How to position compacted mass?

[colinbeaton1]

My goal is to fight the max amount of torque with the weight by having different densities at different places and by keeping the z center of mass low.
I need to find the answer out in terms of the three angles, and the final answer must not have a substituted angles such as 45 degrees.
I need an ideal density function in terms of x,y,z. In terms of height and radius from vertical axis.
The main function is the cone, and the details are here
http://www.wolframalpha.com/input/?i=cone+moment+of+inertia&lk=4&num=5&lk=4&num=5
At the bottom of the cone there is a spherical cap, details here
http://www.wolframalpha.com/input/?i=partial+sphere

Use mass m, and radius r, the angles, and the equations and please show me all the work so I can fully understand for later use.

 

[colinbeaton1]

angle three can be equal to 0 or more

 

[billy_joule]

My goal is to fight the max amount of torque with the weight by having different densities at different places

What does this mean?
Your attachment looks like the cone may be hanging like a pendulum? Is the diagram on the right what you think the force will cause?

 

[colinbeaton1]

Sorry about the picture, the force is on the wrong side. Just pretend the force is on the other side.

What I want to do is create a 3d cone with a 3d spherical cap, and in order to take on the most force times an arm or torque, I want to have a the perfect densities at different locations in the mass.

I want an optimization formula. Gravity is in the picture too.

 

[billy_joule]

What I want to do is create a 3d cone with a 3d spherical cap, and in order to take on the most force times an arm or torque

It's still not clear what this means.
"take the most torque" is vague and would normally be interpreted as 'What geometry and material is required to minimise deformation or failure due to torque" or similar.

Are you trying to minimise the angular acceleration due to an applied torque? Obviously you want to maximise the moment of inertia, but without knowing your constraints (which are unclear; fixed volume? fixed mass? etc) or what exactly you are trying to do it's difficult.
Can you provide a complete free body diagram?

For a cone of homogeneous density the MOI about it's peak is maximised when h →∞ and r→0 (where h = height of cone & r = radii of it's base)
For a cone of heterogeneous density and fixed dimensions the MOI about it's peak is maximised when the greatest density is concentrated at the base.
These are trivial conclusions and can be proved via calculus but should be intuitive; concentrate the mass as far from the axis of rotation as possible.

 

[colinbeaton1]

I want the mass to be able to counteract a maximum amount of torque, T1.

1) The cone with sphere end, the object, is unmoving until a torque, T1, acts on it.
2) We take mass of object, m, and play with the density, inertia, and center of mass to optimize an equation to find what values of what we need to optimize the objecct itself.
3)For example, we could make the density of mass increase in the object of mass m as we go down the object, to make the center of mass lower, to create more counter torque. We could also make it more or less dense with respect to the radius.
4)Basically we have 3 or 4 or more densities of mass at different places in the object, summing up to mass m

 

[colinbeaton1]

Also, the next part to take note of which I do not know is how to weight the masses differently because the actual time we need the mass is when it has T1 acting upon it.

 

[billy_joule]

I want the mass to be able to counteract a maximum amount of torque, T1.

1) The cone with sphere end, the object, is unmoving until a torque, T1, acts on it.
2) We take mass of object, m, and play with the density, inertia, and center of mass to optimize an equation to find what values of what we need to optimize the objecct itself.
3)For example, we could make the density of mass increase in the object of mass m as we go down the object, to make the center of mass lower, to create more counter torque. We could also make it more or less dense with respect to the radius.
4)Basically we have 3 or 4 or more densities of mass at different places in the object, summing up to mass m

You'll end up with a point mass, m, of infinite density at the bottom centre of the rounded base. The rest of the cone will have a density of zero (or negative if you forget a constraint in your working).
All roads lead to Rome;

concentrate the mass as far from the axis of rotation as possible.

https://en.wikipedia.org/wiki/Moment_of_inertia#Calculating_moment_of_inertia_about_an_axis

Also, the next part to take note of which I do not know is how to weight the masses differently because the actual time we need the mass is when it has T1 acting upon it.

I can't understand this.

 

[colinbeaton1]

We need a full matrix solution for the problem in the picture. 3d matrix. Eigen values, moments, inertia, forces, etc. I have not done this sort of matrix before, please guide me.

 

[billy_joule]

Is that the entire problem statement word for word? It seems incomplete and there's no mention of variable density, do you have the actual diagram? Where is the spring connected to the cone? ie does h,r,a affect the torque about the rotation axis due to the spring?

You'll need to do a good free body diagram that includes all the given information.

It still seems fairly trivial, it's clear maximising for T will lead to h →∞, r →∞ and a→0 (and l → 0).