The rolling element

Steel balls sold by the bottle as 4.5 mm air pistol ammunition are the cheapest precision spheres available, and their size is what makes them useful here. A 4.5 mm ball sits inside the 6 mm radial section of a 6804 and still leaves enough plastic on either side to print, which is the constraint that decides the rest of the design. Figure 9.4.2 shows the source.

Fitting the balls into the envelope

Placing the ball centres on the mid-radius of the section,

\[ r_m = \frac{d + D}{4} \]

puts them 13 mm from the axis and leaves the same thickness of material inside and outside. The radial section \((D - d)/2\) measures 6 mm, of which the ball takes 4.5 mm, so 1.5 mm is left to divide between two race walls and two running clearances.

The race groove is an arc of radius

\[ r_g = \frac{d_w}{2} + c \]

struck about the ball centre, with \(c\) the radial clearance between ball and race. Setting \(c\) to 0.05 mm fixes everything else.

import numpy as np
from mechanicskit import ltx

d, D, B = 20.0, 32.0, 7.0        # bore, outside diameter, width of a 6804
d_w, c = 4.5, 0.05               # ball diameter and radial race clearance

r_m = (d + D) / 4                # ball centre radius
r_g = d_w / 2 + c                # race groove radius
t_o = D / 2 - (r_m + r_g)        # wall outside the groove
t_i = (r_m - r_g) - d / 2        # wall inside the groove
f = r_g / d_w                    # race conformity

ltx(r"r_m &=", r_m, r"\ \text{mm}",
    r"\\ r_g &=", r_g, r"\ \text{mm}",
    r"\\ t_\mathrm{o} = t_\mathrm{i} &=", t_o, r"\ \text{mm}",
    r"\\ f &=", f, aligned=True, precision=3)

Both walls come out at 0.70 mm, which is two perimeters at a 0.35 mm extrusion width and the thinnest section anywhere in the bearing. The conformity \(f = r_g / d_w\) lands on 0.511, a little below the 0.515 to 0.53 that catalogue deep groove bearings use [2]. A groove that conforms more closely to the ball spreads the contact over a larger area and lowers the pressure the plastic sees, at the price of more sliding within the contact patch, and for a race that is weak and turns slowly that is the right direction to err in. The value was not aimed at: it follows from rounding the clearance to 0.05 mm.

import matplotlib.pyplot as plt

w_cage = 3 * 0.4                 # three perimeters at a 0.4 mm nozzle
Z = np.arange(6, 21)
w = 2 * r_m * np.sin(np.pi / Z) - d_w
Z_max = np.pi / np.arcsin(d_w / (2 * r_m))

fig, ax = plt.subplots(figsize=(6.5, 4))
ax.plot(Z, w, 'o-', color='#236B8E', lw=1.4, ms=4, label='Web at the pitch circle')
ax.axhline(0, color='k', lw=0.8)
ax.axhline(w_cage, color='0.5', ls='--', lw=0.9, label='Three perimeters at 0.4 mm')
ax.axvline(Z_max, color='0.5', ls=':', lw=0.9)
ax.annotate(f'$Z_\\mathrm{{max}} = {Z_max:.1f}$', (Z_max, 4.5),
            xytext=(-6, 0), textcoords='offset points', ha='right', fontsize=9)
ax.plot(9, 2 * r_m * np.sin(np.pi / 9) - d_w, 'o', ms=9, mfc='none',
        mec='#B5321E', mew=1.6)
ax.annotate('nine balls, $w = 4.39$ mm',
            (9, 2 * r_m * np.sin(np.pi / 9) - d_w),
            xytext=(12, 10), textcoords='offset points', fontsize=9, color='#B5321E')
ax.set_xlabel('Number of balls $Z$')
ax.set_ylabel('Cage web $w$ (mm)')
ax.set_xticks(np.arange(6, 21, 2))
ax.set_ylim(-0.7, 9.2)
ax.legend(fontsize=9)
plt.tight_layout()
plt.show()

One sketch, four profiles

Every part except the ball pockets is a body of revolution, so the whole bearing is drawn in a single sketch on a plane through the axis and revolved four times, once per part. Drawing it this way puts all the running clearances in one place where they can be read against each other, rather than spreading them over four files. Figure 9.4.4 is that sketch.

Four closed regions of that sketch are revolved separately, in the same way as the gear tooth is built by selecting contours rather than by trimming. Figure 9.4.5 redraws them from the dimensions above, which is also a check that the sketch closes.

from scipy.optimize import brentq

z_m = B / 2                                        # ball centre height
r_si, r_so, r_cb = 12.0, 14.0, 15.0                # slot walls and cap counterbore
z_s = z_m + np.sqrt(r_g**2 - (r_so - r_m)**2)      # split plane of the outer race
z_l = z_m + np.sqrt(r_g**2 - (r_cb - r_m)**2)      # lower end of the cap lip
z_e = z_m - np.sqrt(r_g**2 - (r_so - r_m)**2)      # groove exit, lower side

def arc(r0, z0, R, a0, a1, n=240):
    # Points on an arc of radius R about (r0, z0), angles measured from the r axis
    a = np.linspace(a0, a1, n)
    return np.column_stack([r0 + R * np.cos(a), z0 + R * np.sin(a)])

def ang(r0, z0, r, z):
    return np.arctan2(z - z0, r - r0)

g = lambda r, z: ang(r_m, z_m, r, z)               # angle on the race groove

outer = np.vstack([[[D / 2, 0], [D / 2, z_s], [r_cb, z_s], [r_cb, z_l]],
                   arc(r_m, z_m, r_g, g(r_cb, z_l), g(r_so, z_e)),
                   [[r_so, 0]]])

cap = np.vstack([[[D / 2, B], [D / 2, z_s], [r_cb, z_s], [r_cb, z_l]],
                 arc(r_m, z_m, r_g, g(r_cb, z_l), g(r_so, z_s)),
                 [[r_so, B]]])

inner = np.vstack([[[d / 2, 0], [r_si, 0], [r_si, z_e]],
                   arc(r_m, z_m, r_g, g(r_si, z_e), g(r_si, z_s) - 2 * np.pi),
                   [[r_si, B], [d / 2, B]]])

# Cage: a circle 0.50 mm clear of the groove, flared to a 1.00 mm skirt by R8 arcs.
R_c, R_f, r_n = r_g - 0.50, 8.0, r_m + 0.5
centre = lambda th: (r_m + (R_c + R_f) * np.cos(th), z_m + (R_c + R_f) * np.sin(th))
th_t = brentq(lambda th: np.hypot(*np.subtract(centre(th), (r_n, 0.0))) - R_f, -1.2, -0.2)
a_f, b_f = centre(th_t)
P_t = (r_m + R_c * np.cos(th_t), z_m + R_c * np.sin(th_t))

flare = arc(a_f, b_f, R_f, ang(a_f, b_f, r_n, 0.0), ang(a_f, b_f, *P_t))
mirror = lambda p: np.column_stack([2 * r_m - p[:, 0], p[:, 1]])
cage = np.vstack([flare, arc(r_m, z_m, R_c, th_t, np.pi - th_t), mirror(flare[::-1])])

fig, ax = plt.subplots(figsize=(6.6, 6.4))
for poly, colour, name in [(outer, '#2E6E8E', 'Outer ring'), (cap, '#9EC6D8', 'Cap'),
                           (inner, '#7A9E3F', 'Inner ring'), (cage, '#E0A33E', 'Cage')]:
    ax.fill(poly[:, 0], poly[:, 1], color=colour, ec='k', lw=0.8, label=name, zorder=2)
ax.add_patch(plt.Circle((r_m, z_m), d_w / 2, fc='none', ec='k', ls='--', lw=1.2, zorder=4))
ax.plot(r_m, z_m, 'k+', ms=8, zorder=4)

ax.plot([r_so, 17.8], [z_s, z_s], color='0.3', ls=':', lw=0.9, zorder=5)
ax.annotate('split plane, $z = 5.57$', (17.8, z_s), xytext=(0, 4),
            textcoords='offset points', ha='right', fontsize=8.5)
ax.annotate('ball', (r_m, z_m + d_w / 2), xytext=(0, 7), textcoords='offset points',
            ha='center', fontsize=8.5, zorder=5)
ax.annotate('race groove', xy=(r_m + r_g, z_m), xytext=(17.75, z_m), ha='right',
            va='center', fontsize=8.5,
            arrowprops=dict(arrowstyle='-', lw=0.8, color='0.3',
                            shrinkA=0, shrinkB=2))

ax.set_aspect('equal')
ax.set_xlim(9.3, 17.8)
ax.set_ylim(-0.7, 7.7)
ax.set_xlabel('Radius from the axis (mm)')
ax.set_ylabel('Height (mm)')
ax.legend(fontsize=9, loc='upper center', bbox_to_anchor=(0.5, -0.11), ncol=4,
          frameon=False)
plt.tight_layout()
plt.show()

The inner ring

The inner ring is a sleeve from the \(\varnothing 20\) bore out to a land at \(\varnothing 24\), with the race groove cut into that land. The groove leaves the land 1.43 mm above the bottom face, reaches \(\varnothing 21.40\) at mid-width and returns to the land at 5.57 mm. Figure 9.4.6 shows the contour highlighted in the master sketch.

The outer ring

The outer ring mirrors it, running from \(\varnothing 32\) in to \(\varnothing 28\) with the groove reaching \(\varnothing 30.60\) at mid-width. Above the groove its bore opens out to a counterbore at \(\varnothing 30\), one millimetre below the outside surface, which is the seat the cap drops into. Figure 9.4.7 shows it.

The cap

The outer race has to be split, because the groove is an undercut and nothing can be printed into it from above. The split is taken at 5.57 mm, the height where the groove arc crosses \(\varnothing 28\), and everything above that plane becomes a separate cap. The cap is stepped rather than flat: its outer part fills the \(\varnothing 30\) counterbore, which locates it radially, and a short lip reaches down past the split to carry the top of the groove. Figure 9.4.8 shows the profile.

The cage

The cage section is a circle of radius 1.80 mm about the ball centre, which keeps it 0.50 mm clear of the race groove the whole way round, carried down to the bottom face by two tangent arcs of radius 8 mm that close to a 1.00 mm skirt. Revolved, this is a ring with a skirt running in the 2.00 mm slot between the two race rings and clearing the slot walls by 0.05 mm where it is widest. Figure 9.4.9 highlights it.

A sphere centred on the ball centre is then cut from that ring and circular patterned nine times. The sphere is larger than the 3.60 mm section it cuts through, so each pocket opens on both sides and the ball is held by the web around its equator rather than enclosed. That is also what makes the cage flexible enough to assemble, as we come to below.

Revolving the four contours and cutting the pocket pattern gives Figure 9.4.10, which is the first point at which the design can be inspected rather than read.

The two offsets that absorb the process

A printed hole comes out undersize and a printed boss oversize, because the extrudate shrinks as it cools and because the nozzle path is offset by half a line width in both cases. The error is worst on small diameters, where corner compensation and the widened first layer are a larger fraction of the feature. Rather than chase this through slicer settings, the model carries two offsets that are turned until the assembly runs.

The first is on the spherical pocket in the cage. Shrinkage closes the pocket onto the ball and the cage seizes; offsetting the pocket face outwards opens it again. Figure 9.4.11 shows the field, set here to 0.05 mm.

The second is the bore of the outer ring above the race, the surface the rest of the assembly has to pass on its way in. A small bore pulls inwards as it cools, and offsetting that face outwards restores the fit, as in Figure 9.4.12.

Neither offset has a value that can be derived. They are set by printing the parts, assembling them, feeling the drag and changing the number, and the result holds for one printer with one material at one set of temperatures.

Assembly

The bearing goes together in one order only, and every step after the first is a snap fit whose interference follows from the geometry above.

Start by pressing the nine balls into the cage from the open side of the pockets. They have to turn freely without falling out, which is what the pocket offset is for, and the result is Figure 9.4.13.

Next push the inner ring into the ring of balls. The balls reach \(\varnothing 21.50\) at their innermost point and the inner ring’s land measures \(\varnothing 24\), so this step drives 1.25 mm of radial interference past every ball at once. What gives is the cage: with nine spheres cut out of a 3.60 mm section it is a thin and very compliant ring, and it stretches far enough to let the land through before the balls drop into the groove. This is the step that decides whether a given print assembles at all, and it wants to be done slowly and squarely. Figure 9.4.14 is the result.

Then press that assembly into the outer ring. The throat here is the \(\varnothing 30\) counterbore against balls standing out to \(\varnothing 30.50\), so the interference is 0.25 mm on radius, a fifth of what the inner ring demanded, and the outer ring is the stiffer of the two parts. Figure 9.4.15 shows the bearing at this stage, complete except for the cap and with the race still open at the top.

Finally press the cap on, where it seats in the counterbore and closes the top of the race, giving Figure 9.4.16. Cyanoacrylate on the counterbore is optional and depends on where the bearing will be used: it is the only thing besides the press fit that holds the bearing together axially.

Run the bearing dry for a while first, then grease it. General purpose grease is fine.

Printing

Dry the filament before printing. The cage in Figure 9.4.13 was printed wet, and the whiskers bridging its pockets are the result. On a part whose running surfaces are a fraction of a millimetre across, a string in the wrong place is the difference between a bearing that turns and one that does not.

Print the four parts sequentially rather than side by side, one object finished before the next is started, using the print sequence setting in Figure 9.4.17. Printing them in parallel puts a travel move between parts on every layer, and every travel move is an opportunity to leave a blob on a race.

Set the seam to a random position and turn the scarf joint on, as in Figure 9.4.18. The seam is the small step left where a perimeter closes on itself, and on these parts almost every wall is a running surface. Randomising the position scatters those steps around the circumference instead of stacking them into a ridge up one side, and the scarf joint ramps the perimeter in and out over a length rather than stopping it abruptly. Figure 9.4.19 is the sliced result.

When to buy instead

Every dimension in this bearing came out of one inequality: the ball, two race walls and two running clearances have to fit inside the radial section. Turned round, that gives the largest ball a given envelope will take,

\[ d_w = \frac{D - d}{2} - 2t - 2c \tag{9.4.2}\]

with \(t\) the race wall and \(c\) the clearance. At \(t = 0.70\) mm and \(c = 0.05\) mm the 6804’s 6 mm section returns 4.5 mm, which is where the ball in this chapter came from.

Neither \(t\) nor \(c\) is a property of the bearing. Both are set by the nozzle, and neither shrinks when the envelope does. On the 6804 they consume a quarter of the radial section. On the 12×18×4 bearings the course project buys by the packet, Figure 9.4.20, the section is 3 mm and they consume half of it, so 9.4.2 gives a ball of 1.5 mm.

Ball spacing is not the constraint at that size: 9.4.1 allows 31 balls on the 7.5 mm pitch radius. What fails first is the cage. Subtracting the 0.50 mm race clearance from both sides of the ball leaves a cage section of \(d_w + 2c - 1.0\), and asking for three perimeters there puts the floor at a ball of 2.1 mm and a radial section of 3.6 mm. Figure 9.4.21 draws both limits against the envelopes worth comparing. The 12×18×4 sits below the floor: its cage would be 0.60 mm across, a single extruded line, with nine spherical pockets cut through it.

t_wall = 0.70
section = np.linspace(1.5, 8.0, 200)
ball = section - 2 * t_wall - 2 * c
cage = ball + 2 * c - 1.0
sec_floor = w_cage + 2 * t_wall + 1.0    # section at which the cage reaches three perimeters

fig, ax = plt.subplots(figsize=(6.5, 4.2))
ax.plot(section, ball, color='#236B8E', lw=1.6, label='Largest printable ball $d_w$')
ax.plot(section, cage, color='#C98A2B', lw=1.6, label='Cage section $d_w + 2c - 1$')
ax.axhline(w_cage, color='0.5', ls='--', lw=0.9, label='Three perimeters at 0.4 mm')
ax.axvspan(1.5, sec_floor, color='#B5321E', alpha=0.10)
ax.annotate('buy it', ((1.5 + sec_floor) / 2, 5.4), ha='center', fontsize=10,
            color='#B5321E')
ax.annotate('print it', (5.8, 5.4), ha='center', fontsize=10, color='#236B8E')

for name, dd, DD in [('MR105', 5, 10), ('12×18×4', 12, 18), ('6804', 20, 32), ('608', 8, 22)]:
    sec = (DD - dd) / 2
    ax.plot(sec, sec - 2 * t_wall - 2 * c, 'v', ms=7, color='k')
    ax.annotate(name, (sec, sec - 2 * t_wall - 2 * c), xytext=(0, 8),
                textcoords='offset points', ha='center', fontsize=8.5)

ax.set_xlabel('Radial section $(D-d)/2$ (mm)')
ax.set_ylabel('Diameter (mm)')
ax.set_xlim(1.5, 8.0)
ax.set_ylim(-0.5, 6.5)
ax.legend(fontsize=9, loc='lower right')
plt.tight_layout()
plt.show()

There is a second reason to buy at that size, and it has nothing to do with geometry. The bought bearing arrives with ground races, a full hardened ball complement and a shield on each face, which is a specification the printed one does not attempt. What it also arrives with is a preservative grease heavy enough to dominate the running torque at the loads an RC car puts through a 12 mm bore, and whether flushing it for a light oil is worth doing is a measurement this course can make rather than a piece of received practice to repeat.

What the design still lacks

The bearing turns, carries a light radial load and costs almost nothing. What it does not have is a rating. The load a rolling bearing will take is governed by the contact between ball and race, and the standard treatment of that contact assumes both bodies are steel. Here one of them is a polymer with a modulus around one sixtieth of steel’s, a yield stress that falls off well below any temperature that troubles steel, and a layered structure whose strength depends on the direction the part was printed in. The catalogue formulas do not transfer, and the honest position is that we have a geometry that works and no number attached to it.

That is where the next version of this chapter has to go.