[Psychology] Mechanical Assemblies Phần pot

270 10 ASSEMBLY OF COMPLIANTLY SUPPORTED RIGID PARTS

FIGURE 10-15. Geometry of a

Two-Point Contact.

The variable c is called the clearance ratio. It is the di￾mensionless clearance between peg and hole. Figure 10-16

shows that the clearance ratio describes different kinds of

parts rather well. That is, knowing the name of the part

and its approximate size, one can predict the clearance

ratio with good accuracy. The data in this figure are de￾rived from industry recommended practices and ASME

standard fit classes ([Baumeister and Marks]).

Equation (10-2) shows that as the peg goes deeper into

the hole, angle 0 gets smaller and the peg becomes more

parallel to the axis of the hole. This fact is reflected in the

long curved portion of Figure 10-12.

Figure 10-17 plots the exact version of Equation (10-2)

for different values of clearance ratio c. Note particularly

the very small values of 9 that apply to parts with small

values of c. Intuitively we know that small 9 implies dif￾ficult assembly. Combining Figure 10-17 with data such

as that in Figure 10-16 permits us to predict which kinds

of parts might present assembly difficulties.

The dashed line in Figure 10-17 represents the fact that

there is a maximum value for 9 above which the peg cannot

even enter the hole. This value is given by

(10-4)

It turns out in practice that the condition in Equa￾tion (10-4) is very easy to satisfy and that in fact a smaller

maximum value for 9 usually governs. This is called the

wedging angle 9W. Wedging and jamming are discussed

next.

10.C.4. Wedging and Jamming

Wedging and jamming are conditions that arise from the

interplay of forces between the parts. To unify the discus￾sion, we use the definitions in Figure 10-9, Figure 10-10,

and Figure 10-18. The forces applied to the peg by the

compliances are represented by Fx, Fz

, and M at or about

the tip of the peg. The forces applied to the peg by its

contact with the hole are represented by f\, fa, and the

friction forces normal to the contacted surfaces. The co￾efficient of friction is JJL. (In the case of one-point contact,

there is only one contact force and its associated friction

force.) The analyses that follow assume that these forces

are in approximate static equilibrium. This means in prac￾tice that there is always some contact—either one point or

two—-and that accelerations are negligible. The analyses

also assume that the support for the peg can be described

as having a compliance center.

FIGURE 10-16. Survey of Dimensioning Prac￾tice for Rigid Parts. This figure shows that for a

given type of part and a two-decade range in di￾ameters, the clearance ratio varies by a decade or

less, indicating that the clearance ratio can be well

estimated simply by knowing the name of the part.

10.C. PART MATING THEORY FOR ROUND PARTS WITH CLEARANCE AND CHAMFERS 271

FIGURE 10-17. Wobble Angle Versus Dimensionless

Insertion Depth. Parts with smaller clearance ratio are

limited to very small wobble angles during two-point con￾tact, even for small insertion depths. Since successful as￾sembly requires alignment errors between peg and hole

axes to be less than the wobble angle, and since smaller

errors imply more difficult assembly, it is clear that assem￾bly difficulty increases as clearance ratio (rather than clear￾ance itself) decreases.

FIGURE 10-18. Forces and Moments on a Peg Sup￾ported by a Lateral Stiffness and an Angular Stiff￾ness. Left: The peg is in one-point contact in the hole.

Right: The peg is in two-point contact.

and

respectively. These formulas are valid for 9 <$C tan ' (//).

A force-moment equilibrium analysis of the peg in one￾point contact shows that the angle of the peg with respect

to the hole's axis is given by

where

SQ and #o, the initial lateral and angular error between peg

and hole, are defined in Figure 10-9, while Lg, the distance

from the tip of the peg to the mathematical support point,

is defined in Figure 10-10.

We can now state the geometric conditions for stage 1,

the successful entry of the peg into the hole and the avoid￾ance of wedging, in terms of the initial lateral and angular

errors. To cross the chamfer and enter the hole, we need

10.C.4.a. Wedging

Wedging can occur if two-point contact occurs when the

peg is not very far into the hole. A wedged peg and hole

are shown in Figure 10-19. The contact forces f\ and /2

are pointing directly toward the opposite contact point and

thus directly at each other, creating a compressive force

inside the peg. The largest value of insertion depth I and

angle 9 for which this can occur are given by

272 10 ASSEMBLY OF COMPLIANTLY SUPPORTED RIGID PARTS

FIGURE 10-19. Geometry of Wedging Condition. Left: The peg is shown with the smallest 9 and largest i for which wedg￾ing can occur, namely I = i^d. The shaded regions, enclosing angle 20, are the friction cones for the two contact forces. The

contact force can be anywhere inside this cone. The two contact forces are able to point directly toward the opposite contact

point and thus directly at each other. This creates a compressive force inside the peg and sets up the wedge. This can happen

only if each friction cone contains the opposite contact point. Right: Once t > /j,d, this can no longer happen. Contact force f-\

is at the lower limit of its friction cone while f-2 is at the upper limit of its cone, so that they cannot point right at each other.

where W is the sum of chamfer widths on the peg and

hole, and

If parts become wedged, there is generally no cure (if

we wish to avoid potentially damaging the parts) except to

withdraw the peg and try again. It is best to avoid wedging

in the first place. The conditions for achieving this, Equa￾tion (10-8) and Equation (10-9), can be plotted together as

in Figure 10-20. This figure shows that avoiding wedging

is related to success in initial entry and that both are gov￾erned by control of the initial lateral and angular errors.

We can see from the figure that the amount of permitted

lateral error depends on the amount of angular error and

vice versa. For example, we can tolerate more angular er￾ror to the right when there is lateral error to the left because

this combination tends to reduce the angular error during

chamfer crossing. Since we cannot plan to have such op￾timistic combinations occur, however, the extra tolerance

does us no good, and in fact we must plan for the more

pessimistic case. This forces us to consider the smallest

error window.

Note particularly what happens if Lg = 0. In this case

the parallelogram in Figure 10-20 becomes a rectangle and

all interaction between lateral and angular errors disap￾pears. The reason for this is discussed above in connection

with Figure 10-14. This makes planning of an assembly

the easiest and makes the error window the largest.

FIGURE 10-20. Geometry Constraints on Allowed Lateral

and Angular Error To Permit Chamfer Crossing and Avoid

Wedging. Bigger W, c, and e, and smaller \JL make the par￾allelogram bigger, making wedging easier to avoid. Not only

must the error angle between peg and hole be less than the

allowed wobble angle, as shown in Figure 10-17, but the

maximum angular error is also governed by the coefficient of

friction if wedging is to be avoided. If Lg is not zero, then

if there is also some initial lateral error, this error could be

converted to angular error after chamfer crossing. So, avoid￾ing wedging places conditions on both initial lateral error

and initial angular error. The interaction between these con￾ditions disappears if Lg = 0. This fact is shown intuitively in

Figure 10-14.

10.C.4.b. Jamming

Jamming can occur because the wrong combination of

applied forces is acting on the peg. Figure 10-21 states

that any combinations of the applied forces Fx, Fz

, and M

which lie inside the parallelogram guarantee avoidance

10.C. PART MATING THEORY FOR ROUND PARTS WITH CLEARANCE AND CHAMFERS 273

of jamming. The equations that underlie this figure are

derived in Section 10.J.4. To understand this figure, it is

important to see the effect of the variable A. This variable

is the dimensionless insertion depth and is given by

As insertion proceeds, both t and X get bigger. This in

turn makes the parallelogram in Figure 10-21 get taller,

expanding the region of successful assembly. The region

is smallest when A. is smallest, near the beginning of as￾sembly. We may conclude that jamming is most likely

when the region is smallest. (Since the vertical sides of

the region are governed by the coefficient of friction /i,

the parallelogram does not change width during insertion

as long as /z is constant.)

If we analyze the forces shown on the right side of

Figure 10-18 to determine what Fx, Fz

, and M are for the

case where KQ is small, we find that

Fx = — F arising from deformation of Kx

M = LgF = -LgFx

Dividing both sides by rFz

yields

(10-lla)

which says that the combined forces and moments on the

peg Fx/ Fz

andM/rFz

must lie on a line of slope— (L g / r)

passing through the origin in Figure 10-21. If Lg/r is

big, this line will be steep and the chances of FX/FZ and

M/rFz

falling inside the parallelogram will be small. Sim￾ilarly, if M/rFz

and FX/FZ are large, the combination of

these two quantities will define a point on the line that

is far from the origin and thus likely to lie outside the

parallelogram.

On the other hand, if Lg/r is small so that the line is

about parallel to the sloping sides of the parallelogram

when A is small, then the chance of the applied forces

falling inside the parallelogram will be as large as pos￾sible and will only increase as A increases. Similarly if

M/rFz

and FX/FZ are small, they will define a point on

the line that is close to the origin and thus be likely to lie

inside the parallelogram. When A is small and jamming

is most likely, the slope of sides of the parallelogram is

approximately /z. Thus, if Lg/r is approximately equal

to JJL, then the line, and thus applied forces and moments,

have the best chance to lie inside the parallelogram. Since

JJL is typically 0.1 to 0.3, we see that the compliance center

should be quite near, but just inside, the end of the peg to

avoid jamming.

Instead of considering a single lateral spring support￾ing the peg at the compliance center, let us imagine

that we have attached a string to the peg at this point.

FIGURE 10-21. The Jamming Diagram. This dia￾gram shows what combinations of applied forces and

moments on the peg Fx/ Fz and M/r Fz will permit as￾sembly without jamming. These combinations are rep￾resented by points that lie inside or on the boundary

of the parallelogram. A is the dimensionless insertion

depth given in Equation (10-10). When A is small, in￾sertion is just beginning, and the parallelogram is very

small, making jamming hard to avoid. As insertion pro￾ceeds and A gets bigger, the parallelogram expands

as its upper left corner moves vertically upward and

its lower right corner moves vertically downward. As

the parallelogram expands, jamming becomes easier

to avoid.

274 10 ASSEMBLY OF COMPLIANTLY SUPPORTED RIGID PARTS

FIGURE 10-22. Peg in Two-Point

Contact Pulled by Vector F. This

models pulling the peg from the

compliance center by means of a

string.

See Figure 10-22. This again represents a pure force F

acting on the peg. In this case, F can be separated into

components along Fx and Fz

to yield

(10-12)

so that

(10-13)

which is similar to Equation (10-11). In this case, we can

aim the string anywhere we want but we cannot indepen￾dently set Fx and Fz

. But, by aiming the force, which

means choosing 0, we can make Fx as small as we want,

forcing the peg into the hole. As Lg —>• 0, we can aim </>

increasingly away from the axis of the hole and still make

M and Fx both very small.

In Chapter 9, a particular type of compliant support

called a Remote Center Compliance, or RCC, is described

which succeeds in placing a compliance center outside it￾self. The compliance center is far enough away that there

is space to put a gripper and workpiece between the RCC

and the compliance center, allowing the compliance cen￾ter to be at or near the tip of the peg. Thus Lg —>• 0 if an

RCC is used.

Figure 10-23 shows the configuration of the peg, the

hole, and the supporting stiffnesses when Lg = 0. In this

case, Kx hardly deforms at all. This removes the source

of a large lateral force on the peg that would have acted

at distance Lg from the tip of the peg, exerting a con￾siderable moment and giving rise to large contact forces

during two-point contact. The product of these contact

FIGURE 10-23. When Lg is

Almost Zero, the Lateral

Support Spring Hardly De￾forms Under Angular Er￾ror. Compare the deformation

of the springs with that in Fig￾ure 10-13, which shows the

case where L a » 0.

forces with friction coefficient /z is the main source of

insertion force. Drastically reducing these contact forces

consequently drastically reduces the insertion force for a

given lateral and angular error. Section 10.J derives all

these forces and presents a short computer program that

permits study of different part mating conditions by cal￾culating insertion forces and deflections as functions of

insertion depth. The next section shows example experi￾mental data and compares them with these equations.

10.C.5. Typical Insertion Force Histories

We can get an idea of the meaning of the above relations

by looking at a few insertion force histories. These were

obtained by mounting a peg and hole on a milling machine

and lowering the quill to insert the peg into the hole. A

6-axis force-torque sensor recorded the forces. The peg

was held by an RCC. The experimental conditions are

given in Table 10-1.

TABLE 10-1. Experimental Conditions for

Part Mating Experiments

Support: Draper Laboratory Remote Center Compliance

Lateral stiffness = Kx = 1 N/mm (40 Ib/in.)

Angular stiffness = K® = 53,000 N-mm/rad (470 in.-lb/rad)

Peg and hole: Steel, hardened and ground

Hole diameter = 12.705 mm (0.5002 in.)

Peg diameter = 12.672 mm (0.4989 in.)

Clearance ratio = 0.0026

Coefficient of friction = 0.1 (determined empirically from

one-point contact data)

M = -FxLg