3.1.3. Study of the spectrum Figure 4: Influence of the damping y on the energy. B. G. Jean-Marc, Y. S. A. Joresse, T. Augustin, J. Nonlinear Sci. Appl., 16 (2023), 123-144 Figure 6: Tip angle comparison. Figure 9: Asymptote of the spectrum for y = 1. Figure 10: Asymptote of the spectrum for y = 5. In Figures 9 and 10, we plot the spectrum with respect to the above asymptote. We plot the spectrum for y = 1, we observe the asymptote at the spectral abscissa —0.5. We have the representation of the spectrum for y = 5 with respect to the asymptote at the spectral abscissa —2.5. Figure 7: Influence of the variation of the damping y on the spectrum. B. G. Jean-Marc, Y. S. A. Joresse, T. Augustin, J. Nonlinear Sci. Appl., 16 (2023), 123-144 B. G. Jean-Marc, Y. S. A. Joresse, T. Augustin, J. Nonlinear Sci. Appl., 16 (2023), 123-144 Figure 15: Tip position comparison. Figure 16: Tip angle comparison. Figure 14: Influence of the damping y on the energy. Figure 17: Representation of spectrum for y(x) = (1+ x)? Now, by implementing the spectrum matrix L, we represent the spectrum associated with the sys- tem for different values of y. On the Figures 17, 18, and 19, we have represented a distribution of 4N eigenvalues for y(x) = (1+ x)}*, y(x) = (14+ x)’, and y(x) = (1+x)4, respectively. We observe that when m = EI = 1, the location of spectrum moves rapidly on the left-hand side of the complex plane. Figure 19: Representation of spectrum, when y(x) = (1+ x)*. Figure 18: Representation of spectrum, when y(x) = (1+ x). Figure 23: Displacement of the whole beam y(x, t). Figure 21: Asymptote of the spectrum for y = (1+ x). B. G. Jean-Marc, Y. S. A. Joresse, T. Augustin, J. Nonlinear Sci. Appl., 16 (2023), 123-144 Figure 25: Displacement of the whole beam y(x, t). Figure 24: Displacement of the whole beam y(x, t). Figure 22: Asymptote of the spectrum for y = (1+ x4. Figure 26: Influence of the damping y on the energy. Figure 29: Representation of spectrum for y = 5. On the Figures 29 and 30, we have represented a distribution of 4N eigenvalues for y = 5 and y = 10. Figure 27: Displacement of the free end. Figure 30: Representation of spectrum for y = 10. B. G. Jean-Marc, Y. S. A. Joresse, T. Augustin, J. Nonlinear Sci. Appl., 16 (2023), 123-144 Figure 32: Asymptote of the spectrum for y = 10. Figure 36: Influence of the damping y on the energy. Figure 34: Deflection y(x, t). Figure 39: Representation of spectrum for y(x) = (1+ x). Figure 41: Representation of spectrum for y(x) = (1+ x4. Figure 42: Asymptote of the spectrum for y(x) = (1+ x). Figure 40: Representation of spectrum for y(x) = (1+ x). Figure 38: Tip angle comparison. Figure 43: Asymptote of the spectrum for y(x) = (1+ x). Figure 44: Asymptote of the spectrum for y(x) = (1+ x4. First, we plot the spectrum for y(x) = (1+ x)* (see Figure 42) and y(x) = (1+ x)> (see Figure 43). Values are seen in the area to the right of the asymptote. Finally, we represent the asymptote for y(x) = (1+ x)*. We observe no point to the left of it. It is important to note in this case that the value of the abscissa of the asymptote is the same as that found for y = 1. This is justified by the choice of the parameters EI(x),m(x), and y(x). We can see that with these values a simplification of the expressions occurs in (3.1). This leads to the case where y = 1, hence the observation of scattered eigenvalues as in Figure 7.
