Active vibration control of GPLs reinforced FG metal foam plates with piezoelectric sensor and actuator layers

Active vibration control of GPLs reinforced FG metal foam plates with piezoelectric sensor and actuator layers Accepted Manuscript Active vibration control of GPLs reinforced FG metal foam plates with[.]

Active vibration control of GPLs-reinforced FG metal foam plates with piezoelectric

sensor and actuator layers

Nam V Nguyen, Jaehong Lee, H Nguyen-Xuan

DOI: https://doi.org/10.1016/j.compositesb.2019.05.060

Reference: JCOMB 6849

To appear in: Composites Part B

Received Date: 20 February 2019

Revised Date: 4 April 2019

Accepted Date: 5 May 2019

Please cite this article as: Nguyen NV, Lee J, Nguyen-Xuan H, Active vibration control of

GPLs-reinforced FG metal foam plates with piezoelectric sensor and actuator layers, Composites Part B

(2019), doi: https://doi.org/10.1016/j.compositesb.2019.05.060

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Active vibration control of GPLs-reinforced FG metal foam plates

with piezoelectric sensor and actuator layers

Nam V Nguyena, Jaehong Leeb, H Nguyen-Xuanc,b,∗

a Faculty of Mechanical Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

b Department of Architectural Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, Seoul 143-747, South

struc-Keywords: Polygonal finite element method, Piezoelectric materials, FG metal foam plate,Graphene platelets reinforcement, Active control

where lGP L, wGP L and tGP L are the average length, width and thickness of GPLs, respectively;132

Meanwhile, EGP Land Emare the Young’s modulus of GPLs and metal matrix, respectively Then,133

the Poisson’s ratio and the mass density of the GPLs reinforced metal matrix are defined by the134



 εE



146

where u and ˙u represent the mechanical displacement and velocity field vectors; Meanwhile, fs





, u0 =





, u2 =

trian-gular and quadrilateral ones such that Ω =Sn e

e=1Ωeand Ωi∩Ωj = ∅, i 6= j Now, the approximate218

displacement function uh(x) for FG plate element can be expressed as

(33)233

{n,7n} =X

ˆi,ˆj

hˆ

gˆi,7ˆ1 j−4, ˆgˆi,7ˆ2 j−3, ˆgˆ3i,7ˆj−2i+X

ˆi

,

(47)281

in which lˆi = xˆj − xˆk is the length of the ˆith edge of element while the indices ˆi, ˆj, ˆk are found282

in Eq (42) As evidently demonstrated in Eqs (44 − 46), although the quadratic serendipity283

shape functions are adopted to determine the assumed bending strain field, the total number of284

DOFs per element does not increase when compared with the formulations in [50] This means285

the mid-side nodes of element will be eliminated in the computational process

 dφ



=

FQ

Now, by substituting the second equation into the first one of Eq (52), one obtains

sKφu

sds− GvKφφ

aK−1 φφ



sKφu

sd˙

s (63)345

Finally, by substituting Eq (63) into Eq (59) yields

open circuit conditions which is given as ∆ = ωopen −ωclosed

ω closed is also considered and presented Fig.445

11 depicts the first six mode shapes of the piezoelectric FG metal foam plate with PD-S, GPL-S,446

porosity coefficient e0 = 0.3 and weight fraction of GPLs ΛGP L = 1.0 wt %

0 ≤ t ≤ t1,

 1 − t/t10

where q0 = 0.1 MPa, γ = 330s−1 The time history F (t) is plotted in Fig 13

max E

min E

Symmetric porosity distribution

min E Asymmetric porosity distribution

(b) Dispersion patterns of GPLs

Figure 2: Porosity distributions and dispersion patterns of GPLs [17].

23

0.5 0

-0.5

-0.5

-0.2 0 0.2 0.4 0.6 0.8 1

(a) Vertex node

-0.2

1

0 0.2 0.4

0.5

0.6 0.8

1 1

0

0.5 0

-0.5

-0.5

0 0.2 0.4 0.6 0.8 1

(b) Mid-side node

Figure 5: The quadratic serendipity shape functions of a pentagonal element.

Figure 6: A schematic view of a FG metal foam plate integrated piezoelectric sensor and actuator.

Time (second)

-3 -2 -1 0 1 2 3

Figure 8: Dynamic response of the SSSS FG square plate with respect to various velocity feedback gains G v

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PD-A and GPL-A

PD-A and GPL-U

(a) Closed-circuit

Porosity coe,cient e 0

30 35 40 45 50 55

(b) Open-circuit

Figure 9: Effect of porosity coefficient on the natural frequencies of piezoelectric FG metal foam plate with Λ =

26

PD-A and GPL-A

PD-A and GPL-U

(a) Closed-circuit

GPL weight fraction, $ GP L (%)

32 35 40 45

50 53

(b) Open-circuit

Figure 10: Effect of GPL weight fraction on the natural frequencies of piezoelectric FG metal foam plate with e 0 = 0.5.

27

Figure 12: A geometry and polygonal mesh of a piezoelectric FG metal foam square plate with complicated shape hole.

29

Figure 13: Time history of load factor.

(a) Step load

Time (second) #10-3

-5 -4 -3 -2 -1 0 1 2 3

(b) Triangular load

Figure 14: Effect of the porosity coefficient on dynamic responses of the SSSS piezoelectric FG metal foam plate with PD-S and GPL-S (Λ GP L = 1.0 wt %).

Forced vibration Free vibration

(a) Sinusoidal load

Time (second) #10-3

-8 -6 -4 -2 0 2 4

(b) Explosive blast load

Figure 15: Effect of the weight fraction and dispersion pattern of GPLs on dynamic responses of the SSSS tric FG metal foam plate with PD-A (e 0 = 0.3).

piezoelec-31

(a) Step load

Time (second) #10-3

-15 -10 -5 0 5

(b) Triangular load

# 10-3-9

Time (second)

(c) Sinusoidal load

# 10-3-17

-15 -10 -5 0 5

(d) Explosive blast load

Figure 16: Effect of the velocity feedback control gain G v on dynamic responses of the CCCC piezoelectric FG metal foam plate subjected to various dynamic loads.

Table 2: The first natural frequencies (Hz) of the SSSS piezoelectric FG square plate with various mesh levels.

level Present Analytical [68] Error (%) Present Analytical [68] Error (%)

Table 1: Material properties of the core and piezoelectric layers (ε 0 = 8.85 × 10−12F/m).

Table 3: The first three natural frequencies (Hz) of the SSSS piezoelectric FG square plate with different conditions.

(1,2) 1056.726 1046.812 1049.356 916.249 906.103 907.918 787.201 776.889 779.313 (2,2) 1667.141 1647.337 1652.929 1446.531 1426.750 1430.642 1239.573 1221.231 1226.615

(1,2) 1071.133 1062.196 1066.390 934.374 925.467 929.406 810.511 801.917 807.162 (2,2) 1689.272 1668.387 1679.191 1474.415 1453.265 1463.819 1275.207 1255.391 1269.525

(1,2) 1906.338 1948.947 1952.530 1661.371 1691.326 1691.992 1410.556 1439.945 1445.353 (2,2) 2744.670 2965.790 2974.440 2409.402 2577.999 2580.078 2031.625 2186.014 2197.887

(1,2) 1930.704 1976.195 1981.321 1692.145 1725.670 1728.702 1448.483 1483.560 1492.484 (2,2) 2776.269 3001.279 3015.610 2449.726 2622.761 2693.452 2079.403 2242.320 2265.315

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Table 4: The first natural frequencies (Hz) of the SSSS piezoelectric FG metal foam plate reinforced by GPLs.

pattern fraction (%) condition e0= 0.0 e0= 0.2 e0= 0.4 e0= 0.6 e0= 0.2 e0= 0.4 e0= 0.6 GPL-S 0.0 Closed 185.954 184.975 184.692 185.618 180.38 173.628 164.676

Open 187.325 186.442 186.272 187.341 181.889 175.333 166.693 0.5 Closed 226.503 224.939 223.977 224.051 219.380 210.550 198.541

Open 227.663 226.181 225.319 225.519 220.660 212.004 200.282 1.0 Closed 260.176 258.178 256.726 256.210 251.833 241.340 226.838

Open 261.217 259.293 257.932 257.531 252.982 242.650 228.418 GPL-A 0.5 Closed 211.677 210.361 209.757 210.401 204.348 195.498 183.965

Open 212.937 211.712 211.217 211.999 205.753 197.109 185.905 1.0 Closed 231.014 229.307 228.288 228.488 222.456 212.161 198.880

Open 232.228 230.612 229.705 230.046 223.819 213.734 200.789 GPL-U 0.5 Closed 211.560 210.346 209.900 210.802 205.052 197.155 186.657

Open 212.810 211.684 211.343 212.376 206.430 198.714 188.504 1.0 Closed 234.182 232.775 232.200 233.094 226.877 217.998 206.171

Open 235.351 234.027 233.551 234.568 228.166 219.457 207.904

Table 5: The first six natural frequencies (Hz) of the SSSS piezoelectric FG square plate with different conditions.

pattern fraction (%) condition e0= 0.0 e0= 0.3 e0= 0.5 e0= 0.7 e0= 0.3 e0= 0.5 e0= 0.7 GPL-S 0.0 Closed 805.134 794.049 787.516 781.828 769.823 738.376 693.824

Open 808.156 797.319 790.953 785.415 773.385 742.511 698.969 1.0 Closed 1104.946 1082.612 1066.042 1046.436 1051.926 1003.302 932.303

Open 1107.121 1084.965 1068.525 1049.053 1054.502 1006.338 936.221 GPL-A 1.0 Closed 1005.851 988.82 977.757 966.788 951.711 904.34 839.762

Open 1008.595 991.816 980.932 970.144 955.058 908.339 844.892 GPL-U 1.0 Closed 1019.125 1003.897 994.72 986.542 972.206 930.438 871.078

Open 1021.761 1006.753 997.726 989.686 975.317 934.061 875.618

35