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.