Geotechnical Engineering

DHARM

824 GEOTECHNICAL ENGINEERING

z

z

D

D

1

2 2

2

1

=

−

F

H

GG

I

K

exp JJ

π

...(Eq. 20.37)

‘Logarithmic Decrement’ is defined as

δ = ln

z

z

D

D

1

2 2

2

1

=

−

π

...(Eq. 20.38)

In words, logarithmic decrement is defined as the natural logarithm of the ratio of any
two successive amplitudes of same sign in the decay curve obtained in free vibration with
damping.
δ is approximately 2πD, when D is small. Eq. 20.38 also indicates that, in viscous damp-
ing, the ratio of amplitudes of any successive peaks is a constant. It follows that the logarith-
mic decrement may be obtained from any two peak amplitudes z 1 and z1+n from the equation

δ =

(^11)
n 1
z
z n
ln

• ...(Eq. 20.39)
  20.2.9 Forced Vibrations with Damping
  A system which undergoes forced vibrations, and in which viscous damping is present, may be
  analysed by the Mass-spring-dashpot model shown in Fig. 20.13.
  k c
  P sin to w
  Fig. 20.13 Forced vibration with damping
  The equation of motion for this system may be written as follows:-
  Mz cz kz.&&++.&. = Po sin ωt ...(Eq. 20.40)
  This may be rewritten as
  &&z c.& .sin
  M
  z k
  M
  z
  P
  M
  ++= ωo t ...(Eq. 20.41)
  or &&&zz z sin
  P
  M
  ++n^2 =αωo t ω ...(Eq. 20.42)
  where α =
  c
  M
  and ωn^2 =
  k
  M
  .
  The particular solution is a steady state harmonic oscillation having a frequency equal
  to that of the excitation, and the displacement vector lags the force vector by some angle. Let
  us therefore assume that the particular solution is
  z = A sin (ωt – φ) ...(Eq. 20.43)