# cantilever_modal

Family 17: the first natural frequency of a clamped-free box beam (built-in eigenvalue solver).

Anchor: Euler-Bernoulli, f1 = 1.8751^2 / (2 pi L^2) sqrt(E I / (rho A)), with the root simply built in. A stubby beam falls
below it through shear and rotary inertia (Timoshenko) and the fixed root adds a little stiffness: that departure is what
the data adds. The bending is about the weak axis (w > t) so that the first mode stays the same one over the whole space.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `t_over_L` | 0.02 - 0.15 |
| `w_over_t` | 1.2 - 4.0 |
| `nu` | 0.2 - 0.4 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `f_star` | 0.3 % / 0.92 % / 0.92 % | max 7.8e-08 % | max - |

## Checks

- points: 100 solved, 0 failed
- three-mesh studies: 20 (every 5th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: f_star vs t_over_L ok (rho 1.0)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `f_over_beam` | 1.005 | 1.017 | 1.025 |

## Columns of points.csv

parameters, the answers, `<answer>_band` (relative), the hand-theory ratios, `mirror_err`, `equilibrium_err` (where a single reaction exists), `<answer>_loo_err` / `_neighbour_flag` (leave-one-out prediction from all other points), `<answer>_recheck_diff` / `_flag_resolved` (finer re-solve of a flagged point), `has_gci` / `has_similarity` / `has_calculix`, `nodes`, `solve_s`, `studio` (version).

Solved with NovaSolver Studio through the same commands an AI client uses over MCP; the first section says which solver and mesh. The claim is verification (the equations are solved within the band), not validation against tests.

## License

This data is licensed under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/
Credit: NovaSolver, https://novasolver.jp/studio/verified/
Provided as is, without warranty. Design decisions remain the user's responsibility.