Answer Happy

Problem 1: Use Table 6.2 to calculate the first three longitudinal natural frequencies of a 0.5 meter long steel bar with a diameter of 0.01 meters for the free-free, fixed-free, and fixed-fixed configurations. Compare and discuss the results. Use Excel to plot the first mode shape of each configuration. Compare and discuss the results.
TABLE 6.2 VARIOUS CONFIGURATIONS OF A UNIFORM BAR OF LENGTH / IN LONGITUDINAL VIBRATION ILLUSTRATING THE NATURAL FREQUENCIES AND MODE SHAPESa Frequency (rad/s) or characteristic equation Configuration Mode shape NTC wn = -, n = 0,1,2,... nπxb COS I Free-free X wn (2n-1) mc 21 Sin (2n-1) mx 21 Fixed-free X NTC NTTX sin Fixed-fixed X 쭈 Fixed-spring m XX Anc sin 쏙 W₁ = Fixed-mass "Note that the last two examples require a numerical procedure (suggested in the Toolbox problems at the end of the chapter) in order to calculate the values of the natural frequen- cies. Here c = VE/p. The free mode shape is a constant. Tuine = λμ cot an Wn = cot an = n = 1,2,... - (KA) EA - An m 2/ , n = 1,2,... РА sin