CN109783836B - Nonlinear model establishing and verification analysis method of L-shaped piezoelectric energy collector - Google Patents

Nonlinear model establishing and verification analysis method of L-shaped piezoelectric energy collector Download PDF

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CN109783836B
CN109783836B CN201811158513.XA CN201811158513A CN109783836B CN 109783836 B CN109783836 B CN 109783836B CN 201811158513 A CN201811158513 A CN 201811158513A CN 109783836 B CN109783836 B CN 109783836B
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晏致涛
聂小春
颜志淼
王灵芝
孙毅
刘欣鹏
李妍
杨小刚
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Chongqing University of Science and Technology
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Abstract

The invention discloses a nonlinear model building and verification analysis method of an L-shaped piezoelectric energy collector, which builds a physical model of the L-shaped piezoelectric energy collector, a piezoelectric patch load circuit and a coordinate distribution schematic diagram; deducing a control equation of an energy acquisition system of the collector, a circuit equation of a piezoelectric piece load circuit and a control equation of the energy acquisition system; setting parameters and establishing and verifying an L-beam structure finite element model; the influence of external load resistance on the self-vibration frequency and the damping ratio of the verified L-beam structure finite element model is analyzed, and the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the conditions of first-order and second-order main resonance is analyzed. Has the advantages that: the rationality of the proposed theoretical model is verified by adopting modal analysis and transient dynamic analysis of finite element ANSYS; the frequency and damping of the energy harvesting system are greatly affected by the load resistance.

Description

Nonlinear model establishing and verification analysis method of L-shaped piezoelectric energy collector
Technical Field
The invention relates to the technical field of L-shaped piezoelectric energy collectors, in particular to a nonlinear model building and verification analysis method of an L-shaped piezoelectric energy collector.
Background
In recent years, the problem of utilization of renewable energy has received more and more attention. Among them, vibration energy in the environment is one of the most common renewable energy sources. The piezoelectric energy harvester can convert vibration energy in the environment into electric energy, and the converted electric energy can be used for a self-powered device. Such as micro-electromechanical systems, wireless sensors, and structural health monitoring, among others.
Since vibrational energy is a ubiquitous form of energy in the environment, energy harvesters based on substrate excitation have attracted considerable attention from researchers. Previous research is mainly focused on a single-cantilever piezoelectric energy collector, the piezoelectric energy collector can only collect energy when external excitation frequency is close to the natural vibration frequency of a structure, when the external excitation frequency is far away from the natural vibration frequency, the energy collected by a system can be rapidly reduced, and the excitation frequency in the environment is also continuously changed. Therefore, the proposal and design of the broadband energy harvester have important significance. Therefore, researchers have explored the working performance of piezoelectric energy collectors from different aspects to broaden the energy collection bandwidth of the piezoelectric energy collectors. In the literature [13] Twifel, westermann, survey on broadband technology for interference energy harnessing journal of organic Material systems and structures.24 (11) 1291-1302.The Author(s) 2013, twifel and Westermann have studied an energy harvesting system with multiple piezoelectric cantilever structures juxtaposed, with mass blocks of different masses attached to the ends of each beam, and thus with different natural frequencies of each piezoelectric cantilever, the system can harvest energy at multiple external excitation frequencies. In the document [14] Karami, analytical Modeling and Experimental version of the simulations of the Zigzag micro Structures for E energy Harvesting M.Amin Karami1 e-mail, arami @ vt. EduDaniel J.InmanCenter for Intelligent Materials Systems and Structures, virginia Tech,310 Durham Hall, blacksburg, VA 24061, karami designed a Zigzag Microstructure energy harvesting system in which the first 5 th order natural frequencies of the structure approached each other as the length of the system beam structure increased, to achieve a structure capable of harvesting energy at multiple external excitation frequencies. A non-linear electromagnetic ENERGY HARVESTING system that can expand the frequency bandwidth of ENERGY HARVESTING by non-linear hardening phenomena is studied in the literature ENERGY harvesing FROM vibrancons WITH a non-linear induction excitation process of the ASME 2009 International Design Technical references and computers and Information in Engineering on-site etc/CIE 2009 August 30-September 2,2009, san diego, usa, barton. A self-parametric piezoelectric energy harvesting system consisting of a bottom main structure and a piezoelectric cantilever beam with concentrated mass at the ends is proposed in the literature [19] nonlinear industries of an automatic piezoelectric energy harvester, journal of organic Material Systems and structures 1-18 \/the Author(s) 2016, yan. Such a structure may be provided by 2:1 internal resonance phenomenon to realize broadband acquisition of energy and control vibration displacement of the main structure. In the document [17] broadband design of hybrid piezoelectric energy Harvester, tan et al have designed a hybrid energy harvesting system based on galloping and substrate vibration to achieve the broadband effect of energy harvesting. The study shows that the area of the wide frequency band is determined by the boundary of the quenching phenomenon, and the frequency band of the system acquisition can be expanded to infinity when the minimum acquisition power determined by the boundary of the quenching phenomenon is acceptable. In the document [20] a vibration energy harvesting device with a bidirectional response and frequency sensitivity, published 8 January 2008 Online at stacks, iop.org/SMS/17/015035, challa et al propose a semi-active cantilever piezoelectric energy harvester that increases or decreases the natural frequency of the system mainly by the attraction or repulsion of the magnets to expand the working frequency range of the energy harvesting system. In addition to the above mentioned studies, an L-beam structure based on the principle of internal resonance is also an ideal model for achieving broadband energy harvesting. IN the document [21] the following AND empirical STUDY OF modified INTERACTION IN A TWO-DeGREE-OF-FREE DOM STRUCTURURE, haddow AND Barr investigated the nonlinear properties OF L-shaped beam STRUCTUREs by theory AND experiment. IN the document [23] formed resonance of a beam SYSTEM with automatic coupling effects and the document [24] SIMULTANEOUS COMMUNICATION RESONANCES IN AN AUTOPARAMINATIVE RESONANT SYSTEM, robert and Cartwell also investigated the phenomenon of internal resonance of L-beams and found that the SYSTEM is capable of exciting a larger response when a first-second order dominant resonance occurs. Based on the above findings, balachandran and Nayfeh have also investigated experimentally and theoretically considering the dynamic response of second order nonlinear L-beams in the literature [25] nonlinear mechanisms of Beam-Mass Structure, B.BALACHANDRAN and A.H.NAYFIH Engineering Science and Mechanics Department scientific Institute and State UnivefMtv Blacksburg, virginia, U.S.A. in the literature [26] an Experimental Investigation of comprehensive Responses of a Two-depth-of-free Structure. The results show that the theoretical predicted position of the hopplev bifurcation is more consistent with the test results, and energy exchange will occur between the two-order coupling modes of the system when the external excitation frequency is respectively close to the two-order frequency of the structure. On the basis of the above-mentioned research considering only the planar motion of the L-beam, in the literature [27], analytical and experimental information of an automated beam structure, warminski et al derived the motion equation of the L-beam structure considering the out-of-plane motion of the structure, and the results indicate that the L-beam structure may interact between two modes in the plane and the structure may also have out-of-plane motion. Georgiades et al, in [28], methods of analysis for an L-shaped beam: proportions of mechanisms Research Communications 47 (2013) 50-60; and the literature [29] Linear Module Analysis of L-Shaped Beam Structures derives a linear motion equation considering the out-of-plane motion of the L-Shaped Beam, and a large number of parameter analyses are performed on the basis of the linear motion equation. Based on the research on the dynamic response of the L-shaped beam structure, erutk et al, on the basis of the document [30], consider the piezoelectric material into the structure, propose a linear distribution parameter piezoelectric energy collector model based on the L-shaped beam structure, and analyze the output voltage, power and the displacement response of the structure vertex of the collector. Cao et al in [31] internal response for nonlinear vibration energy harvesting, the E EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS, based on the above-mentioned [21] on the basis of the L-shaped beam structure motion equation derived by Haddow and Barrr, obtains the nonlinear motion equation of the L-shaped piezoelectric energy harvester by directly introducing the electromechanical coupling term and the circuit equation, then derives the approximate analytic solution of the system response under a primary resonance of a second order by adopting a multi-scale method, and discusses the influence of the excitation amplitude, the mechanical damping and the external excitation frequency on the system output response. In the document [32] A Broadband Internally-Resonant Vibrity Energy Harvester, chen et al designed a magnetic nonlinear L-beam piezoelectric Energy Harvester in which both the magnetic force acting on the structure and the transverse vibrational displacement of the structure are assumed to be in the form of a cubic polynomial function. The self-vibration frequency of the structure can be adjusted through the distance between the magnets, so that the second-order frequency ratio of the structure is kept about 1.
In addition, harne et al, in [33], legacy nonlinear simulation-based photonic in an L-shaped resonance simulation system, also analyzed the performance of the L-shaped beam energy collector through internal resonance and saturation phenomena, and the result shows that the energy collector can effectively improve the energy collecting effect of the system. Recently, liu et al in [34] piezoelectric energy harvesting L-shaped structures have conducted experimental studies on L-shaped piezoelectric energy harvesters under the action of substrate excitation, and the results show that the energy harvesting bandwidth of the structure is much larger than that of cantilever beam energy harvesters, however, the second-order frequency of the experimental model is not twice as high as the first-order frequency, and the energy exchange between the second-order modes of the structure cannot be excited. Therefore, the study only reflects that the voltage and power response of the system changes with the change of the external excitation, and the output response of the voltage and power can reach the maximum value when the excitation frequency is close to the first two self-oscillation frequency of the system.
Although the performance research of the L-beam piezoelectric energy harvester has received wide attention in recent years, the research on such piezoelectric energy harvester is quite limited. To the best of the applicant's knowledge, the geometric non-linear mathematical model of the L-beam piezoelectric energy harvester is still lacking, and the geometric non-linearity of the piezoelectric material is generally not considered in the existing piezoelectric energy harvester models.
Disclosure of Invention
Aiming at the problems, the invention provides a nonlinear model establishing and verifying analysis method of an L-shaped piezoelectric energy collector, and an electromechanical coupling distribution parameter model considering the geometric nonlinearity of a structural layer and a piezoelectric layer is derived by using a Hamilton principle and a Gauss law. And (5) verifying the reasonability of the proposed theoretical model by adopting modal analysis and transient dynamic analysis of finite element ANSYS. The specific technical scheme is as follows:
a nonlinear model building and verification analysis method of an L-shaped piezoelectric energy collector is characterized by comprising the following key technologies:
s1: constructing an L-shaped energy collector physical model and a piezoelectric patch load circuit; s2: establishing a coordinate system, and obtaining an L-shaped energy collector coordinate distribution schematic diagram corresponding to the L-shaped energy collector physical model based on the L-shaped energy collector physical model; s3: deducing a control equation of an energy acquisition system of the L-shaped energy collector by adopting a Hamilton principle; s4: obtaining a circuit equation of the piezoelectric patch load circuit by adopting Gauss law, and obtaining a control equation of the reduced energy acquisition system by combining the circuit equation; s5: setting physical parameters and geometrical parameters of the L-shaped energy collector physical model, acquiring the physical parameters and geometrical parameters of the L-shaped beam structure from the physical parameters and geometrical parameters of the L-shaped energy collector physical model, and establishing a finite element model of the L-shaped beam structure by adopting large finite element general software ANSYS; s6: verifying the natural vibration frequency, time-course response and internal resonance response of the L-beam structure finite element model to obtain a verified L-beam structure finite element model; s7: and analyzing the influence of the external load resistance on the self-vibration frequency and the damping ratio of the verified L-beam structure finite element model, and analyzing the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the conditions of first-order and second-order main resonance.
Through the design, a geometrical nonlinear model of the L-shaped beam piezoelectric energy collector is deduced by adopting a Hamilton principle and a Gauss law, a finite element model of the L-shaped beam structure is established by adopting large finite element general software ANSYS, the first two-order natural vibration frequency of the theoretical model is verified through the modal analysis of ANSYS, the time-course response of the structure is obtained through the transient dynamic analysis of ANSYS, and the internal resonance response of the L-shaped beam structure is verified by combining Fourier transform.
Further, when the L-shaped energy harvester physical model and the piezoelectric patch load circuit are constructed in the step S1, the following steps are provided: the L-shaped piezoelectric energy collector comprises an L-beam structure and a first concentrated mass M 1 Second lumped mass M 2 (ii) a The L-beam structure comprises a horizontal beam and a vertical beam;
first lumped mass M 1 Fixed at the corner where the horizontal beam and the vertical beam intersect; second lumped mass M 2 The vertical beam is positioned on the vertical beam and can slide up and down; the piezoelectric piece load circuit comprises piezoelectric pieces and a load resistor R, wherein the piezoelectric pieces are respectively adhered to the upper surface and the lower surface of the horizontal beam and connected with the load resistor R to form a parallel circuit; in step S2, the coordinate distribution diagram of the L-shaped energy harvester introduces three rectangular coordinate systems: o1x1y1; o2x2y2; o3x3y3, three coordinate systems are used to describe the motion of the horizontal and vertical beams; the horizontal beam and the vertical beam are regarded as three parts, wherein the horizontal beam is a first beam section, and the vertical beam is regarded as a second beam section and a third beam section.
Further, in step S3, the hamilton principle is used to derive a control equation of the energy collection system of the L-shaped energy collector as follows:
Figure GDA0003927254570000031
Figure GDA0003927254570000032
Figure GDA0003927254570000033
N 1 、N 2 and N 3 Respectively axial loads of the first beam section, the second beam section and the third beam section;
Figure GDA0003927254570000034
Figure GDA0003927254570000035
Figure GDA0003927254570000036
and, the linear boundary conditions are:
Figure GDA0003927254570000041
arc length corresponding to beam section i (i =1,2, 3) is represented by s i Showing the axial and lateral displacements on the beam sections by u, respectively i (s i T) and v i (s i And t) represents; theta i (s i T) represents the angular displacement of each beam section before and after deformation at the section i; l i Represents the length of the ith beam segment;
wherein u is i (s i ,t)、v i (s i T) and θ i (s i The geometric non-linear relationship between t) can be expressed as:
Figure GDA0003927254570000042
Figure GDA0003927254570000043
let v be a simple formula i (s i ,t)＝ vi ；
m b1 And m b2 Respectively representing the mass per unit length of the horizontal beam and the vertical beam; subscripts s and p denote the structural layer and the piezoelectric layer, respectively; subscripts 1,2 of s and p denote the horizontal and vertical beam section structure layers and piezoelectric layers, respectively;ρ represents the density of the material; h and b represent the height and width of the beam segment, respectively; j. the design is a square 1 And J 2 Respectively representing lumped masses M 1 And M 2 The moment of inertia of (a); EI (El) 1 And EI 2 Respectively representing the bending stiffness of the horizontal and vertical beam sections; the symbols "'" and "·" denote the pairs s, respectively i And t is derived; v (t) is the voltage generated by the piezoelectric sheet due to deformation; c. C s And c a Respectively the equivalent viscous strain and the air damping coefficient of the cantilever beam; i is i Is the cross-sectional moment of inertia of the beam; m 1 Is the first lumped mass; m 2 Is the second lumped mass. The specific process of deducing the control equation of the energy acquisition system of the L-shaped energy collector by adopting the Hamiltonian principle comprises the following steps: wherein: the Hamiltonian equation is:
Figure GDA0003927254570000044
(1) (ii) a Wherein: t, V and W nc Respectively representing the kinetic energy, the potential energy and the virtual work done by the external force; the arc length corresponding to the section i (i =1,2, 3) of the beam segment is denoted by si, and the axial and transverse displacements on the respective beam segments are denoted by u, respectively i (s i T) and v i (s i And t) represents; theta i Representing the corner displacement of each beam section before and after deformation at the section i; in connection with the literature paper 25 mentioned in the background, the kinetic energy T and the potential energy V can be expressed as:
Figure GDA0003927254570000045
Figure GDA0003927254570000046
wherein m is b1 And m b2 Respectively represents the mass per unit length of the horizontal beam and the vertical beam, and the expressions are respectively: m is b1 ＝b s1 ρ s1 h s1 +2b p1 ρ p h p ，m b2 ＝b s2 ρ s2 h s2 . Subscripts s and p denote the structural and piezoelectric layers, respectively, and subscripts 1 and 2 denoteHorizontal and vertical beam sections are shown, p represents the density of the material, h and b represent the height and width of the beam section, respectively, J 1 And J 2 Respectively representing lumped masses M 1 And M 2 Of rotational inertia, EI 1 And EI 2 Respectively representing the bending stiffness of the horizontal and vertical beam sections, and the expressions are respectively:
Figure GDA0003927254570000051
Figure GDA0003927254570000052
wherein E s And E p The young's modulus of elasticity of the structural layer and the piezoelectric layer, respectively.
It is known from the article Z.Yan, H.Taha, T.Tan, nonlinear characteristics of an autoparametric visualization system, J.Sound V.ib.390 (2017) 1-22 i (s i ,t)、v i (s i T) and θ i (s i The geometric non-linear relationship between t) can be expressed as:
Figure GDA0003927254570000053
and
Figure GDA0003927254570000054
therefore, the curvature θ of an arbitrary position i ′(s i T), angular velocity
Figure GDA0003927254570000055
And axial displacement u i (s i T) can be expressed as
Figure GDA0003927254570000056
Figure GDA0003927254570000057
In equations (4) and (5), v is ignored i ′(s i T) order 3 and higher. The virtual work done by non-conservative forces can be expressed as：W ne ＝W ele +W damp (7)
Wherein, W ele And W damp Respectively, as the virtual work due to the electrical and damping forces. The geometric nonlinearity of the piezoelectric sheet is considered here, so the virtual work done by the electric power can be expressed as:
Figure GDA0003927254570000058
wherein M is ele For the effect of the charge on the resulting bending moment, the expression is as follows:
Figure GDA0003927254570000059
wherein V (t) is a voltage generated by the piezoelectric sheet due to deformation. H(s) 1 ) Is a Helvessel step function, e 31 ＝E p d 31 In order to obtain a piezoelectric stress coefficient,
Figure GDA00039272545700000510
is a piezoelectric coupling term expressed as
Figure GDA00039272545700000511
Virtual work W by damping force damp Comprises the following steps:
Figure GDA00039272545700000512
wherein the content of the first and second substances,
Figure GDA00039272545700000513
is a bending moment generated by the strain rate, c in the above formula s And c a Equivalent viscous strain and air damping coefficient of the cantilever beam, I i Is the cross-sectional moment of inertia of the beam.
The control equation of the energy collection system of the L-shaped energy collector can be obtained by substituting equations (2), (3) and (7) into Hamiltonian equation (1).
Further, the circuit equation of the piezoelectric piece load circuit in step 4 is as follows:
Figure GDA00039272545700000514
r is a load resistance value; v (t) is the voltage generated by the piezoelectric sheet due to deformation; h is p Is the thickness of the piezoelectric sheet; h is s1 Is the thickness of the first beam segment;
Figure GDA0003927254570000061
is the dielectric constant component at constant strain; e.g. of the type 31 ＝E p d 31 Is the piezoelectric stress coefficient;
Figure GDA0003927254570000062
Figure GDA0003927254570000063
is a piezoelectric coupling term.
The specific derivation steps of the circuit equation of the piezoelectric patch load circuit are as follows: by gauss's law, an expression of the circuit equation can be obtained:
Figure GDA0003927254570000064
the Gaussian Law is described in the article IEEE 176-1987-IEEE, standard on piezoelectric device. Doi:10.1109/IEEESTD,1988, where D is the electrical displacement vector and n is the external normal vector. Electric displacement D 2 The formula is as follows:
Figure GDA0003927254570000065
taking into account geometrical non-linearities of the piezo-electric sheet
Figure GDA0003927254570000066
Figure GDA0003927254570000067
Is the dielectric constant component at constant strain. Carry equation (17) into (16) to get the chipThe circuit equation of the load circuit. For analyzing the response of the energy collection system, the transverse vibration displacement v is displaced by adopting the Galerkin method i (s i T) separation into a spatial variable phi ij (s i ) And a time variable q j (t)：
Figure GDA0003927254570000068
Figure GDA0003927254570000069
φ ij (s i ) And q is j (t) are the jth order mode shape and modal coordinates of the system, respectively. The mode shape of the energy harvesting system may be expressed as:
Figure GDA00039272545700000610
Figure GDA00039272545700000611
Figure GDA00039272545700000612
coefficient A ij 、B ij 、C ij And D ij The coefficients are coefficients of the j-th order mode of the system, and the coefficients are obtained through boundary conditions and orthogonal conditions. The boundary elements after separation of the variables are as follows:
Figure GDA00039272545700000613
and orthogonal bars:
Figure GDA00039272545700000614
Figure GDA0003927254570000071
the upper typeIn which s and r represent the number of modes of the system, δ rs Is the Kronecker delta function, and delta rs ＝1(r＝s)，δ rs ＝0(r≠s)。
The specific derivation process of the vibration shape is as follows: in order to derive the mode shape function of the L-shaped beam structure, the damping term, the piezoelectric coupling term, and the real number term are removed in equations (11) - (13), resulting in the linear control equation of the system:
Figure GDA0003927254570000072
Figure GDA0003927254570000073
Figure GDA0003927254570000074
displacing the transverse vibration by v i，j Separated into spatial and temporal variables, namely:
Figure GDA0003927254570000075
then (a.4) is substituted into (a.1), resulting in the following expression:
Figure GDA0003927254570000076
suppose that
Figure GDA0003927254570000077
The solution of (a) is in the form of:
Figure GDA0003927254570000078
K 1 is a real constant, substituting (A.6) into (A.5) yields the expression for x:
Figure GDA0003927254570000079
in the above formula, the root of x is: x is the number of 1，2，3，4 ＝±α j ，±iα j . Wherein the content of the first and second substances,
Figure GDA00039272545700000710
thus, the mode shape function of the beam segment 1 can be expressed as:
Figure GDA00039272545700000711
substituting (A.4) into (A.2) yields the following expression:
Figure GDA00039272545700000712
in the same way, suppose
Figure GDA00039272545700000713
The expression of (c) is:
Figure GDA00039272545700000714
wherein, K 2 Is a real constant, substituting (a.10) into (a.9) yields an expression for the parameter y:
Figure GDA00039272545700000715
the root of the above formula (a.11) can be represented as:
Figure GDA00039272545700000716
wherein the content of the first and second substances,
Figure GDA00039272545700000717
the mode shape function of the second beam segment may be expressed as:
Figure GDA00039272545700000718
to derive the mode shape function of the beam section 3, a constant term is introduced into equation (A.3)
Figure GDA00039272545700000719
Then equation (a.3) can be expressed as:
Figure GDA00039272545700000720
v(s)＝v 3 (s 3 )+v 2 (l 2 ) Similar to (a.1), the solution of equation (a.13) is:
Figure GDA0003927254570000081
further, the step S4 combines the circuit equation to obtain a control equation of the reduced energy collection system as follows:
Figure GDA0003927254570000082
Figure GDA0003927254570000083
Figure GDA0003927254570000084
Figure GDA0003927254570000085
represents a vertical acceleration; zeta 1 And ζ 2 Respectively representing the first two-stage mechanical damping ratio of the system; omega 1 And ω 2 Respectively representing the first two-order circle frequency of the system;
Figure GDA0003927254570000086
representing the displacement v of the capacitor in transverse oscillation i (s i T) separation into a spatial variable phi ij (s i ) And a time variable q j (t)：
Figure GDA0003927254570000087
φ ij (s i ) And q is j (t) j order mode shape and modal coordinates of the system, respectively;
the vibration shape of the energy acquisition system is as follows:
Figure GDA0003927254570000088
wherein the content of the first and second substances,
Figure GDA0003927254570000089
coefficient A ij 、B ij 、C ij And D ij Coefficients for the j-th order mode of the system; s and r represent the modal number of the system, δ rs Is the Kronecker delta function, and delta rs ＝1(r＝s)，δ rs ＝0(r≠s)；m k 、n k And η l Are dimensionless coefficients.
The governing equation of the reduced energy harvesting system is obtained by substituting equation (19) into (11), (12), (13) and (18) and taking into account the first two modes of the system, through the quadrature and boundary conditions.
Wherein, the dimensionless coefficient m k 、n k And η l The derivation process is as follows: EI is assumed herein for ease of writing in coefficient expressions 2 ＝EI 3 Is established, therefore m k ，n k And η l The expression of (c) can be expressed as:
Figure GDA00039272545700000810
Figure GDA0003927254570000091
Figure GDA0003927254570000092
Figure GDA0003927254570000093
Figure GDA0003927254570000094
Figure GDA0003927254570000095
Figure GDA0003927254570000096
Figure GDA0003927254570000097
Figure GDA0003927254570000098
Figure GDA0003927254570000099
Figure GDA0003927254570000101
Figure GDA0003927254570000102
m 13 ＝m 12 (B.13)
Figure GDA0003927254570000103
Figure GDA0003927254570000104
Figure GDA0003927254570000105
Figure GDA0003927254570000106
Figure GDA0003927254570000111
m 19 ＝m 18 (B.19) m 20 ＝2m 21 (B.20)
Figure GDA0003927254570000112
Figure GDA0003927254570000113
m 23 ＝2m 22 (B.23)
Figure GDA0003927254570000114
Figure GDA0003927254570000115
Figure GDA0003927254570000116
Figure GDA0003927254570000117
Figure GDA0003927254570000118
Figure GDA0003927254570000119
Figure GDA0003927254570000121
Figure GDA0003927254570000122
Figure GDA0003927254570000123
Figure GDA0003927254570000124
Figure GDA0003927254570000125
Figure GDA0003927254570000126
Figure GDA0003927254570000127
Figure GDA0003927254570000131
n 13 ＝n 12 (B.38)
Figure GDA0003927254570000132
Figure GDA0003927254570000133
Figure GDA0003927254570000134
Figure GDA0003927254570000135
Figure GDA0003927254570000136
n 19 ＝n 18 B.44；n 20 ＝2n 21 B.45
Figure GDA0003927254570000141
Figure GDA0003927254570000142
n 23 ＝2n 22 (B.48)
Figure GDA0003927254570000143
Figure GDA0003927254570000144
Figure GDA0003927254570000145
Figure GDA0003927254570000146
Figure GDA0003927254570000147
further, when an L-Beam structure finite element model is established, beam188 and Mss21 units are respectively adopted to simulate a Beam structure and a concentrated mass, and geometric nonlinearity of the L-Beam structure is considered through an 'NLGEOM, ON' command.
In step S7, when analyzing the influence of the external load resistance on the natural frequency and the damping ratio of the verified L-beam structure finite element model, introducing the following vector relationship:
Figure GDA0003927254570000148
based on the variable relationship of (27), the control equation can be rewritten as equation (28):
Figure GDA0003927254570000149
Figure GDA00039272545700001410
the linear coefficient matrix of the variables in equation (28) is:
Figure GDA0003927254570000151
in the L-type piezoelectric energy harvester system, the frequency and the damping of the system can be determined by the eigenvalue of the matrix B, so as to discuss the influence of the external load resistance on the frequency and the damping of the structure.
The invention has the beneficial effects that: the invention utilizes Hamilton principle and Gauss law to derive an electromechanical coupling distribution parameter model considering the geometric nonlinearity of the structural layer and the piezoelectric layer. The rationality of the proposed theoretical model is verified by adopting modal analysis and transient dynamic analysis of finite element ANSYS; the frequency and damping of the energy harvesting system are greatly affected by the load resistance.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is a schematic diagram of a physical model of an L-shaped energy harvester of the present invention;
FIG. 3 is a schematic diagram of the distribution of coordinates of the L-shaped energy harvester of the present invention;
FIG. 4 is a schematic view of a finite element model of an L-shaped beam structure according to the present invention;
FIG. 5 is a schematic diagram of theoretical prediction results and finite element results of the present invention;
FIG. 6 is a schematic time-course plot of modal amplitude and tip displacement in an unstable region in accordance with the present invention;
FIG. 7 is a schematic representation of the variation of the frequency and damping ratio with resistance of the energy harvesting system of the present invention;
FIG. 8 is a schematic diagram showing the variation of the modal amplitude and the end displacement with the external excitation frequency in the second-order primary resonance analysis of the present invention under different load resistances;
FIG. 9 is a schematic diagram of the variation of the collected energy with the external excitation frequency under different load resistances for the second-order primary resonance analysis according to the present invention;
FIG. 10 is a schematic diagram showing the variation of modal amplitude, end displacement and collected energy with external excitation frequency under different external excitation force amplitudes for second-order primary resonance analysis according to the present invention;
FIG. 11 is a schematic diagram showing the variation of modal amplitude, end displacement and collected energy with external excitation frequency under different load resistances of the second-order primary resonance analysis of the present invention;
FIG. 12 is a schematic diagram showing the variation of the modal amplitude and the end displacement with the external excitation frequency under different load resistances in the first-order primary resonance analysis according to the present invention;
FIG. 13 is a schematic diagram of the variation of energy with external excitation frequency under different load resistances in the first-order primary resonance analysis of the present invention
FIG. 14 is a schematic diagram showing the variation of modal amplitude, end displacement and collected energy with external excitation frequency for different external excitation force amplitudes for the first-order primary resonance analysis of the present invention;
FIG. 15 is a schematic diagram of the variation of modal amplitude, end displacement and energy collection with external excitation frequency for a first-order primary resonance analysis of the present invention at different load resistances;
Detailed Description
The following detailed description of the embodiments and the working principles of the present invention will be made with reference to the accompanying drawings.
A nonlinear model building and verification analysis method of an L-shaped piezoelectric energy collector is disclosed, which is shown by combining a flow chart 1 and comprises steps S1-S7; wherein:
s1: constructing an L-shaped energy collector physical model and a piezoelectric patch load circuit;
FIG. 2 is a schematic diagram of a physical model of an L-shaped piezoelectric energy harvester, and in FIG. 2, the L-shaped piezoelectric energy harvester includes an L-beam structure and a first concentrated mass M 1 Second lumped mass M 2 (ii) a The L-beam structure comprises a horizontal beam and a vertical beam; first lumped mass M 1 Fixed at the corner where the horizontal beam and the vertical beam intersect; second lumped mass M 2 The vertical beam is positioned on the vertical beam and can slide up and down; the natural frequency of the structure is adjusted by sliding up and down to achieve 2. It can also be seen that the piezoelectric patch load circuit comprises a piezoelectric patch and a load resistor R, wherein the piezoelectric patch is respectively adhered to the upper surface and the lower surface of the horizontal beam and connected with the load resistor R to form a parallel circuit;
s2: establishing a coordinate system, and obtaining an L-shaped energy collector coordinate distribution schematic diagram corresponding to the L-shaped energy collector physical model based on the L-shaped energy collector physical model; the schematic diagram of the distribution of the L-shaped energy harvester coordinates can be seen from figure 3.
In step S2, the coordinate distribution diagram of the L-shaped energy harvester introduces three rectangular coordinate systems: o1x1y1; o2x2y2; o3x3y3, three coordinate systems are used to describe the motion of the horizontal and vertical beams;
the horizontal beam and the vertical beam are regarded as three parts, wherein the horizontal beam is a first beam section, and the vertical beam is regarded as a second beam section and a third beam section. The motion of the horizontal and vertical beams uses the euler-bernoulli beam assumption.
S3: derivation of control equation of energy collection system of L-shaped energy collector by adopting Hamilton principle(ii) a Specifically, the method comprises the following steps: the Hamiltonian equation is shown below:
Figure GDA0003927254570000161
wherein: t, V and W nc Respectively representing the kinetic energy, the potential energy and the virtual work done by the external force; arc length corresponding to beam section i (i =1,2, 3) is represented by s i Showing the axial and lateral displacements on the beam sections by u, respectively i (s i T) and v i (s i And t) represents; theta i Representing the angular displacement of each beam segment before and after deformation at section i. The kinetic energy T and potential energy V may be expressed as follows:
Figure GDA0003927254570000162
Figure GDA0003927254570000163
wherein m is b1 And m b2 Respectively represents the mass per unit length of the horizontal beam and the vertical beam, and the expressions are respectively: m is b1 ＝b s1 ρ s1 h s1 +2b p1 ρ p h p ，m b2 ＝b s2 ρ s2 h s2 . Subscripts s and p denote the structural layer and piezoelectric layer, respectively, subscripts 1 and 2 denote the horizontal and vertical beam sections, respectively, ρ denotes the density of the material, h and b denote the height and width of the beam sections, respectively, J 1 And J 2 Respectively representing lumped masses M 1 And M 2 Inertia of rotation of 1 And EI 2 Respectively representing the bending stiffness of the horizontal and vertical beam sections, and the expressions are respectively:
Figure GDA0003927254570000164
Figure GDA0003927254570000165
wherein E s And E p The young's modulus of elasticity of the structural layer and the piezoelectric layer, respectively.
u i (s i ,t)、v i (s i T) and θ i (s i The geometric non-linear relationship between t) can be expressed as:
Figure GDA0003927254570000166
and
Figure GDA0003927254570000167
therefore, the curvature θ of an arbitrary position i ′(s i T), angular velocity
Figure GDA0003927254570000168
And axial displacement u i (s i T) can be expressed as follows:
Figure GDA0003927254570000169
Figure GDA00039272545700001610
wherein the symbols' "and" · "respectively denote the pairs s i And t is derived. In equations (4) and (5), v is ignored i ′(s i T) and higher order terms. The virtual work done by non-conservative forces can be expressed as: w ne ＝W ele +W damp (7) (ii) a Wherein, W ele And W damp Indicated as the virtual work due to the electrical and damping forces, respectively. The geometric nonlinearity of the piezoelectric sheet is considered here, so the virtual work done by the electric power can be expressed as:
Figure GDA0003927254570000171
wherein M is ele For the effect of the charge on the resulting bending moment, the expression is as follows:
Figure GDA0003927254570000172
v (t) is a voltage generated by the piezoelectric sheet due to deformation. H (a), (b) s1 ) Is a Helvessel step function, e 31 ＝E p d 31 In order to obtain the piezoelectric stress coefficient,
Figure GDA0003927254570000178
is a piezoelectric coupling term expressed as
Figure GDA0003927254570000179
Virtual work W by damping force damp Comprises the following steps:
Figure GDA0003927254570000173
Figure GDA0003927254570000174
is a bending moment generated by the strain rate, c in the above formula s And c a Equivalent viscous strain and air damping coefficient of the cantilever beam, I i Is the cross-sectional moment of inertia of the beam. Substituting equations (2), (3) and (7) into hamilton equation (1) yields the control equation for the energy harvesting system:
Figure GDA0003927254570000175
Figure GDA0003927254570000176
Figure GDA0003927254570000177
wherein N is 1 、N 2 And N 3 The axial loads of the beam sections 1,2 and 3, respectively, are expressed as:
Figure GDA0003927254570000181
and, the linear boundary conditions (15) are:
Figure GDA0003927254570000182
Figure GDA0003927254570000183
Figure GDA0003927254570000184
s4: obtaining a circuit equation of the piezoelectric patch load circuit by adopting Gauss law, and obtaining a control equation of the reduced energy acquisition system by combining the circuit equation; specifically, the method comprises the following steps:
an expression of the circuit equation is obtained by gauss's law:
Figure GDA0003927254570000185
d is the electrical displacement vector and n is the external normal vector. Electric displacement D 2 Is expressed as
Figure GDA0003927254570000186
Wherein geometric non-linearity of the piezoelectric sheet is taken into account
Figure GDA0003927254570000187
Figure GDA0003927254570000188
Is the dielectric constant component at constant strain. Substituting equation (17) into equation (16), the circuit equation for the system can be expressed as:
Figure GDA0003927254570000189
in order to analyze the response of the energy collection system, the invention adopts a Galerkin method to displace the transverse vibration v i (s i T) separation into a spatial variable phi ij (s i ) And a time variable q j (t)：
Figure GDA00039272545700001810
Wherein phi is ij (s i ) And q is j (t) are the jth order mode shape and modal coordinates of the system, respectively. The mode shape of the energy harvesting system may be expressed as:
Figure GDA00039272545700001811
wherein the content of the first and second substances,
Figure GDA00039272545700001812
Figure GDA0003927254570000191
the specific derivation of the vibration shape is described in the summary of the invention.
Coefficient A in the above formula ij 、B ij 、C ij And D ij The coefficients are coefficients of the j-th order mode of the system, and the coefficients are obtained through boundary conditions and orthogonal conditions. The boundary elements after separation of the variables are as follows:
Figure GDA0003927254570000192
and orthogonal bars:
Figure GDA0003927254570000193
Figure GDA0003927254570000194
in the above formula, s and r represent the modal number of the system, δ rs Is the Kronecker delta function, and delta rs ＝1(r＝s)，δ rs ＝0(r≠s)。
By substituting equation (19) into (11), (12), (13) and (18) and considering the first two-order modes of the system, the reduced control equation can be obtained through the orthogonal condition and the boundary condition:
Figure GDA0003927254570000195
Figure GDA0003927254570000196
Figure GDA0003927254570000197
in the above formula, the first and second carbon atoms are,
Figure GDA0003927254570000198
indicating vertical acceleration, ζ 1 And ζ 2 Respectively representing the first two-stage mechanical damping ratio, omega, of the system 1 And ω 2 Respectively represent the first two order circle frequencies of the system,
Figure GDA0003927254570000199
representing the capacitance. Dimensionless coefficient m k 、n k And η l The details of the content are described in the summary of the invention, and are not described herein.
S5: setting physical parameters and geometrical parameters of the L-shaped energy collector physical model, acquiring the physical parameters and geometrical parameters of the L-shaped beam structure from the physical parameters and geometrical parameters of the L-shaped energy collector physical model, and establishing a finite element model of the L-shaped beam structure by adopting large finite element general software ANSYS;
the physical and geometrical parameters of the L-shaped energy harvester physical model can be seen from table 1. The physical and geometrical parameters of the L-beam structure are detailed in table 2.
Table 1 physical and geometric parameters of L-shaped energy harvester
Figure GDA0003927254570000201
Table 2 physical and geometrical parameters of the L-beam structure
Figure GDA0003927254570000202
Figure GDA0003927254570000211
In step S5, in building the L-Beam structure finite element model, beam188 and Mss21 units are respectively used to simulate the Beam structure and the lumped mass, and the geometric nonlinearity of the L-Beam structure is considered by the "NLGEOM, ON" command. The L-beam structure finite element model is detailed in FIG. 4.
In this embodiment, the first two frequencies are set to 8.15Hz and 16.49Hz respectively, and the first two frequencies calculated by the finite element software ANSYS are set to 8.3Hz and 16.48Hz respectively, so that the maximum error of the frequencies calculated by the theoretical model and the finite element model is 1.81%.
As can be seen from FIG. 5, the modal amplitude a 1 And a 2 Respectively showing the vibration displacement of the first and second order modes of the end part of the third beam section of the structure. Modal amplitude a of finite element model 1 And a 2 The displacement time-course curve of the end part of the third beam section can be obtained by performing transient dynamic analysis on the model, the corresponding modal amplitude can be obtained by performing fast Fourier transform on the time-course curve, and as can be seen from FIG. 5, the frequency response curves of the theoretical model and the finite element model are compared under the second-order main resonance. The solid and dashed lines in the figure represent the stable and unstable solutions, respectively. Wherein the unstable region predicted by the theoretical model is 16.4Hz-16.435Hz, and the unstable region calculated by the finite element model is 16.4Hz-16.48Hz.
S6: verifying the natural vibration frequency, time-course response and internal resonance response of the L-beam structure finite element model to obtain a verified L-beam structure finite element model;
as can be seen in connection with fig. 6, the modal coordinates calculated by the theoretical model in the unstable region are compared with the tip displacement time-course curves calculated by the finite element model. For the theoretical model, the boundary where the unstable region starts is the external excitation frequency of 16.4Hz, and the response of the structure is non-periodic motion, which is detailed in (a) and (d) of FIG. 6. When the external excitation frequency is gradually increased to 16.43Hz, the response of the structure becomes chaotic motion, as shown in fig. 6 (b) and (e). When the external excitation frequency is increased to 16.435Hz, the response of the structure becomes non-periodic again, as shown in fig. 6 (c) and (f). For finite element models, the structure has similar non-linear behavior in the corresponding unstable region. The displacement time course curve of the structure end part is from the non-periodic motion (figure 6 (g)) when the external excitation frequency is 16.4Hz, to the chaotic motion (figure 6 (h)) when the external excitation frequency is 16.45Hz, and then to the non-periodic motion (figure 6 (i)) when the external excitation frequency is 16.48Hz. In conclusion, fig. 6 shows that the unstable region predicted by the theoretical model is in good agreement with the unstable region calculated by the finite element model.
S7: and analyzing the influence of the external load resistance on the self-vibration frequency and the damping ratio of the verified L-beam structure finite element model, and analyzing the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the conditions of first-order and second-order main resonance. When the influence of the external load resistance on the natural vibration frequency and the damping ratio of the verified L-beam structure finite element model is analyzed, the following variable relation is introduced:
Figure GDA0003927254570000212
based on the variable relationship of (27), the control equation can be rewritten as:
Figure GDA0003927254570000213
Figure GDA0003927254570000214
Figure GDA0003927254570000221
therefore, the linear coefficient matrix of the variables in (28) is:
Figure GDA0003927254570000222
in an L-type energy harvester system, the frequency and damping of the system can be determined by the eigenvalues of the matrix B to discuss the effect of the external load resistance on the frequency and damping of the structure. The frequency and damping ratio of the system as a function of load resistance is shown in figure 7. It can be seen from fig. 6 (c) that the ratio of the first secondary frequency of the system is almost maintained around 1 4 The first-order modal damping ratio of the system reaches the maximum value in ohm; when the load resistance value is R =2.2 × 10 4 The second order modal damping ratio of the system reaches a maximum at ohm. The dependence of the natural frequency and the damping of the coupling system on the load resistance has important significance for the later energy collection and the analysis of the structural vibration displacement. And analyzing the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the condition of second-order main resonance.
The variation of modal amplitude and end vibration displacement with external excitation frequency at different load resistances is shown in fig. 8, where F =0.5m/s2. It can be seen from the figure that as the resistance value of the load increases, the excitation frequency region corresponding to the occurrence of internal resonance moves to the right, which can be explained from the change of the second order frequency of the system in fig. 7 (a), (b) with the increase of the resistance. FIG. 8 shows that the end displacement of the system in the inner resonance region is mainly determined by its first order modal amplitude a 1 The end displacement in the non-resonance region is determined by the second-order modal amplitude a 2 And (6) determining. When the load resistance value R =2.2 × 10 4 ohm & R =4 × 10 4 Modal amplitude a at ohm 1 Can be controlled to a minimum. As can be seen from fig. 7 (d), the load resistance values are the resistance values corresponding to the maximum-second order modal damping ratio of the system. In addition, as can be seen from fig. 8, when the load resistance value R =2.2 × 10 4 ohm & R =4 × 10 4 ohm time, system endCan be controlled to a minimum.
The energy collected by the system and the energy provided by each order of modal vibration vary with the external excitation frequency at different load resistances as shown in fig. 9. Wherein F =0.5m/s 2.The energy generated by the first and second order mode vibration of the system is respectively P 1 And P 2 Is represented by P 1 And P 2 The method is obtained by performing fast Fourier transform on a time-course curve of total energy acquired by the system. FIG. 9 shows that the energy collected by the system is mainly P 2 Providing, when the load resistance value R =2.2 × 10 4 ohm & R =4 × 10 4 The energy collected by the system is larger at ohm, mainly because the two-order modal damping ratio of the system reaches the maximum value under the load resistance value, and the maximum energy collected by the system corresponds to the maximum damping. More importantly, when the resistance value R =4 × 10 4 The system can not only collect the maximum energy when in ohm, but also the vibration displacement of the end part of the system is the minimum. The dotted lines in fig. 8 and 9 indicate that the response of the system in the external excitation frequency region corresponding to the dotted line is unstable, and the specific range of the instability is 16.76Hz to 16.8Hz.
The modal amplitude, end displacement and collected energy at different external excitation frequencies as a function of excitation force are shown in fig. 10. Wherein R =10 5 The width of the inner resonance region in ohm is obviously influenced by the magnitude of the exciting force, and the larger the exciting force is, the larger the inner resonance region is. Meanwhile, it can be seen that the larger the excitation force is, the larger the bandwidth of the collected energy is, and the larger the collected energy value is. In addition, the larger the end displacement of the system.
Fig. 11 shows the variation of modal amplitude, end displacement and collected energy with the magnitude of the excitation force for different load resistances. Wherein f =16.55Hz. Modal amplitude a 1 Increasing with increasing excitation force, however, the modal amplitude a 2 The amplitude of the mode is increased along with the increase of the excitation amplitude, and when the excitation amplitude is larger than 0.2m/s2, the amplitude of the mode is a 2 The increasing tendency of (b) is suppressed. The displacement of the end part and the collected energy increase along with the increase of the exciting force when the load resistance R =2.2 multiplied by 10 4 ohm & R =4 × 10 4 System maximum acquisition at ohm timeThe large energy also controls the vibration displacement to the minimum.
And analyzing the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the condition of first-order main resonance.
The modal amplitude, end displacement, and variation with excitation frequency for different load resistances are shown in fig. 12. Wherein F =1m/s2. FIGS. 12 (a) and 12 (b) show the modal amplitude a of the endpoint at different load resistances 1 And a 2 As a function of the excitation frequency. It can be seen from the figure that the modal amplitude a is over the entire external excitation frequency range 1 The contribution to the displacement of the end point plays a major role. Fig. 13 also shows that, in the internal resonance region, when the load resistance R =4 × 10 4 At ohm, the system can harvest the maximum energy. The broken lines in fig. 12 and 13 both indicate when the load resistance R =10 3 The response of the system at ohm is unstable in the range of 8.23Hz to 8.33Hz of the external excitation frequency; when load resistance R =10 6 The response of the system at ohm is unstable in the range of 8.38Hz to 8.44Hz of the external excitation frequency.
The variation of the end modal amplitude, end displacement and energy harvested by the system with excitation frequency at different excitation forces is shown in fig. 14. Wherein R =10 5 And (4) ohm. Similar to the second-order primary resonance, the resonance region width of the first-order primary resonance also increases with an increase in the excitation force. FIGS. 14 (a) and 14 (b) also show that at first-order primary resonance, the vibrational displacement at the end of the system is dominated by the first-order modal amplitude a of the system 1 Controlling the energy collected by the system to be mainly P 2 Provided is a method. In addition, FIG. 14 also shows that with increasing external excitation force, the response at the first order frequency of the system gradually changes from stable to unstable, similar to the law found previously in paper A.G.Haddow, A.D.S.Barr, D.T.Mook, the ecological and experimental study of mode interaction in a two-degree-of-free structure, J.Sound V.ib.97 (1984) 451-473.
In fig. 15, F =8.29Hz, the modal amplitude, end displacement and collected energy increase with increasing external excitation force F, similar to that analyzed in fig. 11 when the load resistance R =4 × 10 Hz 4 ohm & R =2.2 × 10 4 ohmIn time, the system can collect a large amount of energy. Note that the value is when the load resistance is R =10 3 At ohm, with the gradual increase of the external stimulus F, the response of the system gradually changes from stable to unstable, which is the same as Haddow [21]]The laws found before are similar.
It should be noted that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art should also make changes, modifications, additions or substitutions within the spirit and scope of the present invention.

Claims (4)

1. A nonlinear model building and verification analysis method of an L-shaped piezoelectric energy collector is characterized by comprising the following steps:
s1: constructing an L-shaped energy collector physical model and a piezoelectric patch load circuit;
s2: establishing a coordinate system, and obtaining an L-shaped energy collector coordinate distribution schematic diagram corresponding to the L-shaped energy collector physical model based on the L-shaped energy collector physical model;
s3: deducing a control equation of an energy acquisition system of the L-shaped energy collector by adopting a Hamilton principle;
s4: obtaining a circuit equation of the piezoelectric patch load circuit by adopting Gauss law, and obtaining a control equation of the reduced energy acquisition system by combining the circuit equation;
s5: setting physical parameters and geometrical parameters of the L-shaped energy collector physical model, acquiring the physical parameters and geometrical parameters of the L-shaped beam structure from the physical parameters and geometrical parameters of the L-shaped energy collector physical model, and establishing a finite element model of the L-shaped beam structure by adopting large finite element general software ANSYS;
s6: verifying the natural vibration frequency, time-course response and internal resonance response of the L-beam structure finite element model to obtain a verified L-beam structure finite element model;
s7: analyzing the influence of the external load resistance on the self-vibration frequency and the damping ratio of the verified L-beam structure finite element model, and analyzing the influence of the load resistance, the excitation frequency and the excitation amplitude on system energy acquisition and displacement response under the first-order and second-order main resonance conditions;
when the physical model of the L-shaped energy collector and the piezoelectric patch load circuit are constructed in the step S1, the following steps are set:
the L-shaped piezoelectric energy collector comprises an L-beam structure and a first concentrated mass M 1 Second lumped mass M 2 (ii) a The L-shaped beam structure comprises a horizontal beam and a vertical beam;
first lumped mass M 1 Fixed at the corner where the horizontal beam and the vertical beam intersect;
second lumped mass M 2 The vertical beam is positioned on the vertical beam and can slide up and down;
the piezoelectric piece load circuit comprises piezoelectric pieces and a load resistor R, wherein the piezoelectric pieces are respectively adhered to the upper surface and the lower surface of the horizontal beam and connected with the load resistor R to form a parallel circuit;
in step S2, the coordinate distribution diagram of the L-shaped energy harvester introduces three rectangular coordinate systems: o1x1y1; o2x2y2; o3x3y3, three coordinate systems are used to describe the motion of the horizontal and vertical beams;
the horizontal beam and the vertical beam are regarded as three parts, wherein the horizontal beam is a first beam section, and the vertical beam is regarded as a second beam section and a third beam section.
2.The nonlinear model building and verification analysis method of the L-shaped piezoelectric energy harvester according to claim 1, wherein the circuit equation of the piezoelectric piece load circuit in the step 4 is as follows:
Figure FDA0004029567250000021
r is a load resistance value;
v (t) is the voltage generated by the piezoelectric sheet due to deformation;
h p is the thickness of the piezoelectric sheet;
h s1 is the thickness of the first beam segment;
Figure FDA0004029567250000022
is the dielectric constant component at constant strain;
e 31 ＝E p d 31 is the piezoelectric stress coefficient.
3. The nonlinear model building and verification analysis method of the L-type piezoelectric energy harvester according to claim 2, wherein the step S4 of combining with the circuit equation to obtain the reduced control equation of the energy harvesting system is as follows:
Figure FDA0004029567250000031
Figure FDA0004029567250000032
Figure FDA0004029567250000033
Figure FDA0004029567250000034
represents a vertical acceleration;
ζ 1 and ζ 2 Respectively representing the mechanical damping ratios of the first two steps of the system;
ω 1 and ω 2 Respectively representing the first two-order circle frequency of the system;
Figure FDA0004029567250000035
represents a capacitance;
transverse vibrational displacement v i (s i T) separation into a spatial variable phi ij (s i ) And a time variable q j (t)：
Figure FDA0004029567250000036
Wherein phi is ij (s i ) And q is j (t) j order mode shape and modal coordinates of the system respectively; the mode shape of the energy harvesting system may be expressed as:
Figure FDA0004029567250000037
Figure FDA0004029567250000038
Figure FDA0004029567250000039
wherein the content of the first and second substances,
Figure FDA0004029567250000041
coefficient A ij 、B ij 、C ij And D ij Coefficients for the j-th order mode of the system;
s and r represent the modal number of the system, δ rs Is the Kronecker delta function, and delta rs ＝1(r＝s)，
δ rs ＝0(r≠s)；
m k 、n k And η l Are dimensionless coefficients.
4. The nonlinear model building and verification analysis method of the L-shaped piezoelectric energy harvester according to claim 1, characterized in that when a finite element model of the L-shaped Beam structure is built, beam188 and Mss21 units are respectively adopted to simulate the Beam structure and the concentrated mass, and the geometric nonlinearity of the L-shaped Beam structure is considered through an 'NLGEOM, ON' command.
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