Forced Vibration SDOF Calculator

• •Use the Forced Vibration SDOF Calculator when solving homework or exam problems that require quick numerical verification of your hand calculations — instant feedback helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The Forced Vibration SDOF Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Steady-state amplitude ratio, phase angle, and frequency response for harmonically forced SDOF systems All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Inputs support both metric (SI) and imperial unit systems, with unit conversion handled automatically — simply select your preferred unit from the dropdown next to each field. Results are computed instantly in the browser without sending data to a server, ensuring both speed and privacy. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

Forced vibration occurs when a SDOF system is driven by a time-varying force. For harmonic forcing F(t) = F₀·cos(ωt), the steady-state response is x(t) = X·cos(ωt − φ), where X is the amplitude and φ is the phase lag. The amplitude ratio (dimensionless) is X/X_st = 1/√((1−r²)² + (2ζr)²), where r = ω/ω_n is the frequency ratio and X_st = F₀/k is the static deflection. The phase angle is φ = atan(2ζr/(1−r²)). The frequency response curve shows X/X_st vs r for various ζ values: at r = 0 (DC), X = X_st; at r = 1 (resonance), X is large and peak amplitude is X/X_st = 1/(2ζ) for light damping; for r → ∞, X → 0. For ζ < 0.707, there's a peak near r = 1 (resonance). For ζ ≥ 0.707, no peak — the response rolls off monotonically. Phase is −0° at r = 0, −90° at r = 1, and approaches −180° at high r. Transmissibility T = |X/Y_input| for base excitation has a peak near r = 1 then rolls off; it crosses 1 at r = √2 (below this, amplification; above, isolation). Vibration isolation targets r > √2 to achieve T < 1.

• •Rotating machinery diagnostic: forced vibration analysis identifies unbalance (at running speed), misalignment (at 2× running speed), and bearing faults (at specific characteristic frequencies).
• •Base isolation design: buildings with seismic isolators are tuned with very low natural frequency so the operating frequency range (earthquake ground motion) is in the isolation zone.
• •Automotive suspension: suspension resonance is tuned to about 1-1.5 Hz to avoid amplification at typical road frequencies while maintaining good handling.
• •Wind turbine tower analysis: verify that blade passing frequency (3× rotor speed) does not coincide with tower natural frequency to avoid resonant vibration.
• •Active/passive vibration control: reduce amplitude at specific frequencies using tuned mass dampers, active feedback, or viscoelastic damping materials.