Vibration power flow characteristics of the whole-spacecraft with a nonlinear energy sink

Abstract

The power flow analysis approach is applied for quantitatively evaluating the dynamic performance of a nonlinear energy sink in the frequency domain. A two-degree-of-freedom model of the whole-spacecraft coupled to a nonlinear energy sink is considered. Analytical approximations and numerical integrations are performed for the integrated system. The time-averaged input and dissipated and absorbed powers of the system are formulated. Power absorption ratio is proposed as the vibration reduction performance indicator of the nonlinear energy sink and is compared with kinetic energy of the oscillator. Results show that the power absorption ratio of nonlinear energy sink is high at all frequencies except at an antiresonance region. The effects of varying nonlinear energy sink parameters on its vibration absorption performance are discussed. As observed, the nonlinear energy sink performs well with the increase in viscous damping of nonlinear energy sink in the high-frequency range. However, the nonlinear energy sink performs poorly in the low-frequency range in the same case. Therefore, power flow analysis can provide great insight for nonlinear energy sink design in frequency domain and is suitable for practical engineering application.

Keywords

Power flow whole-spacecraft nonlinear energy sink frequency domain vibration absorption

Introduction

Vibration control strategies have received much attention in many engineering fields, such as civil, marine, and aerospace.¹ Especially in spacecrafts, payloads such as antennas, optical mirrors, and scientific equipment easily fail in the vibration environment.^2–5 Active control systems can give good performance in mitigating oscillations. However, in some circumstances, the active control methods are limited by its possible instability and its power requirement.⁶ Therefore, passive vibration control and active–passive integrated vibration control as proven methods are more widely used in space industry than active control.^6,7 Usually, linear vibration absorbers were applied as passive means of control to prevent excessive vibration energy transmission.⁸ However, they are designed to suppress the oscillation vibration according to a specific excitation frequency. ⁹ Linear dynamic vibration absorbers are ineffective and cause the counter effect in other frequencies.

Nonlinear designs are proposed for attenuating broad frequency vibration energy transmitting to the primary system by introducing nonlinear elements.^9,10 As an essentially nonlinear local attachment, the nonlinear energy sink (NES) is a powerful vibration absorber without obviously changing the natural frequencies of systems.^11–15 Vakakis et al.¹⁶ discovered the important phenomenon of targeted energy transfer, in which the vibrational energy from a linear system is directly transferred to a passive NES in an irreversible manner. Based on this mechanism, the NES has been widely designed to suppress the undesirable vibrations.^17–19 NES attached to continuum primary linear structures, such as rod,²⁰ beams,^21–23 drill-string,²⁴ plate,²⁵ elastic strings,²⁶ and hollow rotor systems²⁷ is considered in literatures. According to the NES applications, the NES has been designed to various types, like nonsmooth²⁸ or nonpolynomial.²⁹ In particular, new NES designs based on the rotating NES mass,³⁰ highly asymmetric NES,³¹ modified NES considering negative linear and nonlinear stiffness components,³² a two-degree-of-freedom (2-DOF) NES device,³³ and a asymmetric magnet-based NES³⁴ have been introduced to provide better vibration reduction performance. A mixed multiple scale/harmonic balance method is used to obtain the equations describing the slow- and fast-flow dynamics of NES-controlled systems.³⁵ Two different quasi-periodic response regimes in strongly nonlinear absorbers have been revealed.³⁶ The response regimes of oscillators coupled to an NES excited by the narrow band stochastic force have been researched theoretically and numerically.³⁷ A transformation is proposed to obtain the decay envelope formulas, which can be applied to identify the NES damping.^38,39 The reduction of galloping vibrations for an aeroelastic system through an NES has also been investigated.⁴⁰

In terms of performance indicator of the NES, the displacement response in time domain or the energy dissipation/transition in fixed frequency was often used in the above literatures. Important frequency domain behavior existed in nonlinear systems is usually ignored. The power flow analysis approach can better reflect the nonlinear control performance in the frequency domain, as it combines both forces and velocities as well as their relative phase angle.⁴¹ This method has been largely developed to study linear vibration control systems.^42–47 Recently, the technique is receiving growing attention in nonlinear dynamical systems. Royston and Singh⁴⁸ applied vibration power transmission as a performance indicator in nonlinear mounting systems. Xing and Price⁴⁹ proposed a generalized method for addressing many vibration control problems in engineering. Xiong et al.⁵⁰ examined the power flow characteristics of the integrated system consisting of an equipment, a nonlinear isolator, and a flexible ship excited by sinusoidal wave. Xiong and Cao⁵¹ analyzed the vibration energy dissipation mechanism of a 2-DOF system with irrational nonlinearity. Yang and co-workers^52–56 developed power flow analysis methods to investigate the nonlinear dynamic systems, which include the power flow characteristics of the Duffing oscillator and the 2-DOF systems for vibration isolations and absorptions. However, the NES is completely different from nonlinear absorbers in the above research. The power flow investigated is the spatial-dependent variable, not the time-dependent variable.^54,55 So the properties of the power flow in the NES and the satellite are studied. It is hoped that more energy will flow into the NES. To the best of the authors’ knowledge, power flow characteristics of the NES have not been studied yet.

In the present study, a power flow behavior of the NES is investigated to obtain good designs and applications of the NES. A 2-DOF whole-spacecraft structure with the NES attached is adopted. The harmonic balance method and numerical integrations are used to solve the dynamic governing equations. The instantaneous and time-averaged input and dissipated and absorbed powers of the coupling system are formulated. Power absorption ratio is introduced to indicate the NES performance and is compared with the kinetic energy of the structure. Moreover, the effects of the varying NES parameters on the vibration mitigation performance of the NES are revealed. Finally, conclusions for improving the performance of the NES are provided.

Mathematical modeling

2-DOF structure with NES

A scaled model of the spacecraft studied here is shown in Figure 1. The 2-DOF system with a specific set of parameters is an equivalent model of the scaled model whole-spacecraft.⁵⁵ With the parameters, a well-designed NES reduces effectively the vibration of the whole-spacecraft. As shown in Figure 2, the 2-DOF system is excited by a harmonic displacement u(t) from the supporting base. The mass m₂ represents the adapter. The mass m₁ represents the satellite. To suppress the mass m₁ vibration of the structure, the NES with cubic nonlinear stiffness and linear viscous damping is attached to the mass m₂.

Figure 1.

The scaled spacecraft structure.

Figure 2.

The 2-DOF primary structure with an NES attached.

The dynamic governing equations of the whole-spacecraft vibration reduction system are obtained as

{\begin{cases} m_{1} {\ddot{x}}_{1} + c_{1} ({\dot{x}}_{1} - {\dot{x}}_{2}) + k_{1} (x_{1} - x_{2}) = 0 \\ m_{2} {\ddot{x}}_{2} + c_{1} ({\dot{x}}_{2} - {\dot{x}}_{1}) + c_{2} ({\dot{x}}_{2} - \dot{u}) + c_{3} ({\dot{x}}_{2} - {\dot{x}}_{3}) + k_{1} (x_{2} - x_{1}) + k_{2} (x_{2} - u) + k_{3} {(x_{2} - x_{3})}^{3} = 0 \\ m_{3} {\ddot{x}}_{3} + c_{3} ({\dot{x}}_{3} - {\dot{x}}_{2}) + k_{3} {(x_{3} - x_{2})}^{3} = 0 \end{cases}

(1)

In equation (1), m_i, k_i, and c_i (i = 1, 2) are the masses, stiffness, and damping parameters of the primary structure; m₃, k₃, and c₃ are the mass, the nonlinear stiffness parameter, and the linear damping parameter of the NES.

Analytical approximations and numerical integrations are adopted in this study in investigating the dynamic performance of the NES in terms of power flow. The harmonic balance method is applied to obtain the relationship between power flow variables and system parameters. Meanwhile, numerical integrations of equation (1) based on the fourth-order Runge–Kutta approach are used to examine the analytical results. Both experimental data and numerical simulation via the finite element revealed that the whole-spacecraft vibrates periodically.⁵⁵ Thus, only the periodic responses are sought in the following.

Analytical approximations and numerical results

The harmonic balance method⁵⁶ can be used to obtain the first-order approximations of the dynamic responses of the nonlinear system representing periodic solutions. Thus, the displacement responses of the integrated whole-spacecraft vibration reduction system may be assumed as

{\begin{cases} x_{1} (t) = a_{1} \cos (wt) + a_{2} \sin (wt) \\ x_{2} (t) = b_{1} \cos (wt) + b_{2} \sin (wt) \\ x_{3} (t) = d_{1} \cos (wt) + d_{2} \sin (wt) \\ u (t) = A \cos (wt) \end{cases}

(2)

where a_i, b_i, and d_i (i = 1,2) are the coefficients of the related harmonic responses and A and ω are the amplitude and frequency of the harmonic displacement excitation. Applying a Fourier series expansion transforms the nonlinear term

{(x_{2} - x_{3})}^{3}

in equation (1) into

\begin{array}{l} {(x_{2} - x_{3})}^{3} = \frac{3}{4} (b_{1} - d_{1}) [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] \cos (wt) \\ + \frac{3}{4} [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] (b_{2} - d_{2}) \sin (wt) \end{array}

(3)

Substituting these approximation terms in equation (1) and equating the coefficients of the related harmonic terms, we obtain

{\begin{cases} a_{1} k_{1} - b_{1} k_{1} + a_{2} c_{1} w - b_{2} c_{1} w - a_{1} m_{1} w^{2} = 0 \\ a_{2} k_{1} - b_{2} k_{1} - a_{1} c_{1} w + b_{1} c_{1} w - a_{2} m_{1} w^{2} = 0 \\ - a_{1} k_{1} + b_{1} k_{1} - A k_{2} + b_{1} k_{2} + \frac{3}{4} (b_{1} - d_{1}) [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] k_{3} \\ - a_{2} c_{1} w + b_{2} c_{1} w + b_{2} c_{2} w + b_{2} c_{3} w - c_{3} d_{2} w - b_{1} m_{2} w^{2} = 0 \\ - a_{2} k_{1} + b_{2} k_{1} + b_{2} k_{2} + \frac{3}{4} [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] (b_{2} - d_{2}) k_{3} \\ + a_{1} c_{1} w - b_{1} c_{1} w + A c_{2} w - b_{1} c_{2} w - b_{1} c_{3} w + c_{3} d_{1} w - b_{2} m_{2} w^{2} = 0 \\ - \frac{3}{4} (b_{1} - d_{1}) [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] k_{3} - b_{2} c_{3} w + c_{3} d_{2} w - d_{1} m_{3} w^{2} = 0 \\ - \frac{3}{4} [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] (b_{2} - d_{2}) k_{3} + b_{1} c_{3} w - c_{3} d_{1} w - d_{2} m_{3} w^{2} = 0 \end{cases}

(4)

Thus, the relationship between the responses and the system parameters is provided by equation (4). Based on the Newton–Raphson method, an algorithm is used to solve the abovementioned nonlinear algebraic equations.

Numerical integrations based on the fourth-order Runge–Kutta method are conducted to verify the above approximation results. The variables are defined as

{\begin{cases} y_{1} = x_{1} \\ y_{2} = {\dot{x}}_{1} \\ y_{3} = x_{2} \\ y_{4} = {\dot{x}}_{2} \\ y_{5} = x_{3} \\ y_{6} = {\dot{x}}_{4} \end{cases}

(5)

Using equation (5), we observe that equation (1) is transformed into a set of six first-order differential equations

{\begin{array}{l} {\dot{y}}_{1} = y_{2} \\ {\dot{y}}_{2} = - \frac{k_{1}}{m_{1}} (y_{1} - y_{3}) - \frac{c_{1}}{m_{1}} (y_{2} - y_{4}) \\ {\dot{y}}_{3} = y_{4} \\ {\dot{y}}_{4} = \frac{k_{1}}{m_{2}} (y_{1} - y_{3}) + \frac{c_{1}}{m_{2}} (y_{2} - y_{4}) - \frac{k_{2}}{m_{2}} (y_{3} - u) - \frac{c_{2}}{m_{2}} (y_{4} - \dot{u}) - \frac{k_{3}}{m_{2}} {(y_{3} - y_{5})}^{3} - \frac{c_{3}}{m_{2}} (y_{4} - y_{6}) \\ {\dot{y}}_{5} = y_{6} \\ {\dot{y}}_{6} = \frac{k_{3}}{m_{3}} {(y_{3} - y_{5})}^{3} + \frac{c_{3}}{m_{3}} (y_{4} - y_{6}) \end{array}

(6)

The numerical dynamic response solutions of the nonlinear system can be readily obtained from equation (6) by the fourth-order Runge–Kutta method.

The primary structure parameters are chosen as m₁=60 kg, m₂=12 kg, k₁=1.8677 × 10⁶ N/m, k₂=2.1346 × 10⁶ N/m, c₁=600 N s/m, and c₂=20 N s/m.⁴⁸ The NES parameters can be designed within the proper range, such as m₃=4 kg, k₃=500 N/m³, and c₃=300 N s/m. The harmonic displacement excitation amplitude A is chosen as 0.001 m. Let a, b, and d be the response amplitude of masses m₁, m₂, and m₃, respectively. For this set of parameters, Figure 3 shows the frequency–response curves of the coupled dynamic system. The figure also shows that response amplitude d of the NES has an antiresonance peak behind the first-order natural frequency. The corresponding drop off of power absorption ratio will be determined in the following power flow analysis.

Figure 3.

Verification of the harmonic balance method formulations (m₃=4 kg, k₃=500 N/m³, c₃=300 N s/m, A = 0.001 m).

Power flow analysis

The basic principles of vibration power generation, dissipation, and absorption in the dynamic coupled systems are revealed. The instantaneous input power into the integrated system is the product of the excitation velocity with the corresponding force, that is

P_{in} = (f_{d 1} + f_{k 1}) \dot{u} = \dot{u} [c_{2} ({\dot{x}}_{2} - \dot{u}) + k_{2} (x_{2} - u)]

(7)

where f_d₁ and f_k₁ are the damping and restoring forces acting on the support. Consequently, the time-averaged input power over an excitation cycle is

{\bar{P}}_{in} = \frac{1}{T} \int_{0}^{T} P_{in} d t \approx \frac{1}{2} A w [(b_{1} - A) c_{2} w - b_{2} k_{2}]

(8)

where a first-order harmonic approximation response is used.

For the coupled system, vibration energy is dissipated by the damping in the primary structure and the NES. Thus, the total instantaneous dissipated power is

P_{d} = P_{d 1} + P_{d 2} + P_{d 3} = f_{d 1} ({\dot{x}}_{1} - {\dot{x}}_{2}) + f_{d 2} ({\dot{x}}_{2} - \dot{u}) + f_{d 3} ({\dot{x}}_{2} - {\dot{x}}_{3})

(9)

where P_d₁, P_d₂, and P_d₃ are the dissipated power by damper c₁, damper c₂, and damper c₃, respectively;

f_{d 1} = c_{1} ({\dot{x}}_{2} - {\dot{x}}_{1})

f_{d 2} = c_{2} ({\dot{x}}_{2} - \dot{u})

, and

f_{d 3} = c_{3} ({\dot{x}}_{2} - {\dot{x}}_{3})

are the damping forces in the main structure and in the NES. The time-averaged dissipated power over a cycle is

\begin{array}{l} {\bar{P}}_{d} = \frac{1}{T} \int_{0}^{T} P_{d} d t \approx \frac{1}{2} c_{1} w^{2} [{(a_{1} - b_{1})}^{2} + {(a_{2} - b_{2})}^{2}] + \frac{1}{2} c_{2} w^{2} [{(A - b_{1})}^{2} + b_{2}^{2}] + \frac{1}{2} c_{3} w^{2} [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}] \end{array}

(10)

The instantaneous power absorbed by the NES equals the power dissipated by the damping in the NES and can thus be expressed as

P_{a} = f_{d 3} ({\dot{x}}_{2} - {\dot{x}}_{3})

(11)

The time-averaged absorbed power over a cycle is

{\bar{P}}_{a} = \frac{1}{T} \int_{0}^{T} P_{a} d t \approx \frac{1}{2} c_{3} w^{2} [{(b_{1} - d_{1})}^{2} + {(b_{2} - d_{2})}^{2}]

(12)

Here we introduce vibration power absorption ratio to evaluate the vibration reduction performance of NES. Both the satellite and the NES can absorb vibration energy from the vibration generator. The more energy the NES absorbs, the less energy the satellite is flowed. So power absorption ratio may be defined as the following formulation

R_{a} = \frac{{\bar{P}}_{a}}{{\bar{P}}_{in}}

(13)

Clearly, a large power absorption ratio means a good performance of NES. In comparison, the kinetic energy of a structure is often used as a measure to examine the NES control effectiveness.⁴⁰ The maximum kinetic energy K₁ of mass m₁ is

K_{1} = \frac{1}{2} m_{1} v_{1} \max^{2} = \frac{1}{2} m_{1} {\dot{x}}_{1 \max}^{2} \approx \frac{1}{2} m_{1} \sqrt{a_{1}^{2} w^{2} + a_{2}^{2} w^{2}}

(14)

where a first-order harmonic approximation of the displacement x₁ is used

Results and discussion

In this section, the variations of power flow variables in the frequency domain are obtained and are shown in the following figures. The vertical coordinate of P_a and P_in is a dB scale with a reference level of 10⁻⁶ W. The vertical coordinate of K₁ is a dB scale with a reference level of 10⁻⁶ J. Furthermore, the effects of mass, viscous damping, and cubic nonlinear stiffness of the NES on vibration power flows are investigated. The vibration reduction performance of the NES is evaluated by the power absorption ratio, which is compared with the kinetic energy of the oscillator.

Figures 4 and 5 examine the influence of varying NES mass m₃ on time-averaged power flow as well as power absorption ratio and kinetic energy. With the fixed cubic nonlinear stiffness k₃=10⁷ N/m³ and the viscous damping coefficient c₃ = 300 N s/m of the NES, the mass m₃ of the NES is changed from 4 to 6 kg and then to 8 kg. Figure 4(a) shows that the two peaks of time-averaged input power ${\bar{P}}_{in}$ can be reduced by introducing the NES into the primary system; however, ${\bar{P}}_{in}$ becomes large at other frequencies. When the NES mass is increased, the time-averaged input power ${\bar{P}}_{in}$ around the first resonant frequency decreases gradually, whereas the ${\bar{P}}_{in}$ in the low-frequency range increases. Figure 4(b) shows that the time-averaged absorbed power ${\bar{P}}_{a}$ becomes large over all frequency ranges as the NES mass increases. Moreover, the time-averaged input and absorbed powers of the system are all large at two resonant frequencies, although the response amplitude of the mass m₁ is very small at the second resonance frequency.

Figure 4.

Time-averaged (a) input power and (b) absorbed power of systems with varying mass m₃. NES: nonlinear energy sink.

Figure 5.

Kinetic energy (a) of the oscillator and power absorption ratio (b) with varying mass m₃. NES: nonlinear energy sink.

Figure 5(a) and (b) presents the variations of the power absorption ratio and kinetic energy of the oscillator in the frequency domain, respectively. Figure 5(b) shows that the power absorption ratio of NES is good at all frequencies over the 160 Hz frequency band with the exception of an antiresonance region near 28 Hz. The drop off of the power absorption ratio is acceptable since the NES is antiresonant at this frequency, which can be seen in Figure 3(c). Figure 5 demonstrates that the increase in the NES mass enhances the vibration absorption performance of the NES at all frequencies by increasing power absorption ratio and reducing kinetic energy of the oscillator. However, a relatively light NES is required in engineering design. The NES mass can only increase in a certain range.

Figures 6 and 7 consider the effects of varying NES viscous damping coefficients c₃ on vibration power flow of the system and kinetic energy of the oscillator. The damping coefficient c₃ of the NES changes from 300 to 700 N s/m and then to 1100 N s/m. The other parameters of the NES are set as m₃=4 kg, k₃=10⁷ N/m³. Figure 6(a) shows that introducing the NES can reduce the peak values of time-averaged input power ${\bar{P}}_{in}$ around two resonance frequencies. By contrast, the ${\bar{P}}_{in}$ of the system coupled to an NES becomes larger at other frequencies than that of the system with no NES attached. Figure 6 shows that the increase in NES damping coefficient can decrease the time-averaged input power ${\bar{P}}_{in}$ and increase the time-averaged absorbed power ${\bar{P}}_{a}$ around the second-order resonant frequency. However, there is no such a monotonous relation with ${\bar{P}}_{in}$ and ${\bar{P}}_{a}$ over the first-order resonant frequency region. Furthermore, the change of the damping coefficient slightly affects the natural frequencies of the primary structure.

Figure 6.

Time-averaged (a) input power and (b) absorbed power of systems with varying damping coefficient c₃. NES: nonlinear energy sink.

Figure 7.

Kinetic energy (a) of the oscillator and power absorption ratio (b) with varying damping coefficients c₃. NES: nonlinear energy sink.

Figure 7 plots the variations of power absorption ratio and kinetic energy of the oscillator against the excitation frequency. The figure shows that the power absorption ratio of NES is high at all frequencies except at an antiresonance region. This result is also caused by the antiresonance of the NES response amplitude d. Moreover, the increase in the NES damping coefficient is beneficial in enhancing power absorption ratio and attenuating kinetic energy in the high-frequency range; however, such increase can deteriorate the NES performance in the low-frequency range as shown in Figure 7.

Figures 8 and 9 investigate the effects of varying cubic nonlinear stiffness k₃ of the NES on vibration power flow and dynamic performance of the system. The parameters of the NES are set as m₃=4 kg and c₃= 300 N s/m. The cubic nonlinear stiffness k₃ of the NES changes from 10⁷ to 5 × 10⁷ N/m³ and then to 10⁸ N/m³. When the NES is attached to the primary structure, the two peak values of time-averaged input power ${\bar{P}}_{in}$ are reduced as shown in Figure 8(a). However, the ${\bar{P}}_{in}$ becomes large at other frequencies with the introduction of NES into the system. Figure 8 also shows that the increase in cubic nonlinear stiffness can increase the time-averaged absorbed power ${\bar{P}}_{a}$ and reduce the time-averaged input power ${\bar{P}}_{in}$ around the first-order natural frequency. The nonlinear stiffness has slight effect on the ${\bar{P}}_{a}$ and ${\bar{P}}_{in}$ at the other frequencies.

Figure 8.

Time-averaged (a) input power and (b) absorbed power of systems with varying nonlinear stiffness k₃. NES: nonlinear energy sink.

Figure 9.

Kinetic energy (a) of the oscillator and power absorption ratio (b) with varying nonlinear stiffness k₃. NES: nonlinear energy sink.

Figure 9 presents the variations of power absorption ratio R_a and kinetic energy K₁ of the oscillator in the frequency domain when the nonlinear stiffness k₃ changes. When the nonlinear stiffness k₃ of the NES increases, there is a peak in the curve of power absorption ratio which occurs at the first resonant frequency as shown in Figure 9(b). Moreover, the peak value of power absorption ratio at the first natural frequency gradually becomes high as nonlinear stiffness increases. Correspondingly, the first peak in the kinetic energy of oscillator is reduced by increasing k₃. Hence it can be concluded that the increase in nonlinear stiffness k₃ of the NES can improve its vibration suppression performance at the first resonant frequency.

Conclusions

The power flow analysis approach is for the first time proposed for quantitatively evaluating the dynamic performance of an NES in the frequency domain. A system of 2-DOF whole-spacecraft structure with the NES attached is adopted. Analytical approximation and numerical integration are applied for solving the dynamic equations of the whole-spacecraft vibration reduction system. The effects of the varying NES parameters on the vibration absorption performance of the NES are revealed. The investigation yields the following conclusions: (1) the NES can produce quite high power absorption ratio over a broad frequency spectrum except at an antiresonance region near 28 Hz, (2) the harmonic balance method is verified by numerical integration to solve this mathematical model, (3) the NES vibration reduction performance becomes better at all frequencies as the NES mass increases, (4) the increase in cubic nonlinear stiffness of the NES significantly enhances its performance only around the first resonance frequency of system, and (5) the effects of NES viscous damping on the performance in the high-frequency range are different to that in the low-frequency range.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/ or publication of this article: This paper is supported by the National Natural Science Foundation of China (Project No. 11572182). This work is supported by Scientific Research Fund of Liaoning Provincial Education Department (Project No. L201703). This paper is supported by the National (Outside) Training Project of Liaoning Higher Education Institutions (Project No. 2018LNGXGJWPY-YB008).

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