Nonlinear dynamic behavior of single-layered black phosphorus with an attached mass

The investigation of black phosphorus-based (BP)-based mass sensors provides theoretical support for the development of mass-detection devices. This study examines the non-linear dynamic behavior of a rectangular single-layered BP (SLBP) with an attached mass through the utilization of molecular dynamics (MD) simulations and a nonlinear orthotropic plate model (NOPM) with a concentrated mass. The results indicate that significant deformation of an SLBP with an attached mass necessitates consideration of geometric nonlinearity, although the attached mass does not affect the deformation. Additionally, this paper discusses the impact of the attachment mass and the amplitude of harmonic force on the non-linear forced vibration of the SLBP. It is observed that as the attachment mass increases, the nonlinear vibration resonance frequency decreases, while the peak amplitude increases. Furthermore, the thermal nonlinear vibration of SLBP with an attached mass has been investigated, revealing that an increase in the attached mass leads to a decrease in the nonlinear vibration frequency but an increase in the amplitude of the nonlinear vibration of SLBP with an attached mass. Overall, comparison with MD simulation results, this investigation suggests that NOPM with a concentrated mass effectively describes the nonlinear vibration behavior of an SLBP with an attached mass, providing theoretical support for designing such devices to detect attached masses.


Introduction
Recently, black phosphorus (BP), a layered material, has garnered significant attention from scholars due to its wide-ranging applications in various fields.Its ultra-high charge mobility [1][2][3][4], ultra-high band gap [5][6][7][8], excellent heat conductivity [9][10][11], remarkable in-plane anisotropic optical absorption characteristics [12,13], and special mechanical properties [14][15][16] make it suitable for use in field-effect transistors, semiconductors, thermology, optics, and mechanics, respectively.The mechanical properties are crucial for the dynamic behavior of BP.Li and Li et al [17] reported on the tensile and compressive behaviors of BP, while Yang [18] researched the impact of the thermal environment on its mechanical properties.Mass sensors based on lowdimensional materials have also attracted considerable attentions because of their exceptional mechanical properties [19].
The attachment mass significantly affects the mechanical properties of nanostructure-based with an attached mass, particularly their dynamic behavior.Thus, a thorough investigation of the transverse vibration of nanostructure with an attached mass is extremely necessary.Regarding the linear transverse vibration of nanostructures with an attached mass, Wang et al [20] used the modified Euler-Bernoulli beam theory to investigate the impact of attachment mass on the linear vibration of carbon nanotubes with an attached mass.Lee and Chang [21] researched the linear vibration of carbon nanotubes with an attached mass utilizing beam model that considered the impact of shear deformation and rotary inertia.Duan et al [22] conducted an investigation of linear vibration of pillared graphene with an attached mass using molecular dynamics (MD) simulations, showing that its natural frequency was highly sensitive to the attached mass.Xu et al [23] utilized nonlocal elasticity theory to study the influence of a magnetic field on the linear dynamic behavior of a nanoplate with an attached mass.Zhang et al [24,25] conducted studies on the linear vibration of rectangular singlelayered BP (SLBP) with an attached mass and circular single-layered BP with an attached mass.Zhao et al [26] reported the frequency shift of monolayer diamond-based mass by MD simulations.In terms of nonlinear vibration of nanostructure with an attached mass research, Zhao et al [27] investigated the nonlinear vibration of nanobeam with an attached mass.Li et al [28] examined the effect of moving particles on the non-linear vibration of triple-layered graphene sheets with an attached mass.Cai and Xu [29] reported non-linear vibrations of nanotubes with an attached mass.Dai et al [30] researched nonlinear vibration of graphene with an attached mass, taking into account the impact of geometric nonlinearity.
Despite extensive research of BP, there is currently a lack of studies on the nonlinear vibration characteristics of BP with an attached mass.This research provides valuable theoretical support for the development of BPbased mass detection devices.This paper explores the non-linear vibration of an SLBP with an attached mass using MD simulations and a nonlinear orthotropic plate model (NOPM) with a concentrated mass.An NOPM with a concentrated mass was utilized to study the nonlinear vibration of the SLBP with an attached mass, as presented in section 2. The impact of the attached mass and lateral uniform load on the deformation of SLBP with an attached mass is illustrated in section 4, along with an analysis of how attachment mass, temperature, and amplitude of harmonic force influence the nonlinear vibration in this section as well.Finally, some conclusions are provided in section 5.

Nonlinear vibrational model of SLBP with an attached mass
BP is a highly anisotropic material, which can be considered as orthotropic material.Therefore, we utilized the NOPM with a concentrated mass to investigate the non-linear dynamic characteristics of SLBP with an attached mass.The dynamic von Kármán equations for an orthotropic plate with a concentrated mass can be written as in equation (2), u xy and u yx represent the in-plane Poisson's ratios of SLBP.r denotes mass density of SLBP.While E x and E y are the elastic moduli of SLBP in x-and y-direction, respectively.G stands for the shear modulus of SLBP.M is the attachment mass.( ) q x y t , , signifies an excitation in the z-direction, the symbol h corresponds to the thickness of SLBP, and ( ) u v , is the position of attachment on the SLBP.Additionally, ( ) w = q x y t q t , , cos 0 represents a transverse harmonic excitation, where q 0 and w are the amplitude and angular frequency of the harmonic excitation, respectively.
If an SLBP with an attached mass is clamped along the edges and its in-plane boundary conditions are freely movable, the deflection shape is assumed to be where 2a and 2b are the length and width, respectively, of the SLBP with an attached mass, and t represents time.Equation (3) can fulfill the following out-of-plane boundary conditions: Substituting equation (3) into equation (1a) and using the particular integral with the in-plane movable boundary condition, we obtain where ( ) Equation (1b) is approximately satisfied using the Galerkin method, leading to the following integral in terms of F and W as Substituting equations (3) and (5) into equation (7) and integrating equation (7), we get the nonlinear differential equation in terms of the time function ( ) where and Since an SLBP with an attached mass is a damped system, equation (8) can be rewritten as where c denotes the damping coefficient of the SLBP with an attached mass.We can easily obtain the nonlinear static case of an SLBP with an attached mass from equation (12) as follows: We adopted the multiple time-scale method to study the amplitude-frequency response characteristics of the nonlinear vibration of an SLBP with an attached mass.Let

¯( ) a w b eb w w es
xy yx here e is a small-scale parameter.Substituting equation ( 14) into equation (12), we obtain To obtain the amplitude-frequency response characteristic of the nonlinear vibration of the SLBP with an attached mass, the asymptotic expansions for the solution has the form: When equation ( 16) is substituted into the left hand of equation (15), the coefficients of the same power as e can be obtained as The solution of equation (17a) has the form of where cc represents the conjugate terms, and Substituting equation (18) into equation (17b), we can get The condition for eliminating the secular term is Substituting equation (19) into equation (21) and separating the real and imaginary parts, we can get The corresponding first approximation solution to equation (18) can be written as in equation (22), and then we can obtain the frequency response as Then we can obtain the backbone curve function as The peak amplitude of the SLBP with an attached mass can be written as To investigate the thermally induced nonlinear vibration of SLBP with an attached mass, we adopted the law of energy equipartition.The kinetic energy of the SLBP with an attached mass can be expressed as By using equations ( 23), ( 25) and (27), we can obtain the first nonlinear vibration mode energy E m of the SLBP with an attached mass in the form of According to the law of energy equipartition, each nonlinear vibration mode of an SLBP with an attached mass has the same energy.
( ) where k B represents Boltzmann's constant and T represents the system temperature.Let

MD simulations
The forced nonlinear vibrations of an SLBP with an attached mass were conducted using LAMMPS [31].In this paper, the attachment mass was deposited on a BP atom located at the center position of SLBP.And the atomic structure of SLBP with an attached mass is shown in figure 1(a).The P-P atomic interactions of BP were described by Stillinger-Weber (SW) potential, and all of the parameters of SW potential were calculated by Jiang [32].MD simulations were employed to simulate the nonlinear forced vibration of SLBP with an attached mass.The NVE ensemble was applied during these simulations.A laterally uniformly distributed harmonic force as a type of w F t cos , was applied on the atoms of the SLBP with an attached mass, which was unfixed, as illustrated in figure 1(a).The frequency of the harmonic force was varied to obtain the amplitude-frequency curve of the first nonlinear vibration mode of the SLBP with an attached mass.The thermal nonlinear vibrations of the SLBP with an attached mass were also studied using NVT ensemble is used in MD simulations.The displacement of the central BP atom was recorded every one hundred steps.The natural frequency of the SLBP with an attached mass was determined using a fast Fourier transform.The time step for MD simulation was set to 1 femtosecond.After full relaxation of the system in MD simulations, the outermost red atoms, as illustrated in figure 1(a), were fixed in the z-direction but allowed to move freely in-plane.

Results and discussions
We propose a NOPM model with a concentrated mass at any location on the plate to investigate the nonlinear vibration of the SLBP with an attached mass.The molecular structure and mechanical model of the SLBP with an attached mass are illustrated in figure 1.The elastic moduli of the SLBP along the x-and y-direction are = E 977.42 GPa x and = E 185.87 GPa, y respectively.The shear modulus of SLBP is = G 52.31 GPa.u xy and u yx are the in-plane Poisson's ratios of SLBP, which are 0.054 and 0.0108, respectively [33].In this study, we placed the particles at the mid-point of the SLBP.
To investigate the impact of the attached mass on the deformation of SLBP under a lateral static load, we utilized the NOPM with a concentrated mass and MD simulations to obtain the deflections of the SLBP with an attached mass under laterally uniformly distributed loads.Figure 2 shows the effect of laterally uniformly distributed loads on the deflections of an SLBP with an attached mass when the attached mass is ´- 1 10 kg.

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The black line, blue dashed line and red points represent the results obtained by linear plate model, the NOPM with a concentrated mass, and the MD simulations, respectively.From figure 2, it is evident that the deflections of the SLBP with an attached mass obtained by NOPM with a concentrated mass align well with those obtained by MD simulations.However, there is a significant difference between the results obtained by the linear plate model and MD simulation at large deformations.This indicates that geometric nonlinearity must be taken into account when the deformation of an SLBP with an attached mass is substantial.Figure 3 displays the effect of the attached mass on the deflection of the SLBP with an attached mass obtained by the linear plate model and the  NOPM with a concentrated mass, and the laterally uniformly distributed load is 33.976MPa.From figure 3, the results indicate that the attachment mass has no influence on the deformation of SLBP, As shown by the linear plate model and the NOPM with a concentrated mass.The nonlinear forced vibration of an SLBP with an attached mass was investigated through MD simulations and a NOPM with a concentrated mass.We used MD simulations to get the steady-state amplitude of SLBP with an attached mass in the NVE ensemble.The unfixed atoms of the SLBP with an attached mass illustrated in figure 1(a) were excited by a laterally uniformly distributed harmonic force as w F t cos .The frequency of the harmonic force was then varied to obtain the amplitude-frequency response curve of the first nonlinear vibration mode of the SLBP with an attached mass.Furthermore, we explored the effect of the amplitude of the harmonic force and the attached mass on the amplitude-frequency response curve of the first nonlinear vibration mode of the SLBP with an attached mass.Using equation (24), which was derived from equation (12) through the multiple time-scale method, the amplitude-frequency response curve was obtained by NOPM with a concentrated mass.Figure 4 shows the amplitude-frequency response curve of the SLBP with various attached masses.As the frequency of the harmonic force increases, the amplitude of SLBP with an attached mass also increases.However, when the frequency of the harmonic force reaches a certain value, the amplitude of the SLBP with an attached mass undergoes a downward jump as the frequency continues to increase; Additionally, as the frequency of harmonic force continues to increase, there is a decrease in amplitude, indicating that the system of SLBP with an attached mass exhibits hardening-type nonlinearity.Furthermore, with an increase in the attachment mass, the peak amplitude of SLBP with an attached mass increase.Simultaneously, as the attached mass increases, there is a decrease in nonlinear resonance frequency of SLBP with an attached mass.This suggests that the resonance frequency of SLBP with an attached mass is extremely responsive to the attached mass, making it useful for detecting devices.From figure 4, it can be observed that the results from the NOPM with a concentrated mass closely align with MD results.Figure 5 depicts the amplitude-frequency response curve of the SLBP with an attached mass at different amplitude of harmonic force.As the amplitude of the harmonic force increases, the peak amplitude of the SLBP with an attached mass also increases.The amplitude of the SLBP with an attached mass exhibits a downward jump when the frequency of the harmonic force reaches a certain value.Comparing the MD simulations, figures 4 and 5 show that the NOPM with a concentrated mass is capable of accurately predicting the nonlinear dynamic behavior of the SLBP with an attached mass.
As the temperature of the system increases, the amplitude of the thermal vibration of the SLBP with an attached mass also increases, potentially leading to geometric nonlinearity in the system.Therefore, it is essential to study the thermal nonlinear vibration characteristics of the SLBP with an attached mass.We used MD simulations to get natural frequency of SLBP with an attached mass in the NVT ensemble via Nose-Hoover thermostat.The nonlinear vibration frequency was obtained by NOPM with a concentrated mass.Furthermore, it should be noted that in-plane thermal stress does not exist in the system, due to the in-plane freely moveable boundary condition of the SLBP with an attached mass.
Figure 6 shows the nonlinear thermal vibration frequency of the SLBP with an attached mass obtained through MD simulations and the NOPM with a concentrated mass, with varying the attached mass while maintaining the environmental temperature at 100 K.As the attached mass of the system increases, the thermal nonlinear vibration frequency of the SLBP with an attached mass decrease.The thermal nonlinear vibration frequencies of the SLBP with an attached mass obtained by the NOPM with a concentrated mass closely align with those gotten by MD simulations.Figure 7 depicts the nonlinear vibration frequency of the SLBP with an attached mass versus the environmental temperature of the system gotten through MD simulations, NOPM with a concentrated mass and the linear plate model.The attachment mass is identified using ´- 1 10 kg.

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The results of the NOPM with a concentrated mass closely match those of the MD simulations, showing an increase in the nonlinear vibration frequency as environmental temperature rises.Conversely, significant discrepancies are observed between the results obtained by linear plate model and MD simulations, showing no changes as the temperature increases, indicating the necessity of considering geometric nonlinearity in nonlinear thermal ) on the amplitude-frequency response curve of SLBP with an attached mass predicted by MD simulations and NOPM with a concentrate mass.Dots represent the MD results, and lines represent the NOPM with a concentrated mass.
vibrations.Figures 6 and 7 collectively indicate that the use of the NOPM with a concentrated mass is effective for predicting the non-linear thermal vibration of the SLBP with an attached mass.Figures 6 and 7 collectively demonstrate that NOPM with a concentrated mass is capable of accurately predicting the nonlinear thermal vibration of the SLBP with an attached mass.
Figure 8 illustrates the amplitude of the nonlinear thermal vibration of SLBP with an attached mass versus the system temperature gotten through MD simulations and the NOPM with a concentrated mass.From figure 8, we observe that a minimal difference in the amplitude of the thermal vibration of the SLBP with an attached mass between the results calculated by MD simulations and the NOPM with a concentrated mass.Furthermore, the amplitude of the thermal vibration of an SLBP with an attached mass increases with temperature.Figure 9 depicts the amplitude of the nonlinear thermal vibration of the SLBP with an attached mass versus the attached mass gotten by MD simulations and the NOPM with a concentrated mass.From figure 9, it is apparent that the amplitude of the nonlinear thermal vibration increases with the attachment mass.Additionally, a slight disparity between the results calculated by the MD simulations and the NOPM with a concentrated mass can be observed in figure 9.

Conclusions
In this paper, we employed MD simulations and NOPM with a concentrated mass to investigate the non-linear dynamic behavior of an SLBP with an attached mass.Firstly, we studied the impact of an attached mass and laterally uniformly distributed loads on the deformation of an SLBP with an attached mass.When the deformation of an SLBP with an attached mass is significant, geometric nonlinearity must be taken into consideration.However, the attachment mass had no impact on the deformation of the SLBP with an attached mass.Secondly, we examined the effect of the attached mass and the amplitude of the harmonic force on the nonlinear forced vibration of the SLBP with an attached mass.An increase in the attached mass resulted in a decrease in the resonant frequency of the SLBP with an attached mass, while the peak amplitude of SLBP with an attached mass increased.Similarly, as the amplitude of the transverse harmonic force increases, the peak amplitude of SLBP with an attached mass also increases.The NOPM with a concentrated mass proves effective in predicting the nonlinear forced vibration of the SLBP with an attached mass.Finally, we investigated the nonlinear thermal vibration of the SLBP with an attached mass at a larger amplitude.As expected, an increase in the attachment mass leads to a decrease in the frequency of the nonlinear thermal vibration of SLBP with an attached  mass.Additionally, as the temperature raised, the frequency of the nonlinear thermal vibration of SLBP with an attached mass also increased.The effectiveness of the NOPM with a concentrated mass was demonstrated by accurately predicting the non-linear thermal vibrations of the SLBP with an attached mass.
for the relationship between the temperature and fundamental frequency by using w

Figure 1 .
Figure 1.Models of SLBP with an attached mass.(a) Molecular structure.(b) Mechanics model of SLBP with a concentrated mass.

Figure 2 .
Figure 2. Impact of laterally uniformly distributed loads on the deformation of SLBP with an attached mass ( = = a b 4.14 nm, 3.13 nm).

Figure 3 .
Figure 3. Deformation of SLBP with an attached mass in relation to the attachment mass.

Figure 6 .
Figure 6.Nonlinear thermal vibration frequency of SLBP with an attached mass versus attachment mass.

Figure 7 .
Figure 7. Nonlinear thermal vibration frequency of SLBP with an attached mass versus environmental temperature.

Figure 8 .
Figure 8. Amplitude of nonlinear thermal vibration of SLBP with an attached mass versus environmental temperature.

Figure 9 .
Figure 9. Amplitude of nonlinear thermal vibration of SLBP with an attached mass versus attachment mass.