6.3 Geometrically Nonlinear Analysis of Smart Structures
117
6.3.2.1 Static Analysis
For the static analysis, the smart plate is subjected to a uniformly distributed surface
load with the maximum value 2 × 10
7 Pa. In the simulation, the electrodes of piezoelectric patches are short circuited, meaning that the sensor electric voltage will not
influence the deformation of the plate. The mid-point displacements are computed
by different nonlinear theories, which are presented in Fig. 6.21a. The figure shows
that the load-deflection curves obtained by various models as well as the commercial
software ANSYS are almost identical. This is because the fixed boundary conditions,
the smart plate cannot undergo large rotations, but undergo only moderate rotations.
It can be demonstrated by observation of the rotations in the plate, ϕ 1 and ϕ 2 , which
is calculated by LRT56 theory using SH851URI elements in a loading condition
of 2 × 10
7 Pa, as shown in Fig. 6.22a and b, respectively. The figure shows that
the rotations jump suddenly from zero to a maximum value, then decrease to zero
at the mid-point. The sensor output voltages under different load levels are shown
in Fig. 6.21b. The results show that the linear theory overpredicts the sensor output
voltage, because it does not account for the stress stiffening effects. Since the plate
undergoes only moderate rotations, the results obtained by all nonlinear theories are
in very good agreement.
6.3.2.2 Dynamic Analysis
The dynamic response is investigated under a uniformly distributed step pressure
with the amplitude of 2 × 10
4 N/m
2 . Both the linear and nonlinear displacement
time histories are solved by Newmark method. A time step of 1 × 10
−5 s is used
for the linear case, and 1 × 10
−7 s for the nonlinear case. The dynamic response of
the mid-point displacement and sensor output voltage both for linear and nonlinear
simulations is presented in Fig. 6.23a and b, respectively. The results indicate that
the linear and nonlinear transient response is very similar to each other. This is
because the plate is undergoing small displacements and rotations, meaning that only
weak nonlinear effects are imposed. The present time histories of both displacement
and sensor output voltage agree excellently with those presented by Lentzen and
Schmidt [18]. The present results are obtained by LRT56 and RVK5 models in the
FOSD hypothesis, while those of Lentzen and Schmidt [18] are achieved by FOSD
MRT5 theory. This is because weak nonlinear effects and only moderate rotations
are occurring in the plate. Due to the fact that the linear theory does not take into
account for the von Kármán stress stiffening effect, the time histories of linear theory
have slightly larger amplitudes.
To present nonlinear phenomenon, the magnitude of the surface load increases to
2 × 10
5 N/m
2 . The dynamic response curves are calculated by linear and nonlinear
theories are presented in Fig. 6.24. Now the amplitudes of the displacements are
in the order of the magnitude of the plate thickness. Therefore, a big difference
can be observed between the linear and nonlinear vibrations. This is due to the stress
stiffening effect. Since the linear theory does not account for this effect, it overpredicts
117
6.3.2.1 Static Analysis
For the static analysis, the smart plate is subjected to a uniformly distributed surface
load with the maximum value 2 × 10
7 Pa. In the simulation, the electrodes of piezoelectric patches are short circuited, meaning that the sensor electric voltage will not
influence the deformation of the plate. The mid-point displacements are computed
by different nonlinear theories, which are presented in Fig. 6.21a. The figure shows
that the load-deflection curves obtained by various models as well as the commercial
software ANSYS are almost identical. This is because the fixed boundary conditions,
the smart plate cannot undergo large rotations, but undergo only moderate rotations.
It can be demonstrated by observation of the rotations in the plate, ϕ 1 and ϕ 2 , which
is calculated by LRT56 theory using SH851URI elements in a loading condition
of 2 × 10
7 Pa, as shown in Fig. 6.22a and b, respectively. The figure shows that
the rotations jump suddenly from zero to a maximum value, then decrease to zero
at the mid-point. The sensor output voltages under different load levels are shown
in Fig. 6.21b. The results show that the linear theory overpredicts the sensor output
voltage, because it does not account for the stress stiffening effects. Since the plate
undergoes only moderate rotations, the results obtained by all nonlinear theories are
in very good agreement.
6.3.2.2 Dynamic Analysis
The dynamic response is investigated under a uniformly distributed step pressure
with the amplitude of 2 × 10
4 N/m
2 . Both the linear and nonlinear displacement
time histories are solved by Newmark method. A time step of 1 × 10
−5 s is used
for the linear case, and 1 × 10
−7 s for the nonlinear case. The dynamic response of
the mid-point displacement and sensor output voltage both for linear and nonlinear
simulations is presented in Fig. 6.23a and b, respectively. The results indicate that
the linear and nonlinear transient response is very similar to each other. This is
because the plate is undergoing small displacements and rotations, meaning that only
weak nonlinear effects are imposed. The present time histories of both displacement
and sensor output voltage agree excellently with those presented by Lentzen and
Schmidt [18]. The present results are obtained by LRT56 and RVK5 models in the
FOSD hypothesis, while those of Lentzen and Schmidt [18] are achieved by FOSD
MRT5 theory. This is because weak nonlinear effects and only moderate rotations
are occurring in the plate. Due to the fact that the linear theory does not take into
account for the von Kármán stress stiffening effect, the time histories of linear theory
have slightly larger amplitudes.
To present nonlinear phenomenon, the magnitude of the surface load increases to
2 × 10
5 N/m
2 . The dynamic response curves are calculated by linear and nonlinear
theories are presented in Fig. 6.24. Now the amplitudes of the displacements are
in the order of the magnitude of the plate thickness. Therefore, a big difference
can be observed between the linear and nonlinear vibrations. This is due to the stress
stiffening effect. Since the linear theory does not account for this effect, it overpredicts