REFERENCES
221
8.3 Suppose that the two frequencies in free vibration for the structural
model of Problem 8.1 are computed as wj and w2 and that the modal damping
factors are estimated from experimental data as s, = 0.05 and (2 = 0.07. With
these damping factors, compute the values of the Rayleigh constants ai and a2
in terms of the two frequencies.
8.4 Suppose that the jacket template structure shown in Figure 8.3a is
modeled as a four degree of freedom damped System with the independent coordinates (vi,v2, #i, 02). Here, the first two coordinates are defined in Figure 8.3b;
the angles
and #2 are the rotations of the deck and the mass m2; and J and
J2 are the mass moments of inertia for the deck and for m2. respectively. There
are three loads only: the respective horizontal wave loading pi and p? at uj and
v2, and a wave slamming moment Md on the deck.
(a) Write down the loading vector and the damping matrix.
(b) Identify the éléments of the mass matrix. Include the virtual mass where
appropriate.
(c) Sketch the four diagrams, analogous to the three diagrams of Figure 8.4.
that define the éléments of the stiffness matrix.
(d) For each of the two lumped masses, construct a free body sketch, analogous to the sketches in Figure 8.3c.
(e) Use Newton’s method to dérivé the four équations of motion for this
structural model.
8.5 For the structure described in Problem 8.4, formulate the expressions
for the kinetic energy, the potential energy, and the virtual work of the nonconservative forces. With these three scalar quantities, use Lagrange’s method to
dérivé the four équations of motion for the structure.
REFERENCES
Caughey, T. K., and O’Kelly, M. E. J., Classical Normal Modes in Damped Linear
Dynamic Systems, Journal of Applied Mechanics 32, 1965.
Chopra, A. K., Dynamics of Structures: Theory and Applications to Earthquake
Engineering, second ed., Prentice Hall, Lpper Saddle River, NJ, 2001.
Clough, R.W., and Penzien, J., Dynamics of Structures, second ed., McGraw-Hill,
New York, 1993.
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.
PSl-Plot, version 6, Poly Software International, Sandy. DT, 1999.
Rayleigh, Lord, Theory of Sound 1, Dover, New York, 1945.
SAP 2000, Integrated Structural Analysis and Design Software, Computers and Structures, Inc., Berkeley, CA, 1997.
221
8.3 Suppose that the two frequencies in free vibration for the structural
model of Problem 8.1 are computed as wj and w2 and that the modal damping
factors are estimated from experimental data as s, = 0.05 and (2 = 0.07. With
these damping factors, compute the values of the Rayleigh constants ai and a2
in terms of the two frequencies.
8.4 Suppose that the jacket template structure shown in Figure 8.3a is
modeled as a four degree of freedom damped System with the independent coordinates (vi,v2, #i, 02). Here, the first two coordinates are defined in Figure 8.3b;
the angles
and #2 are the rotations of the deck and the mass m2; and J and
J2 are the mass moments of inertia for the deck and for m2. respectively. There
are three loads only: the respective horizontal wave loading pi and p? at uj and
v2, and a wave slamming moment Md on the deck.
(a) Write down the loading vector and the damping matrix.
(b) Identify the éléments of the mass matrix. Include the virtual mass where
appropriate.
(c) Sketch the four diagrams, analogous to the three diagrams of Figure 8.4.
that define the éléments of the stiffness matrix.
(d) For each of the two lumped masses, construct a free body sketch, analogous to the sketches in Figure 8.3c.
(e) Use Newton’s method to dérivé the four équations of motion for this
structural model.
8.5 For the structure described in Problem 8.4, formulate the expressions
for the kinetic energy, the potential energy, and the virtual work of the nonconservative forces. With these three scalar quantities, use Lagrange’s method to
dérivé the four équations of motion for the structure.
REFERENCES
Caughey, T. K., and O’Kelly, M. E. J., Classical Normal Modes in Damped Linear
Dynamic Systems, Journal of Applied Mechanics 32, 1965.
Chopra, A. K., Dynamics of Structures: Theory and Applications to Earthquake
Engineering, second ed., Prentice Hall, Lpper Saddle River, NJ, 2001.
Clough, R.W., and Penzien, J., Dynamics of Structures, second ed., McGraw-Hill,
New York, 1993.
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.
PSl-Plot, version 6, Poly Software International, Sandy. DT, 1999.
Rayleigh, Lord, Theory of Sound 1, Dover, New York, 1945.
SAP 2000, Integrated Structural Analysis and Design Software, Computers and Structures, Inc., Berkeley, CA, 1997.