# ⚛  L1 Principle — Modal Analysis — natural frequencies and mode shapes

**ID:** `L1-213` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Generalized eigenvalue for structures*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** modal set frequencies shapes
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554010

---

## 🧠 1. Introduction

**Modal Analysis — natural frequencies and mode shapes** is a **linear inverse problem** whose unknown lives in **modal set frequencies shapes** space, within the **Generalized eigenvalue for structures** sub-domain of **Structural Mechanics**.

Measurements consist of mechanical vibrations or strain signals via a **accelerometers** sensing mechanism.

The forward operator applies, in order: L · mass stiffness operator; L · eigenvalue solve operator; L · mode sort operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · mass stiffness → L · eigenvalue solve → L · mode sort → Frequency integration → **y** (detector).

```
y = ∫dω `L.mode_sort` `L.eigenvalue_solve` `L.mass_stiffness` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.mass_stiffness` | L · mass stiffness operator |
| `L.eigenvalue_solve` | L · eigenvalue solve operator |
| `L.mode_sort` | L · mode sort operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Generalized eigenvalue for structures |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | modal_set_frequencies_shapes |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | freq_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `modal` · **Forward operator:** `modal_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 5000 |
| Snr db | dB | 30 |
| N modes | — | 30 |
| Bc error | — | 0 |
| N sensors | — | 16 |
| Mass error | — | 0 |
| Stiffness error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 500000 |
| Snr db | dB | 0 – 40 |
| N modes | — | 3 – 500 |
| Bc error | — | 0.0 – 0.2 |
| N sensors | — | 2 – 256 |
| Mass error | — | 0.0 – 0.1 |
| Stiffness error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 2.0

| Metric | Range |
|---|---|
| Freq rel error | 0.1 – 20.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **freq_rel_error_percent**, with κ = `240` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xb478a40433b67c25196a2762c75b9fb01c8b66a74247debcf7bf2d77cf501c1f`
- **Chain tx hash:** `0xb47d5105dfb72fe174d283062253413dc61047b4dee1496a0d0e2d9be1084d0f`
- **Chain block:** `41554010`

---

## File Mapping

This bundle consists of: `L1-213.md`, `L1-213.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-213.md` | Source of truth — edit this | Human or LLM |
| `L1-213.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-213`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._