Determining Plate direction by having three point of it.

[Kevin20189]

Hi all,

Can anyone help to solve this problem?
Imagine a big plate suspended in the air through three threads with different length.
we know length of each individual thread, and they are 120 degree apart from each other.
How cam we calculate the tilt of the plate and direction of the tilt?

 

[davenn]

Hi all,

Can anyone help to solve this problem?
Imagine a big plate suspended in the air through three threads with different length.
we know length of each individual thread, and they are 120 degree apart from each other.
How cam we calculate the tilt of the plate and direction of the tilt?

Hi there
welcome to PF

what research into this problem have you done so far ?

also ... is this a personal interest or homework/studies question ?

Dave

 

[Kevin20189]

Hi there
welcome to PF

you have marked your thread with an "A" tag indicating post graduate level
So what research into this problem have you done so far ?

also ... is this a personal interest or homework/studies question ?

Dave

Hi Davenn,

Thank you :)
I tried to solve it through vectors, but it's not as easy as it seems.
I'm Control System Engineer and this is a practical question. It's a cap of a silo that we have three sensors around it and we need to calculate the tilt of the cap.Thanks
Regards
Kevin

 

[andrewkirk]

Hi all,

Can anyone help to solve this problem?
Imagine a big plate suspended in the air through three threads with different length.
we know length of each individual thread, and they are 120 degree apart from each other.
How cam we calculate the tilt of the plate and direction of the tilt?

Some clarification is needed as to what is meant by '120 degrees apart from each other'.

First, to all three threads come from the same suspension point?

If so, the angles between threads cannot be 120 degrees because that angle is only achievable if all threads are horizontal, as the threads start to point downwards a bit, the angle between any two reduces.

Perhaps you mean the angle between the projection of the threads on the horizontal plane is 120 degrees. In that case we need one additional parameter (in addition to the length) to locate the lower end of the thread. The angle to the horizontal would do it. So would the distance between attachment points on the plate. I imagine you probably know those three distances.

Essentially, we need to be able to calculate the three vectors representing the locations of the attachment points on the plate, relative to an origin that is the suspension point. Once we have those, we can calculate the angle of the plate.

 

[Kevin20189]

Some clarification is needed as to what is meant by '120 degrees apart from each other'.

First, to all three threads come from the same suspension point?

If so, the angles between threads cannot be 120 degrees because that angle is only achievable if all threads are horizontal, as the threads start to point downwards a bit, the angle between any two reduces.

Perhaps you mean the angle between the projection of the threads on the horizontal plane is 120 degrees. In that case we need one additional parameter (in addition to the length) to locate the lower end of the thread. The angle to the horizontal would do it. So would the distance between attachment points on the plate. I imagine you probably know those three distances.

Essentially, we need to be able to calculate the three vectors representing the locations of the attachment points on the plate, relative to an origin that is the suspension point. Once we have those, we can calculate the angle of the plate.

Thanks for your quick reply.

I think my question isn't clear enough
This might clarify things a little bit more.
Our Plate is circular.
There is three suspension points around the plate where the plate is suspended from and circular distance of suspension points is 120 degree.
We know how long each thread is.
What we want to know is:
1) How much the tilt of circular plate? i.e. the vertical distance between maximum point and minimum point of the plate
2) What is the direction of the tilt? (for this we can assume position of one of the thread suspension is to the north and calculate the direction towards North)

Please let me know if you need any further clarifications

 

[andrewkirk]

The three threads form a tetrahedron of which the base is the triangle formed by the three attachment points on the plate. That base triangle is equilateral, with known side length h. Let the lengths of the three threads be a, b and c and let their vectors be ##\vec a, \vec b,\vec c## relative to an origin at the suspension point. These vectors are unknown at first.

Then to find the unknown vectors we need to find nine parameters. From the equilateralness of the base we have three equations:
$$\|\vec a - \vec b\| =\| \vec b-\vec c\| = \|\vec c-\vec a\| = h$$
We also have the three equations
$$\|\vec a\|=a,\|\vec b\|=b,\|\vec c\|=c$$
The centroid of the base triangle is the centre of mass of the plate, and it must hang directly below the suspension point, so we get the two equations:
$$\frac13(\vec a+\vec b+\vec c)\cdot \vec i = \frac13(\vec a+\vec b+\vec c)\cdot \vec j = 0$$
Finally, as you suggested, we can WLOG assume one of the threads has its horizontal projection in a particular direction, eg:
$$\vec a \cdot \vec i = 0$$

So we have nine equations, which we can solve for the nine unknowns. Then we will know the exact vectors of the threads relative to the chosen coordinate system. We can then use those to find the direction of the plate. Len ##\vec n## be the normal to the plate, which specifies its direction. Then that normal is perpendicular to the three sides of the base. So we have:
$$\vec n\cdot (\vec a-\vec b) = \vec n\cdot (\vec b-\vec c) = \vec n\cdot (\vec c-\vec a)$$
which is three equations, which will allow us to solve for the unknown parameters of the normal. Actually, we only need two equations, since a normal is defined as having length one. One of those equations will be redundant.

There is probably a more elegant approach than this. But this should allow you to find the answer you need to complete your construction.

 

[Kevin20189]

The three threads form a tetrahedron of which the base is the triangle formed by the three attachment points on the plate. That base triangle is equilateral, with known side length h. Let the lengths of the three threads be a, b and c and let their vectors be ##\vec a, \vec b,\vec c## relative to an origin at the suspension point. These vectors are unknown at first.

Then to find the unknown vectors we need to find nine parameters. From the equilateralness of the base we have three equations:
$$\|\vec a - \vec b\| =\| \vec b-\vec c\| = \|\vec c-\vec a\| = h$$
We also have the three equations
$$\|\vec a\|=a,\|\vec b\|=b,\|\vec c\|=c$$
The centroid of the base triangle is the centre of mass of the plate, and it must hang directly below the suspension point, so we get the two equations:
$$\frac13(\vec a+\vec b+\vec c)\cdot \vec i = \frac13(\vec a+\vec b+\vec c)\cdot \vec j = 0$$
Finally, as you suggested, we can WLOG assume one of the threads has its horizontal projection in a particular direction, eg:
$$\vec a \cdot \vec i = 0$$

So we have nine equations, which we can solve for the nine unknowns. Then we will know the exact vectors of the threads relative to the chosen coordinate system. We can then use those to find the direction of the plate. Len ##\vec n## be the normal to the plate, which specifies its direction. Then that normal is perpendicular to the three sides of the base. So we have:
$$\vec n\cdot (\vec a-\vec b) = \vec n\cdot (\vec b-\vec c) = \vec n\cdot (\vec c-\vec a)$$
which is three equations, which will allow us to solve for the unknown parameters of the normal. Actually, we only need two equations, since a normal is defined as having length one. One of those equations will be redundant.

There is probably a more elegant approach than this. But this should allow you to find the answer you need to complete your construction.

Thanks for quick reply