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Double Whitworth Quick Return

This was an experiment of a few different things:

  • A quick return mechanism with a return ratio of 24:1
  • 3d printable herringbone gears
  • The gears can be swapped to change the ratio, without changing any other parts. Ratios of 1:4,6,8,10,12 are possible
  • The whole mechanism was designed for assembly: no more than two parts come together at any stage

I’d been using spiral-faced cams for quick-return purposes and thought maybe there was a better option. The Whitworth quick return (below) is a crank-rocker with one slow stroke and a quick return stroke, constructed by having the distance from crank pivot to rocker pivot “almost” equal to the crank length. The return ratio can be defined as r=\frac{\abs{\theta_{max}-\theta_{min}}}{2\pi}, where \theta_{max} is the crank angle when the rocker reaches it’s maximum, and similar for \theta_{min}.

With a crank length L equal to 90% of the pivot distance y, the WQR has a return ratio of about 6.5. A return ratio of 24:1 requires the ratio L/y to be about 99.2% – a tolerance unachievable with 3d printing and difficult in any other manufacturing method. But the double-crank WQR can be used as an input for the crank-rocker WQR – giving a two-stage (or double) quick return linkage that has a high return ratio while using low-tolerance links. The prototype was a good proof-of-concept but of course, a high return ratio requires very little torque on the output – even slight resistance stops the motion.it was a success

Herringbone gears provide the same smooth operation as helical gears, without the additional axial load produced by the helical teeth. They can be modelled in the same way as helical gears, except with a loft extending from the midplane to each face, in opposite directions. The ones for this project range in module from 1.2 mm to 0.8 mm, and came out pretty well, with smooth operation all around.

I wrote a script to size the gears for this project. Given a set of desired ratios, the script performs a brute-force search through m, z space to find a two-stage gearbox that can swap gears to satisfy all the desired ratios, while fitting in the available space. It then outputs a few of the best options for each ratio. For this project I chose 1:4,6,8,10,12 and the selected gears are: (given gear 1 with z1, meshing gear two with z2, at module m denoted as z1|z2_m)

    \begin{align*} 1:4 &- 20|40_{1.2} \times 20|40_{1.2} \\ 1:6 &- 20|40_{1.2} \times 18|54_{1.0} \\ 1:8 &- 20|40_{1.2} \times 18|72_{0.8} \\ 1:10 &- 17|55_{1.0} \times 22|68_{0.8} \\ 1:12 &- 18|54_{1.0} \times 18|72_{0.8} \end{align*}

The 1:10 gearset doesn’t share any gears with any other set, so I didn’t bother actually printing that set. However all four other ratios run smoothly and are easy to interchange.

The mechanism was designed such that only two parts come together at any point. This was a lot easier to assemble than the last big linkage I printed, but required a lot more fasteners. It was easy to change the gears, as the faceplate could be removed without disturbing the rest of the layout. I think that the DFA design made the skeletal aesthetic a lot “noisier” and less aesthetically pleasing than a fully-aligned design as before.

Overall this was a successful experiment and a good way to further improve my printing skills.

Spur Differential

Helical gears use an involute tooth that’s been swept along a helix. At any cross-section perpendicular to the gear axis, the gear profile is exactly the same as a straight spur gear – so a helical gear can be modelled as a loft between the top & bottom faces. The sketch defining the tooth profile gets offset by an additional amount, corresponding to the desired helix angle.

Both of these are spur differentials, printed about 6 months apart.

The white one was designed when I was new to both 3d printing and gears, so the involute is not quite right and the tolerances are sloppy. There is a lot of play and occasional jamming, and it was constructed using screws fixed with loctite.

The orange/blue one is more recent and uses my helical gear model. There is just enough clearance between all rotating parts to allow smooth rotation without any significant backlash. The nuts are all press-fit and sit flush with external surfaces – the shafts are designed such that the screws can be tightened in place without loctite. Overall a much better model!

Variable Stroke Pt. 1

I’ve been experimenting with variable stroke linkages for a while, with most of my new prototypes being 3d printed. These linkages have two inputs – the crankshaft and the stroke amplitude input – so I’ve been using gears to drive the two from one input.

Traditional gear manufacture involves using off-the-shelf tools to cut gears from blanks, and proper use of the tools will give accurate geometry. But 3d printing the gears requires an exact model of the involute tooth surface, which is a bit more tricky. Solidworks includes a “gear” template, but this is just a placeholder, not an accurate model. Some models can be found online, but these are almost always in imperial units. I wanted a parametric model of a precise involute cylindrical gear, using the metric standard, so I made my own.

There are 5 key numbers that define a given gear:

  • m – module
  • z – number of teeth
  • \alpha – pressure angle
  • \psi – helix angle
  • x – profile shift

I’ll discuss the last two in a later post. For now we can assume they are equal to zero, so that the model is only dependent on the first three.

A metric gear has two primary tooth size measurements aside from m: h_a – the addendum height, and c – the root clearance. These are shown below in a sketch of an “unrolled” tooth. Typically, h_a = 1.0 and c = 0.25. Four important dimensions associated with the gear are:

  • D_p = m \cdot z – the pitch diameter
  • D_a = D_p + 2 \cdot h_a – the tip diameter
  • D_f = D_p - 2 (h_a + c) – the root diameter
  • D_b = D_p \cdot cos\alpha – the base circle diameter

Involute gears use a tooth profile following an involute curve – this is by definition the curve obtained by unrolling a string from a circle. The “string” is always tangent to it’s “base circle” and perpendicular to the curve obtained, as shown below for a base circle of radius r. The position of point O is (x_O, y_O) = (r cos\theta, r sin\theta). Point Q on the involute is then:

    \begin{align*} (x_Q, y_Q) &= (x_O + l sin\theta, y_O - l cos\theta) \\ &= r(cos\theta + \theta sin\theta, sin\theta - \theta cos\theta) \end{align*}

This equation can be used directly in Solidworks as an equation curve defining the tooth face, however it only defines one side of one tooth. Ideally the modelled tooth will be symmetric about the x-axis, so the involute needs to be offset (rotated around the base circle) by the appropriate amount. We also need to know when to start and end the curve.

The rotated involute curve can be formulated by multiplying the involute equation above by a rotation matrix R(\gamma), which rotates the curve counter-clockwise about the origin by \gamma radians. The generic equation for an involute of a base circle with radius r_b, rotated by angle \gamma is:

    \begin{align*} \begin{bmatrix} x_\theta \\ y_\theta \end{bmatrix} &= \begin{bmatrix} cos\gamma & -sin\gamma \\ sin\gamma & cos\gamma \end{bmatrix} \cdot \begin{bmatrix} cos\theta + \theta sin\theta \\ sin\theta - \theta cos\theta \end{bmatrix} \\ &= r_b \cdot \begin{bmatrix} cos(\theta+\gamma) + \theta sin(\theta+\gamma) \\ sin(\theta+\gamma) - \theta cos(\theta+\gamma) \end{bmatrix} \end{align*}

The involute curve needs to be rotated such that the tooth axis lies on the x-axis, as depicted below. This angle \beta can be separated into two components:

  • The half-tooth-width angle \phi = \frac{2\pi}{z} \cdot \frac{1}{4} = \frac{\pi}{2 \cdot z}
  • The involute angle inv(\alpha) = tan\alpha-\alpha

So \beta = \frac{\pi}{2 \cdot z} + tan\alpha - \alpha, and the involute should be rotated clockwise by this angle, so \gamma=-\beta. The equation for this curve is:

    \begin{equation*} \begin{bmatrix} x_\theta \\ y_\theta \end{bmatrix} = \frac{D_b}{2} \cdot \begin{bmatrix} cos(\theta-\beta) + \theta sin(\theta-\beta) \\ sin(\theta-\beta) - \theta cos(\theta-\beta) \end{bmatrix} \end{equation*}

For the opposite of the tooth, the involute “string” is beign unrolled clockwise: simply take \theta\rightarrow-\theta. The curve is then rotated counter-clockwise, so take \beta\rightarrow-\beta.

What are the limits of the curve? From the above diagram, the distance between the origin and Q is \frac{D_Q}{2}. Then l = \sqrt{\frac{D_Q}{2}^2 - \frac{D_b}{2}^2}. But we also have l = \frac{D_b}{2}\cdot\theta, so the angle \theta where point Q is at a diameter D_Q is given by:

    \begin{align*} \theta_Q &= \frac{l}{(\frac{D_b}{2})} \\ &= \frac{\sqrt{D_Q^2-D_b^2}}{D_b} \end{align*}

The maximum point of the curve (at the tip) is then given by \theta_a = \frac{\sqrt{D_a^2-D_b^2}}{D_b}.
The minimum point (at the root) is given by \theta_f = \frac{\sqrt{D_f^2-D_b^2}}{D_b}.
The tooth profile is then formed by tracing the involute curve (x_\theta, y_\theta) between these limits. Except for some gears (z <\approx 40 – see below) the root is actually below the base circle – and the involute is not defined below the base circle.

In this case I model the tooth flank as a straight, radial line. In reality the tooth flank is sometimes undercut, and I will go into this in another post.