Qin Y. Micromanufacturing Engineering and Technology

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mechanisms, the forces applied to the motor,

and the forces generated by the motor. Using

these parameters, a second-order (massspring

damper) system is used to model the behavior of

the model.

Figure 22-22 also shows a diagram of the cho-

sen model, supposing that the central point of the

extender is fixed in space. The symbols are:

x1: displacement of brake 1

Fr: stiffness force of the extendi ng actuator

Fm: force generated by the extending actuator

Fd: damping force of the extending actuator

Fl: force of the applied load

V: voltage supply

By summing the forces shown in the diagram,

the equations of motion are derived according to

Newton’s second law:

¼ Fm  Fr  Fd  Fl

¼ Kv*V  Kr*x1  Kd*

dx1

 Fl

ð6Þ

The parameters in this equation are:

M: mass of brake

Kv: voltage constant of the piezoelectric effect

Kr: stiffness coefficient

Kd: damping coefficient

FIGURE 22-22 Modeling of an inchworm piezo-motor and cycle.

CHAPTER 22 Testing and Diagnosis for Micro-Manufacturing Systems 361

Assuming

dx1

¼ v1 and writing Eqn (6) in a

state-space form, provides:

dx1

dv1









ð7Þ

which represents the first of Eqns (5).

The second equation depends on the outputs

chosen to observe the system. In order to estimate,

for example, position, velocity, load force, the

stiffness and the voltage constant parameter, all

that is needed is the voltage supply V, an input to

the system, and the measurement of mover posi-

tion x1, an observation represented by the follow-

ing equation:

yx½¼10½



ð8Þ

FIGURE 22-23 Force of a load applied to the inchworm.

TABLE 22-5

Number of Measurements (=2) vs Number of Estimated Features (=5)

Estimated State Variables Estimated Parameters Measurement (sensors)

Position

Velocity

Drag force

x (m)

dx/dt (m/s)

Fl(N)

Stiffness

Piezo voltage

coeff.

(N/m)

Kv (N/V)

Voltage

Transducer

Linear encoder

362

CHAPTER 22 Testing and Diagnosis for Micro-Manufacturing Systems

FIGURE 22-24 Piezoelectric parameter variation.

time [s]

Kr [N/m]

x 10

Extended Mechanism’s Stiffness Estimation

Kr (estimated value)

Kr (nominal value)

0.8 0.2 0.4 0.6 0.8010

0.2 0.4 0.6

FIGURE 22-25 Stiffness estimation.

CHAPTER 22 Testing and Diagnosis for Micro-Manufacturing Systems 363

Figure 22-23 shows the applied load force

(dashed line) and its estimate (solid line). The

signal is modulated by the control signal on the

piezo-element. Figure 22-24 shows the estimation

of the piezoelectric coefficient (solid line) and its

true value (dashed line). The variation corre-

sponds to a fault which changes the coefficient

from 7 N/V to 5.25 N/V. As shown in the figure,

the convergence time to the second value is 0.3 s

because the mover is controlled by an ON/OFF

signal and the estimated curve tends to reach the

right value only when the control is ON. This

means that the system needs to be in an excited

state in order to enable possible fault detection.

Figure 22-25 shows the estimatio n of the stiff-

ness coefficient. The very low convergence time

(below 0.1 s) shows the high dynamic capa bility

of the Kalman estimator.

CONCLUSIONS

Testing for micro-manufacturing processes is

characterized by requirements on:

precision;

non-invasiveness;

being contactless;

minimization of the number of sensors;

low complexity.

Non-contact measurement techniques based

on laser sensors and precise movement systems

represent the best choice in order to achieve a

complete testing syst em based on direct measure-

ment. When a great number of signals have to be

acquired in order to improve the monitoring of

the normal working conditions of devices or com-

plex machines, model-based testing can be used to

reduce the number of direct measurements (with-

out a reduction of the number of interested sig-

nals).

Model-based testing and diagnosis for micro-

manufacturing may exploit advantages from the

application of estimations techniques in order to

extract features from modeled systems. The

reduction of the number of sensors is possible by

increasing the information flow from the tested

system. This information could be well repre-

sented by a proper system model. Estimation tech-

niques, such as the Kalman filters, can aid the

extraction of variables and parameters which can-

not be measured in a direct way, allowing a num-

ber of available features greater than the number

of signals measu red.

The application of model-based testing and

diagnostic procedures to the micro-/meso-devices

world represents a very efficient and powerful

way to increase qua lity and reliability, while at

the same time reducing system complexity.

ACKNOWLEDGMENTS

Special thanks to Professor Biagio Turchiano, Full

Professor of Automatic Control and Head of

Department, Dipartimento di Elettrotecnica ed

Elettronica, Politecnico di Bari, Italy, and to Pro-

fessor Mauro Onori, Associate Professor, Evolv-

able Production Systems Group, KTH, Stockholm,

Sweden. Thanks also to colleague Eng. Orlando

Petrone, Mechatronic Group leader of MASMEC

Research and Development Department. Many

issues included in this work are related to results

achieved by MASMEC Srl, Bari, Italy, for the

European Research Project MASMICRO.

REFERENCES

[1] J.S. Oakland, Statistical Process Control, Butter-

worth-Heinemann (1996).

[2] R. Isermann, Model-Based Fault Detection and Diag-

nosis – Status and Applications, IFAC (2004).

[3] Chen, C.H., Pau, L.F., Wang, P.S.P., (eds.), The

Handbook of Pattern Recognition and Computer

Vision (2nd Ed.), World Scientific Publishing Co

(1998) 207–248.

[4] D. Sarid, Scanning Force Microscopy, Oxford Series

in Optical and Imaging Sciences, Oxford University

Press, New York (1991).

[5] V. Venkatasubramanian, R. Rengaswamy, K. Yin,

S.N. Kavuri, A review of process fault detection and

diagnosis, Part I: Quantitative model-based methods,

Computers and Chemical Engineering 27 (2003)

293–311.

[6] L. Ljung, System Identification, Theory for the User,

(2nd ed)., Thomas, Kailath, (Ed.) Prentice Hall PTR

(1999).

[7] G. Bishop, G. Welch, An introduction to the Kalman

Filter, Technical report, Department of Computer

Science, University of North Carolina at Chapel Hill,

Chapel Hill, NC 27599-3175 (May 2003).

364 CHAPTER 22 Testing and Diagnosis for Micro-Manufacturing Systems

[8] Haykin, S., (ed.), Kalman Filtering and Neural Net-

works John Wiley & Sons, Inc (2001).

[9] P. Dewallef and O. L



eonard, On-line measurement

validation and performance monitoring using robust

Kalman filtering techniques, Turbomachinery Group

– University of Li



ege, Belgium.

[10] LaViola Jr., A comparison of unscented and

extended Kalman filtering for estimating quater-

nion motion, Joseph J. Brown University Technol-

ogy Center for Advanced Scientific Computing and

Visualization, PO Box 1910, Providence, RI,

02912, U SA.

[11] N. Hagood, W. Chung, A. von Flotow, Modelling of

piezoelectric actuator dynamics for active structural

control, J. Intell. Mat., Syst., and Struct 1 (July 1990)

327–354.

CHAPTER 22 Testing and Diagnosis for Micro-Manufacturing Systems 365

Micro-Mechanics Modeling

for Micro-Forming Processes

W. Zhuang, J. Cao, S. Wang an d Jianguo Lin

INTRODUCTION

In recent decades, the development towards min-

iaturization of products and devices in industries

such as electr onics, optics, communications, etc.

has increased the demand for metallic parts man-

ufactured at micro-scale. Such parts encompass a

wide variety of geometries, materials, functional-

ities and production processes. Examples of

micro-parts include screws, fasteners, connector

pins, springs, micro-gears and micro-shafts. These

are manufactured by employing a variety of manu-

facturing processes such as machining, folding,

bending, stamping, drawing, molding, lithogra-

phy and forward/backward extrusion [1]. Some

examples of extruded micro-parts are shown in

Fig. 23-1 [2]. Forming is a particularly app ropri-

ate manufacturing technique for these parts, as

often they have a complicated shape and machin-

ing would be time consuming and produce low

yield. Process modeling plays an ever-increasing

role in these areas, for product design and for

reducing lead time and manufacturing costs.

To enable process design t o be undertaken on

a scientific basis, knowledge of the underlying

theory of micro-mechanics is essential. Plastic

deformation can be observed at various length

scales, which can be from the atomic scale where

the atomic arrangement and individual defect

properties of a material are of crucial impor-

tance, up to the macroscopic scale where the

actual mate rial micro-st ructure is not resolve d

and plasticity is described on phenomenological

grounds. A schematic diagram of the length

scales at which plasticity may be addressed –

the nanometer sc ale (ato mistic), the mesoscopic

scale (tens of microns) and the macroscopic

scale – is given in Fig. 23-2.

In conventional metal-forming processes the

size of the workpiece is usually large compared

with the grain size of the metal from which it is

constituted. Standard continuum-plasticity mod-

els are local in the sense that the stress at a mate-

rial point is assumed to be a function of a strain at

that point only. Local theories do not make refer-

ence to the characteristic length scale for disloca-

tions and, therefore, are not able to resolve

dislocation structures. As a consequence, such

models also exhibit no size dependence. The crys-

tals and their inherent directions of preferred slip

(slip-plane) are usually oriented randomly. Thus,

although a workpiece may be deformed by an

external force (or stress) in a clearly defined direc-

tion, internally the crystal deformations are multi-

directional. It is the resolution of the multi-direc-

tions along a common axis that gives rise to the

macro-deformation, resulting in the change in

shape of the workpiece. Because of the large num-

ber of crystals and the randomness of crystal ori-

entation, at the macro-scale a material appears

homogeneous and different samples of the same

material in the same thermo-mechanically treated

condition exhibit the same properties.

CHAPTER

366

At the micro-level, the grain size can be similar

to that of the part being formed. Thus a cross-

section of a workpiece may contain a single-digit

number of grains, compared with the tens or hun-

dreds at the macro-level. Thus the metal is not

homogeneous and the deformation characteristics

are likely to be different, as the resolved crystal

deformation exhibited as the workpiece shape will

be the result of slip on relatively few slip-planes

and a common outcome between different work-

pieces is unlikely. Due to this influence of the

micro-structure on the forming process, the work-

piece for a micro-part can no longer be regarded

as a homogeneous continuum for process-simula-

tion purposes. Thus in the process simulation of

forming micro-parts, it is important to choose

appropriate FE simulation theories, so that the

length scale of material deformation can be con-

sidered. In most of the cases in the forming of

micro-parts, continuum-mechanics theories break

down and crystal plasticity finite-element (CPFE)

methods have to be used. Thus, in this chapter,

numerical procedures for CPFE are introduced.

MICRO-MATERIAL MODELS

Physical Basis for Single-crystal

Deformation

Crystal plasticity is a physically based plasticity

theory that represents the deformation of a metal

at the micro-scale. The flow of dislocations in a

metallic crystal along slip systems is represented

in a continuum framework. Plastic strain is

assumed to be due solely to crystallographic dis-

location slip. Slip is the dominant mechanism for

deformation, which occurs due to dislocat ion

motion. A crystalline material is constructed of a

FIGURE 23-2 Plasticity in metals at various length scales.

FIGURE 23-1 Examples of micro-pins (Shinko, NME) [2].

CHAPTER 23 Micro-Mechanics Modeling for Micro-Forming Processes 367

periodic packing of atoms. A crystal structure

refers to a group of atoms that is situated in

repeating or periodic arrays or a unit cell. For

metals, there are three simple crystal structures,

namely FCC (face-centered cubic), BCC (body-

centered cubic) and HCP (hexagonal close-

packed) structures. Figure 23-3(a) shows the pure

crystal structures for FCC metals.

Observation of single crystals shows that slip

tends to take place most readily in specific direc-

tions (i.e. slip-directions) on certain crystallo-

graphic planes (i.e. slip-planes). A slip-plane

refers to the plane of greatest atomic density and

the slip-direction is the closest packed direction

within the slip-plane. A combination of preferred

slip-planes and slip-directions is called a slip-sys-

tem. For FCC crystals, the most densely packed

planes are the diagonal planes of the unit cell, see

Fig. 23-3(b). The full family of slip systems in an

FCC crystal may be written as < 1



10 > f111g.

There are 12 such systems in an FCC crystal (four

planes each with three directions).

Considered a single crystal of zinc, this is a few

millimeters in width and has been loaded beyond

its yield in tension. The planes that can be seen

are those on which slip has occurred resulting

from many hundreds of dislocations running

through the crystal and emerging at the edge. Each

dislocation contributes just one Burger’svector

of relative displacement, but with many such

dislocations, the displacements become large

(Fig. 23.3(c)).

Crystal Kinematics

The general kinematics of the elastic–plastic defor-

mation of a crystal at finite strains were given by

Taylor (1938) [3],Hill(1966)[4],Rice(1971)[5],

Hill and Rice (1972) [6], Asaro and Rice (1977)

[7] and Asaro (1983) [8]. The total deformation

gradient of finite strain from the reference frame

to the current frame, F

, is defined by:

ð1Þ

where X

and x

denote the reference- and current-

particle positions, respectively. Here, tensor con-

ventions for subscripts are adopted. All indices i, j,

k and l are running from 1 to 3 throughout this

chapter.

In crystal plasticity theory, a crystalline mate-

rial is embedded on its lattice, which undergoes

elastic deformation and rotation. The inelastic

deformation of a single crystal is assumed here

to arise solely from crystalline slip. The material

FIGURE 23-3 (a) FCC-structures in a crystalline material; (b) a particular slip-system ð111Þ½1



10 in an FCC lattice;

and (c) a schematic diagram representing single-slip.

368 CHAPTER 23 Micro-Mechanics Modeling for Micro-Forming Processes

flows through the crystal lattice via dislocation

motion. The total deformation gradient F

given by:

¼ F

ð2Þ

where F

denotes plastic shear of the material to

an intermediate reference configuration in which

the lattice orientation and spacin g are the same as

in the original reference configuration, and where

denotes stretching and rotation of the lattice.

These are shown in Fig. 23-4. The rate of change

of F

is related to the slipping rate

of the ath

slip system by:

1

ð3Þ

where the sum ranges over all activated slip

systems and unit vectors s

and m

are the slip-

direction and the normal to the slip-plane in

the reference configuration, respectively. The

number of slip-systems and their orientations

depend on the crystal lat tice, e.g. an FCC crystal

contains four slip-planes and each slip-plane

has three slip-directions, which results in a =1,

2, ..., 12.

It is convenient to define the vector s

, lying

along the slip-direction of the system a in the

deformed configuration, by:

¼ F

ð4Þ

A normal to the slip-plane which is the recip-

rocal base vector to all such vectors in the slip-

plane is:

¼ m

*1

ð5Þ

The velocity gradi ent in the current state is:

1

¼ D

þ W

ð6Þ

where the symmetric rate of stretching D

and the

anti-symmetric spin tensor W

may be decom-

posed into lattice parts (superscript *) and plastic

parts (superscript

) as follows:

¼ D

þ D

; W

¼ W

þ W

ð7Þ

FIGURE 23-4 Schematic diagram of single crystal kinematics.

CHAPTER 23 Micro-Mechanics Modeling for Micro-Forming Processes 369

Satisfying:

þ W

*1

; D

þ W

ð8Þ

Finally, it is noted that the plastic parts of the

rate of stretching and the rate of spin are given by

the symmetric and skew part of Eqn (8).Ifitis

defined that:

þ s



; W

 s



ð:9Þ

then:

; W

; ð:10Þ

It is worth noting that the plastic pa rt of the rate

of stretching, D

,isthatpartoftherateof

stretchi ng arising from slip in the current lattice

direction s

, in a plane where the current normal

is m

, while the plastic part of the rate of spin,

, is that part of the rate of plastic spin resulting

from the summation of rotation on each slip-

system.

Crystal Plasticity Constitutive

Equations

The main feature of the crystal plasticity consti-

tutive theory will be briefly introduced here. For

each grain, linear elasticity constitutive relations

are given by the generalized Hooke’s law:

¼ L

ijkl

ð11Þ

where L

ijkl

the fourth-order stiffness tensor and

the second-order symmetric rate of stretching

of the lattice. t

represent the Jaumann rates of

Kirchhoff stress formed on axes that spin with the

lattice:

¼ _t

 W

þ t

ð12Þ

where _t

is the material rate of Kirchhoff stress.

The Kirchhoff stress t

is defined as (r

/r)s

where s

is the Cauchy stress and r

and r are

the material density in the reference and current

states. On the other hand, t

is the Jaumann rate

of Kirchhoff stress formed on axes that rotate

with the material:

¼ _t

 W

þ t

ð13Þ

The difference between these two rates is:

 t



ð14Þ

When Eqns (9–11) and Eqn (14) are combined,

the resulting constitutive law becomes:

¼ L

ijkl



ijkl

þ W

 t



ð15Þ

In crystal plasticity, plastic deformation is

assumed to be caused solely by crystalline slip and

crystallinesliptobedrivenbySchmidstress(or

resolved shear stress), t

[3], which is defined by:

¼ m

ð16Þ

where m

and s

are slip-plane normals and

directions for the a th slip-system, respectively.

The rate changes of this Schmid stress is given by

[9]:

¼ m

 D

þ t



ð17Þ

The slipping strain rate

is assumed to be

governed by the resolved shear-stress t

given by

a constitutive equation shown below:







n1

ð18Þ

where

a is the reference strain rate, n is the stress-

sensitivity parameter and g

is the current strain-

hardened state of the crystal. In the limit as n

approaches infinity, this power law approaches

that of a rate-independent mat erial. The current

hardened state g

is defined by:

; b ¼ 1; 2; :::::; 12 for FCC crystals ð19Þ

370 CHAPTER 23 Micro-Mechanics Modeling for Micro-Forming Processes