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
Second-order frequency response: output amplitude and resonance
A lightly damped second-order system can amplify frequencies near its natural frequency even though high frequencies are strongly attenuated.
MINTSI
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Result
Select a target and calculate.
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 Second-order frequency response: output 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|·Âin
| 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.
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.