iv
List of Figures
Figure 1. Three kind of vibrations [31] . 7
Figure 2 Illustration of Gyroscopic effect [40] . 7
Figure 3 Whirl orbit . 8
Figure 4 Mode shapes [41] 9
Figure 5. The Campbell diagram without gyroscopic effect . 10
Figure 6. Campbell diagram with gyroscopic effect . 10
Figure 7. Scheme of a rotor bearing system analysis [42] . 11
Figure 8. Element model of Timoshenko beam [43] . 12
Figure 9 Finite element model of microdrill spindle system . 13
Figure 10 Euler angles of the element . 14
Figure 11 Unbalance force due to eccentric mass of microdrill 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
v
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
vi
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 1
st
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 microdrill 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 2 optimum of drills point 88
Figure 111. Torsional response of between the initial and 2 optimum of drills point . 89
vii
List of Tables
Table 3.1 Structure dimensions and parameters of ZTG04III microdrilling machine
Table 3.2 The geometric features of Union MDS
Table 3.3 Coordinates of nodal points 16 on the zerosurface
Table 3.4 Crosssectional 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 microdrill spindle system
viii
Nomenclature
E,G Young’s modulus, Shear modulus
C ij , Cφ Damping coefficient and torsional damping of bearing; i, j= x, y
I av , Δ Mean and deviatoric moment of area of system element
I p Polar moment of area of system element
I u , I v Second moments of area about principle axes U and V of system element
k s Transverse shear form factor
K ij , Kφ Stiffness coefficient and torsional stiffness of bearing; i, j= x, y
L, A, ρ Length, are and density of system element element
F z , T q Thrust force and torque
N t , N r , N s 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
γ u , γ v Shear deformation angles about U and V axes, respectively
γ x , γ y Shear deformation angles about X and Y axes, respectively
e u , e v Mass eccentricity components of system element in U and V axes
θ u , θ v Angular displacements about U and V axes, respectively
θ x , θ y 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
ix
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 14
th
Hoang Tien Dat
x
Table of Contents
ABSTRACT III
LIST OF FIGURES IV
LIST OF TABLES . VII
NOMENCLATURE . VIII
ACKNOWLEDGEMENTS IX
CHAPTER 1. INTRODUCTION . 1
1.1 RESEARCH MOTIVATION . 1
1.2 LITERATURE REVIEW 2
1.3 OBJECTIVES AND RESEARCH METHODS . 4
1.4. ORGANIZATION OF THE THESIS 5
CHAPTER 2. ROTOR DYNAMICS SYSTEMS 6
2.1 ROTOR VIBRATIONS . 6
2.1.1. Longitudinal or axial vibrations 6
2.1.2. Torsional vibrations . 6
2.1.3. Lateral vibrations 7
2.2 GYROSCOPIC EFFECTS . 7
2.3 TERMINOLOGIES IN ROTOR DYNAMICS . 7
2.3.1. Natural frequencies and critical speeds 7
2.3.2. Whiling 8
2.3.3. Mode shapes 8
2.3.4. Campbell diagram . 9
2.4 DESIGN OF ROTOR DYNAMICS SYSTEMS 11
CHAPTER 3. DYNAMIC EQUATION OF MICRODRILL 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 microdrill 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. Motional equation of flute element (asymmetric part) 27
3.2.5. Motional equation of cylinder element 31
xi
3.2.6. Motional equation of conical element . 33
3.3. BEARING ’ S EQUATION . 35
3.4. ASSEMBLY OF EQUATIONS . 36
CHAPTER 4. MICRODRILL SYSTEM ANALYSIS . 37
4.1 NEWMARK ’ S METHOD TO SOLVE THE GLOBAL EQUATION 37
4.2 CHARACTERISTICS RESULTS OF THE DYNAMIC SYSTEM . 41
4.2.1. The system with unbalance forces . 43
4.2.2. The system with unbalance forces and thrust force . 60
4.2.3. The system with unbalance forces and torque . 65
4.2.4. The system with unbalance forces, thrust force and torque . 69
4.3.1. Lateral or bending and torsional response . 71
4.3.2. Influence of acceleration on bending and torsional response . 75
CHAPTER 5. OPTIMUM DESIGN PROBLEMS 78
5.1 THE OPTIMIZATION PROBLEM 78
5.2 CHOOSING OPTIMUM METHOD 79
5.3 OPTIMUM DESIGN AND SOLUTIONS 80
CHAPTER 6. CONCLUSIONS . 91
CHAPTER 7. DISCUSSION AND FUTURE RESEARCH 94
REFERENCES . 95
1
Chapter 1. Introduction
1.1 Research Motivation
In recent days, the study about rotating machinery has gained more importance within
advance industries such as aerospace, medical machinery, electronic industry. The products
need to get better quality, high speed, high reliability, more precision and lower cost. To get
those requirements we need better analysis tools to optimize them and to get closer to the
limit what the material can withstand.
At high speed, the rotating machinery is more affected by the vibration causing larger
amplitudes, more whirling and resonance. This vibration also causes severe unrecoverable
damages or even beak. Hence, the determination of these rotating dynamic characteristics is
much important. Nowadays 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, hole
quality, and reduced cost are always required in micro drilling.
This research focuses on the analysis the dynamic behavior of microdrill system with
gyroscopic effect and base on finite element method (FEM) to improve the accuracy of the
system. When the gyroscopic effect is taken into account the critical points will be changed
and the forward and backward whirl also appear that makes the stability and resonance
prediction less conservative [1,2]. The spindle system is modeled by the Timoshenko beam
finite element with five degrees of freedom at each node including cylinder, conical and flute
element. The Hamilton’s equations of the system involving both symmetric and asymmetric
elements were progressed. The resulting damping behavior of the system is discussed. The
lateral and torsional responses of drill point were figured out by Newmark’s method.
Furthermore, the dynamic model of a Union microdrilling tool is optimized by using the
interiorpoint method integrated in MATLAB software to minimize the lateral amplitude
response of the drill point with the nonlinear constraints including constant mass, center of
mass and harmonic response. The optimum variables used in this study are the diameters,
lengths of drill segments.
2
1.2 Literature review
The improvement of rotorbearing systems started spaciously very early. Jeffcott
[3]investigated the effect of unbalance on rotating system elements. Ruhl et al. [4] took this
work further, producing an EulerBernoulli finite element model for a turborotor system with
the provision for a rigid disc attachment. The work by Ruhl was later improved upon by
Nelson and McVaugh [5] including the effects of rotary inertia, gyroscopic moments and axial
load, for discsystem element systems. Later, Zorzi and Nelson [6] included the effects of
internal damping to the beam elements. Davis et al. [7] wrote one of the first early works on
Timoshenko finite beam elements for rotordynamic analysis. Thomas and Wilson [8] also
published early work on tapered Timoshenko finite beam elements. Chen and Ku [9]
developed a Timoshenko finite beam element with three nodes for the analysis of the natural
whirl speeds of rotating system elements. Each node has four degrees of freedom, two
translational and two rotational. Mohiuddin and Khulief [10] presented a finite element
method (FEM) for a rotorbearing system. The model accounts for gyroscopic effects and the
inertial coupling between bending and torsional deformations. This appears to be the first
work where inertial coupling has been included simultaneously. However, the researched
model was simple. The Timoshenko beam was improved with five degrees of freedom by
Hsieh et al. [11]. They developed a modified transfer matrix method for analyzing the
coupling lateral and torsional vibrations of the symmetric rotorbearing system with an
external torque. Two years later, Hsieh et al. [12] improved the asymmetric rotorbearing
system with the coupled lateral and torsional vibrations. The coupling between lateral and
torsional deformations, however, was not investigated in spindle systems, especially
microdrilling spindle system.
Xiong et al. [13] studied the gyroscopic effects of the spindle on the characteristics of a
milling system. The method used was finite elements based on Timoshenko beams. It is
considered to be the first analysis of a milling machine in this manner and full matrices are
provided. A study of dynamic stresses in microdrills under highspeed machining was done
by Yongchen et al. [14]. In their paper, a dynamic model of microdrillspindle system is
developed using the Timoshenko beam element from the rotor dynamics to study dynamic
stresses of microdrills. However, the model only has four degrees of freedom each node.
3
These researches show that those dynamic models both have some shortcomings; the coupled
bending and torsional vibration responses of microdrilling spindle systems still
comprehensively needs further study.
A mechanistic model for dynamic forces in microdrilling was studied by Yongpin and
Kornel [15]. The model was only considered with thrust, torque and radial force. Abele and
Recently, some approaches possible to extend longer drill life, hole quality, such as coating
the drill surface [16, 17], designing the drill with geometric optimization, modifying the drill
geometry were proposed. Abele and Fujara [18] developed a method for a holistic
simulationbased twist drill design and geometry optimization. In their study, they just
focused on twist part of the drill. A new fourfacet drill was presented and analyzed by
Lee et al. [19]. Their new drill successfully presents that the cutting forces and torques of the
new drill in drilling can be reduced as compared with the conventional one. Besides geometric
optimization of drill flute, such as cutting lips, rake face, vibration reduction optimum design
is also one of the best choices to improve hole quality as well as drill life.
Many researches with optimization have been done of rotorbearing systems. Rajan et al.
[20] proposed a method to find some optimal placement of critical speeds in the system. A
symmetric model with four disks was studied. After one year, based on the same model in
[16], Ting and Hwang [21] improved with minimum weight design of rotor bearing system
with multiple frequency constraints. Eigenvalue constraints was continuously used to
minimize the weight of rotor system by Chen and Wang [22]. Robust optimization of a
flexible rotor bearing system using Campbell diagram was researched by Ritto et al. [23]. The
idea of the optimization problem is to find the values of a set of parameters (e.g. stiffness of
the bearing, diameter, etc.) for which the natural frequencies of the system are as far away as
possible from the rotational speeds of the machine. Alexander [24]applied gradientbased
optimization for a rotor system with static stress, natural frequencies and harmonic response
constraints. However, all above studied models only were simple symmetric and four degrees
of freedom model. Choi and Yang [25] proposed the optimum shape design of the rotor
system element to change the critical speeds under the constraints of the constant mass.
Genetic algorithms (GAs) were used to minimize the first natural frequency for a sufficient
avoidance resonance region with diameter variables. Shiau et al. [26] studied to minimize the
4
system element weight of the geared rotor system with critical speed constraints using the
enhanced genetic algorithm. Yang et al. [27] used hybrid genetic algorithm (HGA) to
minimize the Q factor of the second mode, the first bending mode. Q–factor of the system is
a measure of the maximum amplitude of vibration that occurs at resonance. A constraint
reduced primaldual interiorpoint algorithm was applied to a case study of quadratic
programming based model predictive rotorcraft control [28]. Rao and Mulkay [29]
demonstrated the interiorpoint methods compared with the wellknown simplex based linear
solver in solving largescale optimum design problems. Nevertheless, all above researchers
have just focused on the symmetric rotor and four degrees of freedom finite element model.
In this study, vibration responses of the coupled lateral and torsional microdrilling
spindle system were analyzed by using the finite element method due to itself unbalance,
thrust force, torque, rotational inertia and gyroscopic effect. The spindle system is modeled by
the Timoshenko beam finite element with five degrees of freedom at each node including
cylinder, conical and flute element. The Hamilton’s equations of the system involving both
symmetric and asymmetric elements were progressed. The resulting damping behavior of the
system is discussed. The lateral and torsional responses of drill point were figured out by
Newmark’s method. Furthermore, the dynamic model of a Union microdrilling tool is
optimized by using the interiorpoint method integrated in MATLAB software to minimize
the lateral amplitude response of the drill point with the nonlinear constraints including
constant mass, center of mass and harmonic response. The optimum variables used in this
study are the diameters, lengths of drill segments.
1.3 Objectives and Research Methods
There are several objectives which to be fulfilled within a logic dynamic analysis. The
microdrill system is divided in to four main parts such as spindle, system element, clamp and
drill. Additionally, the microdrill is separated to five segments, two cylinders, two cones
(symmetric element) and one pretwist part (asymmetric element). The Timoshenko beam
finite element which has a powerful ability to treat the complex structures are used to build a
five degrees of freedom model, two transverse displacements, two bending rotations, and a
torsional rotation. The model is considered itself continuous eccentricity, external axial and
torque forces.
5
The first purpose of this study is to find the most important dynamic characteristics of
the system, whiling orbit, lateral and torsional vibration response. After that, the lateral and
torsional critical speeds will be pointed out from the response plots. To get those results, the
equation of motion is built by Hamilton’s equation and finite element method (FEM) with
gyroscopic effect, shear, and rotational inertia. Newmark method is utilized to receive the
vibration responses in transient and steady state.
The second purpose is to use interiorpoint method to find the optimum parameters to
minimize the lateral response of the drill point. These parameters are the diameters, the
lengths of the drill segments. Nonlinear constraints are imposed on the constant mass, center
of mass and harmonic response of the drill.
1.4. Organization of the thesis
This research proposal is divided into seven chapters. Chapter 1 provides the briefing
and the motivation of this research. Some necessary knowledge of a rotor dynamics system
was presented in chapter 2. Chapter 3 describes how to build the dynamic equation of a whole
microdrill system. Chapter 4 shows the dynamic characteristic results and unbalance
response or transient responses of the system. In chapter 5, the background and the
optimization method, interiorpoint method will be used to treat an optimization problem of
the system. The expected results are the optimum parameters of to minimize the bending
response value of the system with dynamic constraints. The optimum results were provided at
the end of this chapter. Some conclusions will be offered in chapter 6. Finally, chapter 7 will
give some future researches and discussion to improve the present study. References are
attached after the last chapter.
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Nonlinear constrained optimization of the coupled lateral and torsional MicroDrill system with gyroscopic effect
THẠC SĨ Nonlinear constrained optimization of the coupled lateral and torsional MicroDrill system with gyroscopic effect
Nonlinear constrained optimization of the coupled lateral and torsional MicroDrill system with gyroscopic effect
Nonlinear constrained optimization of the coupled lateral and torsional MicroDrill system with gyroscopic effect
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