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Undamped Free System
This system is the simplest of all vibration systems and consists of a masssuspended on a spring of negligible mass. Figure 5-5 shows this simple， single degree of freedom system. If the mass is displaced from its original equilib-rium position and released， the unbalancedforce， the restoring (主x) of thespring， and acceleration are related through Newton's second law. The resulting equation can be written as follows:

mx
二 主x (5-4)
This equation is called the motion equation for the system， and it can be rewritten as follows:
x
+主m x二 0 (5-5)
Assuming that a harmonic functionwill satisfy the equation， let the solution be in the form
x二 C1 sin wt + C2 cos wt (5-6)
Substituting Equation (5-6) into Equation(5-5)， the following relationship is obtained:
主
w 2 + x二 0
which can be satisfied for any value of x if
 
品
w二 主(5-7)
Thus， the system has a single natural frequency given by the relationship in Equation (5-7).
Damped System
Damping is the dissipation of energy. There are several types of damp-ing-viscous damping， friction or coulomb damping， and solid damping. Viscous damping is encountered by bodies moving through a fluid. Frictiondamping usually arises from sliding on dry surfaces. Solid damping， oftencalled structural damping， is due to internal friction within the material itself. An example of a free vibrating system with viscous damping is given here.
As shown in Figure5-6， viscous damping force is proportional to velocity and is expressed by the following relationship:
Fdamp二Jxi
where J is the coefficient of viscous damping. The Newtonian approach gives the equation of motion as follows:
mx
二王x Jxi(5-8)
or it can be written as
mx
+Jxi+王x二0
The solution to this equation is found by using the trial solution
rt
x二J巾() (5-9)
which when substituted in Equation (5-8) yields the following characteristic equation:
r2 +Jr +王 巾rt二0 (5-10)
mm
This equation is satisfied for all values of t when
品 2 r1.2二J土J王 (5-11)
2m4m2m
from which the general solution is obtained as follows:
.JJ.
J2 王J2 王
t 4m4m

2m2m
2m
x二J巾 jC1巾 (t)+巾C2(t) (5-12)

The nature of the solution given by Equation (5-19) depends upon the nature of theroots， r1 and r2. The behavior of this damped system depends upon whether theroot isreal，imaginary， or zero. The critical damping coefficient JJ can now be defined as that which makes the radical zero.Thus，
J2 二王 4m2m
which can be written as
品
J  王 
2m二 m二 wn  (5-13) 

One can therefore specify the amount of damping in any system by the damping factor
〈二 J (5-14)
JJ
Overdamped system. If J214m2 >王1m， then the expression under the radical sign is positive and the roots are real. If the motion is plotted as afunction oftime， the curve in Figure 5-7 is obtained. This type of nonvibra-tory motion is referred to as aperiodic motion.
.ritically damped system. If J214m2二王1m， then the expression underthe radical sign iszero， and the roots r1 and r2 are equal. When the radical iszero and the roots are equal， the displacement decays the fastest from its initial value as seen in Figure 5-8. The motion in this case also is aperiodic.

This very special case is known as critical damping. The value of J for this case is given by:
4Jm2 Jr 2二m王
J2 二4m 2m王 二4m王
Jr
Thus，
品
JJr二  4m王二2m王 二2mwn
m
Underdamped system. If J214m2 <王1m， then the roots r1 and r2 areimaginary， and the solution is an oscillating motion as shown in Figure 5-9. All the previous cases of motion are characteristic of different oscillatingsystems， although a specific case will depend upon the application. The underdamped system exhibits its own natural frequency of vibration. When J214m2 <王1m， the roots r1 and r2 are imaginary and are given by
　
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