact with the ground, and obey the Coulomb friction law with a constant friction coefficient throughout the length of the beam. Since this is a one-dimensional model normal load applied on the beam is assumed to be directly transmitted to the shear layer. The system is analyzed for uniform, and convex and concave normal load distributions, which result in distinct stick-slip transitions along the contact interface. 22

40 x q(x) β E, A, ρ f Shear Layer F(t) L Figure D Microslip model In the remaining parts of this chapter, determination of stick-slip transitions along the contact interface for different normal load distributions is presented and force displacement relationships for constant, convex and concave normal load distributions are derived. The effect of excitation frequency is analyzed and results obtained for different normal load distributions are compared with each other and with a representative point contact model. 23

41 2.2. DETERMINATION OF STICK-SLIP TRANSITIONS An analytical approach is developed to determine the stick-slip transitions of the steady state solution of the frictionally constrained system when subjected to harmonic excitation. In the proposed approach, according to the excitation frequency, single mode of the system is considered in the steady state solution for simplicity; consequently, phase difference among spatially distributed points is neglected Determination of Contact Elastic Force In order to determine stick-slip transition in the microslip model proposed in Figure 2.1, the build-up of elastic force along the contact interface between the beam and the ground is first examined. As a starting point, the microslip model is analyzed assuming the beam is completely stuck and the contact elastic force obtained through this analysis is used to determine the effects of damper inertia. In conjunction with the normal load distribution, it will later be employed to determine the transitions between stick and slip, and to obtain the resulting friction force. The equations of motion for the completely stuck beam can be written as 2 2 u 2 u EA ku = ρaω 2 2 x θ, (2.1) 24

42 with the following boundary conditions u u EA = βu EA = F x x ( 0, θ), sin ( ) x= 0 x= L 0 θ, (2.2) where ω, t, E, ρ, A, L and k are excitation frequency, time, modulus of elasticity, density, cross-sectional area, length and the shear layer stiffness of the beam, respectively, and θ = ωt. Solving the partial differential equation analytically, the contact elastic force acting on the beam can be obtained. The spatial distribution of the resulting contact elastic force for different excitation frequencies is given in Figure 2.2 to illustrate the effects of damper inertia, and is seen to be similar to the mode shape of the constrained beam. Moreover, Figure 2.3 shows the build-up of the contact elastic force, together with three different normal load distributions, while increasing the amplitude of low frequency excitation. It is seen that as the excitation amplitude increases the contact elastic force generated in the shear layer becomes larger than the allowable value, depending on the normal load, and the contact interface starts to slip at a location depending on the distribution of normal load. For the example shown in Figure 2.3, the contact interface starts to slip at the right end of the beam for both the constant and the convex normal load distributions; whereas it starts to slip somewhere around the middle of the beam for the concave normal load distribution. It can also be concluded that for the first vibration mode both the constant and the convex normal load distributions lead to similar stick-slip 25

43 transition along the friction interface, which is a stick region at the left side and a slip region at the right side. The slip region propagates towards the left end of the beam, reverses and repeats. Since showing similar stick-slip transition, they are analyzed together, where the case of constant normal load is considered as a special case of the convex normal load distribution. On the other hand, for the case of concave normal load distribution, the stick-slip transition along the friction interface is composed of two stick regions at the left and the right sides of the beam, and in-between is a slip region, which propagates towards the right and the left ends of the beam, reverses and repeats Two-Region Friction Interface It is concluded in section that, as the amplitude of the excitation force at the right end of the beam increases, convex normal load distribution results in a two-region friction interface, a stuck region on the left side and a slip region on the right. Figure 2.4 shows the microslip model with a two-region friction interface where, L 1 is the length of the stuck region and q( x) is the normal load distribution over the interface. Since it is aimed to obtain hysteresis curves, instead of force input displacement input is used in the model. The nonlinear partial differential equations defining this system are given as 2 2 u 2 u EA 2 k ( u ( x, θ) w ( x )) = ρ A ω 2 0 x L1, (2.3) x θ 26

44 Normalized Contact Elastic Force [μn] Hz 1500 Hz 5000 Hz Hz Normalized Spatial Distribution [L] Figure 2.2 Effect of excitation frequency on contact elastic force 27

45 Normalized Contact Elastic Force [μn] Convex Normal Load Concave Normal Load Constant Normal Load Increasing Excitation Amplitude Normalized Spatial Distribution [L] Figure 2.3 Build up of contact elastic force and its comparison with various normal load distributions 28

46 x q(x) β u(l) = a sin(θ) E, A, ρ L 1 L Figure 2.4 Two-region friction interface due to convex normal load distribution 2 2 u 2 sgn u EA q 2 μ ( x) = ρaω u L x L x θ θ 2 1, (2.4) with the following boundary and compatibility conditions, u EA = βu θ u L θ = a x x= 0 ( 0, ), (, ) sin ( + L1 L1 ) θ, (2.5) + u u u( L1, θ) = u( L1, θ), =, (2.6) x x 29

47 where, μ is the friction coefficient and w( x ) is the displacement of the shear layer at the time of sticking, and for the regions which have not slipped w( x ) = 0. In Eqs. (2.3), (2.4) and (2.6), L 1 is still an unknown; however, it should be noted that the friction force calculated at x = L 1 from Eqs. (2.3) and (2.4) should be the same, therefore equating these friction forces results in the following equation ( ( 1, θ) ( 1) ) μ ( ) k u L w L = q L1, (2.7) from which L 1, the change of stick-slip regions, can be determined. Eqs. (2.3) and (2.4) for θ between 0 and π/2 can be rewritten as follows 2 2 u 2 u EA ku = ρaω 0 x L 2 2 1, x θ (2.8) 2 2 u 2 u EA μq 2 ( x) = ρaω L 2 1 x L, x θ (2.9) where, w( x ) is taken as zero for the points that have not been slipped as pointed out formerly. It should be noted that, Eqs. (2.8) and (2.9) are linear and solution can be obtained in terms of L if u is taken in the following form 1 30

48 ( ) ( θ) ( ) (, θ) u1 x + u2 x, 0 θ L1 u =. (2.10) v1 x + v2 x L1 θ L Since only the first mode of the beam is considered, there is no phase difference between the points throughout the length; hence, inserting Eq. (2.10) into Eqs. (2.8) and (2.9) yields the following results ( ) = sinh ( λ ) + cosh ( λ ) (, θ ) = ( sinh ( γ ) + cosh ( γ )) sin ( θ) ( ) = ( ) + + (, θ ) = ( sin ( α ) + cos( α )) sin ( θ) u x D x D x u x C x C x v x Q x c x c v x C x C x 2 3 4, (2.11) where, k ρ k ρ λ α ω γ ω EA E EA E 2 =, =, =. (2.12) The unknown coefficients in Eq. (2.11) are in terms of L 1 and can be determined using the boundary and the compatibility conditions (2.5) and (2.6), respectively. Using Eq. (2.7) together with Eqs. (2.11) and (2.12) a nonlinear equation for is obtained and given as follows L 1 31

49 ( 1) `( 1)( 1) + ( ) ( 1) α α + βα ( ) μq L Q L + L L Q L Q L EAcos α L L + β α L L ( ) sin ( ) 1 1 kea EA L L 1 ( θ ) a sin = 0 (2.13) where, ( ) ( 1) + cosh ( 1) ( ) + sinh ( ) EAγ sinh γl β γl β = β = γ L EA 1 EA γ cosh γ L L 1 β γ 1 x μ Q( x) = Q` ( ξ ) dξ, Q`( x) = q( ) d EA 0 0 x, (2.14) ξ ξ. (2.15) It should be noted that, Eq. (2.13) is nonlinear and analytical solution for L 1 cannot be obtained; however, it is possible to solve θ analytically for L varying between 0 and L. The solution of Eqs. (2.3) and (2.4) for θ between 0 and π/2 gives the loading curve of the resulting hysteresis loop, and Menq et al. [6] showed that if the loading curve of an elastic element is known and if the normal load distribution is time invariant it is possible to construct the hysteresis curve for cyclic motion from this result. Applying this approach, hysteresis curve for a cycle is constructed, and the effect of excitation amplitude and frequency are analyzed and the results are presented in section

50 Three-Region Friction Interface It is concluded in section that a concave normal load distribution results in a three-region friction interface composed of two stick regions at the left and right sides and a slip region at the middle of the beam. The microslip model for this case is given in L1 2 L ( ) Figure 2.5, where,, and q x are the length of the stuck region on the left side, the beginning of the stuck region on the right side, and the normal load distribution over the interface, respectively. The nonlinear partial differential equations defining this system are given as 2 2 u 2 u EA k 2 ( u( x, θ) w( x) ) = ρaω 0 x L 2 1, (2.16) x θ 2 2 u 2 sgn u EA q 2 μ ( x) = ρaω u L x L x θ θ 1 2 2, (2.17) 2 2 u 2 u EA k 2 ( u( x, θ) w( x) ) = ρaω L 2 2 x L, (2.18) x θ with the following boundary and compatibility conditions, u EA = βu θ u L θ = a x x= 0 ( 0, ), (, ) sin ( ) θ, (2.19) 33

51 x q(x) β u(l) = a sin(θ) E, A, ρ L 1 L 2 L Figure 2.5 Three-region friction interface due to concave normal load distribution u u, =,, = x x + ( 1 θ) u( L1 θ) u L + ( 2 θ) u( L2 θ) u L + L1 L 1 u u, =,, = x x + L2 L2, (2.20) where, w( x ) is the same as in section It should be noted that, in Eqs. (2.16), (2.17), (2.18) and (2.20) and L are unknowns and in order to determine them, the friction L1 2 force calculated at x = L 1 from Eqs. (2.16) and (2.17), and the friction force calculated at 34

52 x = L 2 from Eqs. (2.17) and (2.18) are equated, respectively, resulting in the following equations ( ( 1, θ) ( 1) ) = μ ( 1) ( ( 2, θ) ( 2) ) = μ ( 2) k u L w L q L k u L w L q L. (2.21) Following a similar procedure described in section 2.2.2, Eqs. (2.16), (2.17) and (2.18) becomes 2 2 u 2 u EA ku = ρaω 0 x L 2 2 1, x θ (2.22) 2 2 u 2 u EA μq 2 ( x) = ρaω L 2 1 x L2, x θ (2.23) 2 2 u 2 u EA ku = ρaω L x L. (2.24) x θ These equations are linear and the solutions are in terms of and L. If u is taken in the following form, L1 2 ( ) ( θ ) ( ) (, θ ) ( ) (, θ ) u1 x + u2 x, 0 x L1 u = v1 x + v2 x L1 x L2, (2.25) w1 x + w2 x L2 x L 35

53 and only the first mode is considered, inserting Eq. (25) into Eqs. (2.22) to (2.24) leads to the following solution, ( ) = sinh ( λ ) + cosh ( λ ) (, θ ) = ( sinh( γ ) + cosh( γ )) sin( θ) ( ) = ( ) + + (, θ) = ( sin( α ) + cos( α )) sin( θ) ( ) = sinh ( λ ) + cosh ( λ ) (, θ ) = ( sinh ( γ ) + cosh ( γ )) sin ( θ) u x D x D x u x C x C x v x Q x c x c v x C x C x w x D x D x w x C x C x 2 5 6, (2.26) where, α, γ and λ are given in Eq. (2.12). The unknown coefficients in Eq. (2.26) are in terms of and L, and can be determined by applying the boundary and the L1 2 compatibility conditions given in Eqs. (2.19) and (2.20), respectively. Nonlinear equations to determine and L can then be obtained from Eq. L1 2 (2.21) and they are given as EAγ sinh ( γ L1) + cosh ( γl1) kasin ( θ) EAλ β kd1 sinh ( λl1) cosh ( λl + 1) +,(2.27) β C5sinh L C6 L ( ( )) ( ) ( γ ) + cosh ( γ ) μq( L ) ( ( ) ( )) ( ) ( ) + ( ) sinh λ L L C sinh γl + C cosh γl kasin θ kd = 0 1 ( ) μq L2 cosh λl C5sinh γl C6cosh γl = 0,(2.28) 36

54 where, D1, D3, C 5 and C6 are functions of L1 and L2, and they are given in the appendix. Using a similar approach described in section the complete hysteresis curve can be obtained from the solution for θ between 0 and π/2. The effect of excitation amplitude and frequency are analyzed and compared with the convex normal load case, and the results are presented in section RESULTS In the previous sections, one-dimensional microslip models for two different normal load distributions are presented. In the analyses performed in this section, the normal load distribution for the two-region friction interface is taken as constant, N / L, where N is the total normal load applied, and for the three-region friction interface the following quadratic distribution is used ( L) 4x x q( x) = q0 + q2, (2.29) 2 L where, and q q indicate the maximum and minimum normal loads acting on the q0 0 2 beam, and the total normal load is ( 2 3) N = q q L

55 Stick-Slip Transitions Figure 2.6 shows the change of stuck region length, L 1, as a function of temporal variable θ for the two-region friction interface. As predicted from the analysis of the completely stuck system, slip starts from the right end of the beam and propagates towards the left end. Likewise, Figure 2.7 shows the changes of and L as functions L1 2 of temporal variable θ for the three-region friction interface due to the normal load distribution given in Eq. (2.29). Slip starts around the middle of the beam and it propagates towards the right and the left ends; furthermore, slip region reaches the right end of the beam first and continues to propagate towards the left end resulting in grossslip finally Equivalent Spring Force and Damping Force Hysteresis curves, establishing the relationship between the harmonic input displacement and the resulting net force at the right end of the model, are employed to characterize the effectiveness of the frictional constraint. For the two-region friction interface shown in Figure 2.4, hysteresis curves for different excitation frequencies are given in Figure 2.8, and here ω c denotes the first natural frequency of the completely stuck system. It is seen that, for low excitation frequencies hysteresis curves are close to each other, which is in agreement with the experimental results obtained by Filippi et al. 38

56 [2004], but there is a small rotation in clockwise direction. For higher frequencies, hysteresis curves rotate more in the clockwise direction and the area enclosed by them increases. Equivalent spring force and damping force of a frictional constraint can be obtained from the Fourier coefficients of a hysteresis curve. In this chapter, f s denotes the spring force, being the component in phase with the input displacement, and f c o denotes the damping force, being the 90 out of phase component. Rotation of a hysteresis curve in clockwise direction indicates a decrease in Fourier coefficient f s and increase in area indicates an increase in Fourier coefficient f c. In Figure 2.9 and Figure 2.10, non-dimensionalized Fourier coefficients F s ( f ka) * and c ( s F f * c ka) vs. normalized displacement amplitude ( aa min ) for the two-region friction interface are given for different excitation frequencies, respectively, where, * k is the stiffness of the system when the frequency is zero and a min is the minimum displacement to cause slip, which is μ N/ kl for this case. It is seen that, non-dimensionalized Fourier coefficient F s (normalized stiffness) is constant until the slip starts and after this point it decreases with increasing displacement amplitude; moreover, it decreases with increasing excitation frequency. It should be noted that for ω = 0.5ω c F s curve can become negative, o indicating that equivalent elastic force is 180 out of phase. On the other hand, normalized Fourier coefficient F c (normalized damping) is zero until the slip starts and 39

57 1.0 Complete Stuck 0.8 Slip Normalized L 1 [L] Gross Slip θ [π] Figure 2.6 Temporal change of two-region friction interface Normalized L 1, L Stuck Slip Stuck L 1 /L L 2 /L θ [π] Figure 2.7 Temporal change of three-region friction interface 40

58 Normalized Force [μn] ω c 0.2ω c 0.3ω c 0.5ω c Normalized Displacement [a] Figure 2.8 Effect of excitation frequency on hysteresis: two-region friction interface Normalized Fourier Coefficient F s [k*a] Normalized Displacement Amplitude [a min ] 0.01ω c 0.2ω c 0.3ω c 0.5ω c Figure 2.9 Effect of excitation frequency on Fourier coefficient F s : two-region friction interface 41

59 Normalized Fourier Coeffiecient F c [k*a] Normalized Displacement Amplitude [a min ] 0.01ω c 0.2ω c 0.3ω c 0.5ω c Figure 2.10 Effect of excitation frequency on Fourier coefficient F c : two-region friction interface subsequently, it increases with increasing displacement amplitude, reaches a maximum and becomes stable. In addition to this, although the difference between the curves become insignificant for large displacement amplitudes; for low amplitudes, which is the case for high normal load applications, Fourier coefficient F c increases as the excitation frequency increases. In the harmonic response analysis of a structurally damped system, the damping term is included into the stiffness term resulting in a complex stiffness, which can be written as k (1+ ηi) ; where, k, η and i are the stiffness of the system, the structural damping coefficient and the imaginary number, respectively. Therefore, 42

60 normalized Fourier coefficient F c can be treated as the structural damping coefficient of the microslip friction model; hence, increase in damping of the system. F c expresses an increase in the structural Hysteresis curves and Fourier coefficients for the three-region friction interface are given in Figure 2.11 to Figure 2.13, and similar conclusions as in the case of tworegion friction interface can be drawn from them. In the next section, both friction interfaces are compared with each other and differences between them are presented. 1.5 Normalized Force [μn] ω c 0.2ω c 0.3ω c 0.5ω c Normalized Displacement [a] Figure 2.11 Effect of excitation frequency on hysteresis: three-region friction interface 43

61 Normalized Fourier Coefficient F s [k*a] ω c 0.2ω c 0.3ω c 0.5ω c Normalized Displacement Amplitude [a min ] Figure 2.12 Effect of excitation frequency on Fourier coefficient F s : three-region friction interface Normalized Fourier Coefficient F c [k*a] ω c 0.2ω c 0.3ω c 0.5ω c Normalized Displacement Amplitude [a min ] Figure 2.13 Effect of excitation frequency on Fourier coefficient F c : three-region friction interface 44

62 Effect of Normal Load Distribution In order to examine the effect of normal load distribution on microslip of the friction constraint, comparison of the two-region and the three-region friction interfaces are presented in this section. Both friction interfaces are subject to identical total normal load and are analyzed according to the developed approach. The resulting Fourier coefficients Fc and F s are compared in Figure 2.14 and Figure 2.15, respectively, in which the displacement amplitude for both interfaces is normalized with respect to a min of the two-region friction interface. It is seen that due to the nature of concave normal load distribution specified in this comparison, ( ) q q q =, the three-region friction interface starts to have partial slip at lower vibration amplitude when compared to the two-region friction interface. In other words, the three-region friction interface starts to generate friction damping, and thus lead to attenuated stiffness, at lower vibration amplitude. However, even though the two-region friction interface requires higher vibration amplitude to start slip, as the vibration amplitude increases the resulting friction damping increases rapidly when compared to that produced by the three-region friction interface. At the same time, after starting slip the equivalent stiffness of the two-region friction interface decreases faster than that of the three-region friction interface, and finally both friction damping and stiffness of the two interfaces become comparable at higher vibration amplitude. This result illustrates that, normal load distribution has significant effect on the microslip characteristics of a friction interface, and accurate prediction of its effect on equivalent stiffness and friction damping is important. 45

63 Normalized Fourier Coefficient F c [k*a] region Friction Interface 3-region Friction Interface Normalized Displacement Amplitude [a min ] 6 Figure 2.14 Comparison of F c for both friction interfaces (ω=0.3ω c ) Normalized Fourier Coefficient F s [k*a] region Friction Interface 3-region Friction Interface Normalized Displacement Amplitude [a min ] Figure 2.15 Comparison of F s for both friction interfaces (ω=0.3ω c ) 46

64 Comparison with Point Contact Model A point contact friction model is defined in Figure The resulting friction damping and equivalent stiffness of this point contact model in terms of vibration amplitude will be compared with those of the two-region friction interface model presented earlier. The unknown stiffness values, β, k and k s, are determined in order to satisfy the following three conditions: both models start gross-slip at the same vibration amplitude, and both models have the same equivalent stiffness values in complete stuck and fully slipping states. The resulting Fourier coefficients Fc and F s are compared in Figure 2.17 and Figure 2.18, respectively, in which the displacement amplitude for both interfaces is normalized with respect to a min of the two-region friction interface. For high vibration amplitude, it is seen that the results agree each other well since gross-slip occurs in both models. However, at low vibration amplitude, while the point contact model remains fully stuck partial-slip occurs in the microslip model, and thus increasing friction damping and attenuated stiffness are predicted, which are also observed in experimental results [Filippi et al., 2004]. It is evident that until the gross-slip occurs the effects of microslip are very significant. 47

65 Comparison with Numerical Solution In order to validate the analytical method developed, time integration is performed on the following nonlinear partial differential equation τ 2 2 u 2 u EA τ 2 ( u, x) = ρaω 2 x θ k( u w), for stuck, (2.30) = w sgn μ q ( x ), for slip θ ( ux, ) where, w( x ) is the displacement of the shear layer as defined in section 2.2.2, and the boundary conditions are given in Eq. (2.2). For the numerical solution technique an implicit finite difference scheme is employed and Figure 2.19 shows the change of normalized stuck region for the two-region friction interface obtained from the analytical and steady state numerical solutions at low frequency. It is seen that, both solutions agree each other well in the determination of stick-slip transitions. 48

66 β' k' x = a sin(θ) N k s Figure 2.16 Equivalent point contact model Normalized Fourier Coefficient F s [k*a] region Friction interface Point Contact Model Normalized Displacement Amplitude [a min ] Figure 2.17 Comparison of F s for microslip and point contact models 49

67 Normalized Fourier Coefficient F c [k*a] region Friction Interface Point Contact Model Normalized Displacement Amplitude [a min ] Figure 2.18 Comparison of F c for microslip and point contact models Normalized L 1 [L] Numerical Solution Analytical Solution θ [π] Figure 2.19 Comparison of numerical and analytical solutions 50

68 2.4. CONCLUSIONS A one-dimensional dynamic microslip model is presented in this chapter. An analytical approach is developed to determine the stick-slip transitions of the steady state solution of the frictionally constrained system when subjected to harmonic excitation. In the proposed approach, according to the excitation frequency single mode of the system is considered in the steady state solution for simplicity; consequently, phase difference among spatially distributed points is neglected. The proposed model is analyzed for three different normal load distributions, resulting in two distinct friction interfaces. The tworegion friction interface is composed of a stuck region on the left side and a slip region on the right side of the beam, whereas the three-region friction interface has two stuck regions at the left and the right sides and a slip region in between them. Moreover, the effects of excitation frequency on the resulting hysteresis curves as well as the equivalent stiffness and damping are examined. It is also shown that, even when a friction interface is subjected to identical total normal load, normal load distribution has a significant effect on the equivalent damping and stiffness of the frictionally constrained system. Although only two-region and three-region friction interfaces are discussed in this chapter, other complicated multi-region friction interfaces are possible depending on normal load distribution and excitation frequency. It is possible to extend the developed microslip model to friction interfaces having complicated multiple regions. However, while the number of stick-slip regions increases, the number of unknown variables 51

69 increases, and so does the number of needed nonlinear equations similar to Eqs. (27) and (28). Moreover, in the proposed approach, according to the excitation frequency single mode of the constrained system is employed in the steady state solution for simplicity; consequently, phase difference among spatially distributed points is neglected. In order to determine the phase difference in terms of spatial distribution, it is necessary to include multiple modes of the system into the solution. This issue will be discussed in future investigation. Although the microslip friction model presented in this chapter is based on a beam model, which is simpler than many engineering structures, it provides a better understanding of the effects of normal load distribution and damper inertia on the stickslip transitions. For more complicated engineering structures, since numerical methods are often required to obtain the stick-slip transitions and then friction force distributions, the developed method can be used as a basis of comparison, where the numerical results for a simplified model can be verified by the analytical solutions obtained. 52

70 CHAPTER 3 3. A MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION INDUCED BY NORMAL MOTION 3.1. INTRODUCTION Mechanical systems having frictionally constrained interfaces often involve complex contact kinematics induced by the relative motion between moving components [Padmanabhan, 1994; Yang 1996]. When the relative motion stays on the contact plane, the contact normal load remains constant during the course of motion and the interface experiences stick-slip friction induced by the tangential motion. This type of contact kinematics arises from either the specific design of friction contact [Griffin, 1980; Dowell and Schwartz, 1983] or from the simplification of the analysis [Cameron et al., 1990; Ferri, 1996]. More generally, if the relative motion has normal component perpendicular to the contact plane, the normal motion will cause normal load variation 53

71 and possible intermittent separation of the two contacting surfaces. It can occur in various systems, such as shroud contact interfaces of fan blades and wedge damper contact of turbine blades in turbine jet engines [Menq et al., 1986a; Yang and Menq, 1998a]. In order to determine the forced response of shrouded fan stages, the stick-slipseparation analysis was undertaken by Menq et al. in 1986b, in which the normal motion was assumed to be in phase with the tangential motion and analytical formulas for the resulting contact friction force were derived. A more comprehensive model dealing with normal load variation was developed by Yang et al. in They developed analytical criteria for stick-slip-separation transition when subjected to general time varying normal load and derived analytical formulas for transition angles when the relative motion was simple harmonic motion. The effect of normal load variation was also addressed by Yang and Menq [1998c] for three-dimensional harmonic motion in 1998 and by Chen et al. [2001] for three-dimensional periodic motion in Later in 2003, utilizing the similar criteria developed in [Yang and Menq 1998c], Petrov and Ewins [2003] published their work for one-dimensional tangential motion with normal load variation and described an algorithm to determine transition angles numerically for periodic motion, similar to that reported in [Chen et al.,2001]. They applied their method to a bladed disk system, in which each blade was connected to the neighboring blade through a single nonlinear element. 54

72 It should be noted that, all the above-mentioned friction models are for point contact, in which the entire interface is either in stick, slip, or separation states and partial slip is not possible. This so-called macroslip approach is widely used, and works well if the normal load is small. However, microslip, or partial slip of the friction interface, becomes important and needs to be taken into account when the friction contact pressure is large. An interesting aspect of the microslip approach [Menq et al., 1986c, 1986d; Csaba, 1998; Quinn and Segalman, 2005, Cigeroglu et al., 2006] is the assumption that the friction force is transmitted across a contact area rather than through a point of contact and that a distributed version of Coulomb's law of friction determines which part of the contact surface slips. The effects of microslip on the vibration of frictionally constrained structures and its significance were experimentally verified [Menq et al., 1986d; Filippi et al., 2004; Koh et al., 2005]. It is important to note that, due to its mathematical complexity, most of the models developed for microslip friction are for simple structures and for simple contact kinematics, which leaves real contact problems unaddressed. Menq et al. [1986c] developed a continuous microslip friction model, in which an elastic bar having a uniform normal load distribution and in contact with the rigid ground was studied. A shear layer, which allows elastic deformations before the beginning of slip, was placed in between the rigid ground and the bar. The elastic bar is connected to a spring at the left end and analyzed under the effect of a static force applied at the right end. The developed microslip friction damper is analyzed with a single degree of freedom 55

73 oscillator using Harmonic Balance Method, and the results indicated fifty percent reductions in the resonant response for high normal load distributions. Furthermore, this developed friction model was used to explain turbine blade friction damping data and shroud damping data in reference [Menq et al., 1986d]. Based on the model developed in [Menq et al., 1986c] and described above, Csaba [1998] proposed a microslip friction model with a quadratic normal load distribution, where the shear layer is removed for simplicity. A single blade with a friction damper was analyzed and it was observed that predictions of vibration amplitudes from macroslip model were much higher than predictions of the microslip model. The beam model in [1986c] was also addressed by Quinn and Segalman [2005]; where in order to investigate joint dynamics, authors obtained analytical expressions for quasistatic case excluding the inertial term, and developed a discrete model in order to numerically solve the same problem with the inertial effects included. From the numerical analysis, authors deduced that, for low frequency excitations quasi-static model approximates the system response closely. Cigeroglu et al. [2006] developed a one-dimensional dynamic microslip friction model including the inertia of the damper, based on the beam model developed in [1986c]. Three different normal load distributions resulting in two distinct friction interfaces were considered and the analytical solutions considering the first vibration mode of the elastic bar were developed. The effect of excitation frequency on the 56

74 hysteresis curves and Fourier coefficients was presented and the results obtained were compared with each other. This chapter presents a two-dimensional (2D) microslip friction model, in which the relative motion between the two contacting planes is two-dimensional and can be resolved into two components. The tangential component induces stick-slip friction while the normal component causes normal load variation and possible separation. The model is a distributed parameter model, in which the transition criteria developed in [Yang et al., 1998] are employed to characterize the stick-slip-separation of the contact interface. Since only single harmonic motions are considered, the stick-slip-separation transition angles associated with any point in the contact area can be analytically determined within a cycle of motion. Consequently, the spatial distributions and time variances of these transition angles over the contact interface and the associated 2D contact friction maps can be determined. Along with an iterative multi-mode solution approach utilizing Harmonic Balance Method (HBM), the obtained 2D contact friction maps can be employed to determine the forced response of frictionally constrained structures. In the approach, the forced response is constructed in terms of the free mode shapes of the structure; consequently, it can be determined at any excitation frequency and for any type of normal load distribution. Two examples, a one-dimensional beam like damper and a more realistic blade to ground damper, are employed to illustrate the predictive abilities of the developed model. 57

75 3.2. TWO-DIMENSIONAL MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION In Figure 3.1, planar contact of two bodies is given where the gray region is the contact interface composed of distributed springs, representing normal stiffness and tangential stiffness. The orientation of the contact plane is assumed to be invariant as the amplitude of vibration is relatively small. The global coordinate system is denoted by x y z with respect to which the displacements of bodies A and B are defined while p q r is the contact plane coordinate system where the contact plane is defined as q = 0. Any point in the contact plane coordinate system can be transferred to the global coordinate system by a translation and a rotation as follows x p po y = R q + q. (3.1) o z r r o ] T o o o In this equation, R is the orientation matrix and [ p q r is the position of p q r in x y z. The spatial domain of the contact interface is specified in p q r. Any point within the domain is denoted by [ 0 ] can be determined from the following equation c c T p r and its coordinate in x y z 58

76 xc pc po y c = R 0 + q o. (3.2) z c r c r o The contact preload and its distribution over the contact area can be determined through static analysis. When vibrating, the dynamic motions of the two bodies, associated with ] T c c c any contact point [ x y z in x y z, are denoted by ( x, y, z, t) d and A c c c d B( xc, yc, zc, t), respectively, in which t is the temporal variable. The relative motion in p q r can then be determined from the following relation u( pc, rc, t) v( p,, ) 1 c rc t [ A B. (3.3) = R d d ] wp ( c, rc, t) In general, the relative motion is three-dimensional and has tangential component [ ] T u w and normal component v. For simplicity, this chapter focuses on a twodimensional version, in which while the normal motion v is retained, the two bodies move with respect to each other on the contact plane back and forth along the p direction. In other words, the r component of the relative motion, w, is assumed to be zero, and the relative motion is characterized by the in-plane motion u and out of plane motion v, associated with p and q axis, respectively. The r axis is used to define the contact interface together with the p axis. 59

77 y z o q q body A body B x p Figure 3.1 Planar contact of two bodies Stick, Slip and Separation Transition A contact pair in the distributed contact model is illustrated in Figure 3.2, in which (,, ) u p r t,,,, v( p r t ) k ( p, r ), s ( p, r, t ), ( ) u u n p r and k ( p, r) are the relative 0, motion in the slip direction, relative motion in the normal direction, contact stiffness distribution in the slip direction, slip motion, preload distribution, and normal stiffness v distribution, respectively. The two spatial variables, p and r, are specified on the 60

78 contact plane and within the contact area, and their subscript c is removed for simplicity. The preload distribution n ( ) 0, p r is positive if the contact pair is preloaded. On the other hand, n0 ( p, r) is negative and in proportional to normal stiffness distribution kv ( p, r ) if it has an initial gap. The two-dimensional motion considered in the model is composed of two perpendicular components: tangential motion u( p, r, t ) in the p direction and the normal motion v( p, r, t ) in the q direction. The normal motion causes normal load variation and possible separation, according to the following equation, ( ) n p r t ( ) ( ) ( ) ( ) ( ) v( ) ( ) < ( ) ( ) n p, r + k p, r v p, r, t, if v p, r, t n p, r / k p, r 0 v 0,, =, 0, if v p, r, t n0 p, r / kv p, r (3.4) and the resulting friction force is related to the tangential relative motion according to the following equation, if the slip motion s (,, ) u p r t is known, (,, ) = (, ) (,, ) (,, ) f prt k pr u prt s prt u u. (3.5) It is evident that in order to determine the slip motion and thus the resulting friction force, stick-slip-separation needs to be determined according to the tangential relative motion and normal motion. The stick-slip-separation transition criteria employed in this chapter are based on those developed by Yang et al. in [1998]. 61

79 u ( p, r, θ ) v ( p, r, θ ) k v( p, r ) k u ( p, r ) s u ( p, r, θ ) n o( p, r ) Figure 3.2 Distributed contact model for 2D motion Slip-to-Stick Transition Transition from slip to stick state occurs when slip velocity, s / t =0; and u while slipping, friction force can be determined from (,, ) μ (,, ) f prt =± n prt. (3.6) Differentiating Equations (3.5) and (3.6) with respect to time and equating them, slip velocity is determined as follows 62

80 s u μk ( p, r) (, ) v u v = ± t t ku p r t. (3.7) Using Equation (3.7), transition criteria from positive slip to stick and negative slip to stick are given as v(, ) ( ) ( ) ( ) u μk p r v u μk p, r v = 0, < 0, (3.8) t k p, r t t k p, r t 2 u ( ) ( ) 2 2 v 2 u ( ) ( ) u μkv p, r v u μk p, r v + = 0, + > 0, (3.9) t k p, r t t k p, r t 2 u 2 2 v 2 u respectively. Inequalities in Equations (3.8) and (3.9) are used to guarantee slip to stick transition Stick-to-Slip Transition Friction force at the stick state is given as follows (,, ) = (, ) (,, ) (, ) + (, ) f prt k pr u prt u pr f pr u 0 0, (3.10) where, u0 ( p, r) and f0 ( pr, ) are distribution of displacement and friction force at the beginning of stick state. Stick to slip transition occurs when the friction force reaches to 63

81 the slip load which can be determined by equating Equations (3.6) and (3.10). Transition from stick to positive-slip and from stick to negative-slip are u (, ) (,, ) 0(, ) + 0(, ) μ n ( p, r) + k ( p, r) v( p, r, t) = 0, k p r u p r t u p r f p r u 0 v u v ku( p, r) μkv( p, r) > 0, t t (, ) (,, ) 0(, ) + 0(, ) + μ n ( p, r) + k ( p, r) v( p, r, t) = 0, k p r u p r t u p r f p r 0 v u v ku( p, r) + μkv( p, r) < 0, t t (3.11) (3.12) respectively. Again, inequalities given in Equations (3.11) and (3.12) are used to ensure the stick to slip transition Separation Separation occurs when the friction interfaces loose contact and, beginning and end of separation can be determined from the following criteria v n0 ( p, r) + kv ( p, r) v( p, r, t) = 0, 0 t >, (3.13) v n0 ( p, r) + kv ( p, r) v( p, r, t) = 0, 0 t <, (3.14) 64

82 respectively. At the end of separation, the next state can be determined by the following criteria Stick : v( ) ( ) v( ) ( ) μk p, r v u μk p, r v < < k p, r t t k p, r t, (3.15) u u Positive Slip : Negative Slip : u μk ( p, r) (, ) v v > t ku p r t, (3.16) u μk ( p, r) (, ) v v < t ku p r t. (3.17) It should be noted that, the friction coefficient in this analysis is taken as constant through out the contact interface; however, it can as well be considered as a distributed parameter and the same equations can be used if the distribution of friction coefficient is combined with the normal load distribution Transition Angles While the transition criteria given in Equations (3.8)-(3.17) can be applied to any arbitrary relative motion, the determination of transition angles often requires numerical simulation. Nonetheless, if the forced response is of interest and the relative motion is assumed to be single harmonic motion, stick-slip-separation transition angles can be 65

83 determined analytically based on the derived criteria. Assuming the following form for the relative motions ( ) (,, θ) (, ) sin ( θ), (,, θ) (, ) sin θ ϕ(, ) u pr = a pr v pr = b pr + pr, (3.18) where θ = ωt and, ω and t are the oscillation frequency and time, respectively; transition angles as a function of p and r are determined and expressed in terms of three dimensionless parameters, namely b ( pr, ) = μk ( prbpr, ) (, )/ k( prapr, ) (, ), ϕ ( p, r), and n( pr, ) = μn ( pr, )/ k( prapr, ) (, ). Stick, slip, separation transitions in one cycle of o o u motion can be categorized in three different groups: complete stick, stick-slip without separation and separation. v u When the preload acting on the friction interface is high and the relative motion is small the friction interface sticks all the time and this can be identified by eliminating the criteria given in Equations(3.11) and (3.12), which can be characterized by the following inequality 2 ( ) ( ) ( ) ϕ ( ) 2 n0 p, r > 1 + b p, r 2 b p, r cos p, r. (3.19) b( pr, ) + 2 b( pr, ) cos ϕ ( pr, ) 66

84 Similarly, eliminating the criteria given in Equations (3.13) and (3.14), the condition for no separation can be derived as 2 ( ) ( ) ϕ ( ) 1 + b pr, 2 b pr, cos pr, 2 ( ) ( ) ϕ ( ) ( ) ( ) b pr, + 2 b pr, cos pr, > 2 n0 pr, > b pr, (3.20) Stick-slip but no separation When the amplitude of the relative motion increases to some extent, the friction contact begins to slip but still remains in contact. In this case, the friction contact undergoes alternating stick-to-slip motion, which results in a hysteresis loop consisting of four alternating regions (positive-slip, stick, negative-slip, and stick) separated by four transition angles. Substituting Eq (3.18) to Eqs (3.8) and (3.9), positive-slip to stick and negative-slip to stick transition angles are derived as follows. ( ) ϕ ( ) (, ) sin ϕ (, ) 1 b p, r cos p, r θp ( pr, ) = π ψ ( pr, ) arctan b p r p r ( ) ϕ ( ) (, ) sin ϕ (, ) 1 + b p, r cos p, r θn ( pr, ) = π ψ ( pr, ) + arctan b p r p r (3.21) (3.22) 67

85 where ψ ( pr, ) ( pr) ( pr) 0, if 0 < ϕ, < π =. Transition angles from stick to positive-slip π, if π < ϕ, < 2π and from stick to negative-slip are given as θ θ StP StN,, arccos ( pr) ψ ( pr) ( p, r) 2 n ( p, r) ( pr, ) 2 0 = + ( ) ϕ ( ) (, ) sin ϕ (, ) 1 b p, r cos p, r arctan b p r p r,, arccos ( pr) ψ ( pr) 1 ( p, r) 2 n ( p, r) ( pr, ) 1 0 =+ + ( ) ϕ ( ) (, ) sin ϕ (, ) 1 + b p, r cos p, r + arctan b p r p r 2 (3.23) (3.24) where ( pr) b( pr) 2 b( pr) ϕ ( pr) 1, = 1 +, 2, cos,, ( pr) b( pr) 2 b( pr) ϕ ( pr) 2, = 1 +, + 2, cos, Separation If separation exists, hysteresis loop is composed of 10 possible sequences of alternating stick-slip-separation, which are characterized by six transition angles. From the criteria given in Equations (3.13) and (3.14), start and end of separation angles are derived as 68

86 ( p r) n b ( p, r) ( p, r) θ 0 Sp1, = π + arcsin ϕ θ Sp2 ( p r) ( p, r) ( p, r) 0, arcsin, ( ) p, r, (3.25) n = ϕ ( p r ). (3.26) b Transition angels for positive-slip to stick and negative-slip to stick are given in Equations (3.21) and (3.22), respectively. The transition angels from negative-slip-stick to positive-slip and positive-slip-stick to negative-slip can be obtained from Equations (3.23) and (3.24); however, if the previous state of stick is separation then transition angles from separation-stick to positive-slip and separation-stick to negative-slip are given as ( ) ϕ ( ) (, ) sin ϕ (, ) ( p r) n ( p r) l ( p, r) sin θ Sp2, + 0, θspp ( pr, ) = π ψ ( pr, ) arccos 1 θ SpN 1 b p, r cos p, r arctan b p r p r ( pr) ψ ( pr) ( ) ϕ ( ) (, ) sin ϕ (, ) ( p r) n ( p r) l ( p, r) sin θ Sp2, 0,, =, + arccos b p, r cos p, r + arctan b p r p r, (3.27). (3.28) Since the analytical distribution of transition angles are known, it is possible to determine stick-slip-separation (friction interface) map at any instant. This friction interface map is 69

87 useful to understand how the friction damper works and it can as well be used to estimate wear of the contacting surfaces Stick-slip-separation Map It has been shown that if the relative motion is given and assumed to be single harmonic motion, analytical expressions for transition angles in terms of the two spatial variables are available. These expressions, θ ( p, r), can be visualized as threedimensional (3D) surfaces. The stick-slip-separation map at any instant is composed of stick-slip-separation transition boundaries, which can be determined by intersecting those 3D surfaces with the associated constant θ value. Moreover, the time variance of this friction interface map can be illustrated by changing θ value. In this work, stick-slipseparation map is used to demonstrate the microslip phenomenon, whereas it can as well be employed to estimate component wear caused by the rubbing of contact surfaces. Some of the factors affecting wear of sliding surfaces are duration of sliding, normal load, and friction (tangential) force acting on the contact interface. Normal load and friction force distributions are already determined from the developed friction model and the duration of sliding can be obtained from the stick-slip-separation map, which can be used to estimate the wear of sliding surfaces. 70

88 Distribution of Fourier Coefficients Given the relative motion, which is assumed to be single harmonic motion, transition angles, and thus the resulting friction force over a cycle of motion, are known analytically. If the forced response is of interest, the resulting friction force can be approximated by its Fourier components. f( pr,, θ ) f ( pr, ) f( pr, ) sin ( θ) f ( pr, ) cos( ) + + θ, (3.29) b s c where fb( pr, ) is the distribution of the mean force, fs ( pr, ) spring force, and fc ( pr, ) damping force. For the same purpose, the Fourier coefficients of the variable normal load can be derived. These distribution functions illustrate the spatially distributed dynamic characteristics of the contact friction interface FORCED RESPONSE Finite element models are often used in the forced response analysis of complex structures, which result in many degrees of freedom (DOF). Due to the friction contact, this results in large systems of nonlinear equations, which need to be solved iteratively. This is a computationally expensive and also an inefficient process. Menq and Griffin 71

89 [1985] in 1985 developed a nonlinear forced response analysis method for steady state response of frictionally damped structures using finite element models. In the developed method, using the receptance of the linear system, authors considered only the nonlinear DOF first and determined the harmonic displacement of these DOF by an iterative solution procedure. Using these displacements, the forces acting on these DOF were obtained, and treating them as external forces and together with the excitation forces, authors determined the response of the complete structure. This method reduces the number of nonlinear equations to the number of nonlinear DOF, and it is a very efficient method to analyze frictionally constraint structures, since the nonlinearity comes only from the frictionally constraint DOF. Structural modification techniques were as well applied by Tanrıkulu et al. [1993], Sanliturk et al. [2001b] and Ciğeroğlu and Özgüven [2006] in order to solve large nonlinear equation systems, where the dynamic stiffness matrix of the nonlinear system was determined by applying structural modifications to dynamic stiffness matrix of the linear system. The methods explained above can as well be applied to model microslip friction; however, many friction elements are needed in microslip modeling which results in large number of nonlinear DOF. Moreover, if the bladed disk system is mistuned, since the cyclic symmetry of the structure is destroyed; all the blade-damper sectors have to be included into the forced response analysis resulting in even larger nonlinear equation systems. It should be noted that, the finite element models for bladed disk systems contain many DOF; thus, even for linear forced response analysis, reduction techniques 72

90 are employed [Yang and Griffin, 1997; Bladh et al., 2001, 2001b]. In this work, a modal superposition technique is used, where the motion of the frictionally constrained structure is assembled from its free mode shapes. In this approach, the number of unknowns depends on the number of mode shapes used in the modal expansion process, which decreases the number of nonlinear equations significantly even for microslip models Multi-Mode Solution Method Equation of motion in matrix form for a system with dry friction dampers can be written in the following form e ( ) ( ) MX + CX + KX = F t + F X, (3.30) n in global coordinate system, where,,, F ( t), F ( X) M C K and X are the mass matrix, viscous damping matrix, stiffness matrix, excitation force vector, nonlinear friction force vector and relative displacement vector, respectively. The 2D microslip friction model developed is defined on the contact plane coordinate system; therefore, the nonlinear friction and normal forces obtained from the model are as well in the contact plane e n coordinate system. Displacement vector X in global coordinates can be written as 73

91 X BAxyz B xyz A xyze θ, (3.31) N N i = n n + n n n= 1 n= 1 (,,,, ) φ (,, ) Re φ (,, ) th where φ, B, A and N are the n mode shape of frictionally constrained structure in n n n global coordinate system, th n real and complex modal coefficients for dc and ac components of motion, and number of modes used in the modal expansion process, respectively. Transforming X to contact plane coordinates, the relative motion of friction interface points in contact plane coordinates can be expressed as a function of the modal coefficient vectors (,,,, ) u= ubaprθ, (3.32) (,,,, ) v= vbaprθ, (3.33) where B and A are the real and complex modal coefficient vectors for dc and ac components of motion, respectively. The nonlinear (friction and normal) force vector in contact plane coordinate system can be written similar to Equation (3.29) using the relative displacements given by Equations (3.32) and (3.33) as c (,,,, θ ) (,,, ) (,,, ) sin ( θ) (,,, ) cos( θ) F BApr F BApr + F BApr + F BApr.(3.34) n b s c Using the orthogonality of mode shapes, Equation (3.30) can be simplified to 74

92 ( ) 2 e e Ω ω I+ iωc A= Q+ iq+ Q( BA, ) + iq( BA, ), (3.35) r s c s c Ω B = Q (, ) b B A, (3.36) if mass normalized mode shapes are used. Here e Q s and e Q c are the in phase and out of phase modal force vectors for the excitation forces, Q, b Q s and Q c are the modal force vectors for mean, spring and damping forces, Ω is NxN diagonal matrix of squares of natural frequencies and C r is the modal damping matrix, which is diagonal if the damping is proportional. The modal forcing vectors on the right hand side of Equations (3.35) and (3.36) are (,, ) (,, ) φ (,, ) (,, ) e e e Qs = n φu p q r f n s p q r + u v p q r f n s p q r v dpdqdr De, (3.37) Q = φ ( p, q, r) f ( p, q, r) + φ ( p, q, r) f ( p, q, r) dpdqdr e e e cn un cu vn cv De (, ) = φ (,0, ) (,,, ) + φ (,0, ) (,,, ) Q B A p r f B A p r p r f B A p r dpdr, (3.38) * n un * u vn * v D th where, φ u n and φ v n are the n mode shapes of the frictionally constrained structure in contact plane coordinates, e f and f represent the excitation and nonlinear contact forces in contact plane coordinates, * corresponds to s, c, or b; and u and v indicate the direction of mode shapes and forces along p and q axes, respectively. In addition to this De and D are the domain of integrations for the excitation and contact forces, 75

93 respectively. Since the modal force vectors are in modal coordinates, they can be obtained using the mode shapes and forcing vectors in contact plane coordinate system as given in Equations (3.37) and (3.38). This reduces the order of integration in Equation (3.38) from triple to double integration, since the contact interface D is a 2D plane area in contact plane coordinate system whereas it is a 3D surface in global coordinate system. Equations (3.35) and (3.36) describe a set of nonlinear algebraic equations and the unknown modal coefficient vectors B and A can be solved by an iterative nonlinear solver. It should be noted that, the total number of unknowns in this nonlinear equation set is 3 N, which is equal to the number of terms used in the Fourier series expansion multiplied by the number of mode shapes used in the modal expansion; and once the modal coefficient vectors are obtained, the motion of the frictionally constrained structure can be constructed EXAMPLES Two examples, a one-dimensional bar like damper and a more realistic blade to ground damper, are employed to illustrate the predictive abilities of the developed model. 76

94 D Bar Model This example is of interest because related results are given in the literature. The analytical solution procedure developed in Chapter 2 uses single mode information of a bar like damper to derive analytically the spatial boundary of the stick-slip transition for specific normal load distribution. Owing to the complicity of the stick-slip transition, solutions are limited to the range of first vibration mode and for three different normal load distributions, which are time invariant. On the other hand, the method presented in this chapter is capable of dealing with multi-mode vibration and with normal load that has arbitrary spatial distribution and is time variant. However, for the purpose of comparison, the normal load will be kept time invariant in this example, and the focus will be on the effect of number of modes used in the analysis. A 1D bar model similar to the one in Chapter 2 is given in Figure 3.3, where, ρ β and F( t ) are the modulus of elasticity, cross-sectional area, E, A,, L,, q( x) density, and length of the bar, strain hardening stiffness, normal load distribution and excitation force, respectively. The shear layer in Chapter 2 is replaced by distribution of contact stiffnesses in slip direction, k ( ) equation for this system is u x. For harmonic forcing, partial differential 2 2 u 2 u EA ρ Aω = τ u, x F 2 2 0δ x L x θ ( ) ( ) sin( ) θ, (3.39) 77

95 u u EA = βu ( 0, θ), EA = 0, (3.40) x x x= 0 x= L where u is the displacement of point x, ( ux, ) τ is the friction force distribution acting on the bar, F0 is the amplitude of the harmonic forcing and δ is the Dirac delta function. The motion of the bar for harmonic excitation can be represented by its free mode shapes, which are analytically available for this case, using Equation (3.31). This is a onedimensional bar problem hence, there is no z dependence; in addition to this, since the normal load is time invariant dc component of the motion and the friction force vanishes. Inserting Equation (3.32) into Equation (3.39) and applying the integral orthogonality relations, Equation (3.35) is obtained which can be solved for the unknown complex modal coefficients. An iterative solution procedure is applied to solve the nonlinear algebraic equations given in Equation (3.35) and the motion of the bar is constructed using the determined modal coefficients. The friction interfaces for the bar are determined for different normal load distributions, and the effect of number of modes used in the calculations is as well presented Friction Interface The model given in Figure 3 is analyzed for constant and concave quadratic normal load distributions, which are defined in Chapter 2. It should be noted that the analytical results given in Chapter 2 are derived for displacement input and the method 78

96 presented in this chapter is a forced response method. Therefore, their results can not be compared directly. Figure 3.4 and Figure 3.5 show the build-up of friction force for constant and quadratic normal load distributions at 1000Hz, which is around the first mode of the system. In the figures, 1 and -1 denote positive and negative slip, respectively, and in between them lies the stuck region. Solid lines on the figures are the stick-slip boundary, which can be obtained from the transition angle equations derived in Section 3.2. It is seen that for constant normal load distribution, slip starts from the right end of the bar and propagates towards the left end, which will cause gross-slip if the excitation force is increased further more. For quadratic normal load distribution, slip starts somewhere around the center of the bar and propagates towards the both ends. It first reaches the right end of the beam and then the left end resulting in gross-slip. These results are in agreement with the results obtained in Chapter 2 [Cigeroglu et al., 2006], where the authors divide the contact interface into slip and stick regions and provide the change of length of each region for displacement input and similar normal load distributions Effect of Multiple Modes In order to demonstrate the effect of number of modes used in the analysis, constant and quadratic normal distribution cases are analyzed using single mode, 3 modes and 10 modes of the system. The analyses are performed at 1000Hz and 3100Hz, which are around the first and second modes of the system and the stick-slip boundaries are given in Figure 3.6 and Figure 3.7. It is seen that, for all the cases 3-mode solution and 79

97 the 10-mode solution result in similar friction interfaces. However, it is also seen that, even though single mode solution can estimate the overall behavior of the friction interface, the results obtained may not be accurate. It should as well be noted that, single mode solution predicts the transition from slip to stick occurs at the same time for all the slipping points; however, that transition from slip to stick in multi-mode solution does not occur at the same instant. This is an expected result due to the inertia of the bar and it comes more evident in Figure 3.7. Figure 3.8 shows the slip to stick transition that is predicted by the ten-mode solution and is zoomed into a very small range of θ. x q(x) β E, A, ρ Contact Stiffness F(t) L Figure D bar model 80

98 (a) β x E, A, ρ L q(x) F(t) Temporal Variable θ [rad] Slip + Slip Temporal Variable θ [rad] Slip + Slip (b) Normalized Spatial Variable [x/l] (c) Normalized Spatial Variable [x/l] (d) Temporal Variable θ [rad] Slip + Slip Normalized Spatial Variable [x/l] Figure 3.4 Build-up friction force for constant normal load distribution: (a) model, F( t) for (b)<(c)<(d) 81

99 6 1.0 (a) β x E, A, ρ L q(x) F(t) Temporal Variable θ [rad] Slip + Slip (b) Normalized Spatial Variable [x/l] Temporal Variable θ [rad] Slip + Slip Temporal Variable θ [rad] Slip + Slip (c) Normalized Spatial Variable [x/l] (d) Normalized Spatial Variable [x/l] Figure 3.5 Build-up friction force for quadratic normal load distribution: (a) model, F( t) for (b)<(c)<(d) 82

100 6 (a) a) Temporal Variable θ [rad] Mode 3 Modes 10 Modes - Slip Stuck + Slip Normalized Spatial Variable [x/l] 6 (b) b) Temporal Variable θ [rad] Mode 3 Modes 10 Modes - Slip Stuck + Slip Normalized Spatial Variable Figure 3.6 Effect of number of modes at 1000Hz a) constant b) quadratic normal load distributions: 1 mode, 3 modes, 10 modes 83

101 6 a) Temporal Variable θ [rad] (a) + Slip - Slip - Slip + Slip 1 Mode 3 Modes 10 Modes Normalized Spatial Variable [x/l] 6 b) Temporal Variable θ [rad] (b) + Slip - Slip - Slip + Slip 1 Mode 3 Modes 10 Modes Normalized Spatial Variable [x/l] Figure 3.7 Effect of number of modes at 3100Hz a) constant b) quadratic normal load distributions: 1 mode, 3 modes, 10 modes 84

102 114 (a) Temporal Variable θ [deg] Stuck + Slip Stuck Normalized Spatial Variable [x/l] (b) 106 Temporal Variable θ [rad] Stuck + Slip Stuck Normalized Spatial Variable Figure 3.8 Phase difference along slip-to-stick transition boundary at 1000Hz a) constant b) quadratic normal load distributions 85

103 Blade to Ground Damper The blade to ground damper system analyzed is given in Figure 3.9, where the right side of the platform of the blade is in contact with the ground. In this system, the blade is represented by a finite element model as shown in Figure 3.10, in which B and B` are two symmetric points, where excitation forces are applied and A is the point, where the displacements are calculated. Modal information and the mode shapes of the blade are obtained by a finite element analysis and inserted in Equations (3.35) and (3.36). It is assumed that the system is proportionally damped with a damping ratio of 0.2%. Continuous mode shape functions are determined by applying curve fitting to the ones obtained by the finite element analysis. Unknown forcing vectors in Equations (3.35) and (3.36) are determined by Equation (3.38) using the continuous mode shape functions obtained by curve fitting. An iterative nonlinear solver is used to determine the unknown modal coefficients, from which the motion of the blade can be constructed by using Equation (3.31). In the following sections forced response results and stick-slipseparation maps for the blade to ground damper are presented Forced Response Results Forced response curves for the blade to ground damper system are shown in Figure 3.11 and Figure 3.12, corresponding two distinct forcing directions: x and y directions, respectively. Each forced response curve is associated with a specific preload 86

104 or initial gap, and is around the first resonance of the system. For simplicity, uniform preload distribution over the contact surface is assumed and the total preload is specified in the two figures. It should be noted that, the effects of higher modes ( n> N) on the displacements can be represented by residual stiffnesses, which can be determined through finite element analysis and this information is used in the determination of tangential and normal contact stiffnesses; in addition, these contact stiffnesses make it possible to use lower frequency (higher wavelength) modes to determine microslip on the contact surface. In the analyses of the blade to ground damper, 10 modes of the blade are used. Forced response for free and stuck cases, which are the two linear extreme cases for the system, are as well included in the figures. The nonlinear response of the system is in between these two linear solutions and as the preload increases the peak frequency shifts to the right and the system response approaches to the stuck response, finally becoming completely stuck. It is seen that, there exists an optimum value for the preload, which results in minimum displacement amplitude for each of the two cases. It should be noted that the vibration amplitude in Figure 3.11 is about 20 times greater than that in Figure This is due to the fact that, the first vibration mode of the blade, which is a bending mode around the z -axis, is less sensitive to the forcing in the y direction. It is as well interesting to note that, for this case the stuck response has higher displacement amplitude compared to that of the free response. 87

105 q p r p Contact Area y x Figure 3.9 Blade to ground damper model 88

106 Figure 3.10 Finite element model for the blade (Maybe modified) 89

107 Normalized Response Amplitude [m/n] Frequency [Hz] Figure 3.11 Forced response for excitation force in x direction: free, -5e5, 1000, 7500, 1e4, 5e4, 1e5, 2e5, stuck 90

108 Normalized Response Amplitude [m/n] Frequency [Hz] Figure 3.12 Forced response for excitation force in y direction: free, -2e4, 80, 350, 500, 1000, 1e4, 2e4, stuck 91

109 For high preloads, blade and ground are always in contact, i.e. there is no separation in the friction interface. However, as the preload acting on the friction interface decreases, normal motion of the blade results in separation in the contact interface, which shows itself as a softening effect in the forced response results. On the contrary, if the initial gap between the blade and the ground is decreased, due to the motion in normal direction, blade and ground come into contact resulting in hardening effect. It is possible to observe jump phenomena in case of softening and hardening; therefore, continuation method is used to determine the forced response curves for those cases where there exists an unstable solution branch between two jumps. It should be noted that, unstable solution can not be obtained by time domain simulation, since the system response will converge to one of the two stable solutions unless the initial guess is exactly the unstable solution. Figure 3.13 shows the forced response curve for n 0 = 1000 in Figure 3.11, where the unstable solution is identified by the dashed line. It should be noted that, the actual preload distribution acting on the contact interface depends on the dc component of the motion and the contact stiffness in the normal directions as n( pr, ) = n ( pr, ) k( pr, ) v( pr, ), (3.41) 0 0 v 0 where n ( p r 0, ) is the applied initial preload distribution on the contact interface and v ( p r 0, ) is the distribution of dc component of the motion in the normal direction. 92

110 Friction Interface In order to illustrate the microslip phenomenon, the stick-slip-separation boundaries for the case of n 0 = 1000, Figure 3.13, are examined. Fiction interface maps associated with the excitation frequency at Hz are obtained. Specifically, the maps before and after jump are plotted in Figure 3.14, in which the left column is before the jump and the right column after the jump. It is evident that before the jump the vibration amplitude is significantly greater and the friction interface is not in contact most of the time; whereas, after the jump positive and negative slip states govern most of the friction interface. Therefore, it can be concluded that, jump in the forced response curve is due to the separation of the contact interface caused by the normal load variation. Figure 3.15 shows the transition map when θ =125 o. It is obvious that at this instant the contact interface is governed by three distinct states. In other word, over the contact interface one area is stuck, another area is slipping while the other has separation. This clearly demonstrates the microslip phenomenon. 93

111 Normalized Response Amplitude [m/n] Hz Stable Solution Unstable Solution 306.4Hz Frequency [Hz] Figure 3.13 Stable and unstable solutions for n 0 = 1000 : stable solution, unstable solution 94

112 Normalized Spatial Variable [z] (a) (b) (c) (d) Separation Separation Separation Separation Stuck (e) Slip Separation (f) - Slip + Slip + Slip Separation Normalized Spatial Variable [x'] (g) (h) Separation Figure 3.14 Friction interface for n 0 = 1000 before jump (a) 0, (b) 90, (c) 180, (d) 270 ; after jump (e) 0, (f) 90, (g) 180, (h)

113 1.0 Normalized Spatial Variable [z] Separation + Slip Stuck Normalized Spatial Variable [x'] Figure 3.15 Friction interface for n 0 = 1000 and θ =

114 3.5. CONCLUSION A distributed parameter model is developed to characterize the stick-slipseparation of the contact interface and determines the resulting friction force, including its time variance and spatial distribution, between two elastic structures. A multi-mode solution approach is developed to determine the forced response and stick-slip-separation transitions of the steady state solution of frictionally constrained structures when subjected to harmonic excitation. In the proposed approach, steady state response of the system is constructed by its free mode shapes. The proposed method is applied to a onedimension bar like damper. It is shown that while employing a single mode model, transition boundaries for the bar like damper agrees with the results given in the literature, the developed method identifies the phase difference along the slip to stick transition boundary when a multi-mode model is employed. The proposed method is also applied to a more realistic blade to ground damper model, where the blade is modeled by the finite element method. For this system, due to the complicated geometry analytical mode shapes are not available; hence, continuous functions are fitted to the finite element mode shapes and used in the analysis. Resulting forced response curves, transition maps are obtained, and they clearly show the microslip phenomenon. Typical softening and hardening effects, due to separation of the contact surface, are also predicted for the blade to ground damper. 97

115 Although the relative motion between two contacting bodies is in general threedimensional, for simplicity, this chapter focuses on a two-dimensional version, in which while the normal motion v is retained, the two bodies move with respect to each other on the contact plane back and forth along the p direction. Nevertheless, it is possible to extend the method to general three-dimensional problem so that it can be applied to many real-world systems. It should as well be noted that, in order to apply the proposed method, in-plane and out of plane contact stiffness distributions in the contact interface have to be determined. This issue will be discussed in future investigation. 98

116 CHAPTER 4 4. MODELING MICROSLIP FOR TWO-DIMENSIONAL PERIODIC MOTION AND ITS EFFECTS ON BLADE VIBRATION 4.1. INTRODUCTION In the forced response analysis of frictionally damped structures, Harmonic Balance Method (HBM), where the nonlinear contact forces are represented by Fourier series, is widely used. In general, friction forces are represented by single harmonic Fourier series; whereas, multi-harmonic representations can be found in [Pierre et al., 1985; Wang and Chen, 1993; Kuran and Özgüven, 1996; Cardona et al., 1998; Chen et al., 2000; Petrov and Ewins, 2003]. Even for single harmonic input, depending on the type and strength of nonlinearity, single harmonic representation may not be accurate enough. Therefore, for those cases, multiple-harmonic representation is necessary. 99

117 The objective of this study is to develop a two-dimensional periodic microslip friction model with normal load variation, which can be applied to real design problems. In the following sections, the two-dimensional periodic microslip friction model and the forced response analysis method are described and numerical results for a blade to ground damper are presented MICROSLIP FRICTION MODEL FOR TWO-DIMENSIONAL PERIODIC MOTION WITH NORMAL LOAD VARIATION For the cases where the amplitude of relative motion is low or the normal load on the contact interface is high partial-slip occurs in the friction interface; moreover, due to the contact of two bodies there exists a motion in the normal direction resulting in normal load variation, hence separation of the contact interface. In order to determine the nonlinear contact forces resulting from the friction contact; slip, stick and separation regions in the spatial domain should be determined. The two-dimensional motion is composed of in-plane and out of-plane components associated with p and q axes, respectively and, r axis is used to define the contact interface together with p axis. Consequently, the change of stick, slip and separation regions with respect to time will result in volumes of stick, slip and separation in the p r t coordinate system, from where at any time instant the resulting friction interface can be obtained. 100

118 Transition Angles The transition criteria for two-dimensional microslip friction model are developed in Chapter 3. Those criteria are developed for general motion on the contact interface, therefore; they can as well be applied to periodic motion. For brevity, they are not repeated here. If the relative tangential and normal motions are periodic, they can be represented by Fourier series as follows l 1 1 ( θ) = ( ) ( θ) + ( ) ( θ) u pr,, a pr, cos j b pr, sin j, j j= 0 j= 1 l 2 2 ( θ) = ( ) ( θ) + ( ) ( θ) v pr,, c pr, cos j d pr, sin j, j j= 0 j= 1 l l j j (4.1) where θ = ωt, and ω and t are the oscillation frequency and time, respectively. and l l1 2 are the number of harmonics included in the expansion of tangential and normal motions, respectively, and abcd,,, are the vectors of harmonic coefficients used in the Fourier series expansion, and they are functions of the spatial variables x and z. It can be concluded from the transition criteria given above that, for a general periodic motion, analytical solutions for transition angles do not exist and the criteria given above have to be solved numerically. The problem of determining transition angles can be divided into two groups: with separation and without separation. If separation exists, which can be determined from Eq. (3.13), the state and the transition angle(s) after separation can be obtained from Eqs. (3.14)-(3.17). Depending on the end state of separation the next 101

119 transition angle is determined from Eqs. (3.8), (3.9) or (3.13) whichever transition angle comes first if the previous state is slip, and from Eqs. (3.11), (3.12), or (3.13) whichever transition angle comes first if the previous state is stick. This procedure is repeated until one cycle is completed. If separation does not exist, a similar approach as given above can be followed to determine the transition angles. Starting from an initial stick state with zero initial ( ) ( ) conditions ( u p, r = 0, f p, r = 0), where the relative tangential motion is zero, the 0 0 transition to slip can be obtained from Eqs. (3.11) or (3.12), whichever transition angle comes first and from this point, the next stick state can be determined from Eqs. (3.8) or (3.9) depending on the slip state. Calling this stick transition angle θ pr, the successive transition angles can be determined similar to the case of separation from θ ( pr) to θ ( pr) 0, ( 0, 0, + 2π. It should be noted that, due to the mean value it is possible that a zero of the relative tangential motion does not exist; for a case like this, the simulation 0,, 0, f0 p, r = 0) from can be started with zero initial conditions ( u ( p r) u ( p r) = ( ) the zero of the relative tangential motion without the mean value. mean ) Example Hysteresis Curve for Periodic Motion Using the procedure explained here, stick, slip, separation states for the periodic motion given in Figure 4.1 is determined and friction force for one cycle together with positive and negative slip loads are given in Figure 4.2. Hysteresis curve and amplitude 102

120 of harmonics in the determined friction force are given in Figure 4.3 and Figure 4.4, respectively. The specified tangential and normal motions given in Figure 4.1 have first two and first three harmonics, respectively; whereas, in the resulting friction force, the first four harmonics are dominant. If the periodic tangential and normal motions contain more harmonics, hysteresis curve, Figure 4.5, might be more complicated; in that case, friction force vs. θ plot given in Figure 4.6 can be more explanatory. In this case, the tangential motion is composed of the first three and the fifth harmonics; however, it can be observed from Figure 4.7 that, other harmonics in the friction force are not negligible. For single harmonic motion, the slip state following slip-stick states has to be in the opposite direction of the prior slip state, such as positive slip-stick-negative slip or negative slip-stick-positive slip. However, for periodic motion it is possible to have slip state following the slip-stick state in the same direction as the prior slip state. An example for this case can be obtained by just increasing the initial preload acting on the friction point used previously. Resulting hysteresis curve and friction force curve are given in Figure 4.8 and Figure 4.9, where negative slip state is followed by stick and then another negative slip state. 103

121 ku.u(θ) kv.v(θ) θ [rad] Figure 4.1 2D periodic motion Stick Separation + Slip Friction Force -μ*n(θ) μ*n(θ) - Slip θ [rad] Figure 4.2 Friction force in one cycle 104

122 + Slip Friction Force Separation - Slip Stick Tangential Motion Figure 4.3 Hysteresis curve Friction Force Harmonic Coefficient Harmonic Number Figure 4.4 Harmonic coefficients for the friction force 105

123 Friction Force Tangential Motion Figure 4.5 Hysteresis curve Friction Force -μ*n(θ) μ*n(θ) θ [rad] Figure 4.6 Friction force in one cylcle 106

124 Friction Force Harmonic Coefficient Harmonic Number Figure 4.7 Amplitude of harmonic coefficients Friction Force Tangential Motion Figure 4.8 Hysteresis curve 107

125 + Slip Stick Slip Friction Force -μ*n(θ) μ*n(θ) θ [rad] Figure 4.9 Friction force in one cycle Distribution of Fourier Coefficients follows If multiple-harmonic representation is used, the nonlinear forces can be written as (,,,,,, θ ) b (,,,,, ) + f1s( a, b, c, d, p, r) sin ( θ) + f1 c( a, b, c, d, p, r) cos( θ) + f ( abcd pr) ( θ ) + f ( abcd pr) ( f abcd xz f abcd pr,,,,, sin 2,,,,, cos 2θ 2s 2c ( ) ( θ ) ( ) + f abcd,,,, pr, sin m + f abcd,,,, pr, cos mθ ms 108 mc ) ( ). (4.2)

126 where, m is the number of harmonics used in the Fourier series representation and the coefficients of each harmonic are given as 1 2π fb ( abcd,,,, pr, ) = f( abcd,,,, pr,, θ ) dθ 2π, (4.3) 0 1 π, (4.4) 0 2π f ( abcd,,,, pr, ) = f( abcd,,,, pr,, θ) sin ( jθ) dθ, ( j= 1,, m) js 1 π. (4.5) 0 2π f ( abcd,,,, pr, ) = f( abcd,,,, pr,, θ) cos ( jθ) dθ, ( j= 1,, m) jc If the distributions of transition angles are identified, friction and normal force distributions can be determined from Eqs. (3.6) and (3.10), and consequently, from Eqs. (4.3)-(4.5), the distributions of Fourier coefficients can be obtained FORMULATION The modal superposition method applied in Chapter 3 for the harmonic vibration analysis of dry frictionally damped structures is extended in order to analyze periodic motions and apply multi-harmonic HBM. The method is explained in the following section. 109

127 Multi-Harmonic Solution Method Equation of motion in matrix form for a system with dry friction dampers (or nonlinear elements) can be written in the following form e ( ) ( ) MX + CX + KX = F t + F X (4.6) n where,,, F () t, F ( X) M C K and X are the mass matrix, viscous damping matrix, e n stiffness matrix, excitation force vector, nonlinear force vector and displacement vector, respectively. The motion of the system for harmonic excitation can be written in terms of its mode shapes as follows N l N j i( j ) u( x, z, ) Bn u( x, z) Re A (, ), n n u x z e θ θ = φ + φ n n= 1 j= 1 n= 1 (4.7) N l N j i( j ) v( x, z, θ) = Bnφv( x, z) + Re (, ), n Anφv x z e θ n n= 1 j= 1 n= 1 (4.8) where ( ) ( ) φ x, z, φ x, z, N, i, B and un vn n j A n are the th n mode shapes in x and y directions, number of modes used, imaginary number, th n modal coefficient of the dc component of motion, th n complex modal coefficient for elastic component of motion for the th j harmonic, respectively and l max ( l, ) = l. Displacement vector X can be written as

128 l i( j ) Φ Re( Φ ) X = B+ A e θ j= 1 (4.9) where Φ, B and A are the mode shape matrix, real and complex modal coefficient vectors for dc and elastic components of motion, respectively. Using the orthogonality relation, Eq. (3.30) can be simplified to ( Ω ω I ωc ) 2 ( Ω ( ω) I ( ω) C ) ΩB= Q ( B, A) b i r A = Qe + iq Re e + Q Im Re( B, A) + iqim ( B, A), (4.10) l + il A= Q + iq + Q ( BA, ) + iq ( BA, ) l l l l l r ere eim Re Im if mass normalized mode shapes are used. In Eq. (4.10), Ω is NxN diagonal matrix of squares of natural frequencies and C = r Φ T CΦ, which is a diagonal matrix if the damping is proportional. eb,,re and Im stand for excitation force, dc component, and real and imaginary parts of modal forces, respectively. The modal forcing vectors on the right hand side of Eq. (4.10) are given as (, ) (,,, ) φ (, ) (,,, ) Q* = n φu xz f n * BAxz + u v xz f n * BAxz dxdz v D, (4.11) ( n= 1, 2, N) 111

129 l l l where, * corresponds to,,, and, u and indicate the direction of mode shapes b e Re Im v and forces in x and y directions, respectively, and D corresponds to the domain of contact area. Eq. (4.10) describes a set of nonlinear algebraic equations, and the unknown modal coefficient vectors B and A can be solved by an iterative nonlinear solver. It should be noted that the total number of unknowns in this set is ( 2l+ 1) N and m l. Once the modal coefficients are obtained, the motion of the system can be constructed from Eqs. (4.7) and (4.8) Friction Interface Map If the motion of the system is known, at any instant the stick-slip-separation transition boundary can be determined. The friction interface map can either be determined by descritizing the friction interface or by solving the transition angle equations with continuation method, such as the arclength continuation method. The latter method makes it possible to obtain the boundary very accurately; however, an initial estimate for the interface is needed and it should be fine enough to capture necessary interface boundary details. Fine discretization of the interface will as well result in an accurate friction interface map and it is also a simple process to apply; however, it might be computationally expensive, since analytical solutions for the transition criteria do not exist. In this work, friction interface map is used to demonstrate 112

130 the microslip phenomenon, whereas it can as well be employed to estimate component wear caused by the rubbing of contact surfaces NUMERICAL RESULTS The developed method is applied on a blade to ground damper model, where a finite element model for the blade is used. The blade to ground damper system analyzed is given in Figure 3.9, where the right side of the blade is in contact with the ground. In this system, the blade is represented by a finite element model as shown in Figure 3.10, in which B and B` are two symmetric points, where the excitation forces are applied and A is the point, where the displacements are calculated. Modal information and the mode shapes of the blade is obtained by a finite element analysis and inserted in Eq. (4.10). It is assumed that the system is proportionally damped with a damping ratio of 0.2%, and the contact stiffness distribution is constant throughout the contact interface. It should be noted that, the mode shapes determined by finite element analysis are discrete and therefore, curve fitting is applied to determine continuous mode shapes. Unknown forcing vectors in Eq. (4.10) are determined by Eq. (4.11) using the continuous mode shapes obtained by curve fitting. An iterative nonlinear solver is used to determine the unknown modal coefficients, from which the motion of the blade can be constructed by 113

131 using Eqs. (4.7) and (4.8). In the following sections, tracking plots and friction interface maps for the blade to ground damper is presented Effect of Normal Load Distribution The response of the blade to ground damper system is analyzed for different normal load distributions as given in Figure 4.10, where the total preload acting on the contact interface is unity for all the distributions. Figure 4.11 shows the forced response results for 4 different total preloads together with the completely stuck response. At the highest total load, first three distributions performed much better than the constant normal load case in terms of decreasing vibration amplitudes. Even though the fifth distribution indicates more slip compared to constant normal load case, the reduction in the vibration amplitude is similar to constant distribution. For decreasing total preload, the minimum amplitude obtained from each normal load distribution is different; however, the values are close to each other. On the other hand, the amount of softening is the highest for distribution 3 and the lowest for the constant normal load case. The softening effect also indicates the amount of microslip occurring in the contact interface. It can be concluded that, although the analyzed normal load distributions result in similar minimum vibration amplitudes, the amount of damping obtained for high total load are very different from each other. It should be noted that, for gas turbine engines, working at the optimum total preload value is not practical, since the decrease in total load may 114

132 result in much higher vibration amplitudes. Therefore, in practice higher total preload values are used instead of the optimum value and depending on the distribution, even for high total preloads more damping can be provided to the system (as in the case of distribution 3) Effect of Multiple Harmonics Tracking plots for the blade to ground damper system are given Figure 4.12 and Figure 4.13 for normal load distribution 3 and constant, respectively. The system is excited by single-harmonic excitation forces applied to points B and B`, and the blade response at point A is determined around the first resonance by considering the first three mode shapes of the blade. Tracking plots for free and stuck cases, which are the two linear extreme cases for the system, are as well included in the figures. The nonlinear response of the system is in between these two linear solutions and as the preload increases the system response approaches to the stuck response, finally becoming completely stuck. It should be noted that, since the normal load distribution 3 is zero at the center point it is not possible to obtain initial gap at the center and preserve the shape of the normal load distribution at the same time. Therefore, initial gap case is analyzed by assuming a constant gap between the contact surface and the ground and the result is given in Figure

133 Constant Distribution 1 Distribution 2 Distribution 3 Distribution 4 Distribution 5 Figure 4.10 Normal load distributions 116

134 0.5 N tot =5.0e N tot =1.0e5 Normalized Vibration Amplitude [m/n] Normalized Vibration Amplitude [m/n] Frequency [Hz] Frequency [Hz] 0.3 N tot =2.5e N tot =1.0e4 Normalized Vibration Amplitude [m/n] Normalized Vibration Amplitude [m/n] Frequency [Hz] Frequency [Hz] Figure 4.11 Effect of normal load distribution stuck, constant, distribution 1, distribution 2, distribution 3, distribution 4, distribution 5 117

135 Normalized Vibration Amplitude [m/n] Frequency [Hz] Figure 4.12 Forced response for preload case: distribution 3 stuck, singleharmonic: 1.0e6, 5.0e5, 2.5e5, 1.0e5, 5.0e4, 1.0e4, multi-harmonic: 1.0e6, 5.0e5, 2.5e5, 1.0e5, 5.0e4, 1.0e4 118

136 Normalized Vibration Amplitude [m/n] Free Single-harmonic Multi-harmonic Frequency [Hz] Figure 4.13 Forced response for constant initial gap: single-harmonic, multi-harmonic free, 119

137 When the single and multiple-harmonic results are compared it is observed that, for high preload cases single harmonic solution captures the nonlinear characteristics quite well and the difference between the single and multiple-harmonic solutions is negligible. On the other hand, for the cases where softening or hardening behavior is observed, the difference between the single and multiple-harmonic solutions is significant. The softening or hardening behavior occurs due the separation of the contact interface, and it can be concluded that, if separation occurs single harmonic solution can not capture the nonlinear characteristics accurately. It is observed that for two cases where hardening and softening behavior is observed, single harmonic solution overestimates the vibration amplitude; in addition to this, single harmonic solution estimates a higher frequency shift compared to the multiple-harmonic solution Friction Interface Maps The method proposed in this work is also capable of determining the contact state of the friction interface. The contour plots given in Figure 4.14 to Figure 4.16 are named as friction interface plots. These interface plots show the friction state at the frequency of maximum amplitude for constant, 3 rd and 5 th normal load distributions for a total load of , respectively. It is observed that, most of the area for constant normal load distribution is stuck and simply some region at the center and along the z-axis slips. For the 3 rd normal load distribution, the region of slip is at the center of the contact area, 120

138 Stick Normalized r Slip θ=175 θ=150 Stick Normalized p Figure 4.14 Friction interface plots for constant normal load distribution 121

139 1.0 Stick 0.8 θ=330 Normalized r θ=275 Slip Normalized p Figure 4.15 Friction interface plots for 3 rd normal load distribution 122

140 1.0 Slip-to-Stick Boundary Slip-to-Separation Boundary 0.8 Separation θ=150 θ=150 Normalized r θ=45 Slip Stick Slip θ= Normalized p Figure 4.16 Friction interface plots for 5 th normal load distribution 123

141 where the rest of the interface is stuck. It is observed that the slip region starts somewhere at the center of the interface and propagates outwards. For the 5 th normal load distribution the friction interface is different from the previous two: separation exists as well as the slip region, and the slip and separation regions are around the lower and upper part of the contact interface. It should be noted that, none of the cases experiences gross-slip. It should as well be noted that, the duration of slip given in Figure 4.14 to Figure 4.16 for constant, 3 rd and 5 th normal load distribution is 25, 55 and 105. The amount of slip also indicates the microslip occurring in the friction interface and this is as well in agreement with the forced response results given in Figure Therefore, friction interface maps reveal the microslip phenomena, and they can as well be used to estimate wear caused by the sliding motion CONCLUSION A microslip friction model for two-dimensional periodic motion and a multiharmonic solution method are presented in this chapter. A modal superposition approach is developed to determine the forced response and stick-slip-separation transitions of the frictionally constrained system when subjected to periodic excitation. In this approach, the steady state response of the structure is constructed by superposing its free mode shapes. A microslip friction model for two-dimensional periodic motion with normal load 124

142 variation is developed to determine the friction force and normal load distributions acting on the contact interface. The developed method is applied to a blade to ground damper system and the effect of normal load distribution and multiple harmonics are determined. It is observed that, just like the optimum total preload value, there exists an optimum normal load distribution. Therefore, by tuning the normal load distribution as well as the total preload acting on the contact interface, it is possible to decrease the vibration amplitudes even more or make the damper work more efficiently for a broader range of total preload. Friction interface maps, which describe the stick, slip, separation states on the contact area are determined and they are used to demonstrate the microslip effects. It can be concluded that, in order to capture the characteristics of the friction interface microslip modeling is essential. Moreover, it is observed that, single harmonic solution captures the nonlinear characteristics quite well if the total preload acting on the friction interface is high. However, for lower preload cases, where the friction interface undergoes separation, single harmonic solution estimates the vibration amplitude and the frequency shift higher than the multi-harmonic solution. Therefore, in order to determine system response accurately for the cases of separation, multi-harmonic modeling is crucial. The developed microslip friction method can be applied to any structure where the slip motion is one-dimensional. Moreover, since FE data of the structure can be used 125

143 in the analysis, it is possible to examine practical examples arising in the real design process. It should be noted that, in order to apply the proposed method, in-plane and out of-plane contact stiffness distributions throughout the contact interface has to be determined. This issue will be discussed in future investigation. 126

144 CHAPTER 5 5. THREE-DIMENSIONAL MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION INDUCED BY NORMAL MOTION 5.1. INTRODUCTION Frictionally constrained mechanical systems frequently involve complex contact kinematics due to the relative motion between the constrained interfaces. The relative motion resulting in complex contact kinematics can be decomposed into in-plane tangential and out-of-plane normal components. If the normal component of relative motion is constant, friction interface experiences stick-slip motion due to the tangential component. Contact kinematics with constant normal motion originates either from the specific design of the friction contact [Griffin, 1980; Dowell and. Schwartz, 1983] or from the simplification of the analysis [Cameron et al,. 1990; Ferri, 1996; Toufine et al., 1999; Sanliturk et al., 2001b; Ciğeroğlu and Özgüven, 2006]. On the other hand, normal 127

145 component of relative motion causes normal load variation, which may result in separation of the contacting surfaces as well. In general, tangential component of relative motion can follow a path, which is not a straight line. If the relative motion is periodic, this will form a closed path in the two-dimensional tangential plane. This two-dimensional motion was first studied by Menq et al. [1991] where the friction force is estimated by using an interpolation method, and a special case of two-dimensional motion, circular motion, was analyzed by Griffin and Menq [1991]. Two-dimensional motion was as well analyzed by Sanliturk and Ewins [1996], where the friction contact is characterized by a single stiffness. Later Menq and Yang [1998], developed analytical transition criteria for stick, slip transitions, which were not present in author s earlier work [Menq et al., 1991]. For circular motion, which is a special case of two-dimensional motion, analytical expressions for transition angles and friction forces were derived. In this friction model, flexible element at the contact interface is characterized by a 2x 2 stiffness matrix. Due to the variation of the normal component of relative motion, normal load acting on the friction interface changes. The variations in normal load affects the stick-slip transitions due to tangential motion and it may even result in separation of the contacting surfaces. This phenomenon was investigated by several researches [Menq et al., 1986b; Whiteman and Ferri, 1996; Yang et al., 1998; Yang and Menq, 1998c; Chen et al., 2000; Chen and Menq, 2001; Petrov and Ewins, 2003; Cigeroglu et al., 2007] for one-dimensional and two-dimensional tangential motions. 128

146 Yang and Menq [1998c] developed a three-dimensional point contact friction model where the contact motion was decomposed into in-plane motion and normal motion perpendicular to the contact plane. Authors developed analytical criteria for stick slip and separation transitions and used Harmonic Balance Method to predict the response of a three-dimensional oscillator. Furthermore, three-dimensional friction contact was addressed by Chen et al. [2000] and, Chen and Menq [2001] for periodic motion using similar criteria developed in [Yang et al., 1998]. Friction models described above are for point contact, where the entire friction interface is either in stick, slip or separation states and partial slip of the friction interface is not possible. This is also called as macroslip model, which is widely used, and it works well if the normal load acting on the contact interface is small. However, in microslip approach [Menq et al., 1986c, 1986d; Csaba, 1998; Cigeroglu et al., 2006], which is as well verified by experimental studies [Menq et al., 1986d; Filippi et al., 2004; Koh et al., 2005], friction interface is modeled as an elastic body; therefore, it is possible to have partial slip throughout the contact area. Partial slip or microslip is important especially when the normal load acting on the contact interface is high, which prevents gross slip. In these cases, macroslip model results in stuck interface and estimates no damping. Menq et al. [1986c, 1986d] developed a one-dimensional microslip friction model where authors modeled the friction damper as a one-dimensional elastic bar. Recently, Cigeroglu et al. [2006] has developed a one-dimensional dynamic microslip friction model based on the model given by Menq et al. [1986c]. Authors have considered three 129

147 different normal load distributions and determined analytical results for hysteresis curves considering the first mode of the elastic beam. Cigeroglu et al. [2007], in another recent work, develop a two-dimensional microslip friction model with normal load variation induced by normal motion. The method can be applied to any contact problem, where the relative motion can be decomposed in two components: one-dimensional in-plane motion and normal motion. Authors develop a distributed friction contact model based on the point contact model given in [Yang et al., 1998] and provide the analytical results for the distribution of transition angles. A multi-mode solution method is used and, a one-dimensional beam problem studied in the previous work [Csaba, 1998] and a more complicated blade to ground damper model, where the blade is modeled by finite element method, have been analyzed. Authors obtain the friction interface map, which shows the stick, slip and separation boundaries resulting on the contact area and demonstrate the microslip phenomenon. Due to the normal motion, jump phenomenon is observed in the tracking plot results, and authors indicate that this is caused by the separation of the contacting surfaces. The objective of this chapter is to develop a three-dimensional microslip friction model with normal load variation, which can be used in the analysis of three-dimensional frictional contact problems such as gas turbine engine blade design. The relative motion at the contact surface is decomposed into two-dimensional tangential motion and normal 130

148 motion perpendicular to the tangential plane. In-plane motion causes stick and slip and normal motion causes normal load variation, which may result in separation of the contacting surfaces. In the following sections, analytical criteria of stick, slip and separation are defined and obtained for harmonic motion, and friction interface maps are determined for prescribed motion. Furthermore, a blade to ground damper is analyzed and tracking plots and friction interface maps are given demonstrating microslip effects THREE-DIMENSIONAL MICROSLIP FRICTION MODEL WITH NORMAL LOAD VARIATION Planar contact of two bodies is given in Figure 5.1, in which the gray region characterizes the contact interface composed of distributed springs, representing normal stiffness and tangential stiffnesses. Since the amplitude of vibration is small, the orientation of the contact plane is assumed to be invariant. The displacements of bodies A and B are defined in the global coordinate system denoted by x y z whereas p q r is used for the contact plane coordinate system where the contact plane is defined as q = 0. Contact plane coordinate system and global coordinate system can be related to each other through a translation and a rotation as follows 131

149 x p po y = R q + q. (5.1) o z r r o ] T o o o In this equation, R is the orientation matrix and [ p q r is the position of p q r in x y z. The spatial domain of the contact interface is specified in p q r. Any point within the domain is denoted by [ 0 ] can be determined from the following equation c c T p r and its coordinate in x y z xc pc po y c = R 0 + q o. (5.2) z c r c r o The contact preload and its distribution over the contact area can be determined through static analysis. While vibrating, displacements of contact points on the two bodies are denoted by d ( x, y, z, t) and d ( x, y, z, t), respectively where, [ ] T A c c c B c c c x y z is the c c c vector of contact point coordinates in global coordinate system and t is the temporal variable. The relative motion in contact plane coordinate system can then be determined from the following relation u( pc, rc, t) v( p,, ) 1 c rc t = R [ da d B]. (5.3) wp ( c, rt c, ) 132

150 In Eq. (5.3) u and w are the tangential components of relative motion in p and r directions, respectively, and v is the normal component of relative motion along q direction. It should be noted that, contact interface is defined by p and r axis. y z o q q body A body B x p Figure 5.1 Planar contact of two bodies 133

151 Stick, Slip and Separation Transition Criteria A contact pair in the distributed contact model is illustrated in Figure 5.2, in which ( p, rt, ) u,,,, v( p r t ) K ( p, r), s ( p, rt, ), ( ) u uw n p r and k ( p, r) are the vector of relative motion in tangential direction, relative motion in normal direction, matrix of tangential contact stiffness per unit area, vector of slip motion on the contact interface, 0, v preload (pressure) and normal contact stiffness per unit area, at the point ( p, r ) respectively. The two spatial variables, p and r, are specified on the contact plane and within the contact area, and for brevity their subscript c is removed. The preload distribution n0 ( p, r) is positive if the contact pair is preloaded whereas it is negative and in proportional to distribution of normal stiffness per unit area, kv (, ) p r, if there exists an initial gap. The three-dimensional motion considered in the model is composed of two perpendicular components: tangential motion ( p, rt, ) = [ u w] ) and the normal motion (,, ) ( p r T u in the contact plane v p r t in the q direction. The normal motion causes normal load variation and possible separation, and normal stress is given as follows (,, ) n p r t ( ) ( ) ( ) ( ) ( ) v(, ) ( ) < ( ) ( ) n0 p, r + kv p, r v p, r, t, if v p, r, t n0 p, r / k p r = 0, if v p, r, t n0 p, r / kv p, r. (5.4) 134

152 If the slip motion s ( p, rt, is known, the resulting friction stress can be expressed in uw ) terms of tangential relative motion as ( p, rt, ) = ( pr, ) ( prt,, ) ( prt,, ) f Ku u s uw, (5.5) It is apparent that in order to determine the friction stress, stick-slip-separation states needs to be identified according to relative motion. The transition criteria employed in this chapter for the identification of stick-slip-separation states are based on those developed by Yang and Menq in [1998c]. u( p, r, θ ) v( p, r, θ ) K u ( pr, ) n ( p r) k ( p r ) k v ( p, r) 0, / v, q s ( p, r uw, θ ) r p Figure 5.2 Distributed point contact model for 3D motion 135

153 Slip to Stick Transition Transition from slip to stick state occurs when slip velocity, s / t = 0; and uw when slipping, friction stress amplitude can be determined from ( p, rt, ) = μn( prt,, ) f. (5.6) According to the Coulomb friction law, the slip velocity and the friction stress vectors should be along the same direction, suw = cf ( p, r, t), where c > 0. (5.7) t From Eq. (3.6) one can write, T 2 ( ) ( ) = μ ( 2 f p, rt, f prt,, n prt),,. (5.8) Differentiating Eq. (5.8) with respect to time the following relation is obtained T f 2 n f ( prt,, ) = μ n( prt,, ). (5.9) t t 136

154 Inserting time derivative of Eq. (5.5) in Eq. (5.9) and using Eq. (5.7), the following relation for c is obtained f c = u u t f K f T 2 ( prt,, ) K ( pr, ) μ n( prt,, ) T ( prt,, ) ( pr, ) ( prt,, ) u n t. (5.10) From this result, inserting Eq. (5.10) into Eq. (5.7), the vector of slip velocity can be determined as follows u f = t prt pr prt n T 2 ( prt,, ) Ku ( pr, ) μ n( prt,, ) t t f ( p, rt, T ) f(,, ) K (, ) f(,, ) suw u. (5.11) Inserting time derivative of Eq. (5.5) in Eq. (5.11), the rate of change of vector of friction stress distribution becomes T u 2 n f( prt,, ) K ( pr, ) (,, ) (, ) t μ n prt f u u t = K p r ( p,, T (,, ) (, ) (,, ) ) u f r t. (5.12) t t f prt Ku pr f prt Using Eq. (5.11), the transition criterion from stick to slip becomes 137

155 T u 2 n f( prt,, ) K u ( pr, ) μ n( prt,, ) = 0. (5.13) t t It should be noted that, vector of friction stress distribution used in Eq. (5.13) is an unknown which can be obtained by solving the ordinary differential equation given in Eq. (5.12) using the friction stress at the beginning of the slip state as the initial condition. In order to determine distribution of stick to slip transition, Eq. (5.12) can be integrated numerically by an ODE integrator and the resulting friction stress distribution can be used to determine the time when Eq. (5.13) is satisfied Stick to Slip Transition Friction stress at the stick state is given as follows ( p, rt, ) = ( pr, ) ( prt,, ) ( pr, ) ( pr, ) + 0 f Ku u u0 f, (5.14) where, (, u p r) and ( p, r) 0 f are distributions of displacement vector and friction stress 0 vector at the beginning of stick state. Stick to slip transition occurs when the friction stress amplitude reaches to the slip load (pressure); hence, from Eqs. (5.6) and (5.14) criteria for stick to slip transition can be obtained as 138

156 ( pr, ) ( prt,, ) ( pr, ) + ( pr, ) μn( prt,, ) Ku u u0 f0 n f ( prt,, ) μ > 0. t t = 0, (5.15) Inequality given in Eq. (5.15) is used to ensure the stick to slip transition Separation Separation occurs when the friction interfaces loose contact and, beginning and end of separation can be determined from the following criteria ( ) ( ) ( ) n0 p, r + kv p, r v p, r, t = 0, ( ) ( ) ( ) n0 p, r + kv p, r v p, r, t = 0, v 0 t (5.16) v 0 t (5.17) respectively and the next state after separation is either stick or slip. If the next state is stick the rate of change of friction stress should be less than the rate of change of normal pressure, f u n = K ( pr, ) < μ u. (5.18) t t t Squaring both sides, criteria to have stick state and slip state after separation are 139

157 T 2 2 n μ < u T u Ku( pr, ) Ku( pr, ) t t t T u T u 2 n Ku( pr, ) Ku( pr, ) μ t t t 2 0, (5.19) 0, (5.20) respectively. It should be noted that, the friction coefficient in this analysis is taken as constant through out the contact interface; however, it can as well be considered as a distributed parameter and the same equations can be used if the distribution of friction coefficient is combined with the normal load distribution Transition Angles If the in-plane and out of plane relative motions are harmonic distribution of stick, slip and separation transition angles can be determined by using the transition criteria given in Eqs. (5.12)-(5.20). In order to simplify the analysis let ( pr,, θ) = u ( pr, ) ( pr,, θ), ( prθ) = u ( pr) uw ( prθ) = (, ) (,, θ ). u K u s,, K, s,,, (5.21) v k p r v p r v 140

158 For harmonic relative motion the following forms for u ( p, r, θ ) and (,, ) considered v p r θ are u ( prθ) = a( pr) ( θ) b( pr) ( θ) v( pr,, θ) = d( pr, ) sin θ + ϕ( pr, ).,,, sin, cos, ( ) T (5.22) From Eq. (5.22) the normal stress distribution becomes, ( ) ( ) ( ) θ ϕ( ) n0 pr, + d pr, sin + pr,. (5.23) When slipping, the following form for the distribution of friction stress vector can be assumed θ μ θ ( ψ θ ) ( ψ ( θ )) ( ) = ( ) ( ) f pr,, n pr,, sin pr,, cos pr,,. (5.24) T Inserting Eqs. (5.22)-(5.24) into Eqs. (5.12) and (5.13), a single ODE together with a simple criterion for slip to stick transition can be obtained. These results are given in Appendix B. Transition angle from slip to stick can be obtained by solving the nonlinear ODE given in Eq. (B.1) by considering the initial friction stress vector calculated at the beginning of the slip state as the initial condition, and determining the temporal variable satisfying Eq. (B.2). It should as well be noted that, using the substitution given in Eq. 141

159 (5.24) the number of differential equations in Eq. (5.12) decreased from two to one; therefore, just a single nonlinear ODE is to be solved. Inserting Eqs. (5.22) and (5.23) into Eq. (5.15) a quartic equation, for which analytical solutions are available, in terms of sin ( θ ) or cos( θ ) can be obtained to determine stick to slip transition and it is given in Appendix B. Distribution of transition angles for the beginning and end of separation can be obtained analytically by inserting Eqs. (5.22)-(5.24) into Eqs. (5.16) and (5.17) θ θ SP1 SP2 ( pr, ) ( pr, ) n0 ( p, r) d( p, r) ( p, r) (, ) π + arcsin ϕ( pr, ), if d( pr, ) > 0 =, (5.25) n 0 arcsin ϕ ( pr, ), if d( pr, ) < 0 d p r ( p, r) (, ) n0 ( p, r) d( p, r) n 0 arcsin ϕ ( pr, ), if d( pr, ) > 0 d p r =, (5.26) π + arcsin ϕ( pr, ), if d( pr, ) < 0 respectively. Similarly, the next state after separation can be determined from 142

160 2 2 (, ) sin 2 θ ( ) ( ) 2 SP2, +, cos θsp2(, ) μ d( p, r) sin θ ( p, r) + ϕ( p, r) a pr Stick pr b pr pr < 1, SP2 2 2 (, ) sin 2 θ ( ) ( ) 2 SP2, +, cos θsp2(, ) μ d( p, r) sin θ ( p, r) + ϕ( p, r) a pr Slip pr b pr pr 1. SP2 (5.27) The distribution of transition angles for a cycle of motion can be determined by a stateby-state simulation. Starting from an initial condition, distribution of transition angles of stick and slip can be obtained by solving Eqs. (B.1) and (B.2), and the quartic equation, (B.11), given in Appendix B until convergence occurs. It should be noted that the distribution of transition angles for the beginning and end of separation can be determined analytically and they can be used to determine the distribution of transition angles for stick and slip states in between them in one cycle. If separation does not occur, determination of the distribution of transition angles may take longer time by a state-bystate simulation. In order to accelerate the convergence, the following function is defined in terms of the transition angles ( i) i i 1 F α = α α, (5.28) where i and i 1 indicates the current and previous states and α is the parameter whose steady sate value is required. It should be noted that, when convergence occurs, this function becomes zero. Therefore, secant method can be applied to determine the zero of this function, which will result in the following relation 143

161 ( α α ) 2 i i 1 αi+ 1 = αi α i 2 α i 1+ α i 2. (5.29) This formula can be applied every 3 cycles in order to get better estimates for the transition angles and initial conditions for the ODE. These new values can be used as the initial condition for a new state-by-state simulation, which can be terminated if the difference between the transition angels and/or the friction stress vector drops below a certain limit. It is interesting to note that, the formula given in Eq. (5.29) is the same as Aitken s method for accelerating convergence of series. Since the results of the previous state directly affect the next state, in this work Eq. (5.29) is applied in every 3 cycles in order to decrease the computational time and control the error as well. Transition angle results for state-by-state simulation for a case without separation are given in Table 5.1, with and without using the developed convergence acceleration method. State-by-state simulation needs 78 cycles in order for the friction stress to reach steady state without any modifications, whereas it took only 11 cycles if the proposed convergence acceleration method is used. Similar results are as well obtained for the cases where the contact pair undergoes full slip; therefore, it is evident that, the method proposed decreases the computational time significantly. After obtaining the distribution of transition angles, it is possible to determine stick-slipseparation (friction interface) map at any instant. This friction interface map is useful to 144

162 understand how the friction damper works and it can as well be used to estimate the wear of the contacting surfaces. Convergence Acceleration by No Convergence Acceleration Secant Method Cycle Number Transition Angle State Transition Angle State Stick Stick Slip Slip Fully Slip Fully Slip Stick Slip Stick Slip Stick 3-4 Same for both Slip Stick Slip Stick Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Slip Stick Table 5.1 State-by-state simulation with and without convergence acceleration 145

163 Distribution of Fourier Coefficients In the dynamic response analysis of systems with dry friction dampers, it is common to represent nonlinear forces in terms of Fourier series. If a single harmonic representation is used, nonlinear forces, in this case stresses, in three directions can be represented by f( ab,, pr,, θ ) f ( ab,, pr, ) f ( ab,, pr, ) sin ( θ) f ( ab,, pr, ) cos( θ) + +. (5.30) * * b * s * c where, * indicates p, q and r directions, and 1 2π f* b ( ab,, pr, ) = f* ( ab,, pr,, θ ) dθ 2π, 0 (5.31) 1 2π f* s ( ab,, pr, ) = f* ( ab,, pr,, θ ) sin( θ) dθ π, 0 (5.32) 1 2π f* c ( ab,, pr, ) = f* ( ab,, pr,, θ ) cos( θ) dθ π. 0 (5.33) In Eqs. (5.31)-(5.33), a = a( p, r) and b= b( p, r). If the distributions of transition angles are identified, friction stress and normal stress distributions can be determined from Eqs. (5.4), (5.14) and (5.24), and hence the distributions of Fourier coefficients for each stress component can be obtained from Eqs. (5.31)-(5.33). 146

164 5.3. MULTI-MODE SOLUTION METHOD Equation of motion for a system with dry friction dampers can be written as e ( ) ( ) MX + CX + KX = F t + F X (5.34) n where () ( ) M, C, K, F t, F X e n and X are the mass matrix, viscous damping matrix, stiffness matrix, excitation force vector, nonlinear force vector and displacement vector, respectively. The 3D motion of the system for harmonic excitation can be written in terms of its mode shapes as follows N N u( xyz,,, θ) Bφ ( xyz,, ) Im Aφ ( xyze,, ) θ, (5.35) = + i n 1 n u1n n u1 n= 1 n= 1 N N u( xyz,,, θ) Bφ ( xyz,, ) Im Aφ ( xyze,, ) θ, (5.36) = + i n 2 n u2n n u2 n= 1 n= 1 N N u xyz B xyz A xyze θ, (5.37) (,,, θ) φ (,, ) Im φ (,, ) i = + n 3 n u3n n u3 n= 1 n= 1 where ( ) ( ) ( ) φu1 x, yz,, φu2 xyz,,, φu3 xyz,,, N, i, Bn n n n n th and A are the n mode shapes in th x, y and z directions, number of modes used, imaginary number, n modal coefficient for dc component of motion and th n complex modal coefficient for elastic 147

165 component of motion, respectively. Displacement vector X in global coordinate system can be written as X ( BAxyz,,,, ) Bφ ( xyz,, ) Im Aφ ( xyze,, ) θ, (5.38) N N = i + n n n n n= 1 n= 1 where ( ) th φ n x, yz,, Band A are the n mode shape in global coordinate system, model coefficient vectors for dc and elastic components of motion, respectively. X can be transformed to contact plane coordinates by applying Eq. (5.3). Using the orthogonality relation Eq. (5.34) can be simplified to ( ) 2 e e Ω ω I+ iωc A= Q+ iq+ Q( BA, ) + iq( BA, ), (5.39) r s c s c Ω B = Q (, ) b B A, (5.40) if mass normalized mode shapes are used. Here Ω is NxN diagonal matrix of squares of natural frequencies, C r is the modal damping matrix, which is diagonal if the damping is proportional and ω is the excitation frequency. The forcing vectors on the right hand side of Eqs. (5.39) and (5.40) are * * * ( ) ( ) ( ) ( ) ( ) ( ) Q* =,,,,,,,,,,,, n φu p r F n u B A p r + φw p r F n w B A p r + φv p r F n v B A p r dpdr,(5.41) D 148

166 Q j ( xyz,, ) f ( xyz,, ) + φ ( xyz,, ) 1 (,, ) φ (,, ) (,, ) φ = dxdydz, ( j = c, s ), (5.42) F x y z x y z F x y z e u1n eu u2n j j j D e e + u2 u3n eu3 where, * corresponds to s, c, or b and, uw, and v indicate the mode shapes and forces in p, r and q directions, respectively. F is the nonlinear friction and normal stress distributions in contact plane coordinate system and, F e is the excitation force per unit volume in global coordinate system. D and D e are the domain of integration for contact and excitation forces, respectively. If the excitation forces act on nodes of the finite element model, Eq. (5.42) can be replaced by Q = Φ f, Q = Φ f e, (5.43) e T s e T c s e c where s f e and c f e are the amplitudes of sine and cosine components of excitation force vector. Eqs. (5.39) and (5.40) describe a set of nonlinear algebraic equations composed of 3N equations and 3 N unknowns which can be solved by an iterative nonlinear solver such as Newton solver. The motion of the system can be constructed once the unknown modal coefficient vectors B and A are obtained. It should be noted that, since there exists dc components of forces, solution for displacements contains dc components as well, and the dc component of normal motion 149

167 affects the preload acting on the contact area. The effective preload acting on the contact interface can be obtained as follows n ( pr, ) = n( pr, ) k( pr, ) v( pr, ), (5.44) 0 0 v 0 ( ) v p r) where n p r and ( are the distribution of effective preload and dc component 0, 0, of normal motion, respectively Friction Interface Map If the relative motion in the contact area is known, at any instant, contact state of any point in the friction interface can be determined. Therefore, discritizing the contact area, friction interface map for any instant can be generated. It is as well possible to use arclength continuation method to follow a boundary and only determine the stick, slip, and separation boundaries, which is faster than a fine discretization. However, there are many possible stick, slip and separation sequences and special care must be taken in order to identify them. It should be also noted that, in order to apply continuation method a point on the boundary should be known prior to the application of the process. A coarse discretization will identify possible states occurring in the contact interface and using this information a point on the boundary of interest can be determine by solving for the 150

168 transition angle. After determining a point on the boundary, arclength continuation method can be employed to follow and obtain the complete boundary. Since the transition angles are calculated for much less number of points compared to a fine discretization, continuation method is much faster than the fine discretization of the friction interface. However, it should as well be noted that, it is possible to have a state surrounded by a different state resulting in closed boundaries (islands) in the friction interface. If the size of those islands is small, they might not be identified in the coarse discretization; therefore, if this is the case a fine mesh should be used and continuation method can be applied to increase the accuracy of interface boundaries. Some of the factors affecting wear of sliding surfaces are duration of sliding, contact pressure, and friction stress acting on the contact interface. Contact pressure and friction stress distributions are already determined from the developed friction model and the duration of sliding can be obtained from the stick-slip-separation map. Therefore, friction interface map can be used to estimate the wear of sliding surfaces; however, in this work, it is used to demonstrate the microslip phenomenon NUMERICAL RESULTS The developed microslip friction model is first applied on a contact area where the relative motion over the contact interface is specified, and friction interface maps for 151

169 different instants are given. In addition to this, the developed method is applied to a blade to ground damper and tracking plots and friction interface maps are presented in the following sections Friction Interface for Specified Motion For this analysis, distribution of relative motion over the contact area is defined and friction interface maps for the specified motion are presented in Figure 5.3. The distribution of motion over the contact area is given as u( p, r) = a( p, r) sin ( θ ), (, ) = (, ) sin θ + φu (, ) (, ) = (, ) sin θ + φ (, ) w pr b pr pr v p r d p r n p r,, (5.45) where, ( ) ( ) ( ) ( p + r) ( + p r) ( ) a p r e r d p r cos 0.1, = cos, cos 0.7, = e, b p, r = 1, φ φ u n π, 4 π π ( pr, ) = 2πsin ( p + r pr) ( pr, ) = sin ( p + r pr) (5.46) 152

170 Stick Slip Slip Slip 0.6 Separation a) e) Stick Slip Slip 0.6 Separation Spatial Variable r 0.2 b) Slip Stick Slip 0.2 f) Separation Slip Stick Slip c) g) Separation Slip 0.8 Stick Slip Separation 0.4 Slip d) Spatial Variable p h) Figure 5.3 Friction interface maps for prescribed motion: a) θ=0, b) θ=45, c) θ=90, d) θ=135, e) θ=180, f) θ=225, g) θ=270, h) θ=

171 For simplicity, ( p, r) identity, k ( p, r) 1, n ( p, r) u = v = 0 = 1 1 K and μ =. Friction interface maps for 8 different instants are given in Figure 5.3. The effect of microslip can be seen clearly, since multiple states occur in the contact area. In the four consecutive instants, θ = 315,0,45,90, stick and slip states occur and variation in these states are slow. Separation and slip sates dominate the friction interface for θ =135. For θ =180, contact areas are separated except for a small region at the top right corner, which slips. In the following state friction interface is completely separated and for θ = 270 all three states: stick, slip and separation exist in the contact interface. It is clear that, the developed microslip friction model is capable of determining stick, slip and separation regions for 3D motion if the distribution of the relative motion on the contact interface is known Blade to Ground Damper The blade to ground damper system analyzed is given in Figure 3.9, where the blade is in contact with the ground from the right side. In this analysis, the blade is represented by a finite element model as shown in Figure 5.4, in which B is the point of application of external excitation and A is the point, where the displacements are calculated. For this blade to ground damper system, mode shape information is obtained 154

172 by a finite element analysis. It should be noted that, the mode shapes determined by finite element analysis are discrete and therefore, curve fitting is applied to determine continuous mode shapes and they are inserted in Eq. (5.41) in order to determine the modal forces. Eqs. (5.39) and (5.40) are solved by a nonlinear solver in order to determine the unknown modal coefficients, which are used in Eqs. (5.35)-(5.37) to determine the blade motion. In the following sections, tracking plots and friction interface maps for the blade to ground damper are presented Tracking Plots Tracking plots for the blade to ground damper system are given in Figure 5.5 for different normal load/initial gap cases around the first natural frequency of the system for a forcing in p direction. In these analyses, five modes of the blade are used and the preload (or initial gap) distribution is kept constant all over the contact surface. Tracking plots for the two linear cases, where the blade is free and completely stuck, are given as well and tracking plot for any other preload is in between these two extreme cases. In design point of view, it is important to determine the preload where the minimum vibration amplitude for the specified resonance frequency occurs. It can be observed that, 5 for this example this optimum value of preload is Due to the variation of normal load, softening and hardening effects are visible in the tracking plots, which are caused by the separation of the contact surfaces. For those 155

173 cases jump phenomenon occurs if the frequency scanning is performed from low to high frequency or from high to low frequency and there exists an unstable solution branch in between these two jumps. In Figure 5.5, the unstable solution branches can be seen for 6 5 the cases, where the preloads are 1 10 and In order to determine these unstable solution branches, natural continuation using the phase angle as the scanning parameter and arclength continuation methods are used interchangeably. Natural continuation method is preferred and employed wherever is possible, since it does not increase the number of nonlinear equations Friction Interface Friction interface maps for the blade to ground damper are given in Figure 5.6 and Figure 5.7, where the transition boundaries are determined by the continuation method as described in section The analysis parameters used are: excitation force amplitude ( ) T F = N in p q r coordinate system, excitation frequency ω = 317Hz and preload 6 n0 ( p, r ) = 1 10 N/m 2. For this case, most of the friction interface is governed by slip sate and in Figure 5.6 and Figure 5.7 the propagation of the stick state for two different cases are shown. In Figure 5.6 the propagation of the stick state is from left to right, whereas, it is from right to left in Figure 5.7; moreover, the propagation is faster in the former compared to the latter one. 156

174 Figure 5.4 Finite element model for the blade 157

175 0.20 Displacement / Force [m/n] Frequency [Hz] Figure 5.5 Tracking plots for blade to ground damper system: free, -1x10 6, 2x10 5, 5x10 5, 1.5x10 6, 2x10 6, stuck 158

176 1.0 Stick Normalized Spatial Variable z Slip Normalized Spatial Variable x' Figure 5.6 Propagation of stick-slip boundary: ( ) θ=27.0, ( ) θ=27.1, ( ) θ=27.2, ( ) θ=27.3 (stick on the right, slip on the left) 159

177 1.0 Stick Normalized Spatial Variable z Stick Slip Normalized Spatial Variable x' Figure 5.7 Propagation of stick-slip boundary: ( ) θ=211.0, ( ) θ=212.0, ( ) θ=212.5, ( ) θ=213.0 (stick on the left, slip on the right) 160

178 By applying the developed method and determining the friction interface map, it is possible to observe the friction state of two contacting surfaces having threedimensional relative motion. The results obtained illustrate that, stick, slip, separation may occur at the same time in the contact interface, and microslip modeling is necessary for accurate analysis of friction contact problems associated with multiple contact states CONCLUSION A three-dimensional microslip friction model with normal load variation induced by normal motion is presented in this chapter. Three-dimensional point contact model for time varying normal load is extended for a distributed parameter system, which is used to determine the friction stress and normal stress distributions on the contact interface. A multi-mode solution approach is developed to determine the forced response and stick, slip, separation transitions of the frictionally constrained system when subjected to harmonic excitation. In the proposed approach, steady state response of the system is constructed by its free mode shapes, which are obtained from a finite element software together with a three-dimensional curve fit. The developed microslip friction method is applied for a case where the relative motion of the contact area is specified and effect of microslip phenomenon is presented. 161

179 In addition to this, the developed method is also applied on a more realistic blade to ground damper model, where the blade is modeled by finite element method. Resulting tracking plots and transition maps are given and they clearly show the microslip phenomenon. The three-dimensional relative motion between two contacting surfaces can be decomposed into normal and tangential components with respect to the friction interface. Therefore, the developed method can be applied to any system where the relative motion is three-dimensional. Moreover, the developed method is capable of analyzing practical examples arising in the design process by using a finite element model of the system. It should be noted that, in order to apply the proposed method, contact stiffness distributions on the friction interface has to be determined. This issue will be discussed in future investigation. 162

180 CHAPTER 6 6. FRICTION MODELS FOR STIFFNESS VARIATION 6.1. INTRODUCTION Frictional stiffness considered in the friction models in literature: macroslip [Menq et al., 1986b; Yang et al., 1998; Yang and Menq, 1998c; Petrov and Ewins, 2003] and microslip [Menq et al., 1986c, 1986d; Csaba, 1998; Quinn and Segalman, 2005, Cigeroglu et al., 2006], is constant through out the analysis. it possible to have friction contact stiffness depend on displacements corresponding to coordinate it is attached, resulting in temporal variation of the contact stiffnesses. Variation of friction coefficient and tangential stiffness is considered by Petrov and Ewins [2004] for time domain analysis; however, no transition criteria between stick, slip, separation for general motion have been developed. 163

181 In this chapter, it is aimed to develop transition criteria for two cases: onedimensional tangential motion with stiffness and normal load variation and twodimensional tangential motion with stiffness and normal load variation. In the following sections transition criteria for the two types of motion is developed TWO-DIMENSIONAL MOTION WITH STIFFNESS AND NORMAL LOAD VARIATION A contact pair in the distributed contact model is illustrated in Figure 6.1, in u( p r t ) v( p r t ) k ( p, r, t ), s ( p, r, t ), ( ) which,,,,,, u u n p r and k ( p, r, t) are the relative motion in the slip direction, relative motion in the normal direction, variation of contact stiffness distribution in the slip direction, slip motion, preload distribution, and 0, v variation of normal stiffness distribution, respectively. The two spatial variables, p and r, are specified on the contact plane and within the contact area, and their subscript c is removed for simplicity. The preload distribution n ( ) 0, p r is positive if the contact pair is preloaded. On the other hand, n ( ) 0, p r is negative and in proportional to normal stiffness distribution kv ( p, r, t) if it has an initial gap. The two-dimensional motion considered in the model is composed of two perpendicular components: tangential motion u( p, r, t) in the p direction and the normal motion v( p, r, t ) in the q direction. 164

182 The normal motion and normal stiffness variation cause normal pressure variation and may result in separation of the contact interfaces. Normal pressure variation is defined as follows, (,, ) n p r t n n0 ( pr, ) + kv ( prt,, ) v( prt,, ), if v( prt,, ) kv = n 0, if v( p, r, t) < kv 0 ( p, r) ( p, r, t) 0 ( p, r) ( p, r, t). (6.1) (,, ) u p r θ k k u v ( p, r, θ ) ( p, r, θ ) (,, ) v p r θ s u ( prθ,, ) n ( p r ) 0, Figure 6.1 Distributed contact model for 2D motion with stiffness variation 165

183 The resulting friction force/stress is related to the tangential relative motion according to the following equation, if the slip motion s (,, ) u p r t is known, (,, ) = (,, ) (,, ) (,, ) f prt k prt u prt s prt u u. (6.2) Stick, Slip and Separation Transition Stick-to-Slip Transition When stuck, friction stress is given by (,, ) = (,, ) (,, ) (, ) +δ (, ) f prt k prt u prt u pr pr u 0 0, (6.3) where u0 ( p, r) and δ 0 ( p, r) are the tangential displacement and elongation/contraction of the tangential spring at the beginning of the stick state. Stick to slip transition occurs when the friction stress is equal to the slip load, and in order to guarantee slip, friction force should have higher increase rate with respect to the slip load. Therefore, criteria for stick to slip is given as ( ) ( ) ( ) + δ ( ) γμ ( ) + ( ) ku prt,, u prt,, u0 pr, 0 pr, n0 pr, kv prt,, = 0 ku u k, (6.4) v γ u( p, r, t) u0( p, r) + δ0( p, r) + γku ( p, r, t) μ > 0 t t t 166

184 = = and γ = ± 1, + 1 for positive slip and 1 for where k k( prt,, ) k( prt,, ) v( prt,, ) negative slip. v v v Slip-to-Stick Transition Transition from slip to stick state occurs when slip velocity, s / t = 0 and u 2 2 γ s / t <0; and while slipping, friction stress can be determined from u (,, ) γμ (,, ) f prt = n prt. (6.5) Differentiating Equations (6.2) and (6.5) with respect to time and equating them, slip velocity is determined as follows kv 2 u γku( prt,, ) μ n0 ( pr, ) kv( prt,, ) ku( prt,, ) ku( prt,, s + γμ + ) u = t t 2 t k p r t u (,, ). (6.6) It should be noted that, tangential stiffness, ( ) k p, r, t > 0 and transition criteria for slip to stick transition can be obtained as follows: γ kv (,, ) ( ) ( ) ( ) ( ) 2 u ku prt μ n0 pr, kv prt,, γμ ku prt,, ku prt,, t t =, (6.7) u 167

185 2 2 2 ku kv ku( prt,, ) μ n 2 0 ( pr, ) + kv( prt,, ) μ k 2 u( prt,, ) + t t 2 k u u k γ2 ku p, r, t γku p, r, t n 2 μ 0 p, r t t t t u 2 u ( ) + ( ) ( ) kv 2 u ku kv( p, r, t) μ ku( p, r, t) + γku( p, r, t) 2 ku( p, r, t) < 0 t t t +. (6.8) Inequality given in Equation (6.8) is used to guarantee slip to stick transition. It is possible to have slip acceleration, 2 / 2 su t, as zero at the slip to stick transition time where / 3 su t 3 has to be computed to determine if stick occurs. Time derivative of slip acceleration will be more complicated compared to itself. It should be noted that, if stick occurs rate of change of friction stress should be less than the rate of change of slip load, from which a simpler equation can be obtained to guarantee slip to stick transition. At the beginning of stick state friction stress can be given as follows ( ) = ( ) ( ) ( ) + δ ( ) = γμ ( f prt,, k prt,, u prt,, u pr, pr, n prt,,. (6.9) u 0 0 ) Differentiating Equation (6.3) with respect to time, the following relation is obtained f ku u = u( p, r, t) u0( p, r) + δ0( p, r) + ku ( p, r, t) t t t. (6.10) γ k u n( pr, ) k( prt,, ) ku( prt,, ) k p r t t t u = 0 + v + u (,, ) 168

186 Therefore, in order to guarantee slip to stick transition the following inequality should be satisfied as well k γ t u u n0 pr, kv prt,, ku prt,, ku prt,, t 2 ( ) + ( ) + ( ) γμ ( ) k t v < 0. (6.11) Separation Separation occurs when the friction interfaces loose contact and, beginning and end of separation can be determined from the following relations kv n0 ( p, r) + kv ( p, r, t) = 0, 0 t >, (6.12) kv n0 ( p, r) + kv ( p, r, t) = 0, 0 t <. (6.13) respectively. Assuming the interface is stuck after separation, friction stress can be written as follows (,, ) (,, ) (,, ) f prt = k prt u prt. (6.14) u In order to have stick state the rate of change of friction stress should be less than the rate of change of slip load. Therefore, the state after separation can be determined as follows 169

187 kv ku u kv Stick : μ < u( prt,, ) + ku ( prt,, ) < μ t t t t, (6.15) ku u k Slip : γ u ( p, r, t ) k u ( p, r, t v + ) > μ t t. (6.16) t 6.3. THREE-DIMENSIONAL MOTION WITH STIFFNESS AND NORMAL LOAD VARIATION A contact pair in the distributed contact model is illustrated in Figure 6.2, in which ( p, rt, ) u,,,, v( p r t ) K ( p, rt, ), s ( p, rt, ), ( ) u uw n p r and k p, r, t are the vector of relative motion in tangential direction, relative motion in normal direction, matrix of tangential contact stiffness variation per unit area, vector of slip motion on the contact interface, preload distribution (pressure), and variation of normal contact stiffness 0, v ( ) per unit area, at the point ( p, r ) respectively. The two spatial variables, p and r, are specified on the contact plane and within the contact interface. The preload distribution n0 ( p, r) is positive if the contact pair is preloaded whereas it is negative and in proportional to distribution of normal stiffness per unit area, kv (,, ) p r t, if there exists an initial gap. The three-dimensional motion considered in the model is composed of two perpendicular components: tangential motion ( p, rt, ) = [ u w] T u in the contact plane 170

188 ) and the normal motion (,, ) ( p r v p r t in the q direction. The normal motion causes normal load variation and possible separation, and normal stress is given in Eq. (6.1). If the slip motion s ( p, rt, is known, the resulting friction stress can be expressed in uw ) terms of tangential relative motion as ( p, rt, ) = ( pr, ) ( prt,, ) ( prt,, ) f Ku u suw. (6.17) u( p, r, θ ) ( ) ( ) n0 p, r / kv p, r, θ K u ( prθ,, ) kv v( p, r, θ ) ( p, r, θ ) q s ( p, r uw, θ ) r p Figure 6.2 Distributed contact model for 3D motion with stiffness variation 171

189 Stick to Slip Transition Stick to slip transition occurs when the amplitude of friction stress reaches to the slip load. Friction stress at the stick state s given as follows ( p, rt, ) = ( prt,, ) ( prt,, ) ( pr, ) + δ ( pr, ) f Ku u u0 0, (6.18) where (, u p r) and ( p, r) 0 δ are the distribution of tangential displacement vector and 0 the vector of elongation/contraction in the tangential stiffness at the beginning of stick state. Therefore, stick to slip transition occurs if the following relations are satisfied ( p, rt, ) ( prt,, ) ( pr, ) δ ( pr, ) μ n ( pr, ) k( prt,, ) f = K u u + = + u v f t kv μ > 0 t. (6.19) Slip to Stick Transition According to the Coulomb s law of friction, friction stress is in the direction of the slip velocity. Therefore, friction stress and slip velocity can be related to each other as follows s uw = c f, c > 0. (6.20) t 172

190 When slipping, amplitude of friction stress is equal to the slip load resulting in 2 T 2 ( ) ( ) 2 f = f f = μ n0 p, r + kv p, r, t (6.21) Differentiating Eqs.(6.17) and (6.21) with respect to time, T f 2 kv f = μ n0 ( p, r) + kv ( p, r, t) t (6.22) t f u suw Ku = Ku ( p, rt, ) ( prt,, ) uw ( prt,, ), t t t + t u s u suw Ku = K ( prt,, ) + ( prt,, ) 1 u. t t Ku f t (6.23) Inserting Eq. (6.23) in (6.22) and using the relation given in Eq. (6.20), unkown coefficient c can be determined as follows u K c= f prt Ku prt + f prt Ku prt u prt t t T T u 1 (,, ) (,, ) (,, ) (,, ) (,, ) T (, ) (,, ) } f(,, ) K (,, ) f(,, ) n pr k prt prt prt prt 2 μ 0 + v u. (6.24) Form here slip velocity is rate of change of friction stress can be obtained as 173

191 s t uw T Ku 1 = f( prt,, ) Ku ( prt,, ) u( prt,, ) + t f u t T 2 ( prt,, ) K ( prt,, ) μ n( pr, ) k( prt,, ) u v 0 + v T ( prt,, ) ( prt,, ) ( prt,, ) ( prt,, ) f f K u f k t.(6.25) f K u 1 u T u = Ku( prt,, ) f + Ku( prt,, ) f( prt,, ) Ku( prt,, ) t t t t K f Ku u t T u 1 2 ( prt,, ) ( prt,, ) ( prt,, ) μ n( pr, ) k( prt,, ) T ( prt,, ) ( prt,, ) ( prt,, ) ( prt,, ) f f K u f v 0 + v k. (6.26) t Slip to stick transition occurs when slip velocity, s t = 0, which results in the following stick to slip transition criteria uw u K f Ku f Ku u t t T T u 1 ( p, rt, ) ( prt,, ) + ( prt,, ) ( prt,, ) ( prt,, ) k n ( p, r) k ( p, r, t) f t 2 v μ 0 + v = 0. (6.27) However, it should be noted that, fiction stress vector is still an unknown and can be determined by integrating the ODE set given in Eq. (6.26). 174

192 Separation When the friction interfaces loose contact, separation occurs and the beginning and the end of separation can be determined from Eqs. (6.12) and (6.13). The next state after separation is either stick or slip. If the next state is stick, the rate of change of friction stress should be less than the rate of change of slip load, Stick Slip f k < μ t t f t v k μ t v. (6.28) Rate of change of friction stress vector for stick state at the end of separation can be determined as follows f t = K u ( prt,, ) u, (6.29) t since ( p rt ) ( ) u,, u pr, at the end of separation state. From here, criteria to have sep = 0 stick state and slip state after separation are T u T 2 kv ( prt,, ) ( prt,, ) u K μ u Ku 0 t t t <, (6.30) 2 175

193 T u T 2 kv ( prt,, ) ( prt,, ) u K μ u Ku 0 t t t. (6.31) 2 It should be noted that, friction stress at the end of the separation state is zero and Eq. (6.26) becomes singular and can not be used to determine the friction stress. From Coulomb s law of friction, friction stress vector at the end of separation state can be written as follows u u f ( prt,, ) = μ n0 ( pr, ) + kv ( prt,, ) t t. (6.32) Therefore, differentiating with respect to time and considering that slip load is zero at the end of separation; rate of change of friction stress vector at the end of separation state can be obtained as f k μ v u u = t t t t. (6.33) If slip occurs at the end of separation state, instead of Eq. (6.26), which is singular at that instant, Eq. (6.33) can be used to determine friction stress vector. Since the friction stress vector on the contact interface can be specified by an amplitude and an angle, same form given in Eq. (5.24) can be used which will reduce the number of differential equations from two to one. 176

194 6.4. NUMERICAL EXAMPLE In case of non-planar contact interfaces, contact area changes as a function of normal load. In order to demonstrate this, a hemi-sphere to flat surface contact as shown in Figure 6.3 is studied. N R R>>a u z u z a a r y x Contact Area Figure 6.3 Hemi-sphere to flat surface contact 177

195 According to the Hertz contact theory, contact area between the hemi-sphere and a flat surface is circular and its radius is given as 3 NR* a = 4 E * 1/3, (6.34) where N is the normal load acting on the hemi-sphere and E R * 1 1 = + R1 R2 1 ν 1 ν = + E1 E * (6.35) ν and E are Poisson s ratio and elastic modulus of the material, respectively. This problem can as well be modified by elastic foundation model and according to Johnson [2004], total normal and tangential stiffness of the elastic foundation can be defined as k k v u * E = αv A a, (6.36) * E = αu A a where α v, α u and A are the stiffness coefficients in normal and tangential directions and contact area. Some values for α v are given in Johnson [2004]: 1.7 for axi-symmetric case, 178

196 1.18 for two-dimensional case and it is indicated that using 1.35 will result in an error that is less than 7%. In addition to this, it is suggested to use 1.5α v for α u. Using the stiffnesses defined by the elastic foundation model, this contact problem can be represented by a point contact model as shown in Figure 6.1. Since the normal force on the normal stiffness can be written as 3 NR* N = αve* π 4 E * 1/3 v, (6.37) where v is the normal motion. From Eqs. (6.37) and (6.34) radius of the contact circle can be determined as a function of the normal motion, 3π = α vr v. (6.38) 4 ( ) * a v Therefore, contact stiffnesses as a function of normal motion can be obtained as k k u v 3 3π 3 *2 * α v E R v, if v> 0 = 4 0, otherwise. (6.39) 3 π 3 *2 * α v E R v, if v> 0 = 3 0, otherwise 179

197 Two-dimensional motion with stiffness and normal load variation In order to study the effect of stiffness variation, simple harmonic form is assumed for the tangential and normal relative motions as given below u ( θ) = asin ( θ) ( θ) = + sin ( θ + ϕ) v v b k k u v ( θ ) ( θ ) 0 ( θ) ( θ) k1 v, if v > 0 = 0, otherwise 3 k1 v( θ), if v( θ) > 0 = 2 0, otherwise, (6.40) where k 1 is a function of contact geometry and material of the contacting bodies. In Figure 6.4 hysteresis plots for constant and variable stiffness for a case where separation does not exist are given. Mean stiffness values are used as constant stiffnesses. It is expected to have different hysteresis curves; however, it is observed that transition angles for variable and constant stiffness curves are identical. This is due to the relationship between the tangential and normal stiffnesses, where the variable part cancels and only the ratio between them, which is constant, exists in the transition angle equations. Similar results are also obtained for a case where there is separation and hysteresis curves for two different phase angles are given in Figure

198 10 ϕ = 0 Constant Stiffness + Slip 5 Stick Stick 0 Friction Force Variable Stiffness -Slip ϕ = 90 Constant Stiffness + Slip 0 Stick Stick 5 -Slip Variable Stiffness Tangential Displacement Figure 6.4 Hysteresis curves for no separation case 181

199 4 ϕ = 0 Variable Stiffness 2 Constant Stiffness + Slip Stick 0 Separation - Slip Friction Force ϕ = 90 2 Variable Stiffness Stick Constant Stiffness + Slip 0 Separation - Slip Stick Tangential Displacement Figure 6.5 Hysteresis curves for no separation case 182

200 In Figure 6.6 normalized hysteresis curves for different k 1 are given. It is observed that, due to the special form of the contact stiffnesses, normalized hysteresis curves are identical. k 1 scales-up or scales-down the hysteresis curves and it is only a function of geometry and material. Therefore, by choosing appropriate material and contact radius, it is possible to increase the area enclosed by the hysteresis curves; hence, increase damping provided by the friction contact. 1 k1=1 k1=2 k1=3 k1=5 Normalized Friction Force Normalized Tangential Displacement Figure 6.6 Normalized hysteresis curves for different k1 183

201 Three-dimensional motion with stiffness and normal load variation Hertz contact problem studied in the previous section is applied to the threedimensional relative motion case. Tangential stiffness matrix is assumed to be identity matrix multiplied by the stiffness variation as K u k v ( θ ) ( θ ) ( θ) ( θ) 1 0 k1 v, if v > 0 = 0 1 0, otherwise. (6.41) 3 k1 v( θ), if v( θ) > 0 = 2 0, otherwise For elliptical tangential and simple harmonic normal relative motions given in the following form u asin ( θ ) ( θ ) = bcos( θ ) ( θ ) = + sin ( θ + ϕ) v v d 0, (6.42) Stick, slip, separation of the point contact model is studied and friction force trajectories are determined. Friction force trajectories for variable and constant contact stiffnesses for no separation and separation cases are given in Figure 6.7 and Figure

202 ϕ = 10 0 Slip Load for Variable Stiffness Slip Load for Constant Stiffness Slip 0 Stick Stick Friction Force in u 2 Direction Trajectory for Constant Stiffness Trajectory for Variable Stiffness ϕ = 90 Slip Slip Slip Load for Variable Stiffness 0 Stick Stick Trajectory for Constant Stiffness Slip Slip Load for Constant Stiffness Trajectory for Variable Stiffness Friction Force in u 1 Direction Figure 6.7 Friction force trajectory for no separation case 185

203 ϕ = 0 Variable Stiffness 2 Slip Separation 0 Friction Force in u 2 Direction Constant Stiffness ϕ = 90 Separation Slip Slip Load for Variable Stiffness Slip Load for Constant Stiffness 0 Stick Constant Stiffness 2 Slip Variable Stiffness Friction Force in u 1 Direction Figure 6.8 Friction force trajectory for separation case 186

204 As expected, friction force trajectories are different for constant and variable stiffness cases; however, due to special form of contact stiffnesses, transition angles are identical for variable and constant stiffness cases as it is the case for two-dimensional relative motion. Friction force trajectories for constant contact stiffnesses for no separation case contain the trajectories for variable stiffness. On the other hand, if separation exists, it is observed that trajectories for variable stiffness are larger than the trajectories of constant stiffness case. Similar comments can as well be deduced for the hysteresis curves presented in the previous section CONCLUSION In order to incorporate contact interface variation, which will result in variation of the contact stiffnesses in friction model, transition criteria for two types contact kinematics are derived: one-dimensional motion with stiffness and normal load variation and three-dimensional motion with stiffness and normal load variation. Depending on the application, one of the methods can be used to model the variation in stiffness and normal load, which may arise in contact of curved surfaces. In order to apply these models, variation of the contact stiffnesses has to be obtained; therefore, a hemi-sphere to flat surface contact problem is represented by a point contact model with variable stiffnesses. Variation of the stiffnesses is obtained by considering the Hertz contact solution and the 187

205 stiffness distributions provided by the elastic foundation model. Using these stiffness variations and assuming simple harmonic relative motion, stick slip and separation for two-dimensional and three-dimensional relative motion is studied and compared with the constant stiffness solutions. The application of the developed friction models for the forced response analysis of frictionally constrained structures with variable contact stiffnesses is left as a future work. 188

206 CHAPTER 7 7. WEDGE DAMPER MODELING AND FORCED RESPONSE PREDICTION OF FRICTIONALLY CONSTRAINED BLADES 7.1. INTRODUCTION One of the main problems in the design of gas turbine engines is the high cycle fatigue failure of turbine blades due to resonances in the operating frequency range. In order to prevent blade failures manufacturers use dry friction dampers to suppress vibration amplitudes. A widely used friction damper type is the so-called wedge damper, which is as well referred as underplatform damper. Wedge damper has two inclined surfaces on both sides and forced against the two neighboring blades by centrifugal forces. In order to minimize stresses due to blade vibration, optimal parameter values for the wedge damper and the blade have to be determined. However, dynamic analysis of such systems is complicated due to the contact kinematics and nonlinear nature of dry friction. 189

207 Yang and Menq [1998a, 1998b] developed stick-slip contact kinematics for wedge dampers under two translational degrees of freedom. Authors developed analytical stick-slip transition criteria including the variation of normal load in order to simulate the stick-slip motion precisely. Harmonic balance method was used to predict forced response of bladed disk with wedge dampers, an experimental test beam was analyzed, and the simulation results are validated. A 3D wedge damper model with twodimensional motion on the contact interface was developed by Sanliturk et al. [2001], where authors included the translation of the damper along the axial direction. A twodimensional friction model for constant normal load was used and harmonic balance method is applied to predict forced response. A test case with two blades and a wedge damper was analyzed and the results were compared with the simulations. In addition to wedge dampers, curved shape underplatform dampers were as well studied by several researches [Pfeiffer and Hayek, 1992; Sextro et al., 1997; Csaba, 1999; Jareland, 2001]. An underplatform damper with a curved and inclined surface was analyzed by Panning et al. [Panning et al., 2003] where authors included the damper rotation as well. Different friction models can be utilized in order to determine nonlinear contact forces between two relatively moving bodies. In the analysis of friction damping, onedimensional friction model was used widely [Griffin, 1980; Dowell and Schwartz, 1983; Cameron et al., 1990; Ferri, 1996; Cigeroglu et al. 2006]. This model is useful if the relative motion is one-dimensional. It is possible to have planar motion, for which twodimensional friction models are developed [Sanliturk and Ewins, 1996; Menq and Yang, 190

208 1998]. However, due to the interaction between two bodies the normal load acting on contacting surfaces can vary with normal motion. Yang et al. [1998] developed a onedimensional friction model where the normal load was induced by the normal motions of the mating surfaces. Authors developed analytical transition criteria for stick-slipseparation transition and obtained analytical transition angles for simple harmonic motion. Normal load variation was also addressed by Yang and Menq [1998c] for threedimensional motion (two-dimensional in-plane and one-dimensional out off-plane motion) and Chen and Menq [2001] for three-dimensional periodic motion. Using similar criteria as developed in [Yang et al., 1998], later Petrov and Ewins [2003], for onedimensional motion with normal load variation, described an algorithm to determine transition angles numerically for periodic motion, similar to the one given in [Chen and Menq, 2001]. This chapter presents a wedge damper model and a forced response prediction method for the analysis of bladed disk systems. The proposed model includes six rigid body modes and several elastic modes of the damper; therefore, the damper may undergo three-dimensional translation and three-dimensional rotation, which are constrained by the two adjacent blades only. In the modeling of contact surfaces of the wedge damper, discrete contact points associated with contact stiffnesses are evenly distributed on both contact interfaces. Contact stiffnesses at each contact point are determined by considering the effects of higher frequency modes, which are omitted in the dynamic analysis. The initial preload or gap at each contact pair varies with the engine rpm; therefore, a quasi- 191

209 static contact analysis is performed initially in order to determine the contact area in addition to the initial preload or gap at each contact point due to the centrifugal force. In order to predict forced response of bladed disk system, a friction contact model based on the one-dimensional model with normal load variation in [Yang et al., 1998] is proposed. In the proposed friction model, the three-dimensional relative motion on the contact surface is decomposed into two one-dimensional in-plane components and an out-of-plane component and the one-dimensional friction contact model is employed by assuming these components are independent from each other. Harmonic Balance method is used to represent the resulting nonlinear contact forces resulting in a set of nonlinear algebraic equations. The relative motion at the contact surface is expressed as a modal superposition; therefore, the number of unknowns resulting in the nonlinear equation set is only proportional to the number of modes used in the analysis. As a result, unlike receptance methods, the number of nonlinear equations is independent of the number of contact points used. The developed method is applied to a tuned bladed disk system to obtain its forced response curves and optimal curves. In addition, the effects of normal load on the rigid body motion of the damper are analyzed. Specifically, the effect of the damper s rotational motion on the prediction of the forced response is analyzed. It is shown that the effect of rotational motion is significant, particularly for the in-phase vibration modes. The effect of partial slip in the forced response analysis is investigated. Finally, a test 192

210 case with wedge dampers is studied and forced response predictions are compared with the test data WEDGE DAMPER MODEL In the analysis of bladed disk systems with wedge dampers, the blade is modeled as a constrained structure, whereas the damper is considered as an unconstrained structure. Therefore, the motion of the damper is constrained by the geometric configurations of dampers between adjacent blades. Consequently, the damper undergoes three-dimensional translation and three-dimensional rotation in addition to the elastic deformation. Elastic motion of the damper is necessary if the excitation frequency and/or static forces acting on the damper are high. However, with the proposed approach, it is possible to model the damper as completely rigid or, rigid in certain directions and elastic in others by using the appropriate mode shapes Blade and Wedge Damper Contact Model The interaction between the blade and the wedge damper is modeled by discrete contact points evenly distributed on the two contact surfaces of the blade and damper. At 193

211 each contact pair, contact stiffnesses in the three main directions of motion are determined in order to take into account the effects of higher frequency modes, which can be represented as residual stiffnesses. It is assumed that, residual stiffnesses are only present between contact pairs; hence, they are called as contact stiffnesses. The determination of contact stiffnesses will be explained in the next section. A blade and a wedge damper in contact are given in Figure 7.1 on the left, where the dots represent the contact points, and on the right, the contact stiffnesses between a contact pair in local coordinate system are shown. The contact points on the X Y plane of damper are called as constrained points, which are used to constrain the motion of the damper in Z direction due to space limitations and it should be noted that, the constrained force can be at most on one of the constrained planes. These constraints are due to the physical restrictions in real gas turbine engine, where the damper can move freely under the action of contact and centrifugal forces in a volume between the adjacent blades on the disk. The motion of the blade is expressed in blade coordinate system and the wedge damper is expressed in damper coordinates system. The coordinate systems for the blade and wedge damper are shown in Figure 7.2. The blade coordinate system is on the rotary axis of the disc where the X and Y axes are coincident with the tangential and th i radial orientations of blade i and Z axis is determined by the right hand rule. th i damper coordinate system is determined by three rotations about blade coordinate axes. 194

212 Coordinate systems for other blades and dampers can be obtained by a simple rotation about the Z axis with an amount of blade phase angle. Body 1 v u Y X Contact point n 0 k u /2 k v w Body 2 f n k u /2 Z Constrained points Figure 7.1 Contact model for wedge damper 195

213 Blade j Blade j + 1 Damper j 1 Damper j Z D Y D O D X D Y Z O X Figure 7.2 Blade and damper coordinate systems 196

214 As shown in Figure 7.3, wedge damper has four contact planes, where α and β are used to define the orientations of the left and right contact planes. Coordinate systems for the front and the back constraint planes can be obtained by 90 and -90 rotations about X D axis and the coordinate systems for the right and the left contact planes can be obtained by (90 α) and 90 α rotations about Z D axis. On Figure 3 coordinate system for right plane is shown. Back Contact Plane Left Contact Plane α β w O C r t Y D Z D O D X D Right Contact Plane Front Contact Plane Figure 7.3 Wedge damper contact planes and coordinate systems 197

215 7.3. FRICTION CONTACT MODEL AND FORCED RESPONSE PREDICTION The relative motion at the blade-damper contact interface is three-dimensional and this relative motion is decomposed into in-plane and out-of-plane (normal) components. Furthermore, two major directions for the in-plane component of motion are determined and the in-plane motion is approximated in these directions. Calculation of Fourier coefficients using a three-dimensional friction model as given in [Yang and Menq, 1998c; Chen and Menq, 2001] is time consuming. Main focus of this work is modeling damper motion as an unconstrained body having six degrees of freedom rigid body motion plus elastic motion. Therefore, in order to speed up the forced response calculations these major directions are assumed independent from each other and the onedimensional friction model with normal load variation developed by Yang et al. [1998] is employed. Transition criteria and analytical transition angles for harmonic motion are given in reference [Yang et al., 1998] and for periodic motion these criteria can be solved numerically to determine the transition angles. Therefore, the nonlinear normal force and the friction forces in major directions are obtained which are then expressed in contact plane coordinate system. 198

216 Initial Preload on Contact Surfaces Depending on the engine rpm, the centrifugal force acting on the wedge dampers varies. This results in variation of the contact area and the preload/gap acting on the contact surfaces. In order to determine the initial preload/gap a quasi-static contact analysis is performed for the given normal load. The analysis is performed as such: 1. Initially, contact status of all the contact pairs are assumed completely stuck. 2. The displacements of contact points and the contact forces acting on them are determined using the given contact status. 3. Using Coulomb friction model, the contact status of each contact pair is updated. 4. Check if contact status of each contact pair is changed. 5. If the contact status is changed, go to step Otherwise, output initial preload. This analysis is an important step for the forced response calculations, since the change of contact area and the preload/gap can affect the entire forced response characteristics of the blade and damper system. 199

217 Forced Response Method In the forced response analysis, blade (disk) and the damper are modeled by finite element models. Using receptance methods, the number of unknowns in the forced response analysis method can be decreased to the number of nonlinear (contact points) degrees of freedom multiplied by the number of harmonics. However, if the number of contact points is high, which is necessary for accurate modeling of friction contact, this method is not suitable for forced response analysis due to large matrices involved in the solution procedure. Recently, Cigeroglu et al. [2007] has proposed a modal superposition method for the forced response analysis of bladed disk systems. In this approach, the relative motion between contact surfaces is approximated by modal superposition by using free mode shapes of the structure. This approach is extended for multiple harmonics. In the modal superposition approach the number of unknowns involved in the solution procedure is the number of mode shapes used in the modal expansion process multiplied by the number of harmonics; therefore, it is independent from the number of contact points used. As a result of this, modal superposition approach is suitable for accurate modeling of friction contact with more contact points or for the cases if the tuned approach (cyclic symmetry) can not be used. Equation of motion in matrix form for a system with dry friction dampers can be written in the following form 200

218 ( ) F ( ) M X + C X + K X= F t + X (7.1) e n () ( ) where M, C, K, F t, F X and X are the mass matrix, viscous damping matrix, e n stiffness matrix, excitation force vector, nonlinear force vector and displacement vector, respectively. The motion of the blade and the damper for harmonic excitation can be written in terms of its mode shapes as follows N b m N b j 0 n l n il ( θ ) UB = Anφ j + Re Anφje n= 1 l= 1 n= 1 NR NE m NR NE j 0 n 0 n l n l n UD = Cnψ j + DnΦ j + Re Cnψ j + DnΦj e n= 1 n= 1 l= 1 n= 1 n= 1 ( θ ) il (7.2) where φ, ψ and n j n j n Φ j are the th n mode shape for blade point j, n rigid body mode th shape and elastic mode shape for damper point l l l j, respectively., and are the A n C n D n th l harmonic modal coefficients for blade, damper rigid body modes and elastic modes, respectively; and l = 0 defines the bias (dc) terms. N, N, N and m are the number of blade modes, damper rigid body modes, damper elastic modes and the number of harmonics. In addition to this, i is the complex number and θ is the temporal variable. The relative motion between the j th contact pair can be written in contact plane coordinates as b R E C j C j C j C j Xj = UB UD = BTj UB DTj U D (7.3) 201

219 where C B T j and C D T are the transfer matrices from blade coordinate system to contact j plane coordinates and damper coordinates system to contact plane coordinates for contact point j, respectively. Using the friction contact model and the relative motion given in Eq. (7.3) nonlinear contact forces can be determined in contact plane coordinates as m m C C C l C l n n ns nc l= 1 l= 1 0 f ( θ ) f + f sin ( lθ) + f cos( lθ) (7.4) C 0 where is the vector of dc component of contact forces and, C l C l f f and are the n ns f nc vectors of sine and cosine components of th l harmonic of contact forces. It should be noted that C f n is a function of modal coefficients. Using the orthogonality of mode shapes, Eq. (7.1) for a single blade and damper can be written as follows Ω B A Q 0 0 = Bb 2 ( Ω ( ω) I ( ω) C ) l + i l A = Q + Q + iq l l l l B B e BRe BIm 0 0 = Q, ( j = 1 N ) j Db R j 0 0 ΩDD j = jqd, ( j = N 1 ) b R + Nd = NR + NE 2 l l l ( ΩD ( lω) I+ i( lω) CD) E = QD + iq, ( 1 m) Re D l = Im, (7.5) if mass normalized mode shapes are used. In Eq. (7.5), ΩB and Ω D are NxN b b and NxN d d diagonal matrix of squares of natural frequencies of blade and damper, CB and CD are modal damping matrices of blade and damper, respectively; and they are diagonal 202

220 l l if the damping is proportional. A is the vector of modal coefficients for blade and E is th the vector of modal coefficients for damper for harmonic where E l = C l D l. l ( ) T l Q B and th Q are the vectors of modal forces of l harmonic for blade and damper, l D respectively, and eb,,re and Im stand for excitation force, dc component, and real and imaginary parts of nonlinear modal forces, respectively. It should be noted that, the contact forces acting on the blade j and damper j in contact plane coordinate system are same in magnitude but opposite in signs. If the bladed disk system is tuned, due to the cyclic symmetry, the motion of ( j +1) th blade can be related to the motion of th j blade as iϕ j+ 1 B = e j B, ϕ=2πneo / U U N (7.6) where ϕ is the interblade phase angle with N is the engine order and N is the number of blades; and i is the unit imaginary number. Using this information, the relative displacement on the contact surfaces of damper i can be determined, from which contact forces on the damper i can be obtained. Similar to the displacements, contact forces EO between damper ( j 1) and blade j can be related as i jfn = e ϕ j+ 1 n f (7.7) 203

221 where j f n is the nonlinear contact forces acting on blade i in blade coordinate system. Transferring contact forces between damper j and blade ( j 1) to blade j, the modal coefficients of j th blade and damper can be determined by iteratively solving the nonlinear equation set given in Eq. (7.5) and the blade and damper response can be obtained from Eq. (7.2) NUMERICAL RESULTS Two different cases are analyzed in this section. In the first part, the method is applied to a tuned bladed disk system and forced curves and optimal curves are presented. Moreover, the effects of normal load on the rigid body coefficients of damper and the effects of rotational motion of the damper on forced response results are presented. In order to show the effect of partial slip on the forced response analysis, the bladed disk system is analyzed for different number of contact points. In the second part, a blade to ground damper system is analyzed in order to show the effects multi harmonics and the preload distribution on the forced response. In the results provide below, stuck case indicates that all the contact points that are initially in contact does not slip. Therefore, it is possible to have contact points, which are not in contact due to the applied normal load. These are identified at the quasi-static contact analysis. 204

222 Tuned Bladed Disk System A tuned bladed disk system composed of 65 blades is analyzed in this part. The finite element models of a blade and a damper are given in Figure 7.4. The point of excitation and the point of displacement calculated are indicated by dots as shown in the figure Forced response and optimal curves Tracking plots around the first and second modes of the blade are given in Figure 7.5 and Figure 7.6 for different preload cases. Excitation force is applied at the tip point is in tangential direction for the first mode and in radial direction for the second mode in order to excite those modes. It is observed that, as the preload acting on the damper increases the amplitude of tip point decreases and the resonance frequency of the system increases. When the optimum point is reached, the amplitude is minimum and increasing the preload furthermore results in increase in the vibration amplitude, which converges to the completely stuck magnitude. It is interesting to note that, in Figure 7.5 multiple solutions in the tracking plots exist for some preload cases whereas in Figure 7.6 none of the preload cases results in multiple solutions in the tracking plots. Multiple solutions in the tracking plots are due to the separation of the contact surfaces. In Figure 7.5, the first bending mode of the blade is excited resulting in separation of the contact surfaces due to rotation of the wedge damper about Z axis. However, in Figure 7.6, since the dominant rotation is about X axis, contact surfaces remains in contact through out the analysis. 205

223 Point of excitation Point of displacement calculation a) b) Figure 7.4 Finite element models for a) blade b) wedge damper 206

224 Displacement Amplitude [m] Free Stuck Free Stuck Frequency [HZ] Figure 7.5 Tracking plot for first mode 207

225 Displacement Amplitude [m] Free Stuck Free Stuck Frequency [HZ] Figure 7.6 Tracking plot for second mode 208

226 Multiple solutions in the tracking plots are obtained by continuation method and they show a typical Fold bifurcation which has an unstable solution branch in between two stable solutions resulting in the jump phenomenon. In Figure 7.7 and Figure 7.8, optimal curves and frequency shift curves for the first and second vibration modes of the blade are given. It is observed that the ratio of the stuck case amplitude to the optimal preload case amplitude is approximately 4.4 and 5.5 for first and second modes of vibration, respectively. Due to the fact that, there exist no separation between the wedge damper and the adjacent blades; vibration damping for the second mode is more effective. The frequency shifts observed for the first and second modes are 17.2% and 3.4%, respectively. Using the developed computer code (BDamper) for the analysis of bladed disk systems with wedge dampers, higher modes of the blade can as well be investigated. The tracking plots and, optimum and frequency shift curves for the third and seventh modes of the blade are given in Figure 7.9 and Figure

227 Maximum Amplitude [m] Amplitude Frequency Resonance Frequency [Hz] Preload [N] Figure 7.7 Optimal and frequency shift curves: 1 st mode Maximum Amplitude [m] Amplitude Frequency Preload [N] Resonance Frequency [Hz] Figure 7.8 Optimal and frequency shift curves: 2 nd mode 210

228 Displacement Amplitude [m] Free Stuck a) Frequency [Hz] Displacement Amplitude [m] Free Stuck Free Stuck b) Frequency [Hz] Figure 7.9 Tracking plots a) 3 rd b) 7 th mode 211

229 1900 Maximum Amplitude [m] b) Preload [N] 7000 a) Maximum Amplitude [m] Amplitude Frequency Preload [N] Figure 7.10 Optimum and frequency shift curves a) 3 rd b) 7 th mode 212

230 Effects of normal load and excitation frequency on rigid body motion In Figures 12 and 13 maximum amplitudes of the dc and ac components of rigid body coefficients of the wedge damper are given for the first vibration mode, respectively. For low normal load cases, modal coefficients for translation along X and Y axis and rotation about Z axis are the main contributions to the damper rigid body motion and as the normal load increases, the major contribution comes from the modal coefficient for translation along Y axis. Similarly, the major variable components of rigid body motion are translation along X and Y axis and rotation about Z axis; however, for all normal load range, major variable components of rigid body motion are the same with order of translation along X, rotation about Z and translation along Y. It should be noted that, this rotation about Z axis results in the separation of the contact surfaces which results in multiple solutions or jumps in the tracking plots. The bladed disk system is analyzed using only the translational rigid body modes and translational and rotational rigid body modes of the damper at first vibration mode of the blade. The results with and with out rotational modes for different normal load cases are compared in Figure 14. It is observed that, neglecting the rotation of the damper results in underestimating the maximum vibration amplitudes; in addition to this, frequency shift is over estimated for this case. It is also interesting to note that, a jump phenomenon (multiple solutions) does not exist if the rotational modes of the damper are neglected. Therefore, it can be concluded that, separation of the contact surfaces are associated with the rotation of the damper. 213

231 Maximum Amplitude (Log) [m] Preload [10 4 xn] X Trans. Y Trans. Z Trans. X Rot. Y Rot. Z Rot. Figure 7.11 Effect of normal load on rigid body motion of damper (dc component) 214

232 Maximum Amplitude (Log) [m] X Trans. Y Trans. Z Trans. X Rot. Y Rot. Z Rot. Preload [10 4 xn] Figure 7.12 Effect of normal load on rigid body motion of damper (ac component) 215

233 a) b) Displacement Amplitude [m] c) d) Frequency [Hz] Figure 7.13 Effect of rotational modes for normal load a) 100 b) 200 c) 1000 d) translational and rotational modes, translational modes 216

234 Effects of partial slip on forced response In order to observe the effects of partial slip the forced response analysis is performed for different number of contact points on the left and right contact planes. Figure 7.14 shows the comparison of forced response results for different normal load cases for 9x9, 24x24 and 48x48 contact points. It is observed that, multiple solutions in the forced response are captured better when more contact points are employed in the analysis. Moreover, using 9x9 contact points resulted in underestimation of the vibration amplitude in the cases analyzed. The results for 24x24 and 48x48 contact points are closer to each other 48x48 having the highest vibration amplitude in most of the cases. However, employing more contact nodes results in longer calculation times; therefore, optimum values for the number of contact points can be determined by comparing the forced response results. In Figure 7.15 contact status of four sample contact points on left and right contact planes of the wedge damper are shown for the normal load of 5000N at the maximum amplitude frequency of 308.3Hz. The length of the bar represents the periodic temporal scale. Partial slip on the both contact surfaces can be clearly seen from the figure where contact points undergo different states at different times. 217

235 Displacement Amplitude [m] x9 24x x a) b) x9 24x x c) x x24 48x x x24 48x d) Frequency [Hz] Figure 7.14 Effect of number of contact points for normal load a)100 b) 200 c) 1000 d)

236 a) #1 #2 #3 #4 b) #1 #2 #3 #4 Figure 7.15 Contact status of sample points on a) left b) right contact planes: stick, slip, separation) Comparison with the Test Case Data In order to verify the developed method, prediction for a test blade, used by GE Aircraft Engines in a friction damping experiment, is compared with the test data. Schematic for the experimental set-up is given in Figure 7.16, where two wedge dampers are placed at each side of the test blade, and they are retained by two dummy blades without any airfoils. The normal load on the damper is adjusted by controlling the tension in the damper load wires. Test blade is excited by a pulsating air jet, where the excitation levels are controlled by air jet supply pressure. Strain gages are placed several locations including the airfoil root in order to measure the stress due to vibrations. 219

237 Finite element model of the test case is given in Figure Since the developed computer code is for tuned system analysis, test blade and the dummy blades are considered as a single structure where the interblade phase angle is zero. Comparison of predicted frequency shift curve with the available test data is given in Figure For high damper load cases, predicted frequency shift is lower than the test data however, overall, predicted frequency shift curve is in good agreement with the experimental data. It should be noted that, these predictions are done by using the contact stiffness calculated by the proposed approach as discussed previously. In Figure 7.19 predicted normalized optimal curve and the test data are presented. It is observed that, predictions and test results are in good agreement and optimal damper load can be predicted acceptably without any parameter tuning. The only parameter to be determined is the friction coefficient, which, in this case, was acquired experimentally. Due to the acyclic structure of the test case, complete locking of the contact surfaces is not possible which results in microslip for those cases. This can clearly be seen from the predictions given in Figure Predictions for the test case are as well performed for including more blade modes into the analysis, for which contact stiffnesses are recalculated, and it is observed that predictions obtained from both cases are similar and very close to each other. For brevity, these results are not presented here. 220

238 Air Jet Test blade Damper load wire Damper Dummy blade Figure 7.16 Schematic view of test case 221

239 Z Y X Figure 7.17 Finite element model for the test case Frequrncy [Hz] Test Prediction Damper Load / Excitation Figure 7.18 Frequency shift curve 222

240 Normalized Airfoil Root Stress / Excitation Test Predicted Damper Load / Excitation Figure 7.19 Predicted normalized optimal curve and test data 223

241 7.5. CONCLUSIONS An improved wedge damper model is developed and the effects of wedge dampers on the forced response analysis of frictionally constrained blades are analyzed. In the developed method, wedge damper is modeled as an unconstrained structure having six rigid body motions as well as elastic deformation. Discrete contact points are evenly distributed on contact surfaces, which are associated with contact stiffnesses in three directions of motion. In order to determine the initial preload or gap at each contact point a quasi-static contact analysis is performed initially for each normal load case. A method is proposed to calculate the contact stiffnesses used in the friction model. The suggested method is based on representing the effect of higher vibration modes by springs associated with each contact pair, which makes it possible to capture local deformations at the contact interface. Therefore, contact kinematics can be accurately estimated by using reasonable number of mode shapes in the forced response method, which decreases the computational cost significantly. For the forced response prediction, a friction model with normal load variation induced by normal motion is employed to determine the three dimensional contact forces. Harmonic Balance method is employed to approximate these contact forces in order to calculate the forced response of frictionally constrained bladed disk system. Modal superposition is used to express the relative motion; therefore, the number of unknowns 224

242 in the resulting nonlinear equation set is a function of number of modes and number of harmonics employed in the analysis, and it is independent of the number of contact points. A tuned bladed disk system is analyzed by the method developed and forced response results are presented. Multiple solutions (jumps) in the tracking plots for the first vibration mode are observed which are due to the separation of the contact surfaces associated with the rotation of the damper around the Z axis. The effect of separation is as well observed in the optimal curves where the amount of amplitude decrease compared to the completely stuck case is lower for the first vibration mode. For the first vibration mode, the major contributions to the ac component of motion comes from the translation along X, rotation about Z and translation along Y axes. In addition to this, the effects of rotational modes are as well analyzed and it is observed that, neglecting rotational modes results in underestimation of the vibration amplitude and overestimation of the frequency shift. Moreover, no multiple solutions (jumps) in the tracking plots are observe if the rotational modes are neglected. In order to analyze the effects of partial slip, forced response predictions are performed for three different number of contact points employed on contact surfaces. Utilizing more contact points makes it possible to capture the stick-slip-separation phenomenon of the contact surfaces more accurately, which is observed in contact status 225

243 plots. The method developed can be used to obtain optimum values for the number of contact points in order to meet the accuracy and computational requirements. The effects of multiple harmonics are also investigated on a blade to ground damper example. It is observed that multiple harmonics is necessary only for the case of jump, where the normal load is low. For high normal loads, multiple harmonics and single harmonic solutions are approximately the same; therefore, single harmonic solutions can be used for damper optimization purposes. Finally, predictions for a test case are compared with the test data and it is observed that the simulation results and test results are in good agreement. Similar forced response predictions are obtained by increasing the number of blade modes used in the analysis, which verifies the developed forced response prediction method and contact stiffness calculation method. Utilizing the contact stiffness obtained by proposed method, the only contact parameter left is the friction coefficient and hence, this significantly simplifies the forced prediction process. 226

244 CHAPTER 8 8. ESTIMATION OF CONTACT STIFFNESS 8.1. INTRODUCTION Dry friction dampers are widely used to decrease vibration amplitudes in gas turbine engines in order to prevent blade failures due to high cycle fatigue. In order to predict friction damping and frequency shifts associated with the practice of friction dampers, friction models for the analysis of bladed disks are developed by several researchers [Menq et al., 1986b, 1986c, 1986d; Yang et al., 1998; Yang and Menq, 1998c; Csaba, 1998; Petrov and Ewins, 2003; Quinn and Segalman, 2005, Cigeroglu et al., 2006]. In these models, friction contact is modeled by lumped or distributed stiffnesses associated with each direction of motion where the point at the end of each contact pair slips, sticks or separates depending on the relative motion and the normal load acting on the contact pair. In order to predict forced response vibration levels and frequency shifts connected to the friction contact accurately, which is crucial for the 227

245 determination of fatigue life of engine blades, parameters for the friction contact model has to be determined accurately. Friction coefficient for the contact interface can be determined experimentally; however, it is complicated to determine friction stiffness or the contact stiffness. Mindlin [1949] studied the Hertz contact problem and developed analytical equations for the normal and tangential compliance of the elastic bodies. Surface roughness is also considered in order to determine contact stiffness. Sextro [1999], Popp et al. [2003] applied this method for the analysis of a shrouded test blade where, normal and tangential contact stiffnesses are obtained by considering the surface roughness of the contact surfaces and therefore they are nonlinearly dependent on the normal pressure or normal displacement. The surface roughness approach has to be revised in order to include the effects of wear due to sliding. In the forced response prediction of bladed disks with dry friction dampers, two approaches are used: receptance methods [Menq and Griffin, 1985; Chen and Menq, 1998; Yang and Menq, 1998b; Chen and Menq, 2001; Petrov and Ewins 2003; Petrov, 2004] or mode shape superposition [Cigeroglu et al., 2006, 2007]. For realistic bladed disk examples, finite element models contain many degrees of freedom; therefore, modal analysis is used to determine receptance instead of matrix inversion methods. As a result of this, in both of the methods: receptance or modal superposition, limited number of mode shapes is used where higher modes are neglected, in order to decrease 228

246 computational cost. The number of modes used in the modal analysis is adequate to capture overall system dynamics; however, in order to capture contact characteristics accurately higher vibration modes are essential. In modeling of dry friction in bladed disk systems, usually damper inertia is neglected and it is modeled as a rigid body whose flexibility is lumped to a spring. However, as in the case of wedge damper modeling discussed in Chapter 7, where the damper inertia and damper elastic modes are considered, contact stiffnesses are used to capture the local deformations at the contact interface due to the omission of higher vibration modes CALCULATION METHOD FOR CONTACT STIFFNESS In the wedge damper model given in Chapter 7, multiple contact points are used to simulate the contact dynamics between dampers and blades. Each contact pair is modeled by a point contact friction model, as shown in Figure 1, which contains three contact stiffnesses associated with the normal direction and tangential directions. In order to capture local deformations on the contact interface, high frequency modes have to be included into the modal superposition or receptance approach, which is not practical in terms of calculation times. Therefore, limited number of modes is used in the modal expansion process. Due to this, the local deformations on the contact interface cannot be determined accurately and there will be differences between the exact and approximated 229

247 contact interface displacements. On the other hand, higher vibration modes behave like springs at lower excitation amplitudes; therefore, these omitted higher modes of the bladed disk system can be represented by contact stiffnesses. Contact stiffnesses can be visualized as residual stiffnesses placed between the contact pairs, which are used to include the effects of higher vibration modes as a result, decrease the difference between the approximated and exact displacements. In order to determine contact stiffnesses for a contact pair, normal and tangential forces are to be applied. However, normal stiffness will be affected by the tangential load and tangential stiffness will be affected by the normal load applied. In order to decrease these effects, normal contact stiffness and tangential contact stiffnesses are determined separately. A block-to-block example given in Figure 8.1 is used to demonstrate the methodology where a small block is in contact with a large block fixed from the bottom surface Normal Contact Stiffness In order to determine normal contact stiffness, normal force is applied on the top surface of the small block. Contact point displacements and forces acting on each block is determined from finite element contact analysis. 230

248 D f : Vector of contact forces acting on the small block, B f : Vector of contact forces acting on the large block, D U : Contact point displacement vector for small block, B U : Contact point displacement vector for large block, p : Vector of normal load applied on the small block. Modal information for the blocks is also obtained through finite element analysis. Using modal analysis, displacement for the blocks can be obtained. U D : Displacement vector of small block from modal analysis, U B : Displacement vector of large block from modal analysis. The difference between the finite element results and modal analysis results can be expressed as ΔU = U U B i D i B i D i B i D i ΔU = U U. (8.1) Normal contact stiffnesses for the blocks are defined as k k B B B n i n i n i D n i ΔU ΔU D n i = f = f D n i, (8.2) 231

249 where n denotes normal direction. Substituting Eq. (8.1) in Eq. (8.2) normal contact stiffnesses for each block are determined as k k B B n i n i = B B Un i U n i D n i = U f D n i D fn i U D n i. (8.3) Y Free X Z Fixed Figure 8.1 Block-to-block example 232

250 Tangential Contact Stiffness In addition to the normal load applied to the top surface of the small block tangential traction is as well applied to the small block and contact pair displacements and contact forces are determined from finite element contact analysis. Finite element contact analysis is performed in two steps. In the first step only normal load is applied on the small block and contact analysis is performed; in addition, displacements and contact forces are determined. Therefore, first step is the same as the contact analysis performed in the case of normal contact stiffness calculation. In the second step, tangential traction is applied in one direction and the contact analysis is performed from the previous solution from which new displacements and contact forces are determined. D2 f : Vector of new contact forces acting on the small block, B2 f : Vector of new contact forces acting on the large block, D2 U : New contact point displacement vector for small block, B2 U : New contact point displacement vector for large block, p 2 : Vector of normal load and tangential traction applied on the small block. Using modal analysis, displacement for the blocks can be obtained. U D2 : New displacement vector of small block from modal analysis, 233

251 B2 U : New displacement vector of large block from modal analysis. The difference between the finite element results and modal analysis results can be expressed as ΔU = U U B2 B2 B2 i i i D2 D2 2 i = i D i ΔU U U. (8.4) Normal load is necessary to maintain the contact when tangential traction is present. However, in order to exclude the effects of normal loads the definition for the tangential stiffness is revised where the deformation and contact forces due to the application of normal load are subtracted from the tangential case. Therefore, tangential contact stiffness is defined as follows k k 2 2 ( ΔU ΔU ) = ( f f ) 2 2 ( ΔU ΔU ) = ( f f D i ) B B B B B t i t i t i t i t i D D D D t i t i t i t i t, (8.5) where t denotes tangential direction. Substituting Eqs. (8.1) and (8.4) into Eq. (8.5), tangential contact stiffness for the blocks can be determined as follows 234

252 k k f f B2 B B t i t i t i = B2 B2 B B ( Uti U ti) ( Uti U ti) f f D2 D D t i t i t i = D2 D2 D D ( Uti U ti ) ( Uti U ti) (8.6) Calculation of Rigid Body Motion Since one of the blocks is free, it can have rigid body motion. in order to determine the rigid body motion of the free (small) block a least squares fitting approach is used [Umeyama, 1991; Eggert et al., 1997]. In this method, two 3D point sets are used: one before the normal and tangential forces are applied and one after the application of the loads. The 3D point sets are selected away from the contact zone in order to determine the rigid body motion accurately. The method can be briefly described as follows. Let P i and P f are the 3D point sets for the initial and final positions, respectively. Final state can be written in terms of a translation and a rotation as j j P = RP + t, j = 1 m, (8.7) f i r where R is the rotation matrix, t is the translation vector and m is the number of points r in the data sets. Rotation matrix translation vector are given as follows ( 1,1, det( ) det( )) T R = Vdiag U V U, (8.8) 235

253 t = r μf Rμ i, (8.9) where, M = UDV, by SVD, m j j ( f f)( i i) M = P μ P μ μ μ i f j= 1 1 = m 1 = m m j= 1 m j= 1 P j i P j f,. T, (8.10) Block-to-Block Example Contact stiffnesses for the contact of two blocks given in Figure 8.1 are determined using the methods described before Normal Contact Stiffness Uniform normal forces in y direction are applied to the top surface of the small block and contact analysis is performed. Deformed shape and total displacement plot of the case is given in Figure 8.2 where doted lines are the undeformed edges. Using the procedure defined in section normal contact stiffnesses for the blocks are determined and 3D plot of normal contact stiffness distribution on the contact area is 236

254 given in Figure 8.3 and the total normal contact stiffness distribution in Figure 8.4. In the analysis, only the rigid body modes of the small block are used; whereas, for the large block first 20 mode shapes are utilized. If the contact stiffnesses for each contact pairs are assumed connected to each other at the end, equivalent total contact stiffness for each contact pair can be defined as 1 D B 1 1 ki + ki ki = + D B = D B ki ki ki ki. (8.11) Figure 8.2 Deformed shape and contour plot of total displacement 237

255 a) b) Figure 8.3 Normal stiffness distribution a) small block b) large block 238

256 When the results given in Figure 8.3 are investigated, it is observed that normal contact stiffness increases towards to the corners of the contact area and decreases forming a maximum for each corner. For the small block normal cont stiffness becomes minimum at the corners of the contact area which is also the geometric boundary for the small block; whereas, minimum normal contact stiffness occurs around the center of the contact area for the large block. Contact pressure on the contact interface is given in Figure 8.5, which shows that, contact pressure is minimum around the center and increases towards the corners of the contact area. It can be concluded that, elastic deformations of the small block at the corners are small resulting in less modification at those points even though the contact pressure is maximum. This can be due to the discontinuity of the small area pressing against a larger one. On the other hand, due to low contact pressure at the center of the contact interface, normal contact stiffness for the large block is minimum. In order to observe the effect of number of modes used in the contact stiffness calculation, normal contact stiffnesses for the large block are determined using the first 5, 10 and 20 modes of the block and the 3D plot of resulting normal contact stiffnesses is given in Figure 8.6. It is observed that, as the number of modes used in the calculation increases normal contact stiffness determined also increases. This is an expected result, since increasing the number of modes in the analysis decreases difference between the modal analysis and the exact results. In the limiting case there will be no difference and the contact stiffnesses determined will be infinity indicating direct contact of contact pairs. 239

257 Figure 8.4 Total normal contact stiffness 240

258 Figure 8.5 Contact pressure N=20 N=10 N=5 Figure 8.6 Effect of number of modes used 241

259 Tangential Contact Stiffness After determining the finite element contact solution when only normal load is applied to the top surface of the small block, additional uniform tangential forces are applied to those nodes in positive x direction. The total tangential traction is kept under a certain value in order to prevent gross slip off the small block. Finite element contact analysis is performed from the previous solution and displacements and contact forces are obtained. Applying the procedure developed in section 8.2.2, tangential contact stiffnesses in x direction are obtained and they are given in Figure 8.7 and Figure 8.8 where Eq. (8.11) is used to determine the effective total tangential stiffness. Similar to the normal contact stiffness tangential contact stiffness for the small block is minimum at the corners of the contact area where as it is minimum at the center of the large block. Even though the tangential traction applied to the small box is in positive x direction, tangential contact stiffness obtained for both blocks are symmetric in x direction. Moreover, tangential contact stiffness for the small block is as well symmetric in z direction. The same analysis is also performed by applying the tangential traction in the negative x and the tangential contact stiffnesses obtained for the large block are identical to the ones obtained when the traction is in positive direction. On the hand, there is a small difference in the tangential contact stiffness obtained for the small block due to the estimation of the rigid body motion. The maximum and minimum values of the contact stiffness for both cases are identical. Figure 8.9 shows the percentage difference in tangential contact stiffness for the small block that is less then 9%. Since the 242

260 a) b) Figure 8.7 Tangential contact stiffness a) small block b) large block 243

261 Figure 8.8 Total tangential contact stiffness 244