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

5.8 Boundary Conditions for Transverse Vibrations 105Applying free-end condition Equation (5.36) at x = L yields the following twosets of equations:(A cosh ωL ) (ωL+ cos =−B sinh ωL )ωL+ sinv vv v(A sinh ωLv− sin ωL ) (=−B cosh ωLvv+ cos ωL )vBoth of the preceding two equations cannot hold true for all frequencies. In order todetermine the permissible frequencies, one equation is divided into the other, thuscanceling out the constants A and B. Ridding the resulting equation of fractionalexpressions by cross-multiplication and using the identities cos 2 θ + sin 2 θ = 1and cosh 2 θ + 1 = sinh 2 θ, we obtaincosh ωLvcosωLv =−1We can alter the last equation by using the identitiestan θ √1 − cos θ2 = 1 + cos θ , tanh θ √cosh θ − 12 = cosh θ + 1and we now obtaincot ωL2v=±tanhωL2v(5.37)The frequencies which correspond to the allowable modes of vibration can befound through the use of a microcomputer program which determines the intersectionsof the curves of cot ωL/2v and ± tanh ωL/2v, as shown in Figure 5.8.The frequencies of the permissible modes are given byωL2v = ζ π (5.38)4where ζ = 1.194, 2.988, 5, 7,... with ζ approaching whole numbers for thehigher allowed frequencies. Inserting v = (κωc) 1/2 into Equation (5.38), squaringboth sides, and solving for frequencies f , we obtainf = ζ πκc8L 2The constraint imposed by the boundary conditions leads to a set of discreteallowable frequencies, but the overtone frequencies are not harmonics of thefundamental. When a metal bar is struck in such a manner that the amplitudes ofthe vibration of some of the overtones are fairly strong, the sound produced hasa metallic cast. But these overtones rapidly die out, and the initial sound soonevolves into a mellower pure tone whose frequency is the fundamental. This isa characteristic of the behavior of a tuning fork that emits a short metallic soundupon being struck before emitting a pure tone.The distribution of the nodal points along the transversely vibrating bar is quitecomplex, with three distinct types of nodal points being identified mathematically.