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)
\[\\omega_d = \\omega_n\\sqrt{1-\\zeta^2}\]