REFERENCES
287
10.10
Consider a uniform cable under constant tension, fixed at each end.
and subjected to an arbitrary transverse load per unit length of q(x,t). Use
the normal mode method to predict the transverse cable displacement v(x,t),
assumed in the form of équation (10.82). Follow the same procedure used to
obtain the beam responses to the same loading, which leads to the results given
by équations (10.88) through (10.91). Show that Xn(x), n = 1,2,... for the
cable are orthogonal functions.
10.11 Refer to a text such as Clough and Penzien (1993) or Timoshenko
and Goodier (1951) to dérivé the équation of longitudinal vibrations for a uniform, elastic pipeline. Let u = u(x,t 'j be the longitudinal displacement of a
material point at position x on its longitudinal axis. For a pipeline without
end constraints, deduce that du/dx = 0 on each end. Then, following the same
method used in Section 10.1 to dérivé the free transverse vibration frequencies
for a fixed end cable, to calculate the free longitudinal vibration frequencies for
the unconstrained pipeline. Check that your analytical resuit for
agréés with
équation (10.93).
10.12 Based on the équation of longitudinal vibrations for the pipeline
derived in Problem 10.11, calculate the free vibration frequencies for a uniform
pipeline clamped at each end. Compare your value of uq to u; of équation
(10.93), and discuss briefly the reasons for the différence in the two results.
10.13 The horizontal cross brace modeled in Figure 10.4c is located at z =
Il in water of depth d. The beam is subjected to transverse wave excitation that
is modeled by linear wave theory. For S^(u), assume that the Pierson-Moskowitz
wave spectrum applies, where the significant wave height is 10 m. Choose the
transfer function (r'(u') given by équation (4.30). See Example Problem 10.1 for
the explicit forms of frequencies and mode shapes. Write a computer program
to calculate the midspan values of the displacement response spectrum, the
variance of v and the rms value of v. Specify carefully ail input data. Test the
program by choosing a realistic numerical example. If
is Gaussian, how
are these results interpreted in terms of expected peak displacements?
10.14 For the horizontal beam of Figure 10.4b, dérivé an expression for
the spectral density of the bending moment in terms of the spectral density of
its displacement given by équation (10.120). Where along its length would the
peak rms value of the bending moment occur?
REFERENCES
Barr, R. A., and Johnson, V. E., Evaluation of Analytical and Experimental Methods
for Determining OTEC Plant Dynamics and CWP Loads, Proceedings of the
Offshore Technology Conférence, 1979.
Borgman, L. E., Océan Wave Simulation for Engineering Design. Journal of the
Waterways and Harbors Division, ASCE 95 (4), November 1969.
Camah
an, B., Luther, H. A., and Wilkes, J. O., Applied Numerical Methods, Vol. 2,
Wiley, New York, 1964.
287
10.10
Consider a uniform cable under constant tension, fixed at each end.
and subjected to an arbitrary transverse load per unit length of q(x,t). Use
the normal mode method to predict the transverse cable displacement v(x,t),
assumed in the form of équation (10.82). Follow the same procedure used to
obtain the beam responses to the same loading, which leads to the results given
by équations (10.88) through (10.91). Show that Xn(x), n = 1,2,... for the
cable are orthogonal functions.
10.11 Refer to a text such as Clough and Penzien (1993) or Timoshenko
and Goodier (1951) to dérivé the équation of longitudinal vibrations for a uniform, elastic pipeline. Let u = u(x,t 'j be the longitudinal displacement of a
material point at position x on its longitudinal axis. For a pipeline without
end constraints, deduce that du/dx = 0 on each end. Then, following the same
method used in Section 10.1 to dérivé the free transverse vibration frequencies
for a fixed end cable, to calculate the free longitudinal vibration frequencies for
the unconstrained pipeline. Check that your analytical resuit for
agréés with
équation (10.93).
10.12 Based on the équation of longitudinal vibrations for the pipeline
derived in Problem 10.11, calculate the free vibration frequencies for a uniform
pipeline clamped at each end. Compare your value of uq to u; of équation
(10.93), and discuss briefly the reasons for the différence in the two results.
10.13 The horizontal cross brace modeled in Figure 10.4c is located at z =
Il in water of depth d. The beam is subjected to transverse wave excitation that
is modeled by linear wave theory. For S^(u), assume that the Pierson-Moskowitz
wave spectrum applies, where the significant wave height is 10 m. Choose the
transfer function (r'(u') given by équation (4.30). See Example Problem 10.1 for
the explicit forms of frequencies and mode shapes. Write a computer program
to calculate the midspan values of the displacement response spectrum, the
variance of v and the rms value of v. Specify carefully ail input data. Test the
program by choosing a realistic numerical example. If
is Gaussian, how
are these results interpreted in terms of expected peak displacements?
10.14 For the horizontal beam of Figure 10.4b, dérivé an expression for
the spectral density of the bending moment in terms of the spectral density of
its displacement given by équation (10.120). Where along its length would the
peak rms value of the bending moment occur?
REFERENCES
Barr, R. A., and Johnson, V. E., Evaluation of Analytical and Experimental Methods
for Determining OTEC Plant Dynamics and CWP Loads, Proceedings of the
Offshore Technology Conférence, 1979.
Borgman, L. E., Océan Wave Simulation for Engineering Design. Journal of the
Waterways and Harbors Division, ASCE 95 (4), November 1969.
Camah
an, B., Luther, H. A., and Wilkes, J. O., Applied Numerical Methods, Vol. 2,
Wiley, New York, 1964.