Frequency and vibration responses of a magneto-rheological sandwich micro-beam based on modified couple stresstheory and finite difference method

doi:10.1007/s10483-026-3405-7

Abstract

Abstract:

In this study, the frequency and vibration responses of a sandwich micro-beam are derived based on the modified couple stress theory (MCST). The face sheets are made of pure aluminum, and a magneto-rheological (MR) core is used to control vibrations. The displacement fields are assumed based on the classical beam theory (CBT) and modified classical beam theories. Based on Hamilton’s principle, the governing equations of motion are obtained. To solve these temporal equations, the finite difference method (FDM) with an optimal number of nodes is applied. The effects of various parameters, including viscoelastic properties, magnetic fields, aspect ratio, core-to-face-sheet thickness ratio, MR materials, face sheets, and material length-scale parameters, on the frequency and vibrational response are investigated. In the literature, the effects of different parameters on the frequency response function (FRF) or vibration response are considered. The results obtained from the vibration response and FRF using the FDM show that the viscoelastic property of the MR core causes settling time in the vibration response and a decrease in the excitation frequency of the FRF. Increasing the magnetic field has a negligible effect on the excitation frequency in the FRF, but it increases the vibration response and settling time. The piezoelectric face sheets raise the FRF and suppress the vibration response of the MR sandwich micro-beam compared with the aluminum counterpart.

Key words: frequency and vibration responses, sandwich micro-beam, magneto-rheological (MR) core, modified couple stress theory (MCST), finite difference method (FDM)

2010 MSC Number:

O326

M. JAFARI, M. MOHAMMADIMEHR. Frequency and vibration responses of a magneto-rheological sandwich micro-beam based on modified couple stresstheory and finite difference method. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1581-1602.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3405-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1581

Figures/Tables 15

Fig. 1

Table 1

Table 2

Table 3

Table 4

Mechanical properties of the MR core"

Shear modulus/Pa	Density/(kg· m^-3)
Case 1^[48]: the unit of B is Oersted (Oe) $G ∗ = (0.612 5 + i 0.006 737 5) × 106$ without a magnetic field, $G ∗ = (1 250 + i 13.75) B$ with a magnetic field	3 500
Case 2^[49]: the unit of B is Oersted (Oe) $G ′ = 3.3691 B 2 + 4997.5 B + 0.873 × 106,$ $G ″ = − 0.9 B 2 + 812.4 B + 0.185 5 × 106$	3 500
Case 3^[50]: the unit of B is Oersted (Oe) $G ′ = (− 6.939 5 B 6 − 9.107 7 B 5 + 71.797 B 4 − 93.422 B 3 + 38.778 B 2 + 2.43 B + 2.700 6) × 1010,$ $η = 5.348 5 B 6 − 17.787 B 5 + 22.148 B 4 − 12.185 B 3 + 2.352 2 B 2 + 0.152 6 B + 0.228$	3 500

Table 4

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

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Method	Magnetic field/Oe	Natural frequency mode
Method	Magnetic field/Oe	1	2	3
Present work (ode45)	Without a magnetic field	19.19	53.56	96.88
Present work (FDM)		20.76	64.02	126.80
Ref. [48]		20.98	62.49	124.80
Present work (ode45)	1 000	21.80	65.28	117.24
Present work (FDM)		24.19	77.44	128.63
Ref. [48]		24.30	72.16	138.22
Present work (ode45)	2 000	23.61	77.09	142.32
Present work (FDM)		27.38	87.59	158.35
Ref. [48]		27.11	84.28	158.26

Method	Magnetic field/Oe	Natural frequency mode
Method	Magnetic field/Oe	1	2	3
Present work (ode45)	Without a magnetic field	19.19	53.56	96.88
Present work (FDM)		19.26	51.60	95.33
Ref. [47]		19.16	52.35	98.38
Present work (ode45)	1 000	21.80	65.28	117.24
Present work (FDM)		22.73	59.84	110.33
Ref. [47]		22.75	63.58	114.93
Present work (ode45)	2 000	23.61	77.09	142.32
Present work (FDM)		25.49	71.49	131.83
Ref. [47]		25.73	77.06	138.39

Method	Magnetic field/Oe	Natural frequency mode
Method	Magnetic field/Oe	1	2	3
Present work (ode45)	Without a magnetic field	62.30	256.02	579.11
Present work (FDM)		63.22	253.32	556.05
Ref. [48]		62.72	244.55	546.42
Present work (ode45)	1 000	65.02	263.40	594.01
Present work (FDM)		64.23	258.70	567.71
Ref. [48]		63.91	247.20	549.64
Present work (ode45)	2 000	65.95	265.42	597.78
Present work (FDM)		65.75	262.48	582.80
Ref. [48]		65.03	251.01	555.02