1) A cantilevered beam (below)  is 5 feet long with a cross-section height (h) of 1.5 in. and a depth (b) of 0.75 in. If the beam is made of steel, estimate the natural frequency for transverse (bending) vibration of this beam.

 

 

2) The cantilevered beam shown above is a 3 feet long hollow, round cross-section with an outside diameter of 1.25 in. and a wall thickness of 0.125 in. If the beam is made of aluminum, estimate the natural frequency for axial (longitudinal) vibration of this beam.

 

3) The cantilevered beam shown above is a 3 feet long hollow, round cross-section with an outside diameter of 1.25 in. and a wall thickness of 0.125 in. If the beam is made of aluminum, estimate the natural frequency for torsional vibration of this beam.

 

 

4) The simply-supported beam shown here is 10 m. long with a hollow square cross-section 10 cm. outside width and 10 mm. wall thickness. If the beam is made of steel, estimate the natural frequency for transverse (bending) vibration of this beam.  Use SI units to solve this problem !

Use the tables for Uniform Beams in the Appendix of "Mechanical Vibrations" by J.P. DenHartog, Dover, 1985, on pages 431-432 to solve the four problems above.  

Make a table to compare these cases of "distributed" mass and stiffness to your simplified, single DOF solutions (HW 6).