Macro fiber composite-based active control of nonlinear forced vibration of functionally graded plate

doi:10.1007/s10483-025-3250-9

Abstract

Abstract:

Owing to their high flexibility and directional actuation capabilities, macro fiber composites (MFCs) have attracted significant attention for the active control of structures, especially in the nonlinear vibration suppression applications for large-scale flexible structures. In this paper, an MFC-based self-feedback system is introduced for the active control of geometrically nonlinear steady-state forced vibrations in functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates subject to transverse mechanical loads. Based on the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship, the nonlinear vibration control equations of the plate with MFC sensor and actuator layers are derived by Hamilton's principle. These equations are discretized by the finite element method (FEM), and solved by the Newton-Raphson and direct iterative methods. A velocity feedback control algorithm is introduced, and the effects of the control gain and the MFC actuator position on the nonlinear vibration active control effectiveness are analyzed. Additionally, a nonlinear resonance analysis is carried out, considering the effects of carbon nanotube (CNT) volume fraction and distribution type. The results indicate that the intrinsic characteristics of the structures significantly influence the vibration behavior. Furthermore, the appropriate selections of control gain and MFC position are crucial for the effective active control of the structures. The present work provides a promising route of the active and efficient nonlinear vibration suppression for various thin-walled structures.

Key words: macro fiber composite (MFC), nonlinear resonance, geometrically nonlinearity, active vibration control, amplitude-frequency response

2010 MSC Number:

O322

Peiliang ZHANG, Jianfei WANG. Macro fiber composite-based active control of nonlinear forced vibration of functionally graded plate. Applied Mathematics and Mechanics (English Edition), 2025, 46(5): 869-884.

TrendMD

Figures/Tables 11

Fig. 1

Table 1

Dimensionless linear and nonlinear fundamental frequencies of a fully clamped (CCCC) FGM plate with surface-bonded piezoelectric layers"

$n$	$ω ¯ l$			$ω ¯ nl$
	$ω ¯ l$			$W ¯ max / h f = 0.2$			$W ¯ max / h f = 0.4$			$W ¯ max / h f = 0.6$
	Present	Ref. [40]	Ref. [22]	Present	Ref. [40]	Ref. [22]	Present	Ref. [40]	Ref. [22]	Present	Ref. [40]	Ref. [22]
0	21.01	20.73	20.74	21.25	21.00	21.02	21.99	21.76	21.84	23.21	22.97	23.14
0.2	16.77	17.24	17.21	17.03	17.46	17.44	17.95	18.11	18.13	18.90	19.13	19.21
1	12.87	13.10	13.10	13.16	13.26	13.28	13.62	13.77	13.79	14.39	14.50	14.61
5	10.57	10.74	10.78	10.80	10.86	10.92	11.06	11.25	11.34	11.72	11.86	11.99
$∞$	9.65	9.55	9.55	9.75	9.67	9.68	9.84	10.01	10.06	10.36	10.57	10.65

Table 1

Fig. 2

Table 2

Material and geometrical parameters of the FG-CNTRC plate with MFC actuators and sensors"

Parameter	Value	Parameter	Value
Longitudinal Young's modulus of MFC $E 11 p$ /GPa	30.336	Poisson's ratio of PMMA/GPa	0.34
Transverse Young's modulus of MFC $E 22 p$ /GPa	15.857	Density of PMMA/ $(kg ⋅ m − 3)$	1 150
Shear modulus of MFC $G 12 p$ /GPa	5.515	Length of FG-CNTRC $a$ /mm	210
Poisson's ratio of MFC $ν 12 p$	0.31	Width of FG-CNTRC $b$ /mm	300
Density of MFC $ρ p$ / $(kg ⋅ m − 3)$	7 700	Thickness of FG-CNTRC $h$ /mm	5
Dielectric constant of MFC $Ξ 33$ / $(F ⋅ m − 1)$	$1.5 × 10 − 8$	12% CNT volume fraction efficiency parameter $η 1$	0.137
Piezoelectric strain constant of MFC $d 31 = d 32$ / $(m ⋅ V − 1)$	$− 1.71 × 10 − 12$	12% CNT volume fraction efficiency parameter $η 2$	1.022
Length of MFC $a p$ /mm	85	17% CNT volume fraction efficiency parameter $η 1$	0.142
Width of MFC $b p$ /mm	57	17% CNT volume fraction efficiency parameter $η 2$	1.626
Thickness of MFC $h p$ /mm	0.3	28% CNT volume fraction efficiency parameter $η 1$	0.141
Young's modulus of PMMA/GPa	3.52	28% CNT volume fraction efficiency parameter $η 2$	1.585

Table 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

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