99.96
rad/s
15.9091
Hz
0.025005
m
4
rad/s
25
25.005
12.5
W
90
°
99.96
rad/s
15.9091
Hz
0.025005
m
4
rad/s
25
25.005
12.5
W
90
°
The Resonance Calculator determines the key resonance parameters of a damped harmonic oscillator — the frequency at which maximum amplitude occurs, the peak amplitude itself, the bandwidth, and the quality factor. Resonance is perhaps the single most important concept in wave physics, occurring whenever a system is driven near its natural frequency and responds with dramatically amplified oscillations.
For a damped oscillator with natural frequency ω₀ and damping coefficient γ, the amplitude resonance frequency is not exactly ω₀ but is shifted slightly lower: $$\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$$ This shift arises because damping reduces the effective restoring force at high frequencies. For lightly damped systems (γ ≪ ω₀), the shift is negligible, but for heavily damped systems it can be significant — and if 2γ² ≥ ω₀², no resonance peak exists at all.
The peak amplitude at resonance is $$A_{\max} = \frac{F_0}{2m\gamma\sqrt{\omega_0^2 - \gamma^2}}$$ For light damping, this simplifies to A_max ≈ F₀/(2mγω₀) = Q × A_static, where the quality factor $$Q = \frac{\omega_0}{2\gamma}$$ measures the sharpness of the resonance peak. The half-power bandwidth — the frequency range over which the response exceeds half its peak power — is $$\Delta\omega = 2\gamma = \frac{\omega_0}{Q}$$
Resonance phenomena pervade nature and technology. Acoustic resonance determines the pitch of musical instruments. Orbital resonance stabilizes planetary configurations. Electrical resonance enables radio tuning. Nuclear magnetic resonance (NMR) underlies MRI imaging. Structural resonance caused the famous Tacoma Narrows Bridge collapse and must be carefully avoided in building and aircraft design.
This calculator takes the natural frequency, damping coefficient, driving force amplitude, and mass as inputs, computing the resonance frequency, peak amplitude, bandwidth, Q factor, magnification at resonance, and maximum average absorbed power. Engineers use these results to predict maximum vibration levels, design resonant circuits and filters, specify damping requirements, and evaluate structural safety margins.
The interplay between resonance frequency, Q factor, and bandwidth is central to filter design. A high-Q resonance provides excellent frequency selectivity (narrow bandwidth) but at the cost of large amplitude and long transient response. The maximum power absorbed at resonance is P_max = F₀²/(4mγ), independent of ω₀ — a result with deep implications for impedance matching and energy transfer optimization.
The calculator evaluates resonance properties from the forced oscillator theory:
Resonance Frequency:
$$\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$$
Exists only when ω₀² > 2γ². This is the amplitude resonance; the velocity resonance always occurs at ω₀.
Peak Amplitude:
$$A_{\max} = \frac{F_0}{2m\gamma\sqrt{\omega_0^2 - \gamma^2}}$$
Quality Factor & Bandwidth:
$$Q = \frac{\omega_0}{2\gamma}, \quad \Delta\omega = 2\gamma$$
Maximum Average Power:
$$P_{\max} = \frac{F_0^2}{4m\gamma}$$
This is the power absorbed at velocity resonance (ω = ω₀), where phase lag = 90° and velocity is in phase with the driving force.
The resonance frequency tells you which driving frequency produces maximum displacement. If it's zero, the system is too heavily damped for resonance. The peak amplitude indicates the maximum vibration level — compare it with structural limits. The Q factor and bandwidth together characterize frequency selectivity: high Q means precise tuning but dangerous amplification. The magnification at resonance (≈ Q for light damping) shows how much the dynamic response exceeds the static deflection. Maximum power indicates the peak energy absorption rate — critical for designing energy harvesting systems or avoiding thermal overload in absorbers.
Inputs
Results
With ω₀ = 100 rad/s and γ = 0.5 s⁻¹, Q = 100. The resonance frequency is almost exactly ω₀ (shifted by only 0.0025 rad/s). The amplitude is magnified 100× over static deflection. Bandwidth is 1 rad/s — a very sharp resonance.
Inputs
Results
A 50 kg mass with ω₀ = 10 rad/s and heavy damping (γ = 3), giving Q ≈ 1.67. The resonance frequency drops to 7.48 rad/s — a significant shift. The amplitude magnification is only 1.74×, and the bandwidth is 6 rad/s (a broad, flat response).
Resonance occurs when a system is driven at or near its natural frequency, causing dramatically amplified oscillations. The driving force efficiently transfers energy to the system because the velocity and force become nearly in phase. At exact velocity resonance (ω = ω₀), the phase lag is 90° and power transfer is maximized. Resonance is universal — it occurs in mechanical, electrical, acoustic, optical, and quantum systems wherever a restoring force and inertia create a natural oscillation frequency.
The amplitude resonance frequency ω_res = √(ω₀² − 2γ²) is shifted below ω₀ because damping creates a drag force proportional to velocity. At higher frequencies, the velocity (and thus the drag) is larger, so the amplitude response curve is asymmetric — it drops off faster above ω₀ than below. The peak shifts downward to compensate. This shift is negligible when γ ≪ ω₀ (high Q) but becomes significant for heavily damped systems.
Amplitude resonance (maximum displacement) occurs at ω_res = √(ω₀² − 2γ²), which is below ω₀. Velocity resonance (maximum velocity, maximum power absorption) always occurs at exactly ω₀, regardless of damping. There is also an acceleration resonance at ω = ω₀²/√(ω₀² − 2γ²), which is above ω₀. For lightly damped systems, all three are nearly equal, but they diverge with increasing damping.
Bandwidth and Q factor are inversely related: Δω = ω₀/Q = 2γ. A system with Q = 100 and ω₀ = 1000 rad/s has bandwidth Δω = 10 rad/s (Δf ≈ 1.6 Hz). This means the response is within 3 dB of peak over a 10 rad/s range. High Q means narrow bandwidth (selective but slow to respond); low Q means wide bandwidth (less selective but faster response). This tradeoff is fundamental in filter and receiver design.
Amplitude resonance ceases to exist when the damping is too heavy: specifically when 2γ² ≥ ω₀², or equivalently Q ≤ 1/√2 ≈ 0.707. In this regime, the amplitude response decreases monotonically from its static value — there is no peak. However, velocity resonance at ω₀ always exists regardless of damping (the power peak just becomes very broad and low). Physical examples of non-resonant overdamped systems include door closers and shock absorbers.
Resonance is exploited in many technologies: radio receivers use LC resonance to select a single station from many frequencies; quartz crystal oscillators use mechanical resonance for precise timekeeping; musical instruments rely on acoustic resonance to amplify sound; MRI uses nuclear magnetic resonance to image the body; particle accelerators use electromagnetic cavity resonance to accelerate charged particles; and microwave ovens use the 2.45 GHz resonance of water molecules to heat food.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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