Forced harmonic vibration calculator

Harmonic force on damped SDOF system; steady-state displacement amplitude.

acoustics-forced-response.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

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

Scenario	amplitudemm
	2.53837
	2.03069

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.

amplitude budget2.54 mm✓ passes

BASELINE INPUTS

force1 Nstiffness1000 N/mratio0.8 damping0.1

CALCULATED

amplitude2.5384 mm

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Forced harmonic vibration
# Scope: Harmonic force on damped SDOF system; steady-state displacement amplitude.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

force = 1 N
stiffness = 1000 N/m
ratio = 0.8
damping = 0.1

amplitude := force/stiffness/sqrt((1-ratio^2)^2+(2*damping*ratio)^2) [mm]

check amplitude budget: amplitude <= 3 mm

scenario alternative: force = 0.8 N

Calculated from your source · not saved

Baseline · Baseline

amplitude budget2.54 mm ✓ passes

alternative · alternative

amplitude budget2.03 mm ✓ 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.

amplitude: amplitude = force/stiffness/sqrt((1-ratio^2)^2+(2*damping*ratio)^2) [mm]

Original worked example

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

Given

force: 1 N
stiffness: 1000 N/m
ratio: 0.8
damping: 0.1

Calculated

amplitude: 2.53837 mm

Assumptions and limits

Harmonic force on damped SDOF system; steady-state displacement amplitude.

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: Forced harmonic vibration. Inputs: mass; stiffness; damping; force spectrum. Outputs: displacement response; resonance sensitivity. 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: mass, stiffness, damping, force spectrum
Outputs: displacement response, resonance sensitivity
Modes: calculate, compare, sweep
Drawing: Spectrum and vibration animation

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