Relevant mechanical properties of respective materials.
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Nelson",authors:[{id:"53464",title:"Prof.",name:"Thomas",middleName:null,surname:"Nelius",fullName:"Thomas Nelius",slug:"thomas-nelius"},{id:"53476",title:"Prof.",name:"Stephanie",middleName:null,surname:"Filleur",fullName:"Stephanie Filleur",slug:"stephanie-filleur"},{id:"100431",title:"Dr.",name:"Jonathan",middleName:null,surname:"Nelson",fullName:"Jonathan Nelson",slug:"jonathan-nelson"}]},{id:"20569",title:"Urinary Tract Infections During Pregnancy",slug:"urinary-tract-infections-during-pregnancy",signatures:"Anick Bérard, Fabiano Santos, Ema Ferreira and Sylvie Perreault",authors:[{id:"55191",title:"Dr.",name:"Anick",middleName:null,surname:"Bérard",fullName:"Anick Bérard",slug:"anick-berard"},{id:"55848",title:"MSc",name:"Fabiano",middleName:null,surname:"Santos",fullName:"Fabiano Santos",slug:"fabiano-santos"},{id:"55881",title:"Dr.",name:"Ema",middleName:null,surname:"Ferreira",fullName:"Ema Ferreira",slug:"ema-ferreira"},{id:"55882",title:"Dr.",name:"Sylvie",middleName:null,surname:"Perreault",fullName:"Sylvie Perreault",slug:"sylvie-perreault"}]},{id:"20570",title:"Reducing the Incidence of Catheter-Associated Urinary Tract Infections in the Acute Care Setting Using Evidence-Based Guidelines",slug:"reducing-the-incidence-of-catheter-associated-urinary-tract-infections-in-the-acute-care-setting-usi",signatures:"Diane Gorman",authors:[{id:"55192",title:"Mrs",name:"Diane",middleName:null,surname:"Gorman",fullName:"Diane Gorman",slug:"diane-gorman"}]},{id:"20571",title:"The Formation of Poly-Microbial Biofilms on Urinary Catheters",slug:"the-formation-of-poly-microbial-biofilms-on-urinary-catheters",signatures:"Veronika Hola and Filip Ruzicka",authors:[{id:"48709",title:"Dr.",name:"Veronika",middleName:null,surname:"Holá",fullName:"Veronika Holá",slug:"veronika-hola"},{id:"136557",title:"PhD.",name:"Filip",middleName:null,surname:"Ruzicka",fullName:"Filip Ruzicka",slug:"filip-ruzicka"}]},{id:"20572",title:"Urinary Tract Infections in Psychiatric Patients",slug:"urinary-tract-infections-in-psychiatric-patients",signatures:"Teresita Sáinz Espuñes, Maria Elisa Drago Serrano and Jaime Amadeo Bustos Martínez",authors:[{id:"52242",title:"Dr.",name:"Teresita",middleName:null,surname:"Sainz-Espuñes",fullName:"Teresita Sainz-Espuñes",slug:"teresita-sainz-espunes"},{id:"59716",title:"Dr.",name:"Maria Elisa",middleName:null,surname:"Drago-Serrano",fullName:"Maria Elisa Drago-Serrano",slug:"maria-elisa-drago-serrano"},{id:"59719",title:"Dr.",name:"Jaime",middleName:null,surname:"Bustos-Martínez",fullName:"Jaime Bustos-Martínez",slug:"jaime-bustos-martinez"}]},{id:"20573",title:"Antibiotic Resistance in Urinary Tract Infections: Current Issues and Future Solutions",slug:"antibiotic-resistance-in-urinary-tract-infections-current-issues-and-future-solutions",signatures:"David W. Hilbert",authors:[{id:"55917",title:"Dr.",name:"David",middleName:null,surname:"Hilbert",fullName:"David Hilbert",slug:"david-hilbert"}]},{id:"20574",title:"Identification and Antibiotic Sensitivity of UTI Pathogens Using Raman Spectroscopy",slug:"identification-and-antibiotic-sensitivity-of-uti-pathogens-using-raman-spectroscopy",signatures:"Evdokia Kastanos, Alexandros Kyriakides, Katerina Hadjigeorgiou and Costas Pitris",authors:[{id:"49230",title:"Prof.",name:"Costas",middleName:null,surname:"Pitris",fullName:"Costas Pitris",slug:"costas-pitris"},{id:"49762",title:"Prof.",name:"Evdokia",middleName:null,surname:"Kastanos",fullName:"Evdokia Kastanos",slug:"evdokia-kastanos"},{id:"49763",title:"Mr",name:"Alexandros",middleName:null,surname:"Kyriakides",fullName:"Alexandros Kyriakides",slug:"alexandros-kyriakides"},{id:"49764",title:"BSc.",name:"Katerina",middleName:null,surname:"Hadjigeorgiou",fullName:"Katerina Hadjigeorgiou",slug:"katerina-hadjigeorgiou"}]},{id:"20575",title:"Natural Approaches for Controlling Urinary Tract Infections",slug:"natural-approaches-for-controlling-urinary-tract-infections",signatures:"Mary Anne Roshni Amalaradjou and Kumar Venkitanarayanan",authors:[{id:"61657",title:"Prof.",name:"Kumar",middleName:null,surname:"Venkitanarayanan",fullName:"Kumar Venkitanarayanan",slug:"kumar-venkitanarayanan"},{id:"61765",title:"Dr.",name:"Mary Anne Roshni",middleName:null,surname:"Amalaradjou",fullName:"Mary Anne Roshni Amalaradjou",slug:"mary-anne-roshni-amalaradjou"}]},{id:"20576",title:"Prevention Strategy of Urogenital Infections by Using Lactobacilli with Probiotic Properties",slug:"prevention-strategy-of-urogenital-infections-by-using-lactobacilli-with-probiotic-properties",signatures:"Liliana Pascual and Lucila Barberis",authors:[{id:"59046",title:"Dr.",name:"Lucila",middleName:null,surname:"Barberis",fullName:"Lucila Barberis",slug:"lucila-barberis"},{id:"60021",title:"Dr.",name:"Liliana",middleName:null,surname:"Pascual",fullName:"Liliana Pascual",slug:"liliana-pascual"}]},{id:"20577",title:"Current Management of Urinary Tract Infection in Children",slug:"current-management-of-urinary-tract-infection-in-children",signatures:"Yusuf Kibar",authors:[{id:"51399",title:"Dr.",name:"Yusuf",middleName:null,surname:"Kibar",fullName:"Yusuf Kibar",slug:"yusuf-kibar"}]},{id:"20578",title:"Urinary Tract Infection in Children – Onset of a New Era?",slug:"urinary-tract-infection-in-children-onset-of-a-new-era-",signatures:"Tanja Kersnik Levart and Rajko B Kenda",authors:[{id:"50388",title:"Dr.",name:"Tanja",middleName:null,surname:"Kersnik Levart",fullName:"Tanja Kersnik Levart",slug:"tanja-kersnik-levart"},{id:"136559",title:"Prof.",name:"Rajko",middleName:null,surname:"Kenda",fullName:"Rajko Kenda",slug:"rajko-kenda"}]},{id:"20579",title:"UTI in Children",slug:"uti-in-children",signatures:"Samileh Noorbakhsh and Vida Zarabi",authors:[{id:"45794",title:"Prof.",name:"Samileh",middleName:null,surname:"Noorbakhsh",fullName:"Samileh Noorbakhsh",slug:"samileh-noorbakhsh"},{id:"45801",title:"Dr.",name:"Vida",middleName:null,surname:"Zarabi",fullName:"Vida Zarabi",slug:"vida-zarabi"}]},{id:"20580",title:"Post-Inflammatory Nephropathy",slug:"post-inflammatory-nephropathy",signatures:"Beata Bieniaś, Małgorzata Zajączkowska, Halina Borzęcka, Przemysław Sikora, Marek Majewski, Ewelina Książek and Andrzej Borzęcki",authors:[{id:"52793",title:"Dr",name:"Beata",middleName:null,surname:"Bieniaś",fullName:"Beata Bieniaś",slug:"beata-bienias"},{id:"59721",title:"Dr.",name:"Małgorzata",middleName:null,surname:"Zajączkowska",fullName:"Małgorzata Zajączkowska",slug:"malgorzata-zajaczkowska"},{id:"59722",title:"Dr.",name:"Halina",middleName:null,surname:"Borzęcka",fullName:"Halina Borzęcka",slug:"halina-borzecka"},{id:"59723",title:"Mr.",name:"Przemysław",middleName:null,surname:"Sikora",fullName:"Przemysław Sikora",slug:"przemyslaw-sikora"},{id:"59724",title:"Mr.",name:"Marek",middleName:null,surname:"Majewski",fullName:"Marek Majewski",slug:"marek-majewski"},{id:"59725",title:"Mr.",name:"Ewelina",middleName:null,surname:"Książek",fullName:"Ewelina Książek",slug:"ewelina-ksiazek"},{id:"59726",title:"Mr.",name:"Andrzej",middleName:null,surname:"Borzęcki",fullName:"Andrzej Borzęcki",slug:"andrzej-borzecki"}]},{id:"20581",title:"Studies on Clinical Characteristics, Urovirulence Factor and Host Susceptibility Gene in Pediatric Acute Lobar Nephronia",slug:"studies-on-clinical-characteristics-urovirulence-factor-and-host-susceptibility-gene-in-pediatric-ac",signatures:"Chi-Hui Cheng, Yong-Kwei Tsau and Tzou-Yien Lin",authors:[{id:"51309",title:"Prof.",name:"Tzou-Yien",middleName:null,surname:"Lin",fullName:"Tzou-Yien Lin",slug:"tzou-yien-lin"},{id:"52822",title:"Prof.",name:"Yong-Kwei",middleName:null,surname:"Tsau",fullName:"Yong-Kwei Tsau",slug:"yong-kwei-tsau"},{id:"52824",title:"Dr.",name:"Chi-Hui",middleName:null,surname:"Cheng",fullName:"Chi-Hui Cheng",slug:"chi-hui-cheng"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"70974",title:"Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates",doi:"10.5772/intechopen.89303",slug:"static-and-dynamic-analysis-of-piezoelectric-laminated-composite-beams-and-plates",body:'The content of this chapter is the inheritance and development of the research results of the authors and other authors by published scientific works on composite materials, piezoelectric and structural calculation by piezoelectric composite materials.
Let us consider a block of elastic material in an environment with an electric field of zero, the relationship between stress and strain is followed Hooke’s law, and written as follows [1, 2]:
where {σ} is the mechanical stress vector, {ε} is the mechanical strain vector, and [c] is the material stiffness matrix of beam.
Mechanical-electrical relations in piezoelectric materials have an interactive relationship, strain {ε} will produce eε - polarization, where e is the voltage stress factor when there is no mechanical strain. The imposed electric field E produces the -eE stress in the piezoelectric material according to the reverse voltage effect. Therefore, we have a mathematical model that describes the mechanical-electrical interaction relationship in piezoelectric materials as follows [3, 4, 5, 6]:
where [e] is the piezoelectric stress coefficient matrix, [p] is the dielectric constant matrix, {E} is the vector of applied electric field (V/m), and {D} is the vector of electric displacement (C/m2).
For the linear problem and small strain, strain vector in the piezoelectric structures can be defined as follows:
in which [s] is the matrix of compliance coefficients (m2/N), [d] is the matrix of piezoelectric strain constants (m/V).
In the field of engineering, piezoelectric materials are used by two types. The first type, the piezoelectric layers or the piezoelectric patches act as actuators, called the piezoelectric actuators. In this case, the piezoelectric layers are strained when imposing an electric field on it. The second type, the piezoelectric layers or piezoelectric patches act as sensors, called piezoelectric sensors. In this case, the voltage is generated in piezoelectric layers when there is mechanical strain.
Eq. (5) can be written in the matrix form as follows [4, 6]:
Assuming that the device is pulled along the axis 3, and viewing the piezoelectric material as a transversely isotropic material, which is true for piezoelectric ceramics, many of the parameters in the above matrices will be either zero, or can be expressed through each other. In particular, the non-zero compliance coefficients are s11, s12, s13, s21, s22, s23, s31, s32, s33, s44, s55, s66, in which s12 = s21, s13 = s31, s23 = s32, s44 = s55, s66 = 2(s11 − s12).
Finally, Eq. (6) becomes:
where E1, E2, and E3 are electric fields in the 1, 2, and 3 directions, respectively.
The induction charge equation of piezoelectric sensor layers is derived from Eq. (4) can be written in the matrix form as [4, 6, 7]:
The non-zero piezoelectric strain constants are d31, d32, d15, d24, and d33, in which d31 = d32, d15 = d24. And the non-zero dielectric coefficients are p11, p22, and p33, where p11 = p22. Eq. (8) becomes:
where D1, D2, D3, p11, p22, and p33 are the displacement charge, dielectric constant in the 1, 2, and 3 directions, respectively.
Normally, the voltage is transmitted through the thickness of the actuator layers.
Based on the first-order shear deformation theory (FSDT), the displacement field at any point of the beam is defined as [1, 2]:
where u, w denotes the displacements of a point (x, z) in the beam; u0, w0 are the displacements of a point at the beam neutral axis, and θy is the rotation of the transverse normal about the y axis. The bending and shear strains associated with the displacement field in Eq. (10) are defined as:
in which εx, γxz are the normal strain, and shear strain, respectively.
Using finite element method, we consider 2-node bending elements with 3 degrees of freedom per node (Figure 1).
Two noded beam element.
The displacements of the beam neutral axis are expressed in local coordinate system in the form:
where {qb}e is the vector of vector of nodal displacements of element, [NM] is the matrix mechanical shape functions:
in which [Nu], [Nv], [Nθz] are, in this order, the row vectors of longitudinal, transverse along y, and rotation about z shape functions.
Substituting Eq. (12) into Eq. (11), we obtain:
The electric potential is constant over the element surface:
where n is the element node number.
A voltage ϕ is applied across an actuator of layer thickness tp generates an electric field vector {E}, such that [4, 8, 9, 10]:
in which
where tpk is the thickness of the kth piezoelectric layer.
Substituting Eq. (19) into Eq. (18), the electric field vector {E} can also be defined in terms of nodal variables as:
Using Eqs. (15), and (20), the linear piezoelectric constitutive equations coupling the elastic and electric fields will be completely determined by Eqs. (2) and (3).
Using Hamilton’s principle, we have [11, 12, 13]:
where Te, Ue are the kinetic and potential energy, respectively and We is the work done by external forces. They are determined by:
in which
Substituting Eqs. (15), (2), (20), (22), (23), and (24) into Eq. (21), one obtains:
where
Element mechanical-electrical coupling stiffness matrix:
Element electrical-mechanical coupling stiffness matrix:
Element piezoelectric permittivity matrix:
{f}e, {Q}e are the applied external load and charge, respectively.
In the case of beams subjected to static loads, zero acceleration, from Eqs. (25) and (26), we obtain the static equations of the beam as follows:
Assembling the element equations yields general static equation:
where [Kbb], [Kϕϕ] are the overall mechanical stiffness and piezoelectric permittivity matrices respectively; [Kbϕ] and [Kϕb] are the overall mechanical - electrical and electrical - mechanical coupling stiffness matrices, respectively, and {q}, {ϕ} are respectively the overall mechanical displacement, and electric potential vector.
Substituting Eq. (36) into Eq. (35) yields:
Substituting {q} from Eq. (37) into Eq. (36), we obtain the vector {ϕ}.
From Eqs. (25) and (26), assembling the element equations yields general dynamic equation of motion:
Substituting Eq. (39) into Eq. (38), we obtain:
For free vibrations, from Eq. (40), the governing equation is:
The beam vibrations induce charges and electric potentials in sensor layers. Therefore, the control system allows current to flow and feeds back to the actuators. In this case, if we apply no external charge Q to a sensor, from Eq. (39), we will have:
and
The operation of the amplified control loop implies, the actuating voltage is determined by the following relationship [1, 10, 14]:
where Gd and Gv are the feedback control gains for displacement and velocity.
Substituting Eq. (43) into Eq. (39), the charge in the actuator due to actuator strain in response to the beam vibration modified by control system feedback is:
Substituting (42) into (44) leads to:
Substituting Eq. (45) into (40), we obtain:
in which {q}s ≡ {q}a ≡ {q} is the beam displacement vector, [Kϕϕ]a = [Kϕϕ]s = [Kϕϕ] is the piezoelectric permittivity matrix, and [Kϕb]a = [Kϕb]s = [Kϕb] is the mechanical-electrical coupling stiffness matrix.
Therefore, Eq. (46) becomes:
In the case of considering the structural damping, the equation of motion of the beam is:
where
Eq. (48) can be solved by the direct integration Newmark’s method.
An example for free vibration of laminated beam affected by piezoelectric layers is presented here. The beam is made of four layers symmetrically (0°/90°/90°/0°) of epoxy-T300/976 graphite material with 2.5 mm thickness per layer, and with one layer piezo ceramic materials bonded to the top and bottom surfaces, 2.0 mm thickness per layer as shown in Figure 2 is considered (a = 0.254 m, b = 0.0254 m). The material properties of the piezo ceramic layers and graphite-epoxy are shown in Table 1.
Piezoelectric composite cantilever beam.
Properties | PZT G1195 N | T300/976 |
---|---|---|
E11 [N/cm2] | 0.63 × 106 | 1.50 × 106 |
E22 = E33 [N/cm2] | 0.63 × 106 | 0.09 × 106 |
ν12 = ν13 = ν23 | 0.3 | 0.3 |
G12 = G13 [N/cm2] | 0.242 × 106 | 0.071 × 106 |
G23 [N/cm2] | 0.242 × 106 | 0.025 × 106 |
ρ [kg/m3] | 7600 | 1600 |
d31 = d32 (m/V) | 254 × 10−12 | — |
p11 = p22 (F/m) | 15.3 × 10−9 | — |
p33 (F/m) | 15.0 × 10−9 | — |
Relevant mechanical properties of respective materials.
The direct integration Newmark’s method is used with parameters αR = 0.5, βR = 0.25; integral time step Δt = 0.005 s with total time calculated t = 15 s.
Figures 3 and 4 illustrate the vertical displacement w at the free end of the beam for two cases:
Case 1: With structural damping, and without piezoelectric damping (Gv = 0, Gd = 0).
Case 2: With structural damping, with piezoelectric damping (Gv = 0.5, Gd = 30).
Vertical displacement response (Gv = 0, Gd = 0 − Case 1).
Vertical displacement response (Gv = 0.5, Gd = 30 − Case 2).
Consider laminated composite plates with general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top and bottom surfaces of the plate are bonded to the piezoelectric patches or piezoelectric layers (actuator and sensor). The plate under the load acting on its neutral plane has any temporal variation rule (Figure 5).
Piezoelectric composite plate and coordinate system of the plate (a), and lamina details (b).
Hypothesis: The piezoelectric composite plate corresponds with Reissner-Mindlin theory. The material layers are arranged symmetrically through the neutral plane of the plate, ideally adhesive with each other.
Based on the first-order shear deformation theory, the displacement fields at any point in the plate are [7, 8]:
where u, v and w are the displacements of a general point (x, y, z) in the laminate along x, y and z directions, respectively. u0, v0, w0, θx and θy are the displacements and rotations of a midplane transverse normal about the y-and x-axes respectively.
The components of the strain vector corresponding to the displacement field (49) are defined as:
For the linear strain:
or in the vector form:
and for the nonlinear strain:
where
The equation system describing the stress-strain relations and mechanical-electrical quantities is respectively written as [8, 14]:
where
The in-plane force vector at the state pre-buckling:
The total potential energy of the system is given by:
where W is the energy of external forces, Vp is the entire domain including composite and piezoelectric materials.
Introducing [A], [B], [D], [As], and vectors {Np}, {Mp} as [8]:
where h is the total laminated thickness and combining with (5), (6) the total potential energy equation (8) can be written
where Ω is the plane xy domain of the plate.
Nine-node Lagrangian finite elements are used with the displacement and strain fields represented by Eqs. (49), (53), and (54). In the developed models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior and thus the vector of electrical degrees of freedom is [6, 14]:
in which NPLe is the number of piezoelectric layers in a given element.
The vector of degrees of freedom for the element {qe} is:
where
The dynamic equations of piezoelectric composite plate can be derived by using Hamilton’s principle, accordingly, the vibration equation of the membrane (without damping) with in-plane loads is:
The equation of bending vibrations with out-of-plane loads is:
where [Mss], [Kss] are the overall mass, membrane elastic stiffness matrix respectively, and
The element coefficient matrices are:
When the plate is subjected to in-plane loads only ({R} = {0}), the in-plane stresses can lead to buckling, from Eqs. (63) and (64) the governing differential equations of motion of the damped system may be written as:
Eq. (70) is rewritten as:
The overall geometric stiffness matrix [KG] is defined as follows:
In the case of only tensile or compression plates (w = 0): Solving Eq. (71) helps us to present unknown displacement vector {qss}, and then stress vector:
where [As] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane problem.
In the case of bending plate (w ≠ 0), the stress vector is:
where [Ab] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane bending problem.
Stability criteria [14]:
In the case of plate subjected to periodic in-plane loads and without damping, the elastic stability problems become simple only by solving the linear equations to determine the eigenvalues.
In case of the plate under any in-plane dynamic load and with damping, the elastic stability problems become very complex. This iterative method can be proved effectively and the following dynamic stability criteria are used:
Plate is considered to be stable if the maximum bending deflection is three times smaller than the plate’s thickness:Eq. (71)has the solution (wi)max satisfying the condition
Plate is called to be in critical status if the maximum bending deflection of the plate is three times equal to the plate’s thickness.Eq. (71)has the solution (wi)max satisfying the condition
Plate is called to be at buckling if the maximum deflection of the plate is three times larger than the plate’s thickness:Eq. (71)has the solution (wi)max satisfying the condition
The identification of critical forces is carried out by the iterative method.
Step 1. Defining the matrices, the external load vector and errors of load iterations.
Step 2. Solving Eq. (71) to present unknown displacement vector, {qss} and the stress vector is defined by (72), updating the geometric stiffness matrix [KG].
Step 3. Solving Eq. (71) to present unknown bending displacement vector {qbb}, and then testing stability conditions.
− If for all
− If at least one value
+ In case:
+ In case:
Stability analysis of piezoelectric composite plate with dimensions a × b × h, where a = 0.25 m, b = 0.30 m, h = 0.002 m. Piezoelectric composite plate is composed of three layers, in which two layers of piezoelectric PZT-5A at its top and bottom are considered, each layer thickness hp = 0.00075 m; the middle layer material is Graphite/Epoxy material, with thickness h1 = 0.0005 m. The material properties for graphite/epoxy and PZT-5A are shown in Section 5.1 above. One short edge of the plate is clamped, the other three edges are free. The in-plane half-sine load is evenly distributed on the short edge of the plate: p(t) = p0sin(2πft), where p0 is the amplitude of load, f = 1/T = 1/0.01 = 100 Hz (0 ≤ t ≤ T/2 = 0.005 s) is the excitation frequency, voltage applied V = 50 V. The iterative error of the load εD = 0.02% is chosen.
Consider two cases: with damping (ξ = 0.05, Gv = 0.5, Gd = 15) and without damping (ξ = 0.0, Gv = 0.0, Gd = 15). The response of vertical displacement at the plate centroid over the plate thickness for the two cases is shown in Figure 6.
Vertical displacement response at the plate centroid over the plate thickness.
The results show that the critical load of the plate with damping is larger than that without damping. In the two cases above, the critical load rises by 6.8%.
Analyze the stability of the plate with damping when a voltage of −200, −150, −100, −50, 0, 50, 100, 150 and 200 V is applied to the actuator layer of the piezoelectric composite plate.
Figure 7 shows the time history of the vertical displacement at the plate centroid over the plate thickness when a voltage of 0, 50, 100, 150 and 200 V is applied. The relation between critical load and voltages is shown in Figure 8.
Vertical displacement response at the plate centroid over the plate thickness.
Critical load-voltage relation.
The results show that the voltage applied to the piezoelectric layers affects the stability of the plate. As the voltage increases, the critical load of the plate also increases.
When the amplitude of the load changes from 0.25pcr to 1.5pcr (where pcr is the amplitude of the critical load), a voltage of 50 V is applied to the actuator layer of the plate.
The results show the time history response of the vertical displacement at the plate centroid over the plate thickness as seen in Figure 9.
Time history of the vertical displacement at the plate centroid over the plate thickness when p0 = 0.25pcr, 0.5pcr, 0.75pcr, 1.0pcr, 1.25pcr, and 1.5pcr.
Consider isoparametric piezoelectric laminated stiffened plate with the general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top surface and lower surface of the plate are bonded to the piezoelectric patches (actuator and sensor). The plate subjected to the airflow load acting (Figure 10).
Smart stiffened plate subjected to airflow. (a) Smart stiffened plate and coordinate system and (b) Lamina details.
The dynamic equations of a finite smart composite plate are written as follows:
where
where x-axis is taken along the stiffener centerline and the z-axis is its upward normal. The plate and stiffener element shown in Figure 11.
Modeling of plate and stiffener element.
If we consider that the x-stiffener is attached to the lower side of the plate, conditions of displacement compatibility along their line of connection can be written as:
where tp is the plate thickness and txs is the x-stiffener depth.
The element stiffness and mass matrices are defined as follows [2, 15]:
with [Bxs] is the strain-displacement relations matrix, [Dxs] is the stress-strain relations matrix and le is the element length,
The same as for x-stiffener, the element stiffness and mass matrices of the y-stiffener are defined as follows:
Based on the first order theory, the aerodynamic pressure lh and moment mθ, can be described as [15, 16, 17]:
where
The functions
where F(k) and G(k) are defined as:
Using finite element method, aerodynamic force vector can be described as:
with
where Ae is the element area, [Nw], [Nθ] are the shape functions.
From Eqs. (74) and (84), the governing equations of motion of the smart composite plate-stiffeners element subjected to an aerodynamic force without damping can be derived as:
where
Finally, the elemental equations of motion are assembled to obtain the open-loop global equation of motion of the overall stiffened composite plate with the PZT patches as follows:
where
The solution of nonlinear Eq. (89) is carried out by using Newmark direct and Newton-Raphson iteration method.
A rectangle cantilever laminated composite plate is assumed to be [0°/90°]s with total thickness 4 mm, length of 600 mm and width of 400 mm with three stiffeners along each direction x and y. The geometrical dimension of the stiffener is 5 mm of high and 10 mm of width. The plate and stiffeners are made of graphite/epoxy with mechanical properties: E11 = 181 GPa, E22 = E33 = 10.3 GPa, E12 = 7.17 GPa, ν12 = 0.35, ν23 = ν32 = 0.38, ρ = 1600 kg·m−3. Material properties for piezoelectric layer made of PZT-5A are: d31 = d32 = −171 × 10−12 m/V, d33 = 374 × 10−12 m/V, d15 = d24 = −584 × 10−12 m/V, G12 = 7.17 GPa, G23 = 2.87 GPa, G32 = 7.17 GPa, νPZT = 0.3, ρPZT = 7600 kg·m−3 and thickness tPZT = 0.15876 mm, ξ = 0.05, Gv = 0.5, Gd = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to examine the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative error of the load εD = 0.02% is chosen. The piezoelectric stiffened composite plate is subjected to the airflow in the positive x direction as shown in Figure 10a.
Dynamic response of the piezoelectric stiffened composite plate is shown in Figure 12.
History of the plate at a critical airflow velocity Ucr = 30.5 m/s. (a) Displacement response and (b) Piezoelectric voltage response.
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