236
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
Equations (9.44) and (9.45) are rewritten in matrix form as
m 0
ÿ +
^C1
0
Jg
9
—h.GCi
v
9
Je 1
-hGfci
- moghG + mbghb)
(fcfl - moghG + Tnbghb)
v
e
/ [l-^o-Wp'
1
<9.171
Mpc + (fo) + hG}F
Frequencies and Mode Shapes
Suppose that the monotower is at rest in still water. Then, displaced by a
sudden wind gust that subsequently subsides, the tower undergoes free vibration. The tower’s motion will then be partly translations! and partly rotational,
with an exchange of energy between these modes. This should be expected since
the équations of motion (9.47) are coupled. The two characteristic free vibration frequencies and their corresponding mode shapes are now computed by the
methods in Chapter 8, Section 8.4. Let C = p — 0 in équations (9.47) and form
the characteristic déterminant
det(K - c2M) = 0
(9.48)
With the numerical parameters of Table 9.2 substituted for K and M in the
governing équations (9.47), this déterminant becomes
(3.59 - 0.114cq - 0.615c2) x 109 (0.0772 + 0.0107co) x 1012
= n
(-66.3 + 2.11c0) x 109
(3.60 - O.497co - 0.880c2) x 1012
(9.49)
Assume that the soil foundation stiffoesses k\ and kg are especially sensitive
to the structure’s fondamental frequency ci, which is the smallest positive root
that satisfies this déterminant. It is logical then to let co = uq. With this condition imposed, a trial and success procedure is used to extract the frequencies
from équation (9.49). As a first trial, choose co = 1.41 rad/s, the resuit obtained for this same structure under pure rocking motion. See Example Problem
5.2. The characteristic déterminant then reduces to
c4 - 8.867c2 + 11.08 = 0
(9-50)
l sing the quadratic formula, the smallest positive root of this polynomial is
calculated as ci = 1.23 rad/s. Next, choose cq — 1.24 rad/s, for which the
polynomial from équation (9.49), when solved for the lowest two positive roots,
yields the convergent results to three significant figures, or
C] = 1.24 rad/s (0.197 Hz);
c2 = 2.73 rad/s (0.434 Hz)
(9-51)
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
Equations (9.44) and (9.45) are rewritten in matrix form as
m 0
ÿ +
^C1
0
Jg
9
—h.GCi
v
9
Je 1
-hGfci
- moghG + mbghb)
(fcfl - moghG + Tnbghb)
v
e
/ [l-^o-Wp'
1
<9.171
Mpc + (fo) + hG}F
Frequencies and Mode Shapes
Suppose that the monotower is at rest in still water. Then, displaced by a
sudden wind gust that subsequently subsides, the tower undergoes free vibration. The tower’s motion will then be partly translations! and partly rotational,
with an exchange of energy between these modes. This should be expected since
the équations of motion (9.47) are coupled. The two characteristic free vibration frequencies and their corresponding mode shapes are now computed by the
methods in Chapter 8, Section 8.4. Let C = p — 0 in équations (9.47) and form
the characteristic déterminant
det(K - c2M) = 0
(9.48)
With the numerical parameters of Table 9.2 substituted for K and M in the
governing équations (9.47), this déterminant becomes
(3.59 - 0.114cq - 0.615c2) x 109 (0.0772 + 0.0107co) x 1012
= n
(-66.3 + 2.11c0) x 109
(3.60 - O.497co - 0.880c2) x 1012
(9.49)
Assume that the soil foundation stiffoesses k\ and kg are especially sensitive
to the structure’s fondamental frequency ci, which is the smallest positive root
that satisfies this déterminant. It is logical then to let co = uq. With this condition imposed, a trial and success procedure is used to extract the frequencies
from équation (9.49). As a first trial, choose co = 1.41 rad/s, the resuit obtained for this same structure under pure rocking motion. See Example Problem
5.2. The characteristic déterminant then reduces to
c4 - 8.867c2 + 11.08 = 0
(9-50)
l sing the quadratic formula, the smallest positive root of this polynomial is
calculated as ci = 1.23 rad/s. Next, choose cq — 1.24 rad/s, for which the
polynomial from équation (9.49), when solved for the lowest two positive roots,
yields the convergent results to three significant figures, or
C] = 1.24 rad/s (0.197 Hz);
c2 = 2.73 rad/s (0.434 Hz)
(9-51)