Resonance Curve
y = (1/(sqrt((ω₀² − x²)² +(γ · x)²)))
Open in the app The dials and keys named below are the app's.
What it draws
This is the steady amplitude of a damped oscillator against the frequency it is driven at, which is x. ω₀ is the oscillator's own natural frequency, opening at 3, and γ is how strongly it is damped, opening at 0.5. There is no t anywhere in the expression, so the curve stands still while the clock runs.
The denominator is the distance from the origin to the point (ω₀² − x², γ·x). The first part vanishes when the drive matches the oscillator, at x = ω₀, and only the damping term is left to keep the amplitude finite there. At x = 0 the value is 1/ω₀² = 1/9 ≈ 0.111 whatever the damping, since a steady drive just pushes the mass to one side.
Why the peak
Maximising the response means minimising (ω₀² − x²)² + (γ·x)², which puts the peak at √(ω₀² − γ²/2) ≈ 2.979 and not quite at ω₀ itself. Its height is 1/(γ·√(ω₀² − γ²/4)) ≈ 0.669, six times the value at zero frequency.
Far from resonance the two squares are dominated by x⁴, so the curve dies away as 1/x². At x = 10 it is down to 0.011, a factor of 61 below the peak. Only x² appears, so the plot is even: the mirrored hump on the left is the same resonance at a negative driving frequency.
How sharp the peak is
Damping moves the picture a long way. At the opening value the peak holds half its height over a band 0.44 wide either side; at γ = 0.1 it reaches 3.33 and that band narrows to 0.087. Raise the damping instead and the resonance drains away, the peak down to 0.128 by γ = 3, barely above the 0.111 a steady push gives.
The peak position is real only while γ is under ω₀·√2 ≈ 4.243, and past that the only maximum sits at x = 0. The threshold is exact and all but invisible: at γ = 4 the peak stands 0.62% above the value at zero frequency, a dimple no plot resolves. Past it no drive makes the oscillator swing further than a steady push would.
Try
- Drag γ down to 0.2: the peak climbs to 1.67 and narrows to match, a sharply tuned oscillator.
- Drag γ to 4.2: two peaks are still there on paper, at ±0.424, but they stand 0.02% above the centre, so the plot reads as one low hump.
- Drag ω₀ to 6: the peak moves out to 5.99 and drops to 0.334, since a stiffer oscillator gives less for the same push.
- Change the 1 on top to x, the velocity response: its peak sits exactly at x = ω₀ for every damping, at a height of 1/γ = 2.
- Raise Span of x to 20: the tail is then long enough to read the 1/x² fall.