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Consider a cylindrical shell coupled to an annular plate with the latter subjected to an out-of-plane incident wave with a circumferential dependency of cos (0) propagating radially towards the cylinder/plate interface. It can be shown from the asymptotic expansion of the Bessel functions (see, for example, reference [14]) that the wave amplitude is inversely proportional to the square root of the radius. Since the energy is proportional

Figure 1 Consider a cylindrical shell coupled to an annular plate with the latter subjected to an out-of-plane incident wave with a circumferential dependency of cos (0) propagating radially towards the cylinder/plate interface. It can be shown from the asymptotic expansion of the Bessel functions (see, for example, reference [14]) that the wave amplitude is inversely proportional to the square root of the radius. Since the energy is proportional

Related Figures (10)

Figure 2. Coupling loss factor for different cylinder/source plate thicknesses. ——-, cylinder/ —~—-~, plate/plate.
Figure 2. Coupling loss factor for different cylinder/source plate thicknesses. ——-, cylinder/ —~—-~, plate/plate.
Figure 3. Cylinder and plate elements. All dimensions in mm.
Figure 3. Cylinder and plate elements. All dimensions in mm.
Figure 4. Experimental setup for steady state power balance measurement.
Figure 4. Experimental setup for steady state power balance measurement.
Figure 5. Typical decay record for cylindrical shell. Third octave band at a band centre frequency of 1-25 kHz.
Figure 5. Typical decay record for cylindrical shell. Third octave band at a band centre frequency of 1-25 kHz.
e 6. Internal loss factor of cylinder. ——, steady state power balance method; ——-—, reverberation decay 1e. ler. ——, steady state power balance method; ——-—, reverberation decay Figure 6. Internal loss factor of cylinder. ——, stead >chnique. where E;; is the energy of element 7 due to an input excitation at element /.
e 6. Internal loss factor of cylinder. ——, steady state power balance method; ——-—, reverberation decay 1e. ler. ——, steady state power balance method; ——-—, reverberation decay Figure 6. Internal loss factor of cylinder. ——, stead >chnique. where E;; is the energy of element 7 due to an input excitation at element /.
Figure 7. Internal loss factor of plate. Key as Figure 6.
Figure 7. Internal loss factor of plate. Key as Figure 6.
Figure 8. Energy ratio of the coupled cylinder/plate structure. Following the least square procedure, the sum of the squared errors may be minimised with respect to the internal loss factors and CLFs:
Figure 8. Energy ratio of the coupled cylinder/plate structure. Following the least square procedure, the sum of the squared errors may be minimised with respect to the internal loss factors and CLFs:
Figure 10. Internal loss factor of plate. Key as Figure 9. Figures 9 and 10 show the internal loss factor of the cylinder and plate respectively. T esults given by the im situ method (inversion of matrix) are in close agreement with t teady state power balance method. This finding is consistent with earlier work reports y Bies and Hamid [3] for the case of flat plates and supports the present experiment pproach in determining the CLF. It also suggests that the coupled modal energies of t lements are approximately equal to the uncoupled modal energies. Furthermore, t velding process has little effect on the damping of the structural elements. Figure 11 shov he CLFs of the cylinder/plate structure obtained by equation (8) and measurement. T xperimental results are fairly well predicted by the theory presented in section 2 Ithough the theoretical values are slightly higher than the measured values in t requency range of 800-2500 Hz. The dip in CLF which is predicted in the theoretic nalysis can be observed in the experimental data. It occurs at a frequency of arour 500 Hz compared with a predicted value of 3730 Hz which corresponds to the rir requency of the cylinder. The experimental data also shows some discrepancy with tl redicted CLF above a frequency of 6300 Hz. An attempt to conduct further tests (abo 000 Hz) to confirm the convergence of the experimental results to the theoretical CL f a plate/plate structure was hampered by the limitation in sampling rate of the da cquisition system. However, further examination of the results reveals that tl
Figure 10. Internal loss factor of plate. Key as Figure 9. Figures 9 and 10 show the internal loss factor of the cylinder and plate respectively. T esults given by the im situ method (inversion of matrix) are in close agreement with t teady state power balance method. This finding is consistent with earlier work reports y Bies and Hamid [3] for the case of flat plates and supports the present experiment pproach in determining the CLF. It also suggests that the coupled modal energies of t lements are approximately equal to the uncoupled modal energies. Furthermore, t velding process has little effect on the damping of the structural elements. Figure 11 shov he CLFs of the cylinder/plate structure obtained by equation (8) and measurement. T xperimental results are fairly well predicted by the theory presented in section 2 Ithough the theoretical values are slightly higher than the measured values in t requency range of 800-2500 Hz. The dip in CLF which is predicted in the theoretic nalysis can be observed in the experimental data. It occurs at a frequency of arour 500 Hz compared with a predicted value of 3730 Hz which corresponds to the rir requency of the cylinder. The experimental data also shows some discrepancy with tl redicted CLF above a frequency of 6300 Hz. An attempt to conduct further tests (abo 000 Hz) to confirm the convergence of the experimental results to the theoretical CL f a plate/plate structure was hampered by the limitation in sampling rate of the da cquisition system. However, further examination of the results reveals that tl
discrepancy is consistent with previous work on the experimental investigation of CLF (see, for example, references [3] and [4]) and may be partially attributed to the random nature of the experiment as well as the assumptions involved in the derivation of CLF (for example, the equipartition of energy amongst circumferential modes). Overall, the calculated CLF is considered to be satisfactory as an SEA parameter for the estimation of response levels in a cylinder/plate structure.
discrepancy is consistent with previous work on the experimental investigation of CLF (see, for example, references [3] and [4]) and may be partially attributed to the random nature of the experiment as well as the assumptions involved in the derivation of CLF (for example, the equipartition of energy amongst circumferential modes). Overall, the calculated CLF is considered to be satisfactory as an SEA parameter for the estimation of response levels in a cylinder/plate structure.
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EngineeringSound and VibrationStatistical Energy Analysis
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