Problem $13$

A mass $m$ is attached to the end of a weightless bar of length $L$ that is hinged at $O_1$ and subjected to the harmonic load shown. Assume that:

The pendulum rotates in the vertical plane.

Only small rotations are considered

Damping is neglected.

$L=0\mathrm{.}8m,m=0\mathrm{.}4kg,$ and $p_0=2N.$

Please respond to the following questions:

d$.$ Derive the equations of motion for the pendulum in terms of the angular displacement ${}_1$ of the pendulum away from the vertical equilibrium position.

$P()-r_{0}sin(w_{f}t)$

e$.$ Determine the amplitude of the steady$-$state forced$-$vibration response when the harmonic force is applied at a frequency ${}_f=5ra\frac{d}{s}.$

Now, to reduce its oscillation of the pendulum in Problem #$1,$ another identical pendulum, hinged at $O_2,$ is connected to the original one by a linear spring of spring constant $k$ at a distance of a from the hinge on the ceiling. Assume that:

The two pendulums are hinged at the same height and rotate in the same vertical plane.

The spring is weightless and has its neutral length when the two pendulums are at their vertical positions.

Only small deflections are considered and damping is neglected.

Assume $L=0\mathrm{.}8m,a=0\mathrm{.}4m,m=0\mathrm{.}4kg,$ and $p_0=2N.$

Please respond to the following questions:

f$.$ Use Lagrange's equations to derive the equations of motion for the double$-$pendulum system in terms of the angular displacements ${}_1$ and ${}_2$ of the two pendulums away from their vertical equilibrium position.

g$.$ Determine the natural frequencies and the mode shapes of the system, assuming that $k=4{10}^3\frac{N}{m}.$

h$.$ Discuss the modal characteristics for extremely large and extremely small values of $k.$

i$.$ Determine the amplitude of the steady$-$state forced$-$vibration response of the first pendulum when the harmonic force is applied at a frequency ${}_f=5ra\frac{d}{s}.$ Has the addition of the second pendulum been successful in reducing the response of the first?