- Email: [email protected]

S0022-460X(19)30709-6

DOI:

https://doi.org/10.1016/j.jsv.2019.115146

Reference:

YJSVI 115146

To appear in:

Journal of Sound and Vibration

Received Date: 22 April 2018 Revised Date:

3 December 2019

Accepted Date: 11 December 2019

Please cite this article as: S.M. Hasheminejad, M.M. Mohammadi, A. Jamalpoor, Hydroelastic modeling and active control of transient sloshing in a three dimensional rectangular floating roof tank, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/j.jsv.2019.115146. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Hydroelastic modeling and active control of transient sloshing in a three dimensional rectangular floating roof tank a

b

a

Seyyed M. Hasheminejad *, M. M. Mohammadi , and Ali Jamalpoor a

Acoustics Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114 Iran, Phone: 98912-7371354, Fax: 9821-77240488, *Email: [email protected] b Malek Ashtar University of Technology, Tehran, Iran, Email: [email protected]

Abstract- Partly filled rectangular containers are widely used for liquid transportation and storage in many industrial and civil applications. Undesirable liquid sloshing in these structures can highly degrade their reliable and safe operations.

This paper presents a

rigorous 3D hydro-elasto-dynamic analysis for suppression of transient liquid sloshing in a rigid-walled rectangular parallelepiped container that is equipped with a smart piezosandwich free-floating rectangular panel. The problem formulation is based on the linear water wave theory, the thin piezo-sandwich floating plate model, the pertinent structure/liquid compatibility condition, and the active damping control strategy.

The

controller gain parameters are systematically tuned using a standard MOPSO algorithm with conflicting objective functions.

The key hydro-electro-elastic response parameters are

calculated and discussed for three different external excitations, namely, a bidirectional seismic event, an oblique planar base acceleration, and a distributed impulsive floating roof excitation. Effectiveness of proposed active floating panel control configuration in remarkable suppression of the main hydro-elastic parameters is established, essentially regardless of liquid depth and loading configuration. Limiting cases are considered and validity of model is demonstrated against available data as well as by comparison with the results of a general finite element package.

Keywords: analytical solution; rectangular container; smart floating roof; active anti-slosh system; liquid transport vessels; seismic excitation.

1

1. INTRODUCTION Sloshing is known as the free surface oscillation of liquid in an externally-excited partially filled container.

This important physical phenomenon is a problem of serious

concern that has far reaching applications in a wide field of technologies and engineering disciplines [1,2]. It can undesirably affect the dynamic performance, stability, structural integrity, and safety of many civil and industrial liquid containment systems or structures. Such structures should be designed to resist a wide range of external excitations such as earthquakes, impacts, base motion, and flooding. A close scrutiny of the literature reveals that while extensive research efforts has been focused on the dynamic liquid-tank interaction of cylindrical form [3–5], comparatively fewer contributions have been made on the hydrodynamic characteristics of rectangular-shaped containers [6,7]. On the other hand, numerous authors have investigated the coupled hydro-elasto-dynamic characteristics of flexible plates in contact with finite fluid domains for a wide variety of liquid and plate arrangements since the pioneering works of Rayleigh [8] and Lamb [9]. For example, Yildizdag et al. [10] used a combined isogeometric finite element and boundary element approach to study the hydroelastic vibrations of vertically or horizontally submerged rectangular plates. Kolaei and Rakheja [11] developed a finite element model to examine free liquid vibrations in a tank of arbitrary geometry fitted with a flexible membrane freesurface cover. Canales and Mantari [12] presented an analytical solution based on the Ritz method for free vibration analysis of thick rectangular isotropic bottom plates coupled with a bounded fluid for various boundary conditions. Bendahmane et al. [13] used a hierarchical finite element method to study natural frequencies of variable stiffness composite laminate rectangular plates submerged in fluid for different submersion depths. Design and application of effective liquid slosh suppression techniques can help structural engineers in maintaining stability, structural integrity, and safety of liquid containment systems. Several researchers have considered a wide variety of passive methods to mitigate slosh energy based on various physical devices such as baffles [3], tuned flexible absorbers [14], and floating roofs [4]. The flexible floating panel is a particularly viable and functional option that requires no support from the tank side walls. Although the use of passive elements is very effective, it adds undesirable weight and extra cost to the system. Accordingly, a number of authors have recently implemented various control techniques for suppression of liquid sloshing in partially filled containers. Among them, Hernández and Santamarina [6] adopted an optimal active vibration control technique for suppression of liquid sloshing within a rectangular container by application of force actuators on Reissner– Mindlin baffle plates.

Zang and Huang [7] designed a command smoothing technique to

suppress 3D liquid sloshing within a moving rectangular liquid container. Luo et al. [15]

2

proposed a hybrid method that utilizes an adjustable viscous mass damper to suppress liquid sloshing in cylindrical storage tanks under seismic excitations.

Hasheminejad and

Mohammadi [4] presented a 3D hydro-elastic analysis for active control of transient liquid sloshing in an upright flexible cylindrical shell tank by using a free-floating smart piezosandwich circular panel.

Just recently, Hamaguchi [16] developed a simple 6-dof parallel

linkage vibration reducer mechanism based on a frequency-dependent optimal servo control system for active suppression of liquid sloshing in a cylindrical container that is transferred along a curved path on a horizontal plane. The above brief literature review points to the relative abundance of theoretical solutions for liquid sloshing in a cylindrical floating-roof tank configuration [4,5]. On the other hand, a rigorous analytical hydroelastic model for describing the 3D forced transient liquid sloshing in a rectangular floating roof tank seems to be nonexistent. The first goal of present paper is to fill this important gap in the literature. Such analysis takes full account of the hydroelastic interaction between the flexible floating panel and the oscillating container liquid. This allows for expedient implementation of preliminary parametric studies which can lead to judicious establishment of qualitative trends in the early design stage with a much lower cost than the usual numerical and experimental methodologies. Moreover, the second and primary goal of the current work is to extend the work of Ref. [4] that involves active control of liquid sloshing in a cylindrical floating-roof tank to the rectangular geometry which is of great interest in numerous technological and industrial applications [7,17–19]. This is achieved after elaborate development of the pertinent piezo-sandwich plate model in conjunction with the powerful active damping control (ADC) strategy [20]. In this regard, the key contributions of the proposed manuscript with respect to the previously published models [4,5] may more concisely be stated in two folds: • Formulation of a coupled hydro-elasto-dynamic analytical model for forced transient 3D liquid sloshing in a rectangular parallelepiped floating roof tank subjected to arbitrary time-dependent planar base excitations. • Development and application of a rectangular piezo-sandwich floating plate model in conjunction with a systematically-tuned ADC control strategy for effective suppression of 3D transient sloshing in the rectangular parallelepiped container. The proposed analytical model avoids the intrinsic accuracy and modeling complications associated with the equivalent multi-dof mechanical models that normally involve pendulum or mass-spring analogies [21]. It is of high industrial and academic interest due to its importance in a wide range of technologies and engineering applications [7,17–19]. Additionally, the presented set of accurate space-time solutions provide a practical benchmark for evaluating approximate analytical and/or strictly numerical solutions [16–19]. 3

2. FORMULATION 2.1 Hydrodynamic model Consider a rigid-walled liquid-filled rectangular parallelepiped container of liquid height

H, and base dimensions ( × ), that is fully covered by a freely-floating smart piezocoupled rectangular panel, as depicted in Fig. 1. The container liquid is assumed to be ideal

incompressible with irrotational flow. Two parallel Cartesian coordinate systems (; , , )

and ( ; , , ) are attached at the corner of floating rectangular panel and tank,

respectively. The coupled tank-floating plate system is subjected a planar bidirectional base acceleration, (), with the incidence angle , along with a transverse distributed general

transient mechanical surface load, ( , , ), as shown in Fig. 1. Based on the linear three-

dimensional water wave theory, the hydrodynamic velocity potential is governed by the classical Laplace equation in Cartesian coordinates [1,2]: ∇ Φ( , , , ) = 0.

(1)

with the general solution expediently decomposed in the form [22]: Φ = Φ + Φ ,

(2)

where ∇ is the classical Laplacian, and the so-named “rigid-body” and “sloshing” components are written as [23]:

Φ (, , ) = ! () cos + ! () sin , /

/

Φ ( , , , ) = ( ( )*+ (),*+ ( , )cosh(.*+ ),

(3)

*0 +0

where )*+ () are unknown modal constants, and ; <

.*+ = 12 4 6 + 7 4 9: , ,*+ ( , ) = cos 2 4 6 cos 7 4 9. *3 5

+3 8

*3

+3

5

8

Also, the total pressure field is expanded in the form [1,2]:

/

/

! (),*+ ( , )cosh(.*+ ), =( , , , ) = −?Φ! = −[email protected] () cos + () sin A − ? ( ( )*+ *0 +0

where ? is the sloshing liquid density.

4

(4)

2.2 Structural model

The symmetrically-laminated piezo-coupled rectangular plate ( × ) includes an

isotropic elastic base layer of thickness 2ℎ and material density ? , perfectly bonded to the fully-electroded transversely-poled piezo-skin layers of thickness ℎD and material density ?D ,

as depicted in Fig. 1.

Based on Kirchhoff’s classical plate deformation theory, the

displacement components (E , E , EF ) of the composite thin rectangular panel in the local (; , , ) coordinate system are written as [24]: HGF ( , , ) , H HGF ( , , ) E ( , , , ) = G ( , , ) − , H EF ( , , , ) = GF ( , , ), E ( , , , ) = G ( , , ) −

(5)

where (G ,G ,GF ) denote the base panel mid-surface deformation, and the pertinent straindisplacement relations are given as [24]:

HE HG H GF I = = − , H H H HE HG H GF (6) I = = − , IF = JK = JF = 0, H H H HE HE HG HG H GF J = + = + − 2 . H H H H H H Also, with the thin plate hypothesis, the mid-surface in-plane displacements (G , G ) may be neglected and the strain vector can be written in the reduced form [25]: H EF H EF H EF L = MI I J N = − P 2 Q . H H H H

(7)

HU HU HU R SD = MT T TF N = 1− − − : , H H H

(8)

R

O

Besides, the electric field vector for the piezo-coupled rectangular panel, SD , is given as [25]: R

where the transverse electrical potential, U( , , , ), may be approximated by a half-cosine function as [24]:

2ℎ + ℎD V 1 W ∓ YQ U ( , , ) + \ ] + ℎ + ℎD ^] − ℎ ^U_ (), ℎD 2 ℎ[ (9) in which U ( , , ) is linked to the direct piezoelectric effects, U_ () is the applied external

U( , , , ) = − cos P

electric potential (actuation voltage), and the minus sign refers to the electric potential

distribution of the actuator layer, while the plus sign denotes that of the sensor layer with

5

U_ () set equal to zero.

Besides, the electric potential should satisfy the short circuit

boundary condition in the absence of external applied voltage, i.e. [26], U( , , ±ℎ , ) = 0,

UM , , ±]ℎ + ℎD ^, N = 0.

(10)

Next, the constitutive relations for the isotropic base and transversely isotropic piezo-skin layers are respectively written as [27]: a = b L, cD = bdD L − edRD SD , fD = edD L + gdD SD ,

(11) R

R

D R

where c = Mh , h , h N , cD = Mh , h , h N , fD = Mi , i , iF N , denote the relevant D

D

D

D

D

stress and electric displacement vectors, respectively, and the associated coefficient matrices are defined as

q̅

q̅ 0 0 0 p, bd = nq̅ q̅

0 b = lm< p, D

lmk

k 0 0 0 0 (q̅

− q̅ ) ̅ v

0 0 0 0 0 ̅ 0 0u , gdD = n 0 v

edD = s 0 0 p, t̅\ t̅\ 0 ̅ 0 0 v\\ in which T and o denote the elastic modulus and Poisson’s ratio of the base layer, jk

1 no

o 1

respectively, and bdD , edD , and gdD , refer to the reduced stiffness, reduced piezoelectric, and

enhanced dielectric matrices, respectively, with the associated elements defined as follows: q \ q \ 9 , q̅ = q − 7 9 , q\\ q\\ w;x yxx ̅ = v

, v\\ ̅ = v\\ + t\\ = t\ − 2 w 6, v

/q\\,

q̅

= q

− 7 t̅\

xx

where q

, q , q\ , v

, v\\ , and t\ refer to the piezoelectric material constants. Also, the

resultant moment vector associated with an equivalent single layer plate is expressed as [28]: R

{ = M| | | N = }

~k ~

l~k l~

c d = }

~k

l~k

c d + 2 }

~k ~

~k

cD d ,

(12)

in which c is the stress vector, and by direct implementation of the first two of Eqs. (11) into

Eq. (12), one obtains

4ℎD t̅\ H EF H EF H EF (13) @ 1 1 0AR U , { = P− − − 2 Q + H H H H V \ where = ℎ\ b + ]ℎ + ℎD ^ − ℎ\ bdD , is the equivalent bending stiffness matrix. O

\

Now, the basic Kirchhoff’s plate dynamic equation for the equivalent single layer panel is

stated as [29]:

H H H EK + + ( , , ) = 2]? ℎ + ?D ℎD ^ , H H H 6

(14)

( , , ) = ( , , ) − =( , , = , ) + ( , , ), is the net external

where

transverse force acting on the rectangular panel, in which ( , , ) is the mechanical component (see Fig. 1), =( , , = , ) is the liquid surface pressure, and ( , , ) =

( , )U_ () is the electrical component or the applied actuator voltage, with ( , ) being

its spatial distribution. Also, the resultant shear force components are given as

H| H| H| H| (15) + , = + . H H H H Direct substitution of Eqs. (15) into Eq. (14), and keeping Eq. (13) in mind, one can readily =

obtain the governing panel dynamic equation in its initial form:

H EF H EF H EF ) + + 2( + 2

H H H H (16) 4ℎD t̅\ H EF − ∇ U + 2]? ℎ + ?D ℎD ^ = ( , , ), V H where ∇ = H ⁄H + H ⁄H , and (, = 1,2,6) refer to the nonzero elements of the

bending stiffness matrix, , as defined in the Appendix (see Eq. A-1).

Next, by direct integration of Maxwell’s static electricity equation [25] across the piezo-

coupled panel thickness, i.e., ~k ~

~

k

∇. fD d + l~ kl~ ∇. fD d = 0 , l~

k

one obtains: ̅ 2ℎD v

2V ̅ U − ℎD t̅\ ∇ EF = 0 . (17) ∇ U − v\\ V ℎD In the absence of external forces (i.e., = 0), and after elimination of ∇ U between Eqs. (16) and (17), one obtains the key piezoelectric potential relation in the form:

̅ ℎD v

H EF H EF H EF ) P

+ + 2( + 2

̅ H H H H 4Vt̅\ v\\ (18) ℎD t̅\ −2 ∇ EF + 2]? ℎ + ?D ℎD ^EK − ( , , )Q. ̅ v

Back substitution of Eq. (18) into the governing Eq. (16) after some manipulations, leads to U ( , , ) =

the final form of equation of motion for the piezo-laminated rectangular panel:

H EF H EF H EF ) + + 2( + 2

H H H H ℎD H EF H EF H EF H EF ̅

W ̅

W − Pv

+ Y + v + Y ̅ H H H H H H V v\\ H EF H EF ̅ ( + 2 ) W ̅ ]? ℎ + ?D ℎD ^∇ EF +2v

+ Y + 2v

H H H H −2ℎD t̅\ ∇ EF N + 2]? ℎ + ?D ℎD ^EK = ( , , ),

7

(19)

where, based on the elaborated method of eigen-function expansion in

rectangular

coordinates, the deformation of the liquid-coupled plate may be expressed as a linear combination of the corresponding free-vibration modes of the plate in vacuum. Accordingly,

one can adopt the following explicit general form of solution for the freely-floating rectangular smart panel [30]: /

/

EF ( , , ) = ( ( [ ()[ ( , ) , [0 0

in which [ () (=, = 1,2, … ) are panel modal coordinates,

(20)

[ = (=V/ ), =

]V/ ^, and the free vibrational mode shapes are given as [ ( , ) = cos [ cos ]1 +

sin [ ^]1 + sin ^.

2.3 Floating panel/liquid coupling The knowledge of the dynamics of a flexible structure or the contacting liquid alone is not sufficient to understand the intricate fluid-structure interaction problem.

In particular,

the hydrodynamic pressures induced by a vibrating floating panel interacting with the container liquid medium can significantly alter the dynamic characteristics of the flexible panel which in turn modify the hydrodynamic pressures. hydro-elasto-dynamic problem is developed.

Consequently, a coupled or a

Following the classical methods of linear

hydroelasticity [1,2], the dynamic interaction between the container liquid and the piezoelastic floating roof can exactly be taken into account by imposing the classical structure/fluid kinematical boundary condition which entails equality of the transverse floating panel and normal liquid velocities at the sloshing liquid interface [31], i.e., ( , , , ) HΦ H F

; 0

= EK ( , , ) ,

(21)

where we have used Laplace transform with respect to the time variable, with “s” being the transform parameter, which after incorporation of Eqs. (3) and (20) yields: /

/

/

/

( ( .*+ sinh(.*+ ))d*+ ( ),*+ ( , ) = ( ( ̅[ ( )[ ( , ).

*0 +0

[0 0

(22)

Multiplying the structure/liquid compatibility Eq. (22) by ( , )d d, and integrating on the rectangular surface area, yields: /

/

/

/

( ( +* )d*+ ( ) = ( ( [ ̅[ ( ) ,

*0 +0

[0 0

8

(23)

where i,j =1,2,…, and the matrix entries ] +* , [ ^ are defined in the Appendix (see

Eqs. A-2 and A-6).

Here, it should be noted that the motion of piezo-laminated floating

plate (as governed by Eq. 19) and perturbations in the container liquid domain (as governed by Eq. 1), are coupled through the fluid pressure (Eqs. 4 and 14) in the above kinematical boundary condition (Eq. 21). The simultaneous solution of these equations constitute the coupled fluid-plate hydroelastic problem.

2.4 Active damping control In this subsection, we briefly introduce a second order active damping controller that is

forced by the voltage signal, ¡¢ (), induced within the continuously distributed volume velocity sensor layer [32], i.e.,

d¢ () d D = £ } iF ] , , −]ℎ + ℎD ^, ^d, (24) d d ¥ where ¢ is the electric charge, £ is the leakage resistance of the current amplifier, ¤¢ is the ¡¢ () = £ ¤¢ () = £

output current of piezo-sensor layer, and the transverse component of electrical displacement upon the electroded rectangular panel surface area A is given as [33]:

HU ( , , ) HU ( , , ) D ̅ W iF = −]ℎ + ℎD ^t̅\ ∇ EF + v

+ Y (25) H H ̅ Vv\\ +W Y U ( , , ) . ℎD The induced output voltage ¡¢ () must be fed back to the controller which commands an

input voltage for the piezo-actuator layer.

Thus, the governing controller equation can be

written as a combination of proportional and integral voltage feedbacks in the form [32]: ª

U_ + 2v ¦ U!_ + ¦ U_ = .D ¦ §¡¢ () + .¨ ¦ § } ¡¢ (©) d©,

(26)

where § = −2]ℎ + ℎD ^(2ℎ + ℎD )«ℎD , (.D , .¨ ) are the compensator gain parameters, and ¦ and v refer to the tuned controller frequency and damping ratio, respectively. After Laplace transformation of compensator Eq. (26), the transformed controller coordinate,

Ud_ ( ), can be written as:

Ud_ ( ) = ¬ ( )¡d¢ ( ),

where ¬ ( ) = ¦ §].¨ +

l

.D ^«(

(27)

+ 2v ¦ + ¦ ) is the controller transfer function,

and when a number of vibration modes need to be simultaneously damped out, one can

consider a compound compensator with the associated transfer function ¬( ) = ∑/ 0 ¬ ( )

[32].

9

2.5 Final system response equations. Now, we are ready to assemble the final governing equations. To do this, we shall first substitute the assumed field Eq. (20) into the final form of Eq. (19), which after application /

of Laplace transform along with some manipulations, yields

/

( ( ̅[ ( ) [0 0

−

ℎD

/

/

/

/

H [ H [ H [ ̅ ̅ + ( ( [ ( ) + 2( + 2 ) ( ( [ ( ) H H H H

̅ V v\\

/

[0 0 /

̅

( ( ̅[ ( ) W nv

/

[0 0 /

[0 0 / /

H [ H [ H [ H [ ̅ ( ( ̅[ ( ) W + + v + Y Y

H H H H H H [0 0

/

/

H [ H [ ̅ ( + 2 ) ( ( ̅[ ( ) W ̅ ]? ℎ + ?D ℎD ^ ( ( ̅[ ( ) ∇ [ +2v

+ Y + 2 v

H H H H

[0 0 / /

/

/

−2ℎD t̅\ ( ( ̅[ ( ) ∇ [ p + 2 ]? ℎ + ?D ℎD ^ ( ( ̅[ ( ) [ [0 0

[0 0

[0 0

= d ( , , ) + ( , )Ud_ ( ) + ? @ ( ) cos + ( ) sin A /

/

+? ( ( )d*+ ( ),*+ cosh(.*+ ). *0 +0

(28) d Also direct substitution of the total electric potential U( , , , ) (Eq. 9) along with the

F ( , , ) (Eq. 20) into the output voltage transformed displacement solution expansion E

relation ¡d¢ ( ) (Eqs. 24, 25), and implementation of the result into the controller coordinate

Eq. (27), lead to

/

/

1 P + } ( , )dQ Ud_ ( ) = } ®−]ℎ + ℎD ^t̅\ ( ( ̅[ ( ) ∇ [ ¬( ) £ ¥ ¥ /

/

−]ℎ + ℎD ^t̅\ ( ( ̅[ ( ) ∇ [ + /

[0 0

/

[0 0

̅ ℎD v

H H V ̅ 7 + 9 + v\\ ̅ Q Pv

̅ H H ℎD 4Vt̅\ v\\ /

/

H H H × n( ( ̅[ ( ) W

+ Y [ + 2( + 2 ) ( ( ̅[ ( ) [ H H H H [0 0 / ℎD t̅\

−2

̅ v

/

/

[0 0 /

( ( ̅[ ( ) ∇ [ + 2 ]? ℎ + ?D ℎD ^ ( ( ̅[ ( ) [

[0 0

[0 0

−d ( , , ) − ( , )Ud_ ( ) − ? @ ( ) cos + ( ) sin A /

/

− ? ( ( )d*+ ( ),*+ cosh(.*+ )u¯ d . *0 +0

(29) Next, multiplying Eqs. (28) and (29) by ( , )d d, and integrating on rectangular surface area, one obtains the reduced form of governing equations:

10

/

/

/

/

( ( [ ̅[ ( ) = ( ( +* )d*+ ( ) + ° Ud_ ( ) + ( ),

[0 0

/

/

*0 +0

/

(30a)

/

° Ud_ ( ) = ( ( \[ ̅[ ( ) + ( ( \+* )d*+ ( ) + ( ), [0 0

(30b)

*0 +0

where all the coefficients are given in Appendix (see Eqs. A-2 to A-6). Lastly, the final governing linear system of equations representing the coupled smart floating panel-tank liquid system can be obtained after truncation of Eqs. (23) and (30) with , = 1,2, … , |, ± (where (|, ±) are truncation constants), which can advantageously be put

in the following matrix form:

( ), ² ( )³d( ) = ´ ( )µ

( ) = ² ( )³d( ) + ¶ Ud_ ( ) + · ( ), ´ ( )µ

(31)

( ) + ²\ ( )³d( ) + · ( ). ¶ Ud_ ( ) = ´ \ ( )µ

where the general form of all matrix parameters present in Eqs. (31) are defined in Appendix (see Eqs. A-7 and A-8).

In addition, the system of Eqs. (31) can now be solved for the

( ), ³d( ) and Ud_ ( ). Furthermore, another quantity unknown transformed modal vectors µ

of interest is the overturning moments due to the total lateral hydrodynamic force exerted on

the rectangular container walls that can be found by integrating the moment of hydrodynamic pressure over the wetted inner tank surface areas in the form [34]:

45

( ) = } } M=̅ ( , 0, , ) − =̅ ] , , , ^Nd d = | ( ) + | ( ), | 48

( ) = } } @=̅ ( , , , ) − =̅ (0, , , )Add = | ( ) + | ( ), |

(32)

, | ) and where, after consideration of Eq. (4), the corresponding “rigid body” (| , | ) parts are respectively obtained as: “sloshing” (|

( ) = ; ℳ ( )sin , | <

/ + ( ) = ? ∑/ d | *0 ∑+0 )*+ ( )@(−1) − 1Aℐ *+ ,

/ * ( ) = −; ℳ ( )cos , | ( ) = −? ∑/ d | *0 ∑+0 )*+ ( )@(−1) − 1Aℐ*+ , <

in which ℳ = ? is the total liquid mass, and the coefficients (ℐ *+ , ℐ*+ ) (º, » =

1,2,3, … ) are provided in the Appendix (Eq. A-5). The final time-domain solutions may ultimately be calculated by direct use of Durbin’s numerical inverse Laplace transform algorithm [35] in the interval @0,2¬ A : Æ

2t ¾ª 1 2V 2V ()Â + ( 1Re ÃΛ 7 + Ä 9Å cos 7Ä 9, Λ() = ® ReÁΛ ¬ 2 ¬ ¬ Ç0

7 + Ä − Im ÃΛ

2V 2V 9Å sin 7Ä 9:Å, ¬ ¬

(33)

11

( ), 5 ≤ ¬ ≤ 10, and where is greater than all the real parts of the singularities of Λ ≤ 5000 is the series truncation constant, and ¬ ≥ 2_ with _ denoting the 50 ≤ ±

maximum computation time. This concludes the complete hydro-electro-elasto-dynamic analysis of the problem. Next, we consider some numerical examples.

3. NUMERICAL EXAMPLES Noting the large number of input parameters involved here, while taking into account our computing hardware limitations, we shall consider a particular floating tank configuration. A water-filled square-based rectangular tank ] = = 3m^ for two distinct depth parameters

(⁄ = 1/3,1) is considered. The piezoelectric PZT-5H actuator and sensor layers (ℎD = ℎ =0.0002m, £ = 100kΩ) are assumed to be perfectly bonded to an aluminum base layer

with the associated physical properties as provided in Table 1. A general Mathematica code

was constructed for calculation of the relevant hydroelastic parameters, specifically, the plate/liquid elevation, hydrodynamic pressure, piezoelectric voltages, and overturning moments. Three different types of external excitations are considered, namely, i)

a bidirectional seismic event (El Centro 1940 [36]; see Fig. 2a),

ii)

a planar tank base excitation of the form (see Fig.2b):

iii)

Ï () = () cos , Ð () = () sin ; = V⁄4,

a uniformly distributed blast load on the floating panel of the form (see Fig. 2c):

9: exp(−194.44)( )()(kN⁄m ), ( , , ) = [email protected]() − ( − 0.05)A 11 − 7 0.0018

All numerical integrations were accomplished by employing the Mathematica build-in function "NIntegrate" [37]. The convergence of numerical results was checked in a simple

trial and error manner, i.e., by adding up the truncation constants ]|, ±, ±^, while looking

for stability in the numerical values of computed solutions.

After various numerical

experimentations, it was concluded that uniform convergence could be achieved by taking ]|_ , ±_ , ±_ ^ = (7,7,500) and ¬ = 5 in all computations.

Furthermore, a Multi-

objective Particle Swarm Optimization (MOPSO) evolutionary heuristic algorithm written in Matlab based on the benchmark work by Coello et al. [38] was utilized in order to calculate

the optimal values of controller gain parameters ].D , .¨ ^ (see Table 2). Accordingly, the root mean squares of plate edge transverse displacement, GF ] ≈ , ≈ , ^, and the 12

applied actuator voltage, U( , , , ) = ( )()U_ (), were

concurrently minimized,

with our main Mathematica code linked to Matlab via Mathlink [39].

Also, the following

MOPSO-input variables were adopted in all numerical simulations: original population size=100, number of swarms=1, archive size=30, number of iterations=70, cognitive learning factor q = 0.25, social learning factor q = 0.1, and the inertia weight Ø=0.5.

Before getting to the main numerical results, we shall shortly confirm the overall

correctness of the proposed analytical model. Therefore, in our main Mathematica program, we first let the thicknesses of the floating piezo-composite panel layers approach zero

(ℎD = ℎ = 1 × 10l m), with the following physical properties for the rectangular liquid

domain, (? = 1000kg/m\ ; = = = 3m). The first ten calculated (square of) sloshing frequencies (in rad/sec), as provided in Table 3a, demonstrate good agreements with those calculated by using Eq. (6) in Ref. [40]. Next, we used our main code to calculated the

liquid free surface elevation for a two-dimensional rectangular tank ( = 1m, = 10m, = 0.15m) under a harmonic base excitation Ï () = 0.001 sin (2πt⁄1.5)(m/s ) in

the absence of floating panel (ℎD = ℎ = 1 × 10l m).

The outcome, as presented in Fig.

3a establishes good agreement with the linear results presented in Fig. 4 of Ref. [41]. As a further verification, Tables 3b displays the good agreements found for the first ten calculated free vibration natural frequencies of the freely supported sandwich piezo-coupled (PZT-5H) rectangular panel ]ℎD = ℎ = 0.002m, = = 3m^ with the FEM (Abaqus [42]) results without any form of fluid loading.

In the latter verification, a total of about 17,000

(C3D20RE) 20-node quadratic piezoelectric brick elements were employed to model the piezo-actuator and piezo–sensor skin layers, while about 6000 (C3D20R) 20-node quadratic brick elements were utilized to model the aluminum base layer. As the last verification, the Abaqus/Explicit co-simulation capability was utilized to perform a multidisciplinary Fluid/Structure Interaction (FSI) simulation for a water-filled cubical tank ] = = = 0.5m^ with a freely floating piezo-sandwich (PZT-aluminum) rectangular panel ]ℎD = ℎ =

0.0002m^. The applied FEM model in Abaqus is depicted in Fig. 4. For further applications of Abaqus FSI co-simulation capability in typical structure/liquid interaction problems, the interested reader is referred to Refs. [43,44].

13

Figure 3b demonstrates the good agreements obtained between the analytical results and

the FEM results for the free liquid surface elevation, GF ( , , )|Ý ,Ý , as well as the

hydrodynamic tank pressure, =( , , , )|Þ ,Þ , responses of a water-filled rectangular floating roof tank subjected to a uniformly distributed pulse load of the form: ( , , ) = @() − ( − 0.05)A( )()(kN⁄m ),

for the following selected measurement locations: Point A1: ] = 0.4 , = 0.4 ^, Point A2:

] = 0.01 , = 0.01 ^, Point B1: ] = 0.4 , = 0.4 , = 0.05^, Point B2: ] =

0.01 , = 0.01 , = 0.05^.

Numerical simulations in this final verification were

performed based on the linear Us-Up equation of state for the bulk response with the

adaptive meshing option and the following input parameters: wave speed=45.103m/s,

density=1000kg⁄m , frequency parameter=5, and mesh sweep parameter=3. Also, the three\

dimensional FEM model in Abaqus (see Fig. 4) is composed of about 243000 (C3D20R) 20node quadratic brick fluid elements. Furthermore, the piezo-sandwich plate is segmented into a total of about 6300 (C3D20RE) 20-node quadratic piezoelectric brick elements for modeling the piezo-actuator and piezo-sensor skin layers, and about 2400 (C3D20R) 20-node quadratic brick elements to model the aluminum base layer. At this point, before investigating the effectiveness of active control strategy on suppression of liquid sloshing, we shall briefly describe the recommended experimental setup that includes the basic control mechanism in the proposed smart rectangular floating roof tank system.

The continuously distributed piezoelectric “volume velocity sensor” and

“uniform force actuator” layers are advantageously bonded on the opposite sides of the floating base plate in a matched and collocated configuration [45], as depicted in Fig. 1. Implementation of this configuration in a simple SISO active feedback control system is known to potentially avert the spillover phenomenon besides offering enhanced robustness with respect to variations in model parameters and operating conditions. The main control objective is to utilize the matched sensor-actuator pair layers in the active damping feedback control loop for smart mitigation of flexural vibrations of the floating panel and consequently suppression of the coupled sloshing modes in the rectangular container. The experimental test rig may be designed and built based on a dSPACE real-time rapid control prototyping system [46]. Such system offers the basic hardware platform consisting of a processor and interfaces for input/output signal management, along with the Simulink blocks required to 14

integrate the interfaces into the MATLAB/Simulink environment for controller implementation [47]. Figure 5 depicts the basic schematic diagram of the connections for the proposed concurrent measurement and control system including the smart piezo-sandwich floating panel. The control input is the voltage applied to uniform force piezoelectric actuator layer, while the output of the system is the voltage measured with the collocated volume velocity piezoelectric sensor layer.

The piezo-sensor layer detects and delivers a single (charge)

output proportional to the net volume velocity component of the flexural vibrations of the floating panel (see Eq. 24). Also, in order to match and minimize the vibration field detected by the volume velocity piezo-sensor layer, the uniform force piezo-actuator layer, as driven by the single-channel AD controller, deforms and exerts a uniform transverse force field over the surface of floating base panel (see Eq. 27).

The smart floating roof tank system is

connected to a generic host desktop computer system via a high voltage amplifier (HV) that amplifies the signals from the dSPACE board through the D/A (digital/analog) converter before supplying the piezo-actuator layer. The response time of the high-voltage amplifier must be fast enough in order to avoid deterioration of the dynamic bandwidth of the piezoactuator layer. The electric field needed by the piezo-actuator layer can be provided by a standard high voltage DC power supply [48]. The output voltage of the piezo-sensor layer is amplified by using a standard charge amplifier [49] before being fed back to the dSPACE acquisition system through the A/D (analog/digital) converter. Also, band-pass filters may be utilized to remove aliasing effects introduced by the A/D converter and relieve the effects of unmodeled dynamics (spillover phenomenon) on the controller design.

Furthermore,

rather than using a shaker, a medium size impact hammer (B&K 8206) [49] can expediently be used to produce the excitation force without introducing any extra mass loading to the test structure. For more details on experimental implementation of similar smart piezo-coupled structural systems, the interested reader is referred to Refs. [50,51]. Figures 6a through 6c compare the key uncontrolled and controlled hydro-electro-elastic response parameters of the square-base floating roof-tank system, namely, the panel edge

displacement, GF ] = = 3m, = = 3m, ^, tank liquid pressure, =] = = 3m, = = 3m, 1 = 0.05, ^, overturning moments, | (), | (), and sensor/actuator voltages,

¡¢ (), U_ (), for the three external loading configurations previously described in Fig. 2, and

two distinct depth parameters (⁄ = 1/3,1). Also shown in Figs. 6a and 6b are the time 15

response curves of the uncovered liquid tank. Furthermore, for comparison purposes, the time responses corresponding to the square tank equipped with a relatively thick free-floating aluminum panel ]ℎD = 0, ℎ = 0.0008m^ and subjected to the bidirectional seismic event are

presented in Fig. 6a.

The main observations are as follows. The hydro-electro-elastic

response curves associated with the seismically excited tank exhibit a highly oscillatory behavior primarily due to the very rich frequency content of the seismic excitation in comparison with the other two loading configuration. Also, the sloshing response levels for

(GF , | , | ) generally increase with increasing the liquid height, i.e., proportional with total liquid mass. Conversely, the notable increase in the overall hydrodynamic pressure levels

may be linked to the relative closeness of the floating panel with respect to the tank bottom

for the shallow depth tank (⁄ = 1/3). Moreover, the response curves associated with the

passive (inactive) floating tank control system nearly follow those of the uncovered liquid tank system.

In other words, the inactive thin floating panel appears to have a negligible

effect on the overall sloshing response of the system. On the other hand, the fully active floating panel control system demonstrates remarkable suppression of the key hydro-elastic parameters, especially in moderate times, nearly regardless of liquid depth and loading type. Besides, the actuator input voltage requirement for sloshing control in the shallower tank

(⁄ = 1/3) is observed to be slightly lower for nearly the same level of control action.

Lastly, comparisons of the amplitudes of time response plots in Fig. 6a clearly indicates that the active floating panel control system ]ℎD = ℎ = 0.0002m^ is advantageously capable of

performing practically as good as the tank equipped with a thick, single layer, passive

floating panel ]ℎD = 0, ℎ = 0.0008m^ at a much lower weight penalty, i.e., with about 70% reduction in total calculated panel weight.

Now, in order to briefly deliberate on the role of sloshing modes in the calculated results, the frequency response function (FRF) plots presented in Fig. 7 are obtained by applying the standard Fast Fourier Transform (FFT) algorithm to the uncontrolled and controlled

transverse displacement data (GF ) presented in Figs. 6a-c for the selected external loading

configurations and liquid depths. The numerical values of peak (resonance) frequencies for

the dominant modes are marked in the figure. Once again, the remarkable reduction of response amplitudes associated with the active floating panel is evident. The slight increase

16

in the values of associated resonance frequencies can be linked to the increased stiffness of the smart float. Nevertheless, all these values fall within a reasonable range near the fundamental sloshing frequency as calculated by using basic formula (Eq. 6) of Ref. [40] for a clean rectangular tank, i.e., ω = 0.451Hz for ⁄ = 1/3, and ω = 0.509Hz for

⁄ = 1. This is due to the very high flexibility of the liquid-coupled thin floating panels,

which makes the dominant natural frequency of the panel bulging mode very close to that of the free liquid surface sloshing mode for the clean tank, i.e., there is a very strong interaction or coupling between the two families of modes [52]. Therefore, as far as the dominant

sloshing mode is concerned, presence of the highly flexible floating roof has little influence on the dominant sloshing frequency [53]. For the thick floating panel, on the other hand, the sloshing and bulging modes become practically uncoupled [52], and consequently the numerical value of corresponding peak frequency noticeably deviates from the above-noted fundamental sloshing frequency. For further insights on dynamic characteristics of dominant sloshing modes, the interested reader is referred to the following detailed studies [54,55]. Lastly, in case of using discrete patches or segments of piezoelectric actuators on the host floating panel, the locations of actuator/sensor pair become very important for achieving high actuation/sensation effects, and thus should be carefully chosen. This can be achieved by a simple trial and error procedure, e.g., by trying to place the PZT segments where the maximum strains or transverse displacements occur in the free floating flexible panel [4,50]. On the other hand, one can apply various systematic optimization methodologies for optimal placement of piezoelectric patches. In particular, the optimal electrode configuration for the collocated piezoelectric actuator-sensor segments, as well as the relevant controller gain parameters, can advantageously be obtained by applying the standard multi-objective genetic algorithm (MOGA) or alternatively the particle swarm optimization (MOPSO) technique [20]. This can lead to maximum suppression of structural vibration near certain controlled system resonance frequencies with minimum increase in the expended actuation energy. A more effective real-time control of a wider selection of vibration modes essentially requires a more complex multi-segment sensors/actuator configuration accompanied by sophisticated multi-input multi-output (MIMO) controller design for the smart structure [56,57], which can be the focus of future work.

17

4. CONCLUSIONS A coupled hydro-elasto-dynamic model based on the linear water wave theory, the thin piezo-sandwich floating plate model, the pertinent structure/liquid compatibility condition, and the active damping control strategy, is formulated for effective suppression of forced transient 3D sloshing in a rectangular parallelepiped floating roof container. The extensive numerical simulations include the key hydro-electro-elastic response parameters for three distinct external excitations, namely, a bidirectional seismic event, an oblique planar base acceleration, and a distributed impulsive floating roof excitation. The main observations are as follows. The liquid displacement and overturning moment response levels generally increase with increasing the liquid height, while the hydrodynamic pressure levels notably grow for the shallow depth tank. Also, the hydro-electro-elastic response curves associated with the passive floating tank control system nearly follow those of the uncovered or clean liquid tank system.

In other words, the inactive thin floating panel generally exhibits a

negligible effect on the sloshing response of the system. On the other hand, the proposed fully active floating panel control system exhibits remarkable suppression of the key hydroelastic parameters, especially in moderate times, nearly regardless of tank liquid depth and loading configuration. Furthermore, the actuator input voltage requirement for sloshing control in the shallower tank is observed to be slightly lower for nearly the same level of control action. Moreover, the active floating panel control system is observed to be capable of performing practically as good as the tank equipped with a relatively thick floating panel at a much lower weight penalty. Besides, presence of the highly flexible thin floating roof is seen to have little influence on the dominant sloshing frequency.

For the thick floating

panel, on the other hand, the sloshing and bulging modes become practically uncoupled which causes the dominant natural frequency of the panel bulging mode to noticeably differ from that of the free liquid surface sloshing mode. The proposed analytical model accurately takes account of the hydroelastic interaction between the flexible smart floating panel and the oscillating container fluid. It allows for expedient implementation of parametric studies and judicious establishment of qualitative trends in the early design stage with a much lower cost than the typical numerical and experimental methodologies.

18

REFERENCES [1]

R.A. Ibrahim, Liquid sloshing dynamics: theory and applications, Cambridge University Press, 2005.

[2]

O.M. Faltinsen, A.N. Timokha, Sloshing Cambridge University Press, Cambridge, UK. (2009).

[3]

S.M. Hasheminejad, M.M. Mohammadi, M. Jarrahi, Liquid sloshing in partly-filled laterally-excited circular tanks equipped with baffles, J. Fluids Struct. 44 (2014) 97– 114.

[4]

S.M. Hasheminejad, M.M. Mohammadi, Active sloshing control in a smart flexible cylindrical floating roof tank, J. Fluids Struct. 66 (2016) 350–381.

[5]

T. Matsui, T. Nagaya, Nonlinear sloshing in a floating roofed oil storage tank under long period seismic ground motion, Earthq. Eng. Struct. Dyn. 42 (2013) 973–991.

[6]

E. Hernández, D. Santamarina, Active control of sloshing in containers with elastic baffle plates, Int. J. Numer. Methods Eng. 91 (2012) 604–621.

[7]

Q. Zang, J. Huang, Dynamics and control of three-dimensional slosh in a moving rectangular liquid container undergoing planar excitations, IEEE Trans. Ind. Electron. 62 (2014) 2309–2318.

[8]

Lord Rayleigh, On waves propagated along the plane surface of an elastic solid, Proc. London Math. Soc. 1 (1885) 4–11.

[9]

H. Lamb, On the vibrations of an elastic plate in contact with water, Proc. R. Soc. London. Ser. A, Contain. Pap. a Math. Phys. Character. 98 (1920) 205–216.

[10] M.E. Yildizdag, I.T. Ardic, M. Demirtas, A. Ergin, Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach, Ocean Eng. 172 (2019) 316–329. [11] A. Kolaei, S. Rakheja, Free vibration analysis of coupled sloshing-flexible membrane system in a liquid container, J. Vib. Control. 25 (2019) 84–97. [12] F.G. Canales, J.L. Mantari, Vibrational behavior of isotropic plate structures in contact with a bounded fluid via unified formulation, Chinese J. Aeronaut. 32 (2019) 921–937. [13] A. Bendahmane, S.M. Hamza-Cherif, M.N. Ouissi, Free vibration analysis of variable stiffness composite laminate (VSCL) plates coupled with fluid, Mech. Adv. Mater. Struct. (2019) 1–15. [14] M. Gradinscak, S.E. Semercigil, Ö.F. Turan, A sloshing absorber with a flexible container, in: Struct. Dyn. Vol. 3, Springer, 2011: pp. 315–322. [15] H. Luo, R. Zhang, D. Weng, Mitigation of liquid sloshing in storage tanks by using a 19

hybrid control method, Soil Dyn. Earthq. Eng. 90 (2016) 183–195. [16] M. Hamaguchi, Damping Control of Sloshing in Liquid Container in Cart With Active Vibration Reducer: The Case of a Curved Path on a Horizontal Plane, IEEE/ASME Trans. Mechatronics. 24 (2019) 361–372. [17] M. Moslemi, A. Farzin, M.R. Kianoush, Nonlinear sloshing response of liquid-filled rectangular concrete tanks under seismic excitation, Eng. Struct. 188 (2019) 564–577. [18] Y. Du, C. Wang, N. Zhang, Numerical simulation on coupled ship motions with nonlinear sloshing, Ocean Eng. 178 (2019) 493–500. [19] E. Zhang, Numerical research on sloshing of free oil liquid surface based on different baffle shapes in rectangular fuel tank, Proc. Inst. Mech. Eng. Part D J. Automob. Eng. (2019) 0954407019855569. [20] S.M. Hasheminejad, A. Hakimi, H. Keshavarzpour, Broadband sound transmission loss enhancement of an arbitrary-thick hybrid smart composite plate using multiobjective particle swarm optimization–based active control, J. Intell. Mater. Syst. Struct. (2018) 1045389X1775425. https://doi.org/10.1177/1045389X17754257. [21] Y.-C. Li, H.-L. Gou, Modeling Problem of Equivalent Mechanical Models of a Sloshing Fluid, Shock Vib. 2018 (2018). [22] S.M. Hasheminejad, M. Aghabeigi, Transient sloshing in half-full horizontal elliptical tanks under lateral excitation, J. Sound Vib. 330 (2011) 3507–3525. [23] T. Matsui, Sloshing in a cylindrical liquid storage tank with a floating roof under seismic excitation, J. Press. Vessel Technol. 129 (2007) 557–566. [24] E. Carrera, S. Brischetto, P. Nali, Plates and shells for smart structures: classical and advanced theories for modeling and analysis. 2011, (n.d.). [25] M. Pietrzakowski, Piezoelectric control of composite plate vibration: Effect of electric potential distribution, Comput. Struct. 86 (2008) 948–954. [26] J.-F. Deü, W. Larbi, R. Ohayon, R. Sampaio, Piezoelectric shunt vibration damping of structural-acoustic systems: finite element formulation and reduced-order model, J. Vib. Acoust. 136 (2014) 31007. [27] X. Cao, L. Shi, X. Zhang, G. Jiang, Active control of acoustic radiation from laminated cylindrical shells integrated with a piezoelectric layer, Smart Mater. Struct. 22 (2013) 65003. [28] Q. Wang, S.T. Quek, C.T. Sun, X. Liu, Analysis of piezoelectric coupled circular plate, Smart Mater. Struct. 10 (2001) 229. [29] S.S. Rao, Vibration of continuous systems, Wiley, 2007. 20

[30] M.R. Barati, A.M. Zenkour, Electro-thermoelastic vibration of plates made of porous functionally graded piezoelectric materials under various boundary conditions, J. Vib. Control. 24 (2018) 1910–1926. [31] S.M. Soedel, W. Soedel, On the free and forced vibration of a plate supporting a freely sloshing surface liquid, J. Sound Vib. 171 (1994) 159–171. [32] A. Preumont, Vibration control of active structures: an introduction, Springer, 2018. [33] K. Chandrashekhara, R. Tenneti, Thermally induced vibration suppression of laminated plates with piezoelectric sensors and actuators, Smart Mater. Struct. 4 (1995) 281. [34] S.M. Hasheminejad, M.M. Mohammadi, Effect of anti-slosh baffles on free liquid oscillations in partially filled horizontal circular tanks, Ocean Eng. 38 (2011) 49–62. [35] F. Durbin, Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, Comput. J. 17 (1974) 371–376. [36] F.G. Golzar, R. Shabani, S. Tariverdilo, G. Rezazadeh, Sloshing response of floating roofed liquid storage tanks subjected to earthquakes of different types, J. Press. Vessel Technol. 134 (2012) 51801. [37] S. Wolfram, Mathematica: a system for doing mathematics by computer, AddisonWesley, 1991. [38] C.A.C. Coello, G.T. Pulido, M.S. Lechuga, Handling multiple objectives with particle swarm optimization, IEEE Trans. Evol. Comput. 8 (2004) 256–279. [39] T. Gayley, A MathLink Tutorial, Wolfram Res. (2002). [40] N. Vaziri, M.-J. Chern, A.G.L. Borthwick, Effects of base aspect ratio on transient resonant fluid sloshing in a rectangular tank: a numerical study, Ocean Eng. 105 (2015) 112–124. [41] Y. Su, Z.Y. Liu, Numerical model of sloshing in rectangular tank based on Boussinesq-type equations, Ocean Eng. 121 (2016) 166–173. [42] K. Hibbitt, Sorensen: ABAQUS/Standard-User’s manual. Hibbitt, Karlsson & Sorensen, Inc., United States (Pawtucket). (2001). [43] S.M. Hasheminejad, V. Rabbani, M. Alaei-Varnosfaderani, Active transient elastoacoustic response damping of a thick-walled liquid-coupled piezolaminated cylindrical vessel, Mech. Based Des. Struct. Mach. 44 (2016) 189–211. [44] A. Mohammadi, M.H. Korayem, R. Ghaderi, Investigation of piezoelectric microcantilever performance in constant amplitude mode in different work environments, Mech. Ind. 18 (2017) 504. 21

[45] P. Gardonio, Y.-S. Lee, S.J. Elliott, S. Debost, Analysis and measurement of a matched volume velocity sensor and uniform force actuator for active structural acoustic control, J. Acoust. Soc. Am. 110 (2001) 3025–3031. [46] www.dspace.com. [47] https://www.mathworks.com/products/simulink.html. [48] www.spellmanhv.com. [49] www.bksv.com. [50] S.-B. Choi, Y.-M. Han, Piezoelectric actuators: control applications of smart materials, CRC Press, 2016. [51] S.J. Rupitsch, Piezoelectric Sensors and Actuators—Fundamentals and Applications, Topics in Mining, Metall. Mater. Eng. (Springer, Berlin Heidelberg, 2019). [52] M. Amabili, Vibrations of circular plates resting on a sloshing liquid: solution of the fully coupled problem, J. Sound Vib. 245 (2001) 261–283. [53] F. Sakai, M. Nishimura, H. Ogawa, Sloshing behavior of floating-roof oil storage tanks, Comput. Struct. 19 (1984) 183–192. [54] T. Ikeda, M. Takashima, Y. Harata, Autoparametric resonances of elastic structures coupled with two sloshing modes in a square liquid tank, J. Comput. Nonlinear Dyn. 8 (2013) 11007. [55] T. Ikeda, Y. Harata, Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks, J. Vib. Acoust. 139 (2017) 41009. [56] S.M. Hasheminejad, M. Vahedi, A.H.D. Markazi, Multi-objective robust active vibration control of an arbitrary thick piezolaminated beam, Mech. Adv. Mater. Struct. 22 (2015) 908–924. [57] S. Camperi, M.G. Tehrani, S.J. Elliott, Local tuning and power requirements of a multi-input multi-output decentralised velocity feedback with inertial actuators, Mech. Syst. Signal Process. 117 (2019) 689–708.

22

APPENDIX

The nonzero elements of the bending stiffness matrix, :

2 T ℎ\

= = âq̅

M−ℎ\ + (ℎ + ℎD )\ N + ã, 3 1 − o 2 T ℎ\ o = = âq M−ℎ\ + (ℎ + ℎD )\ N + ã, 3 1 − o 2 1 T ℎ\ (1 − o ) = â (q

− q )M−ℎ\ + (ℎ + ℎD )\ N + ã. 3 2 2(1 − o )

(A-1)

Truncated matrix entries are defined as follows: 48

45

+* = .*+ sinh(.*+ ) } } ,*+ ( , ) ( , )d d, 48

45

*+ = } } ? cosh(.*+ ) ,*+ ( , ) ( , )d d,

\*+

48 45 ̅ ℎD v

H H ̅ 7 + 9 = −? cosh(.*+ ) } } } 1v

̅ H H 4Vt̅\ v\\ ¥ V ̅ ,*+ ( , )Q d ( , )d d, + v\\ ℎD 48

45

° = } } ( , ) ( , )d d,

°

48 45 ̅ ℎ[ v

H H V ̅ 7 + 9 + v\\ ̅ Q ( , ) d ( , )d ä, =− } } } Pv

̅ H H ℎ 4Vt̅\ v\\ D ¥ 48

(A-3)

45

= } } Ád*y ( , , ) + ? @ ( ) cos + ( ) sin AÂ ( , )d ä,

(A-2)

48 45 ̅ ℎ[ v

H H V ̅ ̅ Ád ( , ) =− } ̅ } }¥ Pv

7H + H9 + ℎD v\\ Q , 4Vt̅\ v\\ + ? @ ( ) cos + å ( ) sin AÂ d ( , )d d,

45 ºV ℐ *+ = } } cos 7 9 cosh(.*+ )d d , 48 »V ℐ*+ = } } cos W Y cosh(.*+ )dd ,

23

(A-4)

(A-5)

48

45

[ = } } [ ( , ) ( , )d d,

H H H ) + + 2( + 2

H H H H ℎD H H H H H ̅

W ̅ W ̅ ( + 2 ) W + Y + v + Y + 2v − ̅ Pv

H H H H H H H H V v\\ H H H ̅ ]? ℎ + ?DℎD ^ W + Y + 2 v

+ Y H H H H H EF H EF H EF −2ℎD t̅\ + 2 + Q + 2 ]? ℎ + ?DℎD ^ ( , )[ ( , )æd d, H H H H 48

45

[ = } } â

48

45

\[ = } } } ®−]ℎ + ℎD ^t̅\ P

+

¥

H H + Q H H

(A-6)

̅ ℎD v

H H H H V ̅

7 + 9 + v\\ ̅ Q çP

Pv + Q ̅ H H H H ℎD 4Vt̅\ v\\

+ 2( + 2 )

ℎD t̅\ H H H − 2 W + Y ̅ H H H H v

+ 2 ]? ℎ + ?D ℎD ^èé [ ( , ) äA ( , )d ä.

Also the general form of final coefficient matrices and vectors appearing in Eqs. (31) are: OO

() () () () () () () () () ³d( ) = )d ( ), )d ( ), … , )dë ( ), )d ( ), )d

( ), … , )d ë ( ), … , )dë ( ), )dë ( ), … , )dëë ( ) , O

̅() ( ), … , ̅() ( ), … , ̅() ( ), ̅() ( ), … , ̅() ( ) , ( ) = ̅() ( ), ̅() ( ), … , ̅() ( ), ̅() ( ),

µ ë

ë ë ë ëë O

() () () () () () () () () ·ì ( ) = d ( ), d ( ), … , dë ( ), d ( ), d

( ), … , d ë ( ), … , dë ( ), dë ( ), … , dëë ( ) ,

(A-7)

O

̅ () ( ), °̅ () ( ), … , °̅ () ( ), °̅ () ( ), °

̅ () ( ), … , °̅ () ( ), … , °̅ () ( ), °̅ () ( ), … , °̅ () ( ) , ¶ì ( ) = ° ë

ë ë ë ëë ï ℋ î ⋮ () î ℋë ² ( ) = î î () ℋ î ë ⋮ î () íℋëë () ï ℛ î ⋮ () î ℛë ´ ( ) = î î () ℛ î ë î ()⋮ íℛëë and

()

… ⋱ … ⋮ … ⋱ … … ⋱ … ⋮ … ⋱ …

ℋë ⋮ () ℋëë ()

() ℋëë

⋮

() ℋëëë () ℛë

⋮

() ℛëë

ℛëë ⋮ () ℛëëë ()

where = 1,2,3; = 1,2.

⋯ ⋱

ℋë ⋮ () ℋëë ()

ℋëë ⋯ ⋮ () ℋëëë () ℛë ⋯ ⋮ () ℛëë ⋱ () ℛëë ⋯ ⋮ () ℛëëë ()

… ⋱ … ⋮ … ⋱ … … ⋱ … ⋮ … ⋱ …

ℋëë ö ⋮ õ () ℋëëë õ õ, () õ ℋëëë õ ⋮ õ () ℋëëëë ô () ℛëë ö ⋮ õ () ℛëëë õ õ, () õ ℛëëë õ ⋮ õ () ℛëëëë ô ()

24

(A-8)

LIST OF CAPTIONS

Table 1. Piezo-sandwich panel properties. Table 2. MOPS-Optimized values of controller gain parameters. Table 3. Verification of calculated natural frequencies: a) Sloshing frequencies for a 3D rectangular tank. b) Free vibration frequencies for a sandwich piezo-panel without fluid loading. Fig. 1. Problem configuration. Fig. 2. Applied external excitations: c) Bidirectional seismic event. d) Planar base excitation. e) Distributed surface blast load. Fig. 3. Verification of proposed formulation: a) Liquid free surface elevation for a 2D rectangular tank under a harmonic base excitation. b) Liquid free surface elevation and hydrodynamic tank pressure time responses of a water-filled rectangular floating roof tank for a uniformly distributed pulse load at selected measurement locations. Fig. 4. The applied finite element model. Fig. 5. Schematic block diagram for the propose measurement and control system. Fig. 6. Key uncontrolled and controlled hydro-electro-elastic time response histories of the floating roof-tank system for three external loading configurations and two distinct depth parameters; a) Bidirectional seismic event. b) Planar base excitation. c) Distributed surface blast load. Fig. 7. Identification of the resonance frequencies associated with the dominant sloshing modes of the square tanks via FFT of displacement response plots in Fig. 6a-c.

25

Table 1.

Aluminum base layer kg ? = 2270 7 \ 9 , T = 70 GPa, o = 0.33. m PZT-5H piezo-layers q̅

= 127 × 10û q̅ = 80.2 × 10û Stiffness coefficients q̅\\ = 117.4 × 10û (N/m ) q̅ \ = 84.7 × 10û ?D (kg/m\ ) t̅\ (C/m ) t̅\\ (C/m ) Td

(F/m) Td\\ (F/m)

7750 -6.62 23.4 27.713 × 10lû 30.104 × 10lû

Table 2. Disturbance El-Centro seismic excitation Planar base acceleration Uniform blast load

Table 3a. present 8.032 14.112 20.548 23.142 29.084 30.874 32.567 37.421 41.201 46.07

þ۾ 5094.48 7281.31 678.11

þ۷ 7.451 273.94 10.392

Table 3b.

Ref. [40] 8.020 14.094 20.536 22.970 29.056 30.819 32.486 37.040 41.092 45.942

present 14.133 29.958 42.057 54.292 70.223 93.429 127.46 164.33 201.03 278.91

26

FEM 14.135 29.962 42.101 54.314 70.458 94.512 128.24 165.21 202.17 280.21

Fig. 1

Fig. 2

41

Fig. 3

z

y x Lx

Ly

Piezo-actuator Base Panel Piezo-sensor

Sloshing Liquid

H

Fig. 4

Equipped Structure

Smart Floating Panel

Fme(x,y,t)

Piezo-actuator Base Layer Piezo-sensor

Φa

High Voltage Ampliﬁer

Sloshing Liquid

D/A

Vs Charge Ampliﬁer

Power Supply

Fig. 5

rb(t) Dynamic Signal Analyzer

A/D

D/A

Microcomputer Control Algorithm

thick ﬂoat

Fig. 6a

Fig. 6b

Fig. 6c

H/Lx=1/3 0.012

0.468391

0.444733

0.04

0.602832

0.004

Transverse Displacement

0.06

0.444733

0.008

0

H/Lx=1

0

2

0.492253 0.747196

4

6

8

0

0

2

4

0.08 0.03

0.478502

0.02

0.484991

0.01

0.485735

0

0

2

6

8

0

0

2

4

0.016 0.484032

0.002

uncontrolled controlled

0.512764

0.004 2

8

0.008

0.500161

0

6

Blaﬆ Load

0.507319

0.012

0.001 0

0.503160

0.02 4

8

no ﬂoat uncontrolled controlled

0.484002

0.04

6

Planar Excit.

0.499520

0.06

0.004 0.003

4

6

Frequency (Hz)

Fig. 7

no ﬂoat uncontrolled thick ﬂoat controlled

0.468391

0.02

0.447281

Bidirectional Excit.

8

0

0

2

4

6

Frequency (Hz)

8

CRediT author statement Seyyed M. Hasheminejad: Conceptualization, Methodology, Writing – Original, Writing- Reviewing & Editing, Supervision. M. M. Mohammadi: Methodology, Software, Validation. Ali Jamalpoor: Software, Validation.

1 Author declaration [Instructions: Please check all applicable boxes and provide additional information as requested.] 1. Conflict of Interest Potential conflict of interest exists: We wish to draw the attention of the Editor to the following facts, which may be considered as potential conflicts of interest, and to significant financial contributions to this work: The nature of potential conflict of interest is described below:

No conflict of interest exists. We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

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Author’s name (Fist, Last)

1. Seyyed M. Hasheminejad

Signature

Date

10/28/2019

2. M. M. Mohammadi

10/28/2019

3. Ali Jamalpoor

10/28/2019