Biomedical Engineering Reference
In-Depth Information
The solution is
x
2
(t)
x
2
ω
˙
x
2
cos
ω(t
=
−
t
0
)
+
sin
ω(t
−
t
0
)
k
m
2
.
ω
t
+
sin
ω(t
−
τ)x
1
(τ ) dτ,
ω
=
(2.5)
t
0
From the viewpoint of analytical dynamics, the kinematic disturbance
prescribing the time histories of the motion for part of the system's compo-
nents is a nonstationary (time-dependent)
constraint
imposed on the motion
of the respective components. The imposition of constraints reduces the
number of degrees of freedom of the system and, hence, the number of
equations necessary for the complete determination of the motion of the
system. The system shown in Fig. 2.1 has two degrees of freedom in the
case of the dynamic disturbance and one degree of freedom in the case of
the kinematic disturbance. Using Eq. (2.1), one can calculate the force
F(t)
necessary to sustain the kinematic disturbance
x
1
=
x
1
(t)
of mass
m
1
,
F(t)
=
m
1
x
1
(t)
¨
−
k(x
2
(t)
−
x
1
(t)),
(2.6)
where
x
2
(t)
is the expression of Eq. (2.5). For a kinematic disturbance, the
force
F(t)
is a
constraint force
rather than an
applied force
,asisthecase
for a dynamic disturbance.
The classification of disturbances into dynamic and kinematic ones does
not mean that these disturbances differ in their physical nature. The external
disturbance is always caused by the interaction of the components of the
system under consideration with the environment and this interaction is
characterized by a force. However, in some cases, this force cannot be
effectively measured or modeled, while the displacement time history can
be easier to identify. This is the case, for example, for seismic disturbances.
The motion of the earth's surface can be measured by seismic sensors, while
the forces that cause this motion are not known with great accuracy. The
mass of the ground involved in the seismic motion considerably exceeds
the mass of any building or structure located on this ground, and hence
the motion of the structure relative to the ground does not influence the
motion of the ground. Therefore, to calculate the dynamic response of the
structure to the seismic excitation, it is reasonable to treat this excitation as
a kinematic disturbance.
In some cases, a kinematic disturbance is specified by the time history
of the velocity or acceleration of a component subjected to the excita-
tion, rather than by the time history of the displacement. Then, to deter-
mine the displacement of this component, it is necessary to integrate the
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