NATURAL FREQUENCIES OF LINEAR SYSTEMS
103
side sin# — 0, f(9) — 0, and q(0) — kg0 (the soil foundation reaction moment).
Thus, équation (2.6) reduces to
JqÛ + (ke - TnQgh,G + mbghb)0 = 0
(5.8)
According to équation (2.78), the soil’s restoring moment coefficient is kg =
aQ - ubo, where a0 and bo are soil constants. Here, a> =
Thus, the latter
équation of motion becomes
JqO + (ao — a>obo — moghG + mbghb)0 = 0
(5.9)
Assume a harmonie solution to équation (5.9), or
9 — 0q sin ujQt ; 0 = — cl>o sin ll>q£
(5.10)
where 9q is a constant. When 9 and 6 from équation (5.10) are substituted into
équation (5.9), the resuit is a quadratic équation in ü?o, which can be solved for
the positive root cjq using the quadratic formula. The resuit is
= ryOo + 4Jo(ûo - moghG 4- mbghb)]x/2 - 60}
(5-H)
ZJf)
Figure 5.2 Model of a monopod gravity platform with typical dimensions.