Nonlinear finite element modeling of vibration control ...

Nonlinear finite element modeling of vibration control of piezolaminated composite plates and shells S. Lentzen, R. Schmidt Institute of General Mechanics, RWTH Aachen University, Germany ABSTRACT This paper addresses the theory and finite element analysis of the transient large amplitude vibration response of thin composite structures and its control by integrated piezoelectric layers. A geometrically nonlinear finite shell element for the coupled analysis of piezolaminated structures is developed that is based on the first-order shear deformation (Reissner-Mindlin) hypothesis and the assumptions of small strains and moderate rotations of the normal. The finite element model can be applied to smart structures consisting of a composite laminated master structure with arbitrary ply lay-up and integrated piezoelectric sensor and actuator layers or patches attached to the upper and lower surfaces. Keywords: large amplitude vibrations, smart structures, vibration control, composites, plates, shells

1. I NTRODUCTION In recent years, the subject area of theories and finite element tools for modelling and simulation of smart materials and structures has experienced tremendous growth in terms of research and development. Especially for thin-walled structures, integration of smart materials, like piezoelectric layers or patches that can sense as well as be actuated, is a promising possibility for shape and vibration control. Geometrically linear, small deflection theories for structures with piezoelectric layers and associated numerical methods have been developed by a large number of authors, e.g. by Robbins and Reddy [1] for beams, by Lee [2], Yang et al. [3], Ghosh and Batra [4] for plates, by Lammering [5], Tzou and Tseng [6], Tzou [7], Tzou and Ye [8] for shells, amongst others. Geometrically nonlinear, large deflection problems of structures with integrated smart material layers have been treated considerably less in literature, although these structures are thin and flexible and the deflections often are much larger than the thickness. Structural nonlinearity has been taken into account for interlaminar stress analysis by Icardi and Di Sciuva [9], for large deflection shape control by Yi et al. [10] and Mukherjee and Chaudhuri [11], for active buckling control and post-buckling analysis by Krishna and Mei [12], Chandrashekhara and Bhatia [13] and Wang and Varadan [14], and for piezothermoelastic analysis by Tzou et al. [15], Pai et al. [16], Oh et al. [17,18]. Concerning vibration suppression, geometrically nonlinear problems have been considered by Shi and Atluri [19], Lee and Beale [20], Batra et al. [21,22] and Chróscielewski, Klosowski and Schmidt [23-25], the latter one treating also dynamic stability problems. Nonlinear flutter suppression has been considered by Lai et al. [26], Zhou et al. [27,28], Shen and Sharpe [29]. In our recent papers [30-36] we have developed geometrically nonlinear first and third-order shear deformation theories of smart composite structures with piezoelectric actuator and sensor layers and associated finite element methods for static analysis. The application to shape control problems of beams, plates and shells at large deflections shows the importance of geometrically nonlinear analysis, especially when the sensing capabilities of the piezoelectric layers are investigated and for actuation when certain boundary conditions are applied. In the present paper the aforementioned nonlinear theories and simulation tools are extended for transient large amplitude vibration analysis. More specifically, a geometrically nonlinear finite shell element for the coupled analysis of piezolaminated structures is developed employing the nonlinear shell theory of Schmidt and Reddy [37], which is based on the first-order shear deformation hypothesis and the assumptions of small strains and moderate rotations of the normal (see also Palmerio et al. [38,39] and Kreja et al. [40,41]). As an example, a typical smart composite plate is considered that consists of a laminated master structure and integrated piezoelectric sensor and actuator layers or patches attached to the upper and lower surfaces. This structure was considered earlier by Kioua and Mirza [42] and Lee et al. [43] in the framework of linear shape control. It is

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demonstrated that this problem requires geometrically nonlinear finite element analysis. Then the load is applied impulsively, resulting in transient large amplitude vibrations, which can be suppressed by a suitable control algorithm.

2. NUM ERICAL METHOD 2.1. Electromechanical equations The virtual work principle describes a state of equilibrium between the external virtual work δWe and internal virtual work δWi . In the present work a total Lagrangian formulation is chosen. For this approach the mechanical and electrical quantities are required to refer to the undeformed configuration, denoted by the lower left subscript 0, and integration is performed over the undeformed volume and surfaces. The internal virtual work is the volume integral of the virtual electric enthalpy densitiy δH and it can be written as

δ W = ∫ δ H dV with i V

δ H = 0σ ij 0 δε ij − 0 Di 0 δ Ei ,

(1)

where 0σij are the contravariant 2nd Piola-Kirchhoff stress tensor components, 0εij the covariant Green-Lagrange strain tensor components, 0Di the contravariant electric displacement components and 0Ei the covariant electric field components. The electric field vector referring to the undeformed configuration is calculated as the negative gradient of the electric potential φ along the undeformed surface parameters Θi 0

Ei = −

∂φ . ∂Θi

(2)

The Green-Lagrange strain tensor components with the assumption of first-order shear deformation (Reissner-Mindlin) theory, small strains but moderate rotations are expressed as [37]

ε αβ = ε αβ + Θ3 ε αβ + (Θ3 ) ε αβ 0

1

0

1

2 2

(3)

ε α 3 = ε α 3 + Θ3 ε α 3 ε 33 = 0, with 0 0 10 0 ε αβ = θ αβ + ϕ α ϕ β , 2 1 1 1 0 0 0 1 0 1 1 ε αβ = ν α |β +ν β |α − bαλ ϕ λβ − bβλ ϕ λα + ϕ α bβλ ν λ + ϕ β bαλ ν λ  , 2  2 1 1 1 1 1 ε αβ =  bαλ bβκ ν λ ν κ − bαλ ν λ |β − bβλ ν λ |α  , 2  0 0 1 1 0 1 1 11 1 ε α 3 =  ϕ α +ν α +ν λ ϕ λα  ε α 3 = v λ v λ |α . and 2 2 

(4)

Here the following abbreviations are used: 0 0 0 1 0 θ αβ = ν α |β +ν β |α  − bαβ ν 3 , 2  0

0

0

ϕ αβ = ν α |β − bαβ ν 3

0

and

0

0

ϕ α = ν 3,α + bαλ ν λ ,

where bαβ and bαβ denote the covariant and mixed components of the curvature tensor and

(< )|α

is the covariant

derivative with respect to the surface parameter in direction α. The translation of the midsurface in direction i is denoted 0

1

as ν i , while ν α stands for the rotations of the normal.

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2.2. Constitutive relations The constitutive behavior of the piezolaminated structures is assumed to be linear. The piezoelectric effect can then be described by the direct and converse constitutive equations in vector form as

{0 D} = [e]{0 ε } + [δ ]{0 E} T {0 S } = [c ]{0 ε } + [e] {0 E}

(5)

where {0S} denotes the stress vector, {0ε} the strain vector, {0D} the electric displacement vector and {0E} the electric field vector: σ 11   ε11   22  ε   D1   E1  22  σ   (6)  2     12 {0 S} =  τ  , {0 ε } =  2ε12  , {0 D} =  D  and {0 E} =  E2  .  τ 23   2ε   D3  E   3      23  13  2ε 13   τ  Further [e] = [d][c] and [e]T = [c][d]T, where [c] denotes the elasticity matrix for anisotropic materials, [d] the piezoelectric constant matrix and [δ] the dielectric constant matrix:  c11 c  12 [c] =  c13  0  0

c12 c22

c13 c23

0 0

c23 0

c33 0

0 c44

0

0

c45

0 0  0 ,  c45  c55 

[d ]

T

0 0  = 0  0  d15

d 31  d31  0  and  0 0 

0 0 0 d15 0

0 δ11 0 .  [δ ] =  0 δ 22 0   0 0 δ 33 

(7)

2.3. Tangential stiffness matrix For static analysis of piezolaminated structures in the geometrically nonlinear range, it is necessary to generate a tangential stiffness matrix based on the application of incremental formulations in which the internal virtual work at the searched configuration ‘2’ is expressed in terms of mechanical and electrical quantities in a known configuration ‘1’ and an increment. This procedure is explained in the following. 2.3.1. Internal virtual work The principle of virtual work states that equilibrium is maintained if the internal virtual work equals the external virtual work. After introducing the finite element procedures, the external virtual work can generally be expressed as the global vector of virtual nodal displacements {δq} times the vector of externally applied nodal forces {} plus the global vector of virtual potentials {δφ} times the vector of externally applied charges {} as described e.g. by Gaudenzi et al. [44,45]

δ We = {δ q}

T

{\} + {δφ } {_}. T

(8)

The internal virtual work is calculated as the volume integral of the virtual electric enthalpy density

(

)

1 T T (9) {ε } {S} − {E} {D} . 2 V The virtual electric enthalpy can be expressed in terms of mechanically and electrically induced parts denoted by the subscript m and e, respectively

δ Wi = ∫ δ H dV

with

H=

δ H = {δε } {S } − {δ E} {D} = δ H m − δ H e , T

with

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T

(10)

({S} − {S} ) = δ H = {δ E} ({D} + {D} ) = δ H

δ H m = {δε }

T

δ He

m

mm

e

− δ H me

T

m

e

em

(11)

+ δ H ee .

2.3.2. I ncremental formulation In this work a total Lagrangian approach is chosen for the FE implementation. For this purpose, three configurations are considered, the initial configuration denoted by 0, the calculated configuration denoted by 1 and the searched configuration, denoted by 2. Quantities associated with the searched configuration are expressed as the ones present in the calculated configuration plus an increment. For example the strain can be written as follows:

{ ε } = { ε } + {∆ε }. 2

1

(12)

Generally, the nonlinear strain-displacement relations can be written in matrix form as [38-41]

{ ε } =  [ B ] + 12  A ( v ) ⋅ G   { v}, 2

2

(13)

2

0

where [B0] is the linear part and ½[A(2v)][G] accounts for the nonlinear part. The generalised displacement vector is denoted as {2v}. Introducing the matrix of shape functions {v} = [N]{q}, where {q} is the generalised nodal displacement vector, the strain increment and variation is written as

{∆ε } = ([ B0 ] +  A ( 1v ) ⋅ G  )[ N ]{∆q} +

1  A ( ∆v ) ⋅ G  [ N ]{∆q} = ([ BL ] + [ BNL ]){∆q} 2

{δε } = ([ B0 ] +  A ( v ) ⋅ G  )[ N ]{δ q} +  A ( ∆v ) ⋅ G  [ N ]{δ q} = ([ BL ] + 2 [ BNL ]){δ q}.

(14)

1

The electric field vector {E} is calculated as the gradient of the potential φ, and it can be expressed in matrix form as

{ E} =  B  { φ} , {∆E} =  B  {∆φ} 2

2

φ

{δ E} =  Bφ  {δφ }.

and

φ

(15)

2.3.3. FE -Implementation The integration through the thickness is performed analytically on the constitutive matrices. Introducing the integrated constitutive matrices [H1] for the elastic material, [H2] for the piezoelectric constants and [H3] for the dielectric constants of the piezomaterial between each electrode, the individual parts of the internal virtual work can be written as

∫δ H

dV = {δ q}

T

mm

T

T

1

1

L

T

1

L

Ω

V

= {δ q}

T

∫δ H

∫ ([B ] [H ]{ ε }+ ([B ] [H ][B ] + [G ⋅ N ]

({ F } + ( K  1

dV = − {δ q}

T

me

1

u m

m

+  1K g  m

L

){∆q} + {J })

∫ ([B ] [H ]{ φ }+ [B ] [H ]  B  {∆φ } + [G ⋅ N ] T

T

1

2

L

T

φ

2

L

= − {δ q}

T

)

1

Ω

V

)

 1S  [G ⋅ N ] {∆q} + {J1} d Ω m

)

 1S  [G ⋅ N ]{∆q} + {J 2 } d Ω e

(16)

({ F } +  K  {∆φ } +  K  {∆q}+ {J }) 1

1

e

1

u e

g

e

2

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∫δ H

dV = − {δφ }

T

em

T

T

φ

T

1

T

φ

2

= − {δφ }

T

1

dV = {δφ}

T

1

u

m

T

ee

({ Q} +  K 

e

L

2

Ω

V

∫δ H

∫ ( B  [H ] { ε }+  B  [H ] [B ]{∆q} + {J }) d Ω 3

)

{∆q} + {J 3}

∫ ( B  [ H ] B  { φ} + B  [ H ] B  {∆φ}) dΩ T

φ

T

1

φ

3

φ

3

φ

Ω

V

= {δφ}

T

({ Q} +  K  {∆φ}), 1

1

e

φ e

where Ω is the midsurface and the terms {J1} to {J3} denote nonlinear parts, which cannot be determined directly. It follows that the set of equations to be solved by iteration are then   1K u  +  1K g  +  1K g  m  m  e  T 1   K u  e 

 1K u   {∆q} { 2 \} − {1F } − {1F }  e   m e  ⋅  = 2 . 1 1 1 −  Kφ   {∆φ } { _} − { Q}m + { Q}e  e

(17)

2.4. Geometrically nonlinear transient simulation The set of differential equations of motion, without natural damping, to be solved is

[M ]{q} + {1 \}i = { 2 \}

(18)

{ _} = { _}, 1

2

i

where

{ \} = { F } + { F } { _} = { Q} − { Q} . 1

1

i

1

1

m

1

i

e

1

m

e

It is noticed that when an explicit time integration scheme like the central difference method is applied, the mechanical equation is decoupled from the electrical one. The mass matrix is denoted by [M], which can be calculated as consistent or lumped form. It should be denoted that additional rotational inertia is accounted for due to the Reissner-Mindlin hypothesis.

3. M ODAL FILTERS The theory of discrete modal sensor arrays is based on the modal decomposition of the generated voltage in each individual patch. In order to obtain an eigenvalue problem, the left hand side of Eq.(18) can be rewritten as the following set of differential equations, for the geometrically linear case, as

M 0 

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 1 0 q   Ku  m  + 0  0    1K  T   u e

 1Ku   q  e   . 1   −  Kφ   φ  e

(19)

In modern structural control it is attempted to control the modes rather than the displacements at particular points of the structure. In order to feed back the modal amplitudes or velocities, this information has to be gathered. Discrete modal sensor arrays pose a promising possibility, and despite the spatial aliasing is easy to handle and good in performance.

modal sensor signal + + + α1 φ1 φ2

α2

...

αn

linear combiner

φn

sensors structure Figure 1: Principle of the discrete modal sensor array.

Figure 1 displays the principle of a discrete modal sensor array. By choosing the gain αi in a particular way, the modal sensor will respond only to a mode j. Let gik be the modal output gain of sensor i to a unit amplitude of mode k, then the gains αi are chosen to fulfill n

∑α g i −1

i

ik

= δ jk ,

(20)

where δik is the Kronecker delta. The gains can be obtained by solving the rectangular system of equations

[G ]{α } j = {e} j ,

(21)

where [G] = gki, {α}j contains the gains to sense mode j and {e}j is the unit vector into direction j. The system of equations is solved by multiplying the pseudoinverse of [G], obtained by singular value decomposition, with {e}j.

4. NUM ERICAL EXAMPLE 4.1. Static analysis Next, a numerical example is presented that was initially posed by Kioua and Mirza [42] and later reconsidered by Lee et al. [43], see Figure 2. The linear problem deals with what is referred to in literature as shape control of a composite piezolaminated plate. A graphite/epoxy plate (10 × 10 in2) is hinged on all sides and it is covered on the top and bottom surface with a PZT layer as is shown in Figure 2. The material properties of the graphite/epoxy and the PZT are given in Table 1.

a PZT G1195 T300/976 [0/90/0] s PZT G1195 a Figure 2: Hinged composite piezolaminated plate.

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Initially the plate is loaded by a uniform pressure field of 200N/m2. Then a constant voltage is imposed to both PZT layers, which are polarized in opposite directions, to flatten the plate. The three configurations shown in Figure 3 refer to the applied voltages 0V, 15V, and 27V, respectively. The thickness of each layer of the plate material is 0.138mm and that of the PZT is 0.254mm. The stacking sequence of the composite plate is [0/90/0]S. From Figure 3 it can be seen that the linear result agrees very well with the one obtained by Lee et al. [43], who used a 6-parameter degenerated assumed strain shell element. It is also noticed that this example actually deals with a geometrically nonlinear problem,

centerline deflection (10 4 x w/a)

0

−4

−8 nonlinear Kioua et al. Lee et al. linear −12

0

0.25

0.5 distance (x/a)

0.75

1

Figure 3: Shape control results of the hinged plate.

100 10

5

σ11/E

1000 x w/a

voltage [V]

75

50

25 linear mid−point displacement nonlinear mid−point displacement nonlinear stress 0 0

1

2

3

non−dimensional displacement/stress

Figure 4: Voltage deflection curve of the hinged plate.

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causing the linear FE analysis to be inaccurate. Geometrically nonlinear analysis shows that linear finite element analysis underpredicts the stiffness of the plate by about 10% at a pressure of 200N/m2 and no voltage applied. During the flattening of the plate, i.e. in the configurations obtained by applying the voltages 15V and 27V, respectively, the difference between the deformed configurations obtained by linear and nonlinear analysis becomes small and therefore the latter ones are not indicated in Figure 3.

Table 1: Material properties.

E11 (GPa) E22 (GPa) ν12 (-) G12 (GPa) d31 (10-10 m/V) ρ (kg/m3)

PZT G1195 piezoceramic 63 63 0.3 24.2 2.54 7600

T300/976 graphite/epoxy 150 9 0.3 7.1 1592

Next, only the PZT layers of the same structure are imposed with a voltage. The deflection of the mid-point is shown against the applied voltage in Figure 4. The geometrically nonlinear analysis reveals that the plate behaves stiffer than predicted by linear analysis. This phenomenon is called stress stiffening and is caused by the induced membrane stresses. As an illustration, the stress σ11 in the middle of the plate is shown in Figure 4, too. In linear theory, this example is a pure bending problem and σ11 is expected to remain zero. Both cases are calculated with a [8 × 8] mesh. 4.2. Dynamic analysis In the dynamic analysis the same plate is simulated, only the lower piezoelectric layer is subdivided into 16 equal patches and each patch is used as a sensor, as depicted in Figure 5. actuator PZT G1195

sensor patches

T300/976 [0/90/0] S PZT G1195

3

4

1

2

hinged

symmetry

hinged

symmetry sensor numbering Figure 5: Hinged composite plate with piezoelectric sensor patches.

Now the plate is loaded by a step load of 2000N/m2 uniformly distributed pressure, for 1ms. This causes large amplitude vibrations in the geometrically nonlinear range of deformation as can be seen in Figure 6. Since the plate geometry and the loading are symmetric, only one quarter of the plate is simulated. In the eigenvalue analysis, therefore only the symmetric modes are considered. The first six (symmetric) eigenfrequencies are calculated as 82.542673 Hz, 395.66862 Hz, 431.97131 Hz, 707.40923Hz, 1015.3917Hz, and 1125.6368 Hz.

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We attempt to damp the first mode of vibration for which the following modal voltages and gains are calculated as

φ11 = 3781.4496 V φ21 = 1641.7267 V φ31 = 1675.0417 V φ41 = 742.00021V

α11 = 1.8581188 ⋅10−3 α 21 = 0.8067074 ⋅10 −3 ⇒ . α 31 = 0.8230776 ⋅10−3 α 41 = 0.3646021 ⋅10−3

1.5 linear nonlinear

midpoint deflection [mm]

1

0.5

0

−0.5

−1

−1.5

0

0.05

0.1

time [s]

0.15

0.2

0.25

Figure 6: Vibrations of the hinged plate due to a step load.

Figure 7 compares the modal signal for the first mode predicted by geometrically linear and nonlinear FE analysis, respectively. 0.015 linear nonlinear

0.01

modal signal [V]

0.005

0

−0.005

−0.01

−0.015

−0.02

0

0.05

0.1

time [s]

0.15

0.2

0.25

Figure 7: Modal signal of the free vibration.

It can be noticed that whereas the linear analysis shows a clear signal of the first mode, the nonlinear one predicts the failure of the modal sensor.

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The first mode is now simply damped by applying an actuator voltage proportional to the time derivative of the modal signal with a gain of 0.4V and the results predicted by geometrically linear as well as geometrically nonlinear finite element analysis are displayed in Figure 8. It can be seen that even though the modal sensor fails resolving the first mode of the geometrically nonlinear vibrations, the damping algorithm still seems to be as effective as predicted with geometrically linear finite elements. In both cases the first mode has vanished after 0.2 s. What remains is a vibration consisting of mainly the second and third eigenmodes. linear nonlinear

1

midpoint deflection [mm]

0.6

0.2

−0.2

−0.6

−1 0

0.05

0.1

0.15

0.2

0.25

time [s]

Figure 8: Vibration control of the first mode of the hinged plate.

5. SUMMARY A geometrically nonlinear shell finite element for transient analysis of smart composite layered structures with integrated piezoelectric sensor and actuator layers has been presented. The element accounts for moderately large rotations and transverse shear deformations in the framework of the Reissner-Mindlin hypothesis. A benchmark problem of a composite laminated plate with attached sensor and actuator layers proposed in literature has been used to test the element in static shape control simulation, where the effect of geometrical nonlinearity turned out to be important. For impulsively applied loading large amplitude vibrations are analyzed using the central difference method. For vibration suppression the effectivity of discrete modal sensor arrays is investigated in the geometrically linear and nonlinear approach. It was shown that even though the modal sensor fails resolving the first mode of the geometrically nonlinear vibrations, the damping algorithm is still as effective as predicted by geometrically linear finite element analysis.

REFERENCES 1. Robbins, D.H., Reddy, J.N., Analysis of piezoelectrically actuated beams using a layer-wise displacement theory, Computers & Structures 41 (1991), 265-279. 2. Lee, C.-K., 1992, Piezoelectric laminates: Theory and experiments for distributed sensors and actuators, in: H.S. Tzou and G.L. Anderson (eds.), Intelligent Structural Systems, 75-167, Kluwer Academic Publishers, DordrechtBoston-London 1992.

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