such as natural frequencies, buckling loads, and static equilibriums – provided the
frequency of excitation X is high, and not in sharp external or parametric resonance
with the system. Next we illustrate application of the GNRP and the GARP in the
form of two simple examples.
7.6.3 Example 1: Clamped String with HF Base
Excitation
Fig. 7.8 shows the system: a clamped-free string of mass per unit length qA and
finite bending rigidity EI, subjected to gravity g and time-harmonic horizontal base
excitation of velocity amplitude a and high frequency X. The displacement
amplitude X
−1 a is small, while the acceleration Xa is strong. The linearized partial
differential equation governing small transverse motions u(x, t), and the boundary
conditions, are:
u tt þ cu t þ x
2
0 l
4 u xxxx À ðl À xÞu x
ð
Þ x Xa sinðXtÞ ¼ Àg;
ð7:123Þ
uð0; tÞ ¼ u x ð0; tÞ ¼ u xx ðl; tÞ ¼ u xxx ðl; tÞ ¼ 0;
ð7:124Þ
where subscripts x and t denote partial derivatives with respect to space and time,
c is the viscous damping coefficient, l the string length, and x 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EI=ðqAl 4 Þ
p
is a
characteristic frequency.
Equation (7.123) has the form (7.102), with N = 1, L c = c, L 0 = x 0
2 l
4 ( ) xxxx ,
L 11 ¼ Àa ðl À xÞð Þ x
À
Á
x
; m 1 = 1, f 0 = −g, w 1 = f 1 = 0, and boundary conditions of
the required type. To predict the response under far-from-resonance conditions we
employ the GNRP, and use (7.109)–(7.110) to readily produce an approximation
for the equation of slow motions:
u 0tt þ cu 0t þ x
2
0 l
4 u 0xxxx þ
1
2
a
2
ðl À xÞ ðl À xÞu 0x
ð
Þ xx
À
Á
x
¼ Àg;
ð7:125Þ
which can be rearranged into the following form:
u 0tt þ cu 0t þ
x
2
0 ðxÞl
4 u 0xx
À
Á
xx
¼ Àg;
ð7:126Þ
where
x
2
0 ðxÞ x
2
0 þ
1
2
ða=lÞ
2 1 À ðx=lÞ
ð
Þ
2 :
ð7:127Þ
Here
x
2
0 ðxÞ is seen to represent a distribution of effective bending stiffness (per
unit mass) along the string. It is composed of a first term, x
2
0 , describing the
structural or “real” stiffness of the string, and another one equivalencing the average
432
7 Special Effects of High-Frequency Excitation
frequency of excitation X is high, and not in sharp external or parametric resonance
with the system. Next we illustrate application of the GNRP and the GARP in the
form of two simple examples.
7.6.3 Example 1: Clamped String with HF Base
Excitation
Fig. 7.8 shows the system: a clamped-free string of mass per unit length qA and
finite bending rigidity EI, subjected to gravity g and time-harmonic horizontal base
excitation of velocity amplitude a and high frequency X. The displacement
amplitude X
−1 a is small, while the acceleration Xa is strong. The linearized partial
differential equation governing small transverse motions u(x, t), and the boundary
conditions, are:
u tt þ cu t þ x
2
0 l
4 u xxxx À ðl À xÞu x
ð
Þ x Xa sinðXtÞ ¼ Àg;
ð7:123Þ
uð0; tÞ ¼ u x ð0; tÞ ¼ u xx ðl; tÞ ¼ u xxx ðl; tÞ ¼ 0;
ð7:124Þ
where subscripts x and t denote partial derivatives with respect to space and time,
c is the viscous damping coefficient, l the string length, and x 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EI=ðqAl 4 Þ
p
is a
characteristic frequency.
Equation (7.123) has the form (7.102), with N = 1, L c = c, L 0 = x 0
2 l
4 ( ) xxxx ,
L 11 ¼ Àa ðl À xÞð Þ x
À
Á
x
; m 1 = 1, f 0 = −g, w 1 = f 1 = 0, and boundary conditions of
the required type. To predict the response under far-from-resonance conditions we
employ the GNRP, and use (7.109)–(7.110) to readily produce an approximation
for the equation of slow motions:
u 0tt þ cu 0t þ x
2
0 l
4 u 0xxxx þ
1
2
a
2
ðl À xÞ ðl À xÞu 0x
ð
Þ xx
À
Á
x
¼ Àg;
ð7:125Þ
which can be rearranged into the following form:
u 0tt þ cu 0t þ
x
2
0 ðxÞl
4 u 0xx
À
Á
xx
¼ Àg;
ð7:126Þ
where
x
2
0 ðxÞ x
2
0 þ
1
2
ða=lÞ
2 1 À ðx=lÞ
ð
Þ
2 :
ð7:127Þ
Here
x
2
0 ðxÞ is seen to represent a distribution of effective bending stiffness (per
unit mass) along the string. It is composed of a first term, x
2
0 , describing the
structural or “real” stiffness of the string, and another one equivalencing the average
432
7 Special Effects of High-Frequency Excitation