MODELING , ANALYSIS AND VERIFICATION OF OPTIMAL FIXTURE DESIGN
81
ERNEST TAN YEE TIT B. Eng (Mech.), NUS A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 MODELING , ANALYSIS AND VERIFICATION OF OPTIMAL FIXTURE DESIGN brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by ScholarBank@NUS
Transcript of MODELING , ANALYSIS AND VERIFICATION OF OPTIMAL FIXTURE DESIGN
ERNEST TAN YEE TITB. Eng (Mech.), NUS
A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2003
MODELING , ANALYSIS AND VERIFICATION OF OPTIMAL FIXTURE DESIGN
brought to you by COREView metadata, citation and similar papers at core.ac.uk
provided by ScholarBank@NUS
i
ACKNOWLEDGEMENTS
I am very grateful to the following from whom I received help and guidance for this
research:
1. A/Prof A. Senthil kumar for his valuable direction, insight and giving me the
opportunity to complete my research under him.
2. A/Prof Jerry Y. H. Fuh for his precious time, concern and valuable guidance.
3. Mr Vincent Ling Yun and Mr Kevin Lim Heng Tong, final-year students, for their
contribution in this research. Without them this research would not have been
successful.
4. Dr. Lim Han Seok for his expertise and help in developing the experimental force
sensor system.
5. Mr Lim Soon Cheong who helped me arrange for the experiments.
6. Staff at Workshop 2 who helped produce the fixtures and sensor bodies.
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TABLE OF CONTENTS
Acknowledgements .............................................................................................................i
Table of Contents ...............................................................................................................ii
Summary.............................................................................................................................v
List of Figures....................................................................................................................vi
List of Tables ...................................................................................................................viii
List of Symbols ..................................................................................................................ix
Chapter 1. Introduction................................................................................................1
1.1 Background.............................................................................................................1
1.2 Literature survey.....................................................................................................1
1.3 Objectives ...............................................................................................................6
1.4 Organization of the Thesis......................................................................................6
Chapter 2. Automatic Selection of Clamping Surfaces and Positions using the Force Closure Method .......................................................................................................7
2.1 Theory of Force Closure.........................................................................................7
2.1.1 Force model .....................................................................................................7 2.1.2 Convex hull algorithm .....................................................................................9
2.2 Stages of implementation .....................................................................................12
2.2.1 Inputs..............................................................................................................13 2.2.2 Marking off unavailable grid points on the base plate...................................13 2.2.3 Identify candidate clamping surfaces.............................................................13 2.2.4 Generate spiral mesh......................................................................................14 2.2.5 Visualization ..................................................................................................16 2.2.6 Clamp Sequencing .........................................................................................17
2.3 Summary...............................................................................................................18
Chapter 3. Modeling of Minimum Clamping Force ................................................19
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3.1 Introduction ..........................................................................................................19
3.2 Optimization Equations ........................................................................................19
3.3 Example ................................................................................................................21
3.4 Summary...............................................................................................................22
Chapter 4. Experimental Force Sensor.....................................................................23
4.1 Working Principle of the Sensor ..........................................................................24
4.2 Visual Basic Data Acquisition Program...............................................................27
4.3 Software Requirements.........................................................................................29
4.4 Calibration of Sensors ..........................................................................................29
4.5 Evaluation of Sensor Performance .......................................................................30
4.6 Summary...............................................................................................................31
Chapter 5. Finite Element Modeling of the Workpiece-Fixture Setup..................32
5.1 Description of the Developed FEM model...........................................................32
5.2 Comparison Study ................................................................................................33
5.2.1 Model 1 - Mittal’s FEM Model .....................................................................34 5.2.2 Model 2 - Tao’s FEM Model .........................................................................38
5.3 Summary...............................................................................................................43
Chapter 6. Experimental Verification of the Finite Element Model......................44
6.1 Instrumentation.....................................................................................................44
6.2 Stiffness Test ........................................................................................................45
6.3 Description of Case Study 1 .................................................................................47
6.4 Results & Discussions of Case Study 1................................................................51
6.5 Description of Case Study 2 .................................................................................55
6.6 Results & Discussion of Case Study 2 .................................................................58
Chapter 7. Conclusions and Recommendations.......................................................63
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7.1 Conclusions ..........................................................................................................63
7.2 Recommendations ................................................................................................64
References.........................................................................................................................66
Appendix..........................................................................................................................A1
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SUMMARY
Fixture design is an important manufacturing activity which affects the quality of parts produced.
In order to develop a viable computer aided fixturing tool, the fixture-workpiece system has to be
accurately modeled and analysed. This thesis describes the modeling, analysis and verification of
optimal fixturing configurations by the methods of force closure, optimization, and finite element
modeling (FEM). Force closure has been employed to find optimal clamping positions and
sequencing, while optimization is used for determining the minimum clamping forces required to
balance the cutting forces. The developed FEM is able to determine in detail what are the reaction
forces, workpiece displacement, deformation in the workpiece and fixtures. In order to produce a
more accurate model for predicting the behaviour of the fixture–workpiece system, the developed
FEM includes fixture stiffness, while past models have assumed as rigid bodies.
The reaction forces on the locators are experimentally verified. A sensor-embedded experimental
fixturing setup was developed to verify the modeling and the data was used to compare with the
FEM. Two case studies were conducted and compared in the experiment and in FEM. As a
secondary objective, a prototype fixture-integrated force sensor was developed for use in the
experiment. But it was insufficiently reliable at this stage and the measurement of reaction force fell
back upon the existing Kistler slimline force sensor. It was found that the FEM-predicted reaction
forces trends match well with the experimental data. Therefore this improved finite element model
allowing room for slight error could be used to simulate the behaviour of an actual
fixture-workpiece system during machining.
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LIST OF FIGURES
Figure 1.1. Framework of Computer-Aided Fixture Design ...............................................2
Figure 2.1. Approximation of Friction Cone for Contact Ci ...............................................9
Figure 2.2. Spiral Mesh of clamping surface to find candidate clamping points ..............15
Figure 2.3. Colour Map of Side Clamping Surfaces based on rmax. (Blue is the optimal area and red is the infeasible area.).....................................................................................16
Figure 2.4. Colour Map of Top Clamping Surfaces based on rmax. (Blue is the optimal area and red is the infeasible area.).....................................................................................17
Figure 3.1. Minimum clamping force required vs time predicted by optimization algorithm......................................................................................................................................22
Figure 4.1. Sensor integrated fixture-workpiece system ...................................................23
Figure 4.2. The structure of the sensor ..............................................................................24
Figure 4.3. Uniform load over a small central area of radius r0, edge simply supported. .25
Figure 4.4. Side view of the sensor showing the air gap between the cap and brass plate......................................................................................................................................26
Figure 4.5. Circuit and output connection of the sensor. ...................................................26
Figure 4.6. Frequency output of the sensor. ......................................................................27
Figure 4.7. Instrumentation Layout ...................................................................................29
Figure 5.1. Model 1 after meshing (With reference to Mittal’s model). ...........................34
Figure 5.2. Fixturing layout for model 1............................................................................35
Figure 5.3. Machining profile for model 1 ........................................................................35
Figure 5.4. Reaction force vs time chart obtained by Mittal. ............................................37
Figure 5.5. Results from finite element analysis................................................................37
Figure 5.6. Model 2 after meshing (With reference to Tao’s Model). ..............................39
Figure 5.7. Fixture layout and location for model 2 ..........................................................40
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Figure 5.8. Reaction force vs time obtained in Tao’s experiment. ....................................41
Figure 5.9. Finite element results for Tao’s model (without fixture element stiffness). ...42
Figure 5.10. FEM results from developed model (with fixture element stiffness)............42
Figure 6.1. Schematic of the Fixture Stiffness Test...........................................................45
Figure 6.2. Relationship of force applied vs deflection on supporting element. ...............46
Figure 6.3. Relationship of force applied vs deflection on locating elements...................47
Figure 6.4. Modeling of the workpiece and locations of clamps/locators for Case Study 1......................................................................................................................................48
Figure 6.5. Experimental Setup for Case Study 1..............................................................49
Figure 6.6. Typical dynamic force obtained from experiment. Reaction force is shown at locator 7.......................................................................................................................50
Figure 6.7 A graph of reaction forces of supports and locators vs time of Case Study 1..53
Figure 6.8 A graph of reaction forces of clamps vs time of Case Study 1. .......................54
Figure 6.9. Experimental Setup for Case Study 2..............................................................55
Figure 6.10. Dimension of workpiece and locations of clamps/locators of Case Study 2.57
Figure 6.11. A graph of reaction forces of locators and supports vs time of Case Study 2......................................................................................................................................61
Figure 6.12. A graph of reaction forces of clamps vs time of Case Study 2. ....................62
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LIST OF TABLES
Table 5.1. Comparison of FEM models.............................................................................32
Table 5.2. Modeling Data ..................................................................................................33
Table 5.3. Comparison of features between Mittal’s model and the proposed model.......34
Table 5.4. Modeling data for model 1................................................................................36
Table 5.5. Comparison of features between Tao’s model and the proposed model. .........38
Table 5.6. Modeling data for model 2................................................................................40
Table 6.1. Fixture element stiffness...................................................................................46
Table 6.2. Clamping forces applied in sequence of Case Study 1.....................................49
Table 6.3. Cutting data of Case Study 1. ...........................................................................50
Table 6.4. Clamping forces applied in sequence of Case Study 2.....................................56
Table 6.5. Cutting data of Case Study 2 ............................................................................56
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LIST OF SYMBOLS
fik unit generator of polyhedral friction cone
αik positive factor for the linear combination of unit generators
ai unit normal of clamping face
µi coefficient of static friction between contact i and workpiece
n number of contacts
A matrix of facet normals
x six-dimensional point in the convex hull space
b vector of facet offsets bi
w six dimensional wrench
fikx, fiky, fikz force components of six dimensional wrench
(ri × fik)x , (ri × fik)y , (ri × fik)z moment components of six dimensional wrench
rmax radius of maximally inscribed hypersphere
S, T unit direction vectors of clamping surface
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Chapter 1. INTRODUCTION
1.1 Background
Today’s advanced flexible manufacturing systems contain CNC machines which can
automatically cut parts and change programs on the fly, move parts between machines
automatically, but when it comes to fixturing, a human machinist is required to accurately
locate and clamp the parts and in some cases design the fixture setup. Surely this is a
bottleneck because of the possibility of human error and long lead time for fixture design,
which is a complex task requiring heuristic knowledge from an expert designer. In
designing a fixture, there are two necessary steps, viz., fixture synthesis and fixture
analysis (see Figure 1.1). Fixture synthesis is supported by a CAD representation system
which has access to a parametric fixture element database. Issues such as the setup and
machining operation, fixture element connectivity, selection of fixturing surfaces and
points are considered in the synthesis process. After conceiving a fixture design using
fixture synthesis methods, it has to be verified through fixture analysis to predict, for
example, whether this configuration is stable or will cause improper contact with the
workpiece during machining, etc.
1.2 Literature survey
Fixture analysis can be categorized into four levels [1], viz., geometric, kinematic, force
and deformation. At the geometric analysis level, spatial reasoning is applied to check for
interference between fixture, workpiece and cutting tool. Kinematic analysis checks for
correct location with respect to datum surfaces (to avoid any over-constrained location)
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and whether the fixture contacts are positioned adequately to oppose the cutting forces.
The most commonly adopted method of kinematic analysis is force closure.
Figure 1.1. Framework of Computer-Aided Fixture Design
Force analysis checks that the reaction forces at the fixture contacts are sufficient to
maintain static equilibrium in the presence of cutting forces. Cutting force profiles need to
Fixture Design
Synthesis Analysis
CAD Representation
Fixture Element Connectivity
Setup Information
Bill of Materials
AFD / SFD / IFD
Machining Operation
Selection of Fixturing Surface and points
Parametric Fixture Database
Framework of Computer-Aided Fixture Design
Geometric Analysis
Force Analysis
Deformation Analysis
Kinematic Analysis
Machining Interference
Assembly Interference
FEM
FEM
Minimum Clamping Force
Force Closure
Included in thesis
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be known for this level of analysis. Lastly, especially important for flexible parts,
deformation analysis that determines the elastic or plastic deformation of the part under
the clamping and cutting forces. Mittal [2] developed a dynamic model of the
fixture-workpiece system that is able to describe the elastic effects of fixture-workpiece
contacts, the position, velocity and acceleration of all bodies involved, and the reaction
forces. De Meter[3] developed a linear model for predicting the impact of locator and
clamp placement on workpiece displacement throughout the machining operation and
determining whether the clamping forces are adequate to constrain the part during
machining. Li and Melkote[4] developed a general method for iteratively optimizing the
fixture layout and clamping forces while accounting for workpiece dynamics. The finite
element method (FEM) for fixture analysis has been described in [5] and [6].
Friction plays a dominant and beneficial role in the fixture-workpiece interaction. A
workpiece can be totally restrained by as few as two large contacting surfaces because of
friction, as in a vice. Damping of cutting forces is partly attributed to interfacial friction
between the fixture and workpiece. Therefore it is important to include the frictional
effects in a fixture-workpiece model.
For fairly rigid workpieces, machining forces on the workpiece could cause local elastic
deformations at the points of contact between the locators and clamps, resulting in
workpiece locating error. This is known as contact deformation, and contact stiffness
plays a major role in such a deformation. The ABAQUS/CAE FEM package is able to
model Coulomb frictional contact between the elastic “master” and “slave” surfaces,
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where the “master” surface is defined as the more rigid one of the two. These are modeled
using the contact mechanics theory in partial differential equations defining stress and
elastic strain within the contact pair.
Fixture stiffness has been studied by Rong & Zhu [7]. The deformation of fixture
components and their connections may significantly contribute to machining inaccuracy
of parts and dynamic instability during the machining process. Some factors that affect
fixture stiffness are: fastening force magnitude and the orientation of the fixture
components. The most direct way of determining fixture stiffness is to apply a load to the
fixture assembly and measure the deflection at various points. This gives a deformation
curve, where the stiffness is the gradient. The problem with experimentally determining
the fixture stiffness is that almost infinite combinations of assemblies are possible. This
stiffness is modeled in FEM using a spring element which is placed normal to the
direction of the fixture contact surface.
In this research, the fixture element in the FEM model is modeled as deformable rather
than rigid, which previous researchers have done. One goal of fixture design is to make
the fixture as rigid as possible. However, real fixtures have finite stiffness. Based on
stiffness tests on fixture elements, the stiffness of the locators used is kL = 3.24 x 107 N/m,
which is less stiff than the workpiece. The stiffness of the rectangular workpiece
described in Figure 6.5 is as follows, kz = 2.97 x 1010 N/m, ky = 4.56 x 1010 N/m and kx =
1.14 x 1010 N/m. Clearly in this case, the less stiff fixture would deform much more than
the workpiece when subjected to the same force. Generally, modular fixtures are not as
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rigid as dedicated fixtures, and it is common to see stacking of fixture components, which
reduces the overall stiffness. Including the effect of fixture stiffness in the FEM would
make a difference in cases where the fixture is less stiff than the workpiece.
From a comparison of Tao’s FEM model which does not include fixture stiffness and
the developed FEM model, it was found that when the effect of fixture stiffness is
included into the model, the reaction forces of the analysis are slightly lower than the one
without the fixture stiffness. This comes to a conclusion that the reaction forces are
lowered with the introduction of the fixture stiffness. Therefore the developed FEM
model with fixture stiffness is in fact a safer prediction, leading to higher clamping
intensity required to keep the workpiece stable.
The model is built to simulate the actual physical reaction of a fixturing system and
hence to foresee any potential error in the design. Various engineering properties that
govern the accuracy of the analysis are included into the model. These properties are:
• Contact stiffness,
• Stiffness of locators, clamps and workpiece (element stiffness), and
• Frictional force between contact surfaces
Previous research works on fixture design have never included all the above-mentioned
properties into a single experiment or analysis. Thus, the major aim of this project is to
develop a modeling method that includes all the real time conditions that present in an
actual set up of a fixture-workpiece system.
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1.3 Objectives
The research undertaken involved (1) the use of the force closure method to predict
optimal clamping positions and clamping sequence, (2) using an optimization algorithm
to predict the reaction forces at the fixture contacts, and (3) development of an FEM
model of the fixture-workpiece system that includes fixture stiffness. The force closure
method generates a set of optimal clamping positions based on pre-selected locating and
supporting positions. The optimization algorithm predicts the minimum reaction forces at
the fixture contacts under the external cutting forces and moments. Lastly, the developed
FEM model describes the workpiece and fixture contacts as deformable and interacting
with each other by Coulomb frictional contact. The cutting process is simulated using a
quasi-static cutting force and moment applied along the tool path. To verify the FEM
model, reaction forces predicted at the fixture contacts are compared with the readings
from piezoelectric force sensors in the experiment.
1.4 Organization of the Thesis
Chapter 2 explains the theory and implementation of the force closure method in
automated fixture design, AFD. Chapter 3 discusses an algorithm for the non-linear
optimization of minimum clamping force in the fixture-workpiece system. Chapter 4 is a
report on the developed experimental force sensor. Chapter 5 explains the details of the
developed finite-element model of the fixture-workpiece system and comparison with
two FEM models by Mittal and Tao. Chapter 6 is an experimental verification of the FEM
model with two case studies. Chapter 7 concludes the thesis.
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Chapter 2. AUTOMATIC SELECTION OF CLAMPING SURFACES AND POSITIONS USING THE FORCE CLOSURE METHOD
This section focuses on the selection of optimal clamping points and formulates an
acceptable clamping sequence. Locating and supporting positions and directions have
been automatically selected using the heuristics built in the developed automated fixture
design software[10].
2.1 Theory of Force Closure
Force closure[11] is the balance of forces on the workpiece to determine if static
equilibrium can be achieved. If the applied clamping forces are able to prevent the motion
of the workpiece when it is being acted upon by external machining forces, then there is a
force closure. The fixturing problem is defined by an analytical model which can be
solved mathematically.
The theory of force closure for fixturing is similar to the theory of robotic grasping,
where robotic fingers apply only active forces on an object. In fixturing, only the clamps
apply active forces while the locators and supports are passive elements. Like in a robotic
grasper, friction plays an important part in fixture-workpiece interaction. When a force
model with friction is used, the number of fixturing contacts needed may be reduced.
2.1.1 Force model
Both the contacts and workpiece are regarded as rigid bodies. Each contact is modeled
as an infinite friction cone with the axis along the line of application and zero moment at
the point of contact. Let fi be the contact force acting at the point of contact Ci by the
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fixture and acting in the direction of the contact normal ai , and let µi be the coefficient of
static friction between the two surfaces. Then fi must satisfy the maximum static friction
condition according to Coulomb’s law[12]:
( )ii aff ⋅+= 21 ii µ for i = 1, 2, ..., n ..................................................... (1)
where n = number of fixture contacts ai = contact normal
Since fi lies within the infinite friction cone, it is equivalent to a linear combination of
non-negative unit generating vectors bounding the cone. To improve computational
efficiency, this friction cone is approximated by a four-sided polyhedral convex cone
defined by four unit generators (Figure 2.1. ). Since the goal of the force closure method is
to plot a feasible clamping area and based on the need to keep the complexity down, a
four-sided polygonal cone was chosen for this purpose. An increase in the number of
sides of the polyhedral cone improves accuracy but introduces increased complexity that
is not justifiable by the purpose of the algorithm.
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Figure 2.1. Approximation of Friction Cone for Contact Ci
∑=
=4
1kiki ikff λ λik ≥ 0 for k = 1, 2, 3, 4 ......................................................................(2)
where k = index of each unit generator λik = scalar factor for each unit generator
To find the unit generators, fik, the following vectors are calculated. Find fi1 by rotating
ai by angle tan-1µ about the unit vector RP on the plane of the contact surface. Rotate fi1
about ai by 90° to get fi2. Rotate fi1 about ai by 180° to get fi3 . Rotate fi1 about ai by 270°
to get fi4. Note that each unit generator is represented by a wrench which has six
coordinates.
2.1.2 Convex hull algorithm
For the purpose of determining the clamping stability of a clamping point and clamping
ai surface normal
fi1
fi2 fi3
fi4
ri
tan-1µ
RP
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direction, it was assumed that the fixture elements contact the workpiece at seven points,
namely, three locators, three supports and one clamp which gives a total of seven
contacts. Seven points of contact are used because the model and experimental fixtures
are based on 3-2-1 locating principle with the seventh contact as the first clamping force
required to arrest all the degrees of freedom. In the frictionless case, four and seven
contacts are necessary to achieve force closure for 2-D and 3-D parts respectively. For the
frictional case, three contacts are sufficient for 2-D and four are adequate for 3-D parts
[13]. The actual configuration allows for more than one clamp. Each contact has four unit
generators (square polyhedral cone). Therefore the total number of λik unknowns is 28 (7
x 4). Note that each unit generator is a six-dimensional wrench.
This problem is solved using a class of multi-dimensional geometric methods known as
convex hull algorithms. Among the convex hull algorithms, the Quick Hull Algorithm
developed by the Geometric Center [14] is available in C library source code and is
implemented to solve the fixturing problem.
The primitive (unit) wrench of a unit generator is defined as
( ) ( ) ( )[ ]Tzyx ikiikiikiikzikyikxik frfrfrfffw ×××= ............................................ (3)
Where r = position vector of the contact point with respect to the origin
The bounding (total) wrench of a contact is defined as
i4i3i2i1i wwwww 4321 iiii λλλλ +++= ...................................................................(4)
where λik ≥ 0 for k = 1, 2, 3, 4
Twenty-eight rows of input points (six-dimensional wrenches) are computed from the
given fixturing positions and directions producing the matrix A. This is written to the
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input file for the QuickHull algorithm, i.e.
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
×××
××××××××××××
=
zyxzyx
zyxzyx
zyxzyx
zyxzyx
zyxzyx
)()()(fff::::::
)()()(fff)()()(fff)()()(fff)()()(fff
747747747
141141141
131131131
121121121
111111111
frfrfr
frfrfrfrfrfrfrfrfrfrfrfr
A
747474
141414
131313
121212
111111
..........(5)
where A is a ‘28 x 6’ matrix of facet normals.
For this fixturing application, the specific convex hull is defined such that all points, x,
inside the convex hull must satisfy:
0≤+ bAx ............................................................................................................ (6)
where x = [ x1 x2 x3 x4 x5 x6 ]T is a six-dimensional point in the convex hull space
b = [b1 b2 b3 …b28]T is a 28 component vector of facet offsets from the convex
hull origin (a convex hull is made up of facets)
Each candidate clamping position is associated with a different matrix A. The
QuickHull algorithm computes the vector b from A. The vector b is further evaluated by
the program to check for instability and, if considered stable, to compare with other
candidate clamping points for ranking in their stability. The convex hull includes the
origin only if all the normal offset values are non-positive. A clamping point is therefore
said to be feasible when it is able to achieve equilibrium such that the origin is in the
convex hull, i.e. when all bi are negative.
bi ≤ 0 where i = 1,2,…28 ..................................................................................... (7)
12
By examining the vector b produced by the convex hull algorithm, we can reason about
the stability of the workpiece fixture system as follows:
1. If any bi > 0, the origin must lie outside the convex hull. This means some λιk are
negative, therefore there is non-equilibrium.
2. If all bi < 0, the origin must lie inside the convex hull. This means all λιk are positive,
therefore there is force closure.
3. If one or more offset value bi = 0, the origin must lie on the boundary of the convex
hull. This means that one or more λιk are positive, therefore there is marginal equilibrium.
For evaluating the stability of the force closure, the magnitude of rmax is measured. The
radius of the maximally inscribed hypersphere, rmax, defined as the largest hypersphere
from the origin that can fit into a convex hull. This hypersphere has the greatest distance
possible from the origin to the facets of the convex hull. A large distance (rmax) indicates
that the origin is well inside the convex hull and hence the fixturing configuration is more
stable than for one with a small rmax.
( ) ( ) ( ) ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
×+×+×+++=
==2222224,3,2,1;7,...,2,1
max minzyxikzikyikx
ik
ki fff
br
ikiικιiki frfrfr............... (8)
2.2 Stages of implementation
The following is the sequence of procedures employed to search for optimal clamping
points; starting with defining the inputs and then eliminating the base plate grid points
that are unavailable for use. Candidate clamping surfaces are selected and iterated over in
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a loop. For each candidate clamping surface, a spiral mesh of candidate points is
generated and tested. It is then presented in a visual form where the feasible points are
coloured according to stability. Lastly the algorithm sequences the clamping by ranking
the clamping surfaces and points.
2.2.1 Inputs
The following are given as input to the system,
• absolute locating and supporting points, ri
• normal direction, ai
• coefficient of static friction, µi , defined as 0.4
• origin of each candidate face, for calculating the in-plane vector RP.
2.2.2 Marking off unavailable grid points on the base plate
All the grid points under the “shadow” of the workpiece are inaccessible to the
clamping fixtures and these are marked off. Since support grid points are always in the
shadow of the workpiece, hence they are ignored. This is achieved by raising each grid
point on the baseplate vertically by small increments and using the CAD program
function to test whether the point is within the workpiece body. Grid points which tested
true will be those under the workpiece. This method fails when there are holes in the
workpiece, so user interaction is needed to mark off these grid points manually.
2.2.3 Identify candidate clamping surfaces
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To minimize the computation time, a list of candidate faces for clamping is narrowed
down using the following steps. Firstly, faces to be machined are eliminated because of
cutter collision with fixtures. Secondly, it is a well-known fixturing rule that locating
faces should not be used for clamping, as this would detach the locators from the
workpiece, rendering them useless. Thirdly, as modular fixtures are used, only top and
side faces can be clamped.
2.2.4 Generate spiral mesh
To facilitate testing of all possible clamping points, a mesh at equal intervals on all
possible clamping faces is generated. This task poses a problem of generating these
candidate points despite the irregularities of the planar faces which have curved
boundaries or holes. A spiral search path is used, originating from the centre of the face
for containment computations instead of starting from the corners. Figure 2.2 illustrates
the increasing size of the spiral and shows when the iteration stops. Variables used are
mesh size D = 20 mm, number of loops N, centre point of face and surface unit vectors of
the clamp face S and T.
15
Figure 2.2. Spiral Mesh of clamping surface to find candidate clamping points
Three consecutive tests, namely the containment test, grid availability test and
Quickhull feasibility test, are performed for each mesh point. Grid availability test and
Quickhull feasibility test will be done only if containment test is first successful. For the
containment test, a CAD program function is called to test whether the mesh point lies in
the bounded plane of the candidate face. The grid availability test involves checking
sixteen neighbouring grid points for availability. If none are available at all, this mesh
point cannot be used for clamping. Lastly the Quickhull feasibility test is performed to
check for force closure for each set of contacts. For each locator and support, values of fik
and ri × fik (bounding wrenches) are computed. These are stored in the QuickHull input
file as input coordinates of the matrix A (eq. 5). For each mesh point, a different matrix A
is computed as input. Then the QuickHull library is called to create a convex hull. An
output file of facet normals and a vector of facet offsets, bi (eq. 7) is given. If all bi are
negative, then this mesh point is feasible. rmax is computed from the output file using eq.8.
Mesh points for each face are sorted according to rmax, in descending order and this is
visualized using Matlab.
T
S
16
2.2.5 Visualization
After each face has been computed, the mesh points are colour-coded in the CAD
system. Infeasible points are grayed out. Feasible points are sorted into a spectrum from
blue (most stable) to red (least stable), based on the magnitude of rmax. The user would be
able to observe the feasible coloured areas on each candidate clamping face and use it to
select manually. A simple colour map plot can be obtained in Matlab for the purpose of
visualization. (see Figure 2.3 & Figure 2.4)
Figure 2.3. Colour Map of Side Clamping Surfaces based on rmax. (Blue is the optimal area and red is the infeasible area.)
RED
Clamp C2 applied within the blue optimal clamping area
Clamp C4 applied within the blue optimal clamping area
BLUE BLUE
RED
17
Figure 2.4. Colour Map of Top Clamping Surfaces based on rmax. (Blue is the optimal area and red is the infeasible area.)
2.2.6 Clamp Sequencing
Clamping is done first on the faces with the largest feasible clamping area. The clamp
face is highlighted and user is prompted for the number of clamps to apply. Optimal
clamping point algorithm chooses the mesh point with the highest rmax to be the first
clamping point, and so on. Each time a clamp point is chosen; the program tries to map it
to the nearest grid points. If any of these nearest grid points are successful, the first
successful mapping will be used and the clamp with its mounting adaptors are loaded
from the database into the assembly automatically. The grid point is then marked off as
unavailable. If all the possibilities of grid points are exhausted, the program cycles to the
next best mesh point and repeats the process. Upon the worst case scenario where all
Clamp C8 applied within the blue optimal clamping area
Red areas are infeasible for
clamping
BLUE
RED
18
feasible mesh points cannot be mapped, the user can skip the clamping face or choose a
point manually.
2.3 Summary
In this chapter, an overview of the force closure method in the automatic selection of
clamping points was presented. This force closure method has been effectively integrated
into the AFD system. Three goals were accomplished namely: clamp face selection,
clamping point selection and clamp sequencing. In the next chapter, the minimum
clamping forces at these clamping points are computed with an optimization algorithm.
19
Chapter 3. MODELING OF MINIMUM CLAMPING FORCE
3.1 Introduction
Prediction of clamping force intensity profile is meant for the fixturing operator on the
shop floor to know how much clamping force to apply for each clamp. The necessary
equations are derived from Tao’s paper [6] and integrated into the developed fixturing
program. Required inputs are as follows: position and direction of each fixturing contact,
friction coefficient, cutting force as a function of time, workpiece weight and centre of
gravity of the workpiece. The optimization algorithm minimizes the friction capacity ratio
of the fixture-workpiece system, subject to the constraints of static equilibrium, positive
location and Coulomb friction. This generates a minimum reaction force profile of all the
fixture contacts with respect to time. It is the minimum reaction force required to balance
the cutting forces disturbing the equilibrium at each point of time. If dynamic clamps are
used, the control scheme for the dynamic force intensity follows this profile. If
conventional clamps are used, the operator applies the clamping force for each clamp at a
higher level than the maximum force predicted.
3.2 Optimization Equations
When considering clamping force optimization, representing the friction cone with an
approximated 4-sided polyhedral cone is unsatisfactory. A complete equivalent is needed.
A tetrahedral cone minimally circumscribing (outside) the friction cone is such an
equivalent. The infinite tetrahedral cone is defined by unit generators fik , obeying this
equation:
20
2411
iµ+=⋅ iik af for k = 1, 2, 3. ........................................................................... (9)
The contact force fi is a linear combination of non-negative fik:
fi = αi1 fi1 + αi2 fi2 + αi3 fi3 where 0 <αik < 1 ........................................................... (10)
This constraint must also be obeyed for fi to lie inside the tetrahedral cone:
( ) 041
1321322
2
≥++−++++
i3i2i1 fff iiiiiii
i ααααααµ
µ ............................................. (11)
Contact forces are now resolved to an equivalent 6 dimensional wrench wi:
( ) ( ) ( )⎥⎦⎤
⎢⎣
⎡×+×+×
++=⎥
⎦
⎤⎢⎣
⎡×
=i3ii2ii1i
i3i2i1
i
ii frfrfr
ffffr
fw
321
321
iii
iii
i αααααα
....................................... (12)
All contact forces must be positive, acting towards the workpiece (positive location
constraint):
∑−
≥+
3
1241k
i
i
ik LBµ
α .................................................................................................. (13)
where LB is the lower bound of contact force to keep the fixture in contact with the
workpiece.
External cutting force is a function of time and is defined by the wrench:
( ) ( )( ) ( ) ( )⎥⎦
⎤⎢⎣
⎡×+
=ttt
tt
FrMF
w ........................................................................................ (14)
Where F(t) is the cutting force as function of time, M(t) is the cutting moment as a
function of time and r(t) is the cutting path (position of centre of the cutter as a function of
time).
Workpiece weight is defined by:
21
[ ]Tcc mgxmgymg 000 −−−=gw .............................................................. (15)
where the centre of gravity is defined at ( xc , yc , zc ), acceleration due to gravity g = 9.81
m/s2, mass of workpiece is m.
The complete optimization problem is defined as follows:
Objective function:
Maximize ( )
∑∑
∑=
=
=−
×+
n
i
kik
kik
i1
3
1
3
124α
αµ
iik af.................................................................. (16)
where ( )ii
ii
afaf⋅
×=
ii µ
ϖ is the friction capacity ratio to be maximized, the ratio of friction
component to normal component.
Subject to these constraints:
1. ( )
( )( ) ( ) ( ) 0
1
3
1
1
3
1 =+⎥⎦
⎤⎢⎣
⎡×+
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
×∑∑
∑∑
= =
= =g
iki
ik
wFrM
F
fr
f
tttt
n
i kik
n
i kik
α
α (static equilibrium)
2. Eqn (11) for i = 1, 2, ..., n (tetrahedral cone property)
3. Eqn (13) for i = 1, 2, ..., n (positive location)
4. αik ≥ 0 for i = 1, 2, ..., n and k = 1, 2, 3. (non-negativity)
(n = number of contacts)
3.3 Example
From the plot of the predicted clamping forces vs. time (Figure 3.1), a minimum
clamping force for each clamp is chosen such that it is larger than the maximum clamping
force required by a safety margin. This will ensure that the clamping forces are sufficient
22
to resist the machining forces at all times. The maximum value of clamping force over the
whole profile is taken and a safety margin is added to it. For example, the maximum
expected clamping force for C8 is 1600N and a safety margin of 500N is added to make it
2100N. Refer to Table 6.2 for the actual clamping force used in Case Study 1 as predicted
by this algorithm.
Figure 3.1. Minimum clamping force required vs time predicted by optimization algorithm.
3.4 Summary
This chapter has explained in detail the implementation of a minimum clamping force
algorithm by which the actual clamping forces are selected. The next chapter reports on
the development of an experimental force sensor to be used in the experimental
verification of the FEM model.
0 10 20 30 40 50 60 70400
600
800
1000
1200
1400
1600
1800
2000
C2C4C8
Actual force applied by C8
Actual force applied by C4
Actual force applied by C2
Min
imum
cla
mpi
ng fo
rce
(N)
Minimum clamping forces (N) vs. time (s) predicted by optimization algorithm
time (s)
23
Chapter 4. EXPERIMENTAL FORCE SENSOR
Force sensors are employed in the machining experiment to measure the normal
reaction force at each fixture contact as the cutter exerts a time-varying force on the
workpiece. This experimental force sensor is based on the principle of the capacitance of
an air gap. Each sensor is meant to be an economical replacement for the Kistler
Piezoelectric Slimline Force Sensor, which costs around S$2,000 each. Actual machining
was carried out in the Advanced Manufacturing Lab and the results were recorded using a
baseplate dynamometer and 8 prototype sensors. The dynamometer is used to measure the
machining forces. The actual machining setup is shown in Figure 4.1. The following
sections will discuss the working principle of the sensors, fabrication making of the
sensors, calibration and the data acquisition system.
Figure 4.1. Sensor integrated fixture-workpiece system
24
4.1 Working Principle of the Sensor
Figure 4.2. The structure of the sensor
For high rigidity, the body of the sensor is made of mild steel. As illustrated in Figure
4.2, the cap of the sensor is screw-fastened onto the main body. A brass plate is facing
next to the cap when the cap is tightened. The sensor is put into the contact with the
workpiece at the small circular contact point on the center of the cap. The cap will
experience a deflection when cutting and clamping forces are acted onto the workpiece.
The load-deflection relationship of the sensor’s cap is depicted in Figure 4.3 in a free
body diagram. The following equation (eq. 17) for loading on a circular plates bounded by
a circular boundary can be used to find out the defection, y of the circular plate [15].
( ) ⎥⎦
⎤⎢⎣
⎡−−
++−
=rarra
DWyc ln2
13
16222
υυ
π................................................................. (17)
where
20rqW π= is the uniform load over a very small central circular area of radius r0,
ν is the Poisson ratio,
Contact Point
Cap Brass
25
d is the diameter and a is the radius of the circular plate,
r0 is the radius of a small circular area where the loading is applied.
Figure 4.3. Uniform load over a small central area of radius r0, edge simply supported.
The applied load is measured by the change in capacitance when the cap is deflected,
i.e. change in the value of yc. The relationship between the capacitance, C and the
deflection, yc is given by equation 18:
DAC r 0εε=
----------------------------------------------------------------------------- (18)
where εr is the dielectric constant,
ε0 is the permittivity of the air and is equivalent to 8.85 x 10-12 F/m,
A is the area of the gap, and
D is equal to D0 - yc as shown in Figure 4.4.
W
a r0
20rqW π=
q = load per unit area
26
Figure 4.4. Side view of the sensor showing the air gap between the cap and brass plate.
Figure 4.5. Circuit and output connection of the sensor.
As shown in Figure 4.5, a NE555 silicon monolithic timing circuit is used to produce a
regular clock pulse. In the time delay mode of operation, one external resistor and one
capacitor precisely control the clock pulse frequency. The circuit is negatively-triggered,
i.e. from 1 to 0.
Do, air gap
Do = initial gap between sensor’s cap and brass plate
Brass Plate
NE555
TIMER IC
Output Frequency Counter
27
The frequency output from NE555 timer IC is a function of the circuit resistance and
capacitance, i.e. f = f( R, C ). The frequency output is then transmitted to a frequency
counter. The counter will time the output based on a fixed number of pulses. In Figure
4.6a, when the preset number of pulses, Np is reached, the counter will record the time
taken to reach that Np number of pulses. When the sensor experiences an increase in
applied force as shown in Figure 4.6b, the frequency of the output signal will decrease
hence the time needed to reach Np number of pulses will increase.
Figure 4.6. Frequency output of the sensor.
4.2 Visual Basic Data Acquisition Program
Eight force sensors are attached to a microprocessor-controlled circuit which has a
serial interface. This serial interface allows a computer to communicate with the
microprocessor. Sending “a01000” through the serial interface will set the number of
pulses measured to 1000. Sending “A” tells the microprocessor to measure for example
Sensor 0. The serial interface replies with an eight-digit number that is the time, in
microseconds, for 1000 cycles of the capacitor in Sensor 0. The frequency of the capacitor
can be calculated from this number. It corresponds to the force acting on the sensor at that
a) Initial force = Fo, time taken for Np pulses = to
b) Applied force = Fi , time taken for Np pulses = ti, ∴ Fi > Fo, ti > to
28
time.
A data acquisition program, “Force Sensor Serial Interface”, is written in Visual Basic
6. The platform used is a stand-alone Windows 98 PC. This program sends and receives
signals from the sensor microprocessor circuit and presents a visual display to the user.
Visual Basic is chosen for its ease of programming and powerful integration with
Microsoft Office. The Microsoft Chart ActiveX object is used in the plotting of graphs.
This ActiveX object makes it easy to plot graphs just by specifying the graph type, data
array and other settings. It takes care of the scaling, graphics, colour and other details
which a programmer otherwise has to hard-code from scratch. For the communications
with the serial port, the MSComm ActiveX object was used. This provides for a means to
send and receive a string of text from the serial port. In contrast, the C language is not able
to yield such a program without much programming effort and time. However, one
disadvantage of Visual Basic is performance. There is a slight delay in plotting graphs and
data processing.
29
Figure 4.7. Instrumentation Layout
4.3 Software Requirements
• Communicate with the serial port on COM1 or COM2
• Display graphs of all eight sensors
• Flexibility to read sensors once or continuously, singly or all in sequence.
• Calibrate the sensors to display forces
• Save and load results, calibration data
• Produce results in Excel readable format
4.4 Calibration of Sensors
The following are the steps involved in sensor calibration:
1. Set Np, number of pulses read, for 8 sensors,
(The accuracy and sensitivity of the sensor are affected by the chosen Np.)
MicroprocessorCircuit with Serial
Interface PC serial port COM1/COM2
Windows operating system
Visual Basic Program:
Force Sensor Serial Interface
Eight force sensors
Results & calibration
data
30
2. Read Tm, time for Np pulses at R=0 N, zero load
3. Apply load R, read average Tm for 3 times
4. Check that Tm is within range, 0 < Tm < 59,999,999, otherwise repeat step 3
5. Repeat for different loads
6. Plot R vs Tm for 8 sensors
7. Use curve fitting to find the function, H of the graph, where
R = H(Tm),
From the results, H is a straight-line function, which is in the form,
Tm=M*R + C
To get force R, we express R in terms of the others.
R=(Tm-C)/M
So there are eight different values of both M and C for all the sensors. This is edited and
saved in a calibration file. The procedures and steps for reading and recording the data
during the milling process can be found in Appendix I.
4.5 Evaluation of Sensor Performance
Sensor performance can be measured by its signal-to-noise ratio (SNR). Based on
experimental test runs, the SNR is approximately 1. This means that the fluctuations in
readings due to noise are as great in magnitude as the average sensor readings. This is in
contrast to the SNR of the Kistler Slimline Force Sensor which is at least 2 orders of
magnitude lower. Sampling rate is about 4 Hz per sensor for the experimental sensor. This
is low compared to 100Hz and above for the Kistler sensor. Hence more work needs to be
done on the experimental force sensor before it can produce reliable and accurate
31
readings. Nevertheless, it is still a commendable first effort. (The experiments reported in
Chapter 7 use the Kistler Slimline Force Sensor.)
4.6 Summary
This chapter has described the working theory of an experimental force sensor as well
as its data acquisition hardware and software. This sensor was not utilized in the
experimental setup because of reasons mentioned earlier. The following two chapters, 5
and 6, describe the modeling of the fixture-workpiece system by the finite element
method and its experimental verification respectively.
32
Chapter 5. FINITE ELEMENT MODELING OF THE WORKPIECE-FIXTURE SETUP
5.1 Description of the Developed FEM model
The finite element method is most suitable to analyze the elastic deformation of a
workpiece-fixture system in the presence of clamping and cutting forces. The finite
element model built in this work includes contact stiffness, element stiffness and
frictional force. The differences in comparison to Mittal’s[2] , Tao’s[6] and Lee &
Haynes’[5] models are listed in Table 5.1.
Table 5.1. Comparison of FEM models
Property Mittal’s Model Tao’s Model
Lee & Haynes’ Model
Proposed Model
Deformable Workpiece X X
Frictional Effect X
Contact Stiffness X
Fixture Element Stiffness X X X
The workpiece part model is built using ABAQUS [16] part creation interface. The
workpiece, a 184 x 114 x 92 mm aluminum block, is meshed with C3D8R (Contiuum-3
Dimensional-8 nodes, reduced integration) hexahedral solid element. Each fixture contact
is represented by a 10 x 10 x 3 mm flat square, which approximates the circular contact
surface of the fixtures used in the experiment. Material properties are assigned for the
aluminum workpiece and the steel fixture elements (see Table 5.2). The complete model
33
consists of 3 locators, 3 supports and 3 clamps. For each contact pair, the interaction
model is defined as “friction with hard contact”. The fixture contact surface is defined as
the “master surface”, as opposed to “slave surface”, because it is more rigid than the
workpiece surface. A simple Coulomb law friction model is specified with the coefficient
of friction as 0.4. Each contact is restrained in the tangential directions such that only
displacement in the normal direction is allowed. In this study, only the normal force is
considered while modeling as the frictional (tangential) force is much smaller. A
SPRING2 element is connected to the centre of each contact square and its spring
constant has to be determined experimentally. The reaction forces and the displacement
of each fixture contact are obtained as output from the FEM and is discussed in chapter 7.
The time required to input and prepare the model in Abaqus for meshing and defining the
fixture contacts and interaction properties in the input file is about 20 min. Solver time
ranges from 10 min to 30 min, depending on the number of steps.
Table 5.2. Modeling Data
5.2 Comparison Study
Friction coefficient 0.4 (for aluminum to steel contact) Surface behavior HARD contact Steel Fixture Contact Young’s Modulus, E 207 GPa Poisson’s ratio 0.292 Aluminum Workpiece Young’s Modulus, E 71 GPa Poisson’s ratio 0.334
34
5.2.1 Model 1 - Mittal’s FEM Model
The main purpose for the construction of model 1 is to apply the method used by
Mittal[2] in his study to finite element modeling. Mittal has used a translation spring
element to model the contact stiffness. In model 1, same approach is used to model the
stiffness at the contact with a SPRING1 element. The stiffness of the spring measured
with stiffness tests is used in the analysis. The clamp and locator setup is shown in Figure
5.1 and Figure 5.2 and the cutting forces are shown in Figure 5.3.
Table 5.3. Comparison of features between Mittal’s model and the proposed model.
Mechanical Property Mittal’s Model Proposed FEM Stiffness of clamp/locator No No Contact Stiffness Yes Yes Frictional contact No Yes
Figure 5.1. Model 1 after meshing (With reference to Mittal’s model).
35
Figure 5.2. Fixturing layout for model 1.
Figure 5.3. Machining profile for model 1 All the cutting data is summarized as follows:
Axial cutting force, Fa: 497N
Direction of cutter motion
36
Feed force, Ff: 348N
Torque about X-axis =+9.19Nm at the beginning of the cut and
-40.51Nm at the end of the cut.
Torque about Z-axis remains constant at 9.47Nm throughout the cut.
These cutting forces are assigned to nine different locations along the cutting path.
Other information such as the material property, contact property, etc are tabulated in
Table 5.4.
Table 5.4. Modeling data for model 1
Friction coefficient 0.4 (for steel to steel contact) Spring stiffness 1.1x108 N/m Surface behavior HARD contact Clamping forces 1000 N Cutting speed 18.29m/min End mill diameter 19.05mm Depth of cut 6.35mm Workpiece size 100mm x 100mm x100mm Locators and clamps Spherical Model type Rigid
Mittal’s simulation results on the locators’ reaction forces are shown in Figure 5.4.
Overall, Mittal obtained a higher reaction forces for all the locators.
37
Figure 5.4. Reaction force vs time chart obtained by Mittal.
Simulation Result from FEM
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
00 0.1 0.2 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (sec)
Forc
e (N
)
LALBLCLDLELF
Figure 5.5. Results from finite element analysis
Although a good comparison cannot be made between the finite element results (Figure
5.5) and Mittal’s result (Figure 5.4), the trend of the individual reaction forces are quite
similar. Moreover, the introduction of frictional force between contacts in finite element
-2500
-2000
-1500
-1000
-500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Time (sec)
Forc
e (N
)
M_A
M_B
M_C
M_D
M_E
M_F
Locators
38
model also contributes to the difference in the two results. The main purpose of this model
is to verify the use of spring to represent the material or contact stiffness and from the
results, it can be concluded that it is feasible.
5.2.2 Model 2 - Tao’s FEM Model
A comparison study was made with Tao’s FEM model[6] and the developed FEM
model to see how accurate the predictions are. Tao’s model includes friction but not
fixture element stiffness. Modeling data for Tao’s model is shown in Table 5.6. Modeling
data for model 2 and fixturing layout in Figure 5.6 and Figure 5.7. The experimental result
obtained by Tao is shown in Figure 5.8. The results for two finite element models are
shown in Figure 5.9 and Figure 5.10 respectively. Both FEM results are comparable to
Tao’s experimental result, except for the three locators at the bottom, which have a
slightly higher reaction forces. The reason for this is mainly due to the approximation of
stiffness value for bottom locators. Reaction forces for locator L5 and L4 intersect each
other at an approximate time of 53 seconds, which yield the same intersection point in
Tao’s experiment. The trends of the charts are agreeable with each other.
Table 5.5. Comparison of features between Tao’s model and the proposed model.
Mechanical Property Tao’s Model Proposed FEM Stiffness of clamp/locator No Yes Contact Stiffness Yes Yes/No Frictional contact No Yes
39
Figure 5.6. Model 2 after meshing (With reference to Tao’s Model).
As Mittal’s model is lacking some conformity because his result for the analysis was
not compared to an experimental result, therefore, Tao’s model is chosen as an approach
to further verify the model built using finite element method.
Model 2 is built with two methods. In the first method, only friction coefficient and
stiffness of element are included in the model, contact stiffness is excluded. In the second
method, friction coefficient, element stiffness and contact stiffness are included. The
element stiffness for locator and clamp is represented by the use of SPRING1 element. In
model 1, element stiffness is not included in the model because the actual physical shape
of the fixtures is not specified by Mittal.
40
Figure 5.7. Fixture layout and location for model 2
Table 5.6. Modeling data for model 2
Friction coefficient 0.4 (for steel to steel contact) Spring stiffness 110 MN/m (linear) Surface behavior HARD contact Young’s Modulus, E 6.89 x 1010 Pa Poisson’s ratio 0.33
Clamp P1: 640N Clamping forces Clamp P2: 670N Cutting speed 100mm/min
Fx = -232N Fy = -55N Cutting Forces Fz = 131N
End mill diameter 18mm Depth of cut 3.0mm Workpiece size 122mm x 220mm x 112mm Locators and clamps Flat surface contact Model type Rigid/Deformable
Experimental result obtained by Tao is shown in Figure 5.8. The results for two finite
L0 (110, 10, 0)
L2 (210, 110, 0)
L1 (10, 110, 0)
L5 (10, 122, 60)
L3 (0, 60, 60)
P1 (110, 0, 60)
L4 (210, 122, 60)
P2 (220, 60, 60)
L=Locator C=Clamp
X Z
Y
41
element models are shown in Figure 5.9 and Figure 5.10 respectively. When the effect of
fixture stiffness is included into the model, the reaction forces of the analysis are slightly
lower than the one without the fixture stiffness. This leads to a conclusion that the
reaction forces are lowered with the introduction of the fixture stiffness. Therefore the
developed FEM model with fixture stiffness is in fact a safer prediction, leading to higher
clamping intensity required to keep the workpiece stable.
Figure 5.8. Reaction force vs time obtained in Tao’s experiment.
42
Figure 5.9. Finite element results for Tao’s model (without fixture element stiffness).
Figure 5.10. FEM results from developed model (with fixture element stiffness).
0
100
200
300
400
500
600
700
800
0 17 50 66 83 116 132
Time (s)
Forc
e (N
)
L0
L1
L2
L3
L4
L5
Locators
0
100
200
300
400
500
600
700
0 17 50 66 83 116 132
Time (s)
Forc
e (N
)
L0L1L2L3L4L5
Locators
43
5.3 Summary
In this chapter, a developed FEM model of the fixture-worpiece system which includes
the effect of fixture stiffness was described. Comparisons were made with two previous
models by Mittal and Tao and it was found that inclusion of fixture stiffness produces a
more conservative and hence safer solution. The next chapter reports on the verification
of this FEM model by two experimental case studies.
44
Chapter 6. EXPERIMENTAL VERIFICATION OF THE FINITE ELEMENT MODEL
6.1 Instrumentation
The entire fixture was mounted on a dynamometer which measures the cutting force in
three directions. Modular fixtures were assembled and each contact point ends in a sensor
structure with a circular contact surface. Piezoelectric force sensors measured the reaction
forces on the fixtures during clamping and machining. Signals from the force sensors
were amplified by charge amplifiers and recorded by a data recorder and PC. Only the
locators L4, L6 and L7 and the clamps C0, C2 and C8 had force sensors, a total of six
sensors (refer to fig. 6.4). There were no force sensors on supports S1, S3 and S5 because
from experience, the variations in reaction forces at the supports were insignificant
compared to that of the locators [6]. The experimental force results are converted from a
dynamic data to quasi-static data by taking the maximum of the reaction force at regular
time intervals. Initially before the workpiece was mounted, all reaction forces were
zeroed. Then clamping was done in this order: C8, C2 and C0, according to the clamping
forces shown in Table 6.2. This resulted in a pre-loading of all the contact points,
necessary to withstand the cutting forces. The cutting torque is calculated from the
following formula provided by EMSIM, an end milling simulation software developed by
the MT-AMRI[17]:
T=7116.04* (HP/ SS) ............................................................................................ (19)
Where T is the torque (Nm), HP is the power consumption (horsepower) and SS is the
45
spindle speed. The cutting forces and torque are computed and tabulated in Table 6.3.
Fixture stiffness values were obtained by performing load-deflection tests on the actual
modular fixture assemblies and summarized in Table 6.1. This stiffness refers to the ratio
of applied force over deflection in the normal direction of the fixture contact surface.
6.2 Stiffness Test
It is necessary to determine experimentally the stiffness of the fixture elements which
are represented in the FEM as linear spring stiffness. Stiffness is the ratio of applied force
over deflection in the normal direction of the fixture contact surface. A load was applied
in the direction as shown in Figure 6.1. The dial gauge measured the amount of deflection.
A Kistler slimline force sensor was used to measure the magnitude of the reaction forces
exerted by the fixture. A Sony data recorder and PC setup store the measurements. From
these measurements, a graph of applied force versus fixture deflection is plotted.
Figure 6.1. Schematic of the Fixture Stiffness Test.
The relationship between the force applied and the deflection on the support is shown in
locator
Force sensor
Force applied
Dial gauge
Base plate
46
Figure 6.2. The slope of the curve is the stiffness and it was found to be 1.42x108 N/m.
Force Vs Deflection
gradient = 1.42E+08 N/m
0
5000
10000
15000
20000
0 0.00005 0.0001 0.00015
Deflection/m
Forc
e/N
Figure 6.2. Relationship of force applied vs deflection on supporting element.
The locators and the clamps have the same physical structure and so they are assumed
to have the same stiffness and the relationship of the applied force and deflection is shown
in Figure 6.3. The stiffness of locators and clamps is 3.24x107 N/m. Table 6.1 summarizes
the stiffness of the locators, supports and clamps.
Table 6.1. Fixture element stiffness.
Fixture elements Stiffness
L0, L6, L7, C2, C4, C8 3.24x107 N/m
S1, S3, S5 1.42x108 N/m
47
Force v.s. Deflection
gradient = 3.24E+07 N/m
0
2000
4000
6000
8000
0 5E-05 0.0001 0.0002 0.0002 0.0003
Deflection/m
Forc
e/N
Figure 6.3. Relationship of force applied vs deflection on locating elements.
6.3 Description of Case Study 1
An aluminum block was to be end-milled with a slot feature (Figure 6.4). Cutter used
was a two-flute 10 mm end mill with a spindle speed of 1200 rpm and feed rate of 100
mm/min. The cutting profile was a horizontal pass with depth of cut 2 mm till the centre of
the workpiece, where the cutter descended by 1 mm to increase the subsequent depth of
cut to 3 mm.
48
Figure 6.4. Modeling of the workpiece and locations of clamps/locators for Case Study 1.
0
L0 (184,50,62)
8 mm
S5 (10, 0, 10)
S3 (171,0,10)
Z
X
S1 ( 92,0,106)
L7 (14,34,0) L6 (172,34,0)
C2 (92,34,114)
92
184
114 C4 (0, 50, 62)
Legend L locator S support C clamp
C8 (102,92,25)Cutter
direction
TOP VIEW
L0
Slot with a step is
machined
* All units in mm
L6
X
Y 92
C2
S3S1 S5
L7 C4
FRONT VIEW
C8
Spring attached to ground represents
fixture stiffness
Fixture contact surface
49
Figure 6.5. Experimental Setup for Case Study 1.
All reaction forces were zeroed after the workpiece was mounted. Clamping was first
done with C8 followed by C2 and lastly C4. The clamping forces were recorded as shown
in Table 6.2. A pre-loading of all the contact points is needed to withstand the cutting
forces. The cutting torque is calculated the same way as before. The cutting forces
measured and the torque computed are shown in Table 6.3.
Table 6.2. Clamping forces applied in sequence of Case Study 1.
1st clamp 2nd clamp 3rd clamp
C8 C2 C4
2121 N 751 N 830 N
Kistler slimline force sensor integrated with fixture
Clamps
Support
Locator
Aluminum workpiece
Three-axis force base plate
dynamometer
50
Table 6.3. Cutting data of Case Study 1.
Time t (s) Fx (N) Fy (N) Fz (N) Torque (Nm)
0 – 30 300 50 550 2.15
30 – 42 100 -250 0 2.15
42 – 70 500 100 700 2.15
The experimental force results are converted from a dynamic data to quasi-static data
by taking the maximum of the reaction force at regular time intervals. The variation due to
dynamic effects is small compared to the average magnitude of the reaction force. For
example, for the set of data points in Figure 6.6, the stddev is 3.65 N and range is13 N. In
this case for the reaction force at locator L7 between 0 s and 0.2 s, the local maximum is
taken as 2123 N.
Reaction force(N) at L7 vs time(s)
2100
2105
2110
2115
2120
2125
2130
2135
2140
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
time t(s)
Rea
ctio
n fo
rce(
N)
Maximum of reaction force
Figure 6.6. Typical dynamic force obtained from experiment. Reaction force is shown at locator 7.
51
6.4 Results & Discussions of Case Study 1
These two graphs (&) show the actual reaction force profile with respect to time for
each locator, clamp and support compared with the FEM predicted force profiles. Locator
L0 has an almost constant reaction force of about 820N before 30s which dips to a lower
level of 800N between 30 s and 42 s and rises to around 840N after 42 s. That is because
the cutting force in the x direction increased when the depth of cut increased from 2 mm to
3 mm, accounting for the higher reaction force after 42 s. Reaction force was lowest
between 30 s and 42 s because the end mill was cutting vertically and produced no force in
the x direction. Likewise for L6, L7, C2, C4 and C8, this distinct drop or rise in reaction
forces between 30s and 42s can be explained by the movement of the end mill vertically,
which produced no sideways cutting force. For locator L0, it is observed that the general
trend of the FEM prediction agrees with that of the experimental trend. The FEM profile
has a negative error throughout the whole profile. Also the range of variation in force is
nearly equal for the FEM profile and the experimental profile. Reaction forces for locators
L6 and L7 have opposing trends. Locator L6 has a general decrease in reaction force as
the cutter moves along the length of the tool path but locator L7 has a general increase.
The FEM profiles of L6 and L7 both have similar trends compared with the experimental
data (fig. 6.7). However these two FEM profiles show the largest range of variation in
force. The magnitude of error between the FEM and experimental plots of L6 and L7 lies
below 150N. Clamp C4 is the only fixture on the opposite side of L0, so we expect its
force profile to roughly mirror that of L0 and that is shown to be true. The magnitudes of
52
C4 and L0 are very close as seen in the experimental profiles, around 800N. The FEM
profiles for C4 and L0 are also mirror images about the horizontal and are good
predictions of the actual profiles.
The trends of all the FEM profiles approximate those of the actual reaction forces but
there exists an overall error which can be positive or negative and also the predicted
variations in magnitude are larger. A reason for this discrepancy could be the uncertainty
in chosen values of fixture stiffness and friction coefficient. However, since the general
trends are similar, this improved FEM model could be used to simulate the behaviour of
an actual fixture-workpiece system during machining, with some allowance for error.
53
Reaction Forces at Fixture Contacts vs. Time(Locators and supports)
0
200
400
600
800
1000
1200
1400
-10 0 10 20 30 40 50 60 70 80
Time /s
Rea
ctio
n Fo
rce
/N
L0 EXPT L0-FEM S1-FEM S3-FEM S5-FEM
L6 EXPT L6-FEM L7 EXPT L7-FEM
S3-FEM
L7-FEM
L6 EXPT
L6-FEM
S5-FEML0 EXPT
S1-FEM
L0-FEM
L7 EXPT
Figure 6.7 A graph of reaction forces of supports and locators vs time of Case Study 1.
54
Reaction Forces at Fixture Contacts vs. Time (Clamps)
0
500
1000
1500
2000
2500
3000
3500
-10 0 10 20 30 40 50 60 70 80
Time /s
Rea
ctio
n Fo
rce
/N
C2 EXPT C2-FEM C4 EXPT C4-FEM C8 EXPT C8-FEM
C2-FEM
C4 EXPT
C8-FEM
C2 EXPT
C4-FEM
C8 EXPT
Figure 6.8 A graph of reaction forces of clamps vs time of Case Study 1.
55
6.5 Description of Case Study 2
Another aluminum block of size 184mm x 114mm for the base and a height of 90mm on
the lower side and 92mm on the higher side with a slope of 2mm in height measuring from
length of 69mm to 115mm on the top surface was to be end-milled with a slot feature. The
cutter used was a two-flute 10 mm end mill with a spindle speed of 1200 rpm and feed rate
of 200 mm/min. The cutting profile was a horizontal pass with depth of cut 1 mm from
lower side of the workpiece. The cutter followed the top surface profile of the workpiece,
increasing with the slope until a subsequent depth of cut to 3 mm.
Figure 6.9. Experimental Setup for Case Study 2.
Exactly the same fixture setup and instrumentation was used as in Case Study 1. The
56
sequence of clamping was also the same as in Case Study 1 in the order of C8, followed
by C2 and then C4. The clamping force observed is tabulated in Table 6.4. Torque was
computed using the formula provided by EMSIM. As there was a slope in the center of the
work piece, a simple linear formula was derived for the cutting force at the slope section.
The cutting force that was measured and the torque computed are listed in Table 6.5.
Table 6.4. Clamping forces applied in sequence of Case Study 2
1st clamp 2nd clamp 3rd clamp
C8 C2 C4
2129 N 768 N 832 N
Table 6.5. Cutting data of Case Study 2
Time t (s) Fx (N) Fy (N) Fz (N) Torque (Nm)
0 – 19 80 -2 100 2.15
19 – 38 6.32×(t-19) + 80 1.16×(t-19) -2 13.2×(t-19) + 100 2.15
38 - 60 200 20 350 2.15
The results obtained by the experiment are processed and plotted on graphs. They are
then compared with data obtained using the FEM. The schematic diagram of the model is
shown in Figure 6.10.
57
Figure 6.10. Dimension of workpiece and locations of clamps/locators of Case Study 2.
0
L0 (184,50,62)
8 mm
S5 (10, 0, 10)
S3 (171,0,10)
Z
X
S1 ( 92,0,106)
L7 (14,34,0) L6 (172,34,0)
C2 (92,34,114)
69
184
114 C4 (0, 50, 62)
Legend L locator S support C clamp
C8 (102,92,25) Cutter
direction
TOP VIEW
L0
Slope of height 2 mm
* All units in mm
L6
X
Y 90
C2
S3 S1 S5
L7C4
FRONT VIEW
C8
Spring attached to ground represents
fixture stiffness
Fixture contact surface
69
92
58
6.6 Results & Discussion of Case Study 2
A comparison of the actual reaction force and FEM-predicted reaction force with
respect to time is shown in the two graphs in Figure 6.11 and Figure 6.12 and will be
referred to in this discussion.
All force profiles consist of three distinct linear segments; before 19s, between 19s and
38s and after 38s. The first segment and third segment exhibit the same increasing or
decreasing behaviour, depending on the fixture element. This is linked by a second
segment with a steeper gradient, but not necessarily of the same sign as the gradient of the
first and third segments. The reason for this is the linearly increasing depth of cut from 1
mm to 3 mm as the end mill passes the sloped region of the workpiece.
Locator L0 produces an almost constant reaction force of 815N before 19s, which
linearly increases to about 830N between 19 s and 38s and gradually rises to 840N at
60s.The increase in reaction force was significant between 19 s and 38 s because the end
mill was traversing across a slope of increasing gradient and thus produced the increase.
Reaction force after 38s was higher because the cutting force in the x direction (towards
L0) increased when the depth of cut increased to 3 mm. It is observed that the general
trend of the FEM prediction makes a good comparison with that of the experimental
trend, but in overall the FEM profile predicted reaction forces lower than that of the
experiment as shown in Fig. 6.11. The range of variation in force between the FEM
profile and the experimental profile is in a close agreement.
59
For Locator L6, reaction force is initially at 270N and increases until 19s where the
reaction force decreases a little until 38s, and after 38s, reaction force increases gradually.
Reaction force for locator L7 starts at 500N, and decreases linearly. As the end mill
moves away from L7 and towards L6, the reaction force decreases for L7 and increases
for L6. It can be observed that reaction forces for locators L6 and L7 have the opposing
trends. The FEM profiles of L6 and L7 both have similar trends with that of the
experimental data and the range of variation is also correspondingly small.
For clamp C2, reaction force is initially at 760N before 19s and increases slightly to
800N until 38s and remains at 800N to the end of the cut. This is due to the increase in the
depth of cut across the slope causing the end mill to exert increasing force in the
z-direction (towards C2). For the FEM prediction, it seems to project a rather
conservative increase, from around 750N to 765N. For locator C4, the reaction force
decreases very minimally in the experimental data, about 840N from the start to about
830N till the end. The FEM projects a more generous decrease in the reaction. C4 is the
clamp opposite that of L0, and as the cutter moves towards L0, and away from C4, the
cutter exerts increasing force in the x-direction (towards L0). Thus the reaction forces
decreases for C4 and increases for L0, reflecting a mirror image of each other. Reaction
forces at clamp C8 remains almost constant throughout the milling process. The FEM
data compares well with the experimental data.
Although the FEM profiles shows similarity to those of the actual reaction forces at the
60
locators and clamps, there are errors present that can be positive or negative. The
magnitudes in variations for the finite element modeling are also noticeable. The
uncertainty in chosen values of fixture stiffness and friction coefficient could be reasons
for the discrepancy. Since the general trends compares well with the experimental data,
this improved finite element model allowing room for a slight error could be used to
simulate the behaviour of an actual fixture-workpiece system during machining.
61
Reaction Forces at Fixture Contacts vs. Time FEM simulation and experimental results
(Locators and supports)
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
900.00
1000.00
0 10 20 30 40 50 60 70
Time /s
Rea
ctio
n Fo
rce
/N
L0 EXPT L0-FEM S1-FEM S3-FEM S5-FEML6 EXPT L6-FEM L7 EXPT L7-FEM
S3-FEM
L7-FEM
L6 EXPT
L6-FEM
S5-FEM
L0 EXPT
S1-FEM
L0-FEM
L7 EXPT
Figure 6.11. A graph of reaction forces of locators and supports vs time of Case Study 2.
62
Reaction Forces at Fixture Contacts vs. Time FEM simulation and experimental results
(Clamps)
7.00E+02
9.00E+02
1.10E+03
1.30E+03
1.50E+03
1.70E+03
1.90E+03
2.10E+03
2.30E+03
0 10 20 30 40 50 60 70
Time /s
Rea
ctio
n Fo
rce
/N
C2 EXPT C2-FEM C4 EXPT C4-FEM C8 EXPT C8-FEM
C2-FEM
C4 EXPT
C8-FEM
C2 EXPT
C4-FEM
C8 EXPT
Figure 6.12. A graph of reaction forces of clamps vs time of Case Study 2.
63
Chapter 7. CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions
The developed FEM model is able to provide a realistic simulation of the
fixture-workpiece interaction during machining than Mittal’s[2] and Tao’s[6] models as it
takes into account fixture stiffness, contact friction and elasticity of the workpiece. With
this increased complexity, greater accuracy can be obtained from the model. Inputs such
as the fixture stiffness and friction coefficient are not easily determined. Presence of
coolant and surface profile are some factors that may affect the friction coefficient at the
fixture contacts. This work has presented three techniques of fixture analysis which may
be used by a fixture designer in a complementary manner. The initial stages of choosing
clamping positions and surfaces and clamping sequence are handled well by the force
closure method. Then for fixture analysis, one can use either the optimization method or
FEM, depending on the level of accuracy required. The optimization method is very fast
but gives only information about minimum clamping forces. FEM allows for analyzing
workpiece deformation, location error and fixture deformation. The main drawbacks of
FEM are: the need to determine accurate inputs and longer time for constructing the
model and running the simulation.
64
7.2 Recommendations
In research there is always room for investigation and improvement. The following
research directions are recommended:
• Dynamic modeling of the workpiece-fixture system in FEM
Further research can be pursued in the investigation of dynamic effects.
Abaqus/Explicit is a software module that supports the FEM modeling of dynamic
interactions. Focus should be on the dampening effect of the fixture and predicting
and avoiding the natural frequency of the system. Experimental work can be
conducted to analyse the spectral distribution of vibrations in the workpiece and the
effect on the workpiece quality.
• Integrated Capacitance Force Sensor System
This experimental sensor system has many benefits, most significant of which is
robustness and cost-effectiveness. It is much more durable than Kistler’s slimline
piezoelectric force sensors, which often receive damage in the cable leads. Further
work needs to be done on the microprocessor circuit to account for the drift in
readings because of changes in temperature. Sampling rate has to be raised to an
acceptable level of at least 100Hz if it is to measure the dynamic effects.
• Prediction of Fixture Stiffness
A FEM model of the fixture assembly can be formulated, with the representation of
fastener joints as simplified finite elements. By applying varying loads and
checking the deflection in the same direction, a load-deflection graph can be
plotted. From this graph, the fixture stiffness value for the SPRING element can be
65
determined. If this can be proven to be reasonably accurate, then there is no need
for stiffness tests. A second use for this fixture stiffness model would be to warn the
fixture designer if a fixture assembly is not sufficiently stiff.
66
REFERENCES
[1] S. H. Lee, M. R. Cutkosky, “Fixture Planning with Friction”, Transactions of the
ASME, Vol. 13, August 1991.
[2] R. O. Mittal, Paul H. Cohen, B. J. Gilmore, “Dynamic Modeling of the
Fixture-Workpiece System”, Robotics and Computer-Integrated Manufacturing,
Vol. 8 No. 4, pp. 201-217, 1991.
[3] De Meter E.C., Min-max load model for optimizing machining fixture performance,
Journal of Engineering for Industry, 117, pp 186-193, 1995.
[4] B. Li and S. N. Melkote, Optimal Fixture Design Accounting for the Effect of
Workpiece Dynamics, International Journal of Advanced Manufacturing Technology
(2001) 18:701–707.
[5] J. D. Lee, L. S. Haynes, “Finite Element Analysis of Flexible Fixturing System”,
Transactions of the ASME, Vol. 109, May 1987.
[6] Tao, Z. J., A. Senthil, Kumar, Nee A.Y.C., Automatic Generation Of Dynamic
Clamping Forces For Machining Fixtures, International Journal of Production
Research, 37(12), pp. 2755-2776, 1999.
[7] Kevin Rong, Stephens Zhu, “Computer-Aided Fixture Design”, Marcel Dekker, Inc.,
1999.
[8] IMAO Venlic Block Jig System (BJS) Catalog, IMAO Corporation, Japan, 1990.
[9] Quick Hull Library, The Geometry Center,
http://www.geom.umn.edu/software/qhull/, 2001.
67
[10] A. Senthil, Kumar, Fuh, Y. H., Kow, T. S., An Automated Design And Assembly of
Interference-free Modular Fixture Setup, Computer-Aided Design, 32(2000), pp.
583-596, 1999.
[11] Tao, Z. J., A. Senthil, Kumar, Nee A.Y.C., A Computational Geometry Approach to
Optimum Clamping Synthesis of Machining Fixtures, International Journal of
Production Research, 1998.
[12] Kerr, J., B. Beth, Analysis of Multi-fingered hands, International Journal of Robotics
Research, 4(4) pp. 3-17, 1986.
[13] Mishra B. , N. Silver, Some discussion of static gripping and its stability, IEEE Sys.
Man. Cybernet, 19(4), pp 783-796, 1989.
[14] Quick Hull Library, The Geometry Center,
http://www.geom.umn.edu/software/qhull/, 2001.
[15] Raymond J. Roak & Warren C. Young, “Formulas for Stress and Strain”,
McGraw-Hill International Book Company, 1984
[16] Hibbit, Karlsson & Sorensen, Inc., ABAQUS/Standard 6.2 User Manual, 2001.
[17] DeMeter, E.C., Sayeed Q., R.E. DeVor, R.E. and S.G. Kapoor, S.G. “An
internet-based model for technology integration and access, Part 2: Application to
process modeling and fixture design” Symposium on Agile Manufacturing, ASME
International Mechanical Engineering Congress & Exposition, 1995.
A-1
APPENDIX
A-1. Data acquisition during experiment
1. Edit the calibration file data
2. Setup the workpiece and start the NC program
3. Read Tm, from all eight sensors, continuously before commencing cutting.
Actual time, t, is recorded for each reading.
4. Start cutting
5. Stop reading sensors when cutting is finished
6. Repeat from step 2 for all runs
7. Data processing
A-2. Program Functions in Pseudocode
A-2.1. Reading a single sensor 100 times
Single read button is pressed.
Case
{
Case “0”:
Loop 100 times
{
Send “A” to tell the microprocessor to read sensors 0
Wait a fixed time for the reply.
Store time, t and Tm for sensor 0.
}
Case “7”:
A-2
Loop 100 times
{
Send “H” to tell the microprocessor to read sensors 7
Wait a fixed time for the reply.
Store time, t and Tm for sensor 7.
}
}
Plot results using Microsoft Chart object
A-2.2. Recording all sensors for 100 times
Loop for 100 times
{
Send “AB” to tell the microprocessor to read sensors 0 and 1.
Wait a fixed time for the reply.
Store time, t and Tm for sensors 0 and 1.
Send “CD” to tell the microprocessor to read sensors 2 and 3.
Wait a fixed time for the reply.
Store time, t and Tm for sensors 2 and 3.
Send “EF” to tell the microprocessor to read sensors 4 and 5.
Wait a fixed time for the reply.
Store time t and Tm for sensors 4 and 5.
Send “GH” to tell the microprocessor to read sensor 6 and 7.
Store time t and Tm for sensors 6 and 7.
}
Save results file
Plot results using Microsoft Chart object
A-3
A-2.3. Continuous recording function
Do Loop
{
Send “AB” to tell the microprocessor to read sensors 0 and 1.
Wait a fixed time for the reply.
Store time, t and Tm for sensors 0 and 1.
Send “CD” to tell the microprocessor to read sensors 2 and 3.
Wait a fixed time for the reply.
Store time, t and Tm for sensors 2 and 3.
Send “EF” to tell the microprocessor to read sensors 4 and 5.
Wait a fixed time for the reply.
Store time t and Tm for sensors 4 and 5.
Send “GH” to tell the microprocessor to read sensor 6 and 7.
Store time t and Tm for sensors 6 and 7.
} Loop until stopped by user
Save results file
Plot results using Microsoft Chart object
A-4
Publication Resulting from this Thesis
1. Modeling, analysis and verification of optimal fixturing design, Ernest Y. T. Tan, A
Senthil Kumar*, J. Y. H. Fuh, A. Y. C. Nee, submitted to IEEE Journal of
Automation Science and Engineering, Special Issue on Fixturing, March 2003.
