Problem 7.3 A simple model for the instantaneous penetration depth u(t) during
vibrational piling (Fig. P7.3) is:
€ u þ cusgnð _
uÞ ¼ 1 þ qX
2 sinðXtÞ;
ð7:140Þ
where all variables and parameters are nondimensional, X ) 1, q ( 1, qX = O(1),
and:
c ¼
lEpd
2
Mg
; u ¼
U
l
; t ¼ x ~ t; x
2
¼
g
l
; X ¼
~
X
x
;
ð7:141Þ
where M is the pile mass and d its diameter, E is the elastic modulus of the
surrounding medium, l is the coefficient of kinetic friction between pile and
medium.
(a) Set up an equation governing the slow (i.e. average) component of u(t).
(b) Simplify this equation for the case of a relatively small average piling speed.
(c) Derive and discuss an expression for the vibrational force acting on the pile (i.e.
the static force equivalencing the average effect of the fast vibrations).
(d) Modify (7.140) to account for the case where the coefficient of kinetic friction
is larger for motions directed inwards, i.e. l = l + for _
u > 0, l = l − for _
u < 0,
where l + > l − > 0, and repeat Questions a–c for that case.
Fig. P7.2
444
7 Special Effects of High-Frequency Excitation