224
7 Mathematical Models of Functionally Graded Beams in Temperature Field
behaviour strongly reduces the value of the eigenfrequency. For the homogeneous
beam (cases 1, 4 and 7, 8), the change in the frequency value is of 10 and 12%, respectively. For FG beam (cases 2, 5 and 3, 6)—18 and 25%, respectively. Therefore, the
graded distribution of the beam thickness has an impact on the eigenfrequency value.
(ii) Investigation of Timoshenko Beam Chaotic Dynamics Versus the Combination
of Parameters γ 2 and P
The investigation of the solutions to the dynamic problems, i.e. when the excitation
q = q 0 sin(ω p t) is taken into account, has been carried out for the size-dependent
coefficient γ 2 = 0; 0.3 and different values of P E reported in Table 7.2.
The loading parameters are q 0 = 17000, ω p = 8. The frequency ω p = 8 is essentially higher comparing to the arbitrary eigenfrequency given in Table 7.3. All
obtained results are shown in Tables 7.4, 7.5, 7.6, 7.7, 7.8, 7.9, 7.10 and 7.11:
Table 7.4 (γ 2 = 0.3, P = 1, E = E 0 )—homogeneous beam with the sizedependent behaviour and an initial stiffness;
Table 7.5 (γ 2 = 0.3, P = 2)—functionally graded beam with the size-dependent
behaviour for the case when the stiffer layer is located on the upper side of the beam;
Table 7.6 (γ 2 = 0.3, P = 0.5)—functionally graded beam with the size-dependent
behaviour for the case when the stiffer layer is located on the bottom side of the beam;
Table 7.7 (γ 2 = 0, P = 1, E = E 0 )—homogeneous beam without the sizedependent behaviour and with an initial stiffness;
Table 7.8 (γ 2 = 0, P = 2)—non-homogeneous beam without the size-dependent
behaviour for the case when the stiffer layer is located on the upper side of the beam;
Table 7.9 (γ 2 = 0, P = 0.5)—non-homogeneous beam without the sizedependent behaviour for the case when the stiffer layer is located on the bottom
side of the beam;
Table 7.10 (γ 2 = 0, P = 1, E = 2E 0 )—homogeneous beam without the sizedependent behaviour and with a doubled stiffness;
Table 7.11 (γ 2 = 0.3, P = 1, E = 2E 0 )—homogeneous beam with the sizedependent behaviour and a doubled stiffness.
The following results are reported in the above-listed tables: (a) time history
w(0.5; t); (b) Fourier spectrum based on the Fast Fourier Transform (FFT) S(ω); (c)
2D wavelet spectrum based on Morlet wavelet; (d) Poincaré section w(t + T ) [w(t)];
(e) phase portrait ˙
w [w(t)]; (f) time evolution of the largest Lyapunov exponent (LLE)
based on Wolf’s algorithm [129].
The results obtained for the functionally graded beam with the localization of the
stiffer layer on the upper side (Table 7.5) essentially differ from the results shown in
Tables 7.4, 7.6. This beam exhibits quasi-periodic vibrations at three linearly dependent frequencies and the LLE is negative. Results reported in Tables 7.4, 7.6 imply that
the homogeneous beam and the functionally graded beam with the stiffer layer located
on the bottom side vibrate chaotically. The difference between the results reported
in Tables 7.4, 7.6 is as follows. In the case of the homogeneous beam (Table 7.4),
the transition into chaos takes place at t = 3800, which has been also validated by
7 Mathematical Models of Functionally Graded Beams in Temperature Field
behaviour strongly reduces the value of the eigenfrequency. For the homogeneous
beam (cases 1, 4 and 7, 8), the change in the frequency value is of 10 and 12%, respectively. For FG beam (cases 2, 5 and 3, 6)—18 and 25%, respectively. Therefore, the
graded distribution of the beam thickness has an impact on the eigenfrequency value.
(ii) Investigation of Timoshenko Beam Chaotic Dynamics Versus the Combination
of Parameters γ 2 and P
The investigation of the solutions to the dynamic problems, i.e. when the excitation
q = q 0 sin(ω p t) is taken into account, has been carried out for the size-dependent
coefficient γ 2 = 0; 0.3 and different values of P E reported in Table 7.2.
The loading parameters are q 0 = 17000, ω p = 8. The frequency ω p = 8 is essentially higher comparing to the arbitrary eigenfrequency given in Table 7.3. All
obtained results are shown in Tables 7.4, 7.5, 7.6, 7.7, 7.8, 7.9, 7.10 and 7.11:
Table 7.4 (γ 2 = 0.3, P = 1, E = E 0 )—homogeneous beam with the sizedependent behaviour and an initial stiffness;
Table 7.5 (γ 2 = 0.3, P = 2)—functionally graded beam with the size-dependent
behaviour for the case when the stiffer layer is located on the upper side of the beam;
Table 7.6 (γ 2 = 0.3, P = 0.5)—functionally graded beam with the size-dependent
behaviour for the case when the stiffer layer is located on the bottom side of the beam;
Table 7.7 (γ 2 = 0, P = 1, E = E 0 )—homogeneous beam without the sizedependent behaviour and with an initial stiffness;
Table 7.8 (γ 2 = 0, P = 2)—non-homogeneous beam without the size-dependent
behaviour for the case when the stiffer layer is located on the upper side of the beam;
Table 7.9 (γ 2 = 0, P = 0.5)—non-homogeneous beam without the sizedependent behaviour for the case when the stiffer layer is located on the bottom
side of the beam;
Table 7.10 (γ 2 = 0, P = 1, E = 2E 0 )—homogeneous beam without the sizedependent behaviour and with a doubled stiffness;
Table 7.11 (γ 2 = 0.3, P = 1, E = 2E 0 )—homogeneous beam with the sizedependent behaviour and a doubled stiffness.
The following results are reported in the above-listed tables: (a) time history
w(0.5; t); (b) Fourier spectrum based on the Fast Fourier Transform (FFT) S(ω); (c)
2D wavelet spectrum based on Morlet wavelet; (d) Poincaré section w(t + T ) [w(t)];
(e) phase portrait ˙
w [w(t)]; (f) time evolution of the largest Lyapunov exponent (LLE)
based on Wolf’s algorithm [129].
The results obtained for the functionally graded beam with the localization of the
stiffer layer on the upper side (Table 7.5) essentially differ from the results shown in
Tables 7.4, 7.6. This beam exhibits quasi-periodic vibrations at three linearly dependent frequencies and the LLE is negative. Results reported in Tables 7.4, 7.6 imply that
the homogeneous beam and the functionally graded beam with the stiffer layer located
on the bottom side vibrate chaotically. The difference between the results reported
in Tables 7.4, 7.6 is as follows. In the case of the homogeneous beam (Table 7.4),
the transition into chaos takes place at t = 3800, which has been also validated by