Question

The system in Fig.consists of three bars and a beam (with negligible masses) and a block (mass m). Determine the circular frequency of the free vertical vibrations.

Answer 1

The system consisting of the truss and the beam is equivalent to a system consisting of two springs in parallel (both springs undergo the same elongation when the block is displaced). To determine the spring constant kB of the beam, we subject the beam to the force FB = 1 which acts at the location of the block. This force produces the deflection

Thus, the spring constant is given by

In order to find the spring constant kT of the truss, we apply the force FT = 1 at bar ①. This force causes the displacement

With

we obtain

Now, we replace the two springs in parallel by an equivalent single spring. Its spring constant k∗ is given by

Thus, the eigenfrequency is