If there is no external force applied on the system, , the system will experience free vibration. Motion of the system will be established by an initial disturbance (i.e. initial conditions).

Furthermore, if there is no resistance or damping in the system, , the oscillatory motion will continue forever with a constant amplitude. Such a system is termed undamped and is shown in the following figure,

The equation of motion derived on the introductory page can be simplified to,

with the initial conditions,

This equation of motion is a second order, homogeneous, ordinary differential equation (ODE). If the mass and spring stiffness are constants, the ODE becomes a linear homogeneous ODE with constant coefficients and can be solved by the Characteristic Equation method. The characteristic equation for this problem is,

which determines the 2 independent roots for the undamped vibration problem. The final solution (that contains the 2 independent roots from the characteristic equation and satisfies the initial conditions) is,

The natural frequency w_n is defined by,

and depends only on the system mass and the spring stiffness (i.e. any damping will not change the natural frequency of a system).

Alternatively, the solution may be expressed by the equivalent form,

where the amplitude A₀ and initial phase f₀ are given by,

To obtain the time solution of any free SDOF system (undamped or not), use the SDOF Calculator.

The displacement plot of an undamped system would appear as,

Please note that an assumption of zero damping is typically not accurate. In reality, there almost always exists some resistance in vibratory systems. This resistance will damp the vibration and dissipate energy; the oscillatory motion caused by the initial disturbance will eventually be reduced to zero.