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 wn 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.