Geotechnical Engineering

DHARM

822 GEOTECHNICAL ENGINEERING

M

k c

t

z —— > —

c

2m

k

m

2

Overdamped

—— > — critically damped2mc mk

2

z

Z 1

t 1

t 2

z 2

t

(a) Mass spring

dashpot system

(b) Different damping conditions—

overdamped and critically

damped systems

(c) Underdamped system

Fig. 20.12 A mathematical model for free vibrations with damping

This can be rewritten as

&&z c.&

M

z

k

M

++z = 0 ...(Eq. 20.24)

putting α =

c

M

&&zz++αωn^2 .z = 0 ...(Eq. 20.25)

Let the solution to Eq. 20.25 be in the form

z = eλt ...(Eq. 20.26)

λ being a constant to be determined.

Substituting this in Eq. 20.25, we get

(λ^2 + αλ + ωn^2 )eλt = 0 ...(Eq. 20.27)

or λ^2 + αλ + ωn^2 = 0 ...(Eq. 20.28)

The roots of this equation are

λ 1 = – ααω

22

2

+ F^2

HG

I

KJ

− n ...(Eq. 20.29 (a))

λ 2 = – ααω

22

2

− F^2

HG

I

KJ

− n ...(Eq. 20.29 (b))

Three possible types of damping arise from these roots.

These are:

Case – 1 : Roots are real and negative if

α ω

2

2

F 2

HG

I

KJ

> n

or c

M

k

2 M

2

F

HG

I

KJ

>

The general solution is z = Ce 12 λλ^12 tt+Ce ...(Eq. 20.30)
Since both λ 1 and λ 2 are negative, z will decrease exponentially with time without any
change in sign as shown in Fig. 20.12 (b). The motion is not periodic and the system is said to
be overdamped.