Nonlinear constrained optimization of the coupled lateral and torsional Micro Drill system with gyroscopic effect

Nonlinear constrained optimization of the coupled lateral and torsional Micro Drill system with gyroscopic effect Nonlinear constrained optimization of the coupled lateral and torsional Micro Drill system with gyroscopic effect luận văn tốt nghiệp thạc sĩ

國立交通大學 機械工程學系 碩士論文 Nonlinear Constrained Optimization of the Coupled Lateral and Torsional Micro-Drill System with Gyroscopic Effect Student：Hoang Tien Dat Advisor：Prof An-Chen Lee July 14th, 2015 i Nonlinear Constrained Optimization of the Coupled Lateral and Torsional Micro-Drill System with Gyroscopic Effect 研究生：黃進達 Student：Hoang Tien Dat 指導教授：李安謙 Advisor：An-Chen Lee 國立交通大學 機械工程學系 碩士論文 A thesis Submitted to Department of Mechanical Engineering College of Engineering National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering July 14th, 2015 Hsinchu, Taiwan, Republic of China ii Nonlinear Constrained Optimization of the Coupled Lateral and Torsional Micro-Drill System with Gyroscopic Effect Student：Hoang Tien Dat Advisor：An-Chen Lee Department of Mechanical Engineering National Chiao Tung University Abstract Micro drilling tool plays an extremely important role in many processes such as the printed circuit board (PCB) manufacturing process, machining of plastics and ceramics The improvement of cutting performance in tool life, productivity and hole quality is always required in micro drilling In this research, a dynamic model of micro-drill tool is optimized by the interior-point method To achieve the main purpose, the finite element method (FEM) is utilized to analyze the coupled lateral and torsional micro-drilling spindle system with the gyroscopic effect The Timoshenko beam finite element with five degrees of freedom at each node is applied to perform dynamic analysis and to improve the accuracy of the system containing cylinder, conical and flute elements Moreover, the model also includes the effects of continuous eccentricity, the thrust, torque and rotational inertia during machining The Hamilton’s equations of the system involving both symmetric and asymmetric elements were progressed The lateral and torsional responses of drill point were figured out by Newmark’s method The aim of the optimum design is to find some optimum parameters, such as the diameters and lengths of drill segments to minimize the lateral amplitude response of the drill point Nonlinear constraints are the constant mass and mass center and harmonic response of the drill The FEM code and optimization environment are implemented in MATLAB to solve the optimum problem Keywords: Finite element analysis, Nonlinear constrained optimization, Micro-drill spindle, Gyroscopic effect iii List of Figures Figure Three kind of vibrations [31] Figure Illustration of Gyroscopic effect [40] Figure Whirl orbit Figure Mode shapes [41] Figure The Campbell diagram without gyroscopic effect 10 Figure Campbell diagram with gyroscopic effect 10 Figure Scheme of a rotor bearing system analysis [42] 11 Figure Element model of Timoshenko beam [43] 12 Figure Finite element model of micro-drill spindle system 13 Figure 10 Euler angles of the element 14 Figure 11 Unbalance force due to eccentric mass of micro-drill 18 Figure 12 Relations between shear deformation and bending deformation 19 Figure 13 Nodal points on the zero surface 28 Figure 14 Conical element 33 Figure 15.Bearings stiffness and bearing model 35 Figure 16 Finite element model of spindle system and MDS drill 42 Figure 17 Top point response orbit of drill point 43 Figure 18 Drill point response orbit at the steady state 43 Figure 19 Amplitude of drill point response 44 Figure 20 Amplitude of drill point response at the initial transient time 44 Figure 21 Amplitude of drill point response at the steady state 45 Figure 22 x deflection of drill point 45 Figure 23 x deflection of drill point at the initial transient time 46 Figure 24 x deflection of drill point at the steady state 46 Figure 25 y deflection of drill point 46 Figure 26 y deflection of drill point at the initial transient time 47 Figure 27 x deflection of drill point at the steady state 47 Figure 28 Torsional response of drill point 47 Figure 29 Torsional response of drill point at the initial transient time 48 Figure 30 Torsional response of drill point at the steady state 48 Figure 31 Drill point response orbit 49 Figure 32 Drill point response orbit at the steady state 49 Figure 33 Amplitude of drill point response 50 Figure 34 Amplitude of drill point response at the initial transient time 50 Figure 35 x deflection of drill point 50 Figure 36 x deflection of drill point at the initial transient time 51 Figure 37 x deflection of drill point at the steady state 51 iv Figure 38 y deflection of drill point 51 Figure 39 y deflection of drill point at the initial transient time 52 Figure 40 y deflection of drill point at the steady state 52 Figure 41 Torsional response of drill point 52 Figure 42 Torsional response of drill point at the initial transient time 53 Figure 43 Torsional response of drill point at the steady state 53 Figure 44 Drill point response orbit 53 Figure 45 Amplitude of drill point response 54 Figure 46 Drill point response orbit 54 Figure 47 Drill point response orbit at the steady state 55 Figure 48 Amplitude of drill point response 55 Figure 49 Drill point response orbit 56 Figure 50 Amplitude of Drill point response 56 Figure 51 Drill point response orbit at the steady state 56 Figure 52 Amplitude of Drill point response 57 Figure 53 x, y deflection of drill point 57 Figure 54 Torsional response of drill point 57 Figure 55 Drill point response orbit 58 Figure 56 Drill point response orbit at the steady state 58 Figure 57 Amplitude of drill point 59 Figure 58 Torsional response of drill point 59 Figure 59 A shaft under buckling load 60 Figure 60 Amplitude of drill point at steady state ( Fz =-1 N) 61 Figure 61 Amplitude of drill point at steady state ( Fz =-2.5 N) 62 Figure 62 Amplitude of drill point at steady state ( Fz =-3.5 N) 62 Figure 63 Amplitude of drill point at steady state ( Fz =-4.5 N) 63 Figure 64 Amplitude of drill point at steady state ( Fz =-6 N) 63 Figure 65 Amplitude of drill point at steady state ( Fz =-7.5 N) 64 Figure 66 Whirling orbit of drill point ( Fz =-8.5 N) 64 Figure 67 Amplitude of drill point at steady state ( Fz =-8.5 N) 65 Figure 68 Variation of the buckling loads with amplitude of drill point 65 Figure 69 Response orbit of drill point 66 Figure 70 Amplitude of drill point 66 Figure 71 Torsional response of drill point 67 Figure 72 Amplitude of drill point at the steady state 67 Figure 73 Torsional response of drill point at the steady state 67 Figure 74 Torsional response of drill point 68 Figure 75 Torsional response of drill point at the steady state 68 Figure 76 Variation of the torque with torsional deflection of drill point 69 v Figure 77 Orbit of drill point at the steady state 70 Figure 78 Torsional response of drill point 70 Figure 79 Bending response versus and the rotational speed of the system 71 Figure 80 Torsional response versus the rotational speed of the system 72 Figure 81 Response orbit of drill point 72 Figure 82 Transient orbit of drill point near the first critical speed 73 Figure 83 Amplitude of drill point near the first critical speed 73 Figure 84 x deflection of drill point near the first critical speed 73 Figure 85 y deflection of drill point near the first critical speed 74 Figure 86 Torsional response of drill point near the first critical speed 74 Figure 87 Orbit response of drill point near the second critical speed 74 Figure 88 Torsional response of drill point near the second critical speed 75 Figure 89 Amplitude of drill point near the second critical speed 75 Figure 90 Transient bending responses for the various accelerations (linear plot) 76 Figure 91 Transient bending responses for the various accelerations (log10 plot) 76 Figure 92 Zoom in of transient bending responses for the various accelerations at the 1st critical speed 77 Figure 93 Transient torsional responses for the various accelerations at the critical speed (linear plot) 77 Figure 94 Zoom in of transient torsional responses for the various accelerations at the critical speed (linear plot) 78 Figure 95 The micro-drill dimensions and clamped schematic 81 Figure 96 The historic of objective function of the bending response in the first numerical example 82 Figure 97 Orbit response of the initial drill point at the steady state 83 Figure 98 Amplitude response of the initial drill point 83 Figure 99 Orbit response of the optimum drill point at the steady state 83 Figure 100 Amplitude response of the optimum drill point 84 Figure 101 Bending response of the optimum drill point 84 Figure 102 Torsional response of the optimum drill point 84 Figure 103 Bending response of between the initial and optimum of drill point 85 Figure 104 Torsional response of between the initial and optimum of drill point 85 Figure 105 The historic of objective function of the bending response in the second numerical example 86 Figure 106 Orbit response of the optimum drill point at the steady state 86 Figure 107 Amplitude response of the optimum drill point 87 Figure 108 Bending response of between the initial and optimum of drill point 87 Figure 109 Torsional response of between the initial and optimum of drill point 88 Figure 110 Bending response of between the initial and optimum of drills point 88 Figure 111 Torsional response of between the initial and optimum of drills point 89 vi List of Tables Table 3.1 Structure dimensions and parameters of ZTG04-III micro-drilling machine Table 3.2 The geometric features of Union MDS Table 3.3 Coordinates of nodal points 1-6 on the zero-surface Table 3.4 Cross-sectional properties of flute part of MDS Table 4.1 Dimensions of Union MDS (element 10) Table 4.2 The parameters of the finite element model of the micro-drill spindle system vii Nomenclature E,G Young’s modulus, Shear modulus Cij, Cφ Damping coefficient and torsional damping of bearing; i, j= x, y Iav, Δ Mean and deviatoric moment of area of system element Ip Polar moment of area of system element Iu, Iv Second moments of area about principle axes U and V of system element ks Transverse shear form factor Kij, Kφ Stiffness coefficient and torsional stiffness of bearing; i, j= x, y L, A, ρ Length, are and density of system element element Fz, Tq Thrust force and torque Nt, Nr, Ns Shape functions of translating, rotational and shear deformation displacements, respectively z Axial distance along system element element T, P, W Kinetic, potential energy and work q DOF vector od fixed coordinates (u, v) Components of the displacement in U and V axis coincident with principal axes of system element (x,y) Components of the displacement in X and Y in fixed coordinates γu, γv Shear deformation angles about U and V axes, respectively γx, γy Shear deformation angles about X and Y axes, respectively eu, ev Mass eccentricity components of system element in U and V axes θu, θv Angular displacements about U and V axes, respectively θx, θy Angular displacements about U and V axes, respectively Φ Spin angle between basis axis and X about Z axis ϕ, θ, ψ Euler’s angles with rotating order in rank Ω Operating speed φ Torsional deformation Subscript and Superscript {.}, {'} To be referred to as derivatives of time and coordinate s, c, f Superscript for cylinder, conical, flute element t Superscript for transpose matrix viii Acknowledgements This research was carried out from the month of March 2014 to June 2015 at Mechanical Engineering Department, National Chiao Tung Univeristy, Taiwan I would like to thank and greatly appreciate my respected advisor, Professor An - Chen Lee, for his patient guidance, support and encouragement throughout my entire work He always gives me the most correct direction to solve the problems in my studies In addition, I also would like to thank all my lab mates, especially Mr Nguyen Danh Tuyen for his discussion, kind help and valuable feedback I also gratefully acknowledge other teachers and my classmates Finally, I would also like to thank my parents, my wife, my daughter and best friends for their support throughout my studies, without which this work would not be possible National Chiao Tung University Hsinchu, Taiwan, July 14th Hoang Tien Dat ix Table of Contents ABSTRACT III LIST OF FIGURES IV LIST OF TABLES VII NOMENCLATURE VIII ACKNOWLEDGEMENTS IX CHAPTER INTRODUCTION 1.1 RESEARCH MOTIVATION 1.2 LITERATURE REVIEW 1.3 OBJECTIVES AND RESEARCH METHODS 1.4 ORGANIZATION OF THE THESIS CHAPTER ROTOR DYNAMICS SYSTEMS 2.1 ROTOR VIBRATIONS 2.1.1 Longitudinal or axial vibrations 2.1.2 Torsional vibrations 2.1.3 Lateral vibrations 2.2 GYROSCOPIC EFFECTS 2.3 TERMINOLOGIES IN ROTOR DYNAMICS 2.3.1 Natural frequencies and critical speeds 2.3.2 Whiling 2.3.3 Mode shapes 2.3.4 Campbell diagram 2.4 DESIGN OF ROTOR DYNAMICS SYSTEMS 11 CHAPTER DYNAMIC EQUATION OF MICRO-DRILL SYSTEMS 12 3.1 FINITE ELEMENT MODEL OF THE SYSTEM 12 3.1.1 Timoshenko’s beam 12 3.1.2 Finite element modeling of micro-drill spindle 13 3.2 MOTIONAL EQUATIONS OF SYMMETRIC AND ASYMMETRIC ELEMENTS 14 3.2.1 Hamilton’s equation of the system 15 3.2.2 Shape functions 19 3.2.3 Finite equation of motions 22 3.2.4 3.2.5 Motional equation of flute element (asymmetric part) 27 Motional equation of cylinder element 31 x -9 1.4 x 10 1.2 amplitude(m) 0.8 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 t(sec) 0.25 0.3 0.35 Figure 100 Amplitude response of the optimum drill point -6 x 10 Bending deformation (m) 2.5 1.5 0.5 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 101 Bending response of the optimum drill point -7 4.5 x 10 Torsional deformation (m) Torsional response (rad) 3.5 2.5 1.5 0.5 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 102 Torsional response of the optimum drill point 84 -5 1.8 x 10 1.6 Bending response (m) 1.4 1.2 Original 0.8 0.6 0.4 Optimum 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 103 Bending response of between the initial and optimum of drill point -6 1.8 x 10 1.6 (rad) response Torsional (m) response Torsional 1.4 1.2 0.8 Original 0.6 Optimum 0.4 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 104 Torsional response of between the initial and optimum of drill point Figure.103 shows the clear illustration the deviation between the responses of the original and optimum drill With optimum drill response, the first critical speed value varies form 2.073 x 105 (rpm) to 2.293 x 105 (rpm) far away from the operating speed That is the 85 reason why the amplitude of the optimum drill at Ω=50,000 (rpm) is smaller than the original drill’s amplitude Additionally, the first peak value deceases from 1.8 x10-5 (m) to 2.604 x10-6 (m) Moreover, the second peak on Fig.104 varies from 3.84 x105 (rpm) to 4.05 x105 (rpm), its value is also larger from 1.674 x10-6 (rad) to 4.227 x10-7 (rad) than the original drill’s value - The second study case: In this numerical example, we handle the optimal design with all above constraints, excepting center of mass constraint The system was applied a 50,000 rpm operating speed Current point Function evaluations due to Fz=-2.5 (N), Tq=1.5e-3 (Nmm) After 10 iterations and 153 number of function Current Point Total Function Evaluation 0.01stopped because the size of the current step40is less than the evaluations, the optimal process selected value of the step size tolerance and constraints are satisfied to within 0.005 20 the defined Constraint violation Function value value of the constraint tolerance The optimal design process proceeded along the plot in 0 5 Fig.105 Number of variables: Iteration -8 Current Function Value: -8.56929 Maximum x 10 Constraint Violation -8 -8.5 10 Iteration Iteration -4 Size: 6.17829e-10 First-order Optimality: 10Step Figure 105 The historic of objective x function of the bending response in the second numerical example 4000 -9 x 10 First-order optimality Step size -9 2 Iteration 10 2000 o y rbit(m) -1 -2 -3 -3 -2 -1 x orbit (m) -9 x 10 Figure 106 Orbit response of the optimum drill point at the steady state 86 0 Iteration -9 x 10 amplitude(m) 0 0.05 0.1 0.15 0.2 t(sec) 0.25 0.3 0.35 Figure 107 Amplitude response of the optimum drill point -5 x 10 1.8 Bending response (m) X: 2.159e+05 Y: 1.831e-05 X: 2.073e+05 Y: 1.799e-05 1.6 1.4 Original 1.2 Optimum 0.8 0.6 0.4 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 x 10 Figure 108 Bending response of between the initial and optimum of drill point 87 -6 1.8 x 10 1.6 X: 3.84e+05 Y: 1.674e-06 Torsional response (m) Torsional response (rad) 1.4 1.2 X: 3.926e+05 Y: 8.865e-07 Original Optimum 0.8 0.6 0.4 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 109 Torsional response of between the initial and optimum of drill point -5 x 10 1.8 1.6 Bending response (m) X: 2.159e+05 Y: 1.831e-05 X: 2.073e+05 Y: 1.799e-05 Optimum 1.4 Original 1.2 0.8 0.6 X: 2.293e+05 Y: 2.604e-06 0.4 Optimum 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 x 10 Figure 110 Bending response of between the initial and optimum of drills point 88 -6 1.8 x 10 1.6 X: 3.84e+05 Y: 1.674e-06 (rad)(m) response Torsional response Torsional 1.4 1.2 Original X: 3.926e+05 Y: 8.865e-07 Optimum 0.8 X: 4.05e+05 Y: 4.227e-07 0.6 0.4 Optimum 0.2 0 0.5 1.5 2.5 Omeg (rpm) 3.5 4.5 5 x 10 Figure 111 Torsional response of between the initial and optimum of drills point Figure 108 shows the clear illustration the deviation between the responses of the original and second optimum drill In this second run, the first critical speed value is varied form 2.073 × 105 rpm to 2.159 × 105 rpm far away from the operating speed Nevertheless, the first peak value is slightly increased from 1.8 × 10-5 m to 1.831 × 10-5 m Similarly, the highest peak on Fig 109 is varied from 3.84 × 105 rpm to 3.926 × 105 rpm, the maximum optimum torsional displacement is just smaller a little from 1.674 × 10-6 m to 8.865 × 10-7 m than the original one In the second case, although the harmonic response constraint is satisfied    7.2% the center of mass zc = 14.05 mm is larger zc = 13.85 mm The bending amplitude of drill point is deceased form 8.336 x 10-9 m to 2.68 x 10-9 m at the operating speed In addition, the deviatoric lateral response percentage between the optimum and original drill is also larger than the one in the first case The simultaneous responses of two cases are plotted as in Figs 110 and 111 It is can be noted that the center of mass constraint is more sensitive to the lateral amplitude response than the harmonic response constraint If the maximum allowable 89 value of the harmonic response constraint is decreased more, the design problem converges quickly However, the maximum lateral amplitude at the operating speed seems not to be decreased appreciably 90 Chapter Conclusions The aim of the present research to develop a micro-drill spindle system modelled as Timoshenko finite beam elements with five degrees of freedom at each node has been achieved The finite element method (FEM) is utilized to analyze the coupled lateral and torsional micro-drill spindle system with gyroscopic effect Bending and torsional response of drill point were figured out by Newmark’s method The dynamic model of micro-drill tool of Union company is optimized by the interior-point method, due to applied thrust force and torque It has been shown through simulation, the results are in good agreement with some referred experiments and proposed goals Furthermore, some more researches into damping, unbalance, thrust force and torque effects have been performed The findings of the research can be summarized as follows:  The Hamilton’s equation of the system of both symmetric and asymmetric elements was built in Chapter The complex geometry of a twist drill is modelled by specifying the Cartesian positions of the specific points on the surface of the drill based on the defined functions The drill model is a Union micro-drill labelled as MDS 0.1 mm diameter At the end of the Chapter, the assembly of equation of the whole system was found  Results in Chapter after applying Newmark’s method show the vibration responses of the drill point concluding lateral and torsional responses in the operating speed Ω = 50,000 rpm In the first study case, only unbalance force was applied on the system The transient results point out that the system can be stable after a short approximated time ts = 0.14s At start-up stage, the trajectory of drill point is in a complex whirl The response orbit at the steady state is a rough circle The deviation between maximum and minimum amplitude at this state effects to the quality of hole in manufacturing The amplitude quickly decreases until the steady state that the amplitude varies as sine or cosine waves It can be seen that the vibration amplitudes in the x direction is slightly higher than those in the y direction Nevertheless, the amplitudes in tow those directions are approximately equal to each other at steady state The torsional vibration is much smaller than the vibration amplitude due to 91 unbalance  In the first study case, only unbalance force and thrust force were applied on the system The amplitude of drill point linear increases with the variation of buckling loads slightly The amplitude near critical buckling load peaks quickly In the third study case, the results show that the applied torque has intensive effect to torsional response and slight effect to bending response The most dangerous effects of combined unbalance, torsional, and axial loads are pointed out in the final study case Therefore, it is very necessary to analysis the coupled lateral and torsional vibration with external forces in a micro-drill system  In the other studies in Chapter 4, from damping analysis of the bearing, the results shown that damping has significant effects to the response of the system If damping value is increased, the amplitude of drill point is decreased and the torsional magnitude is decreased slightly In addition to the stable time is increased; If damping value is decreased, the amplitude of drill point is increased and the torsional magnitude is increased slightly In addition to the stable time is decreased It is advisable to focus on the bearing damping values before further analysis to see more its effect  At the final of Chapter 4, the transient motions showing the response of the system from startup and goes through some critical speed were provided A constant acceleration is used to see the difference between steady state response and transient response of the system From those results, the critical speeds corresponding to the lateral and torsional response were received The whirling orbits at the speed nearby the critical speeds are very complicated and varied quickly If the acceleration is increased the spent time to pass the first critical speed is decreased and the value of bending amplitude at the first peak is decreased Similarly, the value of the critical torsional deflection is decreased the acceleration is increased Based on the above considerations, an economical and safe method for accelerating a rotor to pass through critical speeds is for large accelerations only to be employed in the vicinity of the resonant speeds and small accelerations to be employed in the other speed regions Besides, although there is an asymmetric drill flute, the × response is not observed because the asymmetric effect is very slight  The new micro drill tool was found in Chapter after applying interior – point method in Fmincon of Matlab The system was applied a 50,000 rpm operating speed due to Fz=-2.5 (N), Tq=1.5e-3 (Nmm) In the first study case, the bending amplitude of drill point is deceased form 8.336 x 10-9 (m) to 6.756 x 10-10 (rad) while keeping constant mass and satisfying the condition of mass center without deviatoric bending 92 response constraint Furthermore, the positions of lateral and torsional critical speeds were changed that effect to the vibration amplitudes (Figs 103-104) In the second case, although the harmonic response constraint is satisfied the center of mass zc = 14.05 mm is larger zc = 13.85 mm in the first run That is the reason why the deviatoric bending amplitude between optimum and original drill point is smaller than this one of the first case The bending amplitude of drill point is deceased form 8.336 x 10-9 (m) to 2.68 x 10-9 (m) while keeping constant mass and satisfying the deviatoric bending response constraint without condition of mass center It is can be noted that the center of mass constraint is more sensitive to the lateral amplitude response than the harmonic response constraint 93 Chapter Discussion and Future Research In the future, the fully coupled lateral, torsional and axial vibrations of unbalanced system element with a transverse crack, thrust force and torque should be carried out Some new profile of cross section drill should be added to the model More understanding regarding the support connection with the bearings is needed and more models should be tested to see how to get the best results More knowledge about stress analysis how the drill can be broken is to be 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Republic of China ii Nonlinear Constrained Optimization of the Coupled Lateral and Torsional Micro- Drill System with Gyroscopic Effect Student：Hoang Tien Dat Advisor：An-Chen Lee Department of Mechanical.. .Nonlinear Constrained Optimization of the Coupled Lateral and Torsional Micro- Drill System with Gyroscopic Effect 研究生：黃進達 Student：Hoang Tien Dat 指導教授：李安謙... is utilized to analyze the coupled lateral and torsional micro- drilling spindle system with the gyroscopic effect The Timoshenko beam finite element with five degrees of freedom at each node