# A novice’ approach to a Spherical Pendulum Clock

#### ItsAllRelative

##### Registered User
Yup, a spherical Pendulum…and I don’t just mean the shape of the bob!

A quick introduction :

I am creating this thread to follow my conception, exploration, and construction of a Spherical Pendulum clock. I am no physics professor (or student even), but I do hope to keep the thread very scientifically and mathematically informed. I plan to use Wolfram Alpha for data visualization and interaction and Solidworks for CAD renderings along the way. Through this site I hope to connect with other avid horologists, share unique/unusual ideas, and explore a new mechanical concept. As the project progresses from concept to in-the-metal experimentation I will share my thought process’ and I absolutely welcome any input through questions, comments, concerns etc.

The best place to start is by identifying the Spherical Pendulum which could get VERY scientific but I want to keep the intro relatively concise so here goes.

A Spherical Pendulum is free to move any direction in its 3 dimensional space and has 2 degrees of freedom. The bob of the pendulum moves along the surface of a sphere and its location is defined by Spherical coordinates. A spherical pendulum is capable of demonstrating the same behaviors as a simple pendulum (single plane oscillations), a conical pendulum (swinging in circles tracing a cone) and many more behaviors dependent upon its initial starting and driving conditions.

Some initial research has brought forward many unique aspects to this concept and I’ve listed some of the more interesting bits below as food for thought. Some of these may help contribute to an accurate mechanical clock, and some may be detrimental to the project.

*Two degrees of freedom capable of expressing resonance
*Conservation of angular momentum-constant precession of an elliptic orbit
*Stable or Chaotic motions are achievable

….Bring on the Science!

#### Phil Burman

##### Registered User
Bring on the pictures, what does/will a Spherical Pendulum look like? What potential advantage does the concept have over a conventional clock pendulum, and I guess the big question is WHY?

Phil

#### Allan Wolff

##### Moderator
NAWCC Member
I'll be watching with great interest. It's great to see novel approaches with the science to back it up. I don't have a mental image of what you are describing so pictures will be appreciated.

#### ItsAllRelative

##### Registered User
Phil and Allan thanks for the quick interest in the subject...

A diagram of a spherical pendulum taken from Wiki:

As you can see from the diagram, its simply a pendulum suspended from a pivot so that it's able to move freely in any direction (Imagine a weight or bob hung on the end of a string tied off at the top). A photo or diagram with an optimized real world mechanical suspension setup is nearly impossible to find. Optimized for what?? The predicted behaviors of the pendulum that you want to induce. The fist thing that comes to mind for a suspension is simply the string, however this has its downfall, as does something like a cardan joint (universal joint), a Spherical bearing, or a gimbal mount of some kind (I'll cover why in a bit). Lets do a thought experiment with the basic behaviors the system may exhibit.

Imagine looking down on the bob hung from a string tied off at the top with a simple X,Y graph:

The Datum of the graph will represent the pendulum when at rest, hanging vertically. Lets pull the bob to about 15 degrees keeping the string taught and release. If we were to plot the position of the pendulum on the graph (regardless of which direction you initially pulled it in) it will fall in a straight line passing through the center, reaching its peak on the opposite side of the graph to where you released it, and continue oscillating back and forth until it comes to rest. This will all happen in a single plane, or as viewed from above on our chart it will be a straight line extending equally in two opposite directions from the center. Simple so far....

Now lets repeat the experiment but instead of simply dropping the pendulum, give it a slight push in the counterclockwise direction. Again viewed from above, this time the pendulum will trace out an ellipse for each complete oscillation. The first thing that comes to mind is the elliptic orbits of planets around the sun:

This is actually a little bit different than what we will see. Because of the physics of the pendulum system we will actually see our ellipses precess (rotate around the center point) equally on both sides of the pendulums swing. Our pendulum will trace out a path more like this:

At this point, we can use mechanical physics to explore the exact path of the pendulum. Physical variables include pendulum length, mass, angular momentum imparted (exclude driving power/impulse for now) etc, and in our results we will see differences in the ellipse' defining traits (perihelion, aphelion semi/major axis, etc), the rate of precession and more.

#### ItsAllRelative

##### Registered User
With the above info in mind....I bought a cheap pair of "spherical" bearings sometimes referred to as "self aligning" bearings. These bearings have some unique movement capabilities and this was simply for me to mount an adjustable length pendulum by my computer to help me visualize some of the behaviors myself.

The two bearings with a pen behind for an idea of size:
(the larger bearing still has a brass insert pressed in that held the pendulum rod)

Another photo with the bearings showing their range of motion with the small pendulum behind:

The bob has a small set screw on the backside and the rod is about 24" long.
Both bearings were cleaned at my work with kerosene and oiled with a variety of lubes for me to test. Unfortunately no matter what was used as a lubricant the system still had way to much friction and would only swing for a few oscillations before dragging to a stop. This obviously rules out a form of bearing for support in a clock...But it does give me some info I might not need for a while. What happens to the behavior of the pendulum system when friction, air resistance, etc start taking over? Does the precession stay constant and the ellipses loose amplitude? Vice Versa? Both? How can this play into the creation of my mechanical clock??

For more of my own "physical" visualization as I dig into the mathematics of the system I will be hanging a pendulum in my little study room in the basement of my house. I will use the string and weight method this time. I plan on using a longer length, a heavier bob and a suspension based off a type of "ring" suspension I saw somewhere online that I cant seem to find. Its simply an open hook on the top holding a ring with the string connected. From what I remember when reading about the design was that it was supposed to remove the "favored" positions that the suspended weight would take with an otherwise simply tied or clamped string. I will use an Eye hook screwed into the ceiling with a round key-chain type ring on the bottom (pictured with double eye-hook as that's all I have at the moment), although this method is also probably unsuitable for clockwork with the amount of sliding friction standard with its design.

#### ItsAllRelative

##### Registered User
As far as why I'm interested in exploring this system needs only a few words. A resonant, stable system of constant motion is achievable. I'll leave it at that for now!

#### tok-tokkie

##### Registered User
A 'silk' thread suspension would have less (least?) friction. Only the circular component would involve sliding.

#### Phil Burman

##### Registered User
For it to function as a precision clock pendulum you will need to have a suitable method of:

1) Impulsing it (to keep it running).

2) Taking a time signal from the pendulum, in order to indicate the time.

Both of which could probably most accurately be done electrically but quite how is not obvious.

The method of suspension would need to be as near to frictionless as possible. Possibly something similar to a knife edge but instead with a vee point siting in a conical depression.

Understanding the circular error of such a pendulum looks like a bit of a challenge. Amplitude would need to be down around the 1.5 degree mark to keep it reasonably isochronal.

For it to be of interest as a precision clock pendulum it would need to have significant advantages over a conventional pendulum, I can’t see any at the moment. For other types of clock it might be of interest from an art or a craft perspective.

Just some thoughts to consider.

Phil

#### Phil Burman

##### Registered User
As far as why I'm interested in exploring this system needs only a few words. A resonant, stable system of constant motion is achievable. I'll leave it at that for now!
Sounds a lot like a conventional pendulum.

Phil

#### tok-tokkie

##### Registered User
Phil Burman's suggestion of a knife edge type suspension makes me think of V jewels. There is a recent thread on where to source jewels. They all offer V jewels. You would need a column with the V jewel at the top. Pendulum would need a stirrup at the top with a needle resting in the V jewel.
Alternatively pin on the column & V jewel on the stirrup. That has the advantage that dust will not settle in the V jewel.

#### ItsAllRelative

##### Registered User
Phil, I agree with your comments on the importance of reducing friction throughout the system and the challenge of circular error stacked on-top of all the other hurdles I will have to overcome during the creation of this timepiece. This forum will help serve as a "notebook" of sorts for me so that as the ideas progress, concepts are designed in CAD, and machining starts I have a very solid reference document. I'm sure the lists of things that won't work are MUCH longer than the list of ideals.

I don't believe a system like this necessarily has any advantage to a standard regulator, but I simply want to explore the possibilities of the system. The single largest hurdle I see aside from friction is the problem of air resistance. In the book "My Own Right Time" by Phillip Woodward he spends a lot of time covering the "Q" or (Quality) of mechanical clocks and the role that it plays in their accuracy and stability. He makes it very clear that a high Q value isn't the only factor of a precision regulator but is a common trait of them. He then goes on to cite that a regulator in open air can achieve 10,000-30,000 compared to the Shortt clock swinging in a vacuum that can achieve 110,000.

Without making a solid scientific comparison (or having a working clock of this type to do so with), I have to assume that I'm already a step behind the leaders due to the extended path of the pendulum that would be inherent in a design like this. However, I can't discard the conical pendulum type behaviors possible which would MAYBE have an upside here, an (ideally) constant speed meaning a near constant resistance. This again can be argued though that the role resistance plays is proportional to the speed of the pendulum, i.e. the faster the pendulum the more resistance it incurs.

These are exactly the thoughts and types of issues I want to explore with this system from a theoretical standpoint and also comparatively to other clocks.

#### ItsAllRelative

##### Registered User
On the suspension...

These are all very clever ideas! The V jewel was an earlier idea I had, but not the inverted bit in regards to dust prevention which I can't believe didn't cross my mind.....If you could've seen my face when I read that I'm sure a Lightbulb would've lit above my head as I was reading.

If the pendulum is supported in a V jewel (or alternatively an inverted jewel on a pin) the pendulum will be directly underneath when at rest. This would not be acceptable because the pendulum could come into contact with the column during operation. A way around this would be putting the v jewel on a ledge of some type and building a "C" or "O"shape into the top of the pendulum rod to allow clearance. This though would remove the ability for the pendulum to rotate on the vertical or Z axis, unless the support was rotating at the same rate of course. Some of the scientific journals on spherical pendulum behaviors describe a rotating or slewing center and some don't. I have even found a few journals published that explore the behaviors of a spherical pendulum impulsed through a linearly or rotary impulsed suspension.

At one point I considered making the top of the pendulum rod a ring shape with a sharp knife edge around the underside of the ring. The ring would "roll" around a circular track and the ring would look a bit like a spinning coin. This was ruled out though due to the amount of friction a system like this would have. Even though only a small portion of the knife edge would be in contact at a time, it would experience sliding friction as it rolled as well as sliding friction as the knife edge ring was forced to follow a circular track with and elliptical orbit....talk about getting into some complex designs to overcome this.

Food for thought though...this clock appears to operate as a conical pendulum clock and has incorporated the escapement into the suspension design. I have emailed the Wife of the maker as it appears he passed away looking for any info she might have on the timekeeping abilities, though I haven't gotten a response yet.

The silk suspension is more along the lines with my current thought process. A very similar "wire" suspension is used for foccault pendulums and this site has lots of good information and links.
My Foucault Pendulum

#### tok-tokkie

##### Registered User
That Stuart Pratt clock is fantastic.

I tried a 'zero torque' suspension in my gravity clock. This is the suspension used for seismometers. I used Dynema thread (=UHMWPE thread) sold as FireLine fishing thread. It was just 0.2mm diameter but rated 10 lbs. I was very disappointed by the results as the Q of the pendulum was much reduced. I suspect the line was flattening slightly as it wrapped around the cylinder and that was sapping energy from the pendulum. My advice is use a really hard wire instead of something soft like silk or Dynema.

#### novicetimekeeper

##### Registered User
Harmonographs often use double knife edge suspensions, that may give you the low friction you want, though how you impulse it I don't know.

#### ItsAllRelative

##### Registered User
Well after a bit of a rendering marathon I have a few photos of a VERY ROUGH general concept of the way I would like the clock to look.

The clock will be freestanding(if it features a seconds length pendulum) or table standing. The photos depict the freestanding version. If I go with a half seconds length pendulum the supporting pillars will be stretched reaching to the length of the pendulum. The Main design features 3 main pillars supporting a heavy duty support plate with a leveling system based off old school theodolites or modern tripod leveling mounts(this would actually need a central collar in order to allow the pendulum suspension and rod through). I do plan on incorporating X and Y axis bubble levels as well as lateral micrometer feeds to dial in the suspension. The central ring (wooden and represented at different heights in two of the renders) would be my interpretation of the weight, keeping with the symmetry of the clock, and in the photos the pendulum rod is represented as a quartz tube based off the designs by David Walter.

Please remember many of these ideas are just that, and everything may be subject to change. I am very new to solidworks and entirely self taught so far so go easy on these!

#### ItsAllRelative

##### Registered User
tok-that's great info to have with the Dynema. Do you suppose if the system had a rigid rod and support but say an inch long "through section" of line as the suspension that might eliminate some of its energy sucking traits?? This could be a soft elastic material, silk, or a more rigid wire or even a thin enough diameter solid that acted elastically like the springs of a regulator.

Also, Novice timekeeper, I have seen harmonographs before but only in photos without paying much attention to detail. The suspension you pointed out is a perfect example of a gimbal type design. The interesting piece to consider here is that any "linear" suspension could be incorporated into this type of setup....opens a lot of potential doors.

#### ItsAllRelative

##### Registered User
A quick update…

I was searching for a way to better visualize the motions the system could create and the next place I went was here:
Wolfram Demonstrations Project
This website is by Wolfram who created Wolfram Mathematica, Alpha, Demonstrations, the Wolfram language and much much more. These guys are the leaders in the business of computational engines and were very involved with none other than the late great Steve Jobs

A good read and a 1min clip of Steve introducing Wolfram co-founder Theodore Gray with a short story.
Steve Jobs: A Few Memories—Stephen Wolfram Blog

Anyways, this Wolfram demonstrations site allows you to download a CDF player (computable document format) for free and interact with any of the posted projects. A search for “Pendulum” yields about 113 results with any imaginable application (double, triple, torsion, damped, cyloidal, driven and too many more to list), while a more appropriate search for “Spherical Pendulum” yields an easier to swallow 9 examples. I explored a few of these specifically including both “Damped” models and the driven model. They were O.K. but all suffered the same fate. The biggest problem is that they don’t provide charts or graphs showing the period of oscillation or a predicted “stop” time of the system when you adjust the damping coefficient. I would need a baseline measurement with no angular momentum, I.E. a simple pendulums damping coefficient, before I could simply pull one from thin air and apply it to a spherical system. If I had those time charts, I could find a coefficient that allowed say 8 hours of oscillation in a seconds length pendulum and apply that or more to a spherical system to at least be in the ballpark. Right now, I’m driving blind, so on to something more tangible for the evening….

I hung a rigid hook in a joist and machined a quick new bob at work before I left today. The bob is 316 stainless roughly 1.75” in diameter and 3.9” in length. This piece had been consigned to the scrap bucket, but was calling my name with its clean chamfers, perfect size, and even a .201 hole already drilled. I tapped it with 1/4-20 and drilled a through hole in a short cap screw. This would allow the cable suspending it to be tied off inside the bob and guided through the center of the top.

I have read a variety of nightmares from users about using cat-gut or fishing line on their fusee clocks so I paid a visit to one of my bigger influences at work. His name is Paul and he’s a “hobby” fisherman and one of the best machinists I have the pleasure of working with. I only say hobby because he does, but truly he’s a pro. He’s been at it his whole life and gave me great advice with my inquiries. He told a story of friend who had their boat dragged 6 miles or so shown on their GPS system with a tuna tugging and pulling on the line before they finally got it aboard. I picked his brain about what they use for line and he gave me incredible answers explaining the flex and spring properties of monofilament vs. the tightly woven thin synthetic line with high feedback. He explained the animals behaviors with trying to break the line against rocks, sudden jumps, and even the different types of rods used for different situations. If a grown man can bend a 5ft rod nearly in half with the line over their shoulder and an animal pulling him and his boat for a few miles, it qualifies for holding this bob for now!

Paul hooked me up with a spool of this line and said I should have no problems. We looked at the line under the microscope at work as he was giving me this great info and its incredible what goes into it. This line is rated 85pounds and measures .016-.018 in diameter in a set of micrometers without squishing it too much…although it squishes to about .011 or .012 When I got home I put the lens extenders on my camera and tried to grab a photo of it. Its not the greatest as I no longer have my macro specific lens but shows some incredible detail.

I tied off the bob on both ends with a scaffold knot, which was not quite as easy to tie as the photo makes it appear, however I did not incorporate an eye with my design. I now have a “string” suspension and HEAVY bob that exhibits the behavior I was looking for.

This setup allowed me to physically mess around with high and low amplitudes and the imparted angular momentum. I’d compare this part to watching the tourbillon of a watch, with just the right recipe you see a beautiful blend of continuous circular precession with a very distinct oscillation occurring in the same space at the same time. I was watching and counting swings of the system with my phone on stopwatch mode nearby and I will start documenting which precession rates and amplitudes are visually ideal.

The interesting stuff to remember later…..

Regardless of the knot I chose for this setup, I considered that the string would always be tied off 90degrees tangent to the hook it was tied around. This showed itself in my setup and introduced the same favored plane of swing to my pendulum as that experienced by makers searching for an ideal Foucault suspension.

Another and more interesting bit, and definitely something to consider with the design of impulse and suspension:
The angular momentum of the system entirely disipated much before the amplitude of swing did. This could be due to the favored plane of oscillation, and I believe it is, but this changes the way I need to look at impulsing the clock. Considering this piece of info with the articles I’m digesting on resonance in spherical pendulums I need to treat the angular momentum and amplitude of swing as two separate entities that both need a system of management.

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#### MartinM

##### Registered User
We always just called that knot a, "noose".

#### ItsAllRelative

##### Registered User
Well Its been a while since I’ve posted but I wanted to give an update on this project. I should say its a bit challenging managing my time with all of the “New Fronts” I chose to take on at once. Each of the subjects(CAD, Horology, Physics, etc.) is an Entity of its own and demands careful time to explore and understand. I will try to contain the information in my posts here into those categories, expanding on the unique things I’ve learned, how they connect, and how they will potentially shape my project.

General-
I have composed a list of “systems” that I have explored pertaining to the pendulum. Each system has a unique mechanical construction with unique operating principles, and represents a physical solution or expression of complex mathematical functions.

Simple Pendulum
Resonant Pendulums
Paraconical Pendulum
Foucault Pendulum
Bravais Pendulum
Conical Pendulum
Lalannes Pendulum

I have to say at this point….Describing the clock I’m aiming to build as a “Spherical Pendulum” doesn’t quite feel right anymore. Several of the above examples are also spherical pendulums that represent specific physical behaviors that a spherical pendulum can produce. I’m going to re-examine the system I’m aiming to create and come up with a more accurate name, one pertaining more to the behaviors of my system (elliptical precessions, ).

Understanding these systems has given me a great deal of knowledge, but also keeps my thought process trained on the design of this clock. I’m trying to determine the implications that varying degrees of freedom in a spherical pendulum system can create. Think: The pendulum is free to swing any direction in an X or Y plane, but what about freedom of the rotational Z Plane? How has this or can this affect a spherical pendulum, specifically a spherical pendulum experiencing precession of its elliptic oscillations?

Physics/Sciences
-
I am beginning to learn the basics of classical mechanics, specifically expressing systems through Lagrangian and Hamiltonian functions. I’ve also started exploring the numerous attributes of ellipses, including terminology, mathematical functions, and their connections to orbits (orbital mechanics)…talk about overwhelming. A few books I’m planning to purchase are listed at the end of this post…

I have created a handful of VERY basic models representing generalized categories of suspensions that have been used historically in both Horological and Scientific Instruments. This list is by no means all-encompassing. Each Category has its own list of identifying features, pros and cons inherent to its design, and a historical relevance. Many of these suspensions have been used in multiple versions of the pendulum systems listed above and what failed in one case is often the prime candidate for success in another. For now, here are the renders and once I’m done typing the summary of each system I will post them.

Cord Suspension

Gimbal/Universal Suspension
1.linear pivot

2.Point pivots

Point Suspension
1.Ball-Point

2.Jeweled Pivot (or Inverse)

Not included:
Hook Suspension (hook to hook, hook to cord, hook to ring, etc.)
Spherical Bearing introduced in my first mock-up
Spring type (a crazy idea, but plausible non the less!)
..........

Design-
Last but not least, I have to address my initial design ideas. I still believe the clock would be gorgeous in the original conception, but lacking a proper support and rigidity. I also found a clock by Erwin Sattler (Columna Temporis) which was essentially what I saw in my head for my design.

A photo from the Erwin Sattler website:
Welcome to ERWIN SATTLER: Erwin Sattler

I do not wish to emulate or replicate that piece in any way, so the design of my clock is being wholly reconsidered with a bit more of a traditional approach. I still want the “stand alone” idea to be implemented to my design…The clock must be approachable and readable from all sides. How I’m going to do this…not sure yet. I’ve also decided I want the escapement mechanism front and center. The clock should draw you in with its motion, explain itself with its kinetics, and give the time of course. These things will be considered as I move forward. I’m exploring potential changes this could introduce to the existing suspension types, as well as considering how this will factor into an escapement design. Think: Should a “Through” suspension be used with impulsing done above the suspension? How could the clocks design change if the escapement mechanism MUST be mounted below the suspension? Hmmmm……

Literature-
The Pendulum:A case study in physics
By Gregory L. Baker and James A. Blackburn

Seven tales of the pendulum
By Gregory L. Baker

The Pendulum Paradigm: Variations on a Theme and the Measure of Heaven and Earth
By Martin Beech

Countless Scientific papers have proved both enlightening, but also many times with implications far over my ceiling of understanding. I’m still Deciding on which of SEVERAL books on Lagrangian and Hamiltonian mechanics I will invest in.

BONUS if you’re still reading! Look into Heike Kamerlingh Onnes and his Doctoral Thesis. It’s got some GREAT info I’ll be covering here soon. HINT: “unavoidable mechanical asymmetry”

Oh and if you happen to own the book "The Mariners Chronometer: Structure, Function, Maintenance and History", Chapter 8 covers much of the history of the instruments with detailed photos and descriptions as well as factors that led to the evolution of the most widely known version of it. Inspiration is to be had with the variety of mechanical solutions to motion, vibration, etc.

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##### New User
Phil Burman's suggestion of a knife edge type suspension makes me think of V jewels. There is a recent thread on where to source jewels. They all offer V jewels. You would need a column with the V jewel at the top. Pendulum would need a stirrup at the top with a needle resting in the V jewel.
Alternatively pin on the column & V jewel on the stirrup. That has the advantage that dust will not settle in the V jewel.
Do you have a link for the v jewel sources?

Thanks

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