Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 867

Example 20.9: Obtain the velocity of the tup of a forge hammer before impact and that of the
anvil after impact, given the following:
weight of the tup and die = 15.50 kN
stroke of tup = 900 mm
weight of anvil and frame = 342 kN
efficiency of drop = 0.9
coefficient of restitution = 0.5
steam pressure = 700 kN/m^2
area of piston = 0.129 m^2
Determine the amplitudes of vibration of the foundation and anvil if the limiting natu-
ral frequency of the anvil is 253 rad/s, and the two natural frequencies of the combined system
are 272 rad/s and 54.3 rad/s.

Since stream pressure is given, this is a double-acting hammer.

The correction factor, α = 0.9 (the efficiency of drop).

Velocity of the tup before impact

v = α

2 gW pah

W

t

t

(.)+

Substituting α = 0.9, Wt = 15.50 kN, p = 700 kN/m^2 , a = 0.129 m^2 , and h = 900 mm
= 0.9 m,

We have

v = 09

2 9 81 15 50 700 0 129 0 9

15 50

.

.(.. ).

.

×+××

m/s

= 9.88 m/s

Velocity of anvil after impact

va =

()

()

1

1

+

+

e

v

λa

where λa =

M

M

W

W

a

t

a

t

==

342

15 5.

= 22.06

Substituting e = 0.5, λa = 22.06, and v = 9.88 m/s,

we have

va =

(.)

(.)

.

105

12206

988

+

+

× = 0.643 m/s

Amplitude of the vibration of the foundation

A 1 = –

()()

()

ωωωω

ωω ω ω

anan

an n n

va

2

2

22

1

2

2

1

2

2

2

2

−−

−

Substituting ωa = 253 rad/s, ωn 1 = 272 rad/s, ωn 2 = 54.3 rad/s,

and va = 0.643 m/s,

A 1 = –

(.)( )

(.).

.

253 54 3 253 272

253 272 54 3 54 3

0 643

2222

22 3

−−

−

× m = 1.58 mm