0.475651
m
0.442374
rad
25.35
deg
4.7565
9.162896
W
10
0.1
m
0.9
0.05
0.475651
m
0.442374
rad
25.35
deg
4.7565
9.162896
W
10
0.1
m
0.9
0.05
The Forced Oscillation Calculator analyzes the steady-state response of a damped harmonic oscillator driven by a sinusoidal external force. This is one of the most important problems in classical mechanics, with direct applications in structural engineering, electrical circuits, acoustics, and spectroscopy. When a periodic force F₀cos(ωt) acts on a damped oscillator with natural frequency ω₀ and damping coefficient γ, the system eventually settles into a steady oscillation at the driving frequency ω — but with an amplitude and phase that depend critically on how close ω is to ω₀.
The steady-state amplitude is given by the classic resonance formula: $$A = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}$$ This amplitude peaks near ω = ω₀ (resonance) and falls off for frequencies far above or below. The sharpness of this peak depends on the damping γ — light damping produces a tall, narrow peak, while heavy damping yields a broad, low response.
The phase lag between the driving force and the system's response is $$\varphi = \arctan\!\left(\frac{2\gamma\omega}{\omega_0^2 - \omega^2}\right)$$ At low frequencies (ω ≪ ω₀), the system follows the force nearly in phase (φ ≈ 0). At resonance (ω = ω₀), the phase lag is exactly 90° — the velocity is in phase with the force, maximizing power transfer. At high frequencies (ω ≫ ω₀), the response lags by nearly 180°, essentially moving opposite to the force.
The average power absorbed by the oscillator from the driving force is $$\langle P \rangle = \frac{F_0^2 \gamma \omega^2}{m[(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2]}$$ which peaks at resonance. This explains why resonance can be destructive — maximum energy transfer occurs when driving near the natural frequency.
This calculator computes the steady-state amplitude, phase lag, amplitude magnification factor (ratio to static deflection), average absorbed power, and quality factor. These results are essential for predicting vibration levels in machinery, designing vibration absorbers, analyzing electrical filter responses, and understanding spectral line shapes in atomic and molecular physics.
The forced oscillator model is the foundation of resonance theory, with identical mathematics describing driven RLC circuits (voltage replaces force, charge replaces displacement), optical absorption spectra (Lorentzian line shapes), and acoustic resonators. Mastering this system provides deep insight into energy transfer, frequency selectivity, and the universal phenomenon of resonance.
The calculator evaluates the steady-state solution of the equation of motion: $$m\ddot{x} + 2m\gamma\dot{x} + m\omega_0^2 x = F_0\cos(\omega t)$$
Amplitude:
$$A = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}$$
Phase Lag:
$$\varphi = \arctan\!\left(\frac{2\gamma\omega}{\omega_0^2 - \omega^2}\right)$$
Magnification Factor:
$$\frac{A}{A_{\text{static}}} = \frac{A}{F_0/(m\omega_0^2)}$$
Average Power:
$$\langle P \rangle = \frac{1}{2}F_0 A \omega \sin\varphi = \frac{F_0^2 \gamma \omega^2}{m[(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2]}$$
The amplitude tells you how far the system oscillates at steady state. The amplitude ratio (magnification factor) shows amplification relative to a static force — values much greater than 1 indicate near-resonance amplification, which can be dangerous for structures. The phase lag reveals the timing relationship: 0° means in-phase, 90° at resonance, 180° anti-phase. The average power shows energy transfer from the driver to the oscillator, peaking at resonance. The Q factor indicates how sharp the resonance is — high Q means large amplification over a narrow frequency band.
Inputs
Results
A 100-tonne bridge section (ω₀ ≈ 6.28 rad/s, ~1 Hz) driven near resonance at ω = 6.0 rad/s with light damping (γ = 0.1) shows amplitude magnification of ~5.8×. The phase lag is ~19°, and Q ≈ 31, indicating a fairly sharp resonance.
Inputs
Results
At exact resonance (ω = ω₀ = 1000 rad/s) with γ = 5 s⁻¹, the phase lag is exactly 90°. The amplitude is magnified 100× over the static case, equal to Q = 100. Maximum power is transferred to the oscillator.
Forced oscillation occurs when an external periodic force drives a system that has its own natural frequency. After initial transients die out, the system oscillates at the driving frequency (not its natural frequency) with a steady amplitude and phase that depend on the frequency ratio ω/ω₀ and the damping. This is in contrast to free oscillation, where the system vibrates at its natural frequency after an initial disturbance.
At resonance (ω ≈ ω₀), the driving force most efficiently transfers energy to the oscillator. The amplitude reaches its maximum value A_max ≈ F₀/(2mγω₀) = Q × A_static, the phase lag is exactly 90°, and the average absorbed power is maximized. With very light damping, the amplitude at resonance can be enormous — this explains catastrophic resonance failures in bridges, buildings, and machinery when the driving frequency matches a natural mode.
The phase lag φ determines the power transfer: average power is proportional to sin(φ). At φ = 0° (far below resonance), the displacement is in phase with the force but velocity leads by 90°, so little power is absorbed. At φ = 90° (resonance), velocity is in phase with the force, maximizing power transfer. At φ = 180° (far above resonance), the system opposes the force. Phase measurements are used to detect resonance and tune systems.
Increasing damping γ has three effects: (1) the peak amplitude decreases as A_max = F₀/(2mγω₀), (2) the resonance peak broadens (bandwidth Δω = 2γ), and (3) the peak frequency shifts slightly below ω₀ to ω_peak = √(ω₀² − 2γ²). In the limit of very heavy damping, the resonance peak disappears entirely. The quality factor Q = ω₀/(2γ) quantifies this — high Q means sharp resonance, low Q means broad response.
The static deflection A_static = F₀/(mω₀²) = F₀/k is the displacement produced by applying F₀ as a constant (non-oscillating) force. The magnification factor M = A/A_static measures how much the dynamic amplitude exceeds the static case. At resonance with light damping, M ≈ Q, which can be very large. M = 1 at ω = 0 (quasistatic limit) and M → 0 as ω → ∞ (the mass cannot follow rapid oscillations).
Partially. The Tacoma Narrows collapse (1940) involved aeroelastic flutter — a self-excited oscillation where wind energy feeds into a structural mode through aerodynamic coupling. While simple forced resonance (matching wind gust frequency to bridge frequency) was initially blamed, the actual mechanism was more complex: the bridge's torsional mode extracted energy from steady wind. However, forced oscillation theory correctly predicts that low damping (high Q) makes any resonance-like phenomenon more dangerous, which is why modern bridges include dampers.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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