Undamped Consider the motion of the spring/mass system when it is initially disturbed and then allowed to vibrate freely. The displacement of the mass with time, x(t), is measured from the static equilibrium position, i.e. the rest position. If the spring has a linear stiffness k, then P=kx. If at some time t the mass is displaced an amount x(t) in the positive direction shown. Then there will be a force on the mass from the spring of -kx(t). Thus from Newton's second law of motion using a free-body diagram, Equation (1) is called the equation of motion. The equation is unchanged if gravity effects are included.

The solution of the equation of motion gives, where x(0) is the initial displacement from the equilibrium position; x'(0) is the initial velocity. The frequency is called the undamped natural frequency and Thus for an initial displacement but with no initial velocity the motion is sinusoidal with an amplitude x(0) and a frequency ,

NOTE: The undamped natural frequency does not depend on the initial conditions or the amplitude of motion. It only depends on the mass and stiffness. You can check this out by running the interactive program. You should also do the set problem to check your understanding.