L1 ⚛ L1-266 ⊙ Testnet

Structural Vibration Analysis — modal response

Acoustics · Vibration modes of linear elastic structures

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⬇ L1-266.md

# ⚛ 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._

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Read the attached Markdown (L1-266.md).
Regenerate L1-266.json so every field matches the Markdown.
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L2-266-001 Structural Vibration Analysis — modal response Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-266",
  "chain_block": 41554049,
  "chain_hash": "0xfd12c08e0cec74fac5985cf93a1cd17985cecd2c0ba2ad30b5850ff2b079c2ad",
  "chain_tx_hash": "0xc21747de3e9817f079dac9a3059f108b3aa3951ba30c9bcdf3185c4fa928bb54",
  "domain": "Acoustics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 300,
    "metric": "freq_rel_error_percent",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "license_hash": null,
      "name": "primary",
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  "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": "acoustic",
    "difficulty_delta": 3,
    "domain": "Acoustics",
    "integration_axis": "frequency",
    "noise_model": "gaussian",
    "primitives": [
      "L.fem_assemble",
      "L.eigenvalue",
      "L.modal_superposition",
      "int.frequency"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "accelerometers",
    "solution_space": "modal_set_frequencies_shapes",
    "sub_domain": "Vibration modes of linear elastic structures",
    "title": "Structural Vibration Analysis \u2014 modal response"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "modal_vibration_forward"
    ],
    "allowed_omega_dimensions": [
      "N_DOF",
      "N_modes",
      "N_sensors",
      "freq_Hz_max",
      "SNR_dB",
      "stiffness_error",
      "mass_error",
      "damping_error",
      "boundary_error",
      "stiffness_error",
      "mass_error",
      "damping_error",
      "boundary_error"
    ],
    "allowed_problem_classes": [
      "modal_vibration"
    ],
    "center_spec": {
      "epsilon_fn_center": "2.0",
      "forward_operator": "modal_vibration_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_DOF": 5000,
        "N_modes": 30,
        "N_sensors": 16,
        "SNR_dB": 30,
        "boundary_error": 0.0,
        "damping_error": 0.0,
        "freq_Hz_max": 5000,
        "mass_error": 0.0,
        "stiffness_error": 0.0
      },
      "problem_class": "modal_vibration"
    },
    "epsilon_bounds": {
      "freq_rel_error": [
        0.1,
        20.0
      ]
    },
    "omega_bounds": {
      "N_DOF": [
        100,
        200000
      ],
      "N_modes": [
        3,
        200
      ],
      "N_sensors": [
        4,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "boundary_error": [
        0.0,
        0.3
      ],
      "damping_error": [
        0.0,
        0.5
      ],
      "freq_Hz_max": [
        10,
        50000
      ],
      "mass_error": [
        0.0,
        0.1
      ],
      "stiffness_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Vibration modes of linear elastic structures",
  "title": "Structural Vibration Analysis \u2014 modal response"
}

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