**Viscously Damped Free
Vibration**

Viscous damping force is expressed
by the equation

(14)

where c is a constant of
proportionality.

Symbolically. it is designated by a
dashpot, as shown in Fig. 3. From the free body diagram, the equation of motion is .seen
to be

(15)

The solution of this equation has two
parts. If *F(t)* = 0, we have the homogeneous differential equation whose solution
corresponds physically to that of **free-damped vibration**. With *F(t) ¹ 0, *we
obtain the particular solution that is due to the excitation irrespective of the
homogeneous solution. We will first examine the homogeneous equation that will give us
some understanding of the role of damping.

Figure 3 Viscously Damped Free
Vibration

With the homogeneous equation :

(16)

the traditional approach is to assume
a solution of the form :

(17)

where s is a constant. Upon
substitution into the differential equation, we obtain :

which is satisfied for all values of t
when

(18)

Equation (18), which is known as the **characteristic
equation**, has two roots :

(19)

Hence, the general solution is given
by the equation:

(20)

where A and B are constants to be
evaluated from the initial conditions and .

Equation (19)* *substituted into
(20) gives :

(21)

The first term, ,* *is simply an exponentially decaying function of time. The
behavior of the terms in the parentheses, however, depends on whether the numerical value
within the radical is positive, zero, or negative.

When the damping term *(c/2m)**2 *is larger than *k/m*, the exponents in the previous
equation are real numbers and no oscillations are possible. We refer to this case as **overdamped**.

*
*When the damping term *(c/2m)**2 *is* *less than *k/m*, the exponent becomes an
imaginary number, *. *Because

the terms of Eq. (21) within the
parentheses are oscillatory. We refer to this case as **underdamped**.

*
*In the limiting case between the
oscillatory and non oscillatory motion , and the radical is
zero. The damping corresponding to this case is called **critical damping, c****c**.

(22)

Any damping can then be expressed in
terms of the critical damping by a non dimensional number z , called the **damping
ratio**:

*
* (23)

and

(24)

(i) Oscillatory Motion (z < 1.0)
Underdamped Case :

(25)

The frequency of damped oscillation is
equal to :

(26)

Figure 4 shows the general nature of
the oscillatory motion.

Figure 4 Damped Oscillation z
< 1

(ii) Non oscillatory Motion (z >
1.0) Overdamped Case :

(27)

The motion is an exponentially
decreasing function of time as shown in Fig. 5.

Figure 5 Aperiodic Motion z >
1

(iii) Critically Damped Motion (z =
1.0) :

(28)

Figure 6 shows three types of response
with initial displacement x(0).

Figure 6 Critically Damped Motion
z = 1