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(Solved): Problem 8. A horizontal rigid bar of mass M is supported by two viscoelastic elements, each modeled ...



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.



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