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 A_{0} and initial phase f_{0} are given by,
To obtain the time solution of any free SDOF system (undamped or not), use the SDOF Calculator.
