Single Degree of Freedom (SDOF) Systems
Explore the behavior of free vibration in SDOF systems with adjustable parameters.
System Parameters
kg
N/m
Ns/m
m
m/s
System Information
Natural Frequency: 10.00 rad/s
Damping Ratio: 0.250
Damped Natural Frequency: 9.68 rad/s
Damping Type: Underdamped
Simulation
Position vs. Time
Velocity vs. Time
Acceleration vs. Time
Equation of Motion
The equation of motion for an SDOF system in free vibration
\[m\ddot{x} + c\dot{x} + kx = 0\]
Solution Forms
Underdamped (ζ < 1)
Oscillatory motion with decreasing amplitude
\[x(t) = e^{-\zeta\omega_{n} t}\left[A\cos(\omega_{d} t) + B\sin(\omega_{d} t)\right]\]
Critically Damped (ζ = 1)
Non-oscillatory, fastest return to equilibrium
\[x(t) = (A + Bt)e^{-\omega_{n} t}\]
Overdamped (ζ > 1)
Non-oscillatory with slower return
\[x(t) = Ae^{s_{1} t} + Be^{s_{2} t}\]
Key Parameters
Natural frequency (rad/s)
\[\omega_{n} = \sqrt{\frac{k}{m}}\]
Damping ratio (dimensionless)
\[\zeta = \frac{c}{2\sqrt{km}}\]
Relation to natural frequency in Hz
\[\omega_{n} = 2\pi f_{n}\]
Damped natural frequency (rad/s)
\[\omega_{d} = \omega_{n}\sqrt{1-\zeta^{2}}\]