There’s additional support structure around the pulley mount on the dash panel. As it was, the length of the steering column meant that the dash panel would be very easily bent out of shape. The additional structure from the dash support spreads the bending forces, to stiffen the area.

The steering wheel’s connection to the steering column has been changed from a single bolt to three. This means that the manufacture of the column is simpler, there’s now no need to drill and tap the centre of the column, and because the pulley is attached with a keyless fastening, all the adjustment is at the pulley end.

The steering wheel tubes extend up from the steering wheel to allow either bar-end brake levers or normal bike brake levers the correct way round for the rider to hold.

Before I show the complete steering assembly, I wanted to show the detail of the steering pulley, where the driver’s end of the steering cables connects to the steering wheel and how all the parts clamp up around the dash panel to allow the wheel assembly to rotate.

Steering Pulley Assembly

False Colour Assembly

Here you can see the whole assembly, the Steering Pulley (A005_P006_R002) is in red, slotted at the top to accept the ferrule at the end of the cable. Then the cable loops the pulley once and passes though one of the holes in the background, where the side plates of the pedal assembly meet the dash panel (One cable through the panel in the background, and one looped the opposite way round the pulley and passing through one of the hole son the opposing plate — not shown for clarity). The holes also hold the termination of the cable sleeve, allowing some minor adjustment of cable tension at this point, though the major adjustment is at the wheel-end of the cable.

While the pulley geometry I’ve chosen is very similar to available V-Belt pulleys (an SPZ 63mm pulley would have similar external dimensions) I’m not using an off-the-shelf pulley for 2 reasons:

1, The traditional way of attaching a pulley to a shaft is to cut a keyway into the shaft and use a taper bore bush to clamp the pulley around the shaft, but this means that it’s not possible adjust the position of the pulley around the shaft, it has to be set at one point. Using a non-keyway centre fastening means that the angle of the pulley can changed compared to the wheel, and this mans that the wheel can always be set level at straight ahead steering. However, using a non-keyway fastening means it’s not possible to use the widely available tapered bush pulleys, so any standard pulley would still need machining to fit the shaft.

2. Using a non-standard pulley means I can fine-tune the exact diameter that the cable runs round, meaning that I can better control the steering ratio. As I said before, an SPZ 63mm OD pulley has similar dimentions, but they would not give exactly the same steering effect.

Steering Pulley Cutaway

In the cross-sectional diagram, you can see better how the machined pulley centre actually clamps up the bearings of the steering boss. The keyless fastener (Transtorque shown in blue) applies a radial clamping force to both the pulley and the shaft, and also provides some axial resistance too. This means that it can also be used as the single fitting that controls the bearing preload, as well as the axial position of the pulley around the shaft.

The shaft bearings (shown in yellow, a pair of 1201 self aligning ball bearings) are clamped between the flange of the steering shaft, and the pulley. Using M12 washers between the components means that they can both be flat sided — simplifying manufacturing — and still only contact the inner race of the bearings.

To locate the outer races of the bearings, two retaining plates (shown in green — A005_P004_R002) are bolted through the dash panel to surround the bearings and control their location radially. The hole in the dash panel is noticeably larger than the diameter of the steering shaft so it only contacts with the outer race of the bearings.

I’ve had a re-read recently of Frederik Van De Walle’s Master’s Thesis “The Velomobile as a Vehicle for more Sustainable Transportation” (freely available to download from his site); and while I’ve never used the term “Sociotechnical Frame”, I’ve come across a fair amount of misunderstanding about velomobiles while I’ve been developing the Atomic Duck where I tend to say something like:

It’s not a bike or a car, it’s somewhere in between.

Typically, a velomobile gets seen as either a type of large bike, because it uses bicycle components; or as a type of small car, because it is enclosed (velomobiles are often mistaken for being some kind of small electric vehicle). But an observer that views a velomobile as either a type of car or a type of bicycle will automatically tend towards a negative comparisons of it,

i.e. “A velomobile is not as good an automobile as a car”:

• It’s slower than a car
• It has less range than a car
• There is less luggage capacity
• It only carries one person
• It takes more effort to travel, there is no engine.

or “A velomobile is not as good a bike as a bicycle”:

• It’s heavier than a bike
• It’s wider than a bike
• It’s more expensive than a bike

But if velomobiles are seen as a separate vehicle choice, much like a motorbike is seen as being distinct from either a bike or a car, it’s easier to see where the advantages of a velomobile lie.

A velomobile has many of the the advantages that a bicycle has over a car:

• It’s lighter
• It’s smaller
• Costs less to run
• More mechanically simple, so there is less to maintain.
• Emits less harmful emissions
• Healthier for the rider through them exercising

And some of the advantages that the car has over the bicycle:

• Offers protection from the elements
• Is faster (through better aerodynamics)
• Has more luggage carrying capacity
• Has a greater range because of increased rider comfort.

Four Modes of Vehicle

“There are today three vehicle categories: the bicycle, the motorcycle and the automobile. The velomobile is the fourth one: the difference between a bicycle and a velomobile is like the difference between a motorcycle and an automobile (-cycle to –mobile dimension); and the difference between an velomobile and an automobile is like the difference between a bicycle and a motorcycle.” — Frederik Van De Walle, “The Velomobile as a Vehicle for more Sustainable Transportation”

Since I converted a long (>100 pages) document from Microsoft Word format to LaTeX, I’ve now split the file down into the separate chapters for easier editing that are all \include{}‘d or \input{}‘d from a single main file. As suggested by the LaTeX Project Structure Guidelines, this includes moving the separate .tex files into a sub-directory and putting the preamble into its own .sty style file.

Sort of like this:

LaTeX Project Structure Guidelines

To get this to generate a .pdf file properly with the LaTeX-SVG-to-PDF makefile properly needed a couple of adjustments:

• Added a variable for the location of the sub-folder holding the component .tex files (TEX_SRC_DIR) and a for all .tex files in that directory (TEX_SRC) — the default location is ./tex/.
• Adjusted the pdf generation inputs to rebuild if any of the component .tex files change.
• Included any .sty files as inputs for the .pdf file so styles changes will trigger a rebuild.

(see this commit)

I also noticed that if there are any errors when a bibliography is generated, these are not output at the end of the make run so I’ve added a grep for warnings in the bibliography log file.

(see this commit)

As always, the makefile is available to downlod, use, fork, tinker and change on github.

As a special bonus this week, how to typeset a document that looks like it has been written by a crazed cthulhu cultist, including typsetting an elder sign! And yes, you could use the LaTeX-SVG-to-PDF makefile, but it’s overkill for a short, text only document…

This is exactly what should be expected though, the blocking out of the structure is like a first draft of a piece of writing. Once you have a good basis to work from, it’s all a lot if tweaking, adjusting, calculating, redrawing and moving around to finish it.

What you’re looking at is still without the cross-supports for the long members (to make then into I-beams), a rider facing chain cover and the tabs and slots to allow it all to self jig.

Chainline and Chain Guard

The images below show the dash bulkhead at the rear, but not the forward bulkhead that the front of the bottom bracket assembly attaches to, to make it all easier to see. It’s not all hanging out in space really!

Where last week’s images didn’t show the rearward chainguard, it is included here. I’d actually like to extend the guard forward towards the pedal chainring, but I haven’t got a good enough model of a pedal assembly to ensure the guard would lie between the pedal crank and the sprocket, and I’ve no idea how the clearance changes with single, double and triple front chainrings — so that’s on the “to-do” pile.

The chainguard currently covers the top-idler, the lower-front and lower-rear pair of idlers and the run of the idler attached to the arm from the bottom bracket slider (the sliding idler).

A004_R002 Chainline Assembly

A004_R002 Right View with Chain Cover

The assembly structure rises up to meet the dash bulkhead to give structural support to the area where the steering wheel will be mounted and to provide the brackets that will mount the steering cables. Expect this area to evolve as the steering assembly is re-engineered.

Adjustment

A single vertical bolt secures the bottom bracket slider (shown outlined in red), which is plated in on four sides. The bottom bracket shell is connected to the bottom bracket slider with a 1mm aluminium band that is bonded to the inside of the slider, keeping the outside profile of the slider constant.

Sliding Bottom Bracket Adjustment

A004_R002 Left View

The middle idler moves with backwards and forwards with the bottom bracket, so any reduction in the distance between the top idler sprocket and the chainring is equal to the increase in distance between the lower-forward idler and the sliding idler. Adjustment for chain tension (when the chain run is not an exact multiple of chain link length) is in a slide mount for the lower-front idler.

Horizontal adjustment of the bottom bracket is 220mm (0.22m), which is designed to allow rider leg lengths from a 5th percentile female up to a 95th percentile male (rider height 5’1” to 6’3” with seat rake adjustment).

A004_R002 Fully Forwards

A004_R002 Fully Back

Images show for a 52T chainring and 17T idler sprockets. I’ve got some holes to match for a 39T chainring, but the mounting points will have to be necessarily limited.

However, I’m putting up images of the progress as it is for two reasons: 1. It’s been a while since the last Atomic Duck update, but showing progress is almost as important as finishing; and 2. I said in response to a comment last week that I would show progress this week!

If you haven’t see the last post about chainline layout, then this sprocket arrangement might seem a little strange, but it is designed to provide 220mm of horizontal pedal adjustment, without changing the running length of the chain. This should allow for pedal adjustment for almost any adult rider without having to make any manual change to the chain length, or do any manual tensioning.

A004_R002 WIP Sprocket Detail

A004_R002 WIP Chainline Layout

Now I have very little experience with electric drivetrains, so I would love to hear any answer from anyone with more knowledge; but I’ve also had a go at reasoning it out, so you should also tell me if I’m way off base too!

A capacitor bank is a large matrix of capacitors that act like a buffer between the battery and motor/generator on an electric vehicle. Capacitors have low energy capacity compared to batteries, so you wouldn’t want to use them for the only energy storage, but they do have very high charge/discharge rates that don’t affect their energy storage capacity.

As such, by placing a bank of capacitors between the motor and the batteries, the capacitor bank can discharge quickly for acceleration and then be recharged from the motor to provide regenerative braking, while the batteries supply (and receive) much lower currents to drive the motor in steady state operation and supply the energy that is lost from the moving vehicle system (e.g. through aerodynamic drag, heat, friction and noise). This lower current, constant charge and discharge keeps the batteries in much better condition than high current operation, meaning that the batteries will have longer operational lives.

I reasoned that capacitor bank that can hold the kinetic energy of the vehicle at maximum normal cruising speed would be good for a lightweight vehicle.

\[ E_{kinetic} = mv^2 \]

Therefore, a nominal 350kg vehicle with a cruise speed of 60mph (28.82 m/s) would have to have capacitors that can hold 0.25 MJ (251759 J or ≈70 W·h).

The required capacitor size is dependant on voltage:

\[ E_{storage} = \frac{1}{2}CV^2 \]

So capacitance can be calculated with:

\[ C = \frac{2E}{V} \]

At 12V, 0.25 MJ needs a capacitance of 3497 F.

Using seven 500 F (16V) ultracapacitor modules in series would give 3500 F of capacitance at a total mass of almost 40 kg (5.5 kg per module).

When I said capacitors had lower energy density, it really shows here. The ultracapacitors I’ve used for reference have an energy density of 11.5 kJ/kg (3.2 W·h/kg) compared to 108-144 kJ/kg (30-40 W·h/kg) for lead acid batteries. For instance two 30Ah, 12V sealed lead acid batteries can store ten times the energy of this capacitor bank, 2.59 MJ (720 W·h), for just under 15kg.

On the other hand though, the ultracapacitor can operate at up to 7600A (short circuit, max 60 sec) giving a maximum of 91 kJ/s @ 12 V operation. Lead acid batteries can operate at either 7A (slow discharge) or 100A (fast discharge, surge) meaning that they normally only release energy at 0.084-1.2 kJ/s.

As you can see, a capacitor bank that can hold energy required to push a vehicle to it’s cruising speed is very large, and could even have greater mass than the battery storage, but being able to regenerate ‘all’ of a vehicle’s kinetic energy in the capacitors would simplify the control strategy for recharging the battery array by making it unnecessary in normal operation.

Capacitance scales linearly with vehicle mass, but it increases with the square of speed, so lighter and lower speed vehicles have more efficiently sized capacitor bank with this approach.

The required battery storage is more difficult to calculate though, as it’s heavily dependant on vehicle use and efficiency; and I’ve not even taken into account any efficiencies, or having to regulate capacitor voltage.

In the mean time, why not take another look at the Front Axle Assembly A002_R002 and the more recent Laser-cut Lattice Living Hinges and Lattice Hinge Test Results.

Happy New Year!

Probably worth a look at last week’s post for details on the parts of the hinge structure (junctions and spring links).

In short: I’m quite happy that the torsion based formula I calculated gave me some hinges that worked, but it looks like the hinge design needs some more parameters.

For defining hinges I’d currently recommend:

• Calculate the minimum number of links for the material, sheet thickness and link length.
  
  \[ n \geq 0.676125 \times \frac{\Theta G t}{\tau_{yield} l} \]
• Calculate the minimum clearance gap for the links to twist freely — if this is less than the width of the kerf of the laser then only one cut is required.
  \[ k = -t + 2 \sqrt{ \frac{t^2}{2} } \times \cos \left( \frac{\pi}{2} - \frac{\Theta}{n} \right) \]
• Decide the link length from the total hinge width (or the centreline radius of the curve)
  \[ W = tn + k(n+1) \]
• To keep twisting of the bend joint, limit the spring length to less than four times the sheet thickness.
  \[ l \leq 4t \]

Symbol meanings at the bottom of the page.

Obviously, this is not very intuitive as a set of formulas, so I’ll have to generate some graphs/tables that give more succinct design guidance. And some more test pieces to validate them.

Test Results

Of all the test cuts in acrylic, I was able to bend all 4 samples to 90⁰, which suggests that the formula for calculating the number of spring links for the hinge is successful in limiting the material stress. However, the longer the spring connection samples were not very robust.

The long connection length allowed the top edges of the hinge to splay outwards during bending, adding a bending moment to the outer junctions of the hinges and causing several of the samples to break.

As a result of this, I’d recommend limiting the length of the spring connections to less than fours times the thickness of the material sheet.

Link Clearance

When each link twists, because it is square the corner rotates into the space between the links. The greater the rotation of the links, the larger the clearance between the spring connections needs to be to stop two links from contacting.

By adjusting the number of links in the hinge, It’s possible to reduce the twist of each spring connection and keep the clearance required to less than the width of the kerf of the laser that is used to cut the hinge. This means that it can be possible to use just a single cut between each link, reducing laser time and cost.

Link clearance can be calculated by determining the lateral movement of the corner of each junction during twisting:

\[ k = -t + 2 \sqrt{ \frac{t^2}{2} } \times \cos \left( \frac{\pi}{2} - \frac{\Theta}{n} \right) \]

Balancing the number of spring connections and the clearance between them has an effect on the total width of material used as the bend (and therefore on the radius of the resulting curve), so knowing the total hinge width will help to allow an informed decision on whether it’s best to: increase the number of links the keep a single cut clearance; or have fewer links and wider clearances between them to keep the hinge from binding.

\[ W = tn + k(n+1) \]

Nomeclature

\(k\) = Clearence gap (m)
\(l\) = Connected length (m)
\(n\) = Number of links in series
\(t\) = Material thickness (m)
\(G\) = Torsional Modulus of the material (Pa)
\(J’\) = Polar Moment of Inertia for non-circular sections (m⁴)
\(T\) = Torque (Nm)
\(W\) = Total hinge Width (m)
\(\Theta\) = Total bend angle of the piece (\(\Theta = \theta \times n\))
\(\theta\) = Angle of twist per link (radians) (\(90° = \frac{\pi}{2}\) radians)
\(\tau\) = Torsional Stress (Pa)

Flexi-Acrylic Test by Solarbotics, on Flickr

After this style of hinge popped up again, this time on Makezine, I was having a look at the linked project guide and at how they worked and I realised that a bit of mathematical modelling could lead to better designed hinges. This could mean fewer rounds of trial-and-error prototype tests, which would reduce the cost of using lattice hinges in a project, and better fatigue resistance, meaning the hinges could be used for moving parts instead of just for static bends.

Download links for the test specimen files are included at the end of this post.

This style of hinge appeared recently in the work of Snijlab, a Dutch laser cutting workshop when they showcased a folding notebook cover mode from a flat sheet of laser cut plywood. They apparently took some inspiration from MEMS hinges; and work by other designers has used some similar principles.

Others soon started to see the usefulness of a hinge that can be cut into flat sheet materials, .:oomlout:. released a plywood arduino box that can be made from only 3 pieces. Solarbotics and @kngunn showed that the lattice cuts also make acrylic, a notoriously brittle material, flexible enough to bend cold.

How do they work?

In addition to giving a hinge, a set of lattice cuts also allow for in-plane expansion and compression prependicular to the line of the cuts.

Basic Lattice Hinge Shape

Lattice Hinge in Compression

Lattice Hinge in Tension

In tension and compression, there are three repeated parts that allow the distortion to take place: two “Junctions” that do not deform connected by a thin piece that deflects along its length that I’ve dubbed the “Spring Connection”. It’s the elasticity (springiness) of the connection that allows the two ends to move relative to each other, and its this action gives a clue to how the connected system might work as a hinge.

Multiple Torsional Hinges

Junctions and Spring Connection

A Single Spring Connection

Two Parallel Spring Connections

Assuming that the junctions don’t deflect (or that the deflection is very small) it must be the spring connections that deform by twisting to allow the whole plate to bend. This is similar to how a chain, which has solid links, bends by rotating around the pivots that join each link.

This torsional behaviour is shown clearly in the hinge in use where the outside surface resembles a lattice hinge in tension, and the inside resembles the compression image, but the spring connections also show the twist because each junction end is rotated by a different angle.

Lattice Hinge in Bending

Detail of Snijlab's Hinge in Use

So if each link is loaded in torsion, then it is possible to calculate the material stress of each spring connection. And if the torsion of each connection can be calculated then, in reverse, it is possible to design a connection size that limits the torsional stress in the material, to ensure that the material doesn’t fracture in normal use.

Stress Limitation

If the stress in each spring connection is kept below a the torsional yield stress of the material, then the material will always be operating elastically. This means that the deformation will not be permanent; so once any force is removed, the hinge will return to its original, flat shape (Assuming no plastic creep for polymers). This also gives much better fatigue properties than if the yield stress is exceeded, meaning that the joint could be bent and unbent many times without risk of material fracture — particularly important for acrylic.

Calculating Torsional Stress

Warning: here be formulas. Just skip to the end if you want the results!

If it is true that the spring connection operates in torsion only, then it is relatively straight forward to calculate the maximum torsional stress of that region.

Equations for the calculation of torsional stress can be found at roymech.co.uk

Nomeclature

\(T\) = Torque (Nm)
\(l\) = Connected length (m)
\(G\) = Torsional Modulus of the material (Pa)
\(J’\) = Polar Moment of Inertia for non-circular sections (m⁴)
\(\tau\) = Torsional Stress (Pa)
\(t\) = material thickness (m)
\(n\) = number of links in series
\(\theta\) = angle of twist (radians)
\(\Theta\) = total bend angle of the piece (\(\Theta = \theta \times n\))

Calculation

The total torque of one spring connection is given by: \[ T = \frac{\theta{}GJ'}{l} \]

and for a square cross section in the spring link: \[ J' = 0.140625 \times t^4 \]

For one spring connection, the angle of twist is a proportion of the overall bend angle: \[ \theta = \frac{\Theta}{n} \]

and the maximum torsional stress in a square member can be given by:

\begin{align}
\tau_{max} & = \frac{4.808 \times T}{t^3} \\
& = \frac{4.808 \times 0.140623 \times \theta G t^4}{t^3 l} \\
& = 0.676125 \times \frac{\Theta G t}{n l}
\end{align}

Instead of just calculating stress though, it’s also possible to use the stress limit as a known, and calculate the length and number of spring connections required to stay below the limit stress. I’ve chosen yield stress as the maximum allowed stress, as this is the limit of proportionality of the material, and the limit of non-permanent deformation.

Minimum connections and length

The minimum number of connection links can be calculated by rearranging the previous equation: \[ n_{min} = 0.676125 \times \frac{\Theta G t}{\tau_{yield} l} \]

For Acrylic:

\(t\) = 0.003 m (3mm)
\(G\) = \(2 \times 10^9\) Pa (2GPa)
\(\tau_{yield}\) = \(60 \times 10^6\) Pa (60MPa)
\(\Theta\) = \(\pi{}/2\) rad (90deg)

I’ve used typical material data from MatWeb (Acrylic, General Purpose, Molded).

This looks lovely as a graph, but in short:

• For \(l =\) 5mm; \(n_{min} = \) 21.2 (22)
• For \(l =\) 10mm; \(n_{min} = \) 10.6 (11)
• For \(l =\) 20mm; \(n_{min} = \) 5.3 (6)
• For \(l =\) 30mm; \(n_{min} = \) 3.5 (4)

Practical Testing

Of course, this is all mere speculation without any actual testing(!). As I don’t have direct access to a laser cutter, I’ve sent for a set of test pieces to be cut, and I’ve also received a set of courtesy of .:oomlout:. in both 3mm acrylic and 3mm MDF. While I haven’t had time to test these fully, the acrylic is still brittle; I’ve already broken one hinge (oops!) and only the smallest (\(l =\) 5mm; \(n = \) 22) specimen is able to bend to 90 degrees.

This seems to suggest then that the my calculation is not quite accurate, though a few possibilities why come to mind:

• The material data I’m using is not quite the same as the acrylic for cutting.
• The material properties are affected by the heat of cutting.
• The stress is greater in practice because I’ve ignore the kerf of the laser.
• Choosing torsional yield stress as the limit stress is too high for dynamic bending (hence why I’ve shown 60% torsional yield stress in the graph too).

So there’s still a bit more work to do, however in the mean time , here in the zip file with the cutting diagram for these specimens if you’d like to try it your self: DOWNLOAD

Comments, questions, corrections and suggestions are welcomed.