**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.