Multi-particle mass sensing based on a single-walled carbon nanotube resonator

doi:10.1007/s10483-026-3376-6

Abstract

Abstract:

Based on the Timoshenko beam theory, a model for mass resonator sensor to detect multiple particles is developed. For a beam made of single-walled carbon nanotube (SWCNT), the nonlocal effects are incorporated in the governing equations. The approximate analytical solution for the resonance frequency of the system is derived by assuming that the mass of the adsorbed particles is much smaller than that of the system. The mass and position parameters of the multiple adsorbed particles are decoupled to establish an efficient detection method utilizing resonant frequency shifts. The identification process for the doubly clamped beam is systematically analyzed in numerical simulations. In addition, the axial force arising from temperature changes is incorporated into the beam model. The robustness of the proposed particle detection method against noise is analyzed. The model and analytical framework presented in this study provide a theoretical guideline for the design of nanoscale mass resonator sensors and particle mass detection under thermomechanical coupling conditions.

Key words: mass identification, nonlocal effect, carbon nanotube (CNT), multiple particle

2010 MSC Number:

O321

Jie WANG, Yin ZHANG. Multi-particle mass sensing based on a single-walled carbon nanotube resonator. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 883-904.

TrendMD

Figures/Tables 14

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

Comparison of the sensitivities between the Timoshenko beam model and the EBB model when R2=0 and λ=0"

Model	$L / d$	$\| S (ω 1) \| max$	$\| S (ω 2) \| max$	$\| S (ω 3) \| max$	$\| S (ω 4) \| max$
Timoshenko beam	10	26.916 2 $(− 2.88 %)$	59.437 9 $(− 13.14 %)$	103.518 6 $(− 22.45 %)$	147.722 9 $(− 31.64 %)$
	20	27.503 7 $(− 0.75 %)$	65.838 3 $(− 3.79 %)$	123.502 4 $(− 7.49 %)$	190.005 5 $(− 12.08 %)$
	30	27.618 0 $(− 0.34 %)$	67.244 5 $(− 1.73 %)$	128.756 3 $(− 3.55 %)$	203.207 4 $(− 5.97 %)$
	40	27.658 5 $(− 0.19 %)$	67.756 5 $(− 0.99 %)$	130.765 1 $(− 2.04 %)$	208.551 4 $(− 3.49 %)$
EBB	–	27.710 9	68.430 6	133.494 6	216.102 4

Table 1

Fig. 7

Fig. 8

Fig. 9

Table 2

Detection of particles with different masses under the C-C boundary conditions when L=40d, R2=1, and λ=1×10−4"

Particle	Result	Mass	$ξ 1$	$ξ 2$	$ξ 3$
One particle	Exact	$1 × 10 − 3$	0.3	–	–
One particle	Estimated	$1.000 6 × 10 − 3$ $(0.06 %)$	0.300 0	–	–
Two particles	Exact	$1 × 10 − 2$	0.4	–	–
	Estimated	$9.983 1 × 10 − 3$ $(− 0.17 %)$	0.401 1	–	–
	Exact	$1 × 10 − 3$	0.25	0.35	–
	Estimated	$9.987 2 × 10 − 4$ $(− 0.13 %)$	0.250 3	0.350 2	–
Three particles	Exact	$5 × 10 − 3$	0.3	0.4	–
	Estimated	$4.973 3 × 10 − 3$ $(− 0.53 %)$	0.301 3	0.401 8	–
	Exact	$1 × 10 − 3$	0.2	0.3	0.4
	Estimated	$9.957 4 × 10 − 4$ $(− 0.43 %)$	0.201 4	0.300 5	0.400 7
	Exact	$3 × 10 − 3$	0.3	0.4	0.45
	Estimated	$2.984 8 × 10 − 3$ $(− 0.51 %)$	0.303 8	0.396 5	0.457 3

Table 2

Table 3

Detection of particles with different masses under the C-C boundary conditions when L=40d, R2=1, and λ=6.25×10−3"

Particle	Result	Mass	$ξ 1$	$ξ 2$	$ξ 3$
One particle	Exact	$1 × 10 − 3$	0.35	–	–
One particle	Estimated	$1.001 0 × 10 − 3$ $(0.10 %)$	0.349 9	–	–
Two particles	Exact	$6 × 10 − 3$	0.45	–	–
	Estimated	$6.002 8 × 10 − 3$ $(0.05 %)$	0.450 2	–	–
	Exact	$4 × 10 − 3$	0.25	0.45	–
	Estimated	$3.980 7 × 10 − 3$ $(− 0.48 %)$	0.251 4	0.451 6	–
Three particles	Exact	$5 × 10 − 3$	0.28	0.32	–
	Estimated	$4.969 1 × 10 − 3$ $(− 0.62 %)$	0.279 8	0.322 6	–
	Exact	$2 × 10 − 3$	0.23	0.33	0.42
	Estimated	$1.991 0 × 10 − 3$ $(− 0.45 %)$	0.231 3	0.330 7	0.421 3
	Exact	$3 × 10 − 3$	0.30	0.38	0.46
	Estimated	$2.985 1 × 10 − 3$ $(− 0.50 %)$	0.303 8	0.377 7	0.466 2

Table 3

Fig. 10

Fig. 11

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