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Bending and Vibration Analysis of Magneto-Electro Bilaterally Coated Quasi-3D Microbeam Via DQ-FEM

Besma KHOUANI

Besma KHOUANI

“IS2M Laboratory, Mechanical engineering Department, Faculty of Technology, University Abou Beckr BeklaidTlemcen, Algeria
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KHOUANI, Besma
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Ahmed SAIMI

Ahmed SAIMI

“Mechanical engineering Department, Faculty of Science and Technology, University Belhadj BouchaibAin Temouchent, Algeria
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SAIMI, Ahmed
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Ismail BENSAID

Ismail BENSAID

“IS2M Laboratory, Mechanical engineering Department, Faculty of Technology, University Abou Beckr BeklaidTlemcen, Algeria
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BENSAID, Ismail
  
Sep 05, 2025
Bending and Vibration Analysis of Magneto-Electro Bilaterally Coated Quasi-3D Microbeam Via DQ-FEM's Cover Image
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Published Online: Sep 05, 2025
Page range: 337 - 349
Received: Dec 26, 2024
Accepted: Apr 10, 2025
DOI: https://doi.org/10.2478/ama-2025-0041
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© 2025 Besma KHOUANI et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Fig. 1.

Piezo-bilaterally FG microbeam model
Piezo-bilaterally FG microbeam model

Fig. 2.

Comparative analysis of the bending of a doubly S-S microbeam exposed to an even distribution of load
Comparative analysis of the bending of a doubly S-S microbeam exposed to an even distribution of load

Fig. 3.

Comparative analysis of electrical potential of a doubly S-S microbeam exposed to an even distribution of load
Comparative analysis of electrical potential of a doubly S-S microbeam exposed to an even distribution of load

Fig. 4.

Comparative analysis of magnetic potential of a doubly S-S microbeam exposed to an even distribution of load
Comparative analysis of magnetic potential of a doubly S-S microbeam exposed to an even distribution of load

Fig. 5.

Deflection of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))
Deflection of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))

Fig. 6.

Electric potential of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))
Electric potential of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))

Fig. 7.

Magnetic potential of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))
Magnetic potential of MEE bilaterally microbeam with various boundary conditions (kz = 1, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%))

Fig. 8.

Simply supported microbeam midspan deflection with various thickness and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (kz = 1, b = 2h, L = 20h,)
Simply supported microbeam midspan deflection with various thickness and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (kz = 1, b = 2h, L = 20h,)

Fig. 9.

Electric (a) and magnetic (b) potentials at simply supported microbeam midspan with various thickness and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (kz = 1, b = 2h, L = 20h,)
Electric (a) and magnetic (b) potentials at simply supported microbeam midspan with various thickness and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (kz = 1, b = 2h, L = 20h,)

Fig. 10.

Simply supported microbeam midspan deflection with various FG fraction index and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (h = 20μm, b = 2h, L = 20h,)
Simply supported microbeam midspan deflection with various FG fraction index and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (h = 20μm, b = 2h, L = 20h,)

Fig. 11.

Electric (a) and magnetic (b) potentials at simply supported microbeam midspan with various FG fraction index and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (h = 20μm, b = 2h, L = 20h,)
Electric (a) and magnetic (b) potentials at simply supported microbeam midspan with various FG fraction index and MEE inner material mixture percentage BaTiO3 – CoFe2O4 (h = 20μm, b = 2h, L = 20h,)

Fig. 12.

Electric and magnetic potentials distribution of simply supported microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)
Electric and magnetic potentials distribution of simply supported microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)

Fig. 13.

Electric and magnetic potentials distribution of clamped microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)
Electric and magnetic potentials distribution of clamped microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)

Fig. 14.

Electric and magnetic potentials distribution of clamped free microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)
Electric and magnetic potentials distribution of clamped free microbeam with kz = 5, h = 20μm, b = 2h, L = 20h, BaTiO3(50%) – CoFe2O4(50%)

Fig. 15.

Deflection with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h
Deflection with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h

Fig. 16.

Electric potential with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h
Electric potential with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h

Fig. 17.

Magnetic potential with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h
Magnetic potential with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h

Fig. 18.

Natural frequencies with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h
Natural frequencies with respect to length-thickness ratio of simply supported microbeam with kz = 5, b = 2h

Materials Properties of the mixture (BaTiO3 – CoFe2O4)

BaTiO3 – CoFe2O4 epoxy
0%-100% 20%-80% 40%-60% 50%-50% 60%-40% 80%-20% 100%-0%
C11 286 262 238 226 214 190 166 4.889
C13 170 151.6 133.2 124 114.8 96.4 78 2.407
C33 269.5 248 226.5 215.75 205 183.5 162 4.889
C55 45.3 44.84 44.38 44.15 43.92 43.46 43 1.241
q31 580.3 464.24 348.18 290.15 232.12 116.06 0 0
q33 699.7 559.76 419.82 349.85 279.88 139.94 0 0
q15 550 440 330 275 220 110 0 0
e31 0 -0.88 -1.76 -2.2 -2.64 -3.52 -4.4 0
e33 0 3.72 7.44 9.3 11.16 14.88 18.6 0
e15 0 2.32 4.64 5.8 6.96 9.28 11.6 0
s11 0.08 2.3 4.53 5.64 6.75 8.98 11.2 0
s33 0.093 2.59 5.10 6.35 7.6 10.10 12.6 0
d11 0 2.6 4.58 5.38 6.02 7.04 0 0
d33 0 2020 2760 2740 2520 1550 0 0
μ11 590 473 356 297.5 239 122 5 0
μ33 157 127.6 98 83.5 68.8 39.4 10 0
ρ 5300 5400 5500 5550 5600 5700 5800 1180
MLSP l 7.33 7.29 7.24 7.21 7.18 7.10 7 16.93

Comparison of numerical results (b = 2h, L = 20h), 50%-50% BaTiO3 – CoFe2O4

h(μm) Frequency (MHz)
1st mode 2nd mode 3rd mode
[12] present [12] present [12] present
14.42 4.097 4.069 16.811 16.228 39.647 34.853
28.84 1.710 1.701 7.007 6.784 16.466 14.796
h(μm) Midspan Deflection w/h Midspan Electric potential γ(V) Midspan Magnetic potential 𝜁(A)
[12] present [12] present [12] present
14.42 0.0792 0.0793 -1.251 -1.229 0.0125 0.0122
28.84 0.0284 0.0283 -0.896 -0.877 0.0090 0.0087

Natural frequency (MHz) of simply supported MEE FG microbeam (b = 2h, L = 20h,)

BaTiO3 – CoFe2O4
h(μm) kz 0%-100% 20%-80% 40%-60% 50%-50% 60%-40% 80%-20% 100%-0%
20 0 3.0009 2.8571 2.7108 2.6363 2.5613 2.4071 2.2472
0.5 2.5250 2.4124 2.2975 2.2391 2.1802 2.0589 1.9332
1 2.2200 2.1283 2.0348 1.9872 1.9393 1.8404 1.7380
5 1.5182 1.4840 1.4491 1.4313 1.4135 1.3766 1.3386
10 1.3919 1.3731 1.3539 1.3441 1.3344 1.3141 1.2932
15 1.3538 1.3411 1.3282 1.3216 1.3150 1.3013 1.2872
20 1.3352 1.3260 1.3166 1.3118 1.3070 1.2970 1.2867
40 0 1.3578 1.2891 1.2191 1.1836 1.1476 1.0739 0.9974
0.5 1.0979 1.0449 0.9909 0.9634 0.9356 0.8786 0.8194
1 0.9270 0.8847 0.8415 0.8196 0.7973 0.7518 0.7045
5 0.5204 0.5078 0.4950 0.4885 0.4820 0.4687 0.4550
10 0.4603 0.4548 0.4492 0.4464 0.4436 0.4378 0.4318
15 0.4502 0.4469 0.4435 0.4418 0.4401 0.4366 0.4330
20 0.4481 0.4458 0.4435 0.4423 0.4411 0.4387 0.4362
100 0 0.5260 0.4989 0.4713 0.4572 0.4430 0.4139 0.3836
0.5 0.4189 0.3980 0.3768 0.3660 0.3550 0.3326 0.3092
1 0.3475 0.3310 0.3141 0.3055 0.2968 0.2789 0.2604
5 0.1722 0.1677 0.1633 0.1610 0.1587 0.1541 0.1494
10 0.1473 0.1458 0.1442 0.1434 0.1427 0.1411 0.1395
15 0.1449 0.1441 0.1433 0.1429 0.1425 0.1417 0.1408
20 0.1455 0.1450 0.1445 0.1442 0.1440 0.1434 0.1429
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