Damped free vibration of a 1-DOF system

A single-degree-of-freedom system in free vibration is one of the simplest models used to study structural dynamics. When the system is displaced from its equilibrium position by an amount \(x_0\) and given an initial velocity \(v_0\), it responds according to its natural properties of mass, stiffness, and damping. This motion, known as the free response, illustrates how the system oscillates without any external forces acting upon it.

General solution to underdamped free vibration

\[ x(t) = e^{-\xi\omega_n t}\left[x_0\cos(\omega_d t) + \frac{v_0 + \xi\omega_n x_0}{\omega_d}\sin(\omega_d t)\right], \qquad \omega_d = \omega_n\sqrt{1-\xi^2} \]

With damping rato \(\xi = 0.03\), natural frequency \(f_n = 2.5\) Hz, initial displacement \(x_0 = 0.001 \; m\), initial velocity \(v_0 = 0.001 \; m/s\), the amplitude falls by about 17 % in every cycle.

This solution can be put into polar form as

\[ x(t) = \rho_0 e^{-\xi\omega_n t}\cos(\omega_d t - \varphi) \]

With

\[ x_0 = \rho_0 \cos(\varphi), \qquad v_0 = \omega_d \rho_0 \sin(\varphi) - \xi \omega_n x_0 \]

Where amplitude

\[ \rho_0 = \sqrt{x_0^2 + \left(\frac{v_0 + \xi\omega_n x_0}{\omega_d}\right)^2} \]

and phase shift

\[ \varphi = \tan^{-1}\left(\frac{v_0 + \xi \omega_n x_0}{\omega_d x_0}\right) \]

The animation

The video shows an underdamped system with damping ratio \(\xi = 0.03\) in three views: a rotating 3D spiral, the time response with its envelope, and the phase plane.

What you see

• 3D spiral: displacement \(x\), velocity \(v/\omega_n\) and time \(\omega_n t\). The spiral shrinks as the system loses energy.
• Time response: the oscillation stays inside the envelope \(\pm\rho_0 e^{-\xi\omega_n t}\).
• Phase plane: the circles move toward the centre, the equilibrium.