L1 ⚛ L1-213 ⊙ Testnet

Modal Analysis — natural frequencies and mode shapes

Structural Mechanics · Generalized eigenvalue for structures

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⬇ L1-213.md

# ⚛ 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._

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Read the attached Markdown (L1-213.md).
Regenerate L1-213.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

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L2-213-001 Modal Analysis — natural frequencies and mode shapes Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-213",
  "chain_block": 41554010,
  "chain_hash": "0xb478a40433b67c25196a2762c75b9fb01c8b66a74247debcf7bf2d77cf501c1f",
  "chain_tx_hash": "0xb47d5105dfb72fe174d283062253413dc61047b4dee1496a0d0e2d9be1084d0f",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 240,
    "metric": "freq_rel_error_percent",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "freq_rel_error_percent",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "mechanical",
    "difficulty_delta": 3,
    "domain": "Structural Mechanics",
    "integration_axis": "frequency",
    "noise_model": "gaussian",
    "primitives": [
      "L.mass_stiffness",
      "L.eigenvalue_solve",
      "L.mode_sort",
      "int.frequency"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "accelerometers",
    "solution_space": "modal_set_frequencies_shapes",
    "sub_domain": "Generalized eigenvalue for structures",
    "title": "Modal Analysis \u2014 natural frequencies and mode shapes"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "modal_forward"
    ],
    "allowed_omega_dimensions": [
      "N_DOF",
      "N_modes",
      "N_sensors",
      "SNR_dB",
      "stiffness_error",
      "mass_error",
      "bc_error",
      "stiffness_error",
      "mass_error",
      "bc_error"
    ],
    "allowed_problem_classes": [
      "modal"
    ],
    "center_spec": {
      "epsilon_fn_center": "2.0",
      "forward_operator": "modal_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_DOF": 5000,
        "N_modes": 30,
        "N_sensors": 16,
        "SNR_dB": 30,
        "bc_error": 0.0,
        "mass_error": 0.0,
        "stiffness_error": 0.0
      },
      "problem_class": "modal"
    },
    "epsilon_bounds": {
      "freq_rel_error": [
        0.1,
        20.0
      ]
    },
    "omega_bounds": {
      "N_DOF": [
        100,
        500000
      ],
      "N_modes": [
        3,
        500
      ],
      "N_sensors": [
        2,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "bc_error": [
        0.0,
        0.2
      ],
      "mass_error": [
        0.0,
        0.1
      ],
      "stiffness_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Generalized eigenvalue for structures",
  "title": "Modal Analysis \u2014 natural frequencies and mode shapes"
}

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