Problem 3 (34%) Fig. 3 shows a mechanical system. A block is placed
on a frictionless surface and pushed by a plate through a spring and a
damper. The displacements of the block and plate are $x(t)$ and $p(t)$.
1) With a clearly labelled free-body diagram, derive the constitutive
equation of motions for all the elements
2) Derive the transfer function $P(s)/X(s)$ where $P(s)$ and $X(s)$ are the
Laplace transfer of the displacement $p(t)$ and $x(t)$ respectively.
3) Assume the displacement of the plate is sinusoidal with $p(t) = 4sin (2t)$, $m = 1kg$, $k=9.1 N/m$, $b = 3.8$
Ns/m.
a) Please determine the steady state of the displacement $x(t)$ and the force transmitted to the block.
b) Assume the damper is removed from the system ($b=0$ Ns/m). To eliminate the vibration of the block,
please design a dynamic vibration absorber by adding one degree freedom. You need to provide
schematics of the design, constitutive equation of motions for the design, the parameters of all the
mechanical elements in the design and the motion of the additional degree freedom.
