PERFORMANCE OF AN ORIENTING DEVICE

In the preceding section, it was seen that to optimize the design of the orienting system for cup-shaped parts, it was necessary to determine the step height that would provide the maximum efficiency. The information employed to determine the performance of the step orienting device was empirical. It would be most useful in investigations involving the optimization of an orienting system if theoretical expressions were available that described with sufficient accuracy the performance of the orienting device. The present section describes the analysis of one of the passive orienting devices introduced earlier: a V cutout (Figure 3.23) used for the orientation of truncated cone-shaped parts. It can be seen from Figure 3.23 that, with a properly designed device, those parts being fed base uppermost will be rejected, whereas those parts being fed base down will be accepted and allowed to proceed to the outlet chute.

Because the height of the part has no effect on the performance of the device, the only parameters necessary to describe its important characteristics are the radius R of the base and the radius r of the top. The symmetrical orienting device may also be described completely by using only two parameters: the half-angle θ of the cutout and the distance b from the apex of the cutout to the bowl wall.

During vibratory feeding, the part proceeds along the track by a combination of discrete sliding motions either backward or forward or both and, under certain conditions, by a forward hop. All the motions occur sequentially during each cycle of the vibratory motion of the bowl. During each cycle, when the conditions are such that the part hops, there will usually be a distance along the track, denoted by J, where the part does not touch the track. Therefore, J is the smallest gap or slot in the track that will reject all particles that travel with this particular motion.

For vibratory conditions that produce relatively small sliding motions compared to the hop, the motion can be characterized by a series of equal hops, each of distance J.

The object of the design of a V cutout would be to determine the values of the parameters θ and b such that, for a given part (i.e., for given values of R and r) and for a given feeding characteristic (i.e., given J), all the parts fed on their tops would be rejected, and a maximum of those fed on their bases would be accepted.

3.16.1 ANALYSIS

Figure 3.38 shows two limiting conditions for the position of a part resting on its top. In the first position, the center of the part lies at P on the edge of the cutout. Thus, if the part comes to rest momentarily just to the right of P, it will be rejected. The second limiting condition places the center of the part at Q, and the edge of the part is just supported at D by the edge of the cutout. Thus, if the part contacts the track with its center anywhere between P and Q, it will be rejected. Two similar limiting conditions not shown in the figure will exist to the right of the cutout centerline, and these positions may be deduced from the symmetry of the situation.

It is clear that, for a part traveling from left to right (Figure 3.38) in a series of hops, the probability that its center will fall in the space between P and Q (and thus cause rejection) is given by j/J, where j is the distance between P and Q, and J is the length of each hop.

For those parts that negotiate the gap between P and Q, the probability that they will clear the first gap is (1 − j/J) and the probability that they will be rejected in the similar gap lying to the right of the cutout centerline, is ( j/J). Hence, the total probability R_e that a part will be rejected in one of the two gaps is

R_e = j/J + (1 – j/J)j/J = 2( j/J) – ( j/J)² (3.21) For the conditions illustrated in Figure 3.38,

j = 2(R – b)tan θ – r sec θ (3.22) For large cutout angles, Equation 3.22 does not always apply because when the top of the part is supported at D by the right-hand edge of the cutout, point C, diametrically opposite D, may be to the right of the left-hand edge of the cutout. Thus, a part in this situation will be rejected, and an alternative limiting condition will arise. In this case, point P is unchanged, but point Q is found by arranging for the diameter CD of the part to be just supported between the edges of the cutout. From Figure 3.39,

j = (R – b)tan θ – x (3.23)

and

x = r cos α – [(R – b) – r sin α tan θ] (3.24) FIGURE 3.38 Determination of j for small cutout angles.

Bowl wall

b

R B D

R

r P θ

θ θ Q C

A

j r secθ Track

Also,

x = [(R – b) + sin α]tan θ – r cos α (3.25) Eliminating α from Equation 3.24 and Equation 3.25 gives

x = tan θ[r² – (R – b)² tan² θ]^1/2 (3.26) Finally, substituting Equation 3.26 into Equation 3.23 gives

j = tan θ{(R – b) – [r² – (R – b)² tan² θ]^1/2} (3.27) The value of b at which Equation 3.22 becomes invalid and Equation 3.27 must be applied can be found by arranging for CD in Figure 3.39 to lie at right angles to the right-hand edge of the cutout. Under these conditions, α becomes equal to θ, and eliminating x from Equation 3.24 and Equation 3.25 gives

(R – b) = r cos θ cot θ (3.28)

and, thus, from Equation 3.22 or Equation 3.27,

j = r (2 cos θ – sec θ) (3.29) It can readily be shown that, for θ ≥ 45°, Equation 3.27 always applies.

It is convenient to eliminate one variable from the foregoing expressions by dividing through by R and thereby writing the parameters in dimensionless form.

FIGURE 3.39 Determination of j for large cutout angles.

Bowl wall

Track

R

B r D

P

b

r sinα

r cosα r cosα

Q

x

R

C

j A

α

θ θ

[(R−b)−r sin α]

(R−b) tanθ

Defining r₀ = r/R, b0 = b/R, j0 = j/R, and J0 = J/R, the following equation is obtained for the rejection R_e of parts:

R_e = 2( j₀/J₀) – ( j₀/J₀)² (3.30) where

(3.31) unless θ < 45° and b0 > (1 – r0 cos θ cot θ), in which case,

j₀ = 2(1 – b₀)tan θ – r0 sec θ (3.32) These equations are presented in graphical form in Figure 3.40 for a cutout having a half-angle of 30° and for a part where r0 is 0.8.

The figure shows how the theoretical rejection rate R_e varies as the parameter b₀ is changed for a given value of the distance J₀ hopped by the part during each vibration cycle. The figures for the case in which the part is being fed on its base were obtained by setting r₀ equal to unity. A negative b refers to the case in which the apex of the cutout lies outside the interior surface of the bowl wall.

It can be seen from the figure that, as the parameter b₀ is gradually decreased, a condition is eventually reached at which the unwanted parts (those being fed on their tops) will start to be rejected. On further decreases in b₀, a point will be reached at which all these parts will be rejected. Similar situations will arise for the wanted parts but for lower values of b₀. Eventually, a value of b₀ will be reached at which all the parts will be rejected. Clearly, in practice, it will be

FIGURE 3.40 Effect of b0 on rejection of parts: θ = 30 deg, J0 = 0.15.

j₀ =tan {(θ 1−b₀) [− r₀²−(1−b₀) tan^{2 2}θ] }^{1 2/}

b_u−bw

b_w b_u 1.0

0.8 0.6 0.4 0.2 0.0 Rejection Re

−0.2 −0.1 0.0 0.1 0.2

b₀ = b/R

necessary to choose a situation in which all the unwanted parts will be rejected, even at the expense of rejecting some of the wanted parts.

After defining the largest value of b₀ for which all the unwanted parts are rejected as b_u and the smallest value at which all wanted parts are accepted as b_w, it can be stated that the best conditions would be those that resulted in the largest value of (b_u – b_w). This would give the greatest working range for a given part and for given feeding conditions.

It is of interest to study how the magnitudes of b_u and b_w are affected by changes in the design parameters. The value of b_u is obtained by setting R_e equal to unity in Equation 3.30 with the appropriate equation for j₀, Equation 3.31 or Equation 3.32; the value of b_w is obtained by setting R_e equal to zero and r₀ equal to unity.

Thus, after rearrangement, when θ ≥ 45°,

(3.33)

bw = 1 – cos θ (3.34)

when θ ≤ 45°,

b_u = 1 – 0.5 J₀ cot θ – 0.5 r0 cosec θ (3.35)

bw = 1 – 0.5 cosec θ (3.36)

unless b_u < (1 – r₀ cosθ cotθ), in which case it is given by Equation 3.33 and bw

is given by Equation 3.36. These equations are plotted in Figure 3.41 and illustrate the effects of θ and J0.

This theory has been developed for an idealized situation in which the part proceeds along the track by hopping. However, in reality, both hopping and sliding occur. It can be shown that, although this would affect Equation 3.30, and hence the shapes of the curves in Figure 3.40, the values of b_u and b_w would be unchanged.

Values of the working range (b_u – b_w) obtained from Equation 3.33 to Equation 3.36 are plotted against θ in Figure 3.42. It can be seen that, for larger values of J₀, an optimum condition exists that gives the maximum working range. Further, at low values of θ, the magnitude of the working range becomes very sensitive to changes in J₀. Because most vibratory feeders operate at the same frequency, a large value of J₀ implies a high conveying velocity. In practice, it would clearly be desirable to choose conditions that give minimum sensitivity to changes in the feeding parameters and yet give the maximum working range for a reasonably large value of J₀.

The procedure used to obtain the experimental values for b_u and b_w, which are presented in Figure 3.41, is outlined by Boothroyd and Murch [8]. It is seen in Figure 3.41 that the results for b_u show good agreement with the theory over

bu= −1 ² J0 + r02 2 −J

02 1 2

cos θ[ cotθ ( sec θ ) ]^/

the whole range of cutout angles when J0 is set equal to 0.15. The experimental values for bw show good agreement with the theory only at small cutout angles.

For larger cutout angles, the experimental value is always larger.

Ideally, in the design of a V-cutout orienting device, the pertinent data regard-ing the vibratregard-ing motion of the bowl feeder could be used to estimate the value of J0, employing the results presented earlier in this chapter. Subsequently, using Figure 3.42, the half-angle θ of the cutout that gives the best value for the working range could be chosen. From this figure, when J0 ≤ 0.1, the smaller the value of θ, the larger the working range. However, small cutout angles can present a practical problem: Parts that are rejected may not be deflected properly from the bowl track and may interfere with the behavior of the parts that follow. Thus, the angle chosen should be that which gives the best working range and yet provides for adequate deflection of rejected parts.

FIGURE 3.41 Effect of θ on values of bu and bw. (From Boothroyd, G. and Murch, L.E., Performance of an Orienting Device Employed in Vibratory-Bowl Feeders, Transactions of the ASME, Journal of Engineering for Industry, Aug. 1970. With permission.)

1.0

0.5

0.0

−0.5

−1.0

Maximum value of b0 for rejection of all unwanted parts (bu) Theory

Experiment

20 40 60 80

Half angle of cutout θ J⁰ = 0.2

J = 0.1⁰

J0 = 0.0

r0 = 0.8

1.0

0.5

0.0

−0.5

−1.0 Maximum value of b0 for acceptance of all unwanted parts (bw)

20 40 60 80

Half angle of cutout θ

Because it is essential that all the unwanted parts be rejected, the value of b₀ (which defines the position of the cutout apex) must be less than b_u. The value of b_u can be found from either Equation 3.33 or Equation 3.35, whichever is appropriate.

In practice, this method will result in an effective orienting device but not necessarily the most efficient one. If, for the larger cutout angles, the experimental values of b_w were significantly larger than that predicted by theory, the corre-sponding working ranges (b_u – b_w) would be negative. Thus, when the cutout is designed so that all the unwanted parts are rejected, some of the wanted parts will also be rejected. This reduces the output and can result in low efficiency.

In view of these observations, the recommended procedure would be to choose a value of θ less than 45° but large enough to give acceptable levels of deflection of the parts into the bowl and then determine b_u from the appropriate equation. Such an orienting device would have a positive working range when J₀ is less than 0.2 and thus have an efficiency of 100%.

3.17 NATURAL RESTING ASPECTS OF PARTS FOR