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Vibration Assisted Machining
Theory, Modelling and Applications
Lu Zheng
Newcastle University
Newcastle, UK
Wanqun Chen
Harbin Institute of Technology
Harbin, China
Dehong Huo
Newcastle University
Newcastle, UK
This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.
This edition first published 2021
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Library of Congress Cataloging-in-Publication Data
Names: Huo, Dehong, author.
Title: Vibration assisted machining : theory, modelling and applications /
Dehong Huo, Newcastle University, Newcastle, UK, Wanqun Chen, Harbin
Institute of Technology, Harbin, China, Lu Zheng, Newcastle University
Newcastle, UK.
Description: First edition. | Hoboken, NJ : Wiley, 2021. | Series:
Wiley-ASME Press series | Includes bibliographical references.
Identifiers: LCCN 2020027991 (print) | LCCN 2020027992 (ebook) | ISBN
9781119506355 (cloth) | ISBN 9781119506324 (adobe pdf) | ISBN
9781119506362 (epub)
Subjects: LCSH: Machining. | Machine-tools–Vibration. |
Cutting–Vibration. | Machinery, Dynamics of.
Classification: LCC TJ1185 .H87 2021 (print) | LCC TJ1185 (ebook) | DDC
671.3/5–dc23
LC record available at https://lccn.loc.gov/2020027991
LC ebook record available at https://lccn.loc.gov/2020027992
Cover Design: Wiley
Cover Image: © microstock3D/Shutterstock
Set in 9.5/12.5pt STIXTwoText by SPi Global, Chennai, India
10 9 8 7 6 5 4 3 2 1
vii
Contents
Preface xi
1 Introduction to Vibration-Assisted Machining Technology 1

1.1 Overview of Vibration-Assisted Machining Technology 1
1.1.1 Background 1
1.1.2 History and Development of Vibration-Assisted Machining 2
1.2 Vibration-Assisted Machining Process 3
1.2.1 Vibration-Assisted Milling 3
1.2.2 Vibration-Assisted Drilling 3
1.2.3 Vibration-Assisted Turning 5
1.2.4 Vibration-Assisted Grinding 5
1.2.5 Vibration-Assisted Polishing 6
1.2.6 Other Vibration-Assisted Machining Processes 7
1.3 Applications and Benefits of Vibration-Assisted Machining 7
1.3.1 Ductile Mode Cutting of Brittle Materials 7
1.3.2 Cutting Force Reduction 8
1.3.3 Burr Suppression 8
1.3.4 Tool Life Extension 8
1.3.5 Machining Accuracy and Surface Quality Improvement 9
1.3.6 Surface Texture Generation 10
1.4 Future Trend of Vibration-Assisted Machining 10
References 12
2 Review of Vibration Systems 17

2.1 Introduction 17
2.2 Actuators 18
2.2.1 Piezoelectric Actuators 18
2.2.2 Magnetostrictive Actuators 18
2.3 Transmission Mechanisms 18
2.4 Drive and Control 19
2.5 Vibration-Assisted Machining Systems 19
2.5.1 Resonant Vibration Systems 19
2.5.1.1 1D System 20
viii Contents
2.5.1.2 2D and 3D Systems 23

2.5.2 Nonresonant Vibration System 27
2.5.2.1 2D System 29
2.5.2.2 3D Systems 34
2.6 Future Perspectives 35
2.7 Concluding Remarks 36
References 37
3 Vibration System Design and Implementation 45

3.1 Introduction 45
3.2 Resonant Vibration System Design 46
3.2.1 Composition of the Resonance System and Its Working Principle 46
3.2.2 Summary of Design Steps 46
3.2.3 Power Calculation 47
3.2.3.1 Analysis of Working Length Lpu 48
3.2.3.2 Analysis of Cutting Tool Pulse Force F p 49
3.2.3.3 Calculation of Total Required Power 49
3.2.4 Ultrasonic Transducer Design 49
3.2.4.1 Piezoelectric Ceramic Selection 49
3.2.4.2 Calculation of Back Cover Size 51
3.2.4.3 Variable Cross-Sectional, One-Dimensional Longitudinal Vibration Wave
Equation 51
3.2.4.4 Calculation of Size of Longitudinal Vibration Transducer Structure 53
3.2.5 Horn Design 53
3.2.6 Design Optimization 54
3.3 Nonresonant Vibration System Design 55
3.3.1 Modeling of Compliant Mechanism 56
3.3.2 Compliance Modeling of Flexure Hinges Based on the Matrix Method 56
3.3.3 Compliance Modeling of Flexure Mechanism 59
3.3.4 Compliance Modeling of the 2 DOF Vibration Stage 61
3.3.5 Dynamic Analysis of the Vibration Stage 62
3.3.6 Finite Element Analysis of the Mechanism 63
3.3.6.1 Structural Optimization 63
3.3.6.2 Static and Dynamic Performance Analysis 63
3.3.7 Piezoelectric Actuator Selection 65
3.3.8 Control System Design 66
3.3.8.1 Control Program Construction 66
3.3.9 Hardware Selection 66
3.3.10 Layout of the Control System 68
References 69
3.A Appendix 70
4 Kinematics Analysis of Vibration-Assisted Machining 73

4.1 Introduction 73
4.2 Kinematics of Vibration-Assisted Turning 74
4.2.1 TWS in 1D VAM Turning 75
Contents ix
4.2.2 TWS in 2D VAM Turning 78

4.3 Kinematics of Vibration-Assisted Milling 80
4.3.1 Types of TWS in VAMilling 81
4.3.1.1 Type I 81
4.3.1.2 Type II 82
4.3.1.3 Type III 82
4.3.2 Requirements of TWS 83
4.3.2.1 Type I Separation Requirements 83
4.3.2.2 Type II Separation Requirements 85
4.3.2.3 Type III Separation Requirements 87
4.4 Finite Element Simulation of Vibration-Assisted Milling 89
4.5 Conclusion 93
References 93
5 Tool Wear and Burr Formation Analysis in Vibration-Assisted

Machining 95
5.1 Introduction 95
5.2 Tool Wear 95
5.2.1 Classification of Tool Wear 95
5.2.2 Wear Mechanism and Influencing Factors 96
5.2.3 Tool Wear Reduction in Vibration-Assisted Machining 98
5.2.3.1 Mechanical Wear Suppression in 1D Vibration-Assisted Machining 98
5.2.3.2 Mechanical Wear Suppression in 2D Vibration-Assisted Machining 101
5.2.3.3 Thermochemical Wear Suppression in Vibration-Assisted Machining 102
5.2.3.4 Tool Wear Suppression in Vibration-Assisted Micromachining 106
5.2.3.5 Effect of Vibration Parameters on Tool Wear 107
5.3 Burr Formation 108
5.4 Burr Formation and Classification 109
5.5 Burr Reduction in Vibration Assisted Machining 109
5.5.1 Burr Reduction in Vibration-Assisted Micromachining 111
5.6.1 Tool Wear 113
5.6.2 Burr Formation 115
References 115
6 Modeling of Cutting Force in Vibration-Assisted Machining 119

6.1 Introduction 119
6.2 Elliptical Vibration Cutting 120
6.2.1 Elliptical Tool Path Dimensions 120
6.2.2 Analysis and Modeling of EVC Process 120
6.2.2.1 Analysis and Modeling of Tool Motion 120
6.2.2.2 Modeling of Chip Geometric Feature 120
6.2.2.3 Modeling of Transient Cutting Force 124
6.2.3 Validation of the Proposed Method 126
6.3 Vibration-Assisted Milling 127
6.3.1 Tool–Workpiece Separation in Vibration Assisted Milling 128
x Contents
6.3.2 Verification of Tool–Workpiece Separation 131

6.3.3 Cutting Force Modeling of VAMILL 133
6.3.3.1 Instantaneous Uncut Thickness Model 133
6.3.3.2 Cutting Force Modeling of VAMILL 136
6.3.4 Discussion of Simulation Results and Experiments 137
References 143
7 Finite Element Modeling and Analysis of Vibration-Assisted

Machining 145
7.2 Size Effect Mechanism in Vibration-Assisted Micro-milling 147
7.2.1 FE Model Setup 148
7.2.2 Simulation Study on Size Effect in Vibration-Assisted Machining 151
7.3 Materials Removal Mechanism in Vibration-Assisted Machining 152
7.3.1 Shear Angle 152
7.3.2 Simulation Study on Chip Formation in Vibration-Assisted Machining 154
7.3.3 Characteristics of Simulated Cutting Force and von-Mises Stress in
Vibration-Assisted Micro-milling 156
7.4 Burr Control in Vibration-Assisted Milling 158
7.4.1 Kinematics Analysis 159
7.4.2 Finite Element Simulation 160
7.5 Verification of Simulation Models 161
7.5.1 Tool Wear and Chip Formation 162
7.5.2 Burr Formation 163
References 164
8 Surface Topography Simulation Technology for Vibration-Assisted

Machining 167
8.2 Surface Generation Modeling in Vibration-Assisted Milling 171
8.2.1 Cutter Edge Modeling 172
8.2.2 Kinematics Analysis of Vibration-Assisted Milling 173
8.2.3 Homogeneous Matrix Transformation 174
8.2.3.1 Basic Theory of HMT 174
8.2.3.2 Establishment of HTM in the End Milling Process 174
8.2.3.3 HMT in VAMILL 176
8.2.4 Surface Generation 185
8.2.4.1 Surface Generation Simulation 185
8.3 Vibration-Assisted Milling Experiments 187
8.4 Discussion and Analysis 187
8.4.1 The Influence of the Vibration Parameters on the Surface Wettability 188
8.4.2 Tool Wear Analysis 189
References 189
Index 193
xi
Preface
Precision components are increasingly in demand for various engineering industries,

such as biomedical engineering, MEMS, electro-optics, aerospace, and communications.
However, processing these difficult-to-machine materials efficiently and economically is
always a challenging task, which stimulates the development and subsequent application
of vibration-assisted machining (VAM) over the past few decades. Vibration-assisted
machining employs additional external energy sources to generate high-frequency vibra-
tion in the conventional machining process, changing the machining (cutting) mechanism,
thus reducing the cutting force and cutting heat and improving the machining quality.
The effective implementation of the VAM process depends on a wide range of technical
issues, including vibration device design and setup, process parameter optimization,
and performance evaluation. The current awareness on VAM technology is incomplete;
although ample review/research papers have been published, no single source provides
a comprehensive comprehending yet. Therefore, a book is needed to systematically
introduce this emerging manufacturing technology as a subject.
The main objective of this book is to address the basics and the latest advances in the VAM
technology. The first chapter provides a brief introduction to VAM technology, including
VAM process, benefits, and applications, as well as its history and development, so that the
reader would have a general understanding of the subject. The second and third chapters
aim to present a detailed description of the characteristics and design process for vibra-
tion devices. Chapter 2 overviews the current proposed vibration devices in the literature,
and the features of each type vibration devices are critically reviewed. Chapter 3 focuses
on the implementation and design of vibration devices and the corresponding design pro-
cedures are also discussed. Chapters 4 and 5 are dedicated to the effect of vibration and
machining parameters on tool path/tool–workpiece separation and the surface topography
generation. Chapters 4 and 5 are dedicated to the effect of vibration and machining param-
eters on tool path/tool–workpiece separation and its influence on the cutting performance.
Chapter 4 covers the kinematic analysis of VAM, including the tool–workpiece separation
type and the corresponding equations during the processing. Chapter 5 investigates the
mechanisms of tool wear and burr generation under different tool–workpiece separation
situations. Chapter 6 and 7 investigate VAM process through simulation modelling method.
Chapter 6 models the cutting force using both numerical and finite element methods. Finite
element modeling and analysis of VAM are detailed in Chapter 7 to deeply understand the
cutting mechanism of VAM. The last chapter contains the modeling of surface topography
xii Preface
using homogeneous matrix transformation and cutter edge sweeping technology, and the
results are verified by the machining experiments.
This book provides state of the art in research and engineering practice in VAM for
researchers and engineers in the field of mechanical and manufacturing engineering.
This book can be used as a textbook for a final year elective subject on manufacturing
engineering, or as an introductory subject on advanced manufacturing methods at the
postgraduate level. It can also be used as a textbook for teaching advanced manufacturing
technology in general. The book can also serve as a useful reference for manufacturing
engineers, production supervisors, tooling engineers, planning and application engineers,
as well as machine tool designers.
Some of the research findings in this book have arisen from an EPSRC-funded project
“Development of a 3D Vibration Assisted Machining System.” The authors gratefully
acknowledge the financial support of the Engineering and Physical Sciences Research
Council (EP/M020657/1).
The authors wish the readers an enjoyable and fruitful reading through the book.
Newcastle upon Tyne, UK Lu Zheng, Wanqun Chen and Dehong Huo

February 2020
1
Introduction to Vibration-Assisted Machining Technology
1.1 Overview of Vibration-Assisted Machining Technology

1.1.1 Background
Precision components are increasingly in demand in various engineering fields such as
microelectromechanical systems (MEMS), electro-optics, aerospace, automotive, biomedi-
cal engineering, and internet and communication technology (ICT) hardware. In addition
to the aims of achieving tight tolerances and high-quality surface finishes, many applica-
tions also require the use of hard and brittle materials such as optical glass and technical
ceramics owing to their superior physical, mechanical, optical, and electronic properties.
However, because of their high hardness and usually low fracture toughness, the processing
and fabrication of these hard-to-machine materials have always been challenging. Fur-
thermore, the delicate heat treatment required and composite materials in aeronautic or
aerospace alloys have caused similar difficulties for precision machining.
It has been reported that excessive tool wear and fracture damage are the main failure
modes during the processing of such materials, leading to low surface quality and machin-
ing accuracy. Efforts to optimize a conventional machining process to achieve better cut-
ting performance with these materials have never been stopped, and these optimizations
include the cutting parameters, tool materials and geometry, and cutting cooling systems in
the past decades [1–6]. Generally, harder materials or wear-resistant coatings are applied,
and tool geometry is optimized to prevent tool cracking and to reduce wear on wearable
positions such as the flank face [5, 7–10]. Cryogenic coolants are used in the machining
process, and their input pressure has been optimized to achieve better cooling performance
[2, 4, 11]. However, although cutting performance can be improved, the results are often
still unsatisfactory.
Efforts to enhance machining performance have revealed that machining quality can be
improved using the high-frequency vibration of the tool or workpiece. Vibration-assisted
machining (VAM) was first introduced in the late 1950s and has been applied in various
machining processes, including both traditional machining (turning, drilling, grinding, and
more recently milling) and nontraditional machining (laser machining, electro-discharge
machining, and electrochemical machining), and it is now widely used in the precision
manufacturing of components made of various materials. VAM adds external energy to the
Vibration Assisted Machining: Theory, Modelling and Applications,

First Edition. Lu Zheng, Wanqun Chen, and Dehong Huo.
© 2021 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.
2 1 Introduction to Vibration-Assisted Machining Technology
conventional machining process and generate high-frequency, low-amplitude vibration in

the tool or workpiece, through which a periodic separation between the uncut workpiece
and the tool can be achieved. This can decrease the average machining forces and gener-
ate thinner chips, which in turn leads to high processing efficiency, longer tool life, better
surface quality and form accuracy, and reduced burr generation [12–17]. Moreover, when
hard and brittle materials such as titanium alloy, ceramic, and optical glass are involved, the
cutting depth in the ductile regime cutting mode can be increased [18]. As a result, the cut-
ting performance can be improved and unnecessary post-processing can be avoided, which
allows the production of components with more complex shape features [14]. Nevertheless,
there are still many opportunities for technological improvement, and ample scope exists
for better scientific understanding and exploration.
VAM may be classified in two ways. The first classification is according to the dimensions
in which vibration occurs: 1D, 2D, or 3D VAM. The other classification is based on the
vibration frequency range, for example, in ultrasonic VAM and non-ultrasonic VAM.
Ultrasonic VAM is the most common type of VAM. It works at a high vibration frequency
(usually above 20 kHz), and a resonance vibration device maintains the desired vibration
amplitude. Most of its applications are concentrated in the machining of hard and brittle
materials because of the fact that high vibration frequency dramatically improves the
cutting performance of difficult-to-machine materials. Meanwhile non-ultrasonic VAM
uses a mechanical linkage to transmit power to make the device expand and contract,
and this can obtain lower but variable vibration frequencies (usually less than 10 kHz). It
is easier to achieve closed-loop control because of the low range of operating frequency,
which makes it uniquely advantageous in applications such as the generation of textured
surface.
1.1.2 History and Development of Vibration-Assisted Machining

The history of vibration technology in VAM can be traced back to the 1940s. During the
period of World War II, the high demand for the electrically controlled four-way spool valves
mainly used in the control of aircraft and gunnery circuits stimulated the development
of servo valve technology [19]. Because of their wide frequency response and high flow
capacity, electrohydraulic vibrators were successfully developed and applied in VAM in the
1960s with positive effects in enhanced processing quality and efficiency [20]. With the fur-
ther development of technology, electromagnetic vibrators featuring higher accuracy and
a wide range of frequency and amplitude generation were developed based on electromag-
netic technology, and these were successfully applied to various VAM processes [21]. The
need for complex hydraulic lines was eliminated, and greater tolerance for the application
environment was allowed, which also leads to smaller devices. As a result, a transmission
line or connecting body can be attached to the vibrator to achieve a wide range of vibra-
tion frequencies and amplitude adjustments [22]. In the 1980s, the maturity of piezoelectric
transducer (PZT) piezoelectric ceramic technology had brought a new choice for the vibra-
tor. A piezoelectric ceramic stack could be sandwiched under compressive strain between
metal plates, and this has advantages including compactness, high precision and resolu-
tion, high frequency response, and large output force [23]. Various shapes of piezoelectric
ceramic elements can be used to make different types of vibration actuators, which indicate
that the limitations of traditional vibrators were overcome and the application of VAM tech-
nology for precision machining was broadened. In addition, it helped in the development of
multidimensional VAM equipment. Elliptical VAM has received extensive attention since
it was first proposed in the 1990s. Although this process has many advantages compared
to its 1D counterpart in terms of reductions in cutting force and prolongation of tool life,
it requires higher performance in the vibrator, producing a more accurate tool tip trajec-
tory [24–28]. Piezoelectric actuators with high sensitivity can fulfill the requirements of
vibration devices and promote the development of elliptical VAM technology.
1.2 Vibration-Assisted Machining Process

This section briefly introduces commonly used VAM processes, including milling, drilling,
turning, grinding, and polishing. Different vibration device layouts are required to imple-
ment these vibration-assisted processes and to achieve advantages over the corresponding
conventional machining processes.
1.2.1 Vibration-Assisted Milling

Milling is one of the most common machining processes and is capable of fabricating parts
with complex 3D geometry. However, uncontrollable vibration problems during the cutting
process are quite serious and can affect processing stability, especially in the micro-milling
process, leading to excessive tolerance, increased surface roughness, and higher cost.
Vibration-assisted milling is a processing method that combines the external excitation of
periodic vibrations with the relative motion of the milling tool or workpiece to obtain better
cutting performance. In addition to the same advantages as other VAM processes, complex
surface microstructures can also be obtained because of the combination of a unique
tool path and external vibration. Currently, the application of vibration-assisted milling
mainly focuses on the one-dimensional direction. The vibration may be applied in the feed
direction, cross-feed direction, or axial direction, and tool rotational vibrations may also be
applied [14]. Little research has been carried out on 2D vibration-assisted milling because
of the difficulty of developing two-dimensional vibration platforms (motion coupling and
control difficulty), and the vibration mode of these 2D vibration devices mainly involves
elliptical vibration and longitudinal torsional vibration.
1.2.2 Vibration-Assisted Drilling

Problems such as large axial forces and poor surface quality are found in the process of
drilling the hard and brittle materials. Vibration-assisted drilling technology combines the
VAM mechanism with the traditional drilling process, and this can achieve more efficient
drilling, especially for small bore diameters and deep holes. Compared with conventional
drilling, the interaction between the tool and the workpiece is changed, and the drilling tool
edge cutting conditions are improved. Vibration-assisted drilling has found applications in
the high-efficiency and high-quality machining of various parts with difficult-to-machine

holes [29]. Its main merits are as follows:
(1) Reductions in drilling power and drilling torque. The vibration changes the interaction
between the drill tool and the workpiece, and the cutting process changes from contin-
uous cutting to intermittent cutting, leading to lower tool axial force. In addition, the
friction factor between the tool and the workpiece/chips is reduced because of the pulse
torque formed by the vibration. As a result, drilling torque is reduced [30, 31].
(2) Improvement in chip breaking and removal performance. The chip breaking mechanism
is quite different when vibration is added. Fragmented chips can be obtained under
certain vibration and machining parameters. Chip removal performance is much better
compared with the continuous chips produced in conventional drilling [32].
(3) Improvement in the surface quality of the walls of the drilled holes. In the vibration-assisted
drilling process, the reciprocal pressing action of the cutting edge on the inner hole sur-
face is beneficial in reducing surface roughness. Moreover, the improved chip breaking
performance also leads to smoother chip removal, which reduces the scratching of the
drilled hole surface by chips and the surface roughness [33, 34].
(4) Improvement in tool life. The intermittent cutting improves the drilling tool’s cooling
conditions, leading to lower cutting temperature and relieving the built-up edge and
tool chipping effects. As a result, longer tool life can be obtained [35, 36].
As shown in Figure 1.1, according to the direction of vibration, vibration-assisted drilling

can be divided into axial, torsional, and axial–torsional composite vibration drilling. The
vibration direction in axial vibration drilling is consistent with the direction of the drilling
tool axis, while in torsional vibration drilling, it is consistent with the direction of the
drilling tool’s rotation. Axial torsional composite vibration drilling combines the previous
two types.
Tool rotation direction Figure 1.1 Schematic of

vibration-assisted drilling.
Feed direction Axial vibration
Torsional vibration
Workpiece rotation direction

Work
p iece
Feed
direc
tion
l
Tangential direction
too
g
ttin
Cu
ion
e ct
dir
ial
d
Ra
Figure 1.2 Schematic of vibration-assisted turning.
1.2.3 Vibration-Assisted Turning

Turning is a widely used machining method because of its high processing quality, metal
removal rate, and productivity and efficient equipment utilization. However, drawbacks
such as large cutting forces, difficulties in chip removal, and serious tool wear can cause
serious processing problems, such as low machined quality and efficiency and high
cost. Vibration-assisted turning provides a new method for the efficient and high-quality
machining of difficult materials. As shown in Figure 1.2, vibration is applied to the
turning tool mainly in the radial, tangential, and feed directions. Multidimensional
vibration-assisted turning is generally referred as elliptical vibration-assisted turning,
where two of the above three vibration directions are chosen and applied to the turning
tool. One-dimensional vibration-assisted turning represents a large proportion of methods
of vibration-assisted turning proposed so far. Most apply vibration in the feed direction,
and experimental results have proven that this has a significant influence in reducing
cutting forces, cutting temperature, and improving the quality of processing. Currently,
only a few studies have applied vibration in the other directions, and the effects and cutting
mechanisms of involved in material processing need further research.
1.2.4 Vibration-Assisted Grinding

Compared with other machining processes, grinding is increasingly used in the field of
ultraprecision/precision machining because of its better machining accuracy and surface
roughness. However, processing material with grinding wheels is a complex and stochastic
process, where the ground surface may become damaged and low wheel life is caused
by the high grinding forces and high surface cutting temperature (as the grinding wheel
instantaneous temperature can reach 1000 ∘ C). Vibration-assisted grinding process applies
vibration to the grinding wheel or workpiece during the grinding so as to improve the
material removal performance. The vibration can be applied in the tangential, radial, or
Figure 1.3 Schematic of vibration-assisted grinding.

Grinding wheel
Radial direction
Wo
rkp
Ta iec
n ge e
nti
al
d ire
cti
on
Axial vibration
axial direction along the grinding wheel, as is shown in Figure 1.3. Vibration-assisted
grinding in the tangential and radial directions is similar to intermittent grinding, and
tool–workpiece separation can be obtained during the machining process. Although
vibration-assisted grinding in the axial direction involves a continuous grinding process,
the machining process is quite different in conventional grinding and features separation,
impact and reciprocating ironing characteristics, and lubrication effects, which can reduce
grinding wheel blockage, cutting forces, workpiece residual stress, and machined surface
burn. As a result, better processing performance and longer tool life can be obtained. In
addition, it can also effectively reduce the chipping of hard and brittle workpiece materials
and surface or subsurface cracking as well as machined surface quality [37–39]. Although
similar to the mechanism of other VAM processes, the randomness of the size, shape, and
distribution of abrasive grains on the grinding wheel surface and the complexity of the
grinding motion bring great challenges to the study of the mechanisms involved in the
vibration-assisted grinding.
1.2.5 Vibration-Assisted Polishing

At present, various miniature optical lenses are generally fabricated by precision injec-
tion molding with silicon carbide or tungsten carbides molds, and these molds usually
require polishing to achieve optical grade surface quality. However, small mold sizes
and increasingly high precision requirements make the polishing process challenging. In
the conventional polishing process, the high-speed rotation of soft polishing tools such
as wool, rubber, and asphalt polishing heads are often used to process the workpiece
surface. However, when the surface has complex curved shapes and a small curvature
radius, the complicated polishing mechanism and uncontrollable polishing forces severely
limit the processing results. Vibration-assisted polishing can overcome some of the
shortcomings in conventional polishing. Using this method, the polishing head does not
need to be rotated at such high speeds, which helps in ensuring constant polishing force
during the polishing process, which can also be used for smaller size molds. Current
research shows that vibration-assisted polishing can improve the surface roughness of the
polished workpiece and the surface accuracy while achieving high polishing efficiency
[40–42].
1.3 Applications and Benefits of Vibration-Assisted Machining 7
1.2.6 Other Vibration-Assisted Machining Processes

With the advantages of VAM gradually being demonstrated, more machining processes
are being added to the VAM family. Two examples of the newly developed VAM processes
are vibration-assisted boring and vibration-assisted electrical discharge machining. In
order to solve the difficult machining problem of complex deep hole parts with high
length-to-diameter ratios (>20) such as aeroengine fuel nozzles, vibration-assisted boring
has been developed. Compared with the conventional boring process, the tool can be
prevented from colliding with the machined surface during the separation stage, the
plastic critical cutting depth of the brittle material is increased, and cutting edge cracking
and cutting tool flank face reverse bulges are avoided [43, 44]. Its separation and reversal
characteristics can greatly reduce the radial thrust force and effectively improve the
absolute stability of the cutting stiffness. As a result, the machined surface quality is
improved and cutting flutter can be suppressed. Vibration-assisted electrical discharge
machining has also been successfully applied in processing micro-hole parts made of
hard and brittle materials [45–47]. In conventional electrical discharge machining, the
discharge gap between the tool and the workpiece is usually only a few micrometers to
several tens of micrometers and easily causes the deterioration due to the slag discharge
effect and local concentration of processing debris, causing abnormal discharges and
reducing processing efficiency. Compared with the process, vibration-assisted electrical
discharge machining has better processing efficiency, and the results show that the slag
removal effect is ameliorated and electrode wear reduced.
1.3 Applications and Benefits of Vibration-Assisted

Machining
1.3.1 Ductile Mode Cutting of Brittle Materials

When the cutting depth is less than a certain critical value (the critical cutting depth) in
the processing of brittle materials, the cutting process will be transformed from brittle
cutting mode into ductile cutting mode. This removes the workpiece materials by plastic
flow instead of brittle fractures, leading to a crack-free surface. In ductile cutting mode, the
critical cutting depth can be defined as the cutting depth at which a crack appears on the
machined surface. If the undeformed chip thickness is less than the critical cutting depth,
brittle cutting can be reduced in conventional cutting and a better surface finish can be
obtained. However, in the actual processing of brittle materials, their critical cutting depth
is usually in the range of microns or submicron, which reduces the processing efficiency
and increases the manufacturing time. VAM is an effective method used to increase the
critical cutting depth in ductile cutting mode and to improve the economics and feasibility
of the processing of brittle materials. It has been reported that smaller cutting forces can
reduce microcrack propagation on the surface of the brittle parts and can increase the
critical cutting depth for brittle materials under ductile cutting mode. In addition, a large
enough plastic yielding force, but not large enough to cause material rupturing, is also a
necessary condition for the ductile cutting of brittle workpieces. Therefore, it is feasible
to increase the brittle materials critical cutting depth within a reasonable stress range by
using VAM [48–50].
1.3.2 Cutting Force Reduction

A large number of cutting experiments and finite element analysis show that under the
same cutting conditions, the average cutting force of VAM is significantly lower than in the
traditional cutting process, and the cutting force in 2D VAM process is less than that in 1D.
Although the instantaneous peak cutting force of 1D VAM is close to the steady-state cutting
force in conventional machining, a lower average cutting force can be obtained because of
the periodic contact between the tool and workpiece during cutting [51–53]. In 2D VAM
process, the shape of chips and the interaction between them and the tool rake face are
quite different from 1D VAM because of the elliptical cutting tool trajectory, which leads to
lower average cutting force and reduced instantaneous peak cutting force. The cutting force
reduction is manifested in the following ways:
(1) The chip thickness in the 2D VAM can be reduced because of the continuous overlap-
ping of elliptical tool paths. As a result, the cutting forces in different directions can be
reduced.
(2) Under certain conditions such as circular or narrowly elliptical tool paths, the cutting
tool moves faster than the chip flow speed, causing reverse friction between the tool
and the workpiece, and the back cutting force can be reduced or even reversed.
(3) The periodic contact between the cutting tool and workpiece improves the lubrication
conditions during the cutting process and facilitates the dissipation of heat from the
tool, resulting in a reduction in cutting force.
1.3.3 Burr Suppression

Burr formation, similar to chip generation, is a common and undesirable phenomenon in
the machining process and is one of the most important criteria in the evaluation of the
machined surface. VAM can effectively suppress burr formation during processing, and
some researchers have proposed that burr height can be reduced up to 80% compared with
conventional machining [54–56]. Figure 1.4 shows examples of burr reduction in VAM.
Almost no burrs can be found on the machined surface. This phenomenon is mainly due to
the reduced cutting force, which leads to lower transient compressive stress and yield stress
in the cutting deformation area. In addition, unique tool trajectories (such as elliptical
trajectories) can result in discrete small pieces of chips. As a result, burr formation can be
suppressed.
1.3.4 Tool Life Extension

Machining processes are inherently involved in tool wear, which is usually evaluated in
terms of average cutting force, machined surface roughness, and cumulative cutting length.
It has an important impact on surface quality and machining costs. VAM can effectively
1.3 Applications and Benefits of Vibration-Assisted Machining 9
50 µm 10 µm
Near-zero
burr
4000×
(a) (b)
Figure 1.4 SEM images of burr-free structures made using 2D VAM. Single-crystal diamond tool in
hard-plated copper. (a) Microchannel, 1.5 μm deep, and (b) a 8 μm tall regular trihedron made using
a dead-sharp tool with a 70∘ nose angle. Source: Brehl and Dow [14]. © 2008, Elsevier.
improve cutting tool life, especially in the processing of hard materials. Unlike the irregular
wear caused by traditional machining tools, the tool wear in VAM is smooth and inclined. At
lower spindle speeds, due to the lower cutting temperatures, the dominant wear mechanism
is abrasive wear. Because of the mechanical and impact contact between the workpiece and
tool flank surface in VAM, tool life is less than that in the conventional process. At higher
cutting speeds, temperature-activated wear mechanisms occur, such as diffusion, chemi-
cal wear, and thermal wear. On the other hand, because of the intermittent separation of
the workpiece and tool, the temperature in the cutting zone in VAM is lower than that in
conventional process, which tends to increase the tool life. Another reason for reducing
the temperature in VAM is the change in friction coefficient from semi-static to dynamic,
which results in a reduced friction coefficient in the process and a change in the chip for-
mation mechanism. As the cutting speed increases, there is an increase in the degree of
tool–workpiece engagement per tool revolution. As a result, the effect of vibration on the
machining process decreases, and the cutting forces in VAM and conventional milling pro-
cesses become closer to each other. A detailed analysis on how VAM enhances tool life is
provided in Chapter 5.
1.3.5 Machining Accuracy and Surface Quality Improvement

Compared with the conventional machining process, VAM can greatly improve the
machining accuracy and surface quality, and the improvements vary depending on the tool
and workpiece materials, vibration conditions (vibration amplitude, vibration frequency,
and vibration dimensions), tool parameters, and processing parameters such as feed
rate, spindle speed, and cutting depth. If the processing parameters are unchanged, the
surface roughness in 1D and 2D VAM can be reduced by approximately 40% and 85%,
respectively [14]. There are many reasons for this. On the one hand, lower cutting forces
can enhance the stability of the cutting process, which reduces tool run-out in the cutting
depth direction and generates smaller chips. On the other hand, VAM can reduce cutting
tool wear and effectively avoid damage caused to the machined surface by worn tools. The
tool’s self-excited vibration is replaced with regular sine or cos vibration, which reduces the
residual height of the unremoved material. As a result, a better machined surface quality
can be obtained.
1.3.6 Surface Texture Generation

Engineered textured surfaces have the characteristics of regular textural structures and high
aspect ratio, enabling the component surface to serve specific functions such as reducing
adhesion friction, improving lubricity, increasing wear resistance, changing hydrophilic
performance, and enhancing optical properties. Etching methods are commonly used to
produce high precision surface microstructures, but these are costly and time-consuming.
As a more flexible method, it has been proven that VAM in either a single direction or
two directions can form certain surface textures depending on the cutting edge geome-
try and kinematics. Currently, the proposed surface textures mainly include a squamous,
micro-dimple pattern and micro-convex pattern types, and their size ranges from a few
microns to tens of microns, as shown in Figure 1.5. There is an emerging trend to obtain cer-
tain surface performance using VAM. For example, the size of the surface texture features
can be controlled by changing the vibration and processing parameters, leading to variable
surface wettability (Figure 1.5) [12]. The process can also be used to create microchannels
for the microfluidic control of the fluid flow, to name a few. A detailed analysis on how
VAM produces surface texture is provided in Chapter 8.
1.4 Future Trend of Vibration-Assisted Machining

With the development of processing technology, the application of VAM is becoming
increasingly widespread, and research into VAM is becoming more and more intensive,
mainly in the following main aspects:
(1) Development and adoption of new tool materials. The proportion of difficult-to-machine
materials in modern products is increasing, as well as a higher processing quality of
these parts. In order to achieve better cutting performance, in addition to the optimiza-
tion of tool geometry parameters, more attention has been focused on the development
and application of tool materials in VAM, and main research focus is on natural and
synthetic diamond and ultrafine grained carbide materials.
(2) Ultra-high-frequency vibration-assisted machining. Ultra-high-frequency VAM will con-
tinue to be a research focus in VAM in the future. Recent research indicates that the pos-
sibility of grinding wheel ablation can be effectively reduced by adding high-frequency
vibration, which also improves the grinding wheel’s life and the surface quality of the
workpiece. In recent years, research into ultra-high-frequency vibration equipment has
made it possible to reach a maximum vibration frequency of 100 kHz, and at the same
time, its processing performance for brittle and hard materials has also been signifi-
cantly improved.
(3) Precision/ultraprecision application. It has been reported that the dimensional and
geometric accuracy and wear resistance as well as corrosion resistance of the workpiece
can be improved dramatically when low-frequency vibration is applied. However,
only high-frequency VAM, such as ultrasonic VAM, can currently achieve a precision
machining process. For example, a surface roughness of Ra 0.02–0.04 μm can be
obtained by vibration-assisted honing, and surface quality improves by an order of
magnitude in the ultrasonic vibration extrusion process compared to conventional
1.4 Future Trend of Vibration-Assisted Machining 11
(a)
(b) (c)
µm µm
100 1.22
(d) 1.0
250
200 0.5
150 0.0
100 –0.5
50 –1.0
–1.29
0
0 50 100150200250300350
µm µm
364 2.14
300 1.50
1.0
0.5
200 0.0
–0.5
100 –1.00
–1.50
0 –2.07
0 100 200300 400 486
µm µm
475 1.9
400 1.5
1.0
300 0.5
0.0
200 –0.5
–1.0
100 –1.5
0 µm–1.9
0 100 200 300 400 500 600
µm µm
364 3.3
300 2.5
2.0
1.5
200 1.0
0.5
0.0
100 –0.5
–1.0
0 µm–1.64
0 100 200 300 400 490
Figure 1.5 Surface texture produced by vibration-assisted machining: (a) micro-dimple patterns.
Source: Lin et al. [57]. © 2017, IOP Publishing Ltd, (b) micro-convex patterns. Source: Kim and Loh
[58]. © 2010, Springer Nature, (c) squamous patterns. Source: Tao et al. [59]. © 2017, Taylor &
Francis Group, and (d) surface wettability variation with different surface textures. Source: Chen
et al. [12].
extrusion. Ultrasonic vibration machining can not only guarantee the quality of
ultraprecision machining but also allows for higher cutting rates, leading to higher
productivity.
(4) In-depth study of vibration-assisted machining mechanism. Although the cutting mecha-
nism of VAM has been investigated by several researchers, it is still not fully understood.
Current and future research on VAM will focus on several areas, including the effect of
the separation and non-separation of the workpiece and cutting tool on chip formation,
mechanical analysis of the interaction between the cutting tool and workpiece, micro-
scopic studies, and mathematical descriptions of VAM mechanisms, to name a few.
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49 Du Kim, J. and Choi, I.H. (1997). Micro surface phenomenon of ductile cutting in the
ultrasonic vibration cutting of optical plastics. J. Mater. Process. Technol. https://doi.org/
10.1016/S0924-0136(96)02546-0.
50 Zhou, M., Wang, X.J., Ngoi, B.K.A., and Gan, J.G.K. (2002). Brittle-ductile transition
in the diamond cutting of glasses with the aid of ultrasonic vibration. J. Mater. Process.
Technol. 121: 243–251. https://doi.org/10.1016/S0924-0136(01)01262-6.
51 Zhou, M., Eow, Y.T., Ngoi, B.K.A., and Lim, E.N. (2003). Vibration-assisted precision
machining of steel with PCD tools. Mater. Manuf. Processes 18: 825–834. https://doi.org/
10.1081/AMP-120024978.
52 Babitsky, V.I., Mitrofanov, A.V., and Silberschmidt, V.V. (2004). Ultrasonically assisted
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10426914.2016.1198029.
17
Review of Vibration Systems
2.1 Introduction
A well-designed vibration system is quite important in vibration-assisted machining.

A typical vibration system consists of vibration sources (actuators), a vibration trans-
mission/amplification mechanism, and a control system. Generally, the actuators and
transmission/amplification mechanisms were selected to be complementary. Given that
certain demands such as required vibration frequency and amplitude range are proposed
in the design phase, the optimal overall device structure to match those demands is
determined first. Then, the key structural parameters are optimized using the finite
element analysis (FEA) method, and the corresponding dynamic and static characteristics
can be obtained. As a result, these key values in turn influence the choice of the vibration
actuators. According to the system’s operating frequencies, these proposed vibration
devices can be divided to two groups: the nonresonant mode and the resonant mode.
In a nonresonant vibration system, the vibration actuators usually vibrate below its first
natural frequency. In order to increase the stability and reduce the dynamic error in
the vibration stage, flexure hinge structures are widely used because of their superior
dynamic response, low friction, and ease of control. For a resonant system, a sonotrode
(also called a horn or concentrator) vibrates at its natural frequency, transferring and
amplifying a given vibration from a vibration source, which is usually a magnetostrictive or
piezoelectric transducer. This system can achieve a higher operating frequency and greater
energy efficiency compared with a nonresonant system. However, its vibration trajectory
cannot be controlled precisely owing to the nature of resonant vibrations and the phase
lag between excitation and the mechanical response. Compared with resonant systems,
nonresonant systems tend to achieve higher vibration accuracy, and it is easier to achieve
closed loop control of the vibration trajectories under low-frequency conditions.
This chapter provides a general understanding of vibration systems, including actua-
tor types and their selection, transmission/amplification mechanisms, and control system
design. By analyzing design theories and principles, the advantages and disadvantages of
various types of vibration devices are discussed. Moreover, future trends in vibration devices
are also mentioned.
Vibration Assisted Machining: Theory, Modelling and Applications,

First Edition. Lu Zheng, Wanqun Chen, and Dehong Huo.
© 2021 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.
18 2 Review of Vibration Systems
2.2 Actuators
As power output devices, actuators convert other types of energy into mechanical energy
to drive the vibration stage. This contributes not only to the bandwidth of the vibration
frequency and amplitude but also to the accuracy of the motion of the vibration stage.
Currently, two types of actuators, namely, piezoelectric and magnetostrictive actuators, are
mainly selected for vibration systems. This section provides a general understanding infor-
mation about these actuators, including their strengths and weaknesses.
2.2.1 Piezoelectric Actuators

Piezoelectric actuators apply the unique piezoelectric properties of piezoelectric ceramics to
convert high-voltage electrical energy into mechanical energy for high-frequency vibration.
Compared with other types of actuators, they have the characteristics of low cost, simple
structure, small size, fast response, high control precision, the lack of magnetic field and
electrical fields, and no electromagnetic interference or electromagnetic noise, improving
system stability and leading to more flexible designs of vibration devices. Various types of
piezoelectric actuators, including stacked, thin plate, tubular, and bimorph types, have been
developed for different applications [1, 2]. Considering factors such as displacement output,
stiffness, and frequency response, stack-type piezoelectric actuators made of multiple pieces
of piezoelectric ceramic plates mechanically connected in series and electrically connected
in parallel are usually chosen in actual vibration systems.
2.2.2 Magnetostrictive Actuators

In the 1960/1970s, the technology of PZT was not yet fully understood. In addition, a mag-
netostrictive actuator was the best choice for vibration devices. It applies a ripple voltage
to the electromagnetic coil, and its electromagnetic force is used to cause the moving core
to vibrate. Although electromagnetic vibrators are a reliable source of vibration, the major
drawback is low energy efficiency caused by high electrical eddy current losses. These elec-
trical losses are transformed into heat and may damage the vibrator [3]. Therefore, with
electromagnetic vibrators, the cooling issue always needs to be considered, leading to bulky
size [4]. Currently, magnetostrictive actuators are usually applied in vibration systems that
require low vibration frequency and large vibration amplitude.
2.3 Transmission Mechanisms

Two mechanisms using either flexure hinges or an ultrasonic horn are mainly chosen in
the design of vibration system. The history of flexure hinge structures can be traced to the
1960s. With the development of the aerospace and aviation sectors, the resolution and size
of the support in order to achieve small deflection ranges no longer met the requirements.
After exploring various types of elastic support tests, engineers gradually developed flexure
hinges that were characterized by high resolution, small volume, and no mechanical fric-
tion or gaps. Currently, many types of flexure hinges, including circular, elliptical, square
fillet, and single-notch profiles, have been developed and are widely used to guide the dis-
placement of vibration in the nonresonant vibration-assisted systems. In addition, they are
often integrated into a double parallel or parallel four bar linkage so as to reduce the cou-
pling motion because the nonresonant vibration stage is usually designed to work in two or
more dimensions [5]. The ultrasonic horn is an important component of a resonant vibra-
tion system. It is used to transmit the mechanical energy converted from electrical energy
into the workpiece by the transducer. It is a stage of the mechanical amplification of the
power ultrasonic amplitude to improve the ultrasonic processing efficiency.
2.4 Drive and Control
Besides accuracy in the processing and assembly of mechanical components, the control
strategy used also has a great influence on the motion accuracy of the vibration system.
Generally, these control systems can be divided as open or closed loop systems. Because of
the high working frequency involved, an open loop system is the first choice for a resonant
vibration system. To build a proper open loop control system for a piezoelectric actuator, a
mathematical model of the piezo-driven stage needs to be built first because of its hysteresis
and nonlinear and creep properties. Many methods taking into account the intrinsic mech-
anism and dynamic properties of piezoelectric actuators have been developed in recent
decades, including the Preisach, Maxwell, and Prandtl–Ishlinskii models [6–8]. The ref-
erence is calculated from the corresponding control signal according to the reference input
value and it is sent to the piezoelectric actuator through a piezoelectric amplifier to generate
a corresponding displacement. The features of open loop control system include a simple
structure and ease of implementation; however, when the object or control device is dis-
turbed or the characteristic parameters change during the working process, error cannot
be compensated because this affects the accuracy of control. To overcome this drawback,
closed loop control systems are used. Close loop control systems are mainly used in non-
resonant vibration systems, and an industry standard controller is required. The traditional
proportion integration differentiation (PID) control method algorithm has high control pre-
cision, but it is not suitable for uncertain time-varying systems. In contrast, fuzzy adaptive
PID control can effectively identify the mathematical model of the controlled object, adjust
the parameters and structure of the controller in real time according to the given perfor-
mance indicators, and reduce the output error at this stage.
2.5 Vibration-Assisted Machining Systems

2.5.1 Resonant Vibration Systems
As a technology that has been successfully applied commercially, resonant vibration-
assisted machining systems work at the natural frequency of the system and apply the
excitation vibration principle to increase the amplitude of vibration. A typical design for
an ultrasonic vibration-assisted machining system is the resonant rod type, which consists
of three parts, namely, an ultrasonic transducer, an acoustic waveguide booster, and a
Transducer Booster Figure 2.1 Typical design of a resonant vibrator.

Hom
Node point
Tool or workpiece
vibration
horn (see Figure 2.1). In some research, the acoustic waveguide booster and horn are also
called a sonotrode because the functions of the two components are quite similar [9]. The
ultrasonic transducer is the source of vibration for the whole system and converts electrical
energy into mechanical motion in longitudinal or compressive mode under self-excited
vibration [10]. Two types of electromagnetic and piezoelectric transducers are widely
used and were introduced in Sections 2.2.1 and 2.2.2. The high-frequency, low-amplitude
reciprocating harmonic vibration is generated by the ultrasonic transducer and amplified
by the sonotrode to the desired location of a tool or workpiece. The sonotrode works
by resonating with the transducer, and there are strict requirements for its design and
manufacturing. Poor design or fabrication will decrease the energy efficiency, reduce the
cutting performance and vibration system durability, and may even cause serious damage
to the transducer [11–14]. The cutting tool or workpiece is attached to the end of the horn
to obtain the desired vibration. Moreover, the hold point of the whole system is usually
set at the node point with zero displacement in order to maintain its stability and reduce
energy loss. According to the direction of movement, a resonant vibration system can be
divided into three groups: 1D, 2D, and 3D systems.
2.5.1.1 1D System
1D ultrasonic vibration-assisted machining systems are the most common type because
of their simple structure and ease of implementation. They can be divided into resonant
rod and resonant tool types. Many researchers have proposed their own rod type vibrators.
Zhong et al. [15] improved the typical resonant rod-type system and applied it to the turn-
ing process, as shown in Figure 2.2. A tool holder with a notch structure is introduced into
the design to hold the tool firmly in place and to reduce its moving in the other degrees
of freedom. During the machining process, bending occurs at the notch point to prevent
deformation in the rest of the tool holder. Otherwise, the tool holder in proximity to a
parabolic shape will affect the machining performance. To obtain a more accurate measure-
ment of cutting force during the vibration-assisted milling process, a special clamp system
was designed by Shen et al. [16] by integrating the clamping system and a dynamometer, as
shown in Figure 2.3. The results showed that the impact of ultrasonic vibrations on mea-
surement results is reduced effectively. Similarly, Liu et al. [17–19] studied ductile mode
cutting with tungsten carbide, as shown in Figure 2.4. The new clamping system fixes the
vibrator using four bolts, which simplifies the installation procedure of the vibrator and
improves its accuracy.
Figure 2.2 Vibrator proposed by Zhong et al. Source:

Zhong et al. [15].
Piezoelectric
transducer
x
y
Cutting
insert Tool
Notch
holder
Figure 2.3 Vibrator proposed by Shen et al. Source:

Shen et al. [16]. © 2012, Springer Nature.
Figure 2.4 Vibrator proposed by Liu et al.

Sources: Liu et al. [17–19].
PZT
PZT holder
Tool insert
The resonant rod-type 1D resonant vibration system has a simple structure and high reli-
ability; however, the resonance frequency of the system can be easily influenced when a
workpiece or a large mass is attached to the horn. Meanwhile, the issue of the installation
of oversized part is also difficult to solve. Therefore, another type of resonant vibration sys-
tem named resonant tool was developed by integrating the resources of vibration into the
tool holder. A typical design was proposed by Ostasevicius et al. [20]. The milling cutter
assembly is driven by piezoceramic rings that are fixed into a standard Weldon tool holder
and generate resonant tool movement in the vertical direction. Similarly, Alam et al. [21]
Figure 2.5 Vibrator proposed by Alam et al. Source:

Alam et al. [21]. © 2011, Elsevier.
Ultrasonic
assembly
Horn
Drill
Bone
Dynamometer
improved the tool cutting assembly design and obtained a sevenfold increase in vibration
amplitude by using a stepped shape of horn structure (Figure 2.5).
As discussed in the previous sections, the vibration parameters for an ultrasonic vibra-
tion system largely depend on the dimensions and cross-sectional shape of the designed
vibration transmission mechanism consisting of the booster and horn. However, the tra-
ditional approach is based on the application of differential equations where the equilib-
rium of an infinitesimal element is taken into consideration under the influence of elas-
tic and inertia forces. This is time-consuming and inaccurate. To overcome these draw-
backs, FEA is introduced at the design stage of the ultrasonic vibration system, and its
use can increase the accuracy of the vibration system, such as in natural frequency and
the dimensions of the mechanism, which speeds up the development of vibration devices.
Kuo [22] proposed a milling cutter assembly design where the process of harmonic piezo-
electric vibrations was simulated by an FEA dynamic simulation, which optimized the key
dimensions, reduced the influence of stress concentration on the system, and increased
its system efficiency. However, the simulation did not consider a situation where a tool is
attached to the horn, and this leads to a deviation in the system’s natural frequency and
vibration amplitude between the simulation results and operational results. Roy et al. [23]
developed a circular hollow ultrasonic horn for milling cutter assembly and optimized its
outline and cross-sectional shape by using FEA. Compared with conventional ultrasonic
horn designs, such as those with stepped, conical, and exponential shapes, the circular
hollow ultrasonic horn achieved a higher magnification factor and lower axial, radial, and
shear stress, hence improving the system performance and reducing the influence of stress
concentration.
A different type of vibration drilling tool assembly design was proposed by Babitsky et al.
(Figure 2.6). In order to accomplish vibration-assisted drilling, one side of the assembly
was clamped in the three-jaw chunk of the lathe through the intermediate bush and ener-
gized by means of a slip ring assembly fitted to the hollow shaft of the lathe at the end
remote from the chuck [24]. A further 1D ultrasonic vibration system was developed by
Hsu et al. [25], and its working principle is quite similar to the ultrasonic bath. As shown
in Figure 2.7, three commercial Langevin ultrasonic transducers were fixed underneath the
vibration stage and were controlled by the same type of signals, generating vibrations at the
same frequency and phase. As a result, resonance vibration can be obtained in the vibration
stage.
Figure 2.6 Vibrator proposed by Babitsky et al. Source: Babitsky et al. [24]. © 2007, Elsevier.
Face milling cutter
Workpiece
Ultrasonic
Dynamometer
Figure 2.7 Vibrator proposed by Hsu et al. Source: Hsu et al. [25]. © 2007, Springer Nature.
2.5.1.2 2D and 3D Systems

Ultrasonic elliptical vibration-assisted machining is also named 2D ultrasonic vibration-
assisted machining and has received widespread attention since it was first proposed in
1993. Compared to 1D systems, 2D systems can obtain better cutting performance and also
require a higher standard of vibration devices. Because of its simple structure and ease of
implementation, the integrated resonant rod device is the most popular in proposed 2D
vibration devices. There are two main designs: patch and sandwich types. For the patch
type of integrated resonant rod, two sets of piezoelectric plates are attached to the outer wall
of the resonant rod to achieve the same or different modes of resonance. In the sandwich
type, two modes of resonance moment can be obtained by adding another set of piezoelec-
tric rings to the 1D resonant rod. Moriwaki et al. [26] developed a 2D patch-type ultrasonic
vibration-assisted turning system (Figure 2.8) by attaching two pairs of piezoelectric actu-
ators symmetrically in the center of the four sides of a stepped horn. When two sinusoidal
signals with different phases and the same frequency are applied to the piezoelectric actu-
ators, a bending resonance state can be obtained in the vibrator in two mutually perpen-
dicular directions simultaneously and the cutting tool attached at the end of the vibrator
vibrates in elliptical mode. The coupling effect cannot be avoided, as the piezoelectric plates
are placed in parallel. Shamoto et al. [27, 28] optimized the dimensions and shape of the
vibration rod and developed a control system to achieve a more accurate tool trajectory.
Experiments were conducted on hardened stainless steel and the machining accuracy and
Supporting points PZT’s Figure 2.8 Vibrator proposed by Moriwaki

Rake face
(nodal points of vibration) and Shamoto. (a) Ultrasonic vibrator and (b)
Tool tip first resonant mode of bending. Source:
Moriwaki and Shamoto [26].
ϕ15
Flank face
(a)
(b)
Figure 2.9 Vibrator proposed by Liang et al.

Diamond wheel Sources: Liang et al. and Peng et al. [29–31].
Spindle
Elliptic vibration
Workpiece
Ultrasonic
genernator
PZT vibrator
Worktable
machined surface quality improved. A combination of bending and longitudinal vibration

modes was also achieved in a different 2D patch-type ultrasonic vibrator designed by Liang
et al. [29–31], as shown in Figure 2.9. The piezoelectric plates are bonded at the same side
of the metal elastic body, and the workpiece is fixed to the top of the vibrator. However,
the vibration amplitudes involved are quite small, only up to 0.4 μm, because of the mass
issues with the vibrator, and an effect of workpiece mass on the vibration’s amplitude and
frequency is unavoidable.
A typical sandwich-type elliptical ultrasonic vibrator is shown in Figure 2.10 [32, 33].
The two groups of piezoelectric rings are sandwiched together in the drilling cutter assem-
bly, and each group works at a different resonant mode to generate an elliptical tool tip
trajectory. It should be noted that the installation point of this type of vibration device
should be set at the coincidence point of the two resonance modes. Meanwhile, vibrators
for mounting workpiece or nonrotating tools such as for turning and polishing have also
been proposed using the same design principle [34–39]. To achieve better performance,
Börner et al. [40] developed a cross-shaped converter for a 2D ultrasonic vibration-assisted
vibrator (Figure 2.11) and applied it to the milling process. As the high-frequency vibration
Vision Diamond The second group

The first group
microscope core drill PZT’s
X PZT’s
a Back
Z cylinder
b
(a) Y Front cylinder Direction of polarization of PZT
Ellipictal vibration locus Resonant mode of bending vibration in Y direction
A point
B point
Inner
locus
Outer
locus Supporting points (nodes)
Resonant mode of bending vibration in X direction
(c) (b)
Figure 2.10 Vibrator proposed by Liu et al. Source: Liu et al. [32].
Bolt for Electrodes Sample Movement of

preload the converter
Mass Mass
Cross-converter for shifting
Piezoelectric discs Mounting the vibration direction
Transducer
Figure 2.11 Vibrator proposed by Börner et al. Source: Börner et al. [40].
is transmitted to the cross-converter, the extension in horizontal direction will lead to a

compression in the vertical direction. However, only small specimens can be used because
those with large mass may influence the resonance frequency of the vibrator. Tan et al. [41]
built a symmetrically structured ultrasonic elliptical vibration-assisted device (Figure 2.12)
using four pairs of piezoelectric rings. The node point of the device is naturally set at the
center of the flange, which is easy to locate. The device is fixed by two pairs of grip holes
displaced on both sides of the flange to reduce the energy loss and improve the cooling
performance. Compared with the conventional design, the symmetrical structure can com-
pletely balance the internal force, which dramatically reduces error in the vibrator’s motion.
By changing the piezoelectric actuator to a magnetostrictive actuator, Suzuki et al. [42–44]
developed an elliptical vibrating polisher (Figure 2.13) and successfully applied it to pro-
cess micro-aspheric lenses and tungsten carbide die/mold. The vibrator is based on a giant
magnetostrictive material and with coils wound around it. It has four legs and each leg can
be independently controlled for expansion or contraction. The elliptical tool trajectory can
be obtained by appropriately setting the phase difference of the two pairs of opposing coils.
Bending PZT Longitudinal PZT

(a) Round nut ceramics ceramics Vertical direction
Ultrasonic horn Cutting tool
Z
X
O
Y
Elliptical locus Axial direction

Balanced tool Flange Electrode plates
Grip holes
(b)
– + + – – + + –
+ – +
–
(c) 6th resonant mode of bending vibration 3rd resonant mode of longitudinal vibration
Fixed node
Figure 2.12 Vibrator proposed by Tan et al. Source: Tan et al. [41].
Figure 2.13 Vibrator proposed by Suzuki et al.

Sources: Guo et al. [42–44].
Polishing
tool
Magnetostrictive Wire diameter:

vibrator Φ0.16 mm
Turns:
200
Coils
To obtain the highest vibration amplitude, the vibrator is designed to work at a frequency
of 9.2 kHz, which limits its processing performance.
Different from integrated resonant mode vibration devices, a separate type of 2D reso-
nant vibrators uses two independent Langevin vibrators placed in a V or L shape to obtain
a two-dimensional vibration of the tool or workpiece [45–47]. Figure 2.14 shows a typi-
cal separate 2D V-shaped vibrator proposed by Guo et al. [48]. The two Langevin vibrators
are set at an angle of 60∘ to generate a unique tool tip trajectory. The head block is a flex-
ure structure applied at the end of each vibrator to guide motion and reduce movement
error. Each individual vibrator has an added end mass to preload the piezoelectric rings
and adjust the natural frequency of the vibrators. A similar design was also applied in a
vibration-assisted polishing process [49]. Yan et al. [50] developed a 2D “L”-shaped reso-
nant vibrator for grinding, as shown in Figure 2.15. Two independent 1D resonant vibrators
are placed perpendicularly on the sides of the vibrating stage. However, this type of 2D
Figure 2.14 Vibrator proposed by Guo PZT rings Tangential

and Ehmann. Source: Guo and Ehmann direction
[48].
Normal
direction
Insert
End mass
Head block
(fiexure)
PZT rings
Base block
Figure 2.15 Vibrator proposed Ultrasonic generator

Yanyan et al. Source: Yanyan et al. [50]. Transducer
Shoe plate Horn

Transducer
Horn
Locating y
plate x
Ultrasonic generator
Locating plate
resonant vibrator is almost impossible to integrate into a rotating tool such as in a milling
cutter assembly, which limits its application.
In order to obtain complex geometrical shapes, the milling process requires a feed vector
in arbitrary directions with both vertical and horizontal components of feed vector neces-
sary for 3D end milling. Hence, there is a need for three-dimensional vibration assistance.
Figure 2.16 shows a three degree of freedom resonant vibration tool, which can generate
longitudinal and two bending resonance mode vibrations by adding three sets of piezoelec-
tric actuators to a resonance rod [51–53]. The difficulty associated with this design is to
accurately locate the node point of the vibration rod and to achieve three modes of reso-
nance frequencies, which are as close as possible to obtain sufficient vibration amplitude.
Moreover, the cross talk between the three resonance modes is much more prominent than
that in a 2D resonant system. In order to reduce the motion coupling effect, stepped and
the tapered portions are added to the resonant rod, and the overall shape and dimensions
are optimized so as to obtaining optimal performance.
2.5.2 Nonresonant Vibration System

Generally, resonant vibration systems are capable of achieving extremely high operating fre-
quencies, and most of them can even reach ultrasonic vibration frequency levels (≥20 kHz).
Tapered
portion
PZT’s for
bending vibration
PZT sensors PZT’s for
longitudinal vibration
Supported Rake face

Z Slepped
positions portion Elliptical
C
vibration
U V
Longitudinal
mode
Bending mode Tapered

portion
Tool tip Workpiece
Machined region of angle
Figure 2.16 Layout of the 3D vibrator. Sources: Suzuki et al. and Shamoto et al. [51–53].
However, their limitations of fixed working frequency and vibration motion parameters,
heat dissipation, open loop control, and poor dynamic accuracy are also quite obvious. In
addition, the performance of the vibrator heavily relies on the dynamic characteristic of
the vibration horn, which increases the difficulty of vibrator design. To overcome these
shortcomings in the resonant vibrator, much more attention has been paid to nonreso-
nant vibration systems. Nonresonant systems apply forced vibration rather than excitation
vibration as the design principle and produce variable vibration frequencies. However, it
is hard to achieve a high working frequency, which is always less than the natural fre-
quency, because of the issue of structural stiffness. Many of these designs are inspired by
high-precision micro/nano-positioning stages [54, 55], which are discussed below.
The working principle of a nonresonant vibration system can be explained by the
schematic diagram in Figure 2.17. The whole system is driven directly by the preloaded
piezoelectric actuator. In order to accurately transmit motion and reduce parasitic
movement, flexure mechanisms (flexure hinges), which can be simplified into a set of
spring–damper mechanisms, as shown in Figure 2.17, are always chosen as the linkage
between the actuators and end executor. A displacement amplifying mechanism can be
integrated into the flexure hinges if the amplitude is required larger than the displacement
generated by the piezoelectric actuator. Moreover, decoupling issues also need to be
Figure 2.17 Typical working diagram of a

nonresonant vibrator.
actuator
Piezo
End executor
Figure 2.18 Vibrator proposed by Greco et al. t

Source: Hong and Ehmann [57]. d
l H θ
x
R
τ4
b
z h
considered in the design phase of the flexure mechanism when multidimensional motions
are required.
Compared with resonant vibration systems, higher motion accuracy can usually be
achieved with a nonresonant vibration system because of its inherent merits. This makes
it more suitable for the manufacturing of microstructured surfaces. Therefore, 1D nonres-
onant vibration systems are quite rare because of the complex tool trajectories required in
producing unique surface microstructures. A typical design of a 1D nonresonant vibration
system uses a combination of a parallel four-bar flexure hinge structure and a piezoelectric
actuator [56]. Figure 2.18 shows a 1D nonresonant vibration system design proposed by
Long et al. [57, 58]. A single piezoelectric actuator is positioned at the parallel four-bar
flexure hinge structure, which also includes a vibration displacement amplification func-
tion. The structural layout and closed loop control system ensure high motion precision. A
different design was proposed by Suzuki et al. [59] with a complex mechanical structure.
To ensure cutting accuracy, this vibrator aims to achieve high axial mechanical stiffness so
as to reduce elastic deflection during the machining process. A cylindrical roller bearing is
set between the cutting tool and piezoelectric actuators to guide the vibration and support
the bending force acting on the cutter insert, and the twisting force in the machining
process is supported by the pin. Moreover, bending stress is further reduced because of the
flexible tip. Consequently, the shear stress that could damage them cannot be transmitted
to the piezoelectric actuators. Because the output voltage for the control of the vibrator
may reach up to 1000 V, an air-cooling system is also integrated into the vibrator to prevent
overheating.
2.5.2.1 2D System
Compared with 1D systems, the application of 2D systems is more flexible. However, the
coupling effect between the two vibration directions has a great impact on its accuracy.
Two configurations exist among the proposed designs, a vibration tool mainly for turning
and a vibration stage mainly for milling. A flexure hinge structure is often used to guide
motion and to reduce motion error and the coupling effect, although some other designs
have also been reported. Brehl et al. [60–62] developed two types of nonresonant ellipti-
cal vibration-assisted machining devices at two different working frequencies. One of these
vibrators (Figure 2.19a) works at a high vibration frequency of up to 4.5 kHz but a low vibra-
tion amplitude of less than 2 μm and requires a cooling chamber to prevent the vibrator
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MARY,
OR,
"SHE MADE ME DO IT."
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stranger in town."
"Go and make acquaintance with her, Helen," said Cora

Hart, with a sneer. "She is so fashionably dressed and so
elegant looking, she would be a nice companion for you."
"See, she is coming this way!" exclaimed Jane Marvin.

"I do believe she means to speak to us. Let us have some
fun with her, girls!"
"I will have nothing to do with such fun as yours!" said

Helen Arnold. "I think it is downright mean and wicked to
make sport of poor and old people. Mary Willis, you had
better come home with me and be out of mischief."
"Oh, of course she will!" said Jane, scornfully. "I should

like to know why Mary Willis is to do as you say. You are
neither her teacher nor her mistress, if you are the
teacher's niece, and the oldest girl in the class. You stay
here, Mary, and show her that you won't be ordered about
by her."
"Come, Mary," said Helen, again. "You had much better
be going home with me."
Helen spoke rather sharply, and Mary was vexed.
"I shall go home when I please, Miss Helen Arnold! I will

thank you to mind your own business and let me alone."
"Very well!" replied Helen, and she walked away, feeling

both grieved and angry.
She was very fond of Mary Willis, though Mary was

much younger than herself. She helped her in her lessons,
dressed her dolls for her, and taught her how to make
pretty things for them; for Helen was very quick and skilful
in all sorts of work. She was anxious that Mary should be a
good girl and a good scholar, and she did keep her out of a
great deal of trouble which Mary's quick temper and
readiness to be led away would have brought upon her.
For a long time Mary had loved Helen more than any
one else in the world, except her own mother. But when
Jane Marvin came to school, she began, as she said, to put
Mary up to be jealous of Helen. She told Mary that Helen
did not really care for her, and that she only wished to
govern and patronize her, and so show off her own
goodness. She laughed at Helen's plain cheap dress and
what she called her old-fashioned strict ways, and she told
Mary that Helen was a regular little Methodist, and wanted
to make her so. Jenny had never been in a Methodist
church in her life, and knew nothing about them, but she
had heard her father call people Methodists who were
religious and strict in their conduct.
It may seem strange that Mary should listen to such

talk against her best friend; but Mary had a great idea of
being independent and having her own way: and like many
other people of the same sort, she was ready to be made a
fool of by any one who would take the trouble to flatter her.
This Helen never did; nay, I am afraid that in her desire to
be honest, she sometimes went to the other extreme and
found fault with Mary when there was no reason for doing
so.
If Mary had thought a little, she would have seen one

great difference between Helen and Jenny. Helen never
asked her for anything, and if she happened to borrow a
sheet of paper or a steel pen from Mary's store, she was
always careful to return it. Mary's mother was rich and Mary
had a great abundance of pretty and useful things. She
would often have liked to divide with Helen, for she had a
generous disposition: but except at Christmas and on her
birthday, Helen would never accept presents.
Jenny was very different. She not only took all that was
offered her, but she had no scruple in begging for anything
in Mary's desk or play-room to which she took a fancy.
Mary's paper and pens, Mary's thread and needles, Mary's
lunch basket, she used as if they were her own, and she
had already got possession of some of Mary's prettiest and
most expensive toys. Still Mary could see no fault in her
new friend, and she was very much vexed at her mother
because she would hardly ever ask Jenny to tea, and would
never let her go to Mr. Marvin's to stay all night.
Helen went away feeling very much hurt, and Mary

stayed with Jenny and the other girls. The poor woman
walked slowly toward them, and when she came opposite
she crossed over the street to speak to them.
"Can any of you tell me where Mrs. Willis lives?" she

asked, in a sweet, pleasant voice. Her clothes were old-
fashioned and worn but she looked and spoke like a lady.
Mary was starting forward to answer her, when Cora
Hart pulled her back and at the same moment Jane
answered, glibly:
"Mrs. Willis? Yes, ma'am, she lives in that white house

up there on the hill," pointing as she spoke to a farm-house
which stood about half a mile away upon the side of a steep
hill.
"Does she live as far from the village as that?" asked

the stranger. "I thought her house was quite near the
church."
"She did live near the church," said Jane; "but her
house was burned down, and she moved away up there. It
is a beautiful place, but rather far-away from the village,
and the hill is pretty steep. If you go round the corner by
that yellow building with the stairs outside you will be in the
road."
The lady thanked her for her information and turned

into the street which Jenny had pointed out as leading up to
the white house.
"There she goes, trudging along with her bundles," said

Jenny, bursting out laughing. "I hope she will find Mrs. Willis
at home!"
"Mary was for telling her right away," said Cora. "She
would have spoiled all the fun, if I had not stopped her.
What do you suppose she wants with your mother, Mary?"
"I dare say she is some beggar woman," answered

Mary. "We have heaps of them coming to our house all the
time."
"I should think you did!" said Jane, scornfully. "I don't
see how your mother can encourage them. I do despise
beggars!"
"Two of a trade never agree, oh, Martha?" said Cora.
"What do you mean by that, Cora Hart?" asked Jane,

angrily.
But Cora only laughed scornfully and did not answer.
"Well, for my part, I think it was a real mean trick!" said

Julia Davis. "Sending the poor woman all the way up that
steep hill to an empty house. I wish I had just told her the
truth."
"Why didn't you, then?" asked Jane. "Nobody hindered

you, Miss Tell-tale; only just let me catch you getting me
into a scrape, that's all!"
Julia turned away and went into her own gate without
saying a word. She felt very much ashamed of herself, for
she knew she had been a coward—she had been afraid to
do what she knew was right.
Mary also felt uneasy as she went home. She had been
taught her duty towards her neighbor, and she knew right
from wrong. All the time she was eating her nice supper,
she thought of the poor woman with her little lame boy
toiling up the steep road only to find an empty house.
"Helen Arnold would never have done such a thing!" she

said to herself, and she wished over and over again that she
had gone with Helen.
It was a pleasant, dry evening, and after tea, Mrs. Willis

took her work and sat down in a garden chair on the green
lawn. Mary stood by her with her doll, but she did not feel
like playing or talking. Her conscience troubled her more
and more, and she felt very unhappy.
"I have some good news for you, Mary," said Mrs. Willis.
"You have often heard me talk of your godmother, your
father's sister, who married a missionary and went away to
China."
"O yes, mamma!" replied Mary. "I have always wished

to see her so much. You know she sent me that beautiful
box of shells and curiosities."
And then Mary sighed as she thought how Jenny had

begged from her some of the prettiest things, and rarest
curiosities in the box.
"Well, my love, I think you will see her very soon. I had
a letter from her this afternoon, in which she says she
expects to sail the next week for America and will come
directly to us. She has a little boy who is lame, and she is
bringing him home to see if he can be cured."
"How glad I am!" said Mary. "I have so often looked at

her picture—the one you said she painted herself, when she
was teaching you to draw, mamma—and wished she would
come home."
Mrs. Willis smiled and sighed. "You must not expect to

see Aunt Mary looking like her picture," said she. "It was
painted long ago, when we were both young, and she has
been through a great deal since then."
Mrs. Willis sighed, and looked down on the ground. She

was thinking of all that had happened since she had seen
her dear sister—how she had lost her husband and all her
children but Mary. Mary stood leaning on her mother's chair
without speaking, till the sound of the opening gate caused
her to look up. There, coming in at the gate, was that very
poor woman and her little boy, whom Jane Marvin and her
companions had sent "on a fool's errand," as the saying is,
to the empty farm-house on the hill.
Before she could make up her mind what to do, Mrs.

Willis looked up also, and met the gaze of the stranger. With
a scream of delight, she started up and flew to meet her,
kissing her and calling her, her dear, precious sister, her
darling sister Mary!
"Come here, Mary, and see your aunt," said her mother,
turning round.
Mary came forward. She trembled so that she could

hardly stand, and she was very pale, but her mother did not
notice her confusion.
"Now run into the house and tell Jane to get tea ready
as quickly as she can," said. Mrs. Willis. "And this is my
nephew and god-son Willie. But how lame he is, poor little
fellow!"
"He is not always so lame," said Mrs. Lee. "But we have

had a long and hard walk and he is almost tired to death. I
am afraid he will not be able to stir to-morrow."
"I did not expect you till next week, at the earliest,"
said Mrs. Willis. "But how do you come to arrive at this time
of day? The train has been in for two hours."
"I will tell you all about it, presently," replied Mrs. Lee.
"Just now I am anxious to find a resting-place for Willie,
who, I fear, is suffering very much from his knee."
"It does ache!" said poor Willie. "It always hurts me to
go up-hill."
"Well, you shall soon rest it, my dear boy," said Mrs.
Willis. "Where are your trunks, Mary?"
"They will be here to-morrow, I suppose," replied her

sister. "They were left behind by some mistake, but I have
telegraphed and heard that they are safe and on the way. I
must keep out of sight till they come, for between dust and
the long journey I look more like a beggar than a lady."
The travellers were soon washed and brushed and sat

down to a bountiful meal.
"And now tell me how you came here at this time in the
evening?" said Mrs. Willis.
"The matter is easily enough explained," replied Mrs.

Lee. "We sailed a week sooner than we expected and had
an uncommonly short voyage. We came directly on from
Boston, and arrived on the train at four o'clock."
"But where have you been since?" asked Mrs. Willis.
"I am sorry to say we have been made the victims of a

most malicious trick," replied Mrs. Lee. "There were no
carriages at the station, and I thought I could find your
house easily enough. Meeting a party of school-girls, I
asked them the way, that I might be sure, and was told by
one of them that your house had been burned and that you
had moved into a house which she pointed out on the hill-
side. If I had known how far it was I should have gone to
the hotel, where, of course, I should have been set right."
"But as it was, Willie and I toiled all the way up the

steep rough road to find ourselves at an empty, deserted
and half-ruined house. Poor Willie broke down entirely, and
I felt very much like doing the same, for we had travelled all
night and were tired out. I hardly know what we should
have done but for a good-natured teamster, who stopped to
water his horses at the trough by the gate. I made some
inquiries of him, and he not only set me right, but insisted
on bringing me back to the village. Poor Willie is quite
discouraged at his first experience in a Christian land, and
wants to go back to China."
"I do not wonder!" said Mrs. Willis.
"I did not believe there was a girl in the village who
would do such a wicked thing. Who do you suppose it could
have been, Mary?"
Mary, in her corner of the sofa, murmured something,

she hardly knew what. She was wishing that she were in
China, or anywhere else out of the sight of her mother and
aunt. Oh, if she had only gone with Helen! If she had only
been brave enough to defy Jane, and set her aunt right! But
there was no use in wishing.
"She was a short and rather dark girl, with a great deal
of curling black hair, and bold black eyes," said Mrs. Lee.
"There were several others with her, but I did not notice
them so as to be able to know them again."
"It must have been Jane Marvin, I am sure," said Mrs.

Willis.
She turned to Mary as she spoke, and observed her

confusion. Could it be that her daughter had been engaged
in the trick? At that moment her attention was diverted by
poor Willie, who had been trying in vain to eat, and who
now, overcome by fatigue and pain, fainted away in his
chair. Mrs. Willis saw him carried up-stairs and made as
comfortable as he could be, and then returned to the parlor,
where Mary was curled up in the corner of the sofa, crying
as if her heart would break.
"Now, Mary, I want to know all that you can tell me

about this matter!" said Mrs. Willis, seating herself by Mary.
"Tell me the whole truth."
Sobbing so that she could hardly speak, Mary told her

mother the story.
"It is one of the most shameful things I ever heard of!"

said Mrs. Willis. "How could you join in such a piece of
wickedness?"
"I did not say anything, mamma," sobbed Mary.
"No, but by your silence, you consented to what Jane

said, when you might have prevented all this trouble by
speaking."
"I was going to tell Aunt Mary at first, but the girls
pulled me back and would not let me," said Mary, hanging
her head.
"Would not let you!" repeated Mrs. Willis. "How did they
hinder you?"
Mary had no answer ready, and her mother continued:
"Where was Helen Arnold? I should have expected

something better of her."
"She was not there, mamma," replied Mary, eagerly.

"She went away before Jane began. She wanted me to go
with her, but I was vexed and would not. Oh, if I had only
minded her!"
"If you had only minded your own conscience and your
own sense of what was right, you would not have needed
Helen to keep you out or mischief," said Mrs. Willis. "If you
had had one thought of doing as you would be done by, you
would not have allowed a wicked, silly girl to send your aunt
and your poor lame cousin Willie on such an errand."
"I did not know she was my aunt," said Mary.
"That makes no difference, Mary. You knew she was a

woman with a child, and the fact that you thought you were
playing a trick upon a poor person makes your fault worse
instead of better. Nor do I think you mend the matter by
saying that you did not speak a word. You ought to have
spoken, especially when the woman inquired for your own
mother."
"I know it was wicked and mean, mamma," said Mary.

"I have been sorry ever since. I wish Jane Marvin had never
come here!" she added, bursting into tears again. "She is
always making me do bad things and leading me into
mischief!"
"That is sheer nonsense, Mary. Jane could not make you

do anything you did not choose, nor lead you where you did
not choose to go. If you had been so very easily led, you
would have been governed by Helen, whom you have
known three times as long as you have known Jane, and
whom you have every reason to love and trust."
"You have done very wrong, Mary—very wrong,

indeed," continued Mrs. Willis, after a moment's silence. "I
cannot excuse what you have done by throwing the blame
on Jane. Every one of the party who allowed the cruel
imposition to go on was guilty of helping on the cheat. I
shall see that Miss Lyman is informed in the morning of the
way in which her pupils amuse themselves, and you must
expect to take your share of the blame. Now go to bed, and
when you say your prayers, ask God to forgive your mean
and cruel conduct."
"Won't you forgive me, and kiss me, mamma?" sobbed

Mary.
"When I see that you are sensible of your fault, Mary. At

present you seem inclined to throw the blame entirely upon
somebody else, and to think you are to be excused because
'somebody made you' do what you knew was wicked and
cruel."
Mary went away to bed crying bitterly. She had never

been so miserable in all her life. It was not the first time
she had been "made" by Jane to do wrong. She had done
things in Jane's company which she was both afraid and
ashamed to have her mother know; but she had always
excused herself by thinking they were all Jane's faults.
Now, as she thought about the matter, she saw how

useless and vain were all such excuses. If she was so easily
led, why had she not been governed by Helen, whom she
had known more years than she had known Jane months,
who was always ready to give up her own convenience for
her sake, and whom she had never known to do a mean
action? Why was she not as easily led to do right as to do
wrong?
Mary learned more about herself that wretched night

than she had ever known before. She had always known
that she was a sinner—now she felt it, which is quite a
different thing. She thought of all the wrong things she had
done lately—the whispering, and reading story-books in
prayer-time, the playing truant from school and lying to
conceal it—the mysterious private talks about things of
which she ought never to have thought; much less spoken—
the secrets kept from her mother, to whom she used to tell
everything. Mary no longer tried to excuse herself. She felt
her own wickedness, and with real repentance asked her
Heavenly rather to forgive her for Christ's sake. Then
feeling a little comforted, she went to sleep.
She was awakened in the night by her mother sending

for Doctor Arnold. Poor Willie was very ill—so ill that for
several days no one thought he would live. Oh, how
miserable Mary was! She could find no comfort except in
running up and down-stairs and waiting upon her aunt and
Willie. Dr. Arnold had been informed of the cause of Willie's
illness, and the next morning he came into school and told
Miss Lyman the whole story, before the minister and all the
scholars. All the girls concerned in the trick were obliged to
beg Aunt Mary's pardon, and were not allowed any recess
for the rest of the term.
Mr. Marvin took Jane out of school, and every one was
glad when she was gone, for nobody loved her, not even
those who had been the most ready to be governed by her.
I am glad to say, however, that Jane herself was sorry when
she found out how much harm she had done, and that she
had almost caused the death of poor Willie. She went of her
own accord and begged his pardon, when he was well
enough to see her, and she gladly spent hours in reading to
him and amusing him.
But she could not undo the mischief she had done. The
lame knee, which might perhaps have been made well, was
so strained and inflamed by the long rough walk that it
could not be cured, and Willie never walked again without
crutches.
Jane learned a great deal from the gentle little Christian
boy and his kind mother, and I hope she will grow up a
good, useful woman. I think, after all, there was more
excuse for her than for Mary. Jane had never known the
care and teaching of a good mother. Her mother died when
she was a little baby, and she had been brought up by
servants and by her father, who was a foolish and bad man.
She had always heard him laugh at the Bible as an old book
of fables, and at religious people as fools or knaves, and
she naturally took her notion from him.
Mary, on the contrary, had every pains taken with her.

She had been taught her duty towards God and her
neighbor, she had the kindest of mothers, of teachers, and
friends, who all tried to influence her for good.
Girls, when you are ready to excuse yourselves for

doing wrong by saying somebody "made you," think
whether your words are true, and whether if "somebody"
had tried to "make you" do right, you would have been as
easily led. Remember that God sees your heart, and He will
accept no false excuses; and while He is always ready to
give you His Holy Spirit to guide you, you have no right to
let any human being "make you" do wrong.
"My son, if sinners entice thee, consent thou not."

LOUISA, OR, "JUST ONE MINUTE." Frontispiece.
LOUISA,
OR,
"JUST ONE MINUTE!"
"COME, Louisa, are you ready? The car will be here

directly."
"In just one minute," replied Louisa, throwing down the

book she had taken up for "just one minute," while she was
getting ready for school, and hastening to put on her hat
and gloves.
But in that minute the street-car passed. There was not

another car for twelve minutes. Then the drawbridge was
raised for the passage of a ship, which made a delay of ten
minutes more.
The consequence of all these delays was, that though

they walked themselves out of breath, Louisa and her little
sister Anna were ten minutes too late for school, and poor
Anna got a bad mark for no fault of her own except her
good-nature in waiting for her sister.
Louisa was in many respects a good girl. She was

amiable, truthful, and very obliging, yet she made more
trouble and caused more disappointments than any other
person in the family. She was much brighter than her sister
Anna, and yet she "missed" in school three times to Anna's
one. Louisa was truthful, and yet she was not to be trusted:
she was obliging, yet she often disobliged those whom she
tried to help, and if she was not fretful herself, she was very
often the cause of fretfulness in others. All these seeming
contradictions are easily explained. The answer to the riddle
lay in Louisa's favorite phrase, "just one minute."
For instance. An important message was to go to papa's
office and there was nobody to carry it but Louisa. Aunt
Maria had written to say that she was coming to make a
visit and bring her baby; but the measles were prevailing in
D—, and as the baby was a delicate little thing it would not
do to have her exposed to the disease. Papa had gone to his
office in the city before Aunt Maria's letter came.
"I must write a note to papa and ask him to send a

telegraphic dispatch to auntie," said Mrs. Winter; "and you,
Louisa, must carry it, for Anna is not well enough to go out.
Now, can I depend upon you to go straight to papa's
office?"
"Yes, mamma, of course I will!"
Louisa meant what she said, and for once she was
ready for the car when it came along. But, unluckily, to
reach her father's office, she had to pass a toy shop, the
window of which almost always presented some new
attraction, and had many a time delayed Louisa. She did
not mean to stop this time, but only to look at the window
in passing. But behold, there was a grand new baby-house
with the most wonderful rosewood furniture, and such a
kitchen as was never seen in a dolls' house before; and
there was her school-mate Jennie Atridge, looking through
the glass.
"Oh, Louisa, just look here!" she exclaimed, as she saw

Louisa. "Just see what a splendid doll's house! Mamma has
promised me one for my birthday. I wonder if she will buy
this?"
"I have got a doll's house, but it is not furnished," said

Louisa, stopping "just a minute," to look in at the window.
"We are going to buy the furniture next week, if Anna gets
well enough to come into town. She has been sick two days
with a bad cold. I wonder if we could get such a stove as
that?"
"I would rather have a range," said Jennie. "See, there

is a nice one over in that corner."
The "just a minute" lengthened out into ten, while the

girls discussed the furniture, and when Louisa reached the
office she found her father had gone out.
"He has gone over to the South End," said the office-
boy, "and will not be back till noon. It is a pity you did not
come before, for he has not been gone more than five
minutes."
When Mr. Winter came back, he found his wife's note

and sent a message directly. But it was too late. Aunt Maria
had started, and arrived next day to find Anna broken out
with the measles, and another of the children coming down
with the same disease. The baby took it, of course, and was
so ill that its life was despaired of for many days.
Louisa was very sorry, and would gladly have done

anything for her aunt or for baby, but she could not undo
the mischief she had done by "just one minute's" delay.
One would have expected such a severe lesson to do

Louisa some good, but it did not. The truth was that Louisa
had not learned to see that she was in fault. She was
"unlucky," she thought: "it always happened so." She was
sure that she was always ready to do anything that was
wanted of her, and she could not understand why her
mother should go for baby's medicine herself, instead of
sending her, and why Aunt Maria would not let her put into
the post-office box the letter which carried the news that
baby was at last out of danger.
"Miss Louisa, will you watch these cakes for me while I
run out and pick the beans for dinner?" said Mary the cook,
one day.
The girls were going to have a party to celebrate Anna's

birthday, and Mary had been making and frosting some of
the most wonderful cakes in the world. The great table was
covered with cocoanut cake, and chocolate cake, and
almond cake, and Mary had just put into the oven a pan of
macaroons.
"The oven is rather hot, and you must watch it, or the
cakes will burn," said Mary. "Just as soon as they begin to
brown, open the oven door and leave it."
Louisa promised, as usual. She had already looked at

the cakes once or twice, and was just going to look again,
when she heard the express man's wagon stop at the gate.
"I do wonder what he has brought this time?" said

Louisa to herself. "I mean to run to the front door and see.
It will not take more than a minute."
Away she ran, leaving the outside door open, and the
oven door shut. The express man had brought a number of
parcels, some of them containing presents for Anna from
friends in the city, and of course, Louisa had to stop "just a
minute" to see them opened. Meantime a beggar woman
with a large basket came through the side gate and into the
kitchen. No one was there. Louis had deserted her post, and
Mary, supposing that she was watching the cakes, was
looking over the bean vines and gathering all the beans
which were fit to pickle. It was the work of a moment for
the woman to slip the cakes into her big basket and slip
away herself. When Louisa and Mary came back, both at the

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1.2 Vibration-Assisted Machining Process

1.2.1 Vibration-Assisted Milling

1.2.2 Vibration-Assisted Drilling

the high-efficiency and high-quality machining of various parts with difficult-to-machine

As shown in Figure 1.1, according to the direction of vibration, vibration-assisted drilling

Tool rotation direction Figure 1.1 Schematic of

Feed direction Axial vibration

Workpiece rotation direction