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Figure 7.10

If you bend the beam in addition to size y and then let
loose, the load begins to falter under the elastic response of the
rod relative to the equilibrium position O, which it held at a
static deformation of the beam. Since the position of the center of
gravity load is determined by only one coordinate system y , it
will have one degree of freedom.
dy

Let an arbitrary point in time t the speed y  and
dt
d 2 y
 
acceleration y  load is directed toward the positive
dt 2
direction of the axis Oy . Then the mass with such forces: the
force is directed downward gravity Q , inertial force is directed
upward P  m y  and directed upward the force of the elastic
reaction beam P , proportional according to Hooke's law, the
пр
full load displacement  y :
P   c   y  Q  cy .
пр
Using the D'Alembert principle, we can write the equation
0
of equilibrium of forces Y  (fig. 7.10 б):
i
Q  P  P  0 ,
пр
where
Q  ym  Q  cy  0,
or
ym   cy  0.
(7.31)
The resulting equation is a differential equation of free
oscillations of the system with one degree of freedom.
Introducing the notation
c
2
  , (7.32)
m 0
the equation (7.31) as
 
y  2 y  0 . (7.33)
0

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