Calculation
Âout=K·Âin/√{[1−(2πfT0)²]²+[2D·2πfT0]²}
A lightly damped second-order system can amplify frequencies near its natural frequency even though high frequencies are strongly attenuated.
Zacher/Reuter 2024, Abschnitt 5.1.7, Gleichungen 5.18 bis 5.20: Frequenzgang eines P-T2-Glieds
PT2 Frequency Response Calculator: Amplitude and Resonance
A lightly damped second-order system can amplify frequencies near its natural frequency even though high frequencies are strongly attenuated.
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Result
Select a target and calculate.
Steady-state sine response: phase shift -90°
Three periods of input (orange) and output (green) at steady state. Besides the amplitude attenuation, the output lags by φ = −arctan(2·D·ωT₀/(1−(ωT₀)²)) – not part of the calculator's result, but directly readable in the Bode plot.
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Understand the inputs
- Steady output amplitude Âout — Peak amplitude of the steady output sinusoid. Compare with measurement, vibration or component limits; values above K·Âin indicate dynamic resonant amplification.
- Static gain K — Output-to-input amplitude ratio in the very-low-frequency limit. Obtain it from steady operating points or the second-order model.
- Damping ratio D — Dimensionless damping measure. Small values allow strong resonant amplification; obtain it from model parameters or a measured decay curve.
- Natural time T0 — Reciprocal of undamped natural angular frequency ω0. Obtain it from the second-order model or from peak spacing and damping.
- Input amplitude Âin — Distance from mean to positive peak of the sinusoidal input. Divide a peak-to-peak value by two; do not enter an RMS value.
- Excitation frequency f — Complete sinusoidal cycles per second. Obtain it from a signal generator, rotating excitation or spectral analysis; f=0 is the static limiting case.
Example
K=1, D=0.2, T0=1 s, Âin=1 and f=1/(2π)≈0.1592 Hz give 2πfT0=1. The denominator is then 2D=0.4 and output amplitude is 2.5, a clear resonant amplification.
Assumptions and limits
Linear time-invariant second-order system in sinusoidal steady state with positive damping. Dead time, zeros, additional poles, nonlinearity, saturation and startup transients are excluded.
Understand PT2 Frequency Response Calculator: Amplitude and Resonance
This calculator shows how a second-order system transmits periodic input signals. Unlike a first-order system, a lightly damped second-order system can strongly amplify a frequency near its natural frequency—important for vibration, sensors and control loops.
What does this quantity describe?
A second-order proportional system has normalised denominator 1+2DT0s+T0²s². In frequency response, s is replaced by jω. The magnitude of the resulting complex quotient determines output-to-input amplitude ratio.
Formula and variables
Âout=K·Âin/√{[1−(2πfT0)²]²+[2D·2πfT0]²}
Frequency response: G(jω)=K/[1−(ωT0)²+j·2DωT0]Magnitude: |G|=K/√{[1−(ωT0)²]²+(2DωT0)²}ω=2πfÂout=|G|·ÂinPhase: φ(ω)=−arctan(2DωT0/(1−(ωT0)²))
| Symbol / input | Meaning |
|---|---|
| Steady output amplitude Âout | Peak amplitude of the steady output sinusoid. Compare with measurement, vibration or component limits; values above K·Âin indicate dynamic resonant amplification. |
| Static gain K | Output-to-input amplitude ratio in the very-low-frequency limit. Obtain it from steady operating points or the second-order model. |
| Damping ratio D | Dimensionless damping measure. Small values allow strong resonant amplification; obtain it from model parameters or a measured decay curve. |
| Natural time T0 | Reciprocal of undamped natural angular frequency ω0. Obtain it from the second-order model or from peak spacing and damping. |
| Input amplitude Âin | Distance from mean to positive peak of the sinusoidal input. Divide a peak-to-peak value by two; do not enter an RMS value. |
| Excitation frequency f | Complete sinusoidal cycles per second. Obtain it from a signal generator, rotating excitation or spectral analysis; f=0 is the static limiting case. |
Choose the inputs correctly
K is static gain. D is damping ratio from a model or measured decay curve. T0 is natural time, equal to 1/ω0. Âin is peak amplitude relative to mean and f is excitation frequency in complete cycles per second. All model parameters must describe the same operating point.
How to use the calculator
Determine K from steady measurements, D from decay and T0 from model parameters or peak spacing. Obtain the relevant excitation frequency from a signal generator, rotational order or spectrum. Compare calculated output amplitude with vibration, measurement or component limits, paying special attention near 1/(2πT0).
Worked example
With K=1, D=0.2, T0=1 s and Âin=1, frequency f=1/(2π)≈0.1592 Hz gives ωT0=1. Magnitude denominator is 2D=0.4 and Âout=2.5, so the excitation is amplified 2.5 times.
Understand the result and units
As f→0, Âout approaches static value K·Âin. For small D, a resonance peak can occur below natural frequency; when D<1/√2 its maximum is at ωr=√(1−2D²)/T0. Greater damping suppresses the peak. At high frequency, magnitude falls approximately with 1/f². The phase shift is φ=−arctan(2·D·ωT0/(1−(ωT0)²)) and passes exactly −90° at ω=1/T0 regardless of D; the chart below the calculator shows input and output sine with this angle. The complete magnitude and phase plot, also with controller and dead time, is drawn by the Bode plot calculator.
f is entered in hertz and converted internally using ω=2πf radians per second. T0 converts to seconds; ωT0 is dimensionless. Input and output amplitude use one consistent scale.
Typical applications
Useful for resonance estimates, vibration tests, periodic disturbances, sensor and filter models, rotating excitation and learning second-order Bode diagrams.
Assumptions, limits and common mistakes
Valid for a linear time-invariant second-order model in sinusoidal steady state. Startup transient, mean value, phase, dead time, zeros, additional poles, saturation and nonlinear damping are excluded. A very large result may be capped in reality by limits or model error.
Common mistake: Do not use peak-to-peak or RMS values as peak amplitude. Do not confuse hertz and rad/s. Undamped natural frequency 1/T0 is not exactly the maximum-amplitude frequency at every damping ratio. A high resonance result is a warning, not an operating approval.
Frequently asked questions
What is “Second-order output amplitude and resonance under sinusoidal excitation” used for?
Useful for resonance estimates, vibration tests, periodic disturbances, sensor and filter models, rotating excitation and learning second-order Bode diagrams.
Where do the input values come from?
K is static gain. D is damping ratio from a model or measured decay curve. T0 is natural time, equal to 1/ω0. Âin is peak amplitude relative to mean and f is excitation frequency in complete cycles per second. All model parameters must describe the same operating point.
What does the result not cover?
Valid for a linear time-invariant second-order model in sinusoidal steady state. Startup transient, mean value, phase, dead time, zeros, additional poles, saturation and nonlinear damping are excluded. A very large result may be capped in reality by limits or model error.