I've a base cylinder which the volume is given by:
$$V_{cyl} = \pi r_1^2h \tag{1} $$
where $r_1^2$ is the radius of the cylinder and $h$ is its height.
I want to keep the volume of my geometry while adding cut-offs to it, like shown here. To do so I imagined that I could add two correcting factors, but I'm confused on how to approach this. Let's call the correcting factor of the height $\alpha$ and for the radius $\gamma$.
For some mechanical needs I can fully cut the cylinder, so I've added a small height $h_{low}$ to it, where I have the volume of that part to be:
$$V_{low} = \pi r^2_1h_{low} \tag{2}$$
There are two questions that got me highly confused. The first one is how I calculate the volume of the geometry in terms of the variables shown in the pictures: $h$, $h_{low}$, $h_{high}$, $r_1$, $r_2$, $cut_w$. The second is that I want $\alpha$ to be a function of $\gamma$ so that $\gamma$ multiples $r_1$, increasing the overall cylinder radius to compensate the lateral cuts created by $cut_w$. Then $\alpha$ multiples $h$ to compensate the middle perforation, using $\gamma r_1$ already.
Is something like that possible? How I can do this?
I hope my question formulation was good enough, I'll try to improve it based on feedback given. Thank you in advance
EDIT:
What I came up with is to first calculate $\alpha$ using making (3) equal to (1) which will make $\alpha$ be as shown in (4). This is considering only a single perforation like in this case.
$$V_{perCyl} = \pi (r_1^2 - r_2^2)\alpha h + \pi r_2^2h_{low} \tag{3}$$
$$\alpha = \frac{r_1^2h-r_2^2h_{low}}{(r_1^2 - r_2^2)h} \tag{4}$$
Then adding the four cuts I can write the volume (considering the $\gamma$) like shown in (5).
$$V_{finalCyl} = \pi (\gamma r_1^2 - r_2^2)\alpha h+ \pi r_2^2h_{low} - 4 (cut_w(\alpha h - h_{low})(\gamma r1-r2)) \tag{5}$$
Now I need to then make (5) equal to (3) and find $\gamma$ which should take me awhile. If anyone can confirm the logic is correct it'd be nice.
EDIT2:
I used matlab to solve this which yield that gamma can be expressed as in (6)
$$\gamma = \frac{(4cut_wr_2(h_{low} - \alpha h) + \pi\alpha h r_2^2 + \pi\alpha h(r_1^2 - r2^2))}{(4cut_wr_1^2(h_{low} - \alpha h) + \pi \alpha h r_1^2)} \tag{6}$$