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CN115566932A - Performance Optimization Method of Tristable Energy Harvester Under Low Frequency Excitation Based on Vibration Resonance - Google Patents

Performance Optimization Method of Tristable Energy Harvester Under Low Frequency Excitation Based on Vibration Resonance Download PDF

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CN115566932A
CN115566932A CN202211360055.4A CN202211360055A CN115566932A CN 115566932 A CN115566932 A CN 115566932A CN 202211360055 A CN202211360055 A CN 202211360055A CN 115566932 A CN115566932 A CN 115566932A
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靳艳飞
张婷婷
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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Abstract

The invention discloses a performance optimization method of a three-stable-state energy harvester under low-frequency excitation based on vibration resonance, and belongs to the field of energy harvester optimization. The implementation method of the invention comprises the following steps: selecting a standard rectifying circuit as a nonlinear collecting circuit to be connected with the energy harvester so as to realize stable direct current output of the energy harvester and establish a strong nonlinear system model of the electromechanical coupling tristable energy harvesting system; deriving a slow variable equation of the three-stable-state energy harvesting system by adopting a variable separation method; deriving an analytical expression of steady-state solution and direct-current power of the three-steady-state energy harvesting system by combining a harmonic balance method; obtaining an optimized model of the three-stable-state energy harvesting system under low-frequency excitation based on vibration resonance; and analyzing the influence of the low-frequency excitation amplitude, the low-frequency excitation frequency, the stiffness coefficient, the time constant ratio and the electromechanical coupling coefficient on the collection performance of the three-stable-state energy harvesting system by using an optimization model, and optimizing and selecting an optimal combination so as to realize the high-efficiency energy harvesting of the three-stable-state energy harvesting system under the low-frequency excitation.

Description

Performance optimization method of three-stable-state energy harvester under low-frequency excitation based on vibration resonance
Technical Field
The invention relates to a performance optimization method of a three-stable-state energy harvester, relates to a performance optimization method of the three-stable-state energy harvester under low-frequency excitation based on vibration resonance, and belongs to the field of energy harvester optimization.
Background
With the rapid development of microelectronic technology, miniature low-power electronic devices for wireless sensors, environmental control systems, medical implants, wearable devices, and the like are in widespread use. A large number of miniature sensors, microprocessors, low-power transmission modules and other small-sized low-power electronic devices are fully developed in various related industries. For such microelectronic devices, the conventional power supply method is a chemical battery, but the battery has low power density, large volume, difficult material degradation, short life span, and time and labor consuming replacement, which seriously hinders the further development of such devices. The energy harvester has the capability of converting ubiquitous environmental vibration energy into electric energy, so that autonomous power supply for a long-term to infinite life cycle for the microelectronic equipment becomes possible, and the development of the microelectronic technology is greatly promoted.
Energy harvesters achieve the goal of powering microelectronic devices by utilizing the capabilities of active materials (such as piezoelectric, magnetostrictive, and ferroelectric) and electromechanical coupling mechanisms (such as electrostatic and electromagnetic) to generate electrical potentials in response to mechanical stimuli and external vibrations, and by interfacing with appropriate interface circuitry, convert environmental vibrations into direct current that can be used by external electronic devices. Currently, the types of energy harvesters can be classified into electromagnetic, piezoelectric, electrostatic, and hybrid types. Among them, the piezoelectric energy harvester has attracted attention because of its advantages such as simple structure, high power density and good expandability. The traditional vibration energy harvester is based on the fundamental principle of linear resonance, has a very narrow steady-state frequency bandwidth, so that the collection efficiency of the traditional vibration energy harvester in a real environment is very low, the applicability and the practicability of the traditional vibration energy harvester are severely limited, and the energy is difficult to efficiently collect from a vibration energy source with a wide frequency spectrum. To address the issue of bandwidth limitations, nonlinear energy harvesters, obtained by the deliberate introduction of nonlinear magnetic force configurations, have attracted considerable attention. There have been proposed many non-linear energy harvesting structures such as monostable, bistable and tristable strong non-linear structures. The literature, "Dynamics of a coupled nonlinear energy generation under calibrated noise and transient energies, international Journal of Mechanical Sciences 2020, 172" contrasts and researches the properties of monostable, bistable and tristable energy harvesters, and discloses that the collection performance of the tristable energy harvesters is superior to that of monostable and bistable energy harvesters, and higher collection power and conversion efficiency can be obtained. The literature, "Harmonic balance analysis of nonlinear energy generators for performance enhancement, journal of Sound and visualization 2016, 373" also discloses that a tri-stable energy harvester is capable of achieving broadband energy harvesting with higher harvesting potential, and is particularly useful for improving the energy harvesting effect under low-frequency environmental excitation. At present, many documents show that the tri-stable state energy harvester can produce higher output power than the mono-stable state and bi-stable state energy harvesters, which means that research on the tri-stable state energy harvester is of great significance for improving energy harvesting efficiency in a low-frequency environment. However, the current research on the three-stable state energy harvester usually omits the design of the collecting circuit and simplifies the design into a pure resistance circuit, which results in that the output of the energy harvester is a large-voltage and small-current alternating current which cannot directly supply power to the external electronic equipment.
In fact, the energy harvester needs to be connected with a non-linear rectifying circuit to convert the alternating current collected by the energy harvester from the environment into direct current for the external electronic equipment to use. The nonlinear collecting circuit commonly used at present comprises a standard rectifying circuit, a synchronous charge extracting circuit, a synchronous switch energy capture and the like. However, the introduction of the nonlinear collecting circuit will bring complex mutual coupling behavior to the energy harvesting system, which brings new challenges to the dynamics analysis and performance analysis of the energy harvesters. In addition, for energy harvesting systems with complex nonlinear rectification circuits, the nonlinearity introduced by the circuit is often estimated or even ignored to simplify its difficulties for theoretical analysis. Only a few researchers have conducted intensive research on the performance of the multi-stable energy harvester connected with the nonlinear rectifying circuit. The document "Statistical qualification of DC power generated by binary stable semiconductor devices drive by random efficiencies, journal of sound and vibration,2019, 442". The high-efficiency capture capability of the three-stable-state energy harvester from a low-frequency environment is proved at present, and meanwhile, the rapid development of the microelectronic technology causes that the energy capture performance of the energy harvester still cannot meet the application requirement. Therefore, how to realize the optimal design of the energy harvester to improve the energy capture 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, and has been used to optimize the structural design of a nonlinear oscillator. The literature, "Novel visual resonance in multistable systems, chaos,2011,21 (3): l 433" investigated the phenomenon of vibrational resonance in multistable systems and indicated that vibrational resonance can be used for the optimized design of nonlinear oscillators to enhance the output response. Therefore, the performance optimization of the three-stable-state energy harvester under low-frequency excitation based on vibration resonance has very important significance.
Disclosure of Invention
In order to solve the problem of sustainable and efficient power supply of electronic equipment, the invention mainly aims to provide a performance optimization method of a three-stable-state energy harvester under low-frequency excitation based on vibration resonance, wherein a standard rectification circuit is selected as a nonlinear collecting circuit to provide stable direct current power supply for the electronic equipment; deducing to obtain a slow variable equation of the tristable energy harvesting system by adopting a variable separation method; deducing a steady-state solution of the three-steady-state energy harvesting system and an analytical expression of direct current power by combining a harmonic balance method; the tri-stable energy harvesting system is optimized under low-frequency excitation based on vibration resonance, so that the high-efficiency energy harvesting of the tri-stable energy harvester under the low-frequency excitation is realized.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a performance optimization method under low-frequency excitation of a three-stable-state energy harvester based on vibration resonance.A standard rectifying circuit is selected as a nonlinear collecting circuit to realize stable direct current output of the energy harvester, and a strong nonlinear system model of an electromechanical coupling three-stable-state energy harvesting system is established; deducing to obtain a slow variable equation of the tristable energy harvesting system by adopting a variable separation method; deriving an analytical expression of steady-state solution and direct-current power of the three-steady-state energy harvesting system by combining a harmonic balance method; obtaining an optimized model of the three-stable-state energy harvesting system under low-frequency excitation based on vibration resonance; and the output response amplitude gain coefficient, the vibration amplitude, the rectified voltage and the direct current power of the three-stable state energy harvesting system are used as performance indexes, the influence of the low-frequency excitation amplitude, the low-frequency excitation frequency, the stiffness coefficient, the time constant ratio and the electromechanical coupling coefficient on the collection performance of the three-stable state energy harvesting system is analyzed by using the optimization model, the optimal combination of the low-frequency excitation amplitude, the low-frequency excitation frequency, the stiffness coefficient, the time constant ratio and the electromechanical coupling coefficient is selected, the performance enhancement of the three-stable state energy harvesting system under the low-frequency excitation is realized, and the efficient energy harvesting of the three-stable state energy harvesting device under the low-frequency excitation is further realized.
The invention discloses a performance optimization method of a three-stable-state energy harvester under low-frequency excitation based on vibration resonance, which comprises the following steps:
selecting a standard rectification circuit as a collecting circuit to be connected with a three-stable-state energy harvester, converting alternating current collected by the three-stable-state energy harvesting system from low-frequency excitation into stable direct current to realize stable direct current output of the energy harvester, and establishing a strong nonlinear system model of the electromechanical coupling three-stable-state energy harvesting system. And further carrying out non-dimensionalization treatment on a strong non-linear system model of the electromechanical coupling three-stable state energy harvesting system to obtain a non-dimensionalized model of the three-stable state energy harvesting system.
In order to obtain stable direct current output of the energy harvesting system, a standard rectification circuit is considered as a nonlinear interface circuit, and harmonic excitation is used for simulating environment excitation, so that a strong nonlinear dynamic model of the electromechanical coupling tristable energy harvesting system is established:
Figure BDA0003921823830000031
wherein M represents the equivalent mass of the end magnet,
Figure BDA0003921823830000032
defined as the displacement of the end of the cantilever beam,
Figure BDA0003921823830000033
which represents time, C represents an air damping coefficient,
Figure BDA0003921823830000034
representing the voltage across the piezoelectric patch,
Figure BDA00039218238300000314
the electro-mechanical coupling coefficient is represented by,
Figure BDA0003921823830000035
representing harmonic excitation applied to the base, C p The internal capacitance of the piezoelectric patch is represented,
Figure BDA0003921823830000036
representing the current flowing into the rectifier circuit:
Figure BDA0003921823830000037
wherein, C R Representing a filter capacitor in parallel with a load resistor R.
Figure BDA0003921823830000038
Indicating a stable dc voltage output by the rectifier circuit.
Figure BDA0003921823830000039
Is the triple well function of the system, having the following form:
Figure BDA00039218238300000310
in the formula,
Figure BDA00039218238300000311
and
Figure BDA00039218238300000312
linear, cubic, and quintic stiffness coefficients of the energy harvesting system are expressed in order.
Introducing dimensionless transformations
Figure BDA00039218238300000313
Substituting the three-stable state energy harvesting system into the formula (1) to obtain a dimensionless electromechanical coupling model of the three-stable state energy harvesting system, wherein the three-stable state energy harvesting system is represented as follows:
Figure BDA0003921823830000041
Figure BDA0003921823830000042
wherein, ω is 0 Representing the natural frequency of the metal cantilever of the energy harvester,
Figure BDA0003921823830000043
x represents the displacement after dimensionless, which is the length scale of the beam.
Figure BDA0003921823830000044
Representing the non-dimensionalized basis excitation.
Figure BDA0003921823830000045
Is a piezoelectric voltage after dimensionless.
Figure BDA0003921823830000046
Representing the rectified voltage after non-dimensionalization,
Figure BDA0003921823830000047
a non-dimensionalized current is represented,
Figure BDA0003921823830000048
the ratio of the filter capacitance to the internal capacitance of the piezoelectric sheet is shown.
Figure BDA0003921823830000049
Respectively, a non-dimensionalized damping coefficient, a time constant ratio and an electromechanical coupling coefficient;
Figure BDA00039218238300000410
a tristable potential function non-dimensionalized for the system;
Figure BDA00039218238300000411
and
Figure BDA00039218238300000412
and the values of the linear stiffness coefficient, the cubic stiffness coefficient and the quintic stiffness coefficient which are nondimensionalized are respectively related to the distance between the magnet at the tail end of the cantilever beam and the fixed magnet on the frame, and the shape of the potential function of the tristable energy harvesting system and the strength of the geometric nonlinearity of the system are directly reflected.
For the dimensionless pedestal excitation in equation (4), the simulation is performed using harmonic excitation with two different frequencies as shown in equation (6), namely:
Figure BDA00039218238300000413
where fsin (ω t) 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 (Ω t) represents high frequency harmonic excitation in the environment, and F and Ω represent the amplitude and frequency of the high frequency force, respectively. The conditions omega > omega and f < 1 are satisfied between the low-frequency harmonic excitation and the high-frequency harmonic excitation.
Step two, deriving a fast variable equation and a slow variable equation of a dimensionless model of the three-stable-state energy harvesting system based on a variable separation method; aiming at the segmental non-smooth characteristic caused by a standard rectification circuit, smoothing the segmental non-smooth piezoelectric voltage by adopting the fundamental harmonic component of the piezoelectric voltage to obtain an equivalent non-coupled system equation of the dimensionless model of the three-stable state energy harvesting system, and obtaining an optimized model of the three-stable state energy harvesting system under low-frequency excitation based on vibration resonance.
According to the variable separation method, the formula (4) has an approximate solution of the form:
X(t)＝x(t)+Ψ(t) (7)
where x (T) is the time scale T in the output response f Low frequency slow variables of =2 pi/ω, Ψ (t) being a rapidly varying component with a mean value of zero, i.e.,<Ψ(t)>and =0. Symbol(s)
Figure BDA0003921823830000051
Represents the time average over a short time, where T F And =2 pi/omega is a shorter time period. Substituting equation (7) into a dimensionless electromechanical coupling model (4) of the tristable energy harvesting system to respectively obtain equations of a slow-varying component X (t) and a fast-varying component gamma (t) in an output response X (t) of the tristable energy harvesting system, wherein the equations are as follows:
Figure BDA0003921823830000052
Figure BDA0003921823830000053
to solve equation (9), the following definitions are made:
Ψ＝B 11 cos(Ωt)+B 12 sin(Ωt) (10)
in addition, the first derivative of the fast variable with respect to time is calculated according to equation (10)
Figure BDA0003921823830000054
And second derivative
Figure BDA0003921823830000055
Figure BDA0003921823830000056
Wherein, B 11 ，B 12 Is the coefficient to be found. Simple calculation of equation (10) to obtain the fast variable ΨThe expressions, namely:
Figure BDA0003921823830000057
substituting the formula (12) into the formula (9), further arranging according to the formula (10) and the formula (11) according to the same harmonic, and simplifying and sorting to obtain the B 11 ，B 12 The system of equations of (1):
Figure BDA0003921823830000061
for taking smaller B 11 、B 12 Equation (13) is approximated as:
Figure BDA0003921823830000062
obviously, B 11 、B 12 Solved by equation (14), namely:
Figure BDA0003921823830000063
in the following description, k will be 1 +3k 3 x 2 +5k 5 x 4 -Ω 2 Denoted as W, i.e. W = k 1 +3k 3 x 2 +5k 5 x 4 -Ω 2 。
Substituting equation (15), equation (10), and equation (12) into equation (8) converts the equation for the slow variable x (t) to:
Figure BDA0003921823830000064
the piezoelectric voltage of said formula (4) has a piecewise non-smooth nature due to the alternating conduction and blocking action of the rectifier bridge in a standard rectifier circuit. The piezoelectric voltage is approximately replaced by the fundamental harmonic component, and an equivalent non-coupling optimization model is obtained by combining generalized harmonic transformation and a three-stable state energy harvesting system dimensionless equation as shown in the formula (17):
Figure BDA0003921823830000065
in the formula,
Figure BDA0003921823830000066
wherein,
Figure BDA0003921823830000067
and the equivalent damping coefficient of the optimized three-stable-state energy harvester is represented, and the rectifying circuit has an electric damping effect on the original system (4).
Figure BDA0003921823830000068
And the optimized equivalent linear stiffness coefficient is expressed, and not only depends on the stiffness coefficient of the original system and the property of high-frequency force, but also is related to a rectifying circuit.
Figure BDA0003921823830000069
The cubic stiffness coefficient after optimization is expressed and mainly depends on the cubic stiffness coefficient and the quintic stiffness coefficient of the original system and the property of high-frequency force. In addition, the first and second substrates are,
Figure BDA0003921823830000071
and (4) representing the quintic stiffness coefficient after the optimization design, which is consistent with the quintic stiffness coefficient of the original system. Theta represents the diode rectifier bridge blocking angle.
Deriving a fast variable equation and a slow variable equation of a dimensionless electromechanical coupling model of the three-stable state energy harvesting system based on a variable separation method; aiming at the segmented non-smooth characteristic caused by a standard rectification circuit, smoothing the segmented non-smooth piezoelectric voltage by adopting the fundamental harmonic component of the piezoelectric voltage to obtain an equivalent non-coupled system equation of a dimensionless electromechanical coupling model of the three-stable state energy harvesting system, and obtaining an optimized model of the three-stable state energy harvesting system under low-frequency excitation based on vibration resonance.
Step three: and deducing to obtain an analytic expression of a steady-state solution of the system, a system rectification voltage and a system direct current power by using a harmonic balance method aiming at the equivalent non-coupling optimization model of the three-steady-state energy harvesting system under low-frequency excitation in the second step. Further, the output response amplitude gain coefficient, the vibration amplitude, the rectification voltage and the direct current power of the three-stable-state energy harvester are used as performance indexes, the system parameters including the rigidity coefficient, the time constant ratio and the electromechanical coupling coefficient, the influence of the environment excitation amplitude and the frequency on the acquisition performance of the energy harvester are analyzed, the optimal parameter combination when the performance indexes reach the maximum is given, the performance enhancement of the three-stable-state energy harvesting system under the low-frequency excitation is realized according to the optimal parameter combination, and the efficient energy harvesting of the three-stable-state energy harvester under the low-frequency excitation is further realized.
The method comprises the steps of measuring the power generation performance of the tri-stable state energy harvesting system under different parameter combinations by using output rectified voltage of the tri-stable state energy harvesting system shown in a formula (24), direct current power of the system shown in a formula (25) and output response amplitude gain coefficient of the system shown in a formula (26) as performance indexes, analyzing influence rules of system parameters on the dynamic behavior and the collection performance of the energy harvesting system, and selecting an optimal parameter combination for maximizing the performance indexes.
The steady-state approximate solution of the equivalent uncoupled system equation (17) after optimization design is:
x(t)＝b 1 cos(ωt)+b 2 sin(ωt) (19)
wherein,
Figure BDA0003921823830000072
representing the amplitude of the vibrations of the system (17). The harmonic balance method is adopted, the formula (19) and the first-order and second-order derivatives thereof are substituted into the system (17), and the results are arranged according to the same harmonic. When the tri-stable energy harvesting system reaches a steady state, all quantities are zero with respect to time derivative. In addition, the frequency of the high frequency force Fcos (Ω t) is sufficiently large, i.e., W ≈ Ω 2 . Therefore, the steady-state amplitude b 1 、b 2 The derivation yields:
Figure BDA0003921823830000073
Figure BDA0003921823830000074
obtaining a steady state solution of the system (17) by equations (20) (21), wherein the amplitude of the vibration of the system (17) is
Figure BDA0003921823830000075
In addition, the blocking angle theta of the rectifier bridge in the rectifier circuit and the rectified voltage Y R Satisfies the following relationship:
A(cosθ-1)＝-2Y R (22)
further based on kirchhoff's current law, the total charge that flows from the energy harvester is equal to the total charge that flows through the resistor within a half-cycle, i.e.:
Figure BDA0003921823830000081
and (3) combining the formulas (5), (22) and (23) to derive the rectified voltage Y output by the three-stable-state energy harvesting system R And the DC power P is respectively:
Figure BDA0003921823830000082
Figure BDA0003921823830000083
the output response amplitude gain coefficient Q of the three-stable state energy harvesting system is an important quantization index for describing a vibration resonance mechanism, and comprises the following steps:
Figure BDA0003921823830000084
where f is the amplitude of the low frequency harmonic excitation, Q s And Q c The sine and cosine components of the system (17) with respect to the low frequency input fcos (ω t), respectively, are:
Figure BDA0003921823830000085
Figure BDA0003921823830000086
wherein n represents a positive integer, and x (t) is the output displacement of the tristable energy harvesting system after the optimal design.
According to the three-stable-state energy harvester system (17) after optimized design, the output response amplitude gain coefficient and the system vibration amplitude of the three-stable-state energy harvesting system shown in a formula (26), the system rectification voltage shown in the formula (24) and the system direct current power shown in the formula (25) are used as performance indexes, the influence of system parameters including the rigidity coefficient, the time constant ratio and the electromechanical coupling coefficient, the environment excitation amplitude and the environment excitation frequency on the energy harvester acquisition performance is analyzed, the optimal parameter combination when the performance indexes reach the maximum is given, the performance enhancement of the three-stable-state energy harvesting system under the low-frequency excitation is realized according to the optimal parameter combination, and the efficient energy harvesting of the three-stable-state energy harvester under the low-frequency excitation is further realized.
Advantageous effects
1. The invention discloses a performance optimization method of a three-stable-state energy harvester based on vibration resonance under low-frequency excitation.
2. The invention discloses a performance optimization method of a three-stable-state energy harvester under low-frequency excitation based on vibration resonance, which comprises the steps of obtaining a slow variable equation of a dimensionless electromechanical coupling model of a three-stable-state energy harvesting system based on a variable separation method, smoothing segmented non-smooth piezoelectric voltage by adopting fundamental wave components of the piezoelectric voltage to obtain an equivalent non-coupled system equation of the slow variable equation, and obtaining an optimization model of the three-stable-state energy harvesting system under low-frequency excitation based on vibration resonance; based on an optimization model of the three-stable-state energy harvesting system under low-frequency excitation, an analytic expression of a stable-state solution of the optimization model, system rectification voltage and system direct-current power is derived by using a harmonic balance method. And by taking the rectified voltage of the three-stable-state energy harvesting system, the direct current power of the system, the vibration amplitude of the system and the output response amplitude gain coefficient of the system as performance indexes, analyzing the influence of the rigidity coefficient, the electromechanical coupling coefficient, the time constant ratio and the low-frequency force of the system on the acquisition performance of the energy harvester, and giving an optimal parameter combination to maximize the acquisition performance, the performance enhancement of the three-stable-state energy harvester under the low-frequency excitation is realized.
Drawings
FIG. 1 is a schematic diagram of a structure diagram of an electromechanically coupled tristable energy harvester under dual-frequency harmonic drive;
FIG. 2 is a graph of a potential function of a tristable energy harvester system;
FIG. 3 shows the amplitude A of the vibration of the system and the rectified voltage Y R As a function of the low frequency force frequency ω, wherein: FIG. 3 (a) shows the variation of the vibration amplitude A, and FIG. 3 (b) shows the rectified voltage Y R (ii) a change in (c);
fig. 4 is a graph of the vibration amplitude a and dc power P of the system as a function of the low frequency force amplitude f, where: fig. 4 (a) shows the variation of the vibration amplitude a, and fig. 4 (b) shows the variation of the dc power P;
FIG. 5 shows the amplitude A of vibration and the rectified voltage Y of the system R Coefficient of linear stiffness k 1 A graph of variation, wherein: FIG. 5 (a) is a graph showing the change in the vibration amplitude A, and FIG. 5 (b) is a graph showing the rectified voltage Y R (ii) a change in (d);
fig. 6 is a variation of the coefficient output response amplitude gain coefficient Q and the dc power P with the high frequency force amplitude F and the low frequency force amplitude F, where: fig. 6 (a) shows the variation of the gain factor Q of the system output response amplitude, and fig. 6 (b) shows the variation of the dc power P;
FIG. 7 shows the system output response amplitude gain coefficient Q and DC power P at different quintic stiffness coefficient k 3 The following graph of the variation with high frequency force amplitude F, wherein: FIG. 7 (a) is a graph showing the variation of the gain factor Q of the system output response amplitude, and FIG. 7 (b) is a graph showing the variation of the DC power P;
FIG. 8 is a graph of the variation of the system DC power P random electrical coupling coefficient κ and the time constant ratio α;
FIG. 9 is a flowchart of a performance optimization method under low-frequency excitation of a three-stable-state energy harvester based on vibration resonance according to the invention.
Detailed Description
The present invention will be described in detail below with reference to the drawings and examples.
Example 1
The object of the embodiment is a strong nonlinear tristable energy harvester connected with a standard rectifying circuit, and the basic structure of the strong nonlinear tristable energy harvester is shown in figure 1 in the attached drawing of the specification. The three-stable energy capturing structure is constructed by intentionally introducing nonlinear magnetic force, namely magnets with the same specification are arranged at the tail end of the cantilever beam and on the frame. The three-potential well function of the system is constructed by changing the arrangement position and the magnetic pole direction of the magnets in the figure. In addition, the double-layer piezoelectric sheet is attached to the position near the clamping end of the cantilever beam and connected with the standard rectifying circuit on the right side through a wire, and alternating current collected from the environment is converted into direct current to achieve the purpose of supplying power for external electronic equipment. In a standard rectifier circuit, a relatively large filter capacitor C is used R Which enables the voltage output Y of the rectifier circuit during a basic drive cycle R And remain constant. Therefore, the performance optimization method under the low-frequency excitation of the three-stable-state energy harvester based on the vibration resonance can realize stable direct current output and achieve the purpose of supplying power to electronic equipment. In addition, besides providing stable direct current output for electronic equipment, the invention also realizes the optimization design of the three-stable state energy harvester under low-frequency excitation based on vibration resonance, develops detailed analysis on the acquisition performance of the three-stable state energy harvester system, provides optimal parameter combination and provides important reference for the optimization design of the three-stable state energy harvester.
As shown in fig. 9, the performance optimization method of the three-stable-state energy harvester based on the vibration resonance mechanism disclosed in this embodiment under low-frequency excitation includes the following specific implementation steps:
step one, establishing an electromechanical coupling system model of a three-stable state energy harvester connected with a standard rectifying circuit under double-frequency harmonic excitation. According to the schematic diagram of the tristable energy harvester structure shown in FIG. 1, a mathematical model of the system is established in equations (1) - (2). For analysis, non-dimensionalization processing is adopted for the system model, and specific expressions are shown in formulas (4) to (6). In this embodiment, the parameters obtained by non-dimensionalization according to the length and thickness of the cantilever, the placement position of the magnet, and the specification of the piezoelectric sheet are: β =0.1, κ =0.3, α =0.05,k 1 ＝1.2，k 2 ＝-4.2，k 3 =3.2, λ =100. Meanwhile, considering the vibration source property in the environment excitation, in this embodiment, a specified noise excitation, i.e. a dual-frequency harmonic excitation, is applied to the pedestal, and the initially given noise information is: f =0.1, ω =0.1, f =4, Ω =30.
Step two, deriving to obtain a slow variable equation of the three-stable-state energy harvesting system based on a variable separation method; the method adopts the fundamental harmonic component of the piezoelectric voltage to approximately replace the segmented voltage, and combines the generalized harmonic technology and the dimensionless model of the system to deduce the equivalent uncoupled system model of the three-stable state energy harvesting system after the optimization design, and the specific expressions are shown in formulas (7) - (18).
And thirdly, deducing and obtaining a steady-state solution of the system (17) and an analytical expression of system rectification voltage and system direct current power by using a harmonic balance method aiming at the equivalent non-coupled system model (17) after the optimization design. For the purpose of enhancing the acquisition performance of the tristable energy harvester under low-frequency excitation, the rectified voltage (formula 24) of the energy harvesting system, the direct current power (formula 25) of the system, the vibration amplitude of the system and the gain coefficient (formula 26) of the output response amplitude of the system are used as performance indexes, the influence of the rigidity coefficient, the time constant ratio, the electromechanical coupling coefficient and the low-frequency force of the system on the acquisition performance of the energy harvester is analyzed in detail, and the purpose of enhancing the energy harvesting capability is achieved.
Fig. 2 is a potential function shape of the tristable energy harvesting system. As shown in FIG. 2, the potential function of the system has a value defined as X s1 ,X s2 ,X s3 Is defined as X u1 ,X u2 Two unstable saddle points. When the external excitation level meets certain requirements, the cantilever beam end magnet can oscillate back and forth in the three potential wells, and the energy collection potential is high. FIG. 3 shows the system vibration amplitude A and the system rectified voltage Y R A curve of variation with frequency of low frequency forces. As shown in fig. 3 (a), the response curve of the system vibration amplitude with frequency curves to the left, showing softening nonlinearity. The bending means that there is a range of low frequency force frequencies that makes the energy harvesting system have a non-unique solution. As shown in fig. 3 (b), the rectified voltage of the tri-stable energy harvesting system has similar properties with the low frequency variation as shown in fig. 3 (a). In FIG. 3, there are five coexisting branches, which are defined as B 1 ,B 2 ,B 3 ,B 4 ,B 5 . Wherein, B 2 ,B 4 Is an unstable branch, B 1 Corresponding to the off-resonant motion branch in the well with lower energy, B 5 Representing the high-energy vibration branch between the three potential wells, B 3 Corresponding to branches of large orbital periodic motion. The branches collide with each other, and the intersection point is denoted as s i (i =1 \ 82304). At the point of intersection s i Where a jump phenomenon is usually observed, for example, when the frequency ω of the low frequency force decreases from large to small, the magnet at the end of the cantilever beam first follows B 5 Branched motion at s 4 Jump up to B at point 3 Branch off, as the frequency ω continues to decrease, the magnet will be at s 2 Point jump down to B 1 And (4) branching movement. This means that the tristate energy harvester is able to have a higher output level even at lower frequencies.
FIG. 4 further studies the variation of the vibration amplitude of the tri-stable energy harvesting system and the DC power of the system with the low-frequency force amplitude f. The vibration amplitude of the tri-stable energy harvesting system shown in FIG. 4 (a) and the DC power of the system shown in FIG. 4 (b) are in the interval [ f 1 ,f 2 ]And [ f 3 ,f 4 ]Appear onThe jump phenomenon is generated, namely the system can break through the constraint of a potential barrier in the interval and perform trap-to-trap vibration with high energy, so that high direct current power is obtained. As shown in fig. 5, the system exhibits stiffness non-linearity with the linear stiffness coefficient for both the vibration amplitude and the system rectified voltage. Range of existence interval [ c 0 ,c 1 ]Such that the system response has a non-unique solution. The system rectified voltage shown in fig. 5 (a) has a positive correlation with the system vibration amplitude shown in fig. 5 (b), that is, when the system vibration amplitude is large, the resulting rectified voltage is also large.
The output response amplitude gain factor Q is an important quantitative indicator of vibration resonance. As shown in fig. 6, the output response amplitude gain coefficient Q and the system dc power P of the tri-stable energy harvesting system vary with the high frequency force amplitude F and the low frequency force amplitude F. As shown in fig. 6 (a), when the low frequency force amplitude F is small (F <0.08 in the figure), the output response amplitude gain factor Q has two peaks with the increase of the high frequency force amplitude F, which means that the system can generate two vibration resonance phenomena; with the increase of the low-frequency force amplitude f, the two peaks become one peak, and simultaneously, the peak value is reduced, and the high-frequency force amplitude required for reaching the peak value is slightly reduced. As shown in fig. 6 (b), when the system is in vibrational resonance (as shown in the figure, F =4.7, F = 0.136), the harvested dc power is much higher than when vibrational resonance is not in place (as shown in the figure, F =2.3, F = 0.136). According to the properties, the performance optimization of the three-stable-state energy harvester under low-frequency excitation based on vibration resonance has important practical significance.
As shown in fig. 7, the three-stable-state energy harvesting system outputs response amplitude gain Q and system direct current power P at different quintic stiffness coefficients k 3 The lower is a function of the high-frequency force amplitude F. When k is 3 When =2.2, the system dc power P shown in fig. 7 (b) appears to jump twice as the high-frequency force amplitude F increases. When k is 3 Smaller, e.g. k 3 =1.0, only one jump occurs with increasing high-frequency force amplitude F, and the stiffness factor is five times greater (e.g. k in fig. 7 (b)) 3 = 3.0), the jumping phenomenon disappears. As the high frequency force amplitude F increases, the system output shown in FIG. 7 (a) soundsThe tri-stable energy harvesting system will have higher output performance due to the single peak of the amplitude gain Q and the system dc power P as shown in fig. 7 (b) and the vibrational resonance at the peak (F =5 in fig. 7). Fig. 8 further studies the variation of the system dc power P with time constant ratio α and the electromechanical coupling coefficient κ. As shown in fig. 8, the dc power P exhibits a jump phenomenon with an increase in the time constant ratio α, and at the point P 1 A local optimum of the dc power is obtained. The direct current power P shows a trend of increasing first and then slightly decreasing with the increase of the electromechanical coupling coefficient k, and a point P is 3 A local optimum is achieved. Therefore, there is an optimal time constant ratio or electromechanical coupling coefficient to maximize the system output dc voltage.
In conclusion, the performance optimization method based on the vibration resonance for the three-stable-state energy harvesting device under the low-frequency excitation can realize stable direct current output of the three-stable-state energy harvesting device and meanwhile realize the enhancement of the capture performance of the energy harvesting system from the low-frequency excitation. By taking the rectified voltage, the system direct current power, the system vibration amplitude and the system output response amplitude gain coefficient of the three-stable-state energy harvesting system as performance indexes, the influence of the properties of the stiffness coefficient, the time constant ratio, the electromechanical coupling coefficient and the low-frequency force on the acquisition performance of the energy harvester is analyzed in detail, the parameter combination when the performance of the three-stable-state energy harvester is maximized is given, and the optimization of the three-stable-state energy harvester is realized.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (4)

1.基于振动共振的三稳态俘能器低频激励下的性能优化方法，其特征在于：包括以下步骤，1. The performance optimization method under the low-frequency excitation of the tristable energy harvester based on vibration resonance, it is characterized in that: comprise the following steps, 步骤一、选择标准整流电路作为收集电路与三稳态俘能器相连，将所述三稳态俘能系统从低频激励中收集到的交流电转换为稳定的直流电，以实现俘能器稳定的直流输出，并建立机电耦合三稳态俘能系统的强非线性系统模型；进一步针对所述机电耦合三稳态俘能系统的强非线性系统模型采取无量纲化处理，得到三稳态俘能系统的无量纲化模型；Step 1. Select a standard rectifier circuit as the collection circuit and connect it to the tristable energy harvester, and convert the alternating current collected by the tristable energy harvesting system from the low frequency excitation into a stable direct current, so as to realize the stable direct current of the energy harvester. output, and establish a strong nonlinear system model of the electromechanical coupling tristable energy harvesting system; further adopt dimensionless processing for the strongly nonlinear system model of the electromechanical coupling tristable energy harvesting system, and obtain the tristable energy harvesting system The dimensionless model of ; 步骤二、基于变量分离方法推导得到所述三稳态俘能系统的无量纲化模型的快变量方程及慢变量方程；针对标准整流电路引起的分段非光滑特性，采用压电电压的基谐波分量对分段非光滑的压电电压进行平滑处理，得到所述三稳态俘能系统无量纲化模型的等效非耦合系统方程，基于振动共振得到所述三稳态俘能系统在低频激励下的优化模型；Step 2. Deriving the fast variable equation and the slow variable equation of the dimensionless model of the tristable energy harvesting system based on the variable separation method; for the segmental non-smooth characteristics caused by the standard rectifier circuit, the fundamental harmonic of the piezoelectric voltage is used The wave component smoothes the segmented non-smooth piezoelectric voltage to obtain the equivalent uncoupled system equation of the dimensionless model of the tristable energy harvesting system, and obtains the tristable energy harvesting system at low frequencies based on vibration resonance. The optimization model under incentive; 步骤三：针对步骤二所述三稳态俘能系统在低频激励下的等效非耦合优化模型，利用谐波平衡法，推导得到所述系统的稳态解及系统整流电压、系统直流功率的解析表达式；进一步，以三稳态俘能器的输出响应幅值增益系数、振动幅值、整流电压、直流功率作为性能指标，分析包括刚度系数、时间常数比、机电耦合系数的所述系统参数以及环境激励幅值、频率对俘能器采集性能的影响，给出使性能指标达到最大时的最优参数组合，根据所述最优参数组合实现低频激励下三稳态俘能系统的性能增强，进而实现三稳态俘能器在低频激励下的高效俘能。Step 3: For the equivalent non-coupling optimization model of the three-stable energy harvesting system described in step 2 under low-frequency excitation, use the harmonic balance method to derive the steady-state solution of the system and the system rectified voltage and system DC power Analytical expression; further, with the output response amplitude gain coefficient, vibration amplitude, rectified voltage, and DC power of the tristable energy harvester as performance indicators, analyze the described system including stiffness coefficient, time constant ratio, and electromechanical coupling coefficient Parameters and the impact of environmental excitation amplitude and frequency on the energy harvesting performance of the energy harvester, the optimal parameter combination when the performance index reaches the maximum is given, and the performance of the tri-stable energy harvesting system under low-frequency excitation is realized according to the optimal parameter combination Enhanced, and then realize the high-efficiency energy harvesting of the tristable energy harvester under low-frequency excitation. 2.如权利要求1所述的基于振动共振的三稳态俘能器低频激励下的性能优化方法，其特征在于：步骤一实现方法为，2. The performance optimization method under the low-frequency excitation of the tristable energy harvester based on vibration resonance as claimed in claim 1, characterized in that: the implementation method of step 1 is, 为了得到俘能系统的稳定的直流输出，考虑将标准整流电路作为非线性接口电路，并以谐波激励模拟环境激励，建立机电耦合三稳态俘能系统的强非线性动力学模型：In order to obtain a stable DC output of the energy harvesting system, the standard rectifier circuit is considered as a nonlinear interface circuit, and the environmental excitation is simulated by harmonic excitation, and a strong nonlinear dynamic model of the electromechanical coupling tristable energy harvesting system is established:
Figure FDA0003921823820000011
Figure FDA0003921823820000011
 式中，M代表末端磁铁的等效质量，
Figure FDA0003921823820000012
定义为悬臂梁末端的位移，
Figure FDA0003921823820000013
表示时间，C代表空气阻尼系数，
Figure FDA0003921823820000014
表示压电片两端的电压，θp代表机电耦合系数，
Figure FDA0003921823820000015
表示施加给基座的谐波激励，Cp表示压电片的内部电容，
Figure FDA0003921823820000016
表示流入整流电路的电流：In the formula, M represents the equivalent mass of the terminal magnet,
Figure FDA0003921823820000012
Defined as the displacement at the end of the cantilever beam,
Figure FDA0003921823820000013
Indicates the time, C represents the air damping coefficient,
Figure FDA0003921823820000014
Indicates the voltage across the piezoelectric sheet, θ p represents the electromechanical coupling coefficient,
Figure FDA0003921823820000015
Represents the harmonic excitation applied to the base, C p represents the internal capacitance of the piezoelectric sheet,
Figure FDA0003921823820000016
Indicates the current flowing into the rectifier circuit:
Figure FDA0003921823820000021
Figure FDA0003921823820000021
 其中，CR表示与负载电阻R并联的滤波电容；
Figure FDA0003921823820000022
表示整流电路输出的稳定的直流电压；
Figure FDA0003921823820000023
为所述系统的三势阱函数，具有下面的形式：Among them, C R represents the filter capacitor connected in parallel with the load resistance R;
Figure FDA0003921823820000022
Indicates the stable DC voltage output by the rectifier circuit;
Figure FDA0003921823820000023
is the triple potential well function of the system, which has the following form:
Figure FDA0003921823820000024
Figure FDA0003921823820000024
 式中，
Figure FDA0003921823820000025
和
Figure FDA0003921823820000026
依次表示俘能系统的线性、三次和五次刚度系数；In the formula,
Figure FDA0003921823820000025
and
Figure FDA0003921823820000026
In turn, represent the linear, cubic and quintic stiffness coefficients of the energy harvesting system; 引入无量纲变换
Figure FDA0003921823820000027
代入到式(1)中，得到所述三稳态俘能系统的无量纲机电耦合模型，表示为：Introducing dimensionless transformations
Figure FDA0003921823820000027
Substituting into formula (1), the dimensionless electromechanical coupling model of the tristable energy harvesting system is obtained, expressed as:
Figure FDA0003921823820000028
Figure FDA0003921823820000028
Figure FDA0003921823820000029
Figure FDA0003921823820000029
 其中，ω0表示俘能器金属悬臂梁的固有频率，
Figure FDA00039218238200000210
为梁的长度尺度，X表示无量纲化后的位移；
Figure FDA00039218238200000211
表示无量纲化后的基座激励；
Figure FDA00039218238200000212
为无量纲化后的压电电压；
Figure FDA00039218238200000213
表示无量纲化后的整流电压，
Figure FDA00039218238200000214
表示无量纲化电流，
Figure FDA00039218238200000215
表示滤波电容与压电片内部电容之比；
Figure FDA00039218238200000216
分别是无量纲化的阻尼系数、时间常数比和机电耦合系数；
Figure FDA00039218238200000217
为所述系统无量纲化的三稳态势函数；k1，
Figure FDA00039218238200000218
和
Figure FDA00039218238200000219
分别表示无量纲化的线性刚度系数、三次刚度系数和五次刚度系数，其取值与悬臂梁末端磁铁与框架上的固定磁铁之间的距离有关，直接反映三稳态俘能系统势函数的形状以及所述系统的几何非线性的强弱；Among them, ω0 represents the natural frequency of the metal cantilever beam of the energy harvester,
Figure FDA00039218238200000210
is the length scale of the beam, and X represents the displacement after dimensionless;
Figure FDA00039218238200000211
Represents the base excitation after dimensionless;
Figure FDA00039218238200000212
is the dimensionless piezoelectric voltage;
Figure FDA00039218238200000213
Represents the dimensionless rectified voltage,
Figure FDA00039218238200000214
represents the dimensionless current,
Figure FDA00039218238200000215
Indicates the ratio of the filter capacitance to the internal capacitance of the piezoelectric film;
Figure FDA00039218238200000216
are the dimensionless damping coefficient, time constant ratio and electromechanical coupling coefficient;
Figure FDA00039218238200000217
is the dimensionless tristable state function of the system; k 1 ,
Figure FDA00039218238200000218
and
Figure FDA00039218238200000219
represent the dimensionless linear stiffness coefficient, the cubic stiffness coefficient and the quintic stiffness coefficient respectively, and 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 potential function of the tristable energy harvesting system The shape and strength of the geometric nonlinearity of the system; 对于公式(4)中无量纲化后的基座激励，采用如式(6)所示具有两种不同频率的谐波激励进行模拟，即：For the base excitation after dimensionless in formula (4), the harmonic excitation with two different frequencies as shown in formula (6) is used for simulation, namely:
Figure FDA0003921823820000031
Figure FDA0003921823820000031
 其中，f sin(ωt)表示环境中的低频谐波激励，f和ω分别是低频力的振幅和频率；Fsin(Ωt)代表环境中的高频谐波激励，F和Ω分别表示高频力的振幅和频率；低频谐波激励与高频谐波激励之间满足条件Ω＞＞ω，f＜＜1。Among them, f sin(ωt) represents the low-frequency harmonic excitation in the environment, f and ω are the amplitude and frequency of the low-frequency force, respectively; F sin(Ωt) represents the high-frequency harmonic excitation in the environment, F and Ω represent the amplitude of the high-frequency force, respectively and frequency; the condition Ω>>ω, f<<1 is satisfied between the low-frequency harmonic excitation and the high-frequency harmonic excitation. 3.如权利要求2所述的基于振动共振的三稳态俘能器低频激励下的性能优化方法，其特征在于：步骤二实现方法为，3. The performance optimization method under the low-frequency excitation of the tristable energy harvester based on vibration resonance as claimed in claim 2, characterized in that: the realization method of step 2 is, 根据变量分离方法，所述公式(4)具有如下形式的近似解：According to the variable separation method, the formula (4) has an approximate solution of the following form: X(t)＝x(t)+Ψ(t) (7)X(t)=x(t)+Ψ(t) (7) 式中，x(t)是输出响应中具有时间尺度Tf＝2π/ω的低频慢变量，Ψ(t)是均值为零的快速变化的分量，即，<Ψ(t)>＝0；符号
Figure FDA0003921823820000032
表示较短时间上的时间平均，其中TF＝2π/Ω为较短的时间周期；将方程(7)代入所述三稳态俘能系统的无量纲机电耦合模型(4)，分别得到关于所述三稳态俘能系统输出响应X(t)中的慢变成分x(t)及快变成分Γ(t)的方程为：In the formula, x(t) is a low-frequency slow variable with a time scale T f =2π/ω in the output response, Ψ(t) is a fast-changing component with a mean value of zero, that is, <Ψ(t)>=0; symbol
Figure FDA0003921823820000032
Represent the time average on a shorter time, wherein T F =2π/Ω is a shorter time period; Substituting equation (7) into the dimensionless electromechanical coupling model (4) of the tristable energy harvesting system, respectively, about The equations of the slow-varying component x(t) and the fast-varying component Γ(t) in the output response X(t) of the tri-stable energy harvesting system are:
Figure FDA0003921823820000033
Figure FDA0003921823820000033
Figure FDA0003921823820000034
Figure FDA0003921823820000034
 为求解方程(9)，作如下定义，即：To solve Equation (9), the following definition is made, namely: Ψ＝B11cos(Ωt)+B12sin(Ωt) (10)Ψ＝B 11 cos(Ωt)+B 12 sin(Ωt) (10) 另外，根据式(10)计算得到快变量对时间的一阶导数
Figure FDA0003921823820000035
及二阶导数
Figure FDA0003921823820000036
In addition, the first derivative of the fast variable with respect to time is calculated according to formula (10)
Figure FDA0003921823820000035
and the second derivative
Figure FDA0003921823820000036
Figure FDA0003921823820000037
Figure FDA0003921823820000037
 其中，B11，B12是待求系数；对公式(10)作简单计算得到有关快变量Ψ的表达式，即：Among them, B 11 and B 12 are the coefficients to be sought; the expression of the fast variable Ψ is obtained by simple calculation of the formula (10), namely:
Figure FDA0003921823820000041
Figure FDA0003921823820000041
 将公式(12)代入公式(9)，进一步根据公式(10)及公式(11)按照同次谐波排列，化简整理得到关于B11，B12的方程组：Substituting formula (12) into formula (9), further according to formula (10) and formula (11) according to the same order harmonic arrangement, simplifying and sorting out the equations about B 11 and B 12 :
Figure FDA0003921823820000042
Figure FDA0003921823820000042
 对于取较小的B11、B12，公式(13)近似为：For smaller B 11 and B 12 , the formula (13) is approximated as:
Figure FDA0003921823820000043
Figure FDA0003921823820000043
 显然，B11、B12由公式(14)解出，即：Obviously, B 11 and B 12 are solved by formula (14), namely:
Figure FDA0003921823820000044
Figure FDA0003921823820000044
 在后续描述中将k1+3k3x2+5k5x4-Ω2记作W，即W＝k1+3k3x2+5k5x4-Ω2；In the subsequent description, k 1 +3k 3 x 2 +5k 5 x 4 -Ω 2 is recorded as W, that is, W=k 1 +3k 3 x 2 +5k 5 x 4 -Ω 2 ; 将公式(15)、公式(10)和公式(12)代入公式(8)，即将关于慢变量x(t)的方程转换为：Substituting formula (15), formula (10) and formula (12) into formula (8), that is, the equation about the slow variable x(t) is transformed into:
Figure FDA0003921823820000045
Figure FDA0003921823820000045
 由于标准整流电路中整流桥的交替导通与阻断作用，所述公式(4)的压电电压具有分段非光滑性质；采用压电电压的基谐波分量对其进行近似代替，并结合广义谐波变换、三稳态俘能系统无量纲化方程得到等效非耦合优化模型如式(17)所示：Due to the alternating conduction and blocking effects of the rectifier bridge in the standard rectifier circuit, the piezoelectric voltage of the formula (4) has a piecewise non-smooth property; it is approximately replaced by the fundamental harmonic component of the piezoelectric voltage, and combined with The equivalent non-coupling optimization model obtained by generalized harmonic transformation and tri-stable energy harvesting system dimensionless equation is shown in formula (17):
Figure FDA0003921823820000046
Figure FDA0003921823820000046
 式中，In the formula,
Figure FDA0003921823820000051
Figure FDA0003921823820000051
 其中，
Figure FDA0003921823820000052
表示优化后的三稳态俘能器的等效阻尼系数，整流电路对于原系统(4)起到电阻尼效应；
Figure FDA0003921823820000053
表示优化后的等效线性刚度系数，其不仅依赖于原系统的刚度系数及高频力的性质，还与整流电路有关；
Figure FDA0003921823820000054
表示优化后的三次刚度系数，主要依赖于原系统的三次刚度系数及五次刚度系数及高频力的性质；另外，
Figure FDA0003921823820000055
表示优化设计后的五次刚度系数，其与原系统的五次刚度系数一致；θ表示二极管整流桥阻塞角；in,
Figure FDA0003921823820000052
Indicates the equivalent damping coefficient of the optimized tristable energy harvester, and the rectifier circuit has an electrical damping effect on the original system (4);
Figure FDA0003921823820000053
Represents the optimized equivalent linear stiffness coefficient, which not only depends on the stiffness coefficient of the original system and the nature of the high-frequency force, but is also related to the rectifier circuit;
Figure FDA0003921823820000054
Indicates the optimized cubic stiffness coefficient, which mainly depends on the cubic stiffness coefficient and quintic stiffness coefficient of the original system and the properties of high-frequency forces; in addition,
Figure FDA0003921823820000055
Indicates the quintic stiffness coefficient after optimized design, which is consistent with the quintic stiffness coefficient of the original system; θ represents the blocking angle of the diode rectifier bridge; 基于变量分离方法推导得到所述三稳态俘能系统的无量纲机电耦合模型的快变量方程及慢变量方程；针对标准整流电路引起的分段非光滑特性，采用压电电压的基谐波分量对分段非光滑的压电电压进行平滑处理，得到所述三稳态俘能器系统的无量纲机电耦合模型的等效非耦合系统方程，即基于振动共振得到所述三稳态俘能系统在低频激励下的优化模型。Based on the variable separation method, the fast variable equation and the slow variable equation of the dimensionless electromechanical coupling model of the tristable energy harvesting system are derived; for the piecewise non-smooth characteristics caused by the standard rectifier circuit, the fundamental harmonic component of the piezoelectric voltage is used Smoothing the piecewise non-smooth piezoelectric voltage to obtain the equivalent uncoupled system equation of the dimensionless electromechanical coupling model of the tristable energy harvester system, that is, to obtain the tristable energy harvesting system based on vibration resonance Optimization model under low frequency excitation. 4.如权利要求3所述的基于振动共振的三稳态俘能器低频激励下的性能优化方法，其特征在于：步骤三实现方法为，4. the performance optimization method under the low-frequency excitation of the tristable energy harvester based on vibration resonance as claimed in claim 3, is characterized in that: step 3 realization method is, 利用式(24)所示三稳态俘能系统输出整流电压、式(25)所示系统直流功率、式(26)所示系统输出响应幅值增益系数作为性能指标以衡量所述三稳态俘能系统在不同参数组合下的发电性能，分析所述系统参数对俘能系统动力学行为以及收集性能的影响规则，并选取使得性能指标最大化的最优参数组合；Using the output rectified voltage of the tri-stable energy harvesting system shown in formula (24), the system DC power shown in formula (25), and the system output response amplitude gain coefficient shown in formula (26) as performance indicators to measure the tri-stable state The power generation performance of the energy capture system under different parameter combinations, analyzing the influence rules of the system parameters on the dynamic behavior and collection performance of the energy capture system, and selecting the optimal parameter combination that maximizes the performance index; 优化设计后的等效非耦合系统方程(17)的稳态近似解为：The steady-state approximate solution of the equivalent uncoupled system equation (17) after optimal design is: x(t)＝b1cos(ωt)+b2sin(ωt) (19)x(t)=b 1 cos(ωt)+b 2 sin(ωt) (19) 其中，
Figure FDA0003921823820000056
表示所述系统(17)的振动幅值；采用谐波平衡法，将公式(19)及其一阶和二阶导数代入系统(17)，并对结果按照同次谐波排列；当所述三稳态俘能系统达到稳态时，所有量对时间导数均为零；另外，高频力Fcos(Ωt)的频率足够大，即W≈-Ω2；因此，稳态振幅b1、b2推导得到：in,
Figure FDA0003921823820000056
Represent the vibration amplitude of described system (17); Adopt harmonic balance method, formula (19) and its first-order and second-order derivatives are substituted into system (17), and the result is arranged according to the same order harmonic; When described When the tristable energy harvesting system reaches a steady state, all quantities have zero time derivatives; in addition, the frequency of the high-frequency force Fcos(Ωt) is large enough, that is, W≈-Ω 2 ; therefore, the steady-state amplitudes b 1 and b 2 are deduced get:
Figure FDA0003921823820000057
Figure FDA0003921823820000057
Figure FDA0003921823820000058
Figure FDA0003921823820000058
 通过公式(20)(21)得到所述系统(17)的稳态解，其中，所述系统(17)的振动幅值
Figure FDA0003921823820000061
Obtain the steady-state solution of described system (17) by formula (20) (21), wherein, the vibration amplitude of described system (17)
Figure FDA0003921823820000061
 另外，整流电路中整流桥的阻塞角θ与整流电压YR之间满足如下关系：In addition, the blocking angle θ of the rectifier bridge in the rectifier circuit and the rectified voltage Y R satisfy the following relationship: A(cosθ-1)＝-2YR (22)A(cosθ-1)＝-2Y R (22) 进一步基于基尔霍夫电流定律，在半个周期内，从俘能器流出的总电荷量与流过电阻的总电荷相等，即：Further based on Kirchhoff's current law, in half a cycle, the total charge flowing out of the energy harvester is equal to the total charge flowing through the resistor, namely:
Figure FDA0003921823820000062
Figure FDA0003921823820000062
 结合公式(5)(22)(23)，推导得到所述三稳态俘能系统输出的整流电压YR及直流功率P分别为：Combined with formulas (5)(22)(23), the rectified voltage Y R and DC power P output by the tri-stable energy harvesting system are deduced as follows:
Figure FDA0003921823820000063
Figure FDA0003921823820000063
Figure FDA0003921823820000064
Figure FDA0003921823820000064
 所述三稳态俘能系统的输出响应幅值增益系数Q是描述振动共振机制的重要量化指标，有：The output response amplitude gain coefficient Q of the tristable energy harvesting system is an important quantitative index describing the vibration resonance mechanism, which is:
Figure FDA0003921823820000065
Figure FDA0003921823820000065
 其中，f是低频谐波激励的振幅，Qs和Qc分别是所述系统(17)相对于低频输入f cos(ωt)的正弦分量和余弦分量，即：Wherein, f is the amplitude of the low-frequency harmonic excitation, Q s and Q c are the sine and cosine components of the system (17) relative to the low-frequency input f cos(ωt), namely:
Figure FDA0003921823820000066
Figure FDA0003921823820000066
Figure FDA0003921823820000067
Figure FDA0003921823820000067
 其中，n代表正整数，x(t)是优化设计后的三稳态俘能系统的输出位移；Among them, n represents a positive integer, and x(t) is the output displacement of the tristable energy harvesting system after optimal design; 根据所述优化设计后的三稳态俘能器系统(17)，以公式(26)所示三稳态俘能系统的输出响应幅值增益系数、系统振动幅值、所述公式(24)所示系统整流电压、所述公式(25)所示系统直流功率作为性能指标，分析包括刚度系数、时间常数比、机电耦合系数的所述系统参数以及环境激励幅值、频率对俘能器采集性能的影响，给出使性能指标达到最大时的最优参数组合，根据所述最优参数组合实现低频激励下三稳态俘能系统的性能增强，进而实现三稳态俘能器在低频激励下的高效俘能。According to the tri-stable energy harvester system (17) after the optimized design, the output response amplitude gain coefficient, the system vibration amplitude, the formula (24) of the tri-stable energy harvester system shown in formula (26) The rectified voltage of the system shown and the DC power of the system shown in the formula (25) are used as performance indicators to analyze the system parameters including the stiffness coefficient, time constant ratio, electromechanical coupling coefficient, and the environmental excitation amplitude and frequency to the energy harvester. The impact of performance, the optimal parameter combination when the performance index reaches the maximum is given, and the performance enhancement of the tristable energy harvesting system under low frequency excitation is realized according to the optimal parameter combination, and then the tristable energy harvester is realized under low frequency excitation. high efficiency energy capture.
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CN116626274A (en) * 2023-07-20 2023-08-22 中国科学院苏州生物医学工程技术研究所 A Measuring Method for Axial Distribution of Electromechanical Coupling Coefficient of Blood Viscoelastic Sensor
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WO2025225728A1 (en) * 2024-04-26 2025-10-30 日本製鉄株式会社 Electric power generation device
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