Placement Mechanisms, and Robots
5.1 GRAVITY FEED TRACKS
5.1.4 F EED T RACK S ECTION
A compromise is necessary when designing the feed track section. The clearances between the part and the track must be sufficiently large to allow transfer and yet small enough to keep the part from losing its orientation during transfer. In the curved portions of the track, further allowances have to be made to prevent the part from jamming. Figure 5.8 shows a cylindrical part in a curved tubular track. For the part to negotiate the bend, the minimum track diameter dt is given by
dt = c + D (5.20)
and by geometry,
TABLE 5.3
Simulation Results for Equation 5.19; r === 0.99997=
η η
ηη nf nfηηηη
0.1 16 1.6
0.2 8 1.6
0.5 3 1.5
0.8 2 1.6
FIGURE 5.8 Construction to determine minimum diameter of a curved feed track.
R dt C
D L
or
(5.21)
where R is the inside radius of the curved track and L the length of the part. If c is small compared with 2(R + dt), this expression becomes approximately
(5.22)
Substituting for c from Equation 5.20 into Equation 5.22 and rearranging, we obtain
(5.23)
If the parts are sufficiently bent or bowed, it may be difficult to design a curved track that will not allow overlapping of parts and consequent jamming.
This is illustrated in Figure 5.9.
Figure 5.10 illustrates typical track sections used for transferring cylinders, flat plates, and headed parts. An important point to be remembered when design-ing a feed track is that the effective coefficient of friction between the parts and the track may be higher than the actual coefficient of friction between the two
FIGURE 5.9 Blockage in a curved feed track.
(R d c) L ( )
Parts will jam due to overlapping Feed track
materials. Figure 5.11 gives some examples of the effect of the track cross section on the effective coefficient of friction. The increase of 100% in friction given by the example in Figure 5.11d could have very serious consequences in a gravity-feed system. It is also important to design these tracks with removable covers or access holes for the quick removal of jammed parts.
FIGURE 5.10 Various gravity feed track sections for typical parts.
FIGURE 5.11 Relation between effective coefficient of friction μ and actual coefficient of friction μd for various track designs.
(a) Cylinder part
(b) Flat parts
(c) Screws
a a
(a) (b)
120 deg (c) (d)
sin a
⎛⎝ ⎛
⎝
Effective coefficient of fricion equal to actual coefficient of friction
μ = μd Effective coefficient of
fricion equal to actual coefficient of friction
(μ = μd)
Effective coefficient of fricion equal to actual coefficient of friction
(μ = μd)
Effective coefficient of friction twice the actual coefficient of friction
μ = 2μd
Track cross-section Cylindrical Part
Cylindrical Part Wires
5.1.5 DESIGN OF GRAVITY FEED TRACKS FOR HEADED PARTS
Of the many parts that can be fed into a gravity feed track, perhaps the most common are headed parts such as screws and rivets, which are often fed in the manner shown in Figure 5.12. Clearly, if the track inclination is too small or if the clearance above the head is too small, the parts will not slide down the track.
It is not always understood, however, that the parts may not feed satisfactorily if the track has too steep an inclination or if too large a clearance is provided between the head of the part and the track. Also, under certain circumstances, a part may not feed satisfactorily, whatever the inclination or clearance. This anal-ysis of the design of gravity feed tracks for headed parts provides the designer of assembly machines and feeding devices with the information necessary to avoid situations in which difficulty in feeding will occur.
5.1.5.1 Analysis
Figure 5.13 shows a typical headed part (a cap screw with a hexagon socket) in a feed track. It is clear that as the track inclination θ is gradually increased and provided that the corner B of the screw head has not contacted the lower surface of the cover of the track, the screw will slide when θ > arctan μ1, where μ1 is a function of the coefficient of static friction between the screw and the track. On further increase in the track inclination, the condition shown in Figure 5.14 will eventually arise when the corner B of the screw head has just made contact with the lower surface of the cover of the track. Immediately prior to this condition, the center of mass of the screw lies directly below AA, a line joining the points of contact between the screw head and the track. From Figure 5.14, it can be seen that z, the distance from the line AA to the axis of the screw, is given by FIGURE 5.12 Headed parts in a gravity feed track.
FIGURE 5.13 Position of a headed part that does not touch the track cover.
FIGURE 5.14 Position of a headed part when corner B just contacts the lower surface of the track cover.
B A
Center of mass
θ
Center of mass B
h
E
C D
F
A
s z
t
θT
ψ γ
α τ
d
(5.24)
where s is the width of the slot and the diameter of the shank, and d is the diameter of the screw head.
Also, from the triangle ACE (see Figure 5.14),
(5.25)
where is the distance from the center of mass of the screw to a plane containing the underside of the head, α is the angle between the screw axis and a line normal to the track, and θT, which is called the tilt angle, is the track angle at which the top of the screw head just contacts the lower surface of the top cover of the track.
From Figure 5.14,
where D is the depth of the track, h the depth of the screw head, and t the diameter of the top of the screw head. Thus, combining Equation 5.26–Equation 5.28, we obtain
(5.29)
Sliding will always occur if θT > θ > arctan μ1 because, under these circum-stances, the track angle is greater than the angle of friction, and there is no contact between the top of the screw head and the track cover. However, sliding may still occur if θ is larger than θT. This situation is shown in Figure 5.15, where it can be seen that a frictional force occurs between the screw head and the track cover as well as on the lower portion of the track.
z2 d2 s2
For sliding to occur under these conditions,
mg sin θ > F1 + F2 (5.30) where
F1 = μ1N1
and (5.31)
F2 = μ2N2
where μ1 and μ2 are the effective values of the coefficient of static friction between the part and the track at A and B, respectively (see Figure 5.15). Resolving forces normal to the track gives
N1 = N2 + mg cos θ (5.32)
and taking moments about A gives
(5.33) FIGURE 5.15 Forces acting on a cap screw in a feed track.
Center of mass N1
A F1 h t/2
N2 F2
t/2
D
mg
s z
B
θ α
mg[ sin ( θ α− )−zcos (θ α− )]
= ⎛ +
⎝⎜
⎞
⎠⎟ −
⎡
⎣⎢ ⎤
⎦⎥ −
N t
z h F D
2 2
2 cosα sinα
Substituting Equation 5.31 to Equation 5.33 in Equation 5.30 and rearranging the terms, we get
(5.34)
where μ1 is the effective coefficient of friction between the screw head and the bottom portion of the track, and μ2 is the effective coefficient of friction between the screw head and the track cover.
Under the conditions shown in Figure 5.13–Figure 5.15, a portion of the screw head lies below the lower contact surface of the track, and the effective coefficient of static friction between the screw and the track is greater than the actual coefficient of static friction μs for the two materials. It can be shown that, for this situation, the relation between μs and μ1 is given by
(5.35)
In this particular case, the difference between μs and μ1 is small, but in some cases discussed later in this section, this effect is of importance. Because there is point contact between the track cover and the screw head, μ2 = μs.
Thus, at least two conditions govern the motion of the screw in the track.
First, if the track angle is greater than the angle of friction and is less than or equal to the tilt angle θT, the screw slides. Under these circumstances, the greater the tilt angle θT, the larger the range of values of μs for which the screw slides.
Therefore, as can be seen from Equation 5.29, the value of D, the track depth, should be as large as possible.
Second, the maximum track angle is restricted to the value given by Equation 5.34 and, in this case, again, the greater the depth of the track, the greater this maximum value θ. It is of interest now to determine the critical value of track depth below which a screw cannot jam in the track, because this will allow a complete definition of the range of track angles and track depths for which a part will feed satisfactorily.
If the track depth D is gradually increased, a special case of the condition shown in Figure 5.15 will eventually arise. This occurs when the angle between a line joining A and B and a line normal to the track becomes equal to the angle of friction between the screw and the track. For larger values of D, the part will not normally make contact with the upper portion of the track. Under these circumstances, however, the screw will jam in the track if it makes contact across A and B. Thus, the situation arises in which, theoretically, the screw should slide but, if a small perturbation rotates it sufficiently to contact the cover, the screw
sin cos ( ) sin ( ) cos ( )
will lock in the track. This possibility should clearly be avoided in practice. From the geometry of Figure 5.15, the maximum value of D is thus given by
(5.36)
where tan β2 = μ2 = μs.
Figure 5.16 shows a graph of θ plotted versus D/h for a size-8 hexagon socket-type cap screw, where d = 1.712s, h = s, = 1.5s, and t = 1.42s.
Although, in practice, the width of the track s would be larger than the diameter of the screw shank, it should be as small as possible, and in all the examples dealt with in this section, it is considered to be equal to the shank diameter.
The results in Figure 5.16 show, for various values of μs, the ranges of values θ and D/h for which the screw will slide down the track without the possibility of jamming. It can be seen that as the coefficient of friction is increased, the ranges of values of θ and D for feeding to occur decrease. It should also be noted that the line XX, representing the tilt angle θT, passes through the points that give (1) the minimum track angle and minimum track depth and (2) the maximum track angle and maximum track depth.
It is now suggested that a reasonable criterion for the best track inclination and track depth would be one in which these parameters are such that feeding would occur for the widest range of values of μs. This condition occurs when the points defined under (1) and (2) in the preceding paragraph become identical.
Referring to Figure 5.16, this condition occurs when θmax = θmin and Dmax = Dmin,
FIGURE 5.16 Conditions for which a particular screw will slide in a gravity feed track.
(From Redford, A.H. and Boothroyd, G., Designing Gravity Feed Tracks for Headed Parts, Automation, Vol. 17, May 1970, pp. 96–101. With permission.)
1.1 1.2 1.3 1.4 1.5 1.6
Track angle, θ (deg)
X
Track depth Depth of screw head= D
h 0.2
or
Solving Equation 5.37 to Equation 5.40 simultaneously gives the maximum angle of friction β for which the screw will slide without the possibility of jamming.
Substitution of this value in Equation 5.39 and Equation 5.36 gives the corre-sponding optimum values of the track inclination θ (= β1) and the track depth D.
The equations necessary to determine the maximum coefficient of friction μmax(= tan β), the track inclination, and the track depth for the proposed optimum conditions for the following four common types of screws have been developed [3]:
1. A cap-head screw of hexagon socket type 2. A flat-head cap or machine screw in a V track 3. A flat-head cap or machine screw in a plain track 4. A button-head cap or round-head machine screw
In all cases, it has been assumed that the width of the slot in the track is equal to the diameter of the screw shank.
5.1.5.2 Results
Graphs of μmax, D/h, and θ vs. /s for the four types of screws studied are shown in Figure 5.17–Figure 5.20. Each figure shows these values for the two extreme geometries of the particular type of screw. For example, in the case of a cap-head screw of the hexagon socket type, regardless of its size, all values of d/s and t/s fall within the ranges 1.33–1.712 and 1.041–1.42, respectively.
tan(α β+ )= +
FIGURE 5.17 Optimum values of track angle θ and track depth D for maximum coefficient of friction μmax for cap-head or hexagon-socket screws. (From Redford, A.H. and Boothroyd, G., Designing Gravity Feed Tracks for Headed Parts, Automation, Vol. 17, May 1970, pp. 96–101. With permission.)
1.0 2.0 3.0
1.00 1.1 1.2 1.3 1.4 1.5
D/h
/s
20 30 40 50
θ (degrees)
1.0 2.0 3.0
0 /s
/s 1.2
1.0
0.8
0.6
0.40 1.0 2.0 3.0
μmax
d/s = 1.33 d/s = 1.712
h/s = 1.0 h/s = 1.0
t/s = 1.041 (1.5 inch) t/s = 1.42 (No. 8)
D h
s
d t
FIGURE 5.18 Optimum values of track angle θ and track depth D for maximum coeffi-cient of friction μmax for flat-head cap or machine screws in V track. (From Redford, A.H.
and Boothroyd, G., Designing Gravity Feed Tracks for Headed Parts, Automation, Vol. 17, May 1970, pp. 96–101. With permission.)
2.8 2.7 2.6 2.5 2.4 2.3 2.2
D/h
0 1.0 2.0 3.0
/s
/s
/s d
D h
s
0 1.0 2.0 3.0
60 70
50
θ (degrees)
1.3 1.2 1.1 1.0
0.90 1.0 2.0 3.0
μmax
d/s = 1.83 h/s = 0.489 (3/4 inch flat cap) d/s = 2.00 h/s = 0.584 (No. 2 flat mach)
FIGURE 5.19 Optimal values of track angle θ and track depth D for maximum coefficient of friction μmax for flat-head cap or machine screws in plain track. (From Redford, A.H.
and Boothroyd, G., Designing Gravity Feed Tracks for Headed Parts, Automation, Vol. 17, May 1970, pp. 96–101. With permission.)
/s 60
50
400 1.0 2.0 3.0
θ (degrees)
/s 1.2
1.1 1.0 0.9 0.8
0.70 1.0 2.0 3.0
μmax
d/s = 1.83 h/s = 0.487 (3/4 inch flat cap) d/s = 2.00 h/s = 0.584 (No. 2 flat mach.) 2.1
2.0 1.9 1.8 1.7 1.6 1.5 1.4
D/h
0 1.0 2.0 3.0
/s h
s
D d
Clearly, the value of (D/h) cannot be less than unity, and thus for some types of screws, it can be seen from these figures that there is a minimum value of /s for which the analysis is valid. If /s is less than this minimum value, the parts can be fed into a track with a small clearance, and the only restriction on the track angle is that it must be greater than the effective angle of friction.
Although in practice, there will often be no choice as to the type of screw to be used, it is of interest to note the degree of difficulty in feeding equivalent sizes FIGURE 5.20 Optimal value of track angle θ and track depth D for maximum coefficient of friction μmax for button-head cap or round-head machine screws. (From Redford, A.H.
and Boothroyd, G., Designing Gravity Feed Tracks for Headed Parts, Automation, Vol. 17, May 1970, pp. 96–101. With permission.)
1.4 1.3 1.2 1.1
1.00 1.0 2.0 3.0
/s
/s
/s
D/h
D h
d
s
50
40
30
0 1.0 2.0 3.0
θ (degrees)
1.1 1.0 0.9 0.8 0.7
0.60 1.0 2.0 3.0
μmax
d/s = 1.75 h/s = 0.700
d/s = 1.75 h/s = 0.764 1/4 inch button head
of the various screws analyzed. For a given length of screw, the most difficult to feed are the larger sizes of cap-head screws of the hexagon socket type and, in this case, if μs is 0.5, it would be very difficult to feed the screw if its length were greater than five times its diameter. The easiest screws to feed are the flat-head cap or machine screws and, of the two alternative track designs examined, the V track has better feeding characteristics.
One general point is that, in many cases, the optimum track depth is very much larger than would normally be used in practice and, when the track depth is only slightly larger than the depth of the head, difficulty is often encountered in feeding these parts.
5.1.5.3 Procedure for Use of Figure 5.17 to Figure 5.20
The first step in using the data provided in these figures is to determine the value of /s for the screw under consideration. The simplest way is to balance the screw on a knife edge and measure the distance from the center of mass to the underside or top of the screw head, whichever is appropriate. This value is then divided by the shank diameter to obtain the ratio /s.
The optimum values of the track angle θ and the ratio of track depth to screw-head depth are then read off the appropriate curves in the figures. In all cases except the cap-head screw of the hexagon socket type (Figure 5.17), the ranges of these values are quite small. When using this figure, therefore, it should be remembered that the lower line is for the large sizes and the upper line is for the small sizes.
Finally, reference to the lower graph in Figure 5.17 to Figure 5.20 will give the maximum permissible value of μs. If the actual value of μs is greater than this, the screw cannot be fed in a slotted gravity feed track.
Although the gravity feed track is the simplest form of feed track, it has some disadvantages. The main disadvantage is the need to have the feeder in an elevated position. This may cause trouble in loading the feeder and in freeing any block-ages that may occur. In such cases, the use of a powered track may be considered.