Formula used
m x'' + c x' + kx = F₀ cos(ωt + φ) ω₀ = √(k/m) ζ = c/(2√mk) X = F₀ / √((k-mω²)² + (cω)²) δ = atan2(cω, k-mω²) ωᵣ = ω₀√(1-2ζ²), when ζ < 1/√2 Pavg = ½cω²X²
The model assumes linear stiffness and viscous damping. Parameters remain constant during motion. Harmonic forcing acts on one degree of freedom.
How to use this calculator
- Select a direct, base, or imbalance excitation.
- Enter mass, stiffness, damping, and driving frequency.
- Add phase and initial conditions when needed.
- Choose output units and frequency sweep settings.
- Submit the form and review all results.
- Export the response table for further analysis.
Example data
| Application | Mass | Spring constant | Damping | Force | Frequency |
|---|---|---|---|---|---|
| Laboratory oscillator | 2 kg | 800 N/m | 12 N·s/m | 25 N | 2.5 Hz |
| Vehicle approximation | 300 kg | 18,000 N/m | 1,800 N·s/m | 1,200 N | 1.2 Hz |
| Machine isolator | 75 kg | 120,000 N/m | 900 N·s/m | 500 N | 6 Hz |
Driven oscillator concepts
Free and forced oscillation
Free motion follows the system's own natural frequency. Forced motion follows an external periodic input. Both motions can appear during the transient interval.
Damping and resonance
Damping removes mechanical energy from the system. Resonance creates a large response near natural frequency. More damping usually lowers and broadens the peak.
Phase lag
Phase describes timing between force and displacement. The lag changes as driving frequency increases. It approaches opposite phase above resonance.
Practical applications
Applications include machines, vehicles, buildings, and instruments. The model supports vibration isolation and resonance checks. Real systems may require nonlinear analysis.
Frequently asked questions
What is a driven oscillator?
It is an oscillator receiving a periodic external input. The input supplies energy continuously. Damping removes part of that energy.
What causes resonance?
Resonance occurs near the system's natural frequency. Energy transfer becomes especially effective there. Damping limits the resulting response amplitude.
Does resonance equal natural frequency?
They match exactly only in special cases. Damping shifts displacement resonance slightly lower. Other response measures can peak elsewhere.
What does damping ratio mean?
Damping ratio compares actual and critical damping. Values below one are underdamped. Values above one are overdamped.
What is the quality factor?
Quality factor indicates resonance sharpness for light damping. Larger values mean narrower resonance peaks. It equals one divided by twice damping ratio.
Why use atan2 for phase?
Atan2 preserves the correct response quadrant. A simple arctangent can lose sign information. Correct quadrants matter above natural frequency.
What is transient motion?
Transient motion depends on initial conditions. Damping gradually removes this component. Steady motion remains after sufficient time.
Can this calculator model base excitation?
Yes, it converts base displacement into equivalent forcing. The conversion includes spring and damper effects. Results describe the selected linear model.
Can rotating imbalance be analysed?
Yes, imbalance force equals mass times eccentricity times frequency squared. The force therefore increases rapidly with speed. Enter imbalance mass and eccentricity carefully.
Why can amplitude become extremely large?
Very low damping permits a sharp resonance peak. Exact undamped resonance has no finite steady amplitude. Real systems always contain limiting effects.
Are the results suitable for final design?
The calculator supports preliminary engineering analysis. Confirm assumptions, units, and loading conditions independently. Safety-critical designs require qualified professional review.