Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 823

Case – 2 : Roots are equal if α ω

2

2

F 2

HG

I

KJ

= n or

c

M

k

2 M

F^2

HG

I

KJ

=

The general solution is

z = eCCt

c

− 2 M.t() 12 + ...(Eq. 20.31)

This is similar to the overdamped case except that it is possible for the sign to change

once as shown in Fig. 20.12 (b). This is also not a periodic motion and with increase in time,

approaches zero. The value of c for this condition is called the ‘critical damping coefficient’, cc.

Since

c

M

k

M

c

2

F^2

HG

I

KJ

=

cc = 2 kM ...(Eq. 20.32)

Using Eq. 20.13, we may write

cc = 2Mωn ...(Eq. 20.33)

cc is the limiting value for c for the motion to be periodic.

Case – 3 : Roots are complex congugates if

α ω

2

2

F 2

HG

I

KJ

< n

or

c

M

k

2 M

F^2

HG

I

KJ <

By using Eq. 20.32, the roots λ 1 and λ 2 become

λ 1 = ωn()−+ −Di^1 D^2 ...(Eq. 20.34 (a))

λ 2 = ωn()−− −Di 1 D^2 ...(Eq. 20.34 (b))

where D =

c

cc

and is called ‘Damping Ratio’ or ‘Damping Factor’.

Substituting these into Eq. 20.30 and simplifying, the general solution becomes

z = eC tDC tD−ωnDtFH 3 sinωωnn^11 −+^24 cos −^2 KI ...(Eq. 20.35)

where C 3 and C 4 are arbitrary constants.

Eq. 20.35 indicates that the motion is periodic and the decay in amplitude will be pro-

portional to e−ωnDt as shown by the dashed curve in Fig. 20.12 (c). Further Eq. 20.35 indicates

that the frequency of free vibrations with damping is less than the natural frequency for

undamped free vibrations, and that as D → 1, the frequency approaches zero. The relation

between these two frequencies is given by

ωdn = ωn 1 −D^2 ...(Eq. 20.36)
where ωdn = frequency of free vibrations with damping. Fig. 20.12c shows that there is a decre-
ment in the successive peak amplitudes. Using Eq. 20.35, ratios of successive peak amplitudes
may be found.

Let z 1 and z 2 be the amplitudes of successive peaks at times t 1 and t 2 , respectively as

shown in Fig. 20.12 c.