Page 31 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

18 Free Vibration Chap. 2

2.2 EQUATIONS OF MOTION: NATURAL FREQUENCY
Figure 2.2-1 shows a simple undamped spring-mass system, which is assumed to
move only along the vertical direction. It has 1 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
/„, which is a property of the system. We now examine some of the basic concepts
associated with the free vibration of systems with 1 degree of freedom.
Newton’s second law is the first basis for examining the motion of the system.
As shown in Fig. 2.2-1 the deformation of the spring in the static equilibrium
position is A, and the spring force A:A is equal to the gravitational force w acting
on mass m:
A:A «F vv = mg (2.2-1)
By measuring the displacement x from the static equilibrium position, the forces
acting on m are k(A + x) 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:
mx = XF = w - k(A + jc)

and because kA = w, we obtain
mx = -k x (2.2-2)
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 kA from the
equation of motion, and the resultant force on m is simply the spring force due to
the displacement jc.
By defining the circular frequency o)^ by the equation
2 ^ (2.2-3)
m

/f(A -^x)
Static equilibrium
posifion

IM'
Figure 2.2-1. Spring-mass system and free-body diagram.