Dynamic behaviors of graphene platelets-reinforced metal foam piezoelectric beams with velocity feedback control

doi:10.1007/s10483-025-3209-8

Abstract

Abstract:

Graphene platelets (GPLs)-reinforced metal foam structures enhance the mechanical properties while maintaining the lightweight characteristics of metal foams. Further bonding piezoelectric actuator and sensor layers on the surfaces of GPLs-reinforced metal foam beams enables active vibration control, greatly expanding their applications in the aerospace industry. For the first time, this paper investigates the vibration characteristics and active vibration control of GPLs-reinforced metal foam beams with surface-bonded piezoelectric layers. The constant velocity feedback scheme is used to design the closed-loop controller including piezoelectric actuators and sensors. The effects of the GPLs on the linear and nonlinear free vibrations of the beams are numerically studied. The Newmark-β method combined with Newton's iteration technique is used to calculate the nonlinear responses of the beams under different load forms including harmonic loads, impact loads, and moving loads. Additionally, special attention is given to the vibration reduction performance of the velocity feedback control on the responses of the beam.

Key words: active vibration control, piezoelectric material, velocity feedback control, metal foam, nonlinear free vibration

2010 MSC Number:

O341

Jie CHEN, Xinyue ZHANG, Mingyang FAN. Dynamic behaviors of graphene platelets-reinforced metal foam piezoelectric beams with velocity feedback control. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 63-80.

TrendMD

Figures/Tables 20

Fig. 1

Fig. 2

Table 1

Dimensionless fundamental frequency of the GPLs-reinforced metal foam beam (PD-X, GPL-X, C-C beam, ΛGPL=1%, and L/h=20)"

Method	$e 0 = 0$	$e 0 = 0.5$	$e 0 = 0.4$	$e 0 = 0.6$
Present	0.450 5	0.446 8	0.444 3	0.443 6
Ref. [14]	0.450 5	0.446 8	0.444 2	0.443 6

Table 1

Table 2

Nonlinear frequency ratio ωNL/ωL of the GPLs-reinforced metal foam beam (PD-X, GPL-X, C-C beam, L/h=20, ΛGPL=1%, and e0=0.5)"

Method	$w ′ max = 0.2$	$w ′ max = 0.4$	$w ′ max = 0.6$	$w ′ max = 0.8$	$w ′ max = 1.0$
Present	0.448 0	0.461 2	0.482 1	0.509 7	0.542 8
Ref. [40]	0.448 0	0.461 2	0.482 2	0.509 6	0.542 8

Table 2

Table 3

Linear fundamental frequency of the piezoelectric CNTRC beam (h=100 mm, VCN∗=0.17, and L/H=100)"

Method	$h p / h = 1 / 8$	$h p / h = 1 / 12$	$h p / h = 1 / 20$
Present	1.970 0	0.238 6	0.296 6
Ref. [41]	1.960 4	0.237 5	0.295 1

Table 3

Table 4

Nonlinear frequency ratio of the piezoelectric CNTRC beam (h=80 mm, L/H=20, hp/h=1/12, and VCN∗=0.17)"

Method	$w ′ max = 0.5$	$w ′ max = 1.0$	$w ′ max = 1.5$	$w ′ max = 2.0$
Present	1.003 5	1.037 3	1.030 5	1.053 4
Ref. [41]	1.003 4	1.065 8	1.030 0	1.052 9

Table 4

Fig. 3

Table 5

The parameters of the basic piezoelectric copper-matrix beam"

Item	Copper-matrix beam	Piezopatches	Item	Copper-matrix beam	Piezo-patches
Young's modulus, E/GPa	130	126	Thickness, h/mm	5	0.5
Density, $ρ / (kg ⋅ m − 3)$	8 960	7 500	Poisson's ratio, $ν$	0.34	0.3
Length, $L$ /mm	500	500	Piezoelectric stress constant, $e 31 / (C ⋅ m − 2)$	/	-6.5
Width, b/mm	10	10	Dielectric constant, $ξ 33 / (F ⋅ m − 1)$	/	$1.5 × 10 − 8$

Table 5

Fig. 4

Table 6

Fundamental frequency of the GPLs-reinforced metal foam piezoelectric beam (GPL-X, L/h=100, h=5 mm, and ΛGPL=1%)"

$h p / h$	Distribution	Fundamental frequency/Hz
$h p / h$	Distribution	$e 0 = 0$	$e 0 = 0.3$	$e 0 = 0.5$
1/5	PD-X	174.215 2	174.321 1	174.415 5
	PD-U	174.215 2	174.233 4	174.267 2
	PD-O	174.215 2	174.086 1	174.049 1
1/8	PD-X	162.496 7	162.624 3	162.744 2
	PD-U	162.496 7	162.476 0	162.493 1
	PD-O	162.496 7	162.226 6	162.123 2
1/10	PD-X	158.510 5	158.651 2	158.787 2
	PD-U	158.510 5	158.463 0	158.468 1
	PD-O	158.510 5	158.146 2	157.997 7

Table 6

Fig. 5

Fig. 6

Table 7

Nonlinear frequency ratio of the GPLs-reinforced metal foam piezoelectric beam (GPL-X, L/h=100, h=5 mm, ΛGPL=1%, and e0=0.5)"

$h p / h$	Distribution	$ω 0$ /Hz	$W / H$
$h p / h$	Distribution	$ω 0$ /Hz	0.5	1.0	1.5	2.0
1/5	PD-X	174.415 5	1.040 4	1.152 1	1.315 2	1.511 0
	PD-U	174.267 2	1.040 4	1.152 2	1.315 3	1.511 2
	PD-O	174.049 1	1.040 5	1.152 5	1.315 9	1.512 2
1/8	PD-X	162.744 2	1.037 2	1.140 8	1.293 1	1.477 4
	PD-U	162.493 1	1.037 3	1.140 9	1.293 4	1.477 8
	PD-O	162.123 2	1.037 4	1.141 5	1.294 5	1.479 5
1/10	PD-X	158.787 2	1.036 2	1.137 0	1.285 7	1.466 0
	PD-U	158.468 1	1.036 2	1.137 2	1.286 0	1.466 6
	PD-O	157.997 7	1.036 4	1.137 9	1.287 4	1.468 7

Table 7

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

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