THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

94 5. Vibrating BarsThe allowable frequencies for a free–free bar are the same as those for the barfixed at both ends ( fixed–fixed bar). There will be, however, major differences inthe respective wave patterns of the free–free bar and the fixed–fixed bar. RecastingEquation (5.15) by making use of the relation2 cos kx = e −ikL + e ikLwe can express the complex displacements corresponding to the nth mode ofvibration asξ = 2 A n e iω nt cos k n tThe real part of the preceding equation gives the tangible vibrations described byξ n = cos k n x(A n cos ω n t + B n sin ω n t) (5.16)By comparing Equation (5.13) for the fixed–fixed bar with Equation (5.14) for thefree–free bar, it will be seen that antinodes exist at the end points for the latterbar in contrast with the nodes that must exist at the end points of the fixed–fixedbar. The nodal patterns for both bars are shown in Figure 5.3. It is of interest toobserve that when an antinode exists at the center of the bar, the vibrations areFigure 5.3. Typical standing waves for the first three modes of vibration in a fixed–fixedbar and in a free–free bar.