VD-03Vibration & Structural Dynamics

SDOF Frequency Response

Bode magnitude, phase and Nyquist plots of a damped oscillator: receptance, mobility, accelerance.

System

Mass m

Stiffness k

N/m

Damping ratio ζ

c = 31.6 N·s/m · Q = 10

Response type

Natural freq. fₙ

50.33Hz

Damped f_d

50.27Hz

Peak DAF

10×

at 50.2 Hz

Half-power BW

5.03Hz

Bode magnitude - Receptance X/Fm/N

Phase

Nyquist

Static compliance 1/k = 1.00e-5 m/N · phase at fₙ = -90°

§ Theory of operation

Frequency response function

A viscously damped SDOF system driven by a harmonic force $F\cdot e^{i\omega t}$ responds at the same frequency with the complex receptance:

\begin{aligned} H(\omega) = X/F &= \dfrac{1}{k - m\omega^2 + i\,c\,\omega} \\ \text{Mobility}\quad Y(\omega) &= i\,\omega \cdot H(\omega) \\ \text{Accelerance}\quad A(\omega) &= -\omega^2 \cdot H(\omega) \end{aligned}

Below resonance the response is stiffness-controlled (X ≈ F/k, in phase). At resonance it is damping-controlled (phase −90°), and above resonance mass-controlled (X ≈ F/mω², phase → −180°). This is the most important picture in vibration engineering.

Key quantities

\begin{aligned} \text{Damped natural frequency}\quad f_d &= f_n\sqrt{1-\zeta^2} \\ \text{Displacement peak at}\quad f_\mathrm{pk} &= f_n\sqrt{1-2\zeta^2} \\ \text{Peak amplification}\quad \mathit{DAF} &= \dfrac{1}{2\zeta\sqrt{1-\zeta^2}} \approx Q = \dfrac{1}{2\zeta} \\ \text{Half-power bandwidth}\quad \Delta f &\approx 2\,\zeta\,f_n \end{aligned}

The half-power method inverts the last relation: in modal testing the damping ratio is estimated from the width of a resonance peak 3 dB below its maximum.

Nyquist plot

Plotting the real part against the imaginary part sweeps out a near-circle (exactly a circle for mobility with viscous damping). Modal analysis software fits circles to measured FRF data (the “circle-fit” method) because the resonance is where the points are spread furthest apart around the circle.