Rods, beams, plates, and so on are continuous systems, which have an infinite number of degrees of freedom and an infinite number of modes. For simplicity, assume that a cantilever beam is approximated as a single-degree-of-freedom mass–damper–spring system, for which the natural frequency is close to the first mode of the beam. The parameters of the cantilever beam are length = 0.5 m, width = 0.025 m, thickness = 0.005 m, density ρ = 7850 kg/m3, and Young’s modulus = 210 × 109 N/m2.

a. It is known that the equivalent mass for the beam is meq = m/3, where is the actual mass of the beam. Determine the equivalent stiffness keq for the beam. Calculate the natural frequency of the equivalent single-degree-of-freedom system.

b. Figure 9.29 is the measured frequency response of the cantilever beam for the first mode. Determine the natural frequency and the damping ratio based on the given information in the plot.

c. Compare the frequencies obtained in Parts (a) and (b). What is the error if the cantilever beam is approximated as a single-degree-of-freedom system?