New analytical solutions for free vibration of embedded magneto-electro-elastic cylindrical shells with step-wise thickness variations

doi:10.1007/s10483-025-3228-7

Abstract

Abstract:

In recent years, magneto-electro-elastic (MEE) cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting. To aid the design of such energy harvesting devices, an accurate free vibration analysis of embedded MEE cylindrical shells with step-wise thicknesses is performed within the framework of symplectic mechanics. By using the Legendre transformation, a new known vector is defined to transform the higher-order partial differential governing equations into a set of lower-order ordinary differential equations. Therefore, the original vibration analysis is regarded as an eigen problem in the symplectic space, and analytical solutions can be represented by the symplectic series. In numerical examples, the new analytical solutions are compared with the existing results, and good agreement is observed. Furthermore, the effects of critical design parameters on free vibration characteristics are thoroughly investigated. All numerical results can serve as benchmarks for the development of other approximate or numerical methods.

Key words: analytical solution, free vibration, magneto-electro-elastic (MEE) material, cylindrical shell with variable thickness, symplectic method

2010 MSC Number:

O343

Jufang JIA, Huilin YIN, Qinyu YU, Jiabin SUN, Xinsheng XU, Zhenhuan ZHOU. New analytical solutions for free vibration of embedded magneto-electro-elastic cylindrical shells with step-wise thickness variations. Applied Mathematics and Mechanics (English Edition), 2025, 46(3): 447-466.

TrendMD

Figures/Tables 16

Fig. 1

Table 1

Material properties of BaTiO3-CoFe2O4"

Property	Value
Elastic/GPa	$c 11 = c 22 = 226$ , $c 12 = 125$ , $c 13 = c 23 = 124$ , $c 33 = 216$ , $c 66 = 50.5$
Piezoelectric/ $(C ⋅ m − 2)$	$e 31 = e 32 = − 2.2$ , $e 33 = 9.3$
Dielectric/( $C ⋅ V − 1 ⋅ m − 1$ )	$s 11 = s 22 = 5.64 × 10 − 9$ , $s 33 = 6.35 × 10 − 9$
Piezomagnetic/( $N ⋅ A ⋅ m − 1$ )	$q 31 = q 32 = 290.1$ , $q 33 = 349.9$
Magnetoelectric/( $N ⋅ s ⋅ V − 1 ⋅ C − 1$ )	$d 11 = d 22 = 5.367 × 10 − 12$ , $d 33 = 2 737.5 × 10 − 12$
Magnetic/( $N ⋅ s 2 ⋅ C − 2$ )	$ς 11 = ς 22 = − 297 × 10 − 6$ , $ς 31 = 83.5 × 10 − 6$
Density/( $kg ⋅ m − 3$ )	$ρ = 5.55 × 103$

Table 1

Table 2

Table 3

Fig. 2

Table 4

Natural frequencies (Hz) for 2-step MEE cylindrical shells under different boundary conditions"

n	C-C		C-S		S-S
n	$h 2 / h 1 = 0.5$	$h 2 / h 1 = 1$	$h 2 / h 1 = 0.5$	$h 2 / h 1 = 1$	$h 2 / h 1 = 0.5$	$h 2 / h 1 = 1$
1	62.312 6	62.334 8	48.490 2	48.490 6	35.000 3	35.000 7
2	25.381 3	25.739 3	18.515 7	19.115 5	13.004 7	13.662 3
3	19.124 5	22.739 7	15.671 9	20.962 7	15.186 7	19.874 8
4	24.249 9	37.164 2	21.608 6	36.786 7	21.603 8	36.560 6
5	32.312 5	59.024 0	30.967 3	58.913 2	30.967 3	58.640 5
6	44.517 4	86.279 9	43.897 0	86.234 8	43.897 0	86.200 0
7	60.087 8	118.614 1	59.785 9	118.590 8	59.785 9	118.571 0
8	78.505 9	155.961 3	78.344 3	155.947 3	78.344 3	155.934 8

Table 4

Table 5

Natural frequencies (Hz) for 2-step MEE cylindrical shells with different L1/L2"

n	$L 1 / L 2 = 0.5$
n	$m = 1$	2	3	4	5	6
1	62.312 5	136.676 6	222.777 8	311.008 1	395.612 5	471.043 2
2	25.144 2	62.252 3	109.927 2	163.524 6	219.857 3	276.294 4
3	16.703 0	35.354 5	61.883 0	94.501 0	131.363 8	170.631 5
4	21.112 0	31.930 7	46.652 1	64.611 9	88.443 5	115.341 5
5	30.582 1	36.736 2	49.370 9	62.577 0	73.132 1	90.680 3
6	43.640 2	46.751 3	54.432 9	67.179 1	82.852 9	90.740 4
7	59.592 8	61.340 4	65.700 2	73.720 5	85.640 1	100.794 5
8	78.179 5	79.336 5	82.027 2	87.012 6	94.841 3	105.662 7

Table 5

Table 6

Natural frequencies (Hz) for 2-step MEE cylindrical shells with different h2/h1"

n	$h 2 / h 1 = 0.5$
n	$m = 1$	$m = 2$	$m = 3$	$m = 4$	$m = 5$	$m = 6$
1	62.312 6	136.676 9	222.778 3	311.009 3	395.614 4	471.047 3
2	25.381 3	62.299 2	109.959 5	163.558 0	219.886 0	276.327 9
3	19.124 5	36.096 0	62.221 0	94.844 4	131.686 6	170.845 0
4	24.249 9	37.700 4	47.610 3	66.414 1	89.465 5	116.258 7
5	32.312 5	44.579 5	59.627 5	64.890 6	76.493 6	93.481 8
6	44.517 4	51.703 4	67.021 5	85.545 2	88.390 8	95.125 1
7	60.087 8	64.218 5	74.074 8	90.243 6	110.593 7	119.419 2
8	78.505 9	81.130 3	87.366 5	98.416 2	114.347 8	134.110 9

Table 6

Table 7

Mode shapes and the corresponding frequencies (Hz) for 2-step MEE cylindrical shells with different L1/L2"

$L 1 / L 2 = 0.5$	$m$
$L 1 / L 2 = 0.5$	1	2	3	4	5	6

Frequency	78.179 5	79.336 5	82.027 2	87.012 6	94.841 3	105.662 7
$L 1 / L 2 = 1$	$m$
$L 1 / L 2 = 1$	1	2	3	4	5	6

Frequency	78.505 9	81.130 3	87.366 5	98.416 2	114.347 8	134.110 9
$L 1 / L 2 = 2$	$m$
$L 1 / L 2 = 2$	1	2	3	4	5	6

Frequency	79.772 7	88.061 8	106.096 3	132.673 7	156.114 5	157.227 6

Table 7

Table 8

Mode shapes and the corresponding frequencies (Hz) for 2-step MEE cylindrical shells with different h2/h1"

$h 2 / h 1 = 0.5$	$m$
$h 2 / h 1 = 0.5$	1	2	3	4	5	6

Frequency	78.505 9	81.130 3	87.366 5	98.416 2	114.347 8	134.110 9
$h 2 / h 1 = 1$	$m$
$h 2 / h 1 = 1$	1	2	3	4	5	6

Frequency	155.961 3	156.575 7	157.693 0	159.438 9	161.963 9	165.424 2
$h 2 / h 1 = 2$	$m$
$h 2 / h 1 = 2$	1	2	3	4	5	6

Frequency	156.614 3	159.536 0	165.437 3	175.262 3	189.553 7	208.265 8

Table 8

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 9

Table 10

Mode shapes and the corresponding frequencies (Hz) for 3-step MEE cylindrical shells with different h2/h1"

$h 2 / h 1 = 0.5$	$m$
$h 2 / h 1 = 0.5$	1	2	3	4	5	6

Frequency	79.324 6	85.635 9	99.712 3	121.198 1	145.753 2	157.433 6
$h 2 / h 1 = 1$	$m$
$h 2 / h 1 = 1$	1	2	3	4	5	6

Frequency	155.961 3	156.575 7	157.693 0	159.438 9	161.963 9	165.424 2
$h 2 / h 1 = 2$	$m$
$h 2 / h 1 = 2$	1	2	3	4	5	6

Frequency	157.955 3	165.998 3	182.505 3	208.373 3	242.116 4	280.684 7

Table 10

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n	S-S		C-S		C-C
n	Ref. [60]	Present	Ref. [60]	Present	Ref. [60]	Present
1	0.718 8	0.715 5	0.986 3	0.981 8	1.262 9	1.257 1
2	0.410 1	0.408 2	0.490 7	0.488 5	0.601 4	0.598 6
3	0.800 3	0.796 7	0.812 1	0.808 4	0.831 5	0.827 8
4	1.479 3	1.472 6	1.482 1	1.475 4	1.486 3	1.479 6
5	2.372 3	2.361 6	2.373 5	2.362 7	2.374 8	2.364 1
6	3.469 5	3.453 8	3.470 1	3.454 4	3.470 7	3.455 1
7	4.768 5	4.747 0	4.768 9	4.747 4	4.769 3	4.747 8
8	6.268 7	6.240 4	6.268 9	6.240 6	6.269 2	6.240 9
9	7.969 4	7.933 5	7.969 6	7.933 6	7.969 8	7.933 8
10	9.870 7	9.826 1	9.870 8	9.826 2	9.870 9	9.826 4

L/R	n	S-S		C-C		C-S
L/R	n	Ref. [42]	Present	Ref. [42]	Present	Ref. [42]	Present
1	1	0.549 333	0.549 340	0.851 020	0.851 023	0.839 591	0.839 594
	2	0.622 456	0.622 464	0.657 085	0.657 089	0.656 497	0.656 501
	3	0.461 383	0.461 389	0.510 742	0.510 747	0.504 898	0.504 902
	4	0.342 895	0.342 899	0.409 108	0.409 113	0.396 288	0.396 291
	5	0.265 036	0.265 038	0.339 613	0.339 618	0.321 537	0.321 538
	6	0.220 569	0.220 571	0.294 328	0.294 333	0.273 013	0.273 015
	7	0.203 862	0.203 863	0.269 226	0.269 230	0.246 394	0.246 395
	8	0.208 817	0.208 817	0.261 750	0.261 755	0.238 499	0.238 500
5	1	0.176 590	0.176 590	0.236 939	0.236 937	0.221 194	0.221 194
	2	0.072 988	0.072 988	0.124 523	0.124 519	0.105 700	0.105 700
	3	0.041 159	0.041 159	0.073 567	0.073 564	0.059 344	0.059 344
	4	0.041 561	0.041 560	0.057 271	0.057 268	0.047 681	0.047 681
	5	0.053 819	0.053 819	0.062 517	0.062 515	0.054 198	0.054 198
	6	0.064 339	0.064 339	0.072 523	0.072 520	0.064 347	0.064 347
	7	0.077 950	0.077 950	0.083 824	0.083 821	0.077 951	0.077 951
	8	0.096 460	0.096 460	0.100 175	0.100 173	0.096 460	0.096 460
10	1	0.056 538	0.056 537	0.097 703	0.097 659	0.082 455	0.082 436
	2	0.020 750	0.020 749	0.041 392	0.041 361	0.032 011	0.032 001
	3	0.020 546	0.020 544	0.027 756	0.027 740	0.022 860	0.022 856
	4	0.029 146	0.029 144	0.033 748	0.033 732	0.029 148	0.029 146
	5	0.038 195	0.038 195	0.041 385	0.041 373	0.038 196	0.038 196
	6	0.052 000	0.051 999	0.053 611	0.053 604	0.052 000	0.051 999
	7	0.069 888	0.069 888	0.070 682	0.070 679	0.069 888	0.069 888
	8	0.091 165	0.091 165	0.091 582	0.091 580	0.091 165	0.091 165

n	m
n	1	2	3	4	5	6
1	62.407 2	136.820 0	222.943 2	311.181 8	395.818 2	471.294 7
2	26.609 1	63.489 2	110.927 6	164.484 5	220.926 2	277.475 2
3	28.170 5	47.279 3	70.455 5	100.479 1	136.816 5	175.670 4
4	43.335 6	63.282 6	79.294 5	90.163 6	109.734 6	134.334 2
5	63.144 9	80.821 7	107.765 4	121.746 1	126.366 5	139.605 6
6	88.994 0	101.456 5	126.082 6	157.243 6	175.004 2	177.386 2
7	120.624 1	129.457 3	148.041 6	176.034 8	209.400 1	237.244 1
8	157.638 7	164.470 0	178.537 4	200.859 0	230.349 1	264.387 7