140
SINGLE DEGREE OF FREEDOM STRUCTURES
new position B'. Examples of this shift are shown in Figures 2.13. Discuss
qualitatively how this offset can affect the rocking frequency of the structure.
5.7
For the gravity platform of Exemple Problem 5.2, suppose that the
ballast density is changed so that the center of mass and center of buoyancy
arp coincident. Discuss qualitatively how this coincidence can affect both the
rocking frequency and the dynamic stability of this structure.
5.8
Suppose that the only motion for the gravity platform shown in Figure
5.2 is horizontal sliding, v = v(t).
(a) From the free body sketch of this structure, dérivé the équation of motion
for free, undamped horizontal sliding. Model the soil foundation stiffness fci
according to équation (2.76).
(b) Assuming harmonie motion for the horizontal displacement in the form
v = vq sin coot, dérivé an explicit expression for wq similar in form to équation
(5.11).
(c) Based on the soil and structural parameters given in Table 5.1, calculate
cjq and its corresponding period Tq for this platform. Compare your frequencies
for both the lower bound and upper bound values for Gs given in Table 5.1
with the respective natural frequencies for rocking motion computed in Example
Problem 5.2.
(d) If both rocking and sliding motion were taking place simultaneously,
which of these two vibration modes do you think would dominate the structure’s
free vibrations? Explain.
5.9
Your task is to design, construct, and perform free vibration experiments on a desk-top model of a three-legged or four-legged jackup rig. The legs
should be securely fixed to a solid base, and can be either hinged or fixed at the
deck level.
(a) Design your model so that its two latéral periods of free vibration Tq, one
in air and one in water, are sufficiently long to be measured accurately with a
stop watch. Use any theoretical method you wish to predict To for both cases.
(b) Construct your design.
(c) Perform free vibration experiments for the model in air and then with its
legs mostly submerged in water. Measure Tq in both cases as an average value
over several oscillations.
(d) Compare your measured periods for the two types of experiments, and
compare each to the theoretical values you calculated previously. Explain reasons for any discrepancies between your measured and predicted results.
5.10 Based on the numerical values given in Example Problem 5.5 and
the perturbation solution given by équation (5.56), plot the free vibration displacement u( t ) as a function of â>t for a ship in sway motion. Superimpose on
this plot the harmonie displacement for its linear System counterpart (fca =®)’
Discuss the distortion in the free vibration oscillations due to nonlinear cable
restraints.
5.11
Dérivé the differential équation for the horizontal sliding motion only
ne gravity platform shown in Figure 5.2. The structural geometry is given in
SINGLE DEGREE OF FREEDOM STRUCTURES
new position B'. Examples of this shift are shown in Figures 2.13. Discuss
qualitatively how this offset can affect the rocking frequency of the structure.
5.7
For the gravity platform of Exemple Problem 5.2, suppose that the
ballast density is changed so that the center of mass and center of buoyancy
arp coincident. Discuss qualitatively how this coincidence can affect both the
rocking frequency and the dynamic stability of this structure.
5.8
Suppose that the only motion for the gravity platform shown in Figure
5.2 is horizontal sliding, v = v(t).
(a) From the free body sketch of this structure, dérivé the équation of motion
for free, undamped horizontal sliding. Model the soil foundation stiffness fci
according to équation (2.76).
(b) Assuming harmonie motion for the horizontal displacement in the form
v = vq sin coot, dérivé an explicit expression for wq similar in form to équation
(5.11).
(c) Based on the soil and structural parameters given in Table 5.1, calculate
cjq and its corresponding period Tq for this platform. Compare your frequencies
for both the lower bound and upper bound values for Gs given in Table 5.1
with the respective natural frequencies for rocking motion computed in Example
Problem 5.2.
(d) If both rocking and sliding motion were taking place simultaneously,
which of these two vibration modes do you think would dominate the structure’s
free vibrations? Explain.
5.9
Your task is to design, construct, and perform free vibration experiments on a desk-top model of a three-legged or four-legged jackup rig. The legs
should be securely fixed to a solid base, and can be either hinged or fixed at the
deck level.
(a) Design your model so that its two latéral periods of free vibration Tq, one
in air and one in water, are sufficiently long to be measured accurately with a
stop watch. Use any theoretical method you wish to predict To for both cases.
(b) Construct your design.
(c) Perform free vibration experiments for the model in air and then with its
legs mostly submerged in water. Measure Tq in both cases as an average value
over several oscillations.
(d) Compare your measured periods for the two types of experiments, and
compare each to the theoretical values you calculated previously. Explain reasons for any discrepancies between your measured and predicted results.
5.10 Based on the numerical values given in Example Problem 5.5 and
the perturbation solution given by équation (5.56), plot the free vibration displacement u( t ) as a function of â>t for a ship in sway motion. Superimpose on
this plot the harmonie displacement for its linear System counterpart (fca =®)’
Discuss the distortion in the free vibration oscillations due to nonlinear cable
restraints.
5.11
Dérivé the differential équation for the horizontal sliding motion only
ne gravity platform shown in Figure 5.2. The structural geometry is given in