MHB Equation for a distributiong force on a half cylinder

[bugatti79]

Folks,

I have an FEA applicaton where I need to apply a total force acing vertically upward but to be distributed over half a cylinder (1st n 2nd quadrant). This half cylinder i representd by a set of nodes between 0 \le \theta \le 180The force will be maximum at 90 degrees and 0 at 0/180 degrees. I thought that the equation in cylindrical coordinates would work like

y=F*cos(theta)/n where n is the number of nodes.

The force remains constant into the page as it acts over the area represented by the nodes.

Any suggestions?

 

[Ackbach]

Cool problem! I guess I have a few questions:

1. How long is the total cylinder?

2. Are you distributing the force only over the lateral area, or are you also putting some of it on the endcap?

3. You say the force is "acting upward", but you don't mention how the cylinder is oriented. How is the cylinder oriented? For example, if your cylindrical coordinates are $(\rho,\theta,z)$, then is the axis of the cylinder on the $z$ axis?

4. Suppose the cylinder is oriented with its axis on the $z$ axis. How is the force oriented to the cylinder, exactly? You mention that it's a maximum at $90^{\circ}$. How are you measuring that angle? Is it the usual cylindrical angle I mentioned before?

5. What is $y?$

If you could possibly draw a large, clear picture illustrating your setup, that would greatly help me visualize what you're trying to do.

 

[bugatti79]

Cool problem! I guess I have a few questions:

1. How long is the total cylinder?

2. Are you distributing the force only over the lateral area, or are you also putting some of it on the endcap?

3. You say the force is "acting upward", but you don't mention how the cylinder is oriented. How is the cylinder oriented? For example, if your cylindrical coordinates are $(\rho,\theta,z)$, then is the axis of the cylinder on the $z$ axis?

4. Suppose the cylinder is oriented with its axis on the $z$ axis. How is the force oriented to the cylinder, exactly? You mention that it's a maximum at $90^{\circ}$. How are you measuring that angle? Is it the usual cylindrical angle I mentioned before?

5. What is $y?$

If you could possibly draw a large, clear picture illustrating your setup, that would greatly help me visualize what you're trying to do.

I cannot attached my sketch as its above max file size unfortunately. I will try again tomorrow

 

[bugatti79]

Cool problem! I guess I have a few questions:

1. How long is the total cylinder?

2. Are you distributing the force only over the lateral area, or are you also putting some of it on the endcap?

3. You say the force is "acting upward", but you don't mention how the cylinder is oriented. How is the cylinder oriented? For example, if your cylindrical coordinates are $(\rho,\theta,z)$, then is the axis of the cylinder on the $z$ axis?

4. Suppose the cylinder is oriented with its axis on the $z$ axis. How is the force oriented to the cylinder, exactly? You mention that it's a maximum at $90^{\circ}$. How are you measuring that angle? Is it the usual cylindrical angle I mentioned before?

5. What is $y?$

If you could possibly draw a large, clear picture illustrating your setup, that would greatly help me visualize what you're trying to do.

I created a jpeg of size 145KB which is under 195KB for a jpeg but it still won't load up.

1) It is 7 nodes long
2) Distributing the force circumferentially over 7 nodes and axially over 7 nodes thereby giving total number of 49 nodes
3)Axis is z, correct
4)The total force is acting upwards, ie at 90degrees ACW. I want to distribute this force circumferentially so that at angles 0 and 180, the force is 0, then repeat these forces axially 7 times.
5)We could call y our total force Fr.

 

[Ackbach]

I created a jpeg of size 145KB which is under 195KB for a jpeg but it still won't load up.

1) It is 7 nodes long
2) Distributing the force circumferentially over 7 nodes and axially over 7 nodes thereby giving total number of 49 nodes

How are they spaced physically?

3)Axis is z, correct
4)The total force is acting upwards, ie at 90degrees ACW.

What does ACW stand for? Is the force, then, always in the $+z$ direction?

I want to distribute this force circumferentially so that at angles 0 and 180, the force is 0, then repeat these forces axially 7 times.
5)We could call y our total force Fr.

So, $1/7$ of your total force is at each of the seven $z$ values, correct? Then you have seven locations around the circumference, presumably equally distributed, where the force is located. Are you treating each location here as a point force? In other words, is the force only at the node, or is it continually varying through all the degrees? If the force is to be treated as only occurring at the node, then where are the nodes, precisely? Is there one at $\theta=0^{\circ}$? Are they evenly spaced around the circumference?

So, I think I know the answers to your questions above. Proceeding on these assumptions, I am picturing a cylinder, $7$ nodes high and $7$ nodes around. You must distribute a force $F_r$ amongst these nodes such that $F_r/7$ is distributed at each $z$ value, and that $F_r/7$ at each $z$ value is distributed circumferentially such that the force is $0$ at angles $0^{\circ}$ and $180^{\circ}$.

So the trig function that does this is the $\sin(\theta)$ function. At least, it's zero at the right places. The problem is that it goes negative in the third and fourth quadrants. One way to fix this is simply to square it. So, let's take $\sin^{2}(\theta)$ as our starting point for getting the angle correct. We now need to ensure that
$$A\sum_{j=0}^{6}\sin^{2}\left(\dfrac{2\pi j}{7}\right)=\dfrac17.$$
Mathematica says that
$$\sum_{j=0}^{6}\sin^{2}\left(\dfrac{2\pi j}{7}\right)=
2\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right].$$
It follows that
$$A=\frac{1}{14\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right]}.$$
So, taken with the fact that the force must be directed in the $+z$ direction everywhere, we have that the force at any node ($z$ shouldn't enter into it) $\vec{F}$ is given by
$$\vec{F}=\frac{\hat{\mathbf{k}}\sin^{2}\left(\dfrac{2\pi j}{7}\right)}{14\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right]},$$
where $0\le j\le 6$ is the number of the angle corresponding to the node.

Would this work for you?

 

[bugatti79]

How are they spaced physically?
What does ACW stand for? Is the force, then, always in the $+z$ direction?
So, $1/7$ of your total force is at each of the seven $z$ values, correct? Then you have seven locations around the circumference, presumably equally distributed, where the force is located. Are you treating each location here as a point force? In other words, is the force only at the node, or is it continually varying through all the degrees? If the force is to be treated as only occurring at the node, then where are the nodes, precisely? Is there one at $\theta=0^{\circ}$? Are they evenly spaced around the circumference?

So, I think I know the answers to your questions above. Proceeding on these assumptions, I am picturing a cylinder, $7$ nodes high and $7$ nodes around. You must distribute a force $F_r$ amongst these nodes such that $F_r/7$ is distributed at each $z$ value, and that $F_r/7$ at each $z$ value is distributed circumferentially such that the force is $0$ at angles $0^{\circ}$ and $180^{\circ}$.

So the trig function that does this is the $\sin(\theta)$ function. At least, it's zero at the right places. The problem is that it goes negative in the third and fourth quadrants. One way to fix this is simply to square it. So, let's take $\sin^{2}(\theta)$ as our starting point for getting the angle correct. We now need to ensure that
$$A\sum_{j=0}^{6}\sin^{2}\left(\dfrac{2\pi j}{7}\right)=\dfrac17.$$
Mathematica says that
$$\sum_{j=0}^{6}\sin^{2}\left(\dfrac{2\pi j}{7}\right)=
2\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right].$$
It follows that
$$A=\frac{1}{14\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right]}.$$
So, taken with the fact that the force must be directed in the $+z$ direction everywhere, we have that the force at any node ($z$ shouldn't enter into it) $\vec{F}$ is given by
$$\vec{F}=\frac{\hat{\mathbf{k}}\sin^{2}\left(\dfrac{2\pi j}{7}\right)}{14\left[\cos^2\left(\dfrac{\pi}{14}\right)+\cos^2\left(\dfrac{3\pi}{14}\right)+\sin^2\left(\dfrac{\pi}{7}\right)\right]},$$
where $0\le j\le 6$ is the number of the angle corresponding to the node.

Would this work for you?

Hi Ackbach,

This looks good, I think you are moving in the right direction! I have found a pictue online that should clarify my intention. From the picture that we can use as an example.

• There are approximately 22 nodes circumferentially and 9 nodes axially giving a total of 198 nodes to which Fr must be distributed.
• Force vectors are all vertical in the picture butI my case they will be normal to the surface at each point.
  
  [*]My distributed vectors are acting over half thecylinder, ie, the 1^st and 2^nd quadrant only and are 0 inmagnitude at 0 and 180 degrees ad maximum at 90 degrees (Obviously the applied force Fr at the centre of the bearing (not shown in picture) is acting upwards)
  
  
  

 

[Ackbach]

Hmm. This is radically different from what you described before. If you had a cylinder with axis along the $z$ axis, and the force all pointing in the $+z$ direction, then the force would, of necessity, be parallel to the lateral surface of the cylinder. But you're saying it needs to be normal (perpendicular) at every point. That is, your force is going to be in the $\hat{\rho}$ direction.

Is it $7$ nodes by $7$ nodes? Or is it $22$ by $9$?

If you could please give a word-for-word, identical problem statement to that which you are solving, that would be much appreciated.