• ### More Design Priciples

So I have done some engineering analysis and some prototyping work and I now have the next version of the rotary joint that I would like to manufacture and test.  First; How I got to this point.

# Holding Moment Calculations

So I have previously documented a set of calculations that can be done to derive the holding torque for face-to-face contact of two discs.  The resulting formula is:

Where M is the holding moment, µ is the coefficient of friction, F is the applied force, Ro is the Outside Diameter and Ri is the Inside Diameter.

This results in some interesting conclusions:

## µ vs Ro

The holding moment is directly proportional to the coefficient of friction.  This means that achieving the highest coefficient of friction allows the lowest clamping force to hold a given moment.  The moment is also proportional to the outer radius of the disc.  The subtlety here is that the overall size of the joint is given by the diameter not the radius and so it is very beneficial to increase the coefficient of friction against increasing the radius of the disc.

To illustrate this, for a given force (8 kN), a required holding moment (180 Nm) and a coefficient of friction of 1, the disc must have an Ro of 25 mm and an Ri of 20 mm.  If the coefficient of friction is reduced to 0.6 then the disc needs an Ro of 40mm and an Ri of 35mm.  This changes a 50 mm diameter disc to an 80 mm diameter disc, greatly increasing the mass and making manufacture more difficult.  (The difference between Ro and Ri was maintained at 5 mm in both instances to simplify comparison)

## Ri and Ro

The equation for the holding moment of a circular contact is directly proportional to the radius, given that there is no inner diameter.  The holding moment for a disc is more complicated and the relationship between the holding moment and Ro and Ri is not immediately intuitive.

For a given force, the holding moment increases as Ro increases. This is intuitive.

For a given force, the holding moment decreases as Ri DECREASES.  This was not intuitive to me at first.

For example:

So for MORE surface are you get less force???

Okay, so I can now think of this in terms of a certain pressure as a certain radius.  The case with the larger ID (left) has a higher pressure and all of that higher pressure is spread over a larger radius.  The case with the smaller ID (right) has a lower pressure and that pressure applies over a smaller radius than the case on the left.

If anyone knows of a better explanation, I would love to hear it.  Please message me or comment here, I would love to understand this better.

Anyway, the conclusion of this exercise was a table:

µRo (mm)Ri (mm)Desired M (Nm)Required Force (kN)
1.025151808.8
1.02520180
8.0
0.8252018010
0.6252018013.2
0.4252018019.9

(A.N. If anyone wants a deeper explanation of how I got to these figures, message me and I will post a very nerdy update with equations and everything)

# Hand wheel and bolt selection

The required clamping force is applied using a hand wheel or hand lever.  There is a limit to the torque that can be applied by humans and I would like this joint to lock and unlock easily.  The bolt must also hold without damaging the threads.  It would be preferable if the hand wheel or lever would fit within the silhouette of the joint.  Another locking method is the cam-lever.  This does not necessarily need to fit within the silhouette of the joint as it does not lock via rotation.

Bolt torque is calculated as follows:

Where T is the torque, D is the diameter of the bolt, F is the force applied, and K is the coefficient of friction.  The units are typically in metres and Newtons, however, I find that using mm and kN is more intuitive and does not require a correctional factor.  K is typically 0.2 for dry steel on steel and may be as low as 0.12 for very well lubricated bolts.  I am using K = 0.16 for my calculations.

For an M8 bolt to apply 8 kN of force, the required torque is 10.2 Nm.  This is fairly simple to apply with a hex key, but is much more difficult to apply using a 40 mm diameter hand wheel.

Roy Mech has a handy guide to the strength of the average human here: Human Strength.  This gives a mean tightening torque for a human female, on a 40 mm diameter knob, as 3.89 Nm.  This value is considerably lower than the required 10.2 Nm.

The required tightening torque can be reduced by reducing the diameter of the bolt:

For an M6 bolt to apply 8 kN of force, the required torque is 7.6 Nm.  A grade 8.8 M6 bolt can withstand 11.6 kN force without undergoing plastic deformation. Metric Proof Loads.

7.6 Nm is still too high to be comfortable so the diameter of the joint must be increased to reduce the necessary clamping force and therefore the necessary clamping torque.  A larger knob may also help the user apply more torque.

# New Design Parameters

The resultant design has a few significant changes.  The Ro is 50 mm and Ri is 45 mm, giving an overall diameter of 100 mm.  The force required to hold a torque of 180 Nm is 3.8 kN.  An M6 bolt is used to provide the clamping force, requiring a tightening torque of 3.6 Nm.  This is slightly above the stated human torque but I will choose a larger hand wheel to compensate.

• ### Prototype #1 - What I Learned

So I went and made the first prototype rotary joint.  It works surprisingly well being my first attempt and I think it will certainly stand up to testing.  It is very satisfying to hold and Locks very rigid when held in the hand.

The order of operations in terms of component manufacture could definitely be improved.  In the first prototype there are 3 major components: The base, the washer and the cup.

There are several mating surfaces that have tight tolerances to consider.  The washer must have its ID bored and be the correct length before the base is made, as the washer has its OD finished when glued to the base.  This may or may not be necessary as the nylon washer will readily deform to the shape of the base, although it is definitely useful in allowing fine tuning of the washer.

The ID of the cup must be turned after the base and washer have been finished to ensure the correct fit can be achieved.

# What will I change?

First, the washer should not extend to cover the end face of the base and should not be used as both a high friction material in the locked mode and a low friction material in the freely rotating mode.  This will also simplify the production of the washer, with only the facing, boring and parting operations being necessary before gluing the washer to the base.  Fewer critical measurements to make, fewer features to cut, simpler cleanup after glue up.  Good.

Second.  I did not manage to machine the recess in the cup as shown in the picture above.  I am technically capable of this machining operation, although it is not trivial, but I did not consider it necessary for testing and I could always add it later if necessary.

This recess was designed to allow the centre section to flex and make contact with the base, thereby allowing metal to metal contact and therefore good locking.  This approach may have worked, at least, the aluminium could be deformed 0.05 mm when tightened with a hex key.  However, this is very much missing an important engineering subtlety that will be shown in an upcoming project log.  The recess may be added in future iterations for weight-saving purposes but the cup will not be allowed to contact the base near the centre of the screw in future iterations.

I also made the OD of the cup larger as the available stock allowed it and it felt right in the hand.  I liked the larger overall size and I shall likely stick with this choice of OD, although I may design a smaller version in the future.

The base component was basically machined to drawing and will not likely change much in upcoming designs, although the mating face will change slightly.

• ### Some Design Principles

The previously given specification for the joint can be though of as a set of tests which the joint design will be evaluated against.  We can make any joint design we like, but if it wobbles then it fails to meet a fundamental requirement.  To help to pass the specification tests, we can use some fairly basic design principles to guide our decision making.

So lets start by evaluating the forces and torques on the joint.

I would like the arm to have a reach of 3 meters at full extension and I would the payload to be 2 kg.  This is quite a significant extension for such a mass but hopefully we can get somewhere close.  The self-weight of the arm is also important: I am going to assume that this is equivalent to another 2 kg at full extension (4 kg total mass, evenly distributed along the arm).

The maximum static torque is experienced by the joint furthest from the load i.e. the joint attached to the wall (Labelled A below)

The torque at A will be 4 kg x 9.81 m/s^2 x 3 m = 117 Nm.  We will use the value 120 Nm for ease.  This is the value that the joint must be able to comfortably hold.  If the joint is able to hold only 120 Nm then it would begin to slip so a factor of safety should be used.  In this case a factor of safety of 1.5 feels about right so the joint must be able to hold 180 Nm maximum.

From this site:

We can calculate the holding moment between two discs as follows:

Where M is the moment, µ is the coefficient of friction between the surfaces, F is the force pushing one surface onto the other, Ri is the ID of the disc and Ro is the OD of the disc.

It is difficult to find details of the coefficient of friction for aluminium and Nylon.  Tables show the coefficient of friction between Nylon and steel as 0.4 so we will use this value. Ri is 18 mm (0.018 m) and Ro is 25 mm (0.025 m). Friction ref

F is somewhat variable.  High force mechanisms are typically difficult to actuate so lower is better.  We will find the required force to give a moment of 180 Nm.

This gives a relation of M = 0.008675 * F, (M in Nm and F in N)

The required force to hold 180 Nm is 21 kN.

The bolt that is used to apply the holding force is an M8 which according to Bolt Ref has a minimum ultimate tensile of 29.2 kN (for grade 8.8).

Using a bolt torque to force calculator (Bolt Torque ref) a required torque of 20.2 Nm is calculated.

# Discussion

These calculations reveal a lot of the dependency between clamping strategy and the required performance of the underlying components.  We could take them further.  We could calculate the force that can be exerted by a cam mechanism as they are cheap and easy to come by.  We could also investigate the pull-out force that the tapped aluminium could sustain, although I don't think either of these will be hugely useful in my situation.

Instead, I made the first prototype joint and I am going to test it.  At the same time I am going to redesign the joint to use a different face contact and use Aluminium-on-Aluminium for the friction surfaces, instead of Nylon-on-Aluminium.  I am also doing something I should have done before making the prototype: I am going to run these calculations for several designs to understand the sensitivities involved.