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

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

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

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 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 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

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

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

Nomenclature

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 , Iv 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

(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

Subscript and Superscript

{.}, {'} To be referred to as derivatives of time and coordinate

s, c, f Superscript for cylinder, conical, flute element

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

Table of Contents

A BSTRACT III

L IST OF F IGURES IV

L IST OF T ABLES VII

N OMENCLATURE VIII

A CKNOWLEDGEMENTS IX

C HAPTER 1 I NTRODUCTION 1

1.1 RESEARCH MOTIVATION 1

1.2LITERATURE REVIEW 2

1.3OBJECTIVES AND RESEARCH METHODS 4

1.4.ORGANIZATION OF THE THESIS 5

C HAPTER 2 R OTOR DYNAMICS S YSTEMS 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

C HAPTER 3 D YNAMIC EQUATION OF M ICRO -D RILL S YSTEMS 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

3.2.6 Motional equation of conical element 33

3.3 BEARING’S EQUATION 35

3.4 ASSEMBLY OF EQUATIONS 36

C HAPTER 4 M ICRO -D RILL S YSTEM A NALYSIS 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

C HAPTER 5 O PTIMUM DESIGN PROBLEMS 78

5.1 THE OPTIMIZATION PROBLEM 78

5.2 CHOOSING OPTIMUM METHOD 79

5.3 OPTIMUM DESIGN AND SOLUTIONS 80

C HAPTER 6 C ONCLUSIONS 91

C HAPTER 7 D ISCUSSION AND F UTURE R ESEARCH 94

R EFERENCES 95

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 micro-drill 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 micro-drilling tool is optimized by using the interior-point 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,

1.2 Literature review

The improvement of rotor-bearing 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 Euler-Bernoulli finite element model for a turbo-rotor 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 disc-system 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 rotor-dynamic 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 rotor-bearing 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 rotor-bearing system with an external torque Two years later, Hsieh et al [12] improved the asymmetric rotor-bearing system with the coupled lateral and torsional vibrations The coupling between lateral and torsional deformations, however, was not investigated in spindle systems, especially micro-drilling 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 micro-drills under high-speed machining was done

by Yongchen et al [14] In their paper, a dynamic model of micro-drill-spindle system is developed using the Timoshenko beam element from the rotor dynamics to study dynamic stresses of micro-drills However, the model only has four degrees of freedom each node

These researches show that those dynamic models both have some shortcomings; the coupled bending and torsional vibration responses of micro-drilling spindle systems still comprehensively needs further study

A mechanistic model for dynamic forces in micro-drilling 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 simulation-based twist drill design and geometry optimization In their study, they just focused on twist part of the drill A new four-facet 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 rotor-bearing 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 gradient-based 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

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 primal-dual interior-point algorithm was applied to a case study of quadratic programming based model predictive rotorcraft control [28] Rao and Mulkay [29] demonstrated the interior-point methods compared with the well-known simplex based linear solver in solving large-scale 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 micro-drilling 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 micro-drilling tool is optimized by using the interior-point 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 micro-drill system is divided in to four main parts such as spindle, system element, clamp and drill Additionally, the micro-drill is separated to five segments, two cylinders, two cones (symmetric element) and one pre-twist 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

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 interior-point 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 micro-drill 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, interior-point 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

Chapter 2 Rotor dynamics Systems

2.1 Rotor vibrations

Rotor bearing systems did not operate with any high speed many years ago because they had less stability problems Today the rotor bearing systems have to be more efficient, that means that they need to have higher rotational speed Therefore, the turbines may get stability problems Stability problems are the reason why rotor dynamics are important when developing gas turbines, spindle of manufacturing machines The engineer wants to avoid oscillations in a system because oscillations can shorten the lifetime of the machine and make the manufacturing products worse Oscillations can also make the environment around the machine intolerable with heavy vibrations and high sound

Rotor dynamics can be divided into three different types of motion as following types:

 Longitudinal or axial vibration (Fig 1a)

 Torsional vibration (Fig 1b)

 Lateral vibration (Fig 1c)

Axial vibration is defined as an oscillation that occurs along the axis of the rotor Its dynamic behavior is associated with the extension and compression of rotor along its axis Axial vibration problems are not a potential problem and the study related to axial vibrations are very rare in practice To calculate longitudinal vibrations are similar to calculate torsional vibrations except for using the mass instead of the polar mass moment of inertia The damping for longitudinal motion is almost zero, only material damping occurs which is small [30]

Torsional vibration is defined as an angular vibratory twisting of a rotor about its center axial on its angular spin velocity This vibration is potential problem in applications Torsional vibrations there are four important analyses which have to be done, static, real frequencies, harmonic force response and transient In real frequencies analysis the natural frequencies are evaluated in order to decide the critical speeds of the rotor Harmonic force response analysis

is done to see how large the twisting motion on the rotor is when the rotational speed is close

to any of critical speeds In a transient analysis the response of the rotor is calculated after a large torque has been affecting the rotor for a short while The analysis is focused on to see what happens after the torque is released and to see how long time it takes to reduce the oscillations in the rotor If there is an external torque acting on the rotor, which frequency is the same as one of the torsional modes, torsional fatigue can happen with crack progression [30]

2.1.3 Lateral vibrations

Lateral vibration is defined as an oscillation that occurs in the radial plane of the rotor spin axis It causes dynamic bending of the system element in two mutually perpendicular lateral planes The natural frequencies of this vibration are influenced by rotating speed and also the rotating machines can become unstable because of lateral vibration This vibration is more complex than torsional and axial vibration The analysis of lateral vibration conclude with static, harmonic, force response, natural frequencies, eigenvalues, eigenvectors, critical speeds and transient response It is very important to calculate the lateral vibration to improve a dynamic system’s life or system’s quality

Figure 1 Three kind of vibrations [31]

2.2 Gyroscopic effects

Gyroscopic effect is an important term in a rotating system The gyroscopic effect is proportional to the rotor speed and the

polar mass moment of inertia The unit of

it is unit of mass multiple to square unit of

radius When a perpendicular rotation or

precession motion ϕ̇ or nutation motion

θ̇ are applied to the spinning rotor with

speed ψ̇ about its spin axis z, gyroscopic

effect appears (Fig 2) To see clearly how

gyroscopic affects to a rotor dynamic

system, the section 2.3.4 will provide

some illustration

2.3 Terminologies in Rotor dynamics

Figure 2 Illustration of Gyroscopic effect [40]

a, Longitudinal or axial vibration

b, Torsional vibration

c, Lateral vibration

𝛉̇

mode that operating speed is zero If an excitation’s frequency is equal to a natural frequency, resonance will appear That natural frequency is called critical speed The excitation in rotor can come from synchronous excitation or asynchronous excitation The excitation due to unbalance is synchronous with rotating speed It is very dangerous if a rotor operates near or

at critical speed To find these critical speeds, the Campbell diagram was often used, see section 2.3.4

2.3.2 Whiling

When a rotor is operating it is not standing still at the axis of rotation; it is moving in a circular or elliptical motion around the axis of rotation (Fig 3) The motion is called whirling One thing that produces the whirling motion is the centrifugal forces which make the rotor bend Another thing can be that the rotor is not totally axis-symmetric There are two types of whirling, forward and backward Forward whirling is when the whirling rotation is the same direction as the rotational speed This type of whirling is the most dangerous one because it is easier to excite the rotor with forward whirling in resonance than with backward whirling Backward whirling is when the whirling rotation is opposite the rotational speed Which type

of whirling motion the rotor has can also been seen in a Campbell diagram, see section 2.3.4, where a forward whirling is increasing the natural frequency with higher rotational speed and backward whirling is decreasing the natural frequency with higher rotational speed [30]

2.3.3 Mode shapes

When the structure starts vibrating, the components associated with the structure moves together and follow a particular pattern of motion of each natural frequency This pattern of motion is called mode shapes (Fig 4) Knowing the shape of the rotor is much easier to set out the balance weights on the balance planes on the rotor The modes can be torsional, longitudinal or lateral Figure 4 summarizes the way in which products may be visualized as a superposition of single DOF modal components, even though lumped masses and springs are not involved A cantilever beam serves as the example, exhibiting unique deformation patterns called mode shapes The beam can be made to vibrate freely in any of the individual mode shapes, and again, associated with each mode shape is a resonance frequency, modal mass, modal stiffness, modal damping

Figure 3 Whirl orbit

2.3.4 Campbell diagram

Campbell diagram is a graphical presentation of the system frequency versus excitation frequency as a function of rotational speed It is usually drawn to predict the critical speeds of rotor system A sample Campbell diagram is shown in below Figs 5, 6

In general, the eigenvalue got from homogeneous solution are usually complex values and conjugate roots as below

Where, αj, ωjare real value and the natural frequency of the jth mode

One of the effects that separate vibrations of a rotor from other vibrations is the influence

of gyroscopic stiffening effect The influence of gyroscopic stiffening effect is always shown

in rotational parts, for example a rotor, micro-drill (Fig 6) The gyroscopic effect will increase if the rotational speed increase Hence, taking gyroscopic effect into account is very important with rotor bearing system under high speed machine

To prove the important of considering gyroscopic effect into account, we used Ansys software to write APDL code to compare the critical speeds between two cases, with and without gyroscopic by using the model in Ref [21] (Fig 5, 6) The critical speeds with gyroscopic effect are 31,529; 36,682; 51,744 and 91,047 (rpm) whereas the critical speeds without gyroscopic effect are 34,466; 34,466; 61,912 and 61,912 (rpm) It shows that the results between with and without gyroscopic cases are much different Therefore, taking this

Figure 4 Mode shapes [41]

The most important tool, an engineer has when critical speeds should be decided, is a Campbell diagram The diagram has the rotational speed of the rotor on the x-axis and the mode frequencies on the y-axis The modes (Fig 5, 6) are plotted for different rotational speeds The frequencies are not constant over the rotational speed range because most of the modes will be increased or decreased with higher rotational speed Forward whirling, modes are increasing and backward whirling modes are decreasing which is shown in the diagram Torsional and longitudinal modes are constant over the rotational speed range because they are not affected by the gyroscopic effect, the bearings or the stator An extra line is plotted in the above figure and it is the excitation line which represents natural excitations acting on the rotor It can be an external forces or mass unbalance in the rotor The excitation line in is synchronized with the rotational speed The critical speeds are where the excitation line crosses any of the mode lines Therefore in the above diagram, there are four critical speeds Without gyroscopic the natural frequencies of each mode are not depended on rotational speed (Fig 5) Otherwise, the natural frequencies are plotted versus the rotational speed (Fig 6)

Figure 5 The Campbell diagram without gyroscopic effect

Figure 6 Campbell diagram with gyroscopic effect

2.4 Design of rotor dynamics systems

Each manufacturer has categorical machines in different segments of the market Almost all products are redesign of already known concepts Still almost every machine has some unique features since most customers have different requirements Therefore, these machines have to be redesigned to some extent In this case, optimizing an already product is very necessary, especially some machines were required to operate in high accuracy such as micro-drilling, micro-milling systems, etc

Figure [7] gives a rough picture of how a general rotor dynamical analysis is performed for these types of systems

The figure shows a scheme for how the rotor dynamical analyses of the systems are related and performed Some significant input geometries are stator, rotor, bearing, disks In general, the modal analysis gives eigenvalues, eigenvectors, stability range and mode shape of rotor Valuations of deformations, stresses or deflections of rotor were calculated in harmonic analysis Finally, transient analysis gives the response of rotor in a short time In this research

we will focus on study a micro-drilling system including bearing, rotor geometry, such as cylinder, conical and flute part

Figure 7 Scheme of a rotor bearing system analysis [42]

Chapter 3 Dynamic equation of Micro-Drill Systems

This chapter is divided into 4 sections The first section shows the research model, Timoshenko’s beam and finite element model of micro-drill systems Section two describes how to build the equation of motion of each element type with the gyroscopic effect by finite element method Section 3 will deliver the bearing’s equation The global equation of the whole system was shown at the final section

3.1 Finite element model of the system

3.1.1 Timoshenko’s beam

In the Bernoulli-Euler theory of flexural vibrations of beams only the transverse inertia and elastic forces due to bending deflections are considered As the ratio of the depth of the beam to the wavelength of vibration increases the Bernoulli-Euler equation tends to overestimate the frequency The applicability of this equation can be extended by including the effects of the shear deformation and rotary inertia of the beam The equation which includes these secondary effects was derived by Timoshenko (Fig 8) That beam is call Timoshenko’s beam [7]

All most Timoshenko beam finite elements proposed in the previous literatures has a total

of four degrees of freedom, two at each of two nodes A complex element is one with more than four degrees of freedom, having more than two degrees of freedom at a node or more than two nodes [8] In this research we applied the complex element of Timoshenko’s beam that has 5 degree of freedom at each node, two translating degrees and three rotating degrees

Figure 8 Element model of Timoshenko beam [43]

3.1.2 Finite element modeling of micro-drill spindle

The micro-drill system is divided in to four main parts such as spindle, system element, clamp and drill Addition to the micro-drill is separated to five segments, two cylinders, two cones (symmetric element) and one pre-twist part (asymmetric element) (Fig 9c) The Timoshenko beam and finite element which has a powerful ability to treat the complex structures are used to build a five degrees of freedom model (Fig 9a) The model is considered itself eccentricity, external axial and torque forces (Fig 9b) Structure dimensions and parameters of the micro-drilling system are shown in Table 3.1 The model is divided into

15 finite elements in this study which approximately result in a 0.01 percent error in comparison with the results of larger numbers of elements

Table 3 1 Structure dimensions and parameters of ZTG04-III micro-drilling machine

Name of system

element

System symbol

Number of divided elements

Bearing parameters

Bearing positions (mm)

Bearing stiffness (N/m)

Bearing damping (N s/m)

Bearing torsional stiffness

Single row bearing A

L1 and L3 Kx and Ky Cx , Cy, Cφ Kφ

The material properties

Young’s modulus E

Shear modulus G

Mass density ρ

Shear coefficient

k

Single row bearing C

Spindle: E 1 Drill flute :

E 2

Spindle: G 1 Drill flute :

G 2

Spindle: ρ 1 Drill: ρ 2

k

System element D 1

Micro-drill M 10

3.2 Motional Equations of symmetric and asymmetric elements

The general finite form of equation of motion for all rotor-bearing systems is given by,

[M] is symmetric mass matrix

[C] is symmetric damping matrix

[G] is skew-symmetric gyroscopic matrix

[K] is symmetric stiffness matrix

{F} is external force vector

{q} is generalized coordinate vector

All these matrices will be expressed clearly in next section To get the Eq.3.1 we need to handle

as following steps The Euler angles are used to describe the orientation of deflected rotor (Fig 10) The sequence employed here is begun by rotating the initial system of axes parallel to fixed

Figure 10 Euler angles of the element

coordinates into a deflected mode by an angle ∅ counterclockwise about the Z axis Second, the intermediate axes (XYZ)’ rotate about the X’ axis counterclockwise by an angle θ to another intermediate axes (UVW)’ Finally, the (UVW)’ axes are rotated by an angle ψ about the W’ axis to produce the principal axes UVW

3.2.1 Hamilton’s equation of the system

The kinetic energy T of an infinitesimal element is the sum of the translating and rotational motions and may be derived from the equation below [32]

𝚽 = 𝛺𝑡 + 𝜑 + 𝛽; Ω is rotational speed, φ is torsional deformation

β is the angle principal axis U and basis axis of the global coordinates

Iav and Δ is mean and deviatoric moment of inertia, respectively

integral is the polar area effect of rotatory inertia; the third integral represents the gyroscopic effect; the fourth and fifth integral represents the asymmetric effect of rotor Here, there is torsional displacement in Φ that makes T, P and W different with the previous result of the authors [32]

The total potential energy according to the deflection of bending, axial and shear deformations and in the directions of the principal axes is expressed as

2 2

After getting the Eqs 3.6, 3.13, 3.16 we will apply Hamilton’s principle to get the

equations of motion of the micro-drill system

L x

s y

a a

a a dM

 The translations and rotations of a point internal to the element associated with the end- point

displacement are represented by shape functions as

2 3

(0) (0) ( ) ( )

a

R

L a

y

y

x R x

12

1 1

yy y

s y

12 1 1

xx x

2

'

2 3

2

'

2 3

; ; ; ; ;

t t

t t

t t

t t

t t

t t

t t

t t

t t

t t t

t x

t t

t t

t t

t t

t t

t t

t t

sin( ) cos sin + sin cos cos cos sin sin

sin 2( ) cos 2 sin 2 + sin 2 cos 2 2 cos 2 cos 2 2 sin 2 sin 2

cos( ) cos cos -sin sin cos sin sin cos

cos 2( ) cos 2 cos 2 -s

3.2.3 Finite equation of motions

Substituting Eqs.3.31- 3.34 to Eq 3.6 we get the finite equation of kinetic energy as below,

2 2

2

1

2

t t

t t

Substituting Eqs.3.41-3.44 to Eq 3.16 we get the finite equation of work as below,

(3.38)

In this research, the deflections are assumed very small Therefore, the order of magnitude of the high-order nonlinear terms, such as x y, 2

( x) ,  x y which involve square term or multiplication term, are quite small and can be ignore, for simplification To reduce the kinetic equation, we use some sub-matrix as following form:

2 L

u g

g

L 2

2

u 0

To reduce the potential equation, we use some sub-matrix as following form:

0 L 2

0 L 2

0 L

2

0

L 2

0 2

 Apply Lagrange’s equation to build the equation of motion as below,

0 0 0 0 0 1

0 L t

0 t

0 t

0 L t

0 t

0 y11

0 t

0 t 2

0 t 2

0 t 2

0 L t 2

0 t 2

0 y7

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] sin 2 [ ] [ ] cos 2 2

[ ] [ ] cos t [ ] [ ] sin t [ ] [ ] sin t [ ] [ ] cos t [ ]

[ ] [ ] [ ] sin t [ ] [ ] cos t [ ] [ ] cos t [ ] [ ] sin t

[ ] [M ]sin 2 [ ]cos 2 { 0.5 [ ] [ ] [ ]cos 2 [ ]sin 2

[ ] [ ] cos t [ ] [ ] sin t [ ] [ ] sin t [ ] [ ] cos t}

3.2.4 Motional equation of flute element (asymmetric part)

The complex geometry of a twist drill is modeled by specifying the Cartesian positions of the specific points on the surface of the drill The helical geometry of the flutes is specified by rotating the section an amount corresponding to the distance measured axially from the zero surface of the flute In order to calculate the moments of inertia, the cross sectional shaped of the drill will be simplified by ignoring chisel edge and margin as the Figure 13

In a more recent study, drill node loadings were determined from experimentally measured values These researches indicate that the main reasons for drill breakage are excessive twisting or bending The emphasis in the present research is on reducing the possibility of drill breakages This requires a simultaneous study of bending and torsional responses that arise as results of unbalance and the applied drill forces

In this present paper, the drill model is a Union micro-drill labelled as MDS The geometric features are shown in Table 3.2

Table 3.2 The geometric features of Union MDS

Geometric

features

Diameter d (mm)

Flute length

Helix angle δ ( 0 )

Web thickness

t (mm)

Flute/land ratio f

Table 3.3 Coordinates of nodal points 1-6 on the zero-surface

Nodal point Coordinates

, tan (1 04)

11

04

y 180

Where



Therefore, the coordinates (x,y,z) of the point on the z-

surface corresponding to the point (x0 ,y0 ,z0) on the zero

surface are given by:

Also, the symmetrically placed elements in the third

quadrant (x,y,z) are:

' ' '

The flute is an asymmetric part because of the different between second moments of area about principal axes u and v of cross-section Based on the coordinates of nodal points, the cross-section was calculated and drawn by Excel and CAD software, respectively The cross section properties and dimension of flute part of Union MDS are given in table 3.4

Table 3.4 Cross-sectional properties of flute part of MDS

As above, we know that:

𝚽 = 𝛺𝑡 + 𝜑 + 𝛽; Ω is rotational speed, φ is torsional deformation

β is the angle principal axis U and basis axis of the global coordinates

Iav and Δ is mean and deviatoric moment of inertia, respectively

Figure 13 Nodal points on the zero surface

Zero surface

(3.49)

Substituting all Eq.3.51, and β to Eq.3.39-3.41, we get the kinetic, potential energy and work

of flute element as below,

 Kinetic energy of flute element:

(3.52)

N M

Gyroscopic mass matrix Rotatory mass matrix