Acceleration of a spinning sphere

[Bluemoon]

Can anyone please tell me: in what direction does a point on the surface of a sphere accelerate when:
the sphere is spun on an axis which is in-turn spun on an axis orthogonal to the first and that 2nd axis is spun on an axis orthoganal to the previous two:
This relates to a brief discussion that I had with someone about the moulding of Easter eggs(!)
Would the chocolate just slosh about or does it all get uniformly pressed to the inner surface of the sphere?

[swansont]

Rotations do not commute, so orientation depends on what order the rotations occur. That should also be true for angular speeds.

 

[studiot]

6 minutes ago, swansont said:

Rotations do not commute, so orientation depends on what order the rotations occur. That should also be true for angular speeds.

 

Well spotted sir.+1

[steveupson]

There's a series of interactive videos that allow you to view the math, and play with the visuals.  It's a pretty good way to get a handle on the math that is involved.

https://eater.net/quaternions

Edited January 5, 2019 by steveupson

[Bluemoon]

It was an oversight but I should have stated the context to be that all of the axes/gimbles rotate with the same angular velocity.
However: why should non-commutativity preclude an analysis?
Although rotations can not be said to be be generally commutative, the sphere, none-the-less, spins; so surely that can be analysed?

Anyway, I did some brief numerical analysis by hand and it seemed that the chocolate does just slosh about.

steveupson:   that's a wonderful resource {+1;  although each video froze my old laptop for a, Very, long time; the whole lot took me over 4 hours to watch}.
The 4D lecture reminded me of the Charisma Cleo video editor [made by Questech but neither still exist] which was popular with broadcasters in the 1990s.
It would roll-up a 2D screenshot into an apparent sphere (or paper-dart or any other of a large selection of pre- or user- programmed 'objects')
revealing the next camera shot behind it, the spherical image would then roll off into the 'distance'.
I never knew but I now presume that Quaternions would have been the core tool of the algorithm in the processor [which was specified as a 'tailored' supercomputer](?)

 

[Edwina Lee]

I presume that it is powder or melted chocolate that is being spun, so that it is hoped to spread to the inner surface of the mould to form an Easter Egg.

Firstly, it is not a perfect sphere because it is egg shaped.

So, is the combination of 3 orthogonal steady spins a steady spin?

If it is equivalent to a steady spin, then there is no spreading. If not, there is still a need to calculate how to spin so that the chocholate covers the whole egg shell.

(Never tried such analysis myself, so I'll have to think about it.)

[swansont]

14 hours ago, Bluemoon said:

It was an oversight but I should have stated the context to be that all of the axes/gimbles rotate with the same angular velocity.
However: why should non-commutativity preclude an analysis?

You said the rotations happen in-turn. So the order matters in terms of what the final rotation is. If you don't know what the rotation is, how can you analyze it?

 

[Edwina Lee]

My systems analysis skill in action:-

Step 1:-  Fact finding to determine what the problem actually is.

Repeat step 1 until the problem is crystal clear.

I think this is equivalent to the problem we are talking about:- (See attached picture)

 

[Bluemoon]

20 hours ago, Edwina Lee said:

Firstly, it is not a perfect sphere because it is egg shaped.

True, but:

On 1/4/2019 at 7:07 PM, Bluemoon said:

This relates to ... Easter eggs(!)

I idealised it to a sphere as I thought that that would be more analytically tractable though still keeping the molten chocolate.

52 minutes ago, Edwina Lee said:

I think this is equivalent to the problem we are talking about:- (See attached picture)

Yes.

swansont:   I only used the phrase in-turn in the physical sense to describe the mechanical structure [not the temporal sense];
I had supposed that the chocolate would be affected the same way regardless of the phasing of the gimble spins because as I see it:
the rotations, when considered as framed in my OP, are commutative; each axis/(gimble)/rotation is defined relative to another (rather than per the convention) i.e:
the y-axis/(gimble) sits in the z-axis/(gimble); and the x-axis/(gimble) sits in the y-axis/(gimble) and the sphere spins on that x-axis;
and so: the orientation of the sphere after defined rotations of each of Those z, y & x axes/(gimbles) will not depend on the order of their execution.

 

[Edwina Lee]

Analysis:

Looking at it as an engineering problem, a fast spinning M1 and a relatively slower spinning M2 is sufficient to spread the chocolate in the egg mold.

It is clear that the sticky melted chocolate would form a band with just M1. As M2 rotates M1's axis, a shear force would act on the spinning band of chocolate because of the stickiness causing chocolate to smear throughout the shell.

However, from a curiosity viewpoint, a mathematical analysis using Quaternion and rotational matrices to calculate loci and acceleration of each point on the egg would be interesting. Too complicated for my brain at the moment.

Edited January 9, 2019 by Edwina Lee
Extend what I want to say.

[Bluemoon]

Thanks for that practical suggestion Edwina Lee,  I had intuitively thought that keeping everything spinning at equal speeds to be the best starting point as I, too, don't have the brain for a full analysis.

[Edwina Lee]

I expect there is an optimum strategy for M1 & M2's rotations depending on the optimization criterion and the properties of the chocolate & shell surface.

A trial & error experimental approach sounds far more feasible than mathematical prediction.

[anne242]

As was pointed out it s all about  what order the rotations occur

[Mike-from-the-Bronx]

I took a stab at this with my simulation program.
I copied the arrangement offered by Edwina Lee.
The arrangement implies a parent/child hierarchy as suggested by Bluemoon.
In my simulation, the user does not declare the order of processing.  
The hierararchy defines it.  Parent first, then child, grandchild, etc.

Here's a link to a video I created from the results.

http://www.relativitysimulation.com/Documents/Nested_Bowls.mp4