Parameter Settings
Spring Constant k
100.0 N/m
Mass m
1.0 kg
Damping Coefficient c
2.0 N·s/m
Initial Displacement x₀
0.20 m
Forced Excitation
Excitation Force F₀
1.0 N
Excitation Frequency f
1.0 Hz
Natural Frequency fₙ
—
Damping Ratio ζ
—
Damping Type
—
—
ωₙ [rad/s]
—
Damping Ratio ζ
—
fₙ [Hz]
—
Current Displacement [m]
Under
Damping Type
Animation
Displacement Time History x(t)
Governing Equations
$$m\ddot{x} + c\dot{x} + kx = F_0\cos(\Omega t)$$ $$\omega_n = \sqrt{\frac{k}{m}}, \quad \zeta = \frac{c}{2\sqrt{mk}}, \quad \omega_d = \omega_n\sqrt{1-\zeta^2}$$Underdamped ($\zeta < 1$): $x(t) = Ae^{-\zeta\omega_n t}\cos(\omega_d t + \phi)$
Numerical integration uses 4th-order Runge-Kutta (Δt = 1 ms)
CAE Connection: The single-DOF system is the fundamental unit of FEM modal analysis. Natural frequency and damping ratio are directly related to vibration analysis in Nastran/Abaqus. Typical damping ratios in automotive and aerospace vibration design are 0.01–0.05.