[Psychology] Mechanical Assemblies Phần 4 docx

FIGURE 6-20. The Part with the Mislocated Peg in Figure

6-18 Is Assembled to the Part with the Misoriented Hole in

Figure 6-19. Assembly occurs by placing frame D' of part A

directly onto frame E' of part B. Transform T'AF tells in frame

A coordinates where point F is as a result of including both

peg location and hole orientation errors. The last equation can

be read to say: "To go from A to F, first go from A to D', then

from D' to E', then from E' to F." Because we put D' onto E'

when we assembled the parts, the interface transform TOE

is the same as IDE￾FIGURE 6-21. Wedging Conditions for Assembling

Round Pegs and Chamfered Holes. On the left is a

simplified model of peg-hole assembly. D and d are

hole and peg diameters, respectively. SQ and BQ are ini￾tial lateral and angular error of the peg with respect to

the hole. W is the width of the chamfer, /x is the coeffi￾cient of friction, and c is the clearance ratio, defined in

the figure. On the right is a graph showing values of SQ

and do that permit successful assembly, avoiding wedg￾ing the parts or a collision outside the chamfer.

FIGURE 6-22. Illustration of Assembly Process Capability. Top left: A robot puts a peg in a hole on a set of assembled

parts. The chain of frames at the bottom left TA-TD describes the nominal location of the tip of the next part to be assembled,

while the chain of frames T1-T4 describes the nominal location of the receiving part. Transform TO links these two chains.

Bottom left: The nominal design is correct, so that the chains meet and the errors in position and angle fall inside the wedging

diagram, as indicated by the open circle. On the right there are some errors in the fabrication of the parts so that the chains

of frames do not meet exactly. The resulting lateral and angular errors are shown schematically as a black dot just outside the

wedging diagram. Not shown, but also possible, are errors in frames TA-TD representing robot errors, along with an error in

TO representing calibration or other errors that misplace the assembly fixture in robot coordinates.

154

6.D. EXAMPLES 155

FIGURE 6-23. Combination of Wedging Conditions and

Probability Ellipsoid of Position and Angle Error. The area

of the ellipse covered by the parallelogram represents the

probability that assembly will not fail due to wedging.

FIGURE 6-24. Three Planar Parts Assembled by Welding,

and Their Liaison Diagram. The KG is the relative location

of point 1 on part A and point 2 on part C. The thick shaded

lines represent welds.

FIGURE 6-25. First Step in the Assembly, Joining Parts

A and B Using Fixture 1. Parts A and B are placed in the

fixture using pin-hole and pin-slot features. Then they are

welded together. The fixture is shown in heavy lines. The state

of the parts before they are put on the fixture is shown in

dashed lines.

6.D.3. Variation Buildup with Fixtures

In the previous section we looked at error buildup in an

assembly and its effect on assembleability of the next part.

In this section we look at how errors build up when more

than one fixture is used. There are many ways to design

an assembly process using fixtures. Some of these are bet￾ter than others. For example, the fixtures may actually

overconstrain the parts, a point that underlies one of the

thought questions at the end of the chapter. Another ex￾ample is studied here, namely different ways that the parts

can be fixtured, especially when the assembly consists of

several parts, the KC is measured across parts that are not

adjacent to each other, and several fixtures are used one

after the other to build up the assembly.

Someone has proposed a process for assembling the

planar sheet metal parts shown in Figure 6-24. Parts A

and B are welded together using fixture 1, and the sub￾assembly of A and B is then moved to fixture 2 in order

that part C may be welded on. The KC in question is the

relative location of a point on part C with respect to one

on part A. The parts in question do not pass constraint or

location to each other. Their relative positions and angles

are set entirely by the fixtures. We will see as we look

at this proposed process that it is not the optimum way

to accomplish the assembly. The thought questions at the

end of the chapter ask you to consider many alternative

fixturing arrangements.

The first step in the proposed assembly process is

shown in Figure 6-25, in which parts A and B are joined

on fixture 1. The second step is shown in Figure 6-26,

in which the subassembly A-B is carried to fixture 2 and

joined there to part C. Fixture 2 locates the subassembly

using features on part B.

Figure 6-27 uses coordinate frames to show what hap￾pens while assembling these parts. Fixture Fl locates

parts A and B relative to each other, while fixture F2 lo￾cates parts B and C relative to each other.

A coordinate frame representation of the complete as￾sembly and the KC appears in Figure 6-28. It is constructed

by placing the two frames labeled "B" in Figure 6-27 on

top of each other. The figure shows that, in order to find

the relative location of the points on parts A and C that

constitute the KC, we need to trace a chain of frames be￾tween these points that includes both fixtures. This does

not mean that we have to account for the relative location

of the fixtures with respect to each other on the factory

floor. We can see this because there is no direct chain link

156 6 MODELING AND MANAGING VARIATION BUILDUP IN ASSEMBLIES

FIGURE 6-27. Coordinate Frame Representation of the

Two-Step Assembly of Parts A and B Using Fixtures F1

and F2.

between these two frames in Figure 6-28. What we must

do is account for the error that fixture 1 introduces between

parts A and B as well as the error that fixture 2 introduces

between parts B and C, plus the errors inside each part

between the KC points and the features used for fixturing.

Note that this assembly plan locates the first assembly

operation by means of features on parts A and B while

the second step's operations are done by locating the sub￾assembly using features on parts B and C. In cases like this,

we say that a datum transfer or datum shift has occurred

because the second fixture uses different part features than

the first fixture does. If fixture 2 located the subassembly

using the same part A features that fixture 1 used, then

there would be no datum shift and the chain links between

fixture 1 and fixture 2 would not appear in Figure 6-28.

In fact, neither fixture 1 nor part B would even appear in

Figure 6-28! One of the thought questions at the end of

the chapter asks for a drawing of the chain under those

circumstances.

Consider the instance where the subassembly of A and

B is built by a supplier using fixture 1 while C (or a sub￾assembly more complex than just one part) is made by

another supplier. Now consider the problem faced by the

final assembler who buys these subassemblies and puts

them together using fixture 2. If the KC is not achieved,

the final assembler must be aware of the entire chain in

Figure 6-28 in order to carry out an effective diagnosis of

the problem. If the suppliers are far apart, the "length" of

this chain could be hundreds or thousands of miles. On

the other hand, if step 2 used the features on part A, the

final assembler would have an easier diagnosis problem

because most of the chain would be contained within his

plant. Only that part of the chain representing errors within

part A would be outside his plant.

FIGURE 6-28. Left: A chain of frames joins

the ends of the KC. Steps 1 and 2 are indi￾cated by ellipses. Only frame B is in both el￾lipses. Right: For clarity, the arrows represent￾ing the 4x 4 transforms in the chain are shown

separately.

FIGURE 6-26. The Second Step in the Assembly, Adding

Part C to the Subassembly of Parts A and B, Using Fix￾ture 2. The weld joint between parts A and B is shown as a

thick shaded line. The fixture locates subassembly AB using

features on B.

6.D. EXAMPLES 157

FIGURE 6-29. Car Door Dimensions. These are typical di￾mensions, taken from the author's car.

6.D.4. Car Doors

In this section we will do some examples that illustrate the

following:

• The difference between worst-case and statistical tol￾erancing assumptions

• The difference between uniform and Gaussian or nor￾mal statistics

The MATLAB files that support these examples are on the

CD-ROM that is packaged with this book.

Consider the car door sketched in Figure 6-29. We

would like to know the effect on the location (position

and orientation) of the door in three dimensions of mis￾locating the hinges on either the door or the car body

frame. To do this, we need to define the KC and the rel￾evant dimensions. These are shown in Figure 6-30. The

hinges are positioned on the door at coordinate locations

shown in this figure but are assumed possibly mislocated

in dimensions Y and Z with respect to frame 0, which is

the door's base coordinate frame. Errors with respect to X

FIGURE 6-31. Example of the Effect on Door Position

and Orientation Due to Misplacement of the Hinges. The

door is tilted clockwise in the Y-Z plane and counterclock￾wise in the X-Z plane. It is also lifted along Z. The door's

nominal position and orientation are shown in gray while the

varied door is shown in black. Some horizontal and vertical

grid lines have been added to help make the variation easier

to see.

are most likely to occur when the door is mounted to the

car body but are modeled below in MATLAB as though

they occur when the hinges are mounted to the door.

To perform the analysis, we assume that the two

hinges comprise one compound feature as defined in

Section 6.B.2. The origin of this feature is the lower hinge

whose frame a is nominally located at frame 1, while

the other feature component of the compound feature is

the upper hinge located at frame b. The tolerance on each

hinge's location in X, Y, and Z is assumed to be ±4.5 mm

or ±0.1771".

Figure 6-31 shows the door out of position and orien￾tation due to an example set of misplaced hinges.

FIGURE 6-30. Coordinates and KC for a Car

Door. The KC is the length of the vector joining the

origin of the nominal frame 2 and varied origin of

frame 2. Frame 0 is the door's base coordinate frame.

Frame 1 is the nominal location of the lower hinge,

which anchors the compound feature comprising the

two hinges. The actual location of the lower hinge is

frame a while the actual location of the upper hinge is

frame b. For clarity, frames a and b are shown to one

side of the two views of the door.

158 6 MODELING AND MANAGING VARIATION BUILDUP IN ASSEMBLIES

TABLE 6-2. MATLAB Code for Worst-Case Analysis of Door Variation

%door_main_worst

%Door Main Program for Worst Case

door_nominal

ERR_MAX=0;

for jj=l:64

VERRW(jj)=0;

end

q=0;

for i=0:l;

for j=0:l;

for k=0:l;

for 1=0:1;

for m=0:l;

for n=0:l;

V=[ (-l)Ai(-l)":i (-irk(-iri(-irm(-irn]

q=q+l;

door_dev;

door_errs;

door_act;

DT;

q;

ERR;

VERRW(q)=ERR;

if ERR>ERR_MAX

ERR_MAX=ERR;

is = i ;

js = j;

ks=k;

ls = l;

ms =m ;

ns=n;

qs=q;

end

end

end

end

end

end

end

is

js

ks

Is

ms

ns

ERR_MAX