Parameters
Excitation Type
Mass m
1000 kg
Stiffness k
100 kN/m
Damping ratio ζ
0.050
Excitation freq. f_ex
3.00 Hz
Amplitude F₀
1000 N
—
ωₙ [rad/s]
—
Natural freq. fₙ [Hz]
—
Peak displacement [mm]
—
Dynamic amp. factor β
—
ωd [rad/s]
—
Damped period Td [s]
—
Peak acceleration [m/s²]
—
Damping ratio ζ
Time History Response
Frequency Response Function |H(ω)| vs ω/ωₙ
Equation of Motion & Theory
SDOF equation of motion (absolute displacement, base excitation adds ü_g):
$$m\ddot{x} + c\dot{x} + kx = F(t)$$where $c = 2\zeta\sqrt{km}$. Natural angular frequency: $\omega_n = \sqrt{k/m}$, damped: $\omega_d = \omega_n\sqrt{1-\zeta^2}$
Frequency response function: $|H(\omega)| = \dfrac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$, $r=\omega/\omega_n$
At resonance ($r=1$): $|H| \approx \dfrac{1}{2\zeta}$ (maximum dynamic amplification factor)
CAE Applications: Equivalent SDOF model for buildings, bridges, and machinery. Hand-calculation verification of Ansys Mechanical / Abaqus linear dynamic analysis. Foundation of the response spectrum method (SRSS/CQC) used in seismic design codes.