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

Problems for Chapter 5 109that the bar displaces a distance δ, and then is suddenly released, demonstratethat the amplitudes of the subsequent longitudinal vibrations are given byA n = 8δn 2 π sin nπ 2 2Find these amplitudes for this steel bar with a cross-sectional area of5.0(10) −5 m 2 , under the effect of a force of 4500 N.3. A steel bar is free to move at x = 0. It has 0.0002 m 2 cross-sectional area and0.35 m length. A 0.20 kg load is placed at x = 0.35 m.(a) Determine the fundamental frequency of the longitudinal vibration of themass-loaded bar.(b) Establish the position at which the bar may be clamped so as to minimizeinterference with the fundamental mode.(c) Find the ratio of the displacement amplitude of the free end to that of themass-loaded end for the first overtone of the bar.4. A steel bar having a mass of 0.05 kg and 0.25 m length is loaded at one endwith 0.028 kg and 0.056 kg at the other end. Find the fundamental frequencyof the system’s longitudinal vibration and determine the location of the nodein the bar. Also compute the ratio of the displacement amplitudes at the twoends of the bar.5. Redo Problem 4 for an aluminum bar of the same length but with a mass of0.03 kg, subjected to the same end loadings.6. Consider a thin bar of length L and mass M that is rigidly fixed at one endand free at the other. What mass m must be affixed to the free end in order tolower the fundamental frequency of longitudinal vibration by 30% from thefixed-free value?7. A fixed-free bar of length L and mass m has a mechanical reactance, or stiffness,equal to −is/ω in the fixture. Develop an expression for the fundamentalfrequency of the longitudinal vibration.8. A longitudinal force F cos ωt drives a long thin bar at x = 0. The bar is freeto move at x = L.(a) Obtain the equation for the amplitude of the standing waves occurring inthe bar.(b) Obtain the expression giving the input mechanical impedance of the barof length L.(c) Derive the expression for the input mechanical impedance of the same barhaving an infinite length.(d) For the case of part (a) plot the amplitude of the driven end of the baras a function of frequency over the range 100 Hz–3 kHz, if the materialof the bar is aluminum, its length is 1.5 m, its cross-sectional area is1.5 (10) −4 m 2 , and the amplitude of the driving force is 12 N.9. Demonstrate that the dimensional units of v = √ κωc are those of speed. Atwhat frequency will the phase speed of transverse vibrations of a steel rod