Radii!

Premise: horizontals hit harder.

A horizontal is a big bet in the arena. There’s a lot of pull to the destruction-potential side of things. Generally, a horizontal tends to be the geometry that maximizes that variable. But when you look at the rotational energy equation, something stands out:

If I double my moment of inertia, I double my potential energy. But if I double my weapon’s velocity, I quadruple my potential energy. At those speeds, drag torque starts mattering (and was the topic of the previous post), but that’s not the focus here. My point is that the marginal gain from increasing velocity is, in theory, a lot bigger than from increasing radius.

Is it though? Let’s assume my robot will use a bar as a weapon, so we can write $E_{rotacional}$ as a function of radius rather than moment of inertia:

So, treating the bar as an arbitrary cuboid, rotational energy as a function of dimensions is something like:

And the bar’s mass:

Substituting:

Comparing geometries

Alright, this won’t take us very far on its own, but we can plug in numbers to compare the potential energy of two bars: one vertical, one horizontal.

Let’s assume our horizontal will have a radius 3× larger than the vertical, which isn’t far-fetched: combat robots usually aren’t very tall (focus on usually), so verticals tend to have smaller radii, while the only limit on horizontals is weight and how far they can push the weapon shaft from the wheels.

This assumes both spin at the same speed. In reality that’s a lie, since the vertical generally spins much faster than the horizontal, just because it’s way easier to spin a 75g bar at 40k rpm than a 150g one. But for the sake of the analysis, this is the equation for what I’ll call the advantage factor ($f_v$): how much more potential energy one design can store than the other.

Now let’s add the missing factor, the velocity ratio:

So we get $f_{vd} = f_v \cdot d_v^2$. Drag the sliders below (horizontal’s radius, bar depth, radius ratio, velocity ratio) to see where the horizontal wins. The axis is $r_h$ because that’s the one you, the designer, are choosing; $r_v$ is just the proportionally-sized opponent, shown in the readout:

The dashed line at $f_{vd} = 1$ is break-even: anything above it, the horizontal hits harder. To get a feel for it:

• With $r_h/r_v = 3$ and $d_v = 1$ (same speed), the horizontal always has at least 27× more potential energy.
• With $r_h/r_v = 3$ and $d_v = 1/3$ (vertical 3× faster), the advantage drops to 3×. Still wins.
• With $r_h/r_v = 2$ and $d_v = 0.7$ (a modest scenario), the horizontal still hits almost 4× harder.

And it’s not just raw energy: the bite of a slower weapon tends to be larger, so we get more complete energy transfers (real hits) and fewer partial ones (love taps).

Where this leaves me

There are other things worth digging into. Drag at these speeds will affect the steady-state current and the weapon’s spinup time. There’s also the energy-transfer side. But generally, this post is just a simple analysis of how much weapon size affects the robot’s “punch”, if it’s the only thing that changes between geometries.

In other words, this is the post that justifies the viability of a horizontal. I get it, it was a gross simplification to treat everything as a bar, and yes, the moment of inertia of a real vertical (drum, eggbeater) is much bigger than an equivalent bar. But it’s not a throwaway analysis, since we can see that yes, a horizontal has potential to transfer way more energy than a vertical even spinning slower. Even if the geometry difference for a drum means the vertical’s moment of inertia doubles, we’d still be talking about a hit almost 2× as hard. So we can safely assume that a well-designed horizontal has a lot of potential.