'" ::t 2000 1500 2 50.0 1000 N 200.00 500 V ~ 150.00 "0 :l 0 C :';:! ~ 100.00 0.. -500.., cr ~ -1000 :r: 6: 50.00, 1 1-1 500 o. oo l---i--+-_--r--"<--e--t------t--~ ~ <D co co <D ~ Col co cr> >D 0 N N ~ 0) <!> co Col co '" 0) '" ~ C') " '" <!> '" 0 ~ '" 0 N '" co N ~ 0 co N N N C') M M 0 '" 0 0 '" 0 0 0 0 0 0 0 Time, S Time, S Fig. 2 - The amplitude ofthe vibration in under Fig. 3 - Naturalfrequency of4in beam overfu-st 53 damped 4in beam. oscillations. shah soon see, this is very nearly so. The periods of the first 53 oscillations shown in Fig. 2 were easily determined by reviewing the measured data saved in the Excel spreadsheet and noting the times at which the amplitude of the deflections were zero (or nea rly sol. The inverse of these measured periods of oscillation are plotted as frequencies in Fig. 3. IncreaSing slightly as the majàmum deflection decreased with time, the frequency of oscillation for the 4-in beam averaged 220Hz. Assuming that the modulus of elasticity for the 2024-T4 aluminum ahoy is 10.6 million psi, the value of the natural frequency calculated with Eq. 28 is 258Hz, sorne 17% higher than the average measured frequency. Standard references on vibrations offer a plethora of reasons why the measured value of the natura I frequency will inevitably be less than the Model 2000 Analog-to-Digital Converter connected to 2100 Systemfor measwing the bending strains in a cantilevered beam as it vibrates~ October 1994 Page 9

NOTEBOOK theoretical value. 1 Nevertheless, the usual suspects - damping, rotary motion, and shearing forces - have negligible effects on the natural frequency in the present case. And, while the cantilevered beam is simple enough to apply a much more complicated "classical solution" for ca1culating natural frequency, this "exact" method predicts a value of 254Hz; the reduction in the discrepancy between theoretical and actual values to about 15% is hardly worth the effort. Something else, obviously, is rotten in this particular state of Denmark. Repeating the experiment with longer lengths of the same beam produced different ratios of theoretical and measlited natw-a1 frequencies. The results, shown in Table l, suggest that the difference is significantly attenuated by lengthening the beam. Table 2 Spring Constants for CantUevered Beam ~ - Bearn k,lbf/in k,lbf/in Length, in (measured) (ca1culated) 4 58.844 80.872 5 31.840 41.406 6 19.869 23.962 7 12.830 15.090 8 8.749 10.109 9 6.165 7.100 10 4.639 5.176 10.75 3.670 4.166 11.25 3.269 3.635 Table 1 Free Vibration of Cantilevered Beam with Self Mass Only Bearn Theoretical Measured Frequency Length, Frequency, Frequency, Ratio in Hz Hz 4.0 258 220 1.17 5.0 161 141 1.14 11.25 32.5 30.0 1.08 The Reality of Theory AlI expressions for calculating natural frequencies are related to the spring constant, k, of the vibrating body. Readily calculated from theory by rearranging Eq. Il, it is equally easily determined by measurement; simply divide the force produced bya mass hung from a point on the beam by the resulting vertical deflection at that point. The results of such an investigation are given in Table 2 for the present case. These results suggest that smaller forces are required to produce unit deflections than might be expected from theoretical calculations. Although the exact mechanisms are not fully understood, the causes of this phenomenon are clearly related to compliance in the mounting fi.xtures at the fixed end of the beams. Boundary conditions satistying the theoretical calculations require that the beam be constrained from rotation at its root. This is possible only for an infinitely Iigid mount, and even the stiffest of materials have a fmite modulus of elasticity. Accordingly, the deflection of a cantilevered beam will always be greater than that expected from theoretical calculations requiring determinate end fixity. And, consequently, the natural frequency of a cantilevered beam must be lower than the theoretical value. The ratios of the natural frequencies. ri' calculated on the basis of theoretical and measured spring constants. are readily obtained by using Eq. 24: (31) ISee, for example, C.M. Harris and C.E. Crede, Shock and Vibration Handbook, (New York: McGraw-Hill, 1961). Page 10 October 1994