Equation of Motion :
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
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
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
We now apply Newton's second law of
motion to the mass m :
and because kD = w, we obtain :
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
Eq. 6 can be written as
and we conclude that the motion is
harmonic. Equation (8), a homogeneous second order linear differential equation, has the
following general solution :
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
The natural period of the oscillation
is established from , or
and the natural frequency is
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
Note that , depend only on the mass and stiffness of the system, which are
properties of the system.