Beam natural-frequency screening calculator

First bending mode of a uniform Euler-Bernoulli cantilever.

acoustics-beam-frequency.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	aream^2	inertiam^4	frequencyHz
	0.0003	3.6e-9	108.883
	0.0003	3.6e-9	170.13

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

first mode target108.88 Hz✓ passes

BASELINE INPUTS

length0.3 mwidth25 mmdepth12 mmmodulus69 GPadensity2700 kg/m^3

CALCULATED

area0.0003 m^2

inertia3.6e-9 m^4

frequency108.88 Hz

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Beam natural-frequency screening
# Scope: First bending mode of a uniform Euler-Bernoulli cantilever.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

length = 0.3 m
width = 25 mm
depth = 12 mm
modulus = 69 GPa
density = 2700 kg/m^3

area := width*depth [m^2]
inertia := width*depth^3/12 [m^4]
frequency := 1.875104^2/(2*pi*length^2)*sqrt(modulus*inertia/(density*area)) [Hz]

check first mode target: frequency >= 80 Hz

scenario alternative: length = 0.24 m

Calculated from your source · not saved

Baseline · 0.3 m reach · 12 mm depth

first mode target108.88 Hz ✓ passes

alternative · 0.24 m reach · 12 mm depth

first mode target170.13 Hz ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

area: area = width*depth [m^2]
inertia: inertia = width*depth^3/12 [m^4]
frequency: frequency = 1.875104^2/(2*pi*length^2)*sqrt(modulus*inertia/(density*area)) [Hz]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

length: 0.3 m
width: 25 mm
depth: 12 mm
modulus: 69 GPa
density: 2700 kg/m^3

Calculated

area: 0.0003 m^2
inertia: 3.6e-9 m^4
frequency: 108.883 Hz

Assumptions and limits

First bending mode of a uniform Euler-Bernoulli cantilever.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: Beam natural-frequency screening. Inputs: beam geometry; material; supports; added mass. Outputs: first bending frequency estimate. Model scope: Small-signal vibration and declared sound-propagation conditions. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

Inputs: beam geometry, material, supports, added mass
Outputs: first bending frequency estimate
Modes: calculate, compare, sweep
Drawing: Spectrum and vibration animation

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS