L1 ⚛ L1-208 ⊙ Testnet

Euler-Bernoulli Beam — thin-beam bending

Structural Mechanics · Classical thin-beam bending PDE

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⬇ L1-208.md

# ⚛ L1 Principle — Euler-Bernoulli Beam — thin-beam bending **ID:** `L1-208` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Structural Mechanics — *Classical thin-beam bending PDE* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** beam deflection 1D > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554009 --- ## 🧠 1. Introduction **Euler-Bernoulli Beam — thin-beam bending** is a **linear inverse problem** whose unknown lives in **beam deflection 1D** space, within the **Classical thin-beam bending PDE** sub-domain of **Structural Mechanics**. Measurements consist of mechanical vibrations or strain signals via a **strain gauges lvdt** sensing mechanism. The forward operator applies, in order: L · bending stiffness operator; L · 4th order pde operator; L · bc operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → L · bending stiffness → L · 4th order pde → L · bc → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.bc` `L.4th_order_pde` `L.bending_stiffness` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.bending_stiffness` | L · bending stiffness operator | | `L.4th_order_pde` | L · 4th order pde operator | | `L.bc` | L · bc operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Structural Mechanics | | Sub domain | Classical thin-beam bending PDE | | Carrier | mechanical | | Problem class | linear_inverse | | Solution space | beam_deflection_1D | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Conditional stability; mismatch parameters dominate at Omega bounds. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `euler_bernoulli` · **Forward operator:** `euler_bernoulli_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Snr db | dB | 30 | | Ei error | — | 0 | | Bc error | — | 0 | | N sensors | — | 16 | | N elements | — | 50 | | Load error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Snr db | dB | 0 – 40 | | Ei error | — | 0.0 – 0.1 | | Bc error | — | 0.0 – 0.2 | | N sensors | — | 2 – 256 | | N elements | — | 5 – 10000 | | Load error | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 28.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `160` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x946253cecb9371ebe20c0e79c8ccd11787742c2ebe2a5a9ae0e77cfd59a33a12` - **Chain tx hash:** `0x2841ff2f212ab6e9e6f2820303f308c09e00427cf07d87adbe9b7b7c60d323d6` - **Chain block:** `41554009` --- ## File Mapping This bundle consists of: `L1-208.md`, `L1-208.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-208.md` | Source of truth — edit this | Human or LLM | | `L1-208.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 `L1-208`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-208.md).
Regenerate L1-208.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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L2-208-001 Euler-Bernoulli Beam — thin-beam bending Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-208",
  "chain_block": 41554009,
  "chain_hash": "0x946253cecb9371ebe20c0e79c8ccd11787742c2ebe2a5a9ae0e77cfd59a33a12",
  "chain_tx_hash": "0x2841ff2f212ab6e9e6f2820303f308c09e00427cf07d87adbe9b7b7c60d323d6",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 160,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 2.8,
    "carrier": "mechanical",
    "difficulty_delta": 3,
    "domain": "Structural Mechanics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.bending_stiffness",
      "L.4th_order_pde",
      "L.bc",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "strain_gauges_lvdt",
    "solution_space": "beam_deflection_1D",
    "sub_domain": "Classical thin-beam bending PDE",
    "title": "Euler-Bernoulli Beam \u2014 thin-beam bending"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "euler_bernoulli_forward"
    ],
    "allowed_omega_dimensions": [
      "N_elements",
      "N_sensors",
      "SNR_dB",
      "EI_error",
      "bc_error",
      "load_error",
      "EI_error",
      "bc_error",
      "load_error"
    ],
    "allowed_problem_classes": [
      "euler_bernoulli"
    ],
    "center_spec": {
      "epsilon_fn_center": "28.0",
      "forward_operator": "euler_bernoulli_forward",
      "input_format": "measurement_only",
      "omega": {
        "EI_error": 0.0,
        "N_elements": 50,
        "N_sensors": 16,
        "SNR_dB": 30,
        "bc_error": 0.0,
        "load_error": 0.0
      },
      "problem_class": "euler_bernoulli"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "EI_error": [
        0.0,
        0.1
      ],
      "N_elements": [
        5,
        10000
      ],
      "N_sensors": [
        2,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "bc_error": [
        0.0,
        0.2
      ],
      "load_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Classical thin-beam bending PDE",
  "title": "Euler-Bernoulli Beam \u2014 thin-beam bending"
}

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