Finite volume analysis of adaptive beams with piezoelectric sensors and actuators

Applied Mathematical Modelling xxx (2013) xxx–xxx

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Finite volume analysis of adaptive beams with piezoelectric sensors and actuators N. Fallah ⇑, M. Ebrahimnejad Department of Civil Engineering, Faculty of Engineering, The University of Guilan, P.O. Box 3756, Rasht, Iran

a r t i c l e

i n f o

Article history: Received 6 April 2012 Received in revised form 6 June 2013 Accepted 2 July 2013 Available online xxxx Keywords: Finite volume Smart beam Piezoelectric Active vibration control

a b s t r a c t This paper presents a finite volume (FV) formulation for the free vibration analysis and active vibration control of the smart beams with piezoelectric sensors and actuators. The governing equations based on Timoshenko beam theory are discretized using the finite volume method. For the purpose of forced vibration control of beam structures, the negative velocity feedback controller is designed for the single-input, single-output system. To achieve the best effect, the piezoelectric sensors and actuators are coupled with the host structure in different positions and then the performance of the designed control system is evaluated for each position. In the test examples, first the shear locking free feature of the present formulation is demonstrated. This has been performed by doing static and natural frequency analysis of some reference models. Then, the capability of the proposed method for the prediction of uncontrolled forced vibration response and active vibration control of a beam structure is studied. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Smart structure has been an interesting topic of research since the last two decades. This intelligent system composed of an instructing unit incorporated with smart materials such as piezoelectric, shape memory alloy, electro rheological and magneto rheological fluids and so on acting as sensor and actuator on the host structure. The intelligence and accountability of aforementioned system in dealing with structural vibration has made it attractive in many engineering applications. However, utilizing smart materials in civil engineering applications has some limitations and few researches concerning active vibration control of building structures by smart materials have been published so far. However, due to some unique features of smart materials such as fast response to the input signals, adaptability, cheapness, and lightness, a promising future of their civil engineering applications can be expected. Piezoelectric is one of the most popular smart materials which are widely used in vibration control of smart structures. Inherent reciprocal effect of electrical and mechanical domains, which is called electromechanical coupling, provides a feasible use of piezoelectric materials as both sensors and actuators. In sensor patches, the mechanical variables are exchanged into the electrical variables and vice versa, in actuators the electrical variables are exchanged into the mechanical variables. There have been many researches on the subject of piezoelectric sensors and actuators applications [1–3]. These works rather focused on engineering performances of the piezoelectric sensors and actuators in structural vibration control and few studies were carried out on the accuracy and efficiency of the numerical procedures used for the modelling. Finite elements method (FEM) is the most powerful numerical technique to simulate the behaviour of piezoelectric smart structures. Several aspects of piezoelectric smart structures modelling using FEM can be found in many research works. ⇑ Corresponding author. Tel./fax: +98 131 6690271. E-mail address: [email protected] (N. Fallah). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.07.004

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Vibration control of composite beam by taking into consideration of first-order shear deformation theory [4], study of shearbased piezoelectric actuation in vibration control of sandwich beams [5], demonstration of FE modelling accuracy of an active cantilever plate behaviour using an experimental model [6] and finite element model for analysis and design of active vibration control of smart plates [7,8] are examples of these studies. Besides, other numerical techniques such as boundary element method (BEM) and differential quadrature method (DQM) were also utilized for modelling of smart structures [9,10]. Although the above mentioned approaches give proper results, comparison of different numerical methods ability in simulating of complex composite structures and introducing a superior technique which predicts the structural behaviour more factual, seems to be necessary. Finite volume method (FVM), which is a traditional method in fluid mechanics, has been employed for the analysis of solid mechanics problems in the last two decades. Simplicity in concept and formulation, shear locking free feature in the analysis of thin Timoshenko beam model and in the bending analysis of thin Mindlin plates and also having the capability in accurate prediction of structural responses, are the most distinguished characteristics of this procedure [11]. Two common schemes for FVM have been reported in literature; the cell-vertex approach and the cell centred one [12,13]. In the cell-vertex approach, the field variables are computed at vertices of cells. For this purpose a control volume is constructed for each mesh vertex by joining the centre of surrounding cells to the middle of its adjacent sides; whereas in the cell-centred method each element acts as a control volume or cell and field variables are associated with the cell centres. In recent years, the FV methods have been applied to a number of solid mechanics problems. The solutions of solid mechanics problems with material non-linearity [14,15], FV analysis of dynamic fracture phenomena [16–18], analysis of two-and three-dimensional general structural dynamic problems [19,20], stress analysis of geometrically nonlinear solids [21,22], bending analysis of plates [11,23–25] and FV formulation for fluid-structure interaction [26,27] are among the literature concerning on finite volume procedure. The finite volume method has been also extended to model materials with heterogeneous microstructures, including periodic and functionally graded materials [28–32] and also applied to the elastic–plastic analysis of these types of materials [33]. The capability and efficiency of FV method in computational solid mechanics has been examined in the above works. In authors’ knowledge there is no published work concerning the finite volume application for the modelling of smart structures. However, the ability of more accurate structural response prediction by FV can lead to more efficient control of vibrations of smart structures, benefitted by a feedback controller. This paper addresses the finite volume formulation for static and dynamic behaviour of composite beams. At first, the efficiency of FVM in terms of accuracy and non-locking behaviour is demonstrated in some benchmark problems such as the cantilever and simply supported beams. Thereafter, by incorporating the smart layers acting as sensor and actuator into the main structure and using a feedback controller, the capability of present formulation in numerical simulation of active vibration control of smart beams is analysed. The effects of smart layers’ locations for the efficient vibration suppression of beam are studied. For this purpose, four different scenarios of smart layers arrangements are considered and optimal arrangement is determined. 2. Basic formulation 2.1. Piezoelectric constitutive equations The constitutive equations of a piezoelectric material, describing the electromechanical coupling are given as

r ¼ C e  eE; D ¼ eT e þ jE;

ð1Þ

where r, e, D and E are the elastic stress, elastic strain, electric displacement and electric field vectors, respectively, C is the elasticity matrix, e is the piezoelectric matrix and j is the dielectric permittivity matrix. The electrical field is the negative gradient of the electrical potential /, which is applied in the thickness direction. The first above equation is used to determine the stresses resulting from applied voltage to the piezoelectric actuators, while the second one presents the accumulated charges on the unit area of electrodes of sensors created by the structural strains. 2.2. Displacement formulation In the present study, the first order shear deformation theory (FSDT) or Timoshenko model is used to model a layered beam composed of a core and two outer layers of piezoelectric material. In this model the displacement fields are given as

uðx; z; tÞ ¼ zhðx; tÞ;

v ðx; z; tÞ ¼ 0;

ð2aÞ

wðx; z; tÞ ¼ wðx; tÞ;

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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where u, v and w are the mid-plane displacements along the coordinate axes x, y and z, respectively and hðx; tÞ represents the section rotation in xz plane. The electrical potential of smart layers, /ðx; tÞ, is also given as

/ðx; z; tÞ ¼ /ðx; tÞ:

ð2bÞ

This work is based on a linear strain–displacement formulation using the small strain assumption. Thus the strain field consists of the strain–displacement and electric field-electric potential relations are presented as follows

ex ¼ zh;x ; czx ¼ w;x þ h;

ð3Þ

Ez ¼ /;z : Or in the matrix form

 

e

 ¼

E

  u ; L/ /

Lu

0

0

ð4aÞ

where

e ¼ fex czx gT ; E ¼ fEz g; " # @ 0 z @x ; Lu ¼ @ 1 @x

ð4bÞ

u ¼ f w h gT ;  @ L/ ¼  @z : It should be noted that the electric potential / varies in a linear manner across the smart layers. So, the electric field is presented in terms of unknown variable / as

Ez ¼ 

/i ; hi

ð5Þ

where hi and /i are the thickness and applied electric potential of the piezoelectric patches, respectively. The mechanical and electrical boundary conditions are given as

u ¼ u on Cu ; nr ¼ t

on Ct ;

ð6Þ

/ ¼ / on C/ ; nD ¼ Q

on CQ :

where u, t, / and Q are the prescribed displacement, traction, electric potential and surface charge, respectively, on the displacement boundary Cu , traction boundary Ct , electric potential boundary C/ and electric charge boundary CQ of the control volumes. Matrix n contains the components of the unit outward normal to the boundaries of the control volumes:

 n¼

0

nx

znx

0

 ð7Þ

:

3. General discretization Using the principle of virtual displacements, one can obtain a relation, which is a basis for developing the discretized equations. For a layered beam with a core and two piezoelectric layers using the equivalent single layer model, we have

XZ i¼c;p Xi

þ

Z

Xp

deTi ri dXi 

d/T qdXp 

XZ

duT nri dCi 

i¼c;p Ci

Z

d/T nx DdCp ¼ 0;

XZ i¼c;p Xi

duT bdXi þ

XZ i¼c;p Xi

€ dXi þ duT qi u

XZ i¼c;p Xi

_ Xi þ duT ci ud

Z

dET DdXp

Xp

ð8Þ

Cp

where Xc and Xp are the control volumes associated with the core layer and outer piezoelectric layers (sensor layer and actuator layer) and Cc and Cp are their corresponding boundaries, respectively, c is the viscous damping matrix, b is the distrib€ is acceleration vector and u_ is velocity vector. The uted transverse load vector, q is the external electric charge vector, u material density matrix, q, is expressed as

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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q¼



q

0

0

qr 2

 ð9Þ

;

where q is the material density and r is the radius of gyration of the beam cross section. Substituting the constitutive stress–strain equation (1) and the strain–displacement equation (4a) into the above equation yields the displacement formulation as

XZ Xi

i¼c;p

þ

Z

dðLu uÞT ðC i Lu uÞdXi þ

XZ Xi

i¼c;p

€ dXi þ duT qi u

XZ i¼c;p

dðLu uÞT ðe/ÞdXp  Xp

_ Xi þ duT c i ud

Z

Xi

XZ i¼c;p

duT nðC i Lu uÞdCi  Ci

dðL/ /ÞT ðeT Lu u þ jL/ /ÞdXp þ

Xp

Z

Z

duT nðeL/ /ÞdCp 

Cp

d/T qdXp 

Z

XZ i¼c;p

duT bdXi

Xi

d/T nx ðeT Lu u

Cp

Xp

þ jL/ /ÞdCp ¼ 0:

ð10Þ

For both FE and FV methods, the unknown displacements u and electric potential / can be approximated by

^; u ¼ Nuu ^ / ¼ N / /;

ð11Þ

^ where the entries of N u and N / are shape functions corresponding to the mechanical and electrical unknowns, respectively. u ^ also represent the time-dependent displacement and electric potential vectors at the corresponding computational and / points. Substituting Eq. (11) into Eq. (10) results in

XZ Xi

i¼c;p



XZ

i¼c;p

þ

^ ÞdXi þ ðLu duÞT ðC i Lu N u u

Z

Xp

T

du bdXi þ

Xi

d/T qdXp 

Z

XZ i¼c;p

Xi

Z

^ Xp  ðLu duÞT ðeL/ N / /Þd

Xp

€^ dX þ du qi N u u i T

XZ i¼c;p

XZ i¼c;p

Ci

^_ dXi þ du c i N u u T

Xi

^ ÞdCi  duT nðC i Lu N u u Z

Z

^ Cp duT nðeL/ N / /Þd

Cp

^ Xp ^ þ jL/ N / /Þd ðL/ d/ÞT ðeT Lu N u u

Xp

^ Cp ¼ 0: ^ þ jL/ N / /Þd d/T nx ðeT Lu N u u

ð12Þ

Cp

Although in the presence of heterogeneities, delamination caused by stress concentration at the free edge is a distinct possibility in the multilayer beams, but when the main goal of the analysis is to determine the global structural behaviour, the equivalent single-layer laminate theories can accurately predict such responses [34]. It is worth noting that in the standard finite element method the virtual displacement and virtual electric potential are considered as equal to the shape function of the unknown field variables, that is du ¼ N u and d/ ¼ N / , which results in the displacement based finite element formulation. 4. Finite volume approach In finite volume method, the governing equilibrium equations are discretized over the control volumes. The cell centre finite volume scheme [24] is one of the common methods for defining the control volumes in which the domain is discretized into a mesh of elements. Each of these elements act as a control volume or cell and the governing equation is discretized over these volumes as shown in Fig. 1 for the layered beam problems. To obtain a finite volume based discretized equation, one may consider the Heaviside function as the virtual quantities, i.e.

Fig. 1. Discretization of a sandwich beam into the control volumes using the cell centred scheme.

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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du ¼ I

in X and

d/ ¼ I

du ¼ 0

elsewhere

d/ ¼ 0

ð13Þ

in which the virtual displacement and virtual electric potential are equal to zero where displacement and electric potential are prescribed.The above virtual quantities are conveniently chosen in order to achieve the equilibrium equations of the control volumes. Then, we have

" Lu du ¼

0

@ z @x

@ @x

1

#

1 0



 ¼

0 1

0 0



0 1

¼ ^I;

ð14Þ

L/ d/ ¼ 0; Consequently some integrals in Eq. (12) will be vanished and most of the remaining integrations are performed over the boundaries only, hence, the reduction of computational cost is achieved in deriving the stiffness matrix. In the cell centred approach for a given control volume, the computational points are assigned to the centre of cells and the unknown variables at cell faces are approximated according to unknowns of neighbouring cell centres. In the analysis of Timoshenko based beam model, there are two degrees of freedom corresponding to each cell centre which are transverse displacement and section rotation. Substituting Eqs. (13) and (14) into Eq. (12) results in

XZ i¼c;p

þ

^IðC i Lu N u u ^ ÞdXi þ

Xi

^IðeL/ N / /Þd ^ Xp 

Xp

XZ

XZ

qi N u u€^ dXi þ

Xi

i¼c;p

Z

i¼c;p

^_ dXi þ ci N u u Xi

Z

XZ i¼c;p

^ ÞdCi  nðC i Lu N u u

Ci

qdXp 

Z

^ Cp  nðeL/ N / /Þd Cp

Z

^ Cp ¼ 0: ^ þ jL/ N / /Þd nx ðeT Lu N u u

XZ i¼c;p

bdXi

Xi

ð15Þ

Cp

Xp

The above equation presents two linear equations for each control volume which represents the relation of unknowns at the cell centre to those at the centres of the two adjacent cells. Calculations of the integrals involved in Eq. (15) along the beam depth lead to the equilibrium of force and moment resultants acting on both sides of a cell (located at the grid points) which was already used by the first author [35] for the analysis of Timoshenko beams. Eq. (15) corresponding to all cells of the model along with the equations of the boundary conditions presented in Eq. (6) provide the final equations which can be written in compact form as



( € )  C uu ^ u €^ þ 0 0 /

M uu

0

0

( _ )  K uu ^ u _ þ K ^ 0 /u / 0

K u/ K //

    ^ Fu u ¼ ; ^ F/ /

ð16Þ

where M uu , C uu , K uu , K u/ (and K /u ) and K // are the global mass, damping, mechanical stiffness, piezoelectric and permittivity matrices of the composite structure, respectively. F u and F / are the mechanical excitation and electric excitation vectors, respectively. These parameters are determined by assembling the local matrices (for each control volume) which are given as v M cuu ¼

XZ Xi

i¼c;p

XZ

v ¼ C cuu

XZ

v K cu/ ¼

Z

^IðeL/ N / ÞdXp þ

Z

¼

XZ i¼c;p

Cp

Xp

v ¼ K c/u

F cuv ¼

^IðC i Lu N u ÞdXi 

Xi

i¼c;p

v K c//

c i N u dXi ;

Xi

i¼c;p

v K cuu ¼

qi N u dXi ;

Z

nðC i Lu N u ÞdCi ; Ci

nðeL/ N / ÞdCp ; ð17Þ

nx ðeT Lu N u ÞdCp ;

Cp

Z

nx ðjL/ N / ÞdCp ;

Cp

XZ i¼c;p

F c/v ¼ 

Z

bdXi ;

Xi

qdXp ;

Xp

where superscript cv denotes the control volume. Note that the above procedure for forming the equation of motion is different from that of the FEM method. In the FEM, the element matrices are assembled symmetrically into the global matrix. However, in the FV method, the cell (control volume) related equations are stacked together row-by-row to form the global matrix. Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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5. Control system design For the purpose of designing a smart structure which responses to the environmental changes intelligently, it is required to an integrated control system including controller, sensors and actuators. However, the efficiency of the designed controller strongly depends on the position of sensors and actuators as well as their shapes and sizes. Placing the smart elements in their optimal locations can improve the controllability and observability of controller and also can reduce the required efforts to get the designed goals [36]. In this paper, the negative velocity feedback controller is designed by choosing a proper gain matrix to attain the most reduction in structural responses along with the optimum controlling efforts. This control law relates the input electrical potential of actuators to the time gradient of the voltage produced in sensors as follows [8]

/a ðx; tÞ ¼ G

@/s ðx; tÞ ; @t

ð18Þ

where G represents the control gain. It should be mentioned that in the case of single-input single-output (SISO) controllers this parameter has a single value, while in the more complex multi-input multi-output (MIMO) systems with a large number of smart elements, the control gain is expressed by a matrix. Before substituting the above equation in Eq. (16), we first partition equation (16) as follows

2

M uu

6 4 0 0

38 € 9 2 >u > 0 0 > C uu < ^ > = 7 ^€ 6 0 05 / þ 4 0 s > > > > : ; € 0 0 0 ^a /

38 _ 9 2 >u > 0 0 > K uu < ^ > = 7 ^_ 6 0 05 / þ 4 K su s > > > > : ; _ 0 0 K au /^ a

K us K ss K as

38 9 8 9 ^ > K ua > = > = 7 K sa 5 /^s ¼ F s ; > : ^ > : > ; > ; K aa Fa /a

ð19Þ

where M uu , C uu , K uu and F u have been introduced in the previous section. K us (and K su ) and K ss are the piezoelectric and permittivity matrices of the sensor layers, respectively, and K ua (and K au ) and K aa are the above mentioned properties for the actuator layers, respectively. K sa and K as are the null matrices and F s and F a represent the vectors of external applied electrical charge at the sensor and actuators, respectively. According to the second row of Eq. (19) and also the zero electrical charge at the sensor layers, the produced voltage at sensor resulting from the structural deformations is obtained as

^: /^s ¼ ðK ss Þ1 K su u

ð20Þ

Substituting Eq. (20) into Eq. (18) yields

^_ : /^a ¼ GðK ss Þ1 K su u

ð21Þ

Finally, using Eqs. (20) and (21) into the first row of Eq. (19), the equation of motion of a layered cell equipped with the piezoelectric sensors and actuators in the presence of the control system becomes

€^ þ C u ^_ þ K u ^ ¼ Fu; M uu u

ð22aÞ

where

C ¼ C uu þ K ua GðK ss Þ1 K su ;

ð22bÞ

K ¼ K uu  K us ðK ss Þ1 K su :

To obtain the structural responses, Eq. (22a) is rewritten in the standard state space form in which the second order equation turns into the following first order one

x_ ¼ Ax þ B;

ð23aÞ

where state vector x, state matrix A and external excitation vector B, are given as

x¼

   ^ I u ; A ¼ ^_ ðM uu Þ1 K u



0 1

ðM uu Þ C

 ; B¼



0 1

ðM uu Þ F u

ð23bÞ

in which I is an identity matrix. It should be noted that the global damping matrix, C uu , is usually approximated using the Rayleigh damping which is a combination of the global stiffness and mass matrices of the structure as follows [37]

C uu ¼ aM uu þ bK uu ;

ð24Þ

where a and b are the mass and stiffness proportional damping coefficients and can be determined as

a ¼ 2ni



xi xj xi þ xj



 and b ¼ 2ni

1 ; xi þ xj

ð25Þ

where xk and nk are the natural frequency and the damping ratio of kth mode, respectively. Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

N. Fallah, M. Ebrahimnejad / Applied Mathematical Modelling xxx (2013) xxx–xxx

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6. Application and discussion The purpose of the following case studies is to investigate the validity and efficiency of the proposed formulation for the static analysis, natural frequency analysis and vibration control of the beam structures with smart layers. For this purpose, first, FV formulation is verified on a number of benchmark problems and then, using piezoelectric patches incorporating with a negative velocity feedback control algorithm, active control of beam vibrations is studied. In order to achieve more amount of overall damping in vibration suppression, a spatial optimization is performed for finding the best positions of the piezoelectric elements along the host beam. 6.1. Case study I: cantilever beam A flexible cantilever layered beam with aluminium core and two outer bounded, symmetrically located piezoceramic layers are considered for evaluating the proposed finite volume based formulation, see Fig. 2. Both static response and natural frequency calculations are performed. The aluminium core dimensions are 508 mm length and 25.4 mm breadth and the thickness of both piezoceramic layers are 0.1 mm. The material properties of the layers are presented in Table 1. 6.1.1. Static response This test example concerns the static response of the above described cantilever layered beam under a concentrated load at the free end (1 N). The layered beam is discretized to a number of control volumes along its central longitudinal axis. Different numbers of cells are used for the discretization. The variation of normalized transverse displacement along the beam length is obtained and shown in Fig. 3, which is in great agreement with the analytical result given by

w¼

3  2

x3  pl x pl x þ ; 3  l l 6EI kGA l

ð26Þ

where p, l, k, EI and GA represent the concentrated load, beam length, shear correction factor, equivalent bending rigidity and equivalent shearing rigidity, respectively. It is noticeable that Eq. (26) has been concluded from the analytical equation given in Ref. [34] for the laminated composite beams. To demonstrate the capability of the proposed formulation for the analysis of thin to thick beams, different values of thickness for the aluminium core is considered that provides aspect ratio, L/h, in the range of 5–100. The predicted results for the tip displacement are compared with the analytical results and the error of predictions are calculated and shown in Fig. 4. As shown in this figure, by increasing the number of cells, tip vertical displacement converges to the reference results. It is noticeable in Fig. 4 that the proposed formulation can predict accurately the tip vertical displacement of the considered thin beam with L/h = 100. It is well known that the powerful finite element technique cannot handle the analysis of thin beams by using the Timoshenko model easily due to the shear locking phenomena and special treatments have been proposed to overcome this deficiency [39–42]. The shear locking free behaviour of finite volume formulation has been already observed in plate analysis [11,23]. 6.1.2. Natural frequency analysis Calculation of the natural frequencies of the layered beams is a prerequisite for the vibration analysis of the smart beams. This can be accomplished by obtaining the corresponding eigenvalue equation. The undamped natural frequencies can be determined by solving Eq. (22a) in the absence of the damping force and external excitations, i.e.

€^ þ K u ^ ¼ 0: M uu u

ð27Þ

It should be noted that according to Eqs. (4a) and (9), both the shear deformation and rotational inertia effects are con^ is given by the harmonic equation in time sidered in Eq. (27). The standard solution for u

^ ¼ A0 eixt ; u

ð28Þ

Fig. 2. Three-layered smart cantilever beam.

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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N. Fallah, M. Ebrahimnejad / Applied Mathematical Modelling xxx (2013) xxx–xxx Table 1 Material properties of the smart beam (Case study I) [38].

q (kg m3) c11 (GPa) c55 (GPa) e31 (C m2) e33 (108 F m1)

Aluminium

PZT

2769 68.97 27.59 – –

7600 68.97 20.69 8.41 1.05

Fig. 3. Prediction of vertical displacement for the cantilever beam.

Fig. 4. Convergence of the tip displacement predictions for the cantilever beam with different aspect ratios.

where A0 is the part of the nodal displacement matrix called natural modes that is assumed to be independent of time, i is the standard imaginary number and x is the natural frequency. Substituting Eq. (28) into Eq. (27) yields

A0 eixt ðK  x2 M uu Þ ¼ 0:

ð29Þ

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N. Fallah, M. Ebrahimnejad / Applied Mathematical Modelling xxx (2013) xxx–xxx

Then, we have

A0 ðK  x2 M uu Þ ¼ 0:

ð30Þ

The above equation is a set of linear homogeneous equations in terms of displacement mode A0 . Hence, this equation has a nontrivial solution if and only if the determinant of the coefficient matrix of A0 is zero. That is

jK  x2 M uu j ¼ 0:

ð31Þ

Eq. (31) is a standard linear eigenvalue equation and its solutions give the natural frequencies of the system. To demonstrate the capability of the present finite volume formulation for the prediction of the natural frequencies of the layered beams, the following test is performed. In this test, the first and the second natural frequencies of the cantilever Timoshenko beam, shown in Fig. 2, with different aspect ratios are calculated using Eq. (31). The results obtained are presented in Table 2 and are in excellent agreement with the analytical results calculated according to Ref. [34,43]. Error in the above predictions is also shown in Fig. 5. As observed in this figure, by increasing the number of cells, more accurate results can be obtained. It is seen that in the prediction of the first natural frequencies of the thin and thick beams; errors are less than 0.5% even using a few cells. In the prediction of the second natural frequencies, errors are also less than 2.5%. 6.2. Case study II: simply supported beam Before applying the proposed formulation for the active vibration control, a uniformly loaded simply supported layered beam shown in Fig. 6 is considered for the analysis. The beam with total length of L ¼ 400 mm and width of b ¼ 5 mm is composed of an aluminium core with thickness hc and two piezoelectric layers with equal thickness of 1 mm. The material properties of the aluminium and piezoelectric materials are presented in Table 3. 6.2.1. Static response In order to analyse the static behaviour of the above beam, a uniformly distributed transverse load of 1  105 N m1 is applied to the beam. The beam is divided into a number of equal sized cells and the transverse displacement of cell centres is obtained by applying the presented formulation. The variation of vertical displacement along the beam length is shown in Fig. 7. According to this figure, the displacement profile agrees well with the analytical solution given by

w¼

4 

x3 x4  ql2  x x2  ql x 2  þ ; þ l l l 24EI l kGA l

ð32Þ

where q, l, k, EI and GA represent the uniformly distributed transverse load, beam length, shear correction factor, equivalent bending rigidity and equivalent shearing rigidity, respectively. The above equation is concluded for the laminated composite beams from the provided equation in Ref. [34]. In order to investigate the effect of aspect ratio, L/h, on the performance of the present formulation, different aspect ratios from 5 to 100 are considered. The mid-span transverse displacements of both thin and thick beam models are calculated and compared with the reference values. Fig. 8 shows how the error in the predictions reduces by increasing the number of control volumes. It can be seen that an excellent agreement exists between the results obtained and the analytical reference results. This test again reveals the ability of FV formulation in accurate prediction of beam bending responses. 6.2.2. Natural frequency analysis For the above beam, the natural frequency calculations are also performed similar to the previous test. In order to validate the accuracy of the FV formulation in free vibration analysis, the calculated results are compared with the analytical natural frequencies of the simply supported layered Timoshenko beam obtained according to Ref. [34,43]. Results are presented in Table 4 which are in excellent agreement with the analytical solutions.

Table 2 Natural frequencies of the layered cantilever beam with different aspect ratios. Natural frequency

L/h

Analytical solution [34,43]

FV (no. cells = 101)

Error (%)

x1 (rad/s)

5 10 100

1932.3190 986.1303 96.4506

1932.3945 986.1712 96.4547

0.0039 0.0041 0.0043

x2 (rad/s)

5 10 100

10442.0360 5916.5101 604.1553

10443.4963 5917.6335 604.2849

0.0140 0.0190 0.0215

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Fig. 5. Convergence of the natural frequencies predictions for the cantilever beam with different aspect ratios.

Fig. 6. Three-layered simply supported smart beam.

Table 3 Material properties of the smart materials (Case study II) [38].

q (kg m3) c11 (GPa) c55 (GPa) e31 (C m2) e33 (108 F m1)

Aluminium

PZT

2700 72.2 27.77 – –

7750 60.97 21.05 10.43 1.327

Errors in the prediction of the first two natural frequencies, in comparison with the analytical values obtained from [34,43], are also presented in Fig. 9. It can be observed in this figure that a relatively small error value occurred in all predictions. 6.2.3. Active vibration control Active vibration control of a pinned-pinned smart beam structure is studied in this section. The structure consists of a host aluminium beam (500  50  25 mm3) and a set of PZT-5A piezoelectric layers with dimensions of (100  50  8 mm3) working as sensor and actuator. The actuator and sensor are completely bonded to the upper and lower surfaces of the host beam, respectively. To study the effect of sensor and actuator locations on the performances of the control system, similar to the works presented in [7,44] four different arrangements of sensor and actuator sets are considered: collocated cases I and II, and non-collocated cases I and II, shown in Figs. 10 and 11.The material properties of PZT and aluminium core are given in Table 3. A block diagram of the designed controller system is presented in Fig. 12.The dynamic response of the smart beam is studied under a concentrated pulse load of 500 N applied at the mid-span of the beam. Diminishing the vertical displacement of the mid-span is selected as the performance index of the control system. Damping matrix is constructed based on the uniform structural damping ratio ðni Þ with the value of 0.05%. The sensory voltage and applied voltage to the piezoelectric actuators are selected as the input and output variables of the controller, respectively. The uncontrolled displacement, U uc , and controlled displacement, U c , of the beam mid-span for collocated cases I and II are shown in Fig. 13. Also, the corresponding applied actuation voltages of the actuators of both cases are shown in Fig. 14. Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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Fig. 7. Prediction of vertical displacement for the simply supported beam.

Fig. 8. Convergence of the mid-span displacement predictions for the simply supported beam with different aspect ratios.

Table 4 Natural frequencies of the layered simply supported beam with different aspect ratios. Natural frequency

L/h

Analytical solution [34,43]

FV (no. cells = 101)

Error (%)

x1 (rad/s)

5 10 100

6721.0272 3424.0949 246.0525

6721.2352 3424.2238 246.0623

0.0031 0.0038 0.0040

x2 (rad/s)

5 10 100

23283.1585 13069.6964 983.6732

23284.4985 13071.3102 983.8317

0.0058 0.0124 0.0161

For a constant gain, the reduction of structural displacement in collocated case I is more noticeable relative to the collocated case II. The performance comparisons of the considered cases can also be accomplished by calculating the actuator voltage (or control force) required to damp the structural response in a certain settling time. For this purpose the settling time of 2 s

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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Fig. 9. Convergence of the natural frequencies predictions for the simply supported beam with different aspect ratios.

Fig. 10. The model of simply supported beam structure with piezoactuator/sensor: (a) collocated configurations I and (b) collocated configurations II.

Fig. 11. The model of simply supported beam structure with piezoactuator/sensor: (a) non-collocated configurations I and (b) non-collocated configurations II.

External Force Structure

Sensor Voltage

Control Force Piezoelectric Actuators

Control Voltage

Velocity Feedback Controller

Fig. 12. Diagram of the control system.

is considered. As shown in Fig. 14, it is found that the maximum required voltage for suppressing the mid-span displacement in 2 s is 210 V for case I and 712 V for case II. Therefore, it is concluded that the location of sensors and actuators has a considerable effect on the controlling performance of the smart beams. In the non-collocated configurations, which are designed under the same conditions as in the collocated cases, mid-span displacement of the beam and actuation voltage of both non-collocated cases are obtained and shown in Figs. 15 and 16,

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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Fig. 13. Mid-span displacement response of simply supported beam structure with piezoactuator/sensor: collocated configurations.

Fig. 14. Applied control voltage to the simply supported beam structure with piezoactuator/sensor: collocated configurations.

respectively. As shown in Figs. 15 and 16, identical displacement responses of two configurations, with or without active control system, are observed. However, the non-collocated case I requires more control voltage (control force) compared to the non-collocated case II. As can be observed in Figs. 14 and 16, among the above four cases, collocated case II requires the greatest control voltage to achieve the settling time of 2 s, and collocated case I requires the lowest control voltage (see Table 5) for the same settling time. This observation is due to the differences in settling time of the uncontrolled structure, so called natural settling time, and also due to the actuators’ placement. It should be mentioned that the natural settling time represents the time that the structural vibration is suppressed by intrinsic structural damping. Two non-collocated cases have the same natural settling time, but for the case with actuator closer to the mid-span (case II) less control voltage is required and therefore it is preferable configuration relative to the case I. From the results of the above studies, it is observed that different sensor/actuator placements results in different control performances. Also, it is found that the control system is more effective when the actuator is closer to the mid-span of the beam. Further studies can be performed for studying the effect of the selected settling time on the required control voltage. It is reasonable to expect that the required control voltage decreases by increasing the value of the settling time.

Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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Fig. 15. Mid-span displacement response of simply supported beam structure with piezoactuator/sensor: non-collocated configurations.

Fig. 16. Applied control voltage to the simply supported beam structure with piezoactuator/sensor: non-collocated configurations.

Table 5 Performance comparisons of four different piezoelements locations on a simply supported beam (with identical settling time of 2 s).

Maximum actuation voltage (V)

Collocated I

Collocated II

Non-collocated I

Non-collocated II

210

712

538

257

It should be noted that in the practical design, the maximum voltage applying on piezoelectric patches is restricted within the allowable ranges to prevent the electric breakdown. These ranges are predefined based on the material properties and device characteristics. 7. Conclusion A finite volume formulation for static and dynamic analysis of adaptive beams bonded with piezoelectric layers has been presented. The present formulation has been validated for static analysis and natural frequency calculations of thin and thick layered beams in some models such as the cantilever beams and simply supported beams. Thereafter, by incorporation of the Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004

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electro-mechanical effects of the smart layers and using a feedback controller, the capability of finite volume based model for the analysis of controlled response of smart beams has been studied. Comparisons between numerical and analytical displacements and resonance frequencies indicate that the proposed formulation correctly predicts the static and dynamics of layered beams with typical geometry and material properties. This is shown by the good agreement between the numerical and analytical natural frequencies and response functions. Numerical results also have shown that the position of piezoelectric layers has considerable effects on the control performances. Proper placement of smart elements has been identified by studying some indexes such as response reduction value accompanying with the input voltage value.

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Please cite this article in press as: N. Fallah, M. Ebrahimnejad, Finite volume analysis of adaptive beams with piezoelectric sensors and actuators, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.07.004