/ SDOF Frequency Response Analysis Tool Tools Index
JP | EN | ZH
ホーム ライブラリ リクエスト・バグ報告
JPENZH
Vibration Analysis Tool

SDOF Frequency Response Analysis Tool

Visualize the FRF (Frequency Response Function) in real time by adjusting the damping ratio and natural frequency. Gain intuitive understanding of resonance mechanisms and dynamic amplification factors.

$$m\ddot{x} + c\dot{x} + kx = F_0\sin(\omega t)$$
Parameter Settings
Natural Frequency f₀ 10 Hz
Resonance frequency of the system
Damping Ratio ζ 0.05
ζ=0.05 is a typical structural damping value
Frequency Range f_max 100 Hz
Resonance Peak Condition
Maximum response at r = f/f₀ = 1
Peak value ≈ 1/(2ζ)
Resonance Freq. f_r
—
Hz
Dynamic Amplification Q
—
= 1/(2ζ)
Half-Power Bandwidth Δf
—
Hz
Critical Damping Ratio
—
ζ
Frequency Response Function H(f) = X / (F₀/k)
Phase Angle φ(f)
Theory — Frequency Response Function

Amplitude Ratio H(r)

$$H(r)=\frac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$$

Frequency ratio $r = f/f_0$

Phase Angle φ(r)

$$\varphi(r)=\arctan\!\left(\frac{2\zeta r}{1-r^2}\right)$$

At resonance φ = 90° (phase lag)

Resonance Peak Location

$$r_{peak}=\sqrt{1-2\zeta^2}$$

$$H_{max}=\frac{1}{2\zeta\sqrt{1-\zeta^2}}$$

Quality Factor Q and Bandwidth

$$Q = \frac{1}{2\zeta},\quad \Delta f = 2\zeta f_0$$

Higher Q means a sharper, more dangerous resonance peak