L1 ⚛ L1-209 ⊙ Testnet

Timoshenko Beam — shear-deformation thick-beam

Structural Mechanics · Thick-beam bending with shear

⬇ Download bundle 🚀 Submit to mainnet — earn 100 PWM + 5% royalty

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-209.md

# ⚛ L1 Principle — Timoshenko Beam — shear-deformation thick-beam **ID:** `L1-209` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Structural Mechanics — *Thick-beam bending with shear* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** beam w and phi 1D > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554010 --- ## 🧠 1. Introduction **Timoshenko Beam — shear-deformation thick-beam** is a **linear inverse problem** whose unknown lives in **beam w and phi 1D** space, within the **Thick-beam bending with shear** 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 · shear stiffness operator; L · bending operator; L · coupled w phi 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 · shear stiffness → L · bending → L · coupled w phi → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.coupled_w_phi` `L.bending` `L.shear_stiffness` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.shear_stiffness` | L · shear stiffness operator | | `L.bending` | L · bending operator | | `L.coupled_w_phi` | L · coupled w phi operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Structural Mechanics | | Sub domain | Thick-beam bending with shear | | Carrier | mechanical | | Problem class | linear_inverse | | Solution space | beam_w_and_phi_1D | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 3 | ## 📡 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:** `timoshenko` · **Forward operator:** `timoshenko_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Snr db | dB | 30 | | Ei error | — | 0 | | Ga error | — | 0 | | Bc error | — | 0 | | N sensors | — | 16 | | N elements | — | 50 | | K shear error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Snr db | dB | 0 – 40 | | Ei error | — | 0.0 – 0.1 | | Ga error | — | 0.0 – 0.1 | | Bc error | — | 0.0 – 0.2 | | N sensors | — | 2 – 256 | | N elements | — | 5 – 10000 | | K shear error | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 27.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x589cf5d720367181b83396157800b98f39525ff8a8d631f30fa88ced7e795864` - **Chain tx hash:** `0x853802db4a9b5c030b52e3d5e83ceebfc0dff4c22c10c34494b0ae893e40b3dc` - **Chain block:** `41554010` --- ## File Mapping This bundle consists of: `L1-209.md`, `L1-209.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-209.md` | Source of truth — edit this | Human or LLM | | `L1-209.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 `L1-209`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

📝 Polish this artifact in your LLM

Download the files on the right.
Paste the Markdown into ChatGPT / Claude / Gemini and edit freely.
Send your edited Markdown back with the prompt below.
Save the regenerated JSON, then submit via /submit.

Prompt to give your LLM

Read the attached Markdown (L1-209.md).
Regenerate L1-209.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.

⌥ Hierarchy

Upstream lineage and downstream children

↓ L2 Spec 1 child ▾

L2-209-001 Timoshenko Beam — shear-deformation thick-beam Mismatch-Only Spec

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-209",
  "chain_block": 41554010,
  "chain_hash": "0x589cf5d720367181b83396157800b98f39525ff8a8d631f30fa88ced7e795864",
  "chain_tx_hash": "0x853802db4a9b5c030b52e3d5e83ceebfc0dff4c22c10c34494b0ae893e40b3dc",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 200,
    "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": 3.0,
    "carrier": "mechanical",
    "difficulty_delta": 3,
    "domain": "Structural Mechanics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.shear_stiffness",
      "L.bending",
      "L.coupled_w_phi",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "strain_gauges_lvdt",
    "solution_space": "beam_w_and_phi_1D",
    "sub_domain": "Thick-beam bending with shear",
    "title": "Timoshenko Beam \u2014 shear-deformation thick-beam"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "timoshenko_forward"
    ],
    "allowed_omega_dimensions": [
      "N_elements",
      "N_sensors",
      "SNR_dB",
      "k_shear_error",
      "EI_error",
      "GA_error",
      "bc_error",
      "k_shear_error",
      "EI_error",
      "GA_error",
      "bc_error"
    ],
    "allowed_problem_classes": [
      "timoshenko"
    ],
    "center_spec": {
      "epsilon_fn_center": "27.0",
      "forward_operator": "timoshenko_forward",
      "input_format": "measurement_only",
      "omega": {
        "EI_error": 0.0,
        "GA_error": 0.0,
        "N_elements": 50,
        "N_sensors": 16,
        "SNR_dB": 30,
        "bc_error": 0.0,
        "k_shear_error": 0.0
      },
      "problem_class": "timoshenko"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "EI_error": [
        0.0,
        0.1
      ],
      "GA_error": [
        0.0,
        0.1
      ],
      "N_elements": [
        5,
        10000
      ],
      "N_sensors": [
        2,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "bc_error": [
        0.0,
        0.2
      ],
      "k_shear_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Thick-beam bending with shear",
  "title": "Timoshenko Beam \u2014 shear-deformation thick-beam"
}

🎁

Earn PWM by promoting this to mainnet

• 100 PWM one-time grant on mainnet promotion
• 5% upstream royalty on every downstream L4 mining event, forever
• Your wallet is enrolled the moment the founder accepts

🚀 Submit to mainnet →

How: Download the bundle above → polish the Markdown in ChatGPT / Claude / Gemini → submit your version → founder reviews → mainnet.