1. Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell by Marco Amabili Philosophical Transactions A Volume 371(1993):20120474 June 28, 2013 ©2013 by The Royal Society

2. Circular cylindrical shell and cylindrical coordinate system. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society

3. First natural frequency of modes with a different number n of circumferential waves of the 0/30° laminated circular cylindrical shell; calculations with model M=4; mode n=1 is torsional. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society

4. Mode shapes for the radial displacement w of the 0/30° laminated circular cylindrical shell; calculations with model M=4. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society

5. Frequency–response curve of the 0/30° laminated circular cylindrical shell (n=3) obtained with the natural mode reduced-order model (thick lines) versus the results presented in Amabili [25] with model M=2 (37 degrees of freedom) shown in thin lines. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society

6. Frequency–response curve of the 0/30° laminated circular cylindrical shell obtained by using the present model (natural mode reduced-order model, thick lines) versus the results presented in Amabili [25] with models M=2 and M=4 shown in thin lines; n=3. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society

7. Frequency–response curve of the 0/30° laminated circular cylindrical shell (n=3) obtained with the natural mode reduced-order model. Marco Amabili Phil. Trans. R. Soc. A 2013;371:20120474 ©2013 by The Royal Society