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APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
PROBLEMS
9.1
For a water depth of d = 300 m, plot the value of wave number k as
a function of wave frequency w where 0.1 < w < 2 rad/s. Use the relationship
w2 = gk tanh kd. For what range of frequency is k given by
/ g to within 5
percent?
9.2 Calculate the damping matrix C for the fixed leg platform of Section
9.1. Assume
= Ç2 = 0-07. Indicate the units of C.
9.3 Dérivé in detail the resuit of équation (9.28), the mathematical mode!
for an N degree of freedom stalk structure subjected to a horizontal base excitation. Why is the négative sign on the right of équation (9.28) ignored in the
response solution of équation (9.30)?
9.4 For the gravity platform modeled in Section 9.2, calculate X and show
the units for each component vector x,.
9.5 Suppose that the gravity monotower of Section 9.2 is subjected to the
horizontal El Centro earthquake. Calculate the maximum values of v and 0,
the peak absolute value of horizontal displacement for the deck, the peak shear
force at the base, and the peak overturning moment. Assume that the modal
damping for each mode is 0.05.
9.6 Show that if one of the frequencies ut of the characteristic déterminant
of an undamped linear System is a complex number, then the system’s behavior
is divergent.
9.7 For an N degree of freedom linear structure, write a computer program
to calculate the coordinate displacement spectra, the corresponding variances,
and the rms values. Use the results of équations (9.80) and (9.81) where the
following quantifies are specified as input: S^w), X, G(p,w),
and u)*, k =
1,2,... ,1V. As a numerical example, check the results for S(£k,w) and
REFERENCES
Clough, R. W., and Penzien, J., Dynamics of Structures, second ed., McGraw-Hill,
New York, 1993.
Liapunov, A. M., Problème general de la stabilité du mouvement, translated into
French by E. Davaux, Annales de Toulouse (2), 9, 1907. Reprinted by Princeton
University Press, Princeton, NJ, 1949.
Mansour, A. E., and Millman, D. N., Dynamic Random Analysis of Fixed Offshore
Platforms, OTC-2049, Proceedings of the Offshore Technology Conférence, 1974.
Mathematica®, version 4, Wolfram Media, Inc., Champaign, IL, 1999.
Nataraja, R., and Kirk, C. L., Dynamic Response of a Gravity Platform Under
Random Wave Forces, OTC-2904, Proceedings of the Offshore Technology Conférence, 1977.
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
PROBLEMS
9.1
For a water depth of d = 300 m, plot the value of wave number k as
a function of wave frequency w where 0.1 < w < 2 rad/s. Use the relationship
w2 = gk tanh kd. For what range of frequency is k given by
/ g to within 5
percent?
9.2 Calculate the damping matrix C for the fixed leg platform of Section
9.1. Assume
= Ç2 = 0-07. Indicate the units of C.
9.3 Dérivé in detail the resuit of équation (9.28), the mathematical mode!
for an N degree of freedom stalk structure subjected to a horizontal base excitation. Why is the négative sign on the right of équation (9.28) ignored in the
response solution of équation (9.30)?
9.4 For the gravity platform modeled in Section 9.2, calculate X and show
the units for each component vector x,.
9.5 Suppose that the gravity monotower of Section 9.2 is subjected to the
horizontal El Centro earthquake. Calculate the maximum values of v and 0,
the peak absolute value of horizontal displacement for the deck, the peak shear
force at the base, and the peak overturning moment. Assume that the modal
damping for each mode is 0.05.
9.6 Show that if one of the frequencies ut of the characteristic déterminant
of an undamped linear System is a complex number, then the system’s behavior
is divergent.
9.7 For an N degree of freedom linear structure, write a computer program
to calculate the coordinate displacement spectra, the corresponding variances,
and the rms values. Use the results of équations (9.80) and (9.81) where the
following quantifies are specified as input: S^w), X, G(p,w),
and u)*, k =
1,2,... ,1V. As a numerical example, check the results for S(£k,w) and
REFERENCES
Clough, R. W., and Penzien, J., Dynamics of Structures, second ed., McGraw-Hill,
New York, 1993.
Liapunov, A. M., Problème general de la stabilité du mouvement, translated into
French by E. Davaux, Annales de Toulouse (2), 9, 1907. Reprinted by Princeton
University Press, Princeton, NJ, 1949.
Mansour, A. E., and Millman, D. N., Dynamic Random Analysis of Fixed Offshore
Platforms, OTC-2049, Proceedings of the Offshore Technology Conférence, 1974.
Mathematica®, version 4, Wolfram Media, Inc., Champaign, IL, 1999.
Nataraja, R., and Kirk, C. L., Dynamic Response of a Gravity Platform Under
Random Wave Forces, OTC-2904, Proceedings of the Offshore Technology Conférence, 1977.