PROBLEMS
141
the figure, other System parameters are summarized in Table 5.2, and the équations governing its soil foundation stiffness and damping are given by équations
(2.76) and (2.77).
(a) Calculate an upper and lower bound for To, the undamped period of free
horizontal vibration.
(b)
Deduce a damping factor < for the upper and lower bound values of To.
(c) Assume horizontal ground excitation by the El Centro earthquake whose
pseudovelocity response is given by Figure 5.8. Based on the numerical values
of Tq and Ç just calculated, estimate the bounds for the peak values of horizontal displacement, the horizontal shear load on the soil foundation, and the
overturning moment. You may smooth the pseudovelocity curves to estimate
these bounds.
(d) The preceding response calculations do not include the damping effects of
the surrounding water. Deduce whether such additional damping would increase
or decrease the System responses to this same earthquake excitation.
5.12 Use first order perturbation theory to dérivé the amplitude function
g(ü) given by équation (5.85). Using this resuit, deduce équation (5.86).
5.13 Include linear, viscous damping in équation (5.82) and use first order
perturbation theory to dérivé an algebraic resuit analogous to équation (5.86).
Then predict the peak response amplitude for the two curves of Figure 5.10, for
( = 0.05.
5.14 Predict the excitation frequency w =
below which there will be
no amplitude jumps in the System modeled by équation (5.82). Do this in two
ways: first by differentiating équation (5.88) to find a minimum fi; and then by
applying the rule of roots where R = 0 in équation (5.91).
5.15 Use first order perturbation theory to dérivé the one-third subharmonic relationship given by équation (5.95).
5.16 Dérivé équation (5.97) from (5.95). Then show that the frequency
ratio fim above which a one-third subharmonic response can exist is given by
the lowest real root of équation (5.98).
5.17 For the nonlinear System modeled by équation (5.82), the minimum
frequency ratio fim below which a one-third subharmonic response cannot exist is predicted from équation (5.98) or approximately from équation (5.100).
Which of these relationships predicts the lower value of !*..,? Choose some
realistic numerical values of Ks to validate your answer.
5.18
Assume a solution to the linear équation (5.105) of the form
0 = 0oe~‘ 'ÜJ°t cos wjt
where A<|; is a constant and
is given by équation (5.71). Lse this solution to
dérivé the following approximate expression for the damping ratio Ç :
.
1 1
£1
s “ 2tt bl
141
the figure, other System parameters are summarized in Table 5.2, and the équations governing its soil foundation stiffness and damping are given by équations
(2.76) and (2.77).
(a) Calculate an upper and lower bound for To, the undamped period of free
horizontal vibration.
(b)
Deduce a damping factor < for the upper and lower bound values of To.
(c) Assume horizontal ground excitation by the El Centro earthquake whose
pseudovelocity response is given by Figure 5.8. Based on the numerical values
of Tq and Ç just calculated, estimate the bounds for the peak values of horizontal displacement, the horizontal shear load on the soil foundation, and the
overturning moment. You may smooth the pseudovelocity curves to estimate
these bounds.
(d) The preceding response calculations do not include the damping effects of
the surrounding water. Deduce whether such additional damping would increase
or decrease the System responses to this same earthquake excitation.
5.12 Use first order perturbation theory to dérivé the amplitude function
g(ü) given by équation (5.85). Using this resuit, deduce équation (5.86).
5.13 Include linear, viscous damping in équation (5.82) and use first order
perturbation theory to dérivé an algebraic resuit analogous to équation (5.86).
Then predict the peak response amplitude for the two curves of Figure 5.10, for
( = 0.05.
5.14 Predict the excitation frequency w =
below which there will be
no amplitude jumps in the System modeled by équation (5.82). Do this in two
ways: first by differentiating équation (5.88) to find a minimum fi; and then by
applying the rule of roots where R = 0 in équation (5.91).
5.15 Use first order perturbation theory to dérivé the one-third subharmonic relationship given by équation (5.95).
5.16 Dérivé équation (5.97) from (5.95). Then show that the frequency
ratio fim above which a one-third subharmonic response can exist is given by
the lowest real root of équation (5.98).
5.17 For the nonlinear System modeled by équation (5.82), the minimum
frequency ratio fim below which a one-third subharmonic response cannot exist is predicted from équation (5.98) or approximately from équation (5.100).
Which of these relationships predicts the lower value of !*..,? Choose some
realistic numerical values of Ks to validate your answer.
5.18
Assume a solution to the linear équation (5.105) of the form
0 = 0oe~‘ 'ÜJ°t cos wjt
where A<|; is a constant and
is given by équation (5.71). Lse this solution to
dérivé the following approximate expression for the damping ratio Ç :
.
1 1
£1
s “ 2tt bl