A first finite-element model: a clamped steel ruler bends and vibrates changedclaude-science/example-gallery/eng-beam
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99## Prompt
1010
11Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.
11Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview.md). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts.md#permission-cards) before it runs code or installs packages.
1212
1313```text Prompt wrap theme={null}
1414I'm learning the finite-element method. Model a steel ruler clamped at one end: length 300 mm, width 25 mm, thickness 3 mm, E = 200 GPa, density 7850 kg/m³. Split it into 4 standard 2-node beam elements (cubic shape functions, consistent mass matrix) and write the matrices yourself with numpy. First hang 1 N (about 100 g) on the free end: beam theory says the tip drops PL³/(3EI) = 0.800 mm. Check that you get it, and draw the bent ruler over the theory curve. Then let the ruler vibrate: plot its first 3 mode shapes, each labeled with its frequency. Theory gives 27.18, 170.33 and 476.93 Hz; finite-element frequencies land slightly above and get closer with more elements, so show that in a small table for 2, 4, 8 and 16 elements. Give each frequency to 2 decimals, and its error in % worked out from the unrounded frequencies. Explain each step as you would to an undergraduate.
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