Method for judging unstable vibration area of transmission conductor system

Abstract

The invention discloses a method for judging an unstable vibration area of a power transmission conductor system, which comprises the following steps of: establishing a coupled piezoelectric-nonlinear energy trap system, constructing a dynamic model, and establishing a force-electricity coupling control equation set; substituting a complex variable form of displacement and acceleration into a dimensionless force-electricity coupling control equation set, and introducing a variable of relative displacement to obtain a structure control equation; simplifying the structure control equation to obtain a piezoelectric control equation; introducing a disturbance term at the balance point to obtain an incremental linear equation, and substituting the incremental linear equation into the new structure control equation; analyzing whether the piezoelectric-nonlinear energy trap system is stable near the position of the balance point; drawing a relation curve of response amplitude and frequency, and obtaining an HB bifurcation area and an SN bifurcation area; and establishing a system control equation set of the vibration energy capturing device, calculating a stable analytical solution, calculating the damping efficiency of the nonlinear energy trap, and evaluating different damping areas.

Description

Technical Field

[0001] The present invention relates to the field of transmission line vibration research, and in particular to a method for determining an unstable vibration region of a transmission line system. Background Art

[0002] During severe convective weather events, transmission lines often experience large, unstable vibrations. With the development of the Internet of Things (IoT), low-power embedded devices and microsensors are gaining widespread application in structural health monitoring, traffic monitoring, and field measurements. Transmission lines are often equipped with online monitoring sensors (such as wireless sensors, anemometers, attitude sensors, and temperature sensors) to collect real-time data from the lines and their surroundings. In the past, vibrating structures often required the installation of wireless electronic devices such as online monitoring sensors to monitor structural vibration signals in real time. These wireless devices typically rely on chemical batteries for power, but these batteries have drawbacks such as limited capacity, the need for regular replacement, and environmental pollution. Summary of the Invention

[0003] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for determining an unstable vibration region of a transmission line system.

[0004] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: A method for determining an unstable vibration region of a transmission line system is provided, comprising the following steps: S1: A coupled piezoelectric-nonlinear energy sink system is established based on the vibration energy harvesting device and nonlinear energy sink on the transmission line. The dynamic model of the piezoelectric-nonlinear energy sink system is constructed, and the force-electric coupling control equations are established. The dimensionless time scale and displacement response are defined to obtain the dimensionless force-electric coupling control equations. S2: The displacements and accelerations of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are expressed in complex variable form, and considering the resonance between them, the complex variable forms of displacements and accelerations are substituted into the dimensionless force-electric coupling control equations, and the relative displacement variable is introduced to obtain the structural control equations; S3: Introducing the polar coordinate form of the complex amplitude, establishing the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, simplifying the structural control equation, and obtaining the piezoelectric control equation; S4: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; introduce a disturbance term at the equilibrium point to obtain an incremental linear equation, and substitute the incremental linear equation into the new structural control equation; S5: Expand the variables in the formula in step S4, separate the real and imaginary parts of the variables, obtain the Jacobian matrix, and analyze whether the piezoelectric-nonlinear energy sink system is stable near the equilibrium point; S6: Simplify the new structural control equations and construct the formal equations. Based on the Cardano discriminant, obtain the bifurcation boundary conditions of the transmission line system and derive the conditions for the occurrence of SN bifurcation. Plot the relationship between the response amplitude and frequency to obtain the HB bifurcation region and the SN bifurcation region. S7: Establish the system control equations of the vibration energy harvesting device, calculate the stable analytical solution, calculate the vibration reduction efficiency of the nonlinear energy sink, and evaluate different vibration reduction areas.

[0005] Furthermore, step S1 includes: S11: A coupled piezoelectric-nonlinear energy well system is established based on a vibration energy harvesting device and a nonlinear energy well on the transmission line. The vibration energy harvesting device utilizes the piezoelectric effect to convert the vibration generated by the transmission line into a power source to supply power to the piezoelectric-nonlinear energy well system. The vibration energy harvesting device is coupled between the nonlinear energy well and the second-level main structure. The piezoelectric-nonlinear energy well system, the first main structure, and the second-level main structure constitute a transmission line system. S12: Construct a dynamic model of the piezoelectric-nonlinear energy sink system and establish a set of force-electric coupling control equations; ; in, are the masses of the first-level main structure and the second-level main structure on the transmission line, is the mass of the piezoelectric-nonlinear energy sink system, are the displacement responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, are the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, They are the damping between the ground and the first-level main structure, the damping between the first-level main structure and the second-level main structure, and the linear damping of the piezoelectric-nonlinear energy sink system. They are the linear stiffness between the ground and the first-level main structure, the linear stiffness between the first-level main structure and the second-level main structure, and the cubic stiffness of the piezoelectric-nonlinear energy sink system. is the piezoelectric linear stiffness of the vibration energy harvesting device, R is the resistance of the vibration energy harvesting device, is the capacitance of the vibration energy harvesting device, is the voltage obtained by the vibration energy harvesting device, is the piezoelectric coefficient, are the differentials of the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, is the external excitation frequency, A is the external force on the first-level main structure, t For time, It represents the simple harmonic load on the first-level main structure; S13: Defining dimensionless time scales and displacement response ; ; in, i is the displacement response type, and , is the displacement versus time scale The derivative of S14: According to the dimensionless time scale and displacement response , the force-electric coupling control equations are dimensionlessly processed to obtain the dimensionless calculation formula; ; in, are the dimensionless coefficients of the first-level main structure, the second-level main structure, and the third-level main structure, respectively. is the linear stiffness and linear stiffness The ratio between is the linear stiffness and linear stiffness The ratio between is the piezoelectric linear stiffness and linear stiffness The ratio between A 1 is the external force and linear stiffness of the first-level main structure The ratio between M 2 is the mass of the second-level main structure m 2. Quality of the first-level main structure m The ratio between 1, M 3 is the mass of the piezoelectric-nonlinear energy sink system m 3. Quality of the first-level main structure m The ratio between 1, is the piezoelectric coefficient and linear stiffness The ratio between is the dimensionless piezoelectric coefficient related to the first-order principal structure, is the dimensionless resistance associated with the first-level main structure, is the dimensionless capacitance of the vibration energy harvesting device; S15: Substituting the dimensionless calculation formula in step S14 into the dimensional force-electric coupling control equations in step S12 to obtain the dimensionless force-electric coupling control equations; ; Furthermore, step S2 includes: S21: Write the displacement and acceleration of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system in the form of complex variables; ; in, j is an imaginary unit, and , are the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy trap system, and the piezoelectric energy harvesting device, is a conjugate complex variable; S22: Consider the 1:1:1 resonance of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, and the corresponding response frequency is ,in, They are the response frequencies of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, and the response of the piezoelectric-nonlinear energy sink system is divided into the fast-changing part In the slow-varying modulation part, the complex variable forms of displacement and acceleration in step S21 are substituted into the dimensionless force-electric coupling control equations to obtain the slow-varying stationary flow formula: ; in, are the real parts of the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy well system, and the piezoelectric energy harvesting device, respectively. The real part of the complex variable The first derivative of ; S23: Introducing the variable of relative displacement; ; in, is the relative displacement between the first-level main structure and the second-level main structure, is the relative displacement between the second-level main structure and the piezoelectric-nonlinear energy sink system, is the motion displacement of the center of mass of the piezoelectric-nonlinear energy sink system, U is the voltage; S24: Substitute the variables in step S23 into the slowly varying stationary flow formula and simplify it to obtain the structural control equation; ; in, ; ; ; in, It is the intermediate variable of the structural control equation; Furthermore, step S3 includes: S31: In order to obtain the steady-state solution of the piezoelectric-nonlinear energy sink system, the polar coordinate form of the complex amplitude is introduced. The steady-state solution represents the displacement and direction of the displacement; ; in, Relative displacement , relative displacement , motion displacement The first derivative of Relative displacement , relative displacement , motion displacement ,Voltage U The amplitude of Relative displacement , relative displacement , motion displacement ,Voltage U Phase; S32: Based on the polar coordinate form of the complex amplitude, the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are established; ; S33: Combining the response expressions of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy sink system, and the voltage, as well as the polar coordinate form of the complex amplitude, the structural control equation is simplified to obtain the piezoelectric control equation; ; ; in, They are the intermediate variables of the piezoelectric control equation; Furthermore, step S4 includes: S41: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; ; S42: Define the stable state of the piezoelectric-nonlinear energy sink system as , introducing a disturbance term at the equilibrium point, we obtain the linear equation of the increment; ; in, They are the stable states of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, are the slow-varying amplitudes of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, Relative displacement , relative displacement , motion displacement A small increment of Slowly varying amplitudes The first derivative of Small increments The first derivative of ; S43: Substituting the incremental linear equation into the new structural control equation, we obtain: ; in, For small increments Take the conjugate.

[0006] Furthermore, step S5 includes: S51: Expand the variables of the formula in step S43; ; And separate the real and imaginary parts of the variables to get the Jacobian matrix J ; ; in, Small increments The real and imaginary parts of Small increments The real and imaginary parts of Small increments The real and imaginary parts of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of , i is the symbol of the imaginary part; S52: Calculate the Jacobian matrix J The eigenvalue of is used to judge the stability of the equilibrium point of the piezoelectric-nonlinear energy sink system. If the Jacobian matrix J If the eigenvalues of are all negative, the piezoelectric-nonlinear energy sink system is stable near the equilibrium point, otherwise it is unstable.

[0007] Furthermore, step S6 includes: S61: Order , simplify the new structural control equation to obtain the characteristic equation; ; ; in, They are all intermediate operational variables of the characteristic equation; in, ; ; ; ; ; ; ; ; ; ; ; ; ; ; in, It is also the intermediate operational variable of the characteristic equation; S62: Write the characteristic equation as a formal equation; ; ; in, is the intermediate operational variable of the formal equation; S63: Based on the Cardano discriminant, the bifurcation boundary conditions of the transmission line system are obtained; ; when When , the formal equation has only one real root, and the piezoelectric-nonlinear energy sink system has only one equilibrium point; when When , the formal equation has three real roots, and at least two of them are equal, then the piezoelectric-nonlinear energy sink system will undergo SN bifurcation; when When , the formal equation has three unequal real roots, and the piezoelectric-nonlinear energy sink system has three unequal equilibrium points; S64: Derive the conditions for SN bifurcation based on the formal equation; ; in, These are the two conditions for SN bifurcation; S65: As the two-dimensional parameter plane changes, the coefficients of the standard form of the SN bifurcation will degenerate, and additional bifurcation conditions are needed to track the new bifurcation, namely the codimensional two-cusp bifurcation condition; ; S66: Based on the SN bifurcation conditions and the codimensional two-cusp bifurcation conditions, and combined with the force-electric coupling control equations, the amplitude of the HB bifurcation region is obtained, and the relationship curve between the response amplitude and frequency is plotted according to the parameters of the piezoelectric-nonlinear energy sink system, the first-level main structure, and the second-level main structure to obtain the HB bifurcation region and the SN bifurcation region.

[0008] Furthermore, step S7 includes: S71: Construct the system control equations of the vibration energy harvesting device; ; S72: Obtaining a specific solution of the system control equations of the vibration energy harvesting device; ; S73: Combining the system control equations and special solutions of the vibration energy harvesting device to obtain a stable analytical solution The expression of stable analytical solution is the amplitude of the first-order main structure under the condition of no nonlinear energy well; ; ; S74: Using stable analytical solutions Calculating the damping efficiency of nonlinear energy sinks ; ; in, is the amplitude of the first-order main structure coupled with the piezoelectric-nonlinear energy sink system; S75: Based on shock absorption efficiency Evaluate shock absorption area and shock absorption efficiency The amplitude area <-2% is the amplitude amplification area, and the vibration reduction efficiency The amplitude area >2% is the effective vibration reduction area, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area I, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area II.

[0009] The beneficial effects of the present invention are: This invention simplifies the transmission line into a multi-degree-of-freedom oscillator and introduces a nonlinear energy sink (NES) and a piezoelectric energy harvester to achieve broadband vibration suppression. Simultaneously, the piezoelectric device collects the vibration energy transferred to the nonlinear energy sink, effectively suppressing harmful wind-induced vibrations in the transmission line and improving the safe operation of my country's power grid. The piezoelectric-nonlinear energy sink system not only absorbs the conductor's breeze vibration energy but also converts some of the absorbed vibration energy into electrical energy for use by sensors through the piezoelectric effect. While suppressing breeze vibrations in the conductor, it provides green and clean energy for power supply, achieving environmentally friendly vibration reduction while also promoting the development of smart grids and advancing the integrated research of vibration reduction, energy capture, and sensor power supply.

[0010] Based on Newton's second law and Kirchhoff's voltage law, this paper derives the dimensionless electromechanical coupling equations governing the coupling of transmission lines with piezoelectric-nonlinear energy sinks (PNES) and analyzes the vibration suppression effect of PNES on transmission lines. First, the vibration reduction effect and system stability under harmonic excitation are analyzed using the complex variable averaging method (CX-A) and the fourth-order Runge-Kutta (RK) method, and the correctness of the approximate solution is verified by numerical solutions. Next, the saddle point (SN) bifurcation and codimension bifurcations of the system are derived from the transmission line governing equations. An improved algorithm is used to determine the most unfavorable excitation amplitudes for the transmission line under various severe convective meteorological disasters, marking the Hopf (HB) bifurcation. The vibration reduction efficiency is evaluated using the generalized transmissibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 A simplified model diagram of a transmission line-coupled nonlinear energy sink-piezoelectric system.

[0012] Figure 2 This is the amplitude response diagram of the first-level main structure.

[0013] Figure 3 This is the amplitude response diagram of the second-level main structure.

[0014] Figure 4 is the amplitude response diagram of the nonlinear energy sink.

[0015] Figure 5 This is the amplitude response diagram of the vibration energy harvesting device.

[0016] Figure 6 This is the relationship diagram between external excitation frequency and meteorological excitation force. DETAILED DESCRIPTION

[0017] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0018] like Figure 1 As shown, a method for determining an unstable vibration region of a transmission line system includes the following steps: S1: Based on the vibration energy harvesting device and nonlinear energy sink on the transmission line, a coupled piezoelectric-nonlinear energy sink system is established, such as Figure 1 As shown, a dynamic model of the piezoelectric-nonlinear energy sink system is constructed, and a force-electric coupling control equation group is established, and a dimensionless time scale and displacement response are defined to obtain a dimensionless force-electric coupling control equation group. Step S1 specifically includes: S11: A coupled piezoelectric-nonlinear energy well system is established based on a vibration energy harvesting device and a nonlinear energy well on the transmission line. The vibration energy harvesting device utilizes the piezoelectric effect to convert the vibration generated by the transmission line into a power source to supply power to the piezoelectric-nonlinear energy well system. The vibration energy harvesting device is coupled between the nonlinear energy well and the second-level main structure. The piezoelectric-nonlinear energy well system, the first main structure, and the second-level main structure constitute a transmission line system. S12: Construct a dynamic model of the piezoelectric-nonlinear energy sink system and establish a set of force-electric coupling control equations; ; in, are the masses of the first-level main structure and the second-level main structure on the transmission line, is the mass of the piezoelectric-nonlinear energy sink system, are the displacement responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, are the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, They are the damping between the ground and the first-level main structure, the damping between the first-level main structure and the second-level main structure, and the linear damping of the piezoelectric-nonlinear energy sink system. They are the linear stiffness between the ground and the first-level main structure, the linear stiffness between the first-level main structure and the second-level main structure, and the cubic stiffness of the piezoelectric-nonlinear energy sink system. is the piezoelectric linear stiffness of the vibration energy harvesting device, R is the resistance of the vibration energy harvesting device, is the capacitance of the vibration energy harvesting device, is the voltage obtained by the vibration energy harvesting device, is the piezoelectric coefficient, are the differentials of the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, is the external excitation frequency, A is the external force on the first-level main structure, t For time, It represents the simple harmonic load on the first-level main structure; S13: Defining dimensionless time scales and displacement response ; ; in, i is the displacement response type, and , is the displacement versus time scale The derivative of S14: According to the dimensionless time scale and displacement response , the force-electric coupling control equations are dimensionlessly processed to obtain the dimensionless calculation formula; ; in, are the dimensionless coefficients of the first-level main structure, the second-level main structure, and the third-level main structure, respectively. is the linear stiffness and linear stiffness The ratio between is the linear stiffness and linear stiffness The ratio between is the piezoelectric linear stiffness and linear stiffness The ratio between A 1 is the external force and linear stiffness of the first-level main structure The ratio between M 2 is the mass of the second-level main structure m 2. Quality of the first-level main structure m The ratio between 1, M 3 is the mass of the piezoelectric-nonlinear energy sink system m 3. Quality of the first-level main structure m The ratio between 1, is the piezoelectric coefficient and linear stiffness The ratio between is the dimensionless piezoelectric coefficient related to the first-order principal structure, is the dimensionless resistance associated with the first-level main structure, is the dimensionless capacitance of the vibration energy harvesting device; S15: Substituting the formula in step S14 into the dimensioned force-electric coupling control equations in step S12 to obtain a dimensionless force-electric coupling control equations; .

[0019] S2: Write the displacement and acceleration of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system in complex variable form, and consider the resonance between them. Substitute the complex variable forms of displacement and acceleration into the dimensionless force-electric coupling control equations, introduce the variable of relative displacement, and obtain the structural control equation. Step S2 specifically includes: S21: The slowly varying dynamic flow of the piezoelectric-nonlinear energy sink system is analyzed using the complex variable averaging method. The displacements and accelerations of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are expressed in complex variable form. ; in, j is an imaginary unit, and , are the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy trap system, and the piezoelectric energy harvesting device, is a conjugate complex variable; S22: Consider the 1:1:1 resonance of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, and the corresponding response frequency is ,in, They are the response frequencies of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, and the response of the piezoelectric-nonlinear energy sink system is divided into the fast-changing part In the slow-varying modulation part, the complex variable forms of displacement and acceleration in step S21 are substituted into the dimensionless force-electric coupling control equations to obtain the slow-varying stationary flow formula: ; in, are the real parts of the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy well system, and the piezoelectric energy harvesting device, respectively. The real part of the complex variable The first derivative of ; S23: Introducing the variable of relative displacement; ; in, is the relative displacement between the first-level main structure and the second-level main structure, is the relative displacement between the second-level main structure and the piezoelectric-nonlinear energy sink system, is the motion displacement of the center of mass of the piezoelectric-nonlinear energy sink system, U is the voltage; S24: Substitute the variables in step S23 into the slowly varying stationary flow formula and simplify it to obtain the structural control equation; ; in, ; ; ; Please fill in the parameters not defined in the formula; S3: Introducing the polar coordinate form of the complex amplitude, establishing the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, simplifying the structural control equation, and obtaining the piezoelectric control equation. Step S3 specifically includes: S31: In order to obtain the steady-state solution of the piezoelectric-nonlinear energy sink system, the polar coordinate form of the complex amplitude is introduced. The steady-state solution represents the displacement and direction. ; in, Relative displacement , relative displacement , motion displacement The first derivative of Relative displacement , relative displacement , motion displacement ,Voltage U The amplitude of Relative displacement , relative displacement , motion displacement ,Voltage U Phase; S32: Based on the polar coordinate form of the complex amplitude, the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are established; ; S33: Combining the response expressions of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy sink system, and the voltage, as well as the polar coordinate form of the complex amplitude, the structural control equation is simplified to obtain the piezoelectric control equation; ; ; in, They are respectively the intermediate variables of the piezoelectric control equation.

[0020] S4: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; introduce a disturbance term at the equilibrium point to obtain an incremental linear equation, and substitute the incremental linear equation into the new structural control equation. Step S4 specifically includes: S41: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; ; S42: To determine the stability of the equilibrium point, we need to linearize it near the equilibrium point. The stable state of the piezoelectric-nonlinear energy sink system is defined as , introducing a disturbance term at the equilibrium point, we obtain the linear equation of the increment; ; in, They are the stable states of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, are the slow-varying amplitudes of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, Relative displacement , relative displacement , motion displacement A small increment of Slowly varying amplitudes The first derivative of Small increments The first derivative of ; S43: Substituting the incremental linear equation into the new structural control equation, we obtain: ; in, For small increments Take conjugate; S5: Expand the variables in the formula in step S4, separate the real and imaginary parts of the variables, obtain the Jacobian matrix, and analyze whether the piezoelectric-nonlinear energy sink system is stable near the equilibrium point. Step S5 specifically includes: S51: Expand the variables of the formula in step S43; ; And separate the real and imaginary parts of the variables to get the Jacobian matrix J ; ; in, Small increments The real and imaginary parts of Small increments The real and imaginary parts of Small increments The real and imaginary parts of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of , i is the symbol of the imaginary part; S52: Calculate the Jacobian matrix J The eigenvalue of is used to judge the stability of the equilibrium point of the piezoelectric-nonlinear energy sink system. If the Jacobian matrix J If the eigenvalues of are all negative, the piezoelectric-nonlinear energy sink system is stable near the equilibrium point, and the periodic solution is also stable near the equilibrium position. The periodic solution is the displacement of the piezoelectric-nonlinear energy sink system. Otherwise, it is unstable.

[0021] S6: Simplify the new structural control equations and construct formal equations. Based on the Cardano discriminant, obtain the bifurcation boundary conditions of the transmission line system and derive the conditions for SN bifurcation. Plot the relationship between the response amplitude and frequency to obtain the HB bifurcation region and the SN bifurcation region.

[0022] Bifurcation at the fixed point of the piezoelectric-nonlinear energy sink system can affect the steady-state response of the transmission line system. Different types of bifurcations correspond to critical states of different types of motion. Changes in all transmission line system parameters can lead to bifurcations, including those of the transmission line, the NES, and the piezoelectric system.

[0023] Step S6 specifically includes: S61: Order , simplify the new structural control equation to obtain the characteristic equation; ; ; in, They are all intermediate operational variables of the characteristic equation; in, ; ; ; ; ; ; ; ; ; ; ; ; ; ; in, It is also the intermediate operational variable of the characteristic equation; S62: Write the characteristic equation as a formal equation; ; ; in, is the intermediate operational variable of the formal equation; S63: Based on the Cardano discriminant, the bifurcation boundary conditions of the transmission line system are obtained; ; when When , the formal equation has only one real root, and the piezoelectric-nonlinear energy sink system has only one equilibrium point; when When , the formal equation has three real roots, and at least two of them are equal, then the piezoelectric-nonlinear energy sink system will undergo SN bifurcation; when When , the formal equation has three unequal real roots, and the piezoelectric-nonlinear energy sink system has three unequal equilibrium points; S64: Derive the conditions for SN bifurcation based on the formal equation; ; in, These are the two conditions for SN bifurcation; S65: As the two-dimensional parameter plane changes, the coefficients of the standard form of the SN bifurcation will degenerate, and additional bifurcation conditions are needed to track the new bifurcation, namely the codimensional two-cusp bifurcation condition; ; S66: Based on the SN bifurcation conditions and the codimensional two-cusp bifurcation conditions, and combined with the force-electric coupling control equations, the amplitude of the HB bifurcation region is obtained, and the relationship curve between the response amplitude and frequency is plotted according to the parameters of the piezoelectric-nonlinear energy sink system, the first-level main structure, and the second-level main structure to obtain the HB bifurcation region and the SN bifurcation region.

[0024] The parameters of the first and second main structures of this embodiment are specifically shown in Table 1. According to the parameter values of the nonlinear energy sink (NES) and the vibration energy harvesting device in Table 2 and Table 3, the following are obtained: Figure 2-Figure 5 The relationship between the system response amplitude and frequency near the first-order natural frequency is shown in Figure 2. The stable branches of the first-level and second-level main structures calculated using the complex variable averaging method (CA-X) are in good agreement with the numerical results.

[0025] Table 1 Main structural parameter values

[0026] Table 2 Nonlinear energy sink (NES) parameter values

[0027] Table 3 Parameters of vibration energy harvesting device

[0028] S7: Establish the system control equations of the vibration energy harvesting device, calculate the stable analytical solution, calculate the vibration reduction efficiency of the nonlinear energy well, and evaluate different vibration reduction areas. In order to perform vibration suppression analysis on the two-degree-of-freedom main structure coupled with the nonlinear energy well (NES), it is necessary to compare the system response with that without the additional nonlinear energy well (NES). Step S7 specifically includes: S71: Construct the system control equations of the vibration energy harvesting device; ; S72: Obtaining a specific solution of the system control equations of the vibration energy harvesting device; ; S73: Combining the system control equations and special solutions of the vibration energy harvesting device to obtain a stable analytical solution The expression of stable analytical solution is the amplitude of the first-order main structure under the condition of no nonlinear energy well; ; ; S74: Using stable analytical solutions Calculating the damping efficiency of nonlinear energy sinks ; ; in, is the amplitude of the first-order main structure coupled with the piezoelectric-nonlinear energy sink system; S75: The parameters of a piezoelectric-nonlinear energy sink system have a profound influence on the SN bifurcation region at different frequency bands, which in turn is closely related to the effective vibration reduction region. We analyze the vibration reduction efficiency of the bifurcation region and discuss the effects of different parameters on vibration reduction in different frequency bands.

[0029] According to the shock absorption efficiency Evaluate shock absorption area and shock absorption efficiency The amplitude area <-2% is the amplitude amplification area, and the vibration reduction efficiency The amplitude area >2% is the effective vibration reduction area, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area I, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area II.

[0030] like Figure 6 As shown, the low-frequency SN region is completely within the amplitude amplification region, while the high-frequency SN region is completely within the ineffective vibration damping region II. It can be seen that the SN bifurcation in the low-frequency region is the primary detrimental factor, while the middle region is the effective vibration damping region. An HB bifurcation occurs in the red region, where the piezoelectric-nonlinear energy sink system transitions from its initial steady-state periodic motion to irregular, large-scale oscillations. Therefore, the HB vibration damping region has a highly effective vibration damping effect. The HB vibration damping region is relatively small, has a certain triggering threshold, and disappears as the external excitation force gradually increases.

[0031] Instabilities near the natural frequency of transmission lines expand the range of instability near the natural frequency, precisely reflecting the broadband vibration reduction and frequency robustness of the piezoelectric-nonlinear energy sink system. The complex variable averaging method and the numerical solution agree well within the stable regions of the first and second main structures. This invention links bifurcation regions (SN and HB) with vibration reduction efficiency and classifies different vibration reduction regions based on vibration reduction efficiency and bifurcation type, further linking the dynamic behavior of transmission lines with vibration reduction efficiency.

Claims

1. A method for determining an unstable vibration region of a transmission line system, characterized in that: The following steps are involved: S1: A coupled piezoelectric-nonlinear energy sink system is established based on the vibration energy harvesting device and nonlinear energy sink on the transmission line. The dynamic model of the piezoelectric-nonlinear energy sink system is constructed, and the force-electric coupling control equations are established. The dimensionless time scale and displacement response are defined to obtain the dimensionless force-electric coupling control equations. S2: The displacements and accelerations of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are expressed in complex variable form, and considering the resonance between them, the complex variable forms of displacements and accelerations are substituted into the dimensionless force-electric coupling control equations, and the relative displacement variable is introduced to obtain the structural control equations; S3: Introducing the polar coordinate form of the complex amplitude, establishing the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, simplifying the structural control equation, and obtaining the piezoelectric control equation; S4: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; Introduce a disturbance term at the equilibrium point to obtain an incremental linear equation, and substitute the incremental linear equation into the new structural control equation; S5: Expand the variables in the formula in step S4, separate the real and imaginary parts of the variables, obtain the Jacobian matrix, and analyze whether the piezoelectric-nonlinear energy sink system is stable near the equilibrium point; S6: Simplify the new structural control equations and construct the formal equations. Based on the Cardano discriminant, obtain the bifurcation boundary conditions of the transmission line system and derive the conditions for the occurrence of SN bifurcation. Plot the relationship between the response amplitude and frequency to obtain the HB bifurcation region and the SN bifurcation region. S7: Establish the system control equations of the vibration energy harvesting device, calculate the stable analytical solution, calculate the vibration reduction efficiency of the nonlinear energy sink, and evaluate different vibration reduction areas.

2. The method for determining an unstable vibration region of a power transmission line system according to claim 1, wherein: The step S1 comprises: S11: A coupled piezoelectric-nonlinear energy well system is established based on a vibration energy harvesting device and a nonlinear energy well on the transmission line. The vibration energy harvesting device utilizes the piezoelectric effect to convert the vibration generated by the transmission line into a power source to supply power to the piezoelectric-nonlinear energy well system. The vibration energy harvesting device is coupled between the nonlinear energy well and the second-level main structure. The piezoelectric-nonlinear energy well system, the first main structure, and the second-level main structure constitute a transmission line system. S12: Construct a dynamic model of the piezoelectric-nonlinear energy sink system and establish a set of force-electric coupling control equations; ； in, are the masses of the first-level main structure and the second-level main structure on the transmission line, is the mass of the piezoelectric-nonlinear energy sink system, are the displacement responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, are the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, They are the damping between the ground and the first-level main structure, the damping between the first-level main structure and the second-level main structure, and the linear damping of the piezoelectric-nonlinear energy sink system. They are the linear stiffness between the ground and the first-level main structure, the linear stiffness between the first-level main structure and the second-level main structure, and the cubic stiffness of the piezoelectric-nonlinear energy sink system. is the piezoelectric linear stiffness of the vibration energy harvesting device, R is the resistance of the vibration energy harvesting device, is the capacitance of the vibration energy harvesting device, is the voltage obtained by the vibration energy harvesting device, is the piezoelectric coefficient, are the differentials of the velocity responses of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, is the external excitation frequency, A is the external force on the first-level main structure, t For time, It represents the simple harmonic load on the first-level main structure; S13: Defining dimensionless time scales and displacement response ; ； in, i is the displacement response type, and , is the displacement versus time scale The derivative of S14: According to the dimensionless time scale and displacement response , the force-electric coupling control equations are dimensionlessly processed to obtain the dimensionless calculation formula; ； in, are the dimensionless coefficients of the first-level main structure, the second-level main structure, and the third-level main structure, respectively. is the linear stiffness and linear stiffness The ratio between is the linear stiffness and linear stiffness The ratio between is the piezoelectric linear stiffness and linear stiffness The ratio between A 1 is the external force and linear stiffness of the first-level main structure The ratio between M 2 is the mass of the second-level main structure m 2. Quality of the first-level main structure m The ratio between 1, M 3 is the mass of the piezoelectric-nonlinear energy sink system m 3. Quality of the first-level main structure m The ratio between 1, is the piezoelectric coefficient and linear stiffness The ratio between is the dimensionless piezoelectric coefficient related to the first-order principal structure, is the dimensionless resistance associated with the first-level main structure, is the dimensionless capacitance of the vibration energy harvesting device; S15: Substituting the dimensionless calculation formula in step S14 into the dimensional force-electric coupling control equations in step S12 to obtain the dimensionless force-electric coupling control equations; 。 3. The method for determining an unstable vibration region of a power transmission line system according to claim 2, wherein: The step S2 comprises: S21: Write the displacement and acceleration of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system in the form of complex variables; ； in, j is an imaginary unit, and , are the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy trap system, and the piezoelectric energy harvesting device, is a conjugate complex variable; S22: Consider the 1:1:1 resonance of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, and the corresponding response frequency is ,in, They are the response frequencies of the first-level main structure, the second-level main structure and the piezoelectric-nonlinear energy sink system, and the response of the piezoelectric-nonlinear energy sink system is divided into the fast-changing part In the slow-varying modulation part, the complex variable forms of displacement and acceleration in step S21 are substituted into the dimensionless force-electric coupling control equations to obtain the slow-varying stationary flow formula: ； in, are the real parts of the complex variables of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy well system, and the piezoelectric energy harvesting device, respectively. The real part of the complex variable The first derivative of ; S23: Introducing the variable of relative displacement; ； in, is the relative displacement between the first-level main structure and the second-level main structure, is the relative displacement between the second-level main structure and the piezoelectric-nonlinear energy sink system, is the motion displacement of the center of mass of the piezoelectric-nonlinear energy sink system, U is the voltage; S24: Substitute the variables in step S23 into the slowly varying stationary flow formula and simplify it to obtain the structural control equation; ； in, ； ； ； in, It is the intermediate variable of the structural control equation.

4. The method for determining an unstable vibration region of a power transmission line system according to claim 3, wherein: The step S3 comprises: S31: In order to obtain the steady-state solution of the piezoelectric-nonlinear energy sink system, the polar coordinate form of the complex amplitude is introduced. The steady-state solution represents the displacement and direction of the displacement; ； in, Relative displacement , relative displacement , motion displacement The first derivative of Relative displacement , relative displacement , motion displacement ,Voltage U The amplitude of Relative displacement , relative displacement , motion displacement ,Voltage U Phase; S32: Based on the polar coordinate form of the complex amplitude, the displacement response and voltage expressions of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system are established; ； S33: Combining the response expressions of the first-level main structure, the second-level main structure, the piezoelectric-nonlinear energy sink system, and the voltage, as well as the polar coordinate form of the complex amplitude, the structural control equation is simplified to obtain the piezoelectric control equation; ； ； in, They are respectively the intermediate variables of the piezoelectric control equation.

5. The method for determining an unstable vibration region of a power transmission line system according to claim 4, wherein: The step S4 comprises: S41: Couple the piezoelectric control equation into the structural control equation to obtain a simplified new structural control equation; ； S42: Define the stable state of the piezoelectric-nonlinear energy sink system as , introducing a disturbance term at the equilibrium point, we obtain the linear equation of the increment; ； in, They are the stable states of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, are the slow-varying amplitudes of the first-level main structure, the second-level main structure, and the piezoelectric-nonlinear energy sink system, Relative displacement , relative displacement , motion displacement A small increment of Slowly varying amplitudes The first derivative of Small increments The first derivative of ; S43: Substituting the incremental linear equation into the new structural control equation, we obtain: ； in, For small increments Take the conjugate.

6. The method for determining an unstable vibration region of a power transmission line system according to claim 5, wherein: The step S5 comprises: S51: Expand the variables of the formula in step S43; ； And separate the real and imaginary parts of the variables to get the Jacobian matrix J ; ； in, Small increments The real and imaginary parts of Small increments The real and imaginary parts of Small increments The real and imaginary parts of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of Real part and the imaginary part The first derivative of , i is the symbol of the imaginary part; S52: Calculate the Jacobian matrix J The eigenvalue of is used to judge the stability of the equilibrium point of the piezoelectric-nonlinear energy sink system. If the Jacobian matrix J If the eigenvalues of are all negative, the piezoelectric-nonlinear energy sink system is stable near the equilibrium point, otherwise it is unstable.

7. The method for determining an unstable vibration region of a power transmission line system according to claim 6, wherein: The step S6 comprises: S61: Order , simplify the new structural control equation to obtain the characteristic equation; ； ； in, They are all intermediate operational variables of the characteristic equation; in, ; ； ； ； ； ； ； ； ； ； ； ； ； ； in, It is also the intermediate operational variable of the characteristic equation; S62: Write the characteristic equation as a formal equation; ； ； in, is the intermediate operational variable of the formal equation; S63: Based on the Cardano discriminant, the bifurcation boundary conditions of the transmission line system are obtained; ； when When , the formal equation has only one real root, and the piezoelectric-nonlinear energy sink system has only one equilibrium point; when When , the formal equation has three real roots, and at least two of them are equal, then the piezoelectric-nonlinear energy sink system will undergo SN bifurcation; when When , the formal equation has three unequal real roots, and the piezoelectric-nonlinear energy sink system has three unequal equilibrium points; S64: Derive the conditions for SN bifurcation based on the formal equation; ； in, These are the two conditions for SN bifurcation; S65: As the two-dimensional parameter plane changes, the coefficients of the standard form of the SN bifurcation will degenerate, and additional bifurcation conditions are needed to track the new bifurcation, namely the codimensional two-cusp bifurcation condition; ； S66: Based on the SN bifurcation conditions and the codimensional two-cusp bifurcation conditions, and combined with the force-electric coupling control equations, the amplitude of the HB bifurcation region is obtained, and the relationship curve between the response amplitude and frequency is plotted according to the parameters of the piezoelectric-nonlinear energy sink system, the first-level main structure, and the second-level main structure to obtain the HB bifurcation region and the SN bifurcation region.

8. The method for determining an unstable vibration region of a power transmission line system according to claim 7, wherein: The step S7 comprises: S71: Construct the system control equations of the vibration energy harvesting device; ； S72: Obtaining a specific solution of the system control equations of the vibration energy harvesting device; ； S73: Combining the system control equations and special solutions of the vibration energy harvesting device to obtain a stable analytical solution The expression of stable analytical solution is the amplitude of the first-order main structure under the condition of no nonlinear energy well; ； ； S74: Using stable analytical solutions Calculating the damping efficiency of nonlinear energy sinks ; ； in, is the amplitude of the first-order main structure coupled with the piezoelectric-nonlinear energy sink system; S75: Based on shock absorption efficiency Evaluate shock absorption area and shock absorption efficiency The amplitude area <-2% is the amplitude amplification area, and the vibration reduction efficiency The amplitude area >2% is the effective vibration reduction area, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area I, and the vibration reduction efficiency exist The amplitude area is the ineffective vibration reduction area II.

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