Laminated plate MFC transducer
3.2 Generalized Hamilton’s principle of an aniso- aniso-tropic plate with integrated anisoaniso-tropic
trans-ducers
A rectangular anisotropic plate with spatial distributed rectangular MFC transducers is considered in the study, as shown in Figure 3.6. Only one transducer is considered on the plate for convenience of analysis, but it is possible to introduce more transducers of different sizes.
y
x MFC transducers
Plate
-x
-y
Figure 3.6: A rectangular orthotropic plate with integrated MFC transducers
The spatial distribution of the rectangular MFC transducer sizea×b can be expressed by the Heaviside functionH in its local coordinates (¯x,y) [152]:¯
ΛpxΛpy= [H(¯x+a
2)−H(¯x−a
2)]×[H(¯y+b
2)−H(¯y− b
2)]. (3.23) The Heaviside function can be used to describe the spatial distribution of transducers with different shapes so that the local-global coordinates transformation between the transducer and the plate is usually not trivial.
In this analysis, as we assumed that the local coordinates (¯x,y) on the MFC¯ transducer coincide with the global coordinates (x, y). The spatial distribution of the MFC transducer can be rewritten as:
ΛpxΛpy= [H(x+a
2)−H(x−a
2)]×[H(y+b
2)−H(y− b
2)]. (3.24) According to the property of the Heaviside function [173], the following generalized form retains:
Z
Γp
GpdΓp= Z
Γp
GpΛpxΛpydΓp= Z
Γs
GpΛpxΛpydΓs (3.25)
GENERALIZED HAMILTON’S PRINCIPLE 47
whereGp is a generalized term that contains the spatial variablesxandy. Γp
and Γsare the surface dimensions of the host structure and MFC transducer, respectively. The spatial distribution combines the dimensional integration of the host structure and the transducer. Then, the kinetic and stored energies on the plate can be described in a similar way to Equation (3.25).
3.2.1 Potential energy and its variation
The potential energy of the system consists of both the strain energy and the electric energy. It can be expressed as follows:
U∗=−1 augmented stiffness of the MFC transducer, respectively. We can divide the potential energy into two parts: Us∗andUp∗, representing the potential energies in the host plate and the transducer respectively. The potential energy variation of the host plate subject to virtual displacements can be written as:
∆Us∗=−1 The constitutive relations in Equation (3.8) are adopted here to simplify the notations in the following equation. Thereby, ∆Us∗ can be rewritten as follows:
∆Us∗=−
The integration by parts is performed to obtain the virtual energy variations regarding ∆u, ∆v and ∆w. Let the boundary conditions of host plate be either free, simply supported or clamped; the boundary constraints terms from the integration by parts are null. Then, Equation (3.28) can be rewritten as follows:
∆Us∗=−
The potential energy of the transducer is given in the form of Its variation is simply given as
Up∗=−1 Hence, the variation of the stored energy on the transducer is expressed as:
∆Up∗=
The membrane forces and bending moments in Equation (3.32) are different from Equation (3.28) because the piezoelectric terms are included. The integration by parts is carried out here to obtain the variation ofUp∗:
∆Up∗=
The boundary constraints terms of the transducer are not null because there are no clear boundary conditions for the transducer on the host plate.
GENERALIZED HAMILTON’S PRINCIPLE 49
3.2.2 Kinetic energy and its variation
The kinetic energyT∗ of the system can be written as:
T∗= 1
whereρs,ρp are the mass densities of the host structure and the transducer, respectively. The displacement field is expressed as
Substituting Equation (3.35) into Equation (3.34) yields T∗=1
The terms containing ∂w∂x and ∂w∂y are vanished because the rotations do not contribute to the kinetic energy. Then, the variation of kinetic energy can be easily obtained as follows:
∆T∗=
3.2.3 Work due to external loads and its variation
Let’s have a look now the external work due to transverse loads and in-plane loads. As a transverse pointed force f(t) excited at position (x0, y0) on the structure can be a representative of external loads, the induced external work can be written as:
W∗= Z
Γs
f(t)w0δ(x−x0)δ(y−y0)dΓs (3.38)
whereδis the Dirac delta-function. In a large deflection analysis, the work of in-plane loads due to a deflectionwis given as:
V∗=
whereNxx,Nyy andNxy are the pre-buckling loads applied to the mid-surface of the host plate andεxx,εyy andεxy are the strains on the mid-surface due to deflectionw. Since small strain variation is considered in the present study,V∗ can be neglected in the present analysis. Thus, the variation of the work due to external transverse loads is simply as follows:
∆W∗= Z
Γs
f(t)δ(x−x0)δ(y−y0)×∆w0dΓs (3.40)
3.2.4 Governing equations and boundary constraints of the transducer
The generalized Hamilton’s principle for an electro-elastic body can be described as [74]:
Z t2 t1
[∆T∗+ ∆U∗+ ∆W∗]dt= 0. (3.41) The variations of kinetic energy ∆T∗, potential energy ∆U∗ and external work ∆W∗ can be substituted in Equation (3.41). Then, all the terms can be regrouped according to the variations ∆u, ∆v and ∆w. As these three variations can be arbitrary values, their factors should be null. Therefore, the equations of motion of the overall structure can be obtained as follows:
∂Nxxs
The generalized Hamilton’s principle has transformed the system into a weak formulation. We can observe that the mechanical influences of the MFC transducers are described by the spatial distributions in the above equations.
The inverse piezoelectric effect is included in the membrane forces Nijp and
GENERALIZED HAMILTON’S PRINCIPLE 51
bending momentsMijp (see Equation (3.19)). The inverse piezoelectric effect can be deduced from Equations (3.42a) to (3.42c) so that the equivalent force per unit area of the MFC transducer is given as follows:
Nxx=tpe¯∗31E∂ΛpxΛpy We can observe again that the transducer generates both membrane and bending behaviors by its inverse piezoelectric effect. The spatial distributions terms determine the distribution of these equivalent loads. The boundary constraints of the transducer from Equation (3.41) result in the equivalent moment and equivalent transverse shear forces, respectively:
By introducing the spatial distribution of the MFC transducer, the equivalent force per unit area, equivalent moments and transverse shear forces are determined, respectively. These three kinds of loads agree with the elastic equilibrium theory: The transverse shear loads are the first-order derivative of the bending moments and the forceNzz is the second-order derivative of the bending moments. The equivalent loads exist simultaneously and describe the actuation of a piezoelectric actuator through different parameters.