3.3 Hierarchical Principles of Model Building
119
Fig. 3.23 Phase portrait with nonlinear elastic force
m
d
2 r
dt 2 = −kr + F(t)
where
F(t) = −ma(t) = −
md
2 f
dt 2
where F(t) is a given function of time. As in the case of a driving force, with a
corresponding periodic movement of the attachment point in the system, resonance
occurs.
With a more complex geometry, the inertia forces of the system can depend not
only on time t, but also on the coordinate r. If the spring is worn on a rod moving
with an angular velocity ω(t), then the centrifugal inertia is
F = mυ
2
(t)/R
where
υ(t) = ω(t)R
R = R 0 + r , R 0 —spring length in unloaded condition, r—deviation of the load
from the neutral position, r > −R 0 .
119
Fig. 3.23 Phase portrait with nonlinear elastic force
m
d
2 r
dt 2 = −kr + F(t)
where
F(t) = −ma(t) = −
md
2 f
dt 2
where F(t) is a given function of time. As in the case of a driving force, with a
corresponding periodic movement of the attachment point in the system, resonance
occurs.
With a more complex geometry, the inertia forces of the system can depend not
only on time t, but also on the coordinate r. If the spring is worn on a rod moving
with an angular velocity ω(t), then the centrifugal inertia is
F = mυ
2
(t)/R
where
υ(t) = ω(t)R
R = R 0 + r , R 0 —spring length in unloaded condition, r—deviation of the load
from the neutral position, r > −R 0 .