# ⚛  L1 Principle — Structural Vibration Analysis — modal response

**ID:** `L1-266` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Vibration modes of linear elastic structures*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** modal set frequencies shapes
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554049

---

## 🧠 1. Introduction

**Structural Vibration Analysis — modal response** is a **linear inverse problem** whose unknown lives in **modal set frequencies shapes** space, within the **Vibration modes of linear elastic structures** sub-domain of **Acoustics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **accelerometers** sensing mechanism.

The forward operator applies, in order: L · fem assemble operator; L · eigenvalue operator; L · modal superposition 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 · fem assemble → L · eigenvalue → L · modal superposition → Frequency integration → **y** (detector).

```
y = ∫dω `L.modal_superposition` `L.eigenvalue` `L.fem_assemble` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.fem_assemble` | L · fem assemble operator |
| `L.eigenvalue` | L · eigenvalue operator |
| `L.modal_superposition` | L · modal superposition operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Vibration modes of linear elastic structures |
| Carrier | acoustic |
| 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_vibration` · **Forward operator:** `modal_vibration_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 5000 |
| Snr db | dB | 30 |
| N modes | — | 30 |
| N sensors | — | 16 |
| Mass error | — | 0 |
| Freq hz max | — | 5000 |
| Damping error | — | 0 |
| Boundary error | — | 0 |
| Stiffness error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 200000 |
| Snr db | dB | 0 – 40 |
| N modes | — | 3 – 200 |
| N sensors | — | 4 – 256 |
| Mass error | — | 0.0 – 0.1 |
| Freq hz max | — | 10 – 50000 |
| Damping error | — | 0.0 – 0.5 |
| Boundary error | — | 0.0 – 0.3 |
| 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 = `300` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xfd12c08e0cec74fac5985cf93a1cd17985cecd2c0ba2ad30b5850ff2b079c2ad`
- **Chain tx hash:** `0xc21747de3e9817f079dac9a3059f108b3aa3951ba30c9bcdf3185c4fa928bb54`
- **Chain block:** `41554049`

---

## File Mapping

This bundle consists of: `L1-266.md`, `L1-266.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-266.md` | Source of truth — edit this | Human or LLM |
| `L1-266.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 `L1-266`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._