7.4 Chaotic Dynamics of Size-Dependent Graded Timoshenko Beams
223
Table 7.3 The frequencies associated with the studied cases
Case
number
1
2
3
4
5
6
7
8
ω 0
4.9
4.5
5.4
3.9
3.7
4.05
5.9
6.7
material (γ 2 = 0.3, P E = 1, E = 2E 0 ). A comparison of the results obtained for the
same values of P (Fig. 7.8d) implies that the size-dependent behaviour results in a
decrease of the deflection values.
Observe that for the case 2, the deflection curve changes its position, i.e. for
q = 50 it is localized below the deflection of the case 6, for q = 100, it overlaps
with the latter one, whereas, for q = 150, it is above the aforementioned deflection.
In general, the size-dependent behaviour reduces the deflection value for the same
grading parameter. Placing the stiffer layer on the upper side of the beam essentially
influences the deflection value for the same value of the size-dependent coefficient.
7.4.5.3 Dynamic Problems
The investigation of dynamic problems consists of the determination of the eigenfrequencies of the linear counterpart problem and the investigation of the chaotic
dynamics with respect to different combinations of the parameters γ 2 and P.
(i) Determination of Timoshenko Beam Eigenfrequencies
In order to get the eigenfrequencies, we study the linear equations governing the sizedependent behaviour in the functionally graded beams, which are yielded by system
(7.51) if nonlinear terms are neglected. The following linear PDEs are obtained
regarding the functions ψ, w:
k 2 ψ , xx + 3k 4 γ
2
2
ψ , xx − w , xxx
− 12 k s k 3 γ
2
1
ψ + w , x
= ψ , tt ,
k 3
ψ , x + w , xx
+ k 4
γ
2
2
γ
2
1
ψ , xxx − w , xxxx
− q = w, tt .
(7.54)
Equations (7.33) are solved using the algorithm presented earlier. Table 7.3 reports
the results obtained numerically, and the studied cases correspond to those shown in
Table 7.2.
The carried-out analysis yields the following conclusions. The smallest eigenfrequency is exhibited by the FG beam when the stiffer material is located on the beam
bottom side (this is case 5 with a lack of the size-dependent behaviour). The largest
eigenfrequency corresponds to case 8, i.e. the size-dependent behaviour is taken into
account and the stiffness of the homogeneous beam is doubled. The size-dependent
223
Table 7.3 The frequencies associated with the studied cases
Case
number
1
2
3
4
5
6
7
8
ω 0
4.9
4.5
5.4
3.9
3.7
4.05
5.9
6.7
material (γ 2 = 0.3, P E = 1, E = 2E 0 ). A comparison of the results obtained for the
same values of P (Fig. 7.8d) implies that the size-dependent behaviour results in a
decrease of the deflection values.
Observe that for the case 2, the deflection curve changes its position, i.e. for
q = 50 it is localized below the deflection of the case 6, for q = 100, it overlaps
with the latter one, whereas, for q = 150, it is above the aforementioned deflection.
In general, the size-dependent behaviour reduces the deflection value for the same
grading parameter. Placing the stiffer layer on the upper side of the beam essentially
influences the deflection value for the same value of the size-dependent coefficient.
7.4.5.3 Dynamic Problems
The investigation of dynamic problems consists of the determination of the eigenfrequencies of the linear counterpart problem and the investigation of the chaotic
dynamics with respect to different combinations of the parameters γ 2 and P.
(i) Determination of Timoshenko Beam Eigenfrequencies
In order to get the eigenfrequencies, we study the linear equations governing the sizedependent behaviour in the functionally graded beams, which are yielded by system
(7.51) if nonlinear terms are neglected. The following linear PDEs are obtained
regarding the functions ψ, w:
k 2 ψ , xx + 3k 4 γ
2
2
ψ , xx − w , xxx
− 12 k s k 3 γ
2
1
ψ + w , x
= ψ , tt ,
k 3
ψ , x + w , xx
+ k 4
γ
2
2
γ
2
1
ψ , xxx − w , xxxx
− q = w, tt .
(7.54)
Equations (7.33) are solved using the algorithm presented earlier. Table 7.3 reports
the results obtained numerically, and the studied cases correspond to those shown in
Table 7.2.
The carried-out analysis yields the following conclusions. The smallest eigenfrequency is exhibited by the FG beam when the stiffer material is located on the beam
bottom side (this is case 5 with a lack of the size-dependent behaviour). The largest
eigenfrequency corresponds to case 8, i.e. the size-dependent behaviour is taken into
account and the stiffness of the homogeneous beam is doubled. The size-dependent