Last week there was a cosmic conjunction in my thinking! The partial solar eclipse in the UK coincided with my son needing help with his "Gravitational Field Theory". Could SOLIDWORKS help me understand both?
My 18 year old son is revising for his Physics ‘A’ level and
needed some help. I must admit my knowledge of ‘A’ level Physics is pretty
rusty - but these days you can get fantastic revision guides - so armed with the guide I sat down to see if I
could help him.
I soon realised that, not only could I help him, but I could
use SOLIDWORKS Motion to simulate how gravitational forces affect masses
and how bodies move through space. This was very topical in March as the moon happened
to interpose itself between the Sun and the UK causing a partial solar eclipse.
The revision guide reminded me of the gravitational force
equation …
… where Fg is the gravitational force, G is the universal
gravitational constant, M is the mass of one body; m is the mass of a second body
and r is the distance between the bodies.The inverse square law here means
that the force rapidly reduces with distance.
The other equation that I needed to know (and which was less
familiar to me) was the equation that calculates the orbital velocity of a body that maintains a constant orbit. This equation is …
... where Vo is the orbital velocity tangential to the orbital motion.
Now for SOLIDWORKS! The tool to use is Motion. It uses Newton's laws and force equilibriums to simulate the behaviour of rigid bodies. Would this work with the Moon?
Here is where I have to admit a limitation. SOLIDWORKS assembly
models must fit within a 100 km cube. This is so that the geometry is accurate from
kilometres down to nanometres and suits just about everyone – except people
wanting to model the solar system! However, Newton’s laws still apply so I just
needed to scale things down a little.
Therefore I modelled a sphere (I called ‘Earth’) of 100m
diameter and a sphere (I called ‘Moon’) of 24m radius. I realise that these are
not the correct proportions but I liked the look of a big moon! I made both spheres of lead to maximise the gravitational effect and used Mass Properties to calculate the masses.
In an assembly
I positioned the Moon at an 80m radial distance from the centre of the Earth. I
used the little known but very useful ‘For position only’ option in the mate.
This is important because I didn’t want the Moon’s position to be determined by
a mate –but by the forces of gravity and its velocity alone.
For the Moon, I also needed to create 2 axes – one that I
called ‘radial’ and pointed to the centre of the Earth and the other at 90
degrees to it. These axes would allow me to define the gravitational force
direction and the tangential orbital velocity direction in Motion.
I then created a Motion study and applied an ‘Initial
Velocity’ to the Moon using the tangential axis. (I did not apply the ‘Gravity’ Motion
option as that would be uni-directional and incorrect). I ran the study for a
few seconds. The Moon sped off in a straight line into the cosmos as expected –
obeying Newton’s laws. Why do this? The reason is that I needed Motion to
calculate the radial position of the Moon – i.e. ‘r’. This has to be a variable
if I am to get real results and if I want to experiment with non-circular
orbits. By creating it as a Motion result I can then use it as part of the
gravitational force expression.
To create the gravitational force I used the Motion Force
Function Builder with the ‘Expression’ option. This allows the creation of
complex expressions and, crucially, allows a Motion result to be part of the
expression. Hence I can make the force inversely proportional to the square of
the distance between the Earth and the Moon. This is a real time feedback loop
and very powerful!
This is what the Expression looked like ...
Note that the value is negative causing the force to be an attraction and the value of 15,700,000 is a constant equal to G x M x m. The important part is the 'Linear Displacement1' result that is added to the expression.
In the Force property manager I set the force to act from the centre of the moon to the centre of the Earth but being relative to the Moon and in the radial direction (using the axis I created in the Moon part).
An initial simulation (without the correct value for the Initial Velocity of the Moon) let me see either the Moon heading directly towards the Earth due to gravity or else, if the Initial Velocity was too great, heading off into the cosmos.
So what should the correct value of the Initial Velocity be? According to the physics, it should be 0.05 m/s (remembering this is a scale model). How long should it take for one orbit? That's easy - just the orbit circumference divided by the speed of the Moon. This came out as 168.93 minutes. I set this in Motion and used a frame rate of 1 frame per sec.
BINGO! To my great delight the Moon described a perfect circular orbit of 80m taking exactly 168.93 minutes.
This confirmed to me that I had got the physics right and that Sir Isaac Newton was a genius - as if I ever doubted!
For fun I added a decal of the earth to the sphere and the spherical mapping worked a treat. I tilted the Earth at 23.5 degrees and also added an initial angular velocity so the Earth rotates around its axis. I then set up a distant point light to capture the effect of the sun and used an image of the night sky as a background in the scene. You can see the video here ...
You can see the Moon's shadow as it passes over the Earth so in my scale model northern Australia would see the eclipse.
I love SOLIDWORKS Motion! It not only enables me to simulate real-life engineering problems but it is also helping me get my son through his 'A' levels!
By Andy Fulcher
Technical Manager
Solid Solutions Management Ltd