Theory of Longitudinal Vibrations of Viscous Rods

Abstract

Theory of Longitudinal Vibrations $\delta f$ Thin Rods, Taking into Account Damping.—Starting with the general differential equation of wave-motion in one dimension including a viscosity term, expressions are derived for the wave-velocity and logarithmic decrement in the case of free vibrations of the rod. The velocity is practically the same as for undamped vibrations, while the decrement is proportional to the viscosity and to the frequency. The equation is also solved for the case of forced vibrations due to two simple harmonic forces at the ends, equal in amplitude but opposite in phase. If the damping is small and frequency, $\frac{\omega }{2\pi }$, is near the resonance frequency, $\frac{{\omega }_0}{2\pi }$, the expression for the displacement of the end of the rod is very simple: $\xi =\left(\frac{4X_{0}l}{\pi G\delta }\right) cos \theta sin (\omega t-\theta )$, where $tan\theta =\frac{-2\mathrm{}\pi ({\omega }_0-\omega )}{{\omega }_0\delta }$, $\delta$ is the logarithmic decrement per period, $l$ the length of the rod, $X_0$ the maximum value of the periodic stress, and $G$ Young's modulus. It is shown that this expression may also be obtained by reducing the rod to an equivalent system possessing one degree of freedom.