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

106

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)


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)


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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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)


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)


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.

123

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)





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

124

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.

chapter 6 tolerance model for workpiece...

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