Problem 8. A horizontal rigid bar of mass
M
is supported by two viscoelastic elements, each modeled as a Kelvin-Voigt element (consisting of a spring and a damper in parallel), as shown in the figure (a). The bar has a moment of inertia
J
about its center of mass, denoted by
G
. An external force
F(t)
is applied vertically upward at the point
G
, and an external torque
T(t)
is applied about the same point. Due to the exerted force and the torque, the center of mass
G
of the bar experiences a vertical displacement
x(t)
and a rotation
\theta (t)
, as depicted in figure (b).
F_(A)
and
F_(B)
are the reaction forces due to the Kelvin-Voigt elements, and the actual displacements of the left and the right edges of the bar are denoted as
x_(A)
and
x_(B)
respectively. a. Using Newton's force and moment balance, find the equations of motion of the system depicted above. b. The following block diagram represents the input-output relationships of the system above. Find the expressions for
G_(x),G_(\theta )
, and the coupling term
C
. c. Find the necessary condition for the system to be decoupled, that is,
\theta
does not appear in the equation for
x
, or viceversa. Hint: Put
C=0
to find the condition for decoupling.