Enter Oscillator Inputs
Formula Used
Governing equation: m x'' + c x' + k x = F₀ sin(ωt)
Natural angular frequency: ωn = √(k / m)
Natural frequency: fn = ωn / 2π
Critical damping: cc = 2√(km)
Damping ratio: ζ = c / cc
Steady-state amplitude: X = F₀ / √[(k - mω²)² + (cω)²]
Phase lag: φ = tan⁻¹[(cω) / (k - mω²)]
Steady-state displacement: x(t) = X sin(ωt - φ)
Velocity amplitude: V = ωX
Acceleration amplitude: A = ω²X
Static displacement: Xst = F₀ / k
Magnification factor: M = X / Xst
Average absorbed power: Pavg = 0.5 c ω² X²
Peak spring energy: Emax = 0.5 k X²
How to Use This Calculator
- Enter mass, spring constant, damping coefficient, and force amplitude.
- Enter the driving frequency and choose Hz or rad/s.
- Add a time value if you want instantaneous motion values.
- Click the calculate button to generate the report.
- Read amplitude, phase lag, resonance behavior, and energy values.
- Export the computed report as CSV or PDF when needed.
Example Data Table
| Item | Example Value |
|---|---|
| Mass | 2 kg |
| Spring Constant | 50 N/m |
| Damping Coefficient | 3 N·s/m |
| Force Amplitude | 10 N |
| Driving Frequency | 0.7 Hz |
| Time | 1.5 s |
| Natural Frequency | 0.795775 Hz |
| Steady-State Amplitude | 0.575395 m |
| Phase Lag | 49.395158 deg |
| Velocity Amplitude | 2.530721 m/s |
| Acceleration Amplitude | 11.130692 m/s² |
| Average Absorbed Power | 9.606823 W |
Driven Harmonic Oscillator Guide
Forced Vibration Basics
A driven harmonic oscillator is a classic physics model. It describes a mass attached to a spring with damping and an external periodic force. The system responds according to stiffness, inertia, damping, and forcing rate. These variables shape the motion in a predictable way. This makes the model useful in mechanics, electronics, acoustics, and structural dynamics.
This calculator helps you study forced vibration without manual algebra. It evaluates amplitude, phase lag, frequency ratio, and resonance behavior. It also reports energy and average power. Instantaneous displacement, velocity, and acceleration are included at a chosen time. That gives a broader view of steady-state motion from one submission.
Why Resonance Matters
Natural frequency is the preferred vibration rate of the system. Driving frequency is the rate of the external force. When both values become close, resonance may occur. The displacement can rise sharply. In real systems, damping reduces that peak and keeps motion under control. This is why damping is essential in vehicle suspensions, machines, measuring instruments, and buildings.
The phase angle also changes with frequency. At low frequency, motion follows the force more closely. Near resonance, the phase lag grows fast. At higher frequency, the displacement can lag strongly behind the excitation. These phase effects matter in vibration isolation, signal response, and control system interpretation.
What This Calculator Assumes
The model here uses the linear steady-state solution of the forced mass-spring-damper equation. It assumes harmonic forcing and constant system parameters. It does not include nonlinear springs, dry friction, impacts, or changing coefficients. For standard academic and engineering problems, though, the linear model is accurate and efficient.
Use this page to compare damping levels, test resonance sensitivity, and understand response trends. It is useful for students learning oscillation theory and for professionals checking forced response values. The report turns raw inputs into readable outputs quickly. That saves time and improves physical interpretation.
FAQs
1) What is a driven harmonic oscillator?
It is a vibrating system with mass, stiffness, damping, and an external periodic force. The force keeps exciting the system, so the motion depends on both system properties and input frequency.
2) What happens at resonance?
Resonance occurs when the driving frequency approaches the natural frequency. The displacement response can grow very large. Damping lowers the peak and reduces the risk of excessive motion.
3) Why does damping matter?
Damping dissipates energy. It limits response amplitude, changes the phase lag, and reduces resonance sharpness. Higher damping usually means a smoother and safer vibration response.
4) Why can I enter frequency in Hz or rad/s?
Some textbooks and labs use hertz. Others use angular frequency. This calculator accepts both forms so you can work directly with your source data.
5) What does phase lag mean here?
Phase lag measures how much the displacement response falls behind the applied sinusoidal force. It is reported in degrees for easier interpretation.
6) Are these results transient or steady-state?
These outputs represent the steady-state forced response. They do not include the transient part that depends on initial displacement and initial velocity.
7) What is the quality factor Q?
Quality factor indicates how lightly damped the oscillator is. A larger Q means sharper resonance and lower damping. A smaller Q means broader and more damped response.
8) Can I export the calculated report?
Yes. After calculation, the page shows buttons for CSV and PDF export. That makes saving, sharing, and documenting results much easier.