Calculating the Relative COG of Two Masses

[slobberingant]

Homework Statement

Not homework but a work problem. I would like to find a formula to solve the below problem.

I have two masses, M1 and M2, which are located one above the other.
M1 position is fixed. I need to find the position of M2 so that a line from an arbitrary point, A, on M2 will intersect the COG of the combined masses at a nominated angle.
See illustration.

As M2 moves around the combined COG will change.

Relevant Equations

M = combined mass
M1 = mass 1
M2 = mass 2
P1 = position of mass 1
P2 = position of mass 2

COG = 1/M * (M1*P1 + M2*P2)

I can calculate the COG with the above formula however P2 is unknown.
I need to be able to put P2 into another function which will check that the vector A->COG is parallel to the nominated angle. I cannot seem to work this one out algebraically.
Any assistance or guidance would be appreciated.

 

[haruspex]

What distance do you want to use to specify the position of M2?
The distance that A is to the right of the left edge of M1, perhaps?

 

[slobberingant]

That is a good suggestion. Yes, distance from the left edge of M1 is fine.

 

[haruspex]

That is a good suggestion. Yes, distance from the left edge of M1 is fine.

Ok, so if you take the lower left corner of M1 as the origin, and the mass centres are at $(x_1, y_1), (x+x_2, y_2)$, and A is at $(x+x_A, y_A)$, what equation can you write for the tangent of the angle?

 

[Lnewqban]

What do you call a "nominated angle"?
What does that note "masses fixed along this line" mean?

 

[kuruman]

What do you call a "nominated angle"?

I had the same question but I think it means "predefined". It is angle $\alpha$ in the drawing.

 

[Lnewqban]

I had the same question but I think it means "predefined". It is angle $\alpha$ in the drawing.

At least to me, it is not clear whether mass m2 slides along the bottom surface of mass m1 or pivots about a line perpendicular to the paper.

 

[kuruman]

At least to me, it is not clear whether mass m2 slides along the bottom surface of mass m1 or pivots about a line perpendicular to the paper.

I think it slides along the bottom surface. That's how I interpret the note in the drawing "$m_1$ and $m_2$ are fixed along this line."

 

[slobberingant]

Correct. M2 slides along the bottom surface of M1.
The angle is something that we input, not a value to be determined. It ranges from 30 - 60 degrees.

 

[slobberingant]

Ok, so if you take the lower left corner of M1 as the origin, and the mass centres are at $(x_1, y_1), (x+x_2, y_2)$, and A is at $(x, y_A)$, what equation can you write for the tangent of the angle?

I had a go at using the tangent of the angle. The formula tanA = y1/x1 where y1 and x1 are the combined COG. With this I can plug it into the combined COG formula but I still have two unknowns, M2 location and COG location. I feel as though I need a third formula to solve this.

 

[haruspex]

The formula tanA = y1/x1 where y1 and x1 are the combined COG

… $\tan A = y_g/x_g$ where $x_g$ and $y_g$ are the coordinates of the COG relative to A.
(I have already used $x_1, y_1$ to mean the coordinates of the mass centre of $M_1$ relative to its lower left corner.)
Do you understand what all the variables I used in post #4 mean? Can you draw a diagram showing them?

 

[Steve4Physics]

Homework Statement: Not homework but a work problem. I would like to find a formula to solve the below problem.

I have two masses, M1 and M2, which are located one above the other.
M1 position is fixed. I need to find the position of M2 so that a line from an arbitrary point, A, on M2 will intersect the COG of the combined masses at a nominated angle.

Hi @slobberingant. Out of curiosity, since the question relates to your work, may I ask what the two objects are and why their CoG needs to be at a particular angle?

 

[slobberingant]

… $\tan A = y_g/x_g$ where $x_g$ and $y_g$ are the coordinates of the COG relative to A.
(I have already used $x_1, y_1$ to mean the coordinates of the mass centre of $M_1$ relative to its lower left corner.)
Do you understand what all the variables I used in post #4 mean? Can you draw a diagram showing them?

Thank you for the clarification. I believe that I have an understanding of the variables but I need to sit down properly and work it all out. I will come back when I have properly reviewed your responses.

 

[slobberingant]

Hi @slobberingant. Out of curiosity, since the question relates to your work, may I ask what the two objects are and why their CoG needs to be at a particular angle?

The two objects represent a vibrating feeder. M1 represents the main body and M2 represents the drive bracket. The vibrating motors bolt at point A and send vibration through the whole machine at the angle of the drive bracket. The forces must pass through the center of gravity for the vibration to be distributed evenly across the machine.

 

[Lnewqban]

The two objects represent a vibrating feeder. M1 represents the main body and M2 represents the drive bracket. The vibrating motors bolt at point A and send vibration through the whole machine at the angle of the drive bracket. The forces must pass through the center of gravity for the vibration to be distributed evenly across the machine.

Are the modifications to the relative position of m2, as well as the angle of action of the motors-weights just a matter of one-time adjustment, or something to be modified as needed?

Is the direction of the line along m1 is free to move also adjustable?

How precisely could you determine the magnitudes and respective locations of the CoM's of m1 and m2?

Shouldn't we consider the amount and distribution of the mass to be moved over the feeder?

Does the machine look something like these?

 

[slobberingant]

… $\tan A = y_g/x_g$ where $x_g$ and $y_g$ are the coordinates of the COG relative to A.
(I have already used $x_1, y_1$ to mean the coordinates of the mass centre of $M_1$ relative to its lower left corner.)
Do you understand what all the variables I used in post #4 mean? Can you draw a diagram showing them?

I've reworked the diagram to make the origin at the COG of M1. This works well with the software that I am using and simplifies the CCOG (combined COG) formula.
I now have the two formulas for CCOG. If I combine them they still have two unknowns (CCOGy relates to CCOGx). I need a third formula which relates the COG2 with A. However I can't work out how to connect them all.
I've had an attempt using vectors rather than coordinates with some success but I don't think the solution is a simple point in space. I believe that the solution is a line where M2 can be positioned for the angle constraint to be true.
Can you please provide some more insight?

 

[slobberingant]

Are the modifications to the relative position of m2, as well as the angle of action of the motors-weights just a matter of one-time adjustment, or something to be modified as needed?

Just once during the design phase of the machine.

Is the direction of the line along m1 is free to move also adjustable?

Once the center of mass is determined the locations of the two masses are fixed.

How precisely could you determine the magnitudes and respective locations of the CoM's of m1 and m2?

Very. These are modeled up in CAD.

Shouldn't we consider the amount and distribution of the mass to be moved over the feeder?

Yes this is considered but the machine does also need to run empty. Material behaves differently to a fixed mass on a vibrating machine. Generally the machine is made heavy enough to make material influence not a concern.

Does the machine look something like these?

View attachment 349031

Like this one. The motor angle is simplified in this diagram.

 

[Lnewqban]

Very. These are modeled up in CAD.

Then, you can see that the locations of the center of masses 2 and combined move along lines that are parallel to the common surface.
Regardless the angle, the proportion of distances follows the proportions of masses.

 

[haruspex]

View attachment 349048View attachment 349049
I've reworked the diagram to make the origin at the COG of M1. This works well with the software that I am using and simplifies the CCOG (combined COG) formula.
I now have the two formulas for CCOG. If I combine them they still have two unknowns (CCOGy relates to CCOGx). I need a third formula which relates the COG2 with A. However I can't work out how to connect them all.

CCOG must lie on the straight line through COG1, COG2.
Are your X1, X2 vectors or just the x coordinates? If the latter, you need the equivalent for the y coordinates.
Presumably you know where COG2 is in relation to M2, and where A is in relation to M2, so you should be able to figure out where A is in relation to COG2.
I have no idea how you get that equation for $\tan(\alpha)$. The formula should involve the coordinates of A relative to the CCOG.

 

[Steve4Physics]

@slobberingant, we have $m_2$ at $(0, 0)$ and $m_1$ at $(x_1, y_1)$. We know $y_1$.

With (X, Y) the coordinates of point A, the line through A, making the desired angle $\alpha$ with the x-axis, has equation (point-slope form):
$y -Y = (x-X) \tan \alpha$.

If you have expressions for the x and y coordinates of the combined CoG, you substitute them into the above equation. You are then left with a single equation with only one unknown, $x_1$.

Edited - mainly because I'd mixed up masses 1 and 2.

 

[slobberingant]

Thank you all very much for the answers! I think I have it worked out.
With some external help, moving the origin to the COM of m2 simplified the formula (as alpha was relative on this) and made it understandable.