Performance Optimization Method of Tristable Energy Harvester under Low-Frequency Excitation Based on Vibration Resonance

Abstract

This invention discloses a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance, belonging to the field of energy harvester optimization. The implementation method is as follows: a standard rectifier circuit is selected as the nonlinear collection circuit and connected to the energy harvester to achieve a stable DC output; a strongly nonlinear system model of the electromechanical coupled tristable energy harvester system is established; the slow variable equation of the tristable energy harvester system is derived using the variable separation method; the steady-state solution and analytical expression of the DC power of the tristable energy harvester system are derived using the harmonic balance method; an optimization model of the tristable energy harvester system under low-frequency excitation is obtained based on vibration resonance; the influence of low-frequency excitation amplitude, low-frequency excitation frequency, stiffness coefficient, time constant ratio, and electromechanical coupling coefficient on the collection performance of the tristable energy harvester system is analyzed using the optimization model, and the optimal combination is selected to achieve efficient energy harvesting of the tristable energy harvester system under low-frequency excitation.

Description

Technical Field

[0001] This invention relates to a performance optimization method for a tristable energy harvester, specifically a method for optimizing the performance of a tristable energy harvester under low-frequency excitation based on vibration resonance, and belongs to the field of energy harvester optimization. Background Technology

[0002] With the rapid development of microelectronics technology, miniature low-power electronic devices used in wireless sensors, environmental control systems, medical implants, wearable devices, and other applications have become widely used. Numerous miniature sensors, microprocessors, and low-power transmission modules, among other small low-power electronic devices, have seen significant development across various related industries. Traditionally, these microelectronic devices are powered by chemical batteries, but batteries suffer from low power density, large size, difficult-to-degrade materials, and short lifespans. Furthermore, replacement is time-consuming and labor-intensive, severely hindering the further development of such devices. Energy harvesters, however, possess the ability to convert ubiquitous environmental vibration energy into electrical energy, making it possible to provide microelectronic devices with long-term, even indefinite, autonomous power. This will greatly advance the development of microelectronics technology.

[0003] Energy traps generate potential in response to mechanical stimuli and external vibrations by utilizing the capabilities of active materials (such as piezoelectric, magnetostrictive, and ferroelectric) and electromechanical coupling mechanisms (such as electrostatic and electromagnetic). Through connection to appropriate interface circuits, they convert environmental vibrations into direct current usable by external electronic devices, thus powering microelectronic equipment. Currently, energy traps can be categorized into electromagnetic, piezoelectric, electrostatic, and hybrid types. Among these, piezoelectric energy traps have attracted widespread attention due to their simple structure, high power density, and good scalability. Traditional vibration energy traps, based on the principle of linear resonance, possess a very narrow steady-state frequency bandwidth, resulting in very low harvesting efficiency in real-world environments. This severely limits the applicability and practicality of linear energy traps, making it difficult to efficiently harvest energy from wide-spectrum vibration energy sources. To address the bandwidth limitation, nonlinear energy traps, obtained by intentionally introducing nonlinear magnetic structures, have attracted considerable attention. Consequently, numerous nonlinear energy trapping structures have been proposed, such as monostable, bistable, and tristable strongly nonlinear structures. The paper "Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations, International Journal of Mechanical Sciences 2020, 172:105418" compares the properties of monostable, bistable, and tristable energy harvesters, and publicly points out that the tristable energy harvester has better harvesting performance than the monostable and bistable energy harvesters, achieving higher harvested power and conversion efficiency. The paper "Harmonic balance analysis of nonlinear tristable energy harvesters for performance enhancement, Journal of Sound and Vibration 2016, 373:223-235" also publicly points out that tristable energy harvesters can achieve broadband energy harvesting and have higher harvesting potential, especially for improving energy harvesting performance under low-frequency excitation. Currently, numerous studies indicate that tristable energy harvesters can produce higher output power than monostable and bistable energy harvesters, which means that research on tristable energy harvesters is of great significance for improving energy harvesting efficiency in low-frequency environments. However, current research on tristable energy harvesters usually neglects the design of the harvesting circuit and simplifies it into a purely resistive circuit. This results in the energy harvester outputting a high-voltage, low-current alternating current, which cannot directly power external electronic devices.

[0004] In fact, energy harvesters need to be connected to a nonlinear rectifier circuit to convert the alternating current (AC) collected from the environment into direct current (DC) usable by external electronic devices. Commonly used nonlinear harvesting circuits include standard rectifier circuits, synchronous charge extraction circuits, and synchronous switching energy harvesting. However, the introduction of nonlinear harvesting circuits introduces complex coupling behaviors into the energy harvesting system, posing new challenges to the dynamics and performance analysis of the energy harvester. Furthermore, for energy harvesting systems with complex nonlinear rectifier circuits, the nonlinearity introduced by the circuit is often estimated or even ignored to simplify the difficulties it brings to theoretical analysis. Only a few researchers have conducted in-depth studies on the performance of multistable energy harvesters connected to nonlinear rectifier circuits. The paper "Statistical quantification of DC powergenerated by bistable piezoelectric energy harvesters when driven by random excitations, Journal of sound and vibration, 2019, 442:770-786" attempts to use a standard rectifier circuit as a nonlinear harvesting circuit to study the DC output of a bistable energy harvester; unfortunately, this paper does not explore the case of a tristable energy harvester. The efficient energy harvesting capability of tristable energy traps from low-frequency environments has been proven. However, the rapid development of microelectronics technology has also resulted in the energy harvesting performance of energy traps still not meeting application requirements. Therefore, how to optimize the design of energy traps to improve their energy harvesting capability from the environment is a very important research topic. Vibrational resonance is a phenomenon in which the response of a nonlinear system to a weak low-frequency signal can be amplified by a high-frequency signal. It has been used to optimize the structural design of nonlinear oscillators. The literature "Novel vibrational resonance in multistable systems, Chaos, 2011, 21(3): L433" studies the vibrational resonance phenomenon in multistable systems and points out that vibrational resonance can be used to optimize the design of nonlinear oscillators to enhance the output response. Therefore, optimizing the performance of tristable energy traps under low-frequency excitation based on vibrational resonance is of great significance. Summary of the Invention

[0005] To address the issue of sustainable and efficient power supply for electronic devices, the main objective of this invention is to provide a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance. A standard rectifier circuit is selected as the nonlinear collection circuit to provide a stable DC power supply for the electronic device. The slow variable equation of the tristable energy harvester system is derived using a variable separation method. The steady-state solution and analytical expression of the DC power of the tristable energy harvester system are derived using the harmonic balance method. The optimization of the tristable energy harvester system under low-frequency excitation is achieved based on vibration resonance, thereby realizing efficient energy harvesting by the tristable energy harvester under low-frequency excitation.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] This invention discloses a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance. A standard rectifier circuit is selected as the nonlinear collection circuit to achieve stable DC output from the energy harvester, and a strongly nonlinear system model of the electromechanically coupled tristable energy harvester is established. The slow variable equation of the tristable energy harvester is derived using the variable separation method. The steady-state solution and analytical expression of the DC power of the tristable energy harvester are derived using the harmonic balance method. An optimization model of the tristable energy harvester under low-frequency excitation is obtained based on vibration resonance. Using the output response amplitude gain coefficient, vibration amplitude, rectified voltage, and DC power of the tristable energy harvester as performance indicators, the optimization model is used to analyze the influence of low-frequency excitation amplitude, low-frequency excitation frequency, stiffness coefficient, time constant ratio, and electromechanical coupling coefficient on the collection performance of the tristable energy harvester. The optimal combination of these factors is selected to enhance the performance of the tristable energy harvester under low-frequency excitation, thereby achieving efficient energy harvesting by the tristable energy harvester under low-frequency excitation.

[0008] The present invention discloses a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance, comprising the following steps:

[0009] Step 1: Select a standard rectifier circuit as the collection circuit and connect it to the tristable energy harvester to convert the AC power collected from the low-frequency excitation of the tristable energy harvester into stable DC power, thereby achieving a stable DC output from the energy harvester. Establish a strongly nonlinear system model of the electromechanically coupled tristable energy harvester. Further, dimensionless processing is applied to the strongly nonlinear system model of the electromechanically coupled tristable energy harvester to obtain a dimensionless model of the tristable energy harvester.

[0010] To obtain a stable DC output from the energy harvesting system, a standard rectifier circuit is considered as a nonlinear interface circuit, and harmonic excitation is used to simulate environmental excitation. A strongly nonlinear dynamic model of the electromechanical coupled tristable energy harvesting system is established:

[0011]

[0012] In the formula, M represents the equivalent mass of the end magnet. Defined as the displacement at the end of the cantilever beam. Indicates time, and C represents the air damping coefficient. This indicates the voltage across the piezoelectric element. Represents the electromechanical coupling coefficient. C represents the harmonic excitation applied to the base. p This indicates the internal capacitance of the piezoelectric element. This indicates the current flowing into the rectifier circuit:

[0013]

[0014] Among them, C R This represents the filter capacitor connected in parallel with the load resistor R. This indicates the stable DC voltage output by the rectifier circuit. The three-well function of the system has the following form:

[0015]

[0016] In the formula, and The linear, cubic, and quintic stiffness coefficients of the energy-harvesting system are represented, respectively.

[0017] Introducing dimensionless transformation Substituting into equation (1), we obtain the dimensionless electromechanical coupling model of the tristable energy harvesting system, which is expressed as:

[0018]

[0019]

[0020] Where ω0 represents the natural frequency of the metal cantilever beam of the energy trap. Let X be the length of the beam, and let X represent the dimensionless displacement. This represents the dimensionless base excitation. This is the dimensionless piezoelectric voltage. This represents the dimensionless rectified voltage. Represents dimensionless current. This indicates the ratio of the filter capacitor to the internal capacitance of the piezoelectric element. These are the dimensionless damping coefficient, the time constant ratio, and the electromechanical coupling coefficient, respectively. This is the dimensionless tristable state function of the system. and These represent the dimensionless linear stiffness coefficient, cubic stiffness coefficient, and quintic stiffness coefficient, respectively. Their values ​​are related to the distance between the magnet at the end of the cantilever beam and the fixed magnet on the frame, directly reflecting the shape of the potential function of the tristable energy-harvesting system and the strength of the geometric nonlinearity of the system.

[0021] For the dimensionless base excitation in formula (4), the simulation is performed using harmonic excitation with two different frequencies as shown in formula (6), namely:

[0022]

[0023] Where fsin(ωt) represents the low-frequency harmonic excitation in the environment, and f and ω are the amplitude and frequency of the low-frequency force, respectively. Fsin(Ωt) represents the high-frequency harmonic excitation in the environment, and F and Ω are the amplitude and frequency of the high-frequency force, respectively. The low-frequency harmonic excitation and the high-frequency harmonic excitation satisfy the conditions Ω >> ω and f << 1.

[0024] Step 2: Based on the variable separation method, derive the fast variable equations and slow variable equations of the dimensionless model of the tristable energy harvesting system; to address the piecewise nonsmooth characteristics caused by the standard rectifier circuit, use the fundamental harmonic component of the piezoelectric voltage to smooth the piecewise nonsmooth piezoelectric voltage, thereby obtaining the equivalent uncoupled system equations of the dimensionless model of the tristable energy harvesting system; and based on vibration resonance, obtain the optimized model of the tristable energy harvesting system under low-frequency excitation.

[0025] According to the method of separation of variables, formula (4) has an approximate solution of the following form:

[0026] X(t)=x(t)+Ψ(t) (7)

[0027] In the formula, x(t) is the output response with time scale T. f For a low-frequency, slow variable equal to 2π / ω, Ψ(t) is a rapidly changing component with a mean of zero, i.e., <Ψ(t)>=0. (Symbol) This represents the time average over a shorter period of time, where T F =2π / Ω is a relatively short time period. Substituting equation (7) into the dimensionless electromechanical coupling model (4) of the tristable energy harvesting system, we obtain the equations for the slow component x(t) and the fast component Γ(t) in the output response X(t) of the tristable energy harvesting system as follows:

[0028]

[0029]

[0030] To solve equation (9), we define it as follows:

[0031] Ψ = B11 cos(Ωt)+B 12 sin(Ωt) (10)

[0032] In addition, the first derivative of the fast variable with respect to time is calculated according to equation (10). and second derivative

[0033]

[0034] Among them, B 11 B 12 These are the coefficients to be determined. A simple calculation of formula (10) yields the expression for the fast variable Ψ, namely:

[0035]

[0036] Substituting formula (12) into formula (9), and further simplifying and rearranging according to formulas (10) and (11) based on the same harmonic order, we obtain the following about B. 11 B 12 The system of equations:

[0037]

[0038] For taking the smaller B 11 B 12 Formula (13) is approximately:

[0039]

[0040] Obviously, B 11 B 12 Solving from formula (14), we get:

[0041]

[0042] In the following description, k1+3k3x 2 +5k5x 4 -Ω 2 Let it be denoted as W, i.e., W = k1 + 3k3x 2 +5k5x 4 -Ω 2 .

[0043] Substituting equations (15), (10), and (12) into equation (8), the equation concerning the slow variable x(t) is transformed into:

[0044]

[0045] Due to the alternating conduction and blocking of the rectifier bridge in the standard rectifier circuit, the piezoelectric voltage in formula (4) has piecewise non-smooth properties. The fundamental harmonic component of the piezoelectric voltage is used to approximate it, and combined with generalized harmonic transformation and the dimensionless equation of the tristable energy-harvesting system, an equivalent uncoupled optimization model is obtained as shown in formula (17):

[0046]

[0047] In the formula,

[0048]

[0049] in, The equivalent damping coefficient of the optimized tristable energy trap is represented by the rectifier circuit, which has a resistive damping effect on the original system (4). This represents the optimized equivalent linear stiffness coefficient, which depends not only on the stiffness coefficient of the original system and the properties of the high-frequency force, but also on the rectifier circuit. This represents the optimized cubic stiffness coefficient, which mainly depends on the original system's cubic and quintic stiffness coefficients and the properties of high-frequency forces. Additionally, This represents the fifth-order stiffness coefficient after optimization, which is consistent with the fifth-order stiffness coefficient of the original system. θ represents the blocking angle of the diode rectifier bridge.

[0050] Based on the variable separation method, the fast variable equations and slow variable equations of the dimensionless electromechanical coupling model of the tristable energy harvesting system are derived. To address the piecewise nonsmooth characteristics caused by the standard rectifier circuit, the fundamental harmonic component of the piezoelectric voltage is used to smooth the piecewise nonsmooth piezoelectric voltage, thereby obtaining the equivalent uncoupled system equations of the dimensionless electromechanical coupling model of the tristable energy harvesting system. Based on vibration resonance, the optimized model of the tristable energy harvesting system under low-frequency excitation is obtained.

[0051] Step 3: For the equivalent uncoupled optimization model of the tristable energy harvesting system described in Step 2 under low-frequency excitation, the steady-state solution of the system and the analytical expressions for the system rectified voltage and system DC power are derived using the harmonic balance method. Furthermore, using the output response amplitude gain coefficient, vibration amplitude, rectified voltage, and DC power of the tristable energy harvester as performance indicators, the influence of system parameters including stiffness coefficient, time constant ratio, and electromechanical coupling coefficient, as well as the amplitude and frequency of environmental excitation, on the energy harvester's acquisition performance is analyzed. The optimal parameter combination that maximizes the performance indicators is given. Based on this optimal parameter combination, the performance of the tristable energy harvesting system under low-frequency excitation is enhanced, thereby achieving efficient energy harvesting by the tristable energy harvester under low-frequency excitation.

[0052] The output rectified voltage of the tristable energy harvesting system shown in Equation (24), the DC power of the system shown in Equation (25), and the amplitude gain coefficient of the output response of the system shown in Equation (26) are used as performance indicators to measure the power generation performance of the tristable energy harvesting system under different parameter combinations. The influence rules of the system parameters on the dynamic behavior and harvesting performance of the energy harvesting system are analyzed, and the optimal parameter combination that maximizes the performance indicators is selected.

[0053] The steady-state approximate solution of the equivalent uncoupled system equation (17) after optimization is:

[0054] x(t)=b1cos(ωt)+b2sin(ωt) (19)

[0055] in, This represents the vibration amplitude of the system (17). Using the harmonic balance method, formula (19) and its first and second derivatives are substituted into the system (17), and the results are arranged according to the same harmonic order. When the tristable energy-harvesting system reaches steady state, the time derivatives of all quantities are zero. Furthermore, the frequency of the high-frequency force Fcos(Ωt) is sufficiently large, i.e., W≈-Ω. 2 Therefore, the steady-state amplitudes b1 and b2 are derived as follows:

[0056]

[0057]

[0058] The steady-state solution of the system (17) is obtained through formulas (20) and (21), wherein the vibration amplitude of the system (17) is...

[0059] In addition, the blocking angle θ of the rectifier bridge in the rectifier circuit is related to the rectified voltage Y. R The following relationship exists between them:

[0060] A(cosθ-1)=-2Y R (twenty two)

[0061] Furthermore, based on Kirchhoff's current law, within half a cycle, the total charge flowing out of the energy trap is equal to the total charge flowing through the resistor, that is:

[0062]

[0063] Combining formulas (5), (22), and (23), the rectified voltage Y output by the tristable energy harvesting system is derived. R The DC power P is as follows:

[0064]

[0065]

[0066] The output response amplitude gain coefficient Q of the tristable energy harvesting system is an important quantitative indicator describing the vibration resonance mechanism, and has the following characteristics:

[0067]

[0068] Where f is the amplitude of the low-frequency harmonic excitation, and Q s and Q c These are the sine and cosine components of the system (17) relative to the low-frequency input fcos(ωt), respectively:

[0069]

[0070]

[0071] Where n represents a positive integer, and x(t) is the output displacement of the optimized tristable energy harvesting system.

[0072] Based on the optimized tristable energy harvester system (17), the output response amplitude gain coefficient, system vibration amplitude, system rectified voltage (as shown in formula (24), and system DC power (as shown in formula (25)) of the tristable energy harvester system are used as performance indicators. The influence of the system parameters, including stiffness coefficient, time constant ratio, electromechanical coupling coefficient, environmental excitation amplitude, and frequency, on the energy harvester acquisition performance is analyzed. The optimal parameter combination that maximizes the performance indicators is given. Based on the optimal parameter combination, the performance enhancement of the tristable energy harvester system under low-frequency excitation is achieved, thereby realizing the efficient energy harvesting of the tristable energy harvester under low-frequency excitation.

[0073] Beneficial effects

[0074] 1. The present invention discloses a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance. A standard rectifier circuit is selected as the collection circuit and connected to the energy harvester to convert the AC power collected by the tristable energy harvester system from the low-frequency excitation into stable DC power, so as to realize the tristable energy harvester as a stable DC power supply for electronic equipment.

[0075] 2. This invention discloses a performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance. Based on the variable separation method, the slow variable equation of the dimensionless electromechanical coupling model of the tristable energy harvester system is obtained. The piecewise non-smooth piezoelectric voltage is smoothed using the fundamental harmonic component of the piezoelectric voltage to obtain the equivalent uncoupled system equation of the slow variable equation. An optimization model of the tristable energy harvester system under low-frequency excitation is obtained based on vibration resonance. Based on the optimization model of the tristable energy harvester system under low-frequency excitation, the steady-state solution of the optimization model and the analytical expressions for the system rectified voltage and system DC power are derived using the harmonic balance method. Using the rectified voltage, DC power, vibration amplitude, and output response amplitude gain coefficient of the tristable energy harvester system as performance indicators, the influence of the system's stiffness coefficient, electromechanical coupling coefficient, time constant ratio, and the properties of low-frequency forces on the energy harvester's acquisition performance is analyzed. The optimal parameter combination is given to maximize the harvesting performance, thereby enhancing the performance of the tristable energy harvester under low-frequency excitation. Attached Figure Description

[0076] Figure 1 This is a schematic diagram of the structure of a tristable energy trap with electromechanical coupling under dual-frequency harmonic drive;

[0077] Figure 2 It is a graph of the potential function of the tristable energy trap system;

[0078] Figure 3 The system's vibration amplitude A and rectified voltage Y R As the low-frequency force frequency ω changes, where: Figure 3 (a) shows the change in vibration amplitude A. Figure 3 (b) is the rectified voltage Y R The changes;

[0079] Figure 4 This is a graph showing the system's vibration amplitude A and DC power P as a function of the low-frequency force amplitude f, where: Figure 4 (a) shows the change in vibration amplitude A. Figure 4 (b) shows the variation of DC power P;

[0080] Figure 5 The system's vibration amplitude A and rectified voltage Y R A graph showing the variation of the linear stiffness coefficient k1, where: Figure 5 (a) is a graph showing the change in vibration amplitude A. Figure 5 (b) is the rectified voltage Y R The changes;

[0081] Figure 6This describes the changes in the output response amplitude gain coefficient Q and DC power P with respect to the high-frequency force amplitude F and the low-frequency force amplitude f, where: Figure 6 (a) shows the change in the amplitude gain coefficient Q of the system output response. Figure 6 (b) shows the variation of DC power P;

[0082] Figure 7 This is a graph showing the variation of the system output response amplitude gain coefficient Q and DC power P with the high-frequency force amplitude F under different fifth-order stiffness coefficients k3, where: Figure 7 (a) is a graph showing the variation of the system output response amplitude gain coefficient Q. Figure 7 (b) shows the variation of DC power P;

[0083] Figure 8 This is a graph showing the variation of the system's DC power P, random electrical coupling coefficient κ, and time constant ratio α.

[0084] Figure 9 This is a flowchart of the performance optimization method for a tristable energy harvester based on vibration resonance under low-frequency excitation according to the present invention. Detailed Implementation

[0085] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0086] Example 1

[0087] This embodiment describes a strongly nonlinear tristable energy trap connected to a standard rectifier circuit. Its basic structure is shown in the accompanying drawings. Figure 1 A tristable energy-harvesting structure is constructed by intentionally introducing nonlinear magnetic force; that is, magnets of the same specification are placed at the ends of the cantilever beam and on the frame. The three-potential-well function of the system is constructed by changing the placement and pole orientation of the magnets shown in the figure. Furthermore, a double-layer piezoelectric sheet is attached near the clamping end of the cantilever beam and connected to a standard rectifier circuit on the right side via wires to convert AC power collected from the environment into DC power, thereby powering external electronic devices. A relatively large filter capacitor C is used in the standard rectifier circuit. R This enables the rectifier circuit's voltage output Y to be within a basic excitation cycle. R The system maintains a constant output. Therefore, the performance optimization method of the tristable energy harvester based on vibration resonance under low-frequency excitation in this invention can achieve a stable DC output, thus enabling the power supply to electronic devices. Furthermore, in addition to providing a stable DC output to electronic devices, this invention also optimizes the design of the tristable energy harvester under low-frequency excitation based on vibration resonance, conducts a detailed analysis of the acquisition performance of the tristable energy harvesting system, and provides the optimal parameter combination, offering important reference for the optimized design of the tristable energy harvester.

[0088] like Figure 9As shown in the figure, the performance optimization method of the tristable energy harvester based on the vibration resonance mechanism under low-frequency excitation disclosed in this embodiment is implemented in the following specific steps:

[0089] Step 1: Establish an electromechanical coupling system model of a tristable energy harvester connected to a standard rectifier circuit under dual-frequency harmonic excitation. Based on... Figure 1 The schematic diagram of the tristable energy trap structure shown is provided, and the mathematical model of the system is established as shown in formulas (1)-(2). For ease of analysis, the system model is dimensionless, and the specific expressions are shown in formulas (4)-(6). In this embodiment, based on the length and thickness of the cantilever beam, the placement of the magnets, and the specifications of the piezoelectric sheet, the dimensionless parameters are obtained as follows: β=0.1, κ=0.3, α=0.05, k1=1.2, k2=-4.2, k3=3.2, λ=100. At the same time, considering the vibration source properties in the environmental excitation, a specified noise excitation, namely dual-frequency harmonic excitation, is applied to the base in this embodiment. The initial noise information is given as follows: f=0.1, ω=0.1, F=4, Ω=30.

[0090] Step 2: Based on the variable separation method, the slow variable equation of the tristable energy harvesting system is derived; the fundamental harmonic component of the piezoelectric voltage is used to approximate the segmented voltage, and combined with the generalized harmonic technology and the dimensionless model of the system, the equivalent uncoupled system model of the optimized tristable energy harvesting system is derived, and the specific expressions are shown in formulas (7)-(18).

[0091] Step 3: For the optimized equivalent uncoupled system model (17), the steady-state solution of the system (17) and the analytical expressions of the system rectified voltage and system DC power are derived using the harmonic balance method. In order to enhance the acquisition performance of the tristable energy harvester under low-frequency excitation, the rectified voltage of the energy harvesting system (Formula 24), the DC power of the system (Formula 25), the system vibration amplitude, and the gain coefficient of the system output response amplitude (Formula 26) are used as performance indicators. The effects of the system stiffness coefficient, time constant ratio, electromechanical coupling coefficient, and the properties of low-frequency force on the acquisition performance of the energy harvester are analyzed in detail to achieve the purpose of enhancing the energy harvesting capability.

[0092] Figure 2 This describes the shape of the potential function of the tristable energy-harvesting system. For example... Figure 2 As shown, the potential function of the system has a definition of X. s1 ,X s2 ,X s3 The three stable points are defined as X. u1 ,X u2 The two unstable saddle points. When the external excitation level meets certain requirements, the magnet at the end of the cantilever beam can oscillate back and forth in three potential wells, exhibiting high energy harvesting potential. Figure 3 The system vibration amplitude A and the system rectified voltage Y are... R The curve shows the frequency change of low-frequency force. For example... Figure 3 As shown in (a), the system's vibration amplitude response curve bends to the left, exhibiting softening nonlinearity. This bend implies the existence of a certain low-frequency force range where the energy harvesting system has a non-unique solution. Figure 3 As shown in (b), the rectified voltage of the tristable energy harvesting system exhibits the same characteristics as the low-frequency frequency. Figure 3 Similar properties as shown in (a). Figure 3 There are five coexisting branches, defined as B1, B2, B3, B4, and B5. B2 and B4 are unstable branches, B1 corresponds to the low-energy non-resonant motion branch within the well, B5 represents the high-energy vibration branch between the three potential wells, and B3 corresponds to the branch of large orbital periodic motion. The branches collide with each other, and their intersection point is denoted as s. i (i = 1…4). At intersection point s i At this point, a jumping phenomenon can usually be observed. For example, when the low-frequency force frequency ω decreases from large to small, the magnet at the end of the cantilever beam first moves along branch B5, jumps up to branch B3 at point s4, and as the frequency ω continues to decrease, the magnet will jump down to branch B1 at point s2. This means that even at lower frequencies, the tristable energy trap can still have a high output level.

[0093] Figure 4 The vibration amplitude and DC power of the tristable energy harvesting system were further investigated as a function of the low-frequency force amplitude f. Figure 4 The vibration amplitude of the tristable energy harvesting system shown in (a) and as shown in (a) Figure 4 (b) shows that the DC power of the system exhibits a jump phenomenon in the intervals [f1,f2] and [f3,f4], meaning that within these intervals, the system can break through the potential barrier and undergo inter-well oscillations with higher energy, thereby achieving higher DC power. Figure 5 As shown, the vibration amplitude and rectified voltage of the system exhibit hardening nonlinearity with respect to the linear stiffness coefficient. The existence of the interval [c0, c1] results in a non-unique solution for the system response. Figure 5 The rectified voltage of the system shown in (a) is similar to that shown in (a). Figure 5 (b) shows that the system vibration amplitude has a positive correlation, that is, when the system vibration amplitude is large, the rectified voltage obtained is also large.

[0094] The output response amplitude gain coefficient Q is an important quantitative indicator of vibration resonance. For example... Figure 6 As shown, the output response amplitude gain coefficient Q and the system DC power P of the tristable energy harvesting system vary with the high-frequency force amplitude F and the low-frequency force amplitude f. Figure 6As shown in (a), when the low-frequency force amplitude f is small (f < 0.08 in the figure), the output response amplitude gain coefficient Q has two peaks as the high-frequency force amplitude F increases, which means that the system can experience two vibration resonance phenomena; however, as the low-frequency force amplitude f increases, the two peaks become one peak, and the peak value decreases, while the high-frequency force amplitude required to reach the peak value decreases slightly. Figure 6 As shown in (b), when the system experiences vibration resonance (as shown in the figure, F = 4.7, f = 0.136), the harvested DC power is significantly higher than when vibration resonance does not occur (as shown in the figure, F = 2.3, f = 0.136). Based on this property, optimizing the performance of a tristable energy harvester under low-frequency excitation based on vibration resonance has significant practical implications.

[0095] like Figure 7 As shown, the output response amplitude gain Q and system DC power P of the tristable energy harvesting system vary with the high-frequency force amplitude F under different fifth-order stiffness coefficients k3. When k3 = 2.2, as... Figure 7 (b) shows that the DC power P of the system exhibits two jumps with the increase of the high-frequency force amplitude F. When k3 is small, for example, k3 = 1.0, only one jump occurs with the increase of the high-frequency force amplitude F, while for larger fifth-order stiffness coefficients (such as...), the system exhibits a different phenomenon. Figure 7 (b) In k3 = 3.0), the jumping phenomenon disappears. As the high-frequency force amplitude F increases, such as Figure 7 (a) shows the system output response magnitude gain Q and as shown in the figure. Figure 7 (b) shows that the DC power P of the system exhibits a single peak, and at the peak value ( Figure 7 When vibration resonance occurs at F=5, the tristable energy harvesting system will have high output performance. Figure 8 Further investigation is needed into the variation of the system's DC power P with the time constant ratio α and the electromechanical coupling coefficient κ. For example... Figure 8 As shown, the DC power P exhibits a jump phenomenon with the increase of the time constant ratio α, reaching a local optimum at point P1. Conversely, the DC power P initially increases and then slightly decreases with the increase of the electromechanical coupling coefficient κ, reaching a local optimum at point P3. Therefore, there exists an optimal time constant ratio or electromechanical coupling coefficient that maximizes the system's output DC voltage.

[0096] In summary, the performance optimization method of the tristable energy harvester based on vibration resonance under low-frequency excitation, as proposed in this invention, can achieve stable DC output of the tristable energy harvester and enhance the energy harvesting performance of the system under low-frequency excitation. By using the rectified voltage, DC power, vibration amplitude, and output response amplitude gain coefficient of the tristable energy harvester as performance indicators, the influence of stiffness coefficient, time constant ratio, electromechanical coupling coefficient, and the properties of low-frequency forces on the energy harvester's acquisition performance is analyzed in detail. The parameter combination that maximizes the performance of the tristable energy harvester is given, thus optimizing the tristable energy harvester.

[0097] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A performance optimization method for a tristable energy harvester under low-frequency excitation based on vibration resonance, characterized in that: Includes the following steps, Step 1: Select a standard rectifier circuit as the collection circuit and connect it to the tristable energy harvester to convert the AC power collected from the low-frequency excitation of the tristable energy harvester into stable DC power, so as to realize the stable DC output of the energy harvester and establish a strongly nonlinear system model of the electromechanical coupled tristable energy harvester; further, dimensionless processing is performed on the strongly nonlinear system model of the electromechanical coupled tristable energy harvester to obtain the dimensionless model of the tristable energy harvester. Step 2: Based on the variable separation method, derive the fast variable equations and slow variable equations of the dimensionless model of the tristable energy harvesting system; to address the piecewise nonsmooth characteristics caused by the standard rectifier circuit, use the fundamental harmonic component of the piezoelectric voltage to smooth the piecewise nonsmooth piezoelectric voltage, thereby obtaining the equivalent uncoupled system equations of the dimensionless model of the tristable energy harvesting system; and based on vibration resonance, obtain the optimized model of the tristable energy harvesting system under low-frequency excitation. Step 3: For the equivalent uncoupled optimization model of the tristable energy harvesting system described in Step 2 under low-frequency excitation, the steady-state solution of the system and the analytical expressions for the system rectified voltage and system DC power are derived using the harmonic balance method. Furthermore, using the output response amplitude gain coefficient, vibration amplitude, rectified voltage, and DC power of the tristable energy harvester as performance indicators, the influence of system parameters including stiffness coefficient, time constant ratio, electromechanical coupling coefficient, as well as the amplitude and frequency of environmental excitation on the energy harvester's acquisition performance is analyzed. The optimal parameter combination that maximizes the performance indicators is given. Based on the optimal parameter combination, the performance of the tristable energy harvesting system under low-frequency excitation is enhanced, thereby achieving efficient energy harvesting of the tristable energy harvester under low-frequency excitation.

2. The performance optimization method for a tristable energy harvester based on vibration resonance under low-frequency excitation as described in claim 1, characterized in that: The implementation method for step one is as follows: To obtain a stable DC output from the energy harvesting system, a standard rectifier circuit is considered as a nonlinear interface circuit, and harmonic excitation is used to simulate environmental excitation. A strongly nonlinear dynamic model of the electromechanical coupled tristable energy harvesting system is established: In the formula, Represents the equivalent mass of the end magnet. Defined as the displacement at the end of the cantilever beam. Indicates time, Represents the air damping coefficient. This indicates the voltage across the piezoelectric element. Represents the electromechanical coupling coefficient. This indicates the harmonic excitation applied to the base. This indicates the internal capacitance of the piezoelectric element. This indicates the current flowing into the rectifier circuit: in, This represents the filter capacitor connected in parallel with the load resistor R; This indicates the stable DC voltage output by the rectifier circuit. The three-well function of the system has the following form: In the formula, , and These represent the linear, cubic, and quintic stiffness coefficients of the energy harvesting system, respectively. Introducing dimensionless transformation , , Substituting into equation (1), we obtain the dimensionless electromechanical coupling model of the tristable energy harvesting system, which is expressed as: in, This represents the natural frequency of the metal cantilever beam of the energy trap. Let X be the length of the beam, and let X represent the dimensionless displacement. This represents the dimensionless base excitation; The dimensionless piezoelectric voltage; This represents the dimensionless rectified voltage. Represents dimensionless current. This indicates the ratio of the filter capacitor to the internal capacitance of the piezoelectric element; , , These are the dimensionless damping coefficient, the time constant ratio, and the electromechanical coupling coefficient, respectively. Let k1 be the dimensionless tristable state function of the system; and These represent the dimensionless linear stiffness coefficient, cubic stiffness coefficient, and quintic stiffness coefficient, respectively. Their values ​​are related to the distance between the magnet at the end of the cantilever beam and the fixed magnet on the frame, directly reflecting the shape of the potential function of the tristable energy-harvesting system and the strength of the geometric nonlinearity of the system. For the dimensionless base excitation in formula (4), the simulation is performed using harmonic excitation with two different frequencies as shown in formula (6), namely: in, Represents low-frequency harmonic excitation in the environment, f and These are the amplitude and frequency of the low-frequency force, respectively. F and represent high-frequency harmonic excitation in the environment. These represent the amplitude and frequency of the high-frequency force, respectively; the low-frequency harmonic excitation and the high-frequency harmonic excitation satisfy the following condition. , .

3. The performance optimization method for a tristable energy harvester based on vibration resonance under low-frequency excitation as described in claim 2, characterized in that: The second step is implemented as follows: According to the method of separation of variables, the formula (4) has an approximate solution of the following form: In the formula, It is the output response that has a time scale Low-frequency, slow variables It is a rapidly changing component with a mean of zero, that is, ;symbol This represents a time average over a shorter period of time, where For a shorter time period, substitute equation (7) into the dimensionless electromechanical coupling model (4) of the tristable energy harvesting system to obtain the slow component in the output response X(t) of the tristable energy harvesting system. Turn into points as soon as possible The equation is: To solve equation (9), we define it as follows: In addition, the first derivative of the fast variable with respect to time is calculated according to equation (10). and second derivative : Among them, B 11 B 12 These are the coefficients to be determined; a simple calculation of formula (10) yields the relevant fast variables. The expression, that is: Substituting formula (12) into formula (9), and further simplifying and rearranging according to formulas (10) and (11) based on the same harmonic order, we obtain the following about B. 11 B 12 The system of equations: For taking the smaller B 11 B 12 Formula (13) is approximately: Obviously, B 11 B 12 Solving from formula (14), we get: In the following description It is denoted as W, that is ; Substituting equations (15), (10), and (12) into equation (8), the equation concerning the slow variable x(t) is transformed into: Due to the alternating conduction and blocking of the rectifier bridge in the standard rectifier circuit, the piezoelectric voltage in formula (4) has piecewise non-smooth properties. The fundamental harmonic component of the piezoelectric voltage is used to approximate it, and combined with the generalized harmonic transformation and the dimensionless equation of the tristable energy-harvesting system, the equivalent uncoupled optimization model is obtained as shown in formula (17): In the formula, in, The equivalent damping coefficient of the optimized tristable energy trap is represented by the rectifier circuit, which has a resistive damping effect on the original system (4). This represents the optimized equivalent linear stiffness coefficient, which depends not only on the stiffness coefficient of the original system and the properties of the high-frequency force, but also on the rectifier circuit. The optimized cubic stiffness coefficient mainly depends on the original system's cubic and quintic stiffness coefficients and the properties of high-frequency forces; additionally, This represents the fifth-order stiffness coefficient after optimization, which is consistent with the fifth-order stiffness coefficient of the original system. Indicates the blocking angle of the diode rectifier bridge; Based on the variable separation method, the fast variable equations and slow variable equations of the dimensionless electromechanical coupling model of the tristable energy harvesting system are derived. To address the piecewise nonsmooth characteristics caused by the standard rectifier circuit, the fundamental harmonic component of the piezoelectric voltage is used to smooth the piecewise nonsmooth piezoelectric voltage, thereby obtaining the equivalent uncoupled system equations of the dimensionless electromechanical coupling model of the tristable energy harvesting system. That is, the optimized model of the tristable energy harvesting system under low-frequency excitation is obtained based on vibration resonance.

4. The performance optimization method for a tristable energy harvester based on vibration resonance under low-frequency excitation as described in claim 3, characterized in that: The method for implementing step three is as follows: Using the rectified output voltage of the tristable energy harvesting system shown in Equation (24), the DC power of the system shown in Equation (25), and the amplitude gain coefficient of the system output response shown in Equation (26) as performance indicators, the power generation performance of the tristable energy harvesting system under different parameter combinations is measured. The influence rules of the system parameters on the dynamic behavior and harvesting performance of the energy harvesting system are analyzed, and the optimal parameter combination that maximizes the performance indicators is selected. The steady-state approximate solution of the equivalent uncoupled system equation (17) after optimization is: in, The vibration amplitude of the system (17) is represented by the harmonic balance method. Formula (19) and its first and second derivatives are substituted into the system (17), and the results are arranged according to the same harmonic. When the tristable energy-harvesting system reaches steady state, the time derivatives of all quantities are zero. In addition, the high-frequency force The frequency is high enough, that is Therefore, the steady-state amplitudes b1 and b2 are derived as follows: The steady-state solution of system (17) is obtained by formulas (20) and (21), wherein the vibration amplitude of system (17) is... ; In addition, the blocking angle of the rectifier bridge in the rectifier circuit With rectified voltage Y R The following relationship exists between them: Furthermore, based on Kirchhoff's current law, within half a cycle, the total charge flowing out of the energy trap is equal to the total charge flowing through the resistor, that is: Combining formulas (5), (22), and (23), the rectified voltage Y output by the tristable energy harvesting system is derived. R The DC power P is as follows: The output response amplitude gain coefficient Q of the tristable energy harvesting system is an important quantitative indicator describing the vibration resonance mechanism, and has the following characteristics: Where f is the amplitude of the low-frequency harmonic excitation, and Q s and Q c These are the system (17) relative to the low-frequency input. The sine and cosine components are: Where n represents a positive integer, and x(t) is the output displacement of the optimized tristable energy harvesting system; Based on the optimized tristable energy harvesting system (17), the output response amplitude gain coefficient, system vibration amplitude, system rectified voltage (as shown in formula (24), and system DC power (as shown in formula (25)) of the tristable energy harvesting system are used as performance indicators. The influence of the system parameters, including stiffness coefficient, time constant ratio, electromechanical coupling coefficient, environmental excitation amplitude, and frequency, on the energy harvester's acquisition performance is analyzed. The optimal parameter combination that maximizes the performance indicators is given. Based on the optimal parameter combination, the performance enhancement of the tristable energy harvesting system under low-frequency excitation is achieved, thereby realizing the efficient energy harvesting of the tristable energy harvester under low-frequency excitation.

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