118
SINGLE DEGREE OF FREEDOM STRUCTURES
For the second régime, 0.1 <
< 0.5 and the response is quasi-static, or v(f)
is amplified only a little above its static value. For the third régime, w/u>q —» 1,
the response is in résonance and the peak response v(t) is large compared to
its static value, especially for Ç near zéro. In the fourth régime, w/wq - oo
(> 10 for practical purposes) and the peak System response approaches zéro for
ail dumping ratios G The latter three response régimes for |
| as a fonction
of w/wq for several values of < are shown in Figure 5.4. The usual range for
structural damping is 0.02 < C, < 0.10.
For the case of structural excitation by a simple harmonie water wave, however, the response amplitude curves of Figure 5.4 become distorted and thus
should not be used directly. This distortion occurs because the wave loading
amplitude p0 is not independent of the wave frequency w. Recall from équation
(4.32) that po = H Go where H is the given wave height and Go is the transfer
fonction which is dépendent on w.
Impulse Response Function
The impulse response function h(t) is the solution to équation (5.58) for an
impulsive load such as a swift kick or a hammer blow. Define the impulsive load
as
p1(t) = C<5(t)
(5.67)
where C =11b- sec or 1 N-s. The Dirac delta function 6(t) is defined as zéro for
ail time except at t = 0; and in addition has the intégral property
4)'
/ b(t)dt = 1
(5.68)
Jotsing this loading, the impulse response function is derived as follows. Since
h(t) is a solution to équation (5.58) for pj(t) = 5(t), then
mh(t) +
+ fcih(t) = 5(t)
(5.69)
For t > 0, then 6(t) = 0 by définition, and the general solution to the homogeneous form of équation (5.69) is
h(t) = (C3 sin Udt + C4 cos üjdt)e_|,u'ot
(5.70)
where C3 and C4 are arbitrary constants. The damped frequency in tenus of
wo and ( of équations (5.64) is
“d = w0(l - <2)V2
(5.71)
One can verify by substitution using équations (5.64) and (5.71) that équation (5.70) is a solution to équation (5.69). The System is not displaced at
* ~ 80
=
With équation (5.70), then C4 = 0. Next h(0) is found by
differentlating équation (5.70) and setting t = 0. The resuit is
Â(0) = wdC3
(5.72)
SINGLE DEGREE OF FREEDOM STRUCTURES
For the second régime, 0.1 <
< 0.5 and the response is quasi-static, or v(f)
is amplified only a little above its static value. For the third régime, w/u>q —» 1,
the response is in résonance and the peak response v(t) is large compared to
its static value, especially for Ç near zéro. In the fourth régime, w/wq - oo
(> 10 for practical purposes) and the peak System response approaches zéro for
ail dumping ratios G The latter three response régimes for |
| as a fonction
of w/wq for several values of < are shown in Figure 5.4. The usual range for
structural damping is 0.02 < C, < 0.10.
For the case of structural excitation by a simple harmonie water wave, however, the response amplitude curves of Figure 5.4 become distorted and thus
should not be used directly. This distortion occurs because the wave loading
amplitude p0 is not independent of the wave frequency w. Recall from équation
(4.32) that po = H Go where H is the given wave height and Go is the transfer
fonction which is dépendent on w.
Impulse Response Function
The impulse response function h(t) is the solution to équation (5.58) for an
impulsive load such as a swift kick or a hammer blow. Define the impulsive load
as
p1(t) = C<5(t)
(5.67)
where C =11b- sec or 1 N-s. The Dirac delta function 6(t) is defined as zéro for
ail time except at t = 0; and in addition has the intégral property
4)'
/ b(t)dt = 1
(5.68)
Jotsing this loading, the impulse response function is derived as follows. Since
h(t) is a solution to équation (5.58) for pj(t) = 5(t), then
mh(t) +
+ fcih(t) = 5(t)
(5.69)
For t > 0, then 6(t) = 0 by définition, and the general solution to the homogeneous form of équation (5.69) is
h(t) = (C3 sin Udt + C4 cos üjdt)e_|,u'ot
(5.70)
where C3 and C4 are arbitrary constants. The damped frequency in tenus of
wo and ( of équations (5.64) is
“d = w0(l - <2)V2
(5.71)
One can verify by substitution using équations (5.64) and (5.71) that équation (5.70) is a solution to équation (5.69). The System is not displaced at
* ~ 80
=
With équation (5.70), then C4 = 0. Next h(0) is found by
differentlating équation (5.70) and setting t = 0. The resuit is
Â(0) = wdC3
(5.72)