124
6 Nonlinear Analysis of Piezoceramic Laminated Structures
Fig. 6.27 Rotations of the
cylindrical shell under a
pressure of 2 × 10 7 Pa: a
rotation ϕ 1 , b rotation ϕ 2 ,
reprinted from Ref. [5],
copyright 2014, with
permission from ELSEVIER
(a)
0
200
400
0
0.05
0.1
0
5
10
15
Θ
1 (mm)
Θ
2 (rad)
| 1
| (degree)
(b)
0
200
400
0
0.05
0.1
0
5
10
15
Θ
1 (mm)
Θ
2 (rad)
| 2| (degree)
found between the static nonlinear curves, which implies that large rotations have
occurred in the structure. The hoop displacements of MRT5, LRT5 and LRT56 models show first softening behavior, then hardening at large loads. The load-voltage
curves follow this tendency. It is interesting that the radial displacements of LRT56
first increase then decrease.
6.3.4.3 Nonlinear Dynamic Analysis
In this part, the nonlinear dynamic analysis of the semicircular cylindrical shell is
investigated. A step load with the amplitude of F 1 = 50 N is applied on the tip point
at the free end along the positive hoop direction. The dynamic response of the tip
displacement in the hoop and radial directions, as well as the output voltage of the
inner PZT layer is calculated using Newmark method with a time step 1 × 10
−3 s for
the linear case and 1 × 10
−4 s for the nonlinear case, which is respectively shown in
Fig. 6.32a–c.
In the nonlinear static analysis, it has been shown that MRT5 and LRT5 models
perform very similar. Therefore, the time histories obtained by MRT5 and LRT5
models have very small differences. Because of the softening behavior of LRT56,
LRT5 and MRT5 models, larger amplitudes of dynamic vibrations in hoop direction
are achieved than those of linear model. This tendency of dynamic response is in
accordance with that of static response as shown in Fig. 6.31a. In addition, from
6 Nonlinear Analysis of Piezoceramic Laminated Structures
Fig. 6.27 Rotations of the
cylindrical shell under a
pressure of 2 × 10 7 Pa: a
rotation ϕ 1 , b rotation ϕ 2 ,
reprinted from Ref. [5],
copyright 2014, with
permission from ELSEVIER
(a)
0
200
400
0
0.05
0.1
0
5
10
15
Θ
1 (mm)
Θ
2 (rad)
| 1
| (degree)
(b)
0
200
400
0
0.05
0.1
0
5
10
15
Θ
1 (mm)
Θ
2 (rad)
| 2| (degree)
found between the static nonlinear curves, which implies that large rotations have
occurred in the structure. The hoop displacements of MRT5, LRT5 and LRT56 models show first softening behavior, then hardening at large loads. The load-voltage
curves follow this tendency. It is interesting that the radial displacements of LRT56
first increase then decrease.
6.3.4.3 Nonlinear Dynamic Analysis
In this part, the nonlinear dynamic analysis of the semicircular cylindrical shell is
investigated. A step load with the amplitude of F 1 = 50 N is applied on the tip point
at the free end along the positive hoop direction. The dynamic response of the tip
displacement in the hoop and radial directions, as well as the output voltage of the
inner PZT layer is calculated using Newmark method with a time step 1 × 10
−3 s for
the linear case and 1 × 10
−4 s for the nonlinear case, which is respectively shown in
Fig. 6.32a–c.
In the nonlinear static analysis, it has been shown that MRT5 and LRT5 models
perform very similar. Therefore, the time histories obtained by MRT5 and LRT5
models have very small differences. Because of the softening behavior of LRT56,
LRT5 and MRT5 models, larger amplitudes of dynamic vibrations in hoop direction
are achieved than those of linear model. This tendency of dynamic response is in
accordance with that of static response as shown in Fig. 6.31a. In addition, from