Free Vibration SDOF Calculator

• •Use the Free 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 Free Vibration SDOF Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Natural frequency, damping ratio, and free response of a single-DOF spring-mass-damper system with decay envelope plot 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.

Free vibration of a single-degree-of-freedom (SDOF) spring-mass-damper system is governed by m·ẍ + c·ẋ + k·x = 0, where m is mass, c is damping coefficient, k is spring stiffness, and x is displacement. The natural frequency is ω_n = √(k/m) (rad/s) or f_n = ω_n/(2π) (Hz). The damping ratio is ζ = c/(2·√(k·m)), dimensionless. The damped natural frequency is ω_d = ω_n·√(1−ζ²). Response types depending on ζ: undamped (ζ = 0) oscillates forever; underdamped (0 < ζ < 1) oscillates with decaying amplitude x(t) = A·e^(−ζω_n·t)·cos(ω_d·t + φ); critically damped (ζ = 1) returns to equilibrium fastest without oscillation; overdamped (ζ > 1) returns slowly without oscillation. Logarithmic decrement δ = ln(x_n/x_(n+1)) = 2πζ/√(1−ζ²) relates successive peak amplitudes, allowing experimental determination of ζ from a decaying response trace. Free vibration is the natural response of any mechanical system when disturbed from equilibrium and released. It reveals the system's fundamental characteristics (natural frequencies and damping) that govern response to all other inputs. For multi-DOF systems, each natural frequency and mode shape is analyzed separately using eigenvalue analysis. The calculator handles SDOF with user-supplied mass, stiffness, and damping, plotting the free response and computing key metrics.

• •Machine foundation design: analyze free vibration of machine mountings to ensure natural frequencies are far from operating frequencies.
• •Earthquake engineering: building free vibration characteristics determine how they respond to ground motion; design tunes fundamental frequency to avoid resonance with typical earthquake frequencies.
• •Vehicle suspension analysis: wheel-hop frequency (unsprung mass on tire) and body-hop frequency (sprung mass on suspension) are key SDOF frequencies.
• •Musical instrument acoustics: string and membrane vibrations determine pitch and timbre through fundamental frequencies and harmonics.
• •Sensor and accelerometer design: MEMS accelerometers use cantilever beams as SDOF resonators whose deflection is proportional to acceleration.