CHAPTER 6 TOLERANCE MODEL FOR WORKPIECE...
Transcript of CHAPTER 6 TOLERANCE MODEL FOR WORKPIECE...
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CHAPTER 6
TOLERANCE MODEL FOR
WORKPIECE-FIXTURE SYSTEM
6.1 GEOMETRIC ERROR MINIMIZATION
To illustrate the optimal design methodology for fixture design
analysis, consider two examples using the developed software on ANSYS.
The first example is a fairly simple example of a thin part considered for
Multi spindle drilling operation where four holes have to be drilled
simultaneously. This example demonstrates how excessive deflection can be
reduced by optimization by using one of the intelligent optimization
techniques Genetic algorithm (GA), which is mainly useful when thin
castings have to be machined. The second example is a comprehensive one in
which a design of a fixture is sought to handle machining of several holes in a
sequence drilling operation using single spindle.
6.2 MATHEMATICAL EXPRESSION FOR LOCATING ERROR
ANALYSIS
The repeatedly used geometric tolerances include flatness, profile
of surface, angularity, perpendicularity, parallelism and position and are
shown in Table 6.1. In this research, the position and perpendicularity of the
hole on the workpiece in the fixture system is explained in Figure 6.1(a). The
CSp is the theoretical exact fixturing coordinate system which is composed by
theoretical exact locators and theoretical exact locating features.
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Table 6.1 Examples of tolerance definition
Tolerance Symbols DefinitionsStraightness ____ A condition where an element of a surface or an axis is
a straight line. Flatness
A two dimensional tolerance zone defined by two parallel planes within which the entire surface must lie.
Parallelism // The condition of a surface or axis which is equidistant to all points from a datum of reference.
Angularity
The distance between two parallel planes, inclined at a specified basic angle in which the surface, axis, or center plane of the feature must lie.
True Position
A zone within which the center, axis, or center plane of a feature of size is permitted to vary from its true (theoretically exact) position.
Roundness
A circularity tolerance specifies a tolerance zone bounded by two concentric circles within which each circular element of the surface must lie, and applies independently at any plane.
6.2.1 Position of a Hole
The deviation calculation for position type is a little different from that for the other types. The sample points are derived from the cylinder axis instead of from the surface contour. Refer to Figure 6.1(b), it has two datum and one target of the hole. Figure 6.1(a) shows the true position and perpendicularity of the hole (Rong et al 2001).
(a) (b) (c)
Figure 6.1 Position and perpendicularity of a hole
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aipos
i noPPt (6.1)
2Ttmax POSpos
i (6.2)
Where an = the unit vector of the datum plane
oP = the theoretical exact locating point of the axis
iP = any arbitrary point on the substitute axis
ipost = the position value of the point iP
POST = the required position tolerance of the hole (actually the axis)
6.2.2 Perpendicularity of a Hole
The perpendicularity of the hole can be found out by using the
following analytical solution. Refer to Figure 6.1 (c), it has one datum and
one target of the hole.
aiperp
i noPPt (6.3)
2T
tmax perpperpi (6.4)
where
oP = the locating point of the substitute axis,
perpit = the perpendicularity value of point iP
perpT = the required perpendicularity tolerance of the hole
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6.2.3 Perpendicularity of a Plane
The perpendicularity of the plane can be found out by using the
following analytical solution. Refer to Figure 6.2 it has one datum and one
target of a plane.
P-P’O P ns PO P’O CSP na
Figure 6.2 Perpendicularity of plane
0nn as (6.5)
oPPnt isperp
i (6.6)
2T
tmax perpperpi (6.7)
where
an = the unit vector of the normal direction of the
primary datum plane
sn = the unit vector of the normal direction of the theoretical
exact plane
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iP = any arbitrary point on the substitute plane
oP = the centre point of the substitute axis
perpit = the perpendicularity value of point iP
perpT = the required perpendicularity tolerance of the plane
6.2.4 Roundness (Circularity)
The roundness or circularity tolerance defines a tolerance zone
bounded by two concentric circles so that all the surface elements should lie
within this zone. It is generally used to define cylinder shaped surfaces. For
instance, the roundness of a cylinder is illustrated in Figure 6.3(a). The
tolerance zone is depicted as in Figure 6.3(b). The roundness for a cylinder
shaped surface is calculated with the following procedure: (1) calculate the
best fit cylinder for a given set of points on a cylinder (xi, yi, zi), i =1,�, n,
and get the parameters of the axis (a, b, c), the locating point (x0, y0, z0), and
the diameter R using the linear least square estimation method (LLSE) (Rice
1988), and (2) calculate the maximum and minimum distances of all surface
points to the fitted cylinder axis.
(a) (b)
Figure 6.3 Circularity
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The difference between the maximum and minimum distances is
estimated as the roundness. Figure 6.4 shows the tolerance specification of a
drilled hole. Figure 6.5 shows the experimental measurement using CMM.
Figure 6.4 Tolerance specification of a drilled hole
Figure 6.5 Experimental measurement using CMM
6.3 GEOMETRIC MANUFACTURING ERRORS
The purpose of optimal fixture configuration design may use an
alternative expression to evaluate the geometric errors based on the point-wise
deviation (Wang 2002). Select a number of points on the machined features of
the workpiece as the critical points of concern. Those points may belong to a
0.3
0.3
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single feature, for example, as the vertex points of a flat surface. They may
belong to multiple features, for example, as the center points of multiple
drilled holes. For a set of m critical points, a direct evaluation of these variations is
to use the sum of their squared magnitudes, such that (Wang 2002):
yyT (6.8)
TGAG )( 11 (6.9)
where
2
j
m
1js and (6.10)
2
jj
jm
1j RRRI
A (6.11)
The positional error of a hole can be described as a directional
point-wise error which is described as the following. If the plane
perpendicular to the axis of the hole is represented with two orthogonal unit
vectors m1 and m2 at the hole center sj, then the positional error of the hole
can be described as:
2
jTj2
2
jTj1
2j smsmd (6.12)
If multiple holes are to be machined in a single fixture setup, it is
reasonable to use the sum of the positional errors 2jj d of all the holes for a
representation of their overall error.
Another example is about the perpendicularity of a hole to a surface
shown in Figure 6.6. According to the standards, the perpendicularity form
error is being evaluated based on the following calculation of the hole center
point deviations at its end positions s1 and s2:
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2T21
T1 smsmt (6.13)
m1 s1 s2 m2
Figure 6.6 Perpendicularity error of the hole axis
6.4 DRILLING SIMULATION STUDY
6.4.1 Workpiece Solid Model
The workpiece model is the preliminary point of the analysis. This
research currently limits the workpiece geometry to solids with planar
locating surfaces. The workpiece model, created in ANSYS or in any other
solid modeling software that is imported to ANSYS. Figure 6.7 shows
deformable plate workpiece solid model created in ANSYS.
Figure 6.7 Solid model of thin workpiece
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6.4.2 Meshed Workpiece Model
Figure 6.8 shows an eight nodded hexahedral meshed workpiece
model. This hexahedral element with 3DOF at each node and linear
displacement behaviour is selected to mesh the workpiece.
Figure 6.8 Meshed workpiece model
SOLID 45 is used for the 3D modeling of solid structures. The
element is defined by 8 nodes having 3DOF at each node that is translations
in the nodal X, Y, & Z directions. The SOLID 45 element degenerates to a 4
node tetrahedral configuration with 3DOF per node. The tetrahedral
configuration is more suitable for meshing non-prismatic geometry, but is less
accurate than the Hex configuration. ANSYS recommends that no more than
10% of the mesh be comprised of SOLID 45 elements in tetrahedral
configuration. This research provides a methodology for describing the shape
and dimensions of machined surfaces after machining which incorporates the
effects of removed material on the workpiece deformation state and the
dimensions of machined regions. The essence of the methodology is to predict
the location of the series of points that come in contact with the drill bit outer
surface as the drill penetrates through the deformed workpiece. These points
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are in a deformed state and have travelled during machining from their
original location. The identification of the original location of these points
will provide the shape and the dimensions of the hole subsequent to
machining. In the deformed state, these points form a perfect cylinder while
coming into contact with the drill bit as the drill bit penetrates the workpiece.
It is assumed that the drill bit is rigid. The nodes are assumed to get displaced
from their original locations after the element deforms from coordinates xi, yi
to xi�, yi�. There exists a line containing points p� and q� on the drill bit outer
surface, and these points are displaced from positions p and q. The curve
containing p and q become a straight line containing p� and q�. One needs to
determine p and q from the knowledge of the coordinates of p� and q�. The
coordinates of p� and q� are known quantities with respect to a global
reference frame. Figure 6.9 shows an eight node hexahedral element.
Figure 6.9 An eight node hexahedral element (Wardak 1999)
6.5 BOUNDARY CONDITIONS
Locators and clamps define the boundary condition of the
workpiece model. The locators can be modeled as area contact and clamps are
modeled as area pressures.
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6.5.1 Locators
To model a rigid locator with contact area, multiple nodes are fixed
within the contact area. An LCS is created on the workpiece surface at the
center of the locator contact area. All DOF of each of the nodes are
constrained. This model assumes rigid constraints; however in reality locators
are elastic.
6.5.2 Clamps
The clamps are used to fully constrain the workpiece once it is
located. It is common to use multiple clamps and clamping forces that are
generally constant for each clamp. The clamping force, Fcl is applied through
either a hydraulic mechanism, which moves the plunger that comes to in
contact with workpiece or through a mechanical means. Friction is just as
important clamping parameter as it is in locating a comprehensive 3D model
of the entire workpiece-fixture system with contact and target surfaces
defined at the fixture-workpiece contact areas, and is required. The clamping
forces are modeled in ANSYS as surface pressures on contact areas selected
either within a circular area on the workpiece surface for a hydraulic clamp or
mechanical device. Both clamps may also be modeled with a circular contact
area.
6.5.3 Loading
The purpose of this research is not to accurately model the
machining process, but to apply and analyze the torque and forces that are
transferred through the workpiece during machining, to determine the
reactions at the boundary conditions of the workpiece. The forces in a drilling
operation include a torque (T), to generate tool rotation, shear force (V),
created by tool rotation at the cutting edge contact for chip removal, and an
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axial load (P), due to feeding. The forces in drilling are time and position
dependent and oscillatory due to cutter rotation, since the cutting edge of the
tool is not in constant contact with the workpiece at a particular location. The
cutting force increases monotonically during tool entry and then approaches
steady-state. Fluctuations in the cutting force are due to cutting tool tooth
distribution during rotation. In this study, the torque and thrust forces in
feeding are applied as steady-state loads, since initial tool entry is not
considered. In previous FEA fixture design research, loads were applied as
steady-state load. An initial attempt to model the distributed load using a
number of point loads applied at key points, was unsuccessful due to
limitations in ANSYS. The model consisted of placing key points on a local
coordinate system created on the machining surface of the workpiece. The
key points were located at exact R, , and Z positions on the cutting tool
perimeter. At each key point forces were applied to model a drilling
operation. The torque was modeled with tangential forces placed at the outer
radius of the cutting tool contact area. The tangential couple forces were
decomposed into global X and Y components. The axial load was modeled by
applying forces at each key point in the global Z direction. The reason for the
failure of this model is that the key points created on the workpiece surface
are geometric entities and are not part of the finite element mesh, i.e., key
points are not nodes. Due to this limitation in ANSYS, the point load model
was modified to apply loads at existing nodes on the workpiece surface.
Figure 6.10 shows the modified load model for drilling. Note that node i is
slightly offset from the cutting tool perimeter. Because a node may not exist
in the exact location specified by R, , and Z, the node closest to that location
in the local coordinate system is selected and forces were applied as point
loads with global X, Y, Z components. The user may minimize the distance
between a specified coordinate location and an existing node by increasing
the mesh density. The nodes are selected at equivalent intervals on or near
the cutting tool perimeter. At each selected node, global X and Y components
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Y
X
FtiY Fti
Node i
Cutting Tool Perimeter
FtiX
r Cutter
Rotation
of the tangential couple force, Fti and axial load component, Fci were applied.
The applied torque is equal to the sum of the tangential forces multiplied by
the cutting tool radius (r). FtiX and FtiY are the global X and Y components,
respectively, of the tangential force ( Fti). Fci is equal to the total axial load,
Fc, divided by the number of nodes over which it is applied. A simplified
model entails the use of a single point force normal to the surface of the
workpiece to model the cutting tool axial load and a couple to model the
applied torque. A study was conducted to determine whether multiple point
forces applied along the cutting tool perimeter are actually necessary to model
the axial load and assess the validity of the simplified model.
Figure 6.10 Drilling load model
6.6 CUTTING PARAMETERS
Figure 6.11 shows that spiral drill for feed rate
Figure 6.11 Spiral drill for feed rate
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The cutting parameters, and therefore the operating parameters of a
drill for drilling operations are:
d: tool and hole diameter (mm)
vc: cutting speed (m/ min) which gives the rotational speed of the
tool (rev/ min)
rpm4545
143.71001000
dv1000N c
f : feed per rotation in mm/ rev
The resulting performance parameter is:
*vf :feed rate in mm/min
vf = f × N = 0.4 × 4545 = 1818 mm/min
The feed rate is one of the main factors of productivity, as it
conditions chip-to-chip time t = p / vf.
t = p / vf = 6.35/1818 = 0.00349 min. (p: hole depth)
6.6.1 Cutting Force
Thrust force and cutting force are important parameters because
they make it possible to select and invest in a machine whose characteristics
are suited to the operation being carried out to obtain the cutting conditions
that allow the machine's power to be used in the most effective way possible,
so as to ensure optimal material removal rate while taking into account the
capacity of the tool being used. Figure 6.12 shows that spiral drill for axial
thrust.
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Figure 6.12 Spiral drill for axial thrust
The axial thrust F (N) can be estimated with the following formula:
N5002
143.74.07005.02
dfkkF c'
kc : specific cutting force (N/mm2), which depends primarily on
the material being machined (refere Table 6.2)
f : feed per rotation (mm)
d : tool diameter (mm) (the coefficient k' depends on the geometry
of the tip of the tool, consider an average value of 0.5)
6.6.2 Drilling Torque
Drilling torque is expressed as:
Figure 6.13 Spiral drill for drilling torque
The above Figure 6.13 shows that spiral drill for drilling torque
Nmm28.1
8000143.74.0500
8000dfkM
22c
c
Mc : drilling torque in Nmm
kc : specific cutting force in N/mm2
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f : feed per rotation in mm
d : tool diameter in mm
6.6.3 Power of Metal Cutting
Cutting power is expressed as:
Figure 6.14 Spiral drill for power of metal cutting
The above Figure 6.14 shows that spiral drill for power of metal
cutting
kW59525.0240000
100143.74.0500240000
vdfkP ccc
Pc : cutting power in kW
kc : specific cutting force (N/ mm2)
f : feed per rotation (mm)
d : tool diameter (mm)
vc : cutting speed (m/min)
(Compared to steels, the specific cutting force for aluminium alloys
is low (3 times lower) but the cutting speeds used are high (3 to 5 times faster
than for steel). This leads to low thrust forces, comparable power values, but
only at high rotational speeds that are sometimes difficult to attain when a
small-diameter tool is used).
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6.6.4 Specific Cutting Force
The specific cutting force, kc (N/mm2) is mainly a function of:
a) the material being machined
b) the feed
c) the cutting geometry
d) tool wear (an increase of 30 to 40%)
The Table 6.2 gives values of kc for:
f = 0.4 mm, a geometry and a cutting speed that are suited to the
material.
Table 6.2 kc values for aluminium alloy materials
Class of material kc* new tool kc* worn tool
Aluminium alloys: -forgedannealed - cast Si<13
500 700
Aluminium alloys: -forged aged - cast Si>13
750 1,050
6.7 THREE DIMENSIONAL FINITE ELEMENT MODELING
The finite element method involves the approach of obtaining
numerical approximations to the exact solutions of partial differential
equations. In the case of solid mechanics, the finite element method is used to
calculate approximate solutions to the boundary value problems of elasticity.
The finite element method always consists of an algorithm which involves
three steps:
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(1) Pre-Processing: Describing the geometry, material properties
selection, state the boundary conditions of the domain.
Discretize the domain through meshing. This consists of
creating nodes and finite elements within the domain.
(2) Processing: Solve the finite element equations resulting from
the boundary value problem for all of the pertinent field
quantities that are displacements, strains, stresses,
temperatures, plasticity, etc.
(3) Post-Processing: Display the results in a meaningful way, and
verify the correctness of the results.
In this work the GA implemented with a FEM tool, following the
static equilibrium condition for each time step ti as it is written on
Equation (6.14) ( Sales et al 2008 ),
T,...,2,1ifor,trtuK ii (6.14)
where: K is the structural stiffness matrix; u(ti) is the structural
displacement vector; r(ti) is the instantaneous force vector; and i is the ith
time step.
The instantaneous force and displacement vectors may be written as
Tn2n1n
T22212
T12111
tr...trtr............
tr...trtrtr...trtr
R (6.15)
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Tn2n1n
T22212
T12111
tu...tutu............
tu...tututu...tutu
U (6.16)
where n is the total acquisition point quantity. Rewriting Equation (6.14):
RUK (6.17)
The degree of freedom that match the acquisition points present known displacements [U]a, where: bTb2b1b tu...tutuU , and T is
the total quantity of time steps. The unknown displacements are represented by [U]b, where aTa2a1a tu...tutuU . The equilibrium equations
can be divided as follows:
b
a
b
a
bbba
abaa
RR
UU
KKKK
(6.18)
Considering that all external forces [R]a are null:
b
a
b
a
bbba
abaa
R0
UU
KKKK
(6.19)
Taking the first equation set:
0UKUK babaaa (6.20)
So:
ababaaa RUKUK (6.21)
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The displacement matrix [U]a is calculated by an algorithm for
static solution case based on Saturnino (2004).
6.8 PRE-PROCESSING
6.8.1 Geometry and Mesh Generation
The geometry of the plate is subject to the given boundary conditions
as shown in Figure 6.15. Three layers of super elements were generated across the
thickness. Three of the said layers were removed from the interior of the hole.
Figure 6.16 shows experimental fixture setup with sample workpiece.
Figure 6.15 Workpiece Model
Figure 6.16 Experimental fixture setup with sample workpiece
50.8
All Dimensions are in mm.
38.1
76.2
101.6
6.35
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6.8.2 Material Properties
The material of the finite element model was specified as Al 6061
having the following material properties as shown in the below Table 6.3.
Table 6.3 Material properties (Haiyan Deng 2006)
S.No Description Properties
1 Workpiece Al 6061
2 Young�s modulus, E 70000 N/mm2
3 Poison ratio, 0.33
4 Yield strength, y 270 N/mm2
5 Density, 2700kg/m3
6 Static coefficient of friction, 0.375
Thus the above material properties were given as input variables to
ANSYS via the �Material Property� menu option.
6.8.3 Boundary Conditions
Locators and clamps define the boundary condition of the
workpiece model. The locators can be modeled as area contact points and
clamps are modeled as area pressure values.
6.8.4 Contact Elasticity Model Description
Consider a case in which a drilling operation is performed on the
workpiece. First the workpiece is mounted on the locators and the clamps are
applied as an external force. As long as there is no other external force, the
contact force generated at the clamp interface with the workpiece will be the
same as the applied clamp force. The sum of all contact forces will then be
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equal to the applied clamp force. When an external load such as the cutting
force is applied to drill the workpiece surface, the original clamping force will
now become the initial external applied force. Figure 6.17 shows the contact
status of workpiece, locators and clamping elements. This initial force may be
from the control handle of the clamp. During the machining process, the
contact between the workpiece and the fixture elements results in interactive
forces and pressure distribution at the workpiece-fixture interface.
Figure 6.17 Contact status of work piece and fixture elements
6.9 PROCESSING
The governing equations of the finite element method can be
obtained by minimizing the total potential energy of the system.
Equation (6.22) is the resulting system of linear equations.
[K] {uN} = {fN} (6.22)
Governing FE Equation, where [K] is the global stiffness matrix; uN
is the global nodal displacement vector, and fN represents the global nodal
forces. The finite element model developed in section was solved using
ANSYS.
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6.10 POST-PROCESSING
6.10.1 Displacement Field and Deformed Mesh
Figure 6.18 shows the deformed configuration of the finite element
mesh for the plate. This result was obtained using ANSYS post-processing
menu options. Clearly, the linear, small deformations small strain model
produces deformations which are very small compared to the plate
dimensions and as such a magnification factor is required to visually display
the deformed configuration.
Figure 6.18 Deformation of finite element mesh for the workpiece material
6.11 CASESTUDY I
A 3D Casestudy part is taken from Wardak (1999). This
Casestudy I describe the methodology used in this research. A rectangular
plate with lx=38.1mm (1.5inches), Lx=76.2mm (3inches), ly=50.8mm
(2inches), Ly=101.6mm (4 inches) and h=6.35mm (0.25 inches) is drilled
using a drill bit of 19.05mm (0.75 inches) diameter. The machining detail of
Casestudy I simulation is shown in Table 6.4. This problem under
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consideration is solved using the ANSYS software. The initial and optimum
position of the locators and clamps are shown in Tables 6.5 and 6.6.
Table 6.4 Machining details
Sl.No Description Details
1 Diameter of drill 19.05mm
2 Material of the drill HSS
3 Material of the workpiece
Al-7075
4 Cutting speed(V) 1.16 to 1.67m/sec
5 Spindle speed(S) 1500rpm
6 Feed 0.2mm/tooth
7 Horse power 1
Table 6.5 Initial position of fixture element layout (Casestudy I)
Fixture element Co-ordinates(x,y)mm
L1 31.14, 3.19
L2 23.51,3.78
L3 71.97,3.34
L4 16.58,79.38
L5 49.55,72.10
L6 40.40,34.04
C1 41.48,3.93
C2 55.16, 2.72
F1 1000N
F2 1000N
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Table 6.6 Optimal position of fixture element layout (Casestudy I)
Fixture element Co-ordinates(x,y)mmL1 36.08,3.61 L2 21.70,3.98 L3 73.04,3.48 L4 26.69,63.52 L5 48.50,64.57 L6 34.16,37.60 C1 39.26,3.15 C2 58.93,1.51 F1 1000N F2 1000N
Table 6.7 shows GA Parameters for Casestudy I. Figures 6.19 and
6.20 show convergence of GA for fixture layout and the relationship between
the average deformation values and number of generation of this Casestudy
problem. Figure 6.21 shows the relationship between the von Mises stresses
and number of generation.
Table 6.7 GA Parameters for Casestudy I
Variable ValueProbability of Mutation (Pm) 0.05 Probability of Crossover (Pc) 0.8 Population Size (Ps) 20 String length 180 Maximum Number of Iterations 50 Machining Force, Fy (N) 1000 Clamping Force (N) 770
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Figure 6.19 Convergence of GA for fixture layout of Casestudy I
Figure 6.20 Average deformation Vs Generation
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Figure 6.21 von Mises stress Vs Generation nos
Figures 6.22 and 6.23 show the workpiece deformation and von
Mises stress in initial fixture configuration.
Figure 6.22 Deformation of initial fixture configuration (Casestudy I)
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Figure 6.23 von Mises stress of initial fixture configuration (Casestudy I)
Figures 6.24 and 6.25 show the workpiece deformation and von
Mises stress in optimal fixture configuration. Figure 6.26 shows the deformed
configuration of the finite element mesh for the plate.
Figure 6.24 Deformation of optimum fixture configuration (Casestudy I)
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Figure 6.25 von Mises stress of optimum fixture configuration (Casestudy I)
Figure 6.26 Deformed configuration of the finite element mesh for the Workpiece
The magnitude of the maximum deformation was a significant
0.1493 inches (Wardak 1999). The results of GA-APDL drilling optimization
study are shown in Table A 6.1 in �Appendix 6�.
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6.12 CASESTUDY II
This example is a practically simple problem where a fixture is
designed to drill four holes simultaneously in a thin square Aluminum
workpiece (i.e. 100mm x 100mm x 6.35mm). The fixture design process is
first to determine the positions of the fixture elements (locators and clamps)
and the reference data to machine the workpiece.
Table 6.8 Machining details
Sl.No. Description Details
1 Diameter of drill 7.143mm
2 Material of the drill HSS
3 Material of the workpiece Al6061
4 Cutting speed(V) 80-100 m/min
5 Spindle speed(S) 4550rpm
6 Feed 0.4mm/tooth
7 Horse power 1
The machining detail of this simulation is shown in Table 6.8. The
positions of these fixture elements along with machining loads create as
boundary conditions to the finite element model. Figure 6.27 shows the
Simultaneous drilling experimental Fixture Setup. The fixture element
positions are optimized to minimize workpiece deflection based on the FEM
simulation. The constraints on this FEM model are six fixed displacement
constraints on the locator�s locations, the two clamping point locations, and
the four load cases.
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Figure 6.27 Simultaneous drilling experimental fixture setup
Figure 6.28 Multi drilling head
The initial and optimum position of the locators and clamps are
shown in Tables 6.9 and 6.10. Table 6.11 shows GA parameters for Casestudy
II and III. Figure 6.28 shows the multi drilling head used in this Casestudy.
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Table 6.9 Initial position of fixture element layout (Casestudy II)
Fixture element Co-ordinates(x,y)mm
L1 55.99, 3.24
L2 13.60, 3.19
L3 80.32, 3.25
L4 32.39, 77.90
L5 87.04, 71.10
L6 49.34, 26.71
C1 66.98, 3.00
C2 47.09, 3.17
F1 500N
F2 500N
Table 6.10 Optimal position of fixture element layout (Casestudy II)
Fixture element Co-Ordinates(x,y)mm
L1 57.70, 3.01
L2 22.72, 3.18
L3 66.87, 3.14
L4 15.97, 64.43
L5 85.74, 71.81
L6 53.98, 26.22
C1 60.19, 3.02
C2 71.33, 3.08
F1 500N
F2 500N
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Figures 6.29 and 6.30 show the workpiece deformation and von
Mises stress in initial fixture configuration. Figures 6.31 and 6.32 show the
workpiece deformation and von Mises stress in optimal fixture configuration.
Figure 6.29 Deformation of initial fixture configuration (Casestudy II)
Figure 6.30 von Mises stress of initial fixture configuration (Casestudy II)
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Figure 6.33 and 6.34 shows the convergence of GA for fixture
layout and the relationship between average deformation values and number
of generation. Figure 6.35 shows the relationship between the von Mises
stress and number of generation of this Casestudy problem. The results of
GA-APDL simultaneous drilling optimization study are shown in Table A7.1
in �Appendix 7�.
Figure 6.31 Deformation of optimum fixture configuration (Casestudy II)
Figure 6.32 von Mises stress of optimum fixture configuration (Casestudy II)
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Table 6.11 GA Parameters for Casestudy II and III
Variable ValueProbability of Mutation (Pm) 0.05 Probability of Crossover (Pc) 0.80 Population Size (Ps) 10 String length 180 Maximum Number of Iterations 50 Machining Force, Fy (N) 500 Clamping Force (N) 770
Figure 6.33 Convergence of GA for fixture layout of Casestudy II
Figure 6.34 Average deformation Vs generation
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Figure 6.35 von Mises stress Vs Generation of Casestudy II
6.13 CASE STUDY III
This example is concerned with drilling four holes sequentially in
the workpiece. With the reference data and fixture element positions are
determined by the initial fixture design, the constraints for solving the FEM
model and the optimization for deflection are applied. It will be shown that an
optimal configuration can be achieved that further minimizes the objective
function. The FEM meshed workpiece for Casestudy III is as shown in
Figure 6.36. Figure 6.37 shows the sequential drilling experimental Fixture
Setup. Since the loads that are assumed to be the result of drilling holes
through the aluminum, the supports should not be directly under them. For
this reason, the Global Constraint Region (GCR) is defined to be a square area
where the locators are located. The selection of these constraints depend on
practical considerations, for example, the size of a support will limit how
close the supports can be to each other. In order to demonstrate the robustness
of this optimization scheme consider drilling of four holes on the part on the
workpiece simultaneously. The objective is to minimize deflection
considering all these machining loads by finding an optimal location for the
fixture elements for all of these machining loads considered simultaneously.
117
The initial and optimum position of the locators and clamps are
shown in Tables 6.12 and 6.13. Figures 6.38 and 6.39 show the workpiece
deformation and von Mises stress in initial fixture configuration. Figures 6.40
and 6.41 show the workpiece deformation and von Mises stress in optimal
fixture configuration. The results of GA-APDL sequential drilling
optimization study are shown in Table A8.1 in �Appendix 8�.
Figure 6.36 FEM meshed workpiece for Casestudy III
Figure 6.37 Single spindle (sequential) drilling experimental fixture setup
118
Table 6.12 Initial position of fixture element layout (Casestudy III)
Fixture element Co-ordinates(x,y)mm
L1 42.89,3.09
L2 16.73,3.05
L3 67.41,3.09
L4 25.31,66.31
L5 68.32,63.92
L6 35.31,33.00
C1 59.95,3.11
C2 70.80,3.16
F1 500N
F2 500N
Table 6.13 Optimal position of fixture element layout (Casestudy III)
Fixture element Co-ordinates(x,y)mm
L1 62.05,3.11
L2 35.30,3.13
L3 72.52,3.01
L4 27.28,80.88
L5 65.46,74.91
L6 50.51,27.05
C1 42.25,3.04
C2 30.18,3.00
F1 500N
F2 500N
119
Figure 6.38 Deformation of initial fixture configuration (Casestudy III)
Figure 6.39 von Mises stress of initial fixture configuration (Casestudy III)
120
Figure 6.40 Deformation of optimum fixture configuration (Casestudy III)
Figure 6.41 von Mises stress of optimum fixture configuration (Casestudy III)
Figure 6.42 and 6.43 shows convergence of GA for fixture layout
and the relationship between the average deformation values and number of
generation. Figure 6.44 shows the relationship between the von Mises Stress
and number of generation of this Casestudy problem.
121
Figure 6.42 Convergence of GA for fixture layout of Casestudy III
Figure 6.43 Average deformation values Vs Generation
122
Figure 6.44 von Mises stress Vs Generation of Casestudy III
6.14 SUMMARY OF THE RESULTS
From the above two methods of drilling process, it has been found
that the geometric error is minimized only in the simultaneous drilling
process. It is evident that the deformation values in simultaneous drilling
process are 0.029575mm in initial fixture configuration and 0.026693mm in
optimum fixture configuration compared to 0.033022mm in initial fixture
configuration and 0.028257mm in optimum fixture configuration in the
sequential drilling process. It shows that the deformation value is reduced by
6% in simultaneous drilling process compared to sequential drilling process.
Table 6.14 shows the deformation values of initial and optimum
fixture layout of Casestudy I, II and III. Tables 6.15 and 6.16 show the CMM
measurement for multi spindle and single spindle drilled hole workpiece.
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Table 6.14 Deformation values of initial and optimum fixture layout of Casestudy I, II and III
Casestudy I Casestudy II (Simultaneous drilling)
Casestudy III (Sequential drilling)
Deformation value for initial fixture layout (in mm)
Deformation value for optimum fixture layout (in mm)
Deformation value for initial fixture layout (in mm)
Deformation value for optimum fixture layout (in mm)
Deformation value for initial fixture layout (in mm)
Deformation value for optimum fixture layout (in mm)
0.017575 0.008087 0.029575 0.026693 0.033022 0.028257
Table 6.15 CMM Measurement for multi spindle drilled hole workpiece
Hole No Cylindricity Diameter Perpendicular angle
1 0.151 7.066 89:04:20
2 0.050 7.127 88:45:16
3 0.074 7.036 89:45:21
4 0.235 7.318 88:41:10
AVG. 0.1275 7.13675 88:83:91
Table 6.16 CMM Measurement for single spindle drilled hole workpiece
Hole No Cylindricity Diameter Perpendicular angle
1 0.056 7.352 87:25:38
2 0.079 7.094 89:42:00
3 0.293 7.115 89:02:12
4 0.262 7.146 88:24:40
AVG. 0.1725 7.1767 88:48:47
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Table 6.17 Comparison of single spindle drilled hole and multispindle drilled hole
Geometricalanalysis
Multispinle drilled Workpiece No:1
Single point drilled Workpiece
Diameter 7.13675 7.17675
Cylindricity 0.1275 0.1725
Perpendicularity 88:83:91 88:48:47
It is also evident that the geometric error is minimized in
simultaneous drilling process of perpendicularity and circularity is 88:83:91
and 0.1275 compared to 88:48:47and 0.1725 in sequential drilling process and
it is shown in Table 6.17. It shows 1% of increase in accuracy of hole
dimension, 26% increase in cylindricity and 0.4% increase in perpendicularity
in simultaneous drilling process. The von Mises stress at each support
location is compared to the yield stress of the workpiece material, to ensure
that the material does not exhibit plastic deformation during machining. The
von Mises is treated as state variable and is not allowed to exceed the
workpiece material yield strength.