Stiffness and natural vibration of a rotating sandwich metal porous cantilever pre-twisted plate reinforced by graphene

doi:10.1007/s10483-026-3341-8

Abstract

Abstract:

With the continuous increase in performance requirements for power systems in the aerospace and low-altitude economy sectors, designing lightweight and high-strength blade structures with excellent dynamic characteristics has become critical. This paper puts forward a dynamic model for a rotating functionally graded graphene-reinforced (FG-GPR) sandwich metal porous cantilever pre-twisted plate (PTP), aiming to analyze its natural vibration characteristics. To this end, the mixture principle and the revised Halpin-Tsai model are used to determine the parameters of graphene and porosity distributions in the core layer. With the classical plate theory, the Rayleigh-Ritz method, and the polynomials, the dynamic equations are derived to solve for the free vibration mode shapes and frequencies of the rotating FG-GPR sandwich metal porous cantilever PTP. The comparison of natural frequencies and mode shapes with available literature results confirms the precision of the theoretical formulation and numerical computations. The bending stiffnesses are analyzed. Finally, the effects of different graphene/pore distributions, length-to-thickness/width ratios, layer thickness ratios, twist angles, and rotational speeds on the natural frequencies of the system are systematically investigated.

Key words: sandwich pre-twisted plate (PTP), metal porous, graphene-reinforced (GPR) composite, natural vibration

2010 MSC Number:

O326

Chengmin NIE, Fu GUO, Yuxin HAO, Xiaojun GU. Stiffness and natural vibration of a rotating sandwich metal porous cantilever pre-twisted plate reinforced by graphene. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 135-152.

TrendMD

Figures/Tables 19

Fig. 1

Fig. 2

Table 1

Comparison of natural frequencies of the cantilever PTP under the rotating state"

$Ω ⌢$	Mode order	Frequency/Hz			Error 1/%	Error 2/%
$Ω ⌢$	Mode order	Present	Ref. [13] (theoretical)	Ref. [13] (FEM)	Error 1/%	Error 2/%
0 r/min	1st	340.58	343.3	337.8	0.792	0.823
	2nd	1 469.90	1 473.0	1 435.5	0.210	2.396
	3rd	2 110.60	2 117.9	2 097.5	0.345	0.625
1 000 r/min	1st	489.39	482.6	482.6	1.407	1.407
	2nd	1 530.88	1 516.9	1 479.4	0.922	3.480
	3rd	2 330.90	2 285.1	2 265.7	2.004	2.878
Note: Error 1 is defined as the absolute difference between the present result and the theoretical result from Ref. [13], divided by the theoretical value, and Error 2 is defined as the absolute difference between the present result and the FEM result from Ref. [13], divided by the FEM value

Table 1

Table 2

Table 3

Structural and material parameters of the rotating FG-GPR cantilever PTP"

Parameter	Value	Parameter	Value
Length of the plate, $L$ /m	0.3	Young’s modulus of the matrix material, $E m$ /GPa	70
Width of the plate, $b$ /m	0.2	Density of the matrix material, $ρ m / (kg ⋅ m − 3)$	2 707
Thickness of the plate, $h$ /m	0.001	Poisson’s ratio of the matrix material, $ν m$	0.3
Average length of GPL, $L GPL$ /μm	2.5	Young's modulus of GPL, E_GPL/GPa	1 010
Average width of GPL, b_GPL/μm	1.5	Density of GPL, $ρ GPL / (kg ⋅ m − 3)$	1 062.5
Average thickness of GPL, $h GPL$ /nm	1.5	Poisson’s ratio of GPL, $ν GPL$	0.186
Installation angle, $φ / (∘)$	0	Porosity parameter, $N 0$	0.4
Twist angle, $θ / (∘)$	0	GPL addition parameter, $V GPL0$	0.01
Layer thickness ratio	1:1:1

Table 3

Table 4

Fig. 3

Table 5

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Table 6

Natural frequencies of the FG-GPR PTP at different rotational speeds"

GPL pattern	$Ω ⌢$ / $(r ⋅ mi n − 1)$	Frequency/Hz
GPL pattern	$Ω ⌢$ / $(r ⋅ mi n − 1)$	1st	2nd	3rd	4th	5th
No GPL	0	27.710 9	170.387 9	197.437 6	447.755 8	481.155 0
	3 000	67.846 3	208.650 3	238.420 4	463.900 3	512.577 6
	6 000	125.726 3	238.144 3	372.742 4	488.045 7	592.779 1
	9 000	184.706 0	279.724 3	512.210 9	532.023 2	705.237 7
	12 000	244.143 4	322.279 3	558.337 2	682.323 2	836.081 5
	15 000	303.948 5	380.203 5	606.703 2	839.395 2	978.896 3
	18 000	363.801 1	401.210 7	659.614 0	941.694 4	995.138 0
GPL-O	0	28.870 9	177.525 5	209.270 5	467.173 7	507.518 7
	3 000	68.384 9	219.919 9	243.638 2	483.313 0	537.516 5
	6 000	126.139 1	248.130 1	376.212 8	506.982 3	614.454 7
	9 000	184.947 8	287.570 0	521.517 4	543.208 4	723.467 9
	12 000	244.436 8	334.229 6	575.532 1	684.829 6	852.927 5
	15 000	304.158 7	387.341 3	622.253 2	841.969 0	992.443 8
	18 000	363.995 7	434.786 5	674.078 0	998.668 7	1 139.554 0
GPL-X	0	29.113 9	179.023 1	210.164 1	470.976 8	510.289 4
	3 000	68.499 0	220.770 7	244.743 6	487.107 5	540.160 4
	6 000	126.227 8	248.888 2	376.943 8	510.643 0	616.802 4
	9 000	185.007 3	289.026 1	522.754 9	545.976 6	725.627 3
	12 000	244.486 1	336.516 5	578.747 0	685.296 1	853.979 5
	15 000	304.296 2	387.887 7	625.346 2	842.327 3	944.003 2
	18 000	364.034 5	436.243 8	676.889 5	999.057 2	1 140.979 0
GPL-A	0	28.944 0	177.976 4	209.543 3	468.298 4	508.363 0
	3 000	68.419 2	220.180 9	243.970 4	484.447 2	538.319 9
	6 000	125.647 3	248.362 8	376.384 2	507.847 5	615.167 9
	9 000	184.965 7	288.567 8	521.904 0	544.024 5	742.211 2
	12 000	244.486 1	336.516 5	578.747 0	685.296 1	853.979 5
	15 000	304.296 2	387.887 7	625.346 2	842.327 3	994.003 2
	18 000	364.034 5	436.243 8	676.889 5	999.057 2	1 140.979 0
GPL-U	0	28.965 6	178.110 5	209.618 3	468.655 5	508.607 3
	3 000	68.424 5	220.252 3	244.072 2	484.397 3	538.541 2
	6 000	126.171 3	248.424 7	376.496 0	508.372 7	615.366 6
	9 000	184.964 0	288.582 5	522.014 3	544.268 8	724.317 0
	12 000	244.455 8	336.159 9	576.780 4	685.008 2	852.899 2
	15 000	304.175 5	387.550 8	623.627 9	842.123 5	993.049 9
	18 000	364.010 9	436.078 0	675.171 4	998.820 4	1 140.136 7

Table 6

Fig. 12

Fig. 13

References 40

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GPL pattern	Method	Frequency/Hz
GPL pattern	Method	1st	2nd	3rd	4th	5th
GPL-X	Present	28.890	97.630	179.48	330.71	451.65
GPL-X	COMSOL	28.950	97.349	179.87	328.35	447.76
GPL-O	Present	28.960	97.870	179.93	331.54	452.78
GPL-O	COMSOL	28.940	96.480	178.23	330.21	450.96
GPL-A	Present	28.910	97.700	179.62	330.96	451.99
GPL-A	COMSOL	28.940	98.880	178.26	329.66	449.56
GPL-U	Present	28.919	97.720	179.65	331.03	452.09
GPL-U	COMSOL	28.895	97.149	179.52	327.67	446.87

Pore distribution	GPL pattern	Frequency/Hz
Pore distribution	GPL pattern	1st	2nd	3rd	4th	5th
FG-X	No GPL	27.710 8	170.387 7	197.441 6	446.677 2	481.170 5
	GPL-O	28.870 8	177.525 6	209.274 7	466.169 0	507.533 2
	GPL-X	29.113 2	179.023 1	210.167 6	465.210 1	510.313 8
	GPL-A	28.943 9	177.976 7	209.547 7	467.297 3	508.377 9
	GPL-U	28.965 5	178.110 7	209.266 2	467.686 2	508.621 6
FG-O	No GPL	27.206 7	167.302 2	197.112 4	439.091 1	478.140 0
	GPL-O	28.326 8	174.194 3	209.400 1	458.079 7	505.115 3
	GPL-X	28.482 6	175.157 7	208.890 1	460.356 1	505.052 6
	GPL-A	28.364 7	174.429 7	209.266 1	459.672 1	505.080 0
	GPL-U	28.376 6	174.502 7	209.038 0	458.826 7	505.085 6
FG-A	No GPL	27.539 0	169.337 6	197.150 6	444.090 0	479.835 6
	GPL-O	28.376 6	176.502 7	209.238 0	463.826 7	506.502 8
	GPL-X	28.900 0	177.714 7	209.599 5	466.476 7	508.287 2
	GPL-A	28.739 3	176.724 7	209.059 7	464.212 8	506.542 4
	GPL-U	28.589 0	175.787 1	209.155 2	461.895 9	505.875 4
FG-U	No GPL	27.502 4	169.115 3	196.850 7	443.475 6	479.151 6
	GPL-O	28.644 9	176.476 3	208.806 6	462.784 1	505.652 1
	GPL-X	28.856 8	177.453 2	209.178 4	465.953 8	507.372 3
	GPL-A	28.704 3	176.512 6	208.911 0	463.946 4	506.134 6
	GPL-U	28.722 1	176.622 9	208.941 2	464.006 9	506.269 9