Geotechnical Engineering

DHARM

864 GEOTECHNICAL ENGINEERING

Example 20.3: Determine the natural frequency of a machine foundation of base area 2m ×

2m and weight 150 kN, assuming that the soil mass participating in the vibration is 20% of the

weight of foundation. Take Cu = 36,000 kN/m^3.

Weight of foundation = 150 kN

Weight of soil mass participating in the vibration = 20% of 150 kN = 30 kN

Total weight = 150 + 30 = 180 kN

fn =

1

2 π

CA

M

u.

Substituting Cu = 36,000 kN/m^3 , A = 4 m^2 ,

and M =

180

981.

kN sec^2 /m,

fn =

1

2

36 000 4 1

π 180 9 81

,

(/.)

××

cps

= 14.1 Hz

Example 20.4: The exciting force in a constant force-amplitude excitation is 90 kN. The natu-
ral frequency of the machine foundation is 3 Hz. The damping factor is 0.30. Determine the
magnification factor and the transmitted force at an operating frequency of 6 Hz.

Frequency ratio, ξ =

f

fn

=

6

3 = 2, Damping factor, D = 0.30

Magnification factor, η 1 =

1

() 14 −+ξξ^22 D2 2

=

1

()14 4 03 2−+×^222 (.)×

= 0.31

Transmissibility, T =

()

()

14

14

22

22 2 2

+

−+

D

D

ξ

ξξ

= η 1 14 03 2+×(.)^22 ×

= 0.31 × 244. = 0.484

∴ Transmitted force = T × (Exciting force) = 0.484 × 90 kN = 43.56 kN

Example 20.5: In a block test according to IS: 5249–1977 (Revised), a resonant frequency of

18 cps was observed in vertical vibrations. Determine the coefficient of elastic uniform

compression.

A machine weighing 90 kN is to be supported on a block of size 3 m × 4 m × 2 m high.

Determine its natural frequency in vertical vibration.

According to IS:5249–1977 (Revised),

Size of model block = 1.50 m × 0.75 m × 0.70 m high.

Weight of model block = 1.5 × 0.75 × 0.70 × 24.0 kN = 18.9 kN

Mass of block =

18 9

981

.

. kN.sec

(^2) /m = 1.927 kN sec (^2) /m
Natural frequency, fn = resonant frequency = 18 cps (Hz)
Area of test block, At = 1.5 × 0.75 = 1.125 m^2