A gravity graph is a kind of art "tool" that lets you draw nice geometric shapes

[daniel_i_l]

A gravity graph is a kind of art "tool" that let's you draw nice geometric shapes. It has a board the size of a piece of paper that is weighted in the middle, each of the corners is connected to a piece of string and the strings are tied to a small rectangle (paralell to the board) about 1.5 feet from the board. Basiclly it is a board hanging by 4 strings. Then there is a pen that touches the middle of the paper (that is on the board) and it can bob up and down so that it touches the paper even if it's (=the board) moving up and down. Now to draw a shape you just swing the board and as it slows down and stops it draws nice geometrical shapes. For example, if you swing it in a circle you will get a spirle.

Now I wanted to make something like that on the computer. First of all I have to simulate the swinging. I thought that I'd think of the strings as tense springs, and figure out the sum of the forces on each corner (spring+gravity), Then from those forces I'll check the sum of the forces at a right angle to each of the 3 axises of spin (x,y,z) and from that figure out the angular acceleration of each axis with Fx = Ia. Then I sum all of the forces around the center point to see how much it moves (translates) with F = ma. For every frame I'd callcuate the rotation by the angular acceleration and the position by the linear acceleration.

Is this correct? The main problem that I see is that the same forces both spin and move, should the forces that I use to translate be weakened because of the force that is being used to spin? Or is there an easier way to do the whole thing?

Thanks!

 

[haruspex]

It is a complex arrangement to model accurately.

think of the strings as tense springs

I guess you mean horizontal ones, with the board remaining in a fixed horizontal plane. It should not make much difference whether you take them to be always under tension or all relaxed at some equilibrium position. But if that is what you mean, where does gravity come into it?

the same forces both spin and move

That's not a problem. F=ma applies to linear motion independently of τ=Iα applying to rotation.

You will need to model some decay function.

The main problem with simulation of dynamic systems is error accumulation. That can be reduced by applying adjustments based on conservation laws, but I'm unsure how you would do that here, and besides, you don't need to model it accurately.

 

[256bits]

A gravity graph is a kind of art "tool" that let's you draw nice geometric shapes.

Thanks!

It appears what you are describing is a quadrifilar pendulum.
Apparantly, versions of this machine were all the rage in the late 1800's.
Other means to make nice graphs are mechanical wheels, such as the Sprirograph.
https://en.wikipedia.org/wiki/Harmonograph
https://en.wikipedia.org/wiki/Pendulum

The strings would be modeled as taut wires, not as a spring, just as for a simple pendulum.
The board is free to move in the x- and the y-direction, and also to rotate and to revolve around the central axis.
Combining all four motions gives the design on the paper.

You can also look up bifilar pendulum ( a rod hanging from 2 string ), to get better luck finding some mathematics, besides that for the simple. You might be able to adapt to the quadrifilar version.

 

[haruspex]

The strings would be modeled as taut wires, not as a spring,

Please see my comment in post #2. I believe the idea is to model each vertical string as a pair of horizontal springs at right angles. Seems reasonable to me, for small oscillations.

 

[256bits]

Please see my comment in post #2. I believe the idea is to model each vertical string as a pair of horizontal springs at right angles. Seems reasonable to me, for small oscillations.

What is 'springing'?
The horizontal plate is suspended by the wires ( strings ).
The strings are vertical, or nearly so

Basiclly it is a board hanging by 4 strings.

For the plate to oscillate, the wires do not need to stretch or shrink to provide to an impetus of movement.

 

[haruspex]

What is 'springing'?
The horizontal plate is suspended by the wires ( strings ).
The strings are vertical, or nearly soFor the plate to oscillate, the wires do not need to stretch or shrink to provide to an impetus of movement.

The vertical wires provide (mostly) horizontal 2D SHM. Orthogonal pairs of horizontal springs will do the same. In this model, ignore gravity.

 

[256bits]

The vertical wires provide (mostly) horizontal 2D SHM. Orthogonal pairs of horizontal springs will do the same. In this model, ignore gravity.

Springs - the period of oscillation of the plate depends upon the spring constant.
For nearly horizontal springs, the mass is low, the spring constant is high, the the period is short, and the amplitude is small.
Vertical wires can also supply a torsion so that the plate can rotate about the central axis, as well as a rotation at a radius r about the central axis.
Vertical springs could also add movement in the z- direction ( up and down ) which would really complicate the situation.
Nice problem to model.

 

[haruspex]

For nearly horizontal springs, the mass is low, the spring constant is high, the the period is short, and the amplitude is small.

You just have to arrange that the period is the same as for the vertical wire pendulum being modeled.

 

[256bits]

You just have to arrange that the period is the same as for the vertical wire pendulum being modeled.

I don't see a physical way to do that on earth.
In space free fall satellite the horizontal springs would not have to be stiff to remain in the same plane as the plate.

 

[haruspex]

I don't see a physical way to do that on earth.

1. Place the board on a frictionless surface.
2. It's only a model for purposes of analysis/simulation. It does not have to be operated in reality.

 

[256bits]

1. Place the board on a frictionless surface.
2. It's only a model for purposes of analysis/simulation. It does not have to be operated in reality.

There you go!
This bantering discussion has evolved to solve some issues and questions about the simulation.
Nice going!

 

[haruspex]

There you go!
This bantering discussion has evolved to solve some issues and questions about the simulation.
Nice going!

Sorry, I see no evolution. I thought my interpretation of the OP's model was reasonably clear in post #2, but perhaps it was not obvious that gravity was to be ignored.

 

[256bits]

Sorry, I see no evolution. I thought my interpretation of the OP's model was reasonably clear in post #2, but perhaps it was not obvious that gravity was to be ignored.

the original op was a machine that had a hanging plate with vertical rigid strings.
You adjusted the machine to have horizontal strings ( or springs ).

In any event, a simulation can be done with the plate moving along one or both horizontal axis in a sinusoidal fashion. The pen could also be made to move sinusoidally as a simple pendulum.
The accelerations, velocities, and positions of the pen point wrt the paper can be more easily calculated if one chooses the driving equations to make some nice Lissaajous curves. If one feels more adapt at a mechanical adaptation, then fitting the sinusoids to a complex mass-spring-damper system could be accomplished, but already stated, that would be complicated.

It is quite much the same as shining a laser on a moving mirror, whose movement is controlled by simple electronics, or controlling the electron beam on a CRT display,
A mechanical-electronic system anolog.