A first finite-element model: a clamped steel ruler bends and vibrates - Claude.ai Documentation

Three plots: the ruler bent under 1 N, tip 0.800 mm down as theory says; its first 3 mode shapes at 27.18, 170.53, and 480.62 Hz over theory; and frequency errors falling from 2 to 16 elements.

Three plots: the ruler bent under 1 N, tip 0.800 mm down as theory says; its first 3 mode shapes at 27.18, 170.53, and 480.62 Hz over theory; and frequency errors falling from 2 to 16 elements.

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Prompt

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Prompt

I'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.

Reference analysis

A 300 mm steel ruler clamped at one end is cut into 4 beam elements: under 1 N its tip drops 0.800 mm, exactly as beam theory says. Its first 3 vibration modes come out at 27.18, 170.53, and 480.62 Hz against 27.18, 170.33, and 476.93 Hz in theory, and 16 elements bring all 3 within 0.004%.

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