238
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
The modal vectors of the last two équations are shown in Figure 9.6. For
mode 1 corresponding to the lowest frequency, both components of
are positive, which indicates a positive or right displacement for v (the f. j, term) and
a positive or clockwise rotation for 6 (the £21 term), signs which are consistent
with the chosen coordinate directions. The broken line is also possible since
could hâve been chosen as (-1) instead of (+1), with the resuit that both
v and 0 would then be négative. The motions of mode 1 are in phase since
both components of i, hâve the same sign. The motions of mode 2 are out of
phase since the signs of the £2 components are always opposite. In free vibration, the actual motion is a combination of both mode shapes, which dépend
on the structure’s initial displacement and velocity. Once in motion, the mode
shapes continually change as potential and kinetic energies are transferred in
the foundation restraints.
With the frequencies and mode shapes in hand, the deterministic responses
of this monotower to given time historiés of loading, F(t) and Mpc(t), can be
calculated in a straightforward way using the normal mode solutions derived
in Chapter 8 and illustrated for the fixed leg platform at the beginning of this
chapter.
This particular problem gives some interesting insights into the frequency
behavior of multi-degree of freedom Systems. For instance, it is observed that
the fondamental undamped frequency is depressed by about 12 percent, from
1.41 rad/s for its single degree of freedom counterpart of Example Problem 5.2,
to
= 1.24 rad/s for the two degree of freedom analysis. This is characteristic
of linear Systems: as more flexibility is incorporated by allowing more degrees
of freedom, uq decreases. In either model, the effect of foundation damping
and viscous damping due to the surrounding water is to depress the undamped
frequency.
Nataraja and Kirk (1977) analyzed a similar structure. They calculated
a value of 1.02 rad/s for the fundamental frequency of a three-legged gravity
platform modeled to include leg flexibility and the same type of soil foundation
elasticity (fc) and kg) as used here. Three factors in the Nataraja and Kirk
model account for their lower value of uq: their structural mass was somewhat
higher, their model included soil foundation damping, and their model had
added degrees of freedom because of leg flexibility.
Dynamic Stability
There is a vast literature on criteria and methods for determining the dynamic stability of coupled, linear Systems such as the gravity platform. The
< lassical works of Liapunov (1907) and Ziegler (1956) are especially noteworthy.
For présent purposes, however, the criteria are relatively simple. The dynamic
stability of a linear, undamped structure in free vibration about its static equilibrium position can be tested by investigating the nature of the roots - 85
calculated from the characteristic déterminant, équation (9.3). If every
is
real and positive, then the characteristic frequencies given by .<■, - y,'»7
rea^
and positive, and the System undergoes stable, bounded, harmonie oscillations.
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
The modal vectors of the last two équations are shown in Figure 9.6. For
mode 1 corresponding to the lowest frequency, both components of
are positive, which indicates a positive or right displacement for v (the f. j, term) and
a positive or clockwise rotation for 6 (the £21 term), signs which are consistent
with the chosen coordinate directions. The broken line is also possible since
could hâve been chosen as (-1) instead of (+1), with the resuit that both
v and 0 would then be négative. The motions of mode 1 are in phase since
both components of i, hâve the same sign. The motions of mode 2 are out of
phase since the signs of the £2 components are always opposite. In free vibration, the actual motion is a combination of both mode shapes, which dépend
on the structure’s initial displacement and velocity. Once in motion, the mode
shapes continually change as potential and kinetic energies are transferred in
the foundation restraints.
With the frequencies and mode shapes in hand, the deterministic responses
of this monotower to given time historiés of loading, F(t) and Mpc(t), can be
calculated in a straightforward way using the normal mode solutions derived
in Chapter 8 and illustrated for the fixed leg platform at the beginning of this
chapter.
This particular problem gives some interesting insights into the frequency
behavior of multi-degree of freedom Systems. For instance, it is observed that
the fondamental undamped frequency is depressed by about 12 percent, from
1.41 rad/s for its single degree of freedom counterpart of Example Problem 5.2,
to
= 1.24 rad/s for the two degree of freedom analysis. This is characteristic
of linear Systems: as more flexibility is incorporated by allowing more degrees
of freedom, uq decreases. In either model, the effect of foundation damping
and viscous damping due to the surrounding water is to depress the undamped
frequency.
Nataraja and Kirk (1977) analyzed a similar structure. They calculated
a value of 1.02 rad/s for the fundamental frequency of a three-legged gravity
platform modeled to include leg flexibility and the same type of soil foundation
elasticity (fc) and kg) as used here. Three factors in the Nataraja and Kirk
model account for their lower value of uq: their structural mass was somewhat
higher, their model included soil foundation damping, and their model had
added degrees of freedom because of leg flexibility.
Dynamic Stability
There is a vast literature on criteria and methods for determining the dynamic stability of coupled, linear Systems such as the gravity platform. The
< lassical works of Liapunov (1907) and Ziegler (1956) are especially noteworthy.
For présent purposes, however, the criteria are relatively simple. The dynamic
stability of a linear, undamped structure in free vibration about its static equilibrium position can be tested by investigating the nature of the roots - 85
calculated from the characteristic déterminant, équation (9.3). If every
is
real and positive, then the characteristic frequencies given by .<■, - y,'»7
rea^
and positive, and the System undergoes stable, bounded, harmonie oscillations.