1 þ 2a sin
2
ðpyÞ
À
Á € u þ c 1 þ 2pa_ y sinð2pyÞ
ð
Þ _
u
þ 1 þ q A sinðXsÞ þ lu
2
À
Á
u ¼ 0;
ð4:106Þ
€ y þ c 2 _
y þ f r ðyÞ ¼ À
1
2
p sinð2pyÞ€ uu:
ð4:107Þ
Averaged System Following the approach of Sect. 4.7.3, a Van der Pol
transformation (u, _
u) ! (a sinw,
1
2 aX cosw) with a = a(s), w = w(s) =
1
2 Xs + h(s)
with condition _
a sinw + a _
h cosw 0 turns (4.106) into a pair of first-order
equations in a(s) and h(s). With (X
2
−4, a, c 1 , q A , a) ( 1, all right-hand terms of
these equations will be small, implying that a(s) and h(s) are slowly varying as
compared to the rapidly changing phase w(s). Averaging the right-hand sides over
w 2 [0, 2p] the averaged string equations become:
1 þ 2a sin
2
ðpyÞ
À
Á X _
a ¼ À
1
2
c 1 þ 2pa_ y sinð2pyÞ
ð
Þ Xa
À
1
2
q A cosð2hÞa;
1 þ 2a sin
2
ðpyÞ
À
Á Xa _
h ¼ À
1
4
X
2
À 4 þ 2aX
2 sin
2
ðpyÞ
À
Á
a
þ
1
2
q A sinð2hÞa þ
3
4
la
3
:
ð4:108Þ
As for the point mass equation (4.106), employing the Van der Pol transformation and subsequently averaging out rapidly varying terms (c 2 , f r , a, _
a ( 1)
yields:
0
0.16
0.6
0.8
1
1.2
1.4
String amplitude
a
Excitation frequency Ω ( )
Excitation frequency Ω ( )
(a)
r s =0
-2E-4 +2E-4
-2E-3
+2E-3
0
0.16
0.6
0.8
1
1.2
1.4
String amplitude
a
(b)
r s =0
-2E-4
+2E-4
-2E-3
+2E-3
Fig. 4.19 Swept frequency responses a(X(s)) of base excited string – effect of sweep rate r s . (a)
Point mass fixed at y = y 0 ; (b) Point mass sliding. Solid/dashed line: stationary amplitudes (r s = 0)
according to (4.102); dotted line: numerical integration of averaged system (4.99)–(4.100)
with X(s) given by (4.105). Non-zero parameters: a = 0.3, w A = 0.01, c 1 = 0.1, c 2 = 0.35,
f r (y) = j(y – y 0 ), j = 0.03, y 0 = 0.1, X 0 = 0.5, X 1 = 1.5, s 0 = 100
4.7 String with a Sliding Point Mass
255