Equation of Motion : Natural Frequency Figure 2 shows a simple undamped spring-mass system, which is assumed to move only along the vertical direction. It has one degree of freedom (DOF), because its motion is described by a single coordinate x. When placed into motion, oscillation will take place at the natural frequency fn which is a property of the system. We now examine some of the basic concepts associated with the free vibration of systems with one degree of freedom. Figure 2 Spring-Mass System and Free-Body Diagram Newton's second law is the first basis for examining the motion of the system. As shown in Fig. 2 the deformation of the spring in the static equilibrium position is D , and the spring force kD is equal to the gravitational force w acting on mass m (5) By measuring the displacement x from the static equilibrium position, the forces acting on m are and w. With x chosen to be positive in the downward direction, all quantities - force, velocity, and acceleration are also positive in the downward direction. We now apply Newton's second law of motion to the mass m : and because kD = w, we obtain : (6) It is evident that the choice of the static equilibrium position as reference for x has eliminated w, the force due to gravity, and the static spring force kD from the equation of motion, and the resultant force on m is simply the spring force due to the displacement x. By defining the circular frequency w n by the equation (7)   Eq. 6 can be written as (8) and we conclude that the motion is harmonic. Equation (8), a homogeneous second order linear differential equation, has the following general solution : (9) where A and B are the two necessary constants. These constants are evaluated from initial conditions , and Eq. (9) can be shown to reduce to (10) The natural period of the oscillation is established from , or (11) and the natural frequency is (12) These quantities can be expressed in terms of the static deflection D by observing Eq. (5), . Thus, Eq. (12) can be expressed in terms of the static deflection D as (13) Note that , depend only on the mass and stiffness of the system, which are properties of the system.