CN109783836A - The Building Nonlinear Model and verifying analysis method of L-type piezoelectric energy collector - Google Patents
The Building Nonlinear Model and verifying analysis method of L-type piezoelectric energy collector Download PDFInfo
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Abstract
The invention discloses the Building Nonlinear Model of L-type piezoelectric energy collector and verifying analysis method, this method constructs L-type energy collecting device physical model, piezoelectric patches load circuit and coordinate distribution schematic diagram；And derive the governing equation of the governing equation of the energy collecting system of collector, the circuit equation of piezoelectric patches load circuit and energy collecting system；Pass through setup parameter and establishes and demonstrate L girder construction finite element model；Influence of the external load resistors to the natural frequency of vibration and damping ratio of the L girder construction finite element model after verifying is analyzed, and analyzes the influence of load resistance, driving frequency and excitation amplitude to system capacity acquisition and dynamic respond in the case of single order, second order primary resonance.The utility model has the advantages that the reasonability of the theoretical model proposed using the model analysis of finite element and TRANSIENT DYNAMIC ANALYSIS verifying；The frequency of energy collecting system and damping are affected by load resistance.
Description
Technical field
The present invention relates to L-type piezoelectric energy collector technical field, specifically a kind of L-type piezoelectric energy collector
Building Nonlinear Model and verifying analysis method.
Background technique
In recent years, the Utilizing question of renewable energy receives more and more extensive concern.Wherein, the vibrational energy in environment
It is a kind of most commonly seen renewable energy.Vibrational energy in environment can be converted to electric energy by piezoelectric energy collector, and can
Self-power supply device is used to the electric energy for carrying out conversion use.For example, MEMS, wireless sensor and monitoring structural health conditions
Deng other aspects.
Since vibrational energy is the energy form that one of environment is widely present, adopted based on the energy under substrate excitation
Storage causes the extensive concern of researcher.Previous research is concentrated mainly on single-cantilever piezoelectric type energy collector, this
Piezoelectric energy collector can only in the natural frequency of vibration of the external driving frequency close to structure could collecting energy, when external excitation frequency
When rate is far from its natural frequency of vibration, the collected energy of system institute can be reduced rapidly, and the driving frequency in environment is also constantly to become
Change.Therefore, the proposition of wideband energy collecting device and great significance for design.For this purpose, researchers visit in terms of different
The rope working performance of piezoelectric energy collector, to widen the energy acquisition bandwidth of piezoelectric energy collector.In document [13]
Twiefel, Westermann, Survey on broadband techniques forvibration energy
harvestingJournal of Intelligent Material Systemsand Structures.24(11)1291–
In 1302.The Author (s) 2013, in, Twiefel and Westermann have studied a kind of by muti-piece piezoelectric cantilever knot
Energy collecting system made of structure is arranged side by side, every piece of beam end of the system are attached with the different mass block of quality, thus every piece of pressure
The natural frequency of vibration of electric cantilever beam is different, and system can collect energy at multiple external excitation frequencies.In document [14]
Karami, Analytical Modeling andExperimental Verification of the Vibrations of
the Zigzag Microstructure for Energy HarvestingM.Amin Karami1e-mail: arami@
vt.eduDaniel J.InmanCenter for Intelligent Materials Systems and Structures,
In Virginia Tech, 310Durham Hall, Blacksburg, VA 24061, Karami devises saw tooth like microstructures energy
Quantity collection system, in this system, the preceding 5 rank natural frequency of vibration of structure can be mutual with the increase of the length of system girder construction
It is close, realize that structure can collect energy at multiple external excitation frequencies with this.In document ENERGY HARVESTING
FROM VIBRATIONS WITH A NONLINEAR OSCILLATOR Proceedings of the ASME
2009International Design Engineering Technical Conferences&Computers and
Information in Engineering onferenceIDETC/CIE 2009August 30-September 2,2009,
San Diego, USA, Barton has studied a kind of nonlinear electromagnetic energy collecting system, which can be by non-linear hard
Change phenomenon to expand the frequency bandwidth of energy acquisition.In document [19] Nonlinear performances of an
Autoparametric vibration-based piezoelastic energy harvester, Journal of
Intelligent Material Systems and tructures1-18_The Author (s) 2016, Yan proposes one
Kind autoregressive parameter piezoelectric energy acquisition system, the system are made of bottom main structure and end with the piezoelectric cantilever of lumped mass.
This structure can realize the wideband acquisition of energy by the 2:1 internal resonance phenomenon between main structure and cantilever beam structure, together
When also can control the vibration displacement of main structure.In document [17] Broadband design of hybrid piezoelectric
In energy Harvester., Tan et al. devise it is a kind of based on wave with substrate vibration mixed tensor acquisition system come
Realize the wideband effect of energy acquisition.This is studies have shown that the region of wideband is determined by the boundary of extinguishing phenomenon, when extinguishing phenomenon
When the acquisition power of minimum determined by boundary is acceptable, the bandwidth of system acquisition extends to infinity.In document [20] A
vibration energy harvesting device withbidirectional resonance frequency
Tenability. in Published 8January 2008Online at stacks.iop.org/SMS/17/015035,
Challa et al. proposes a kind of half active cantilever piezoelectric energy collector, which mainly passes through the attraction or repulsion of magnet
Power increaseds or decreases the natural frequency of vibration of system, to expand the operating frequency range of energy collecting system.It mentioned is arrived except above-mentioned
Research except, the L-type girder construction based on internal resonance principle be also realize wide band energy acquisition ideal model.In document
[21]THEORETICAL AND EXPERIMENTAL STUDY OF MODAL INTERACTION IN A TWO-DEGREE-
In OF-FREEDOM STRUCTURE, Haddow and Barr pass through the theoretical and experimental study non-linear spy of L-type girder construction
Property.In document [23] Forced vibration of a beam system with autoparametric coupling
Effects and document [24] SIMULTANEOUS COMBINATION RESONANCES IN AN
In AUTOPARAMETRICALLY RESONANT SYSTEM, the internal resonance that Robert and Cartmell also studied L-type beam is existing
As, and it was found that when a second order primary resonance occurs, system can motivate biggish response.Based on above-mentioned discovery, in document
[25]Nonlinear Motions of Beam-Mass Structure,B.BALACHANDRAN and A.H. NAYFEH
Engineering Science and Mechanics Department Virginia Polytechnic Institute
And State UnivefMtv Blacksburg, Virginia, U.S.A. is in document [26] An Experimental
Investigation of Complicated Responses of a Two- Degree-of-Freedom Structure
In, Balachandran and Nayfeh are rung also by the power of the L-type beam of experimental and theoretical study consideration second nonlinear
It answers.The result shows that the position of the theoretical Hopf bifurcation predicted and test result are more coincide, and when external excitation frequency is distinguished
Close to structure a second order frequency when system a second order coupled mode between energy exchange will occur.L is only considered above-mentioned
On the basis of the research of type beam plane motion, document [27] Analytical and experimental
In investigations of an autoparametric beam structure, Warminski etc. has derived consideration knot
The equation of motion of the L-type girder construction of structure motion outside plane, the results showed that may between two mode of L-type girder construction planar
It can interact, structure is also possible to out-of-plane movement occur.In addition, Georgiades etc. is in document [28]
Towards linear modal analysis for an L-shaped beam:Equations of motion
Mechanics Research Communications 47(2013)50–60；With document [29] Linear Modal
Analysis of L-Shaped Beam Structures has derived the linear movement equation for considering the outer surface movement of L-type Liangping,
And quantity of parameters analysis has been done on this basis.Based on the above-mentioned research to L-type girder construction dynamic response, Erutk et al. is in text
[30] are offered on this basis by piezoelectric material in view of proposing the linear distributed parameter piezoelectricity based on L-type girder construction in structure
Energy collecting device model, and analyze the output voltage, power and the dynamic respond on structure vertex of collector.Cao et al. is in text
Offer [31] Internal resonance for nonlinear vibration energy harvesting.THE
In EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS, based on document [21] mentioned above in Haddow and
On the basis of the L-type girder construction equation of motion that Barrr is derived, L has been obtained by being introduced directly into mechanical-electric coupling item and circuit equation
Then the Nonlinear Equations of Motion of type piezoelectric energy collector has derived system under a second order primary resonance using Method of Multiple Scales
The approximate analytic solution of system response, and excitation amplitude, mechanical damping and external excitation frequency are discussed to the shadow of system output response
It rings.In document [32] A Broadband Internally-Resonant Vibratory Energy Harvester,
Chen et al. devises the non-linear L-type beam piezoelectric energy collector of magnetic, wherein acting on the cross of the magnetic force and structure in structure
It is assumed to be the form of cubic polynomial function to vibration displacement.The natural frequency of vibration of structure can pass through the distance between magnet
It adjusts, so that a second order frequency ratio of structure is maintained at 1:2 or so, and solves to have obtained system output using Method of Multiple Scales
The frequency response relationship of voltage and power, correlated results have also obtained the verifying of test data.
In addition, Harne et al. is in document [33] Leveraging nonlinear saturation-based
Phenomena in an L-shaped vibration energy harvesting system is also by internal resonance and saturation
Phenomenon analyzes the performance of L-type beam energy collecting device, the results showed that the energy collecting device can effectively improve system
Adopting can effect.Recently, Liu et al. people is in document [34] Piezoelectric energy harvesting using L-shaped
In structures, experimental study is carried out to the L-type piezoelectric energy collector under substrate incentive action, the results showed that this
The energy acquisition bandwidth of structure is far longer than the bandwidth of cantilever beam energy collecting device, however the second order frequency of the test model is simultaneously
Two times of its non-fundamental frequency, the energy exchange between one second-order modal of structure cannot be excited out.Therefore, the research is only
The voltage and power response for reflecting system change with the variation of external excitation, when driving frequency close to system the one or two from
When vibration frequency, the output response of voltage and power can reach maximum value.
Although the performance study of L-type beam piezoelectric energy collector has received widespread attention in recent years, to this piezoelectricity
The research of energy harvester is also fairly limited.As far as the applicant is aware, the geometry at present about L-type beam piezoelectric energy collector is non-
Linear mathematical model also lacks very much, and does not also consider the several of piezoelectric material usually in existing piezoelectric energy collector model
What is non-linear.
Summary of the invention
In view of the above-mentioned problems, the present invention provides a kind of Building Nonlinear Model of L-type piezoelectric energy collector and verifyings
Analysis method, Hamiton's principle and Gauss law have derived the mechanical-electric coupling point for considering structure sheaf and piezoelectric layer geometrical non-linearity
Cloth parameter model.The reasonable of the theoretical model proposed is verified using the model analysis of finite element and TRANSIENT DYNAMIC ANALYSIS
Property.Specific technical solution is as follows:
A kind of Building Nonlinear Model of L-type piezoelectric energy collector and verifying analysis method, key technology are:
S1: building L-type energy collecting device physical model and piezoelectric patches load circuit；S2: establishing coordinate system, is based on L-type energy
Collector physical model is measured, L-type energy collecting device coordinate distribution schematic diagram corresponding with L-type energy collecting device physical model is obtained；
S3: the governing equation of the energy collecting system of L-type energy collecting device is derived using Hamiton's principle；S4: using Gauss law,
The circuit equation of piezoelectric patches load circuit is obtained, and combined circuit equation obtains the controlling party of the energy collecting system after depression of order
Journey；S5: the physical parameter and geometric parameter of setting L-type energy collecting device physical model, from L-type energy collecting device physical model
Physical parameter and geometric parameter in obtain L-type girder construction parameter physical parameter and geometric parameter, and using large-scale limited
First general software ANSYS establishes L girder construction finite element model；S6: the natural frequency of vibration, the time-histories of verifying L girder construction finite element model
Response and internal resonance response, the L girder construction finite element model after being verified；S7: after analysis external load resistors are to verifying
The influence of the natural frequency of vibration and damping ratio of L girder construction finite element model, and analyze load resistance, driving frequency and excitation amplitude
Influence to system capacity acquisition and dynamic respond in the case of single order, second order primary resonance.
By above-mentioned design, the several of L-type beam piezoelectric energy collector will be derived using Hamiton's principle and Gauss law
What nonlinear model, and large size Universal Finite Element software ANSYS is used to establish the finite element model of L girder construction, pass through ANSYS
Model analysis carry out the preceding two ranks natural frequency of vibration of proof theory model, by the TRANSIENT DYNAMIC ANALYSIS of ANSYS acquire structure when
Journey response and the internal resonance response that L girder construction is verified in conjunction with Fourier transformation.
Further, when constructing L-type energy collecting device physical model and piezoelectric patches load circuit in step S1, if: L-type
Piezoelectric energy collector includes L girder construction, first lumped mass M1, second lumped mass M2；L girder construction includes horizontal beam
And vertical beam；
First lumped mass M1It is fixed on the corner of horizontal beam and vertical beam intersection；Second lumped mass M2It is located at
In vertical beam and its position can slide up and down；Piezoelectric patches load circuit includes piezoelectric patches and load resistance R, piezoelectric patches difference
It is pasted onto the upper and lower surface of horizontal beam, is connected to form parallel circuit with load resistance R；In step s 2, L-type energy collecting device
Coordinate is distributed schematic diagram and introduces three rectangular coordinate systems: O1x1y1；O2x2y2；O3x3y3, three coordinate systems are for describing horizontal beam
With the movement of vertical beam；Horizontal beam and vertical beam are considered as three parts composition, wherein horizontal beam is the first beam section, vertical beam view
For the second beam section and third beam section.
Further, the control of the energy collecting system of L-type energy collecting device is derived in step S3 using Hamiton's principle
Equation processed are as follows:
N1、N2And N3Respectively the first beam section, the second beam section, the xial feed of third beam section；
And linear barrier's condition are as follows:
The corresponding arc length s in the place beam section section i (i=1,2,3)iIt indicates, the axially and transversely displacement in each beam section point
U is not usedi(si, t) and vi(si, t) and it indicates； θi(si, t) and indicate the corner displacement occurred before and after each beam section deforms at the i of section；
liIndicate the length of the i-th beam section；
Wherein, ui(si,t)、vi(si, t) and θi(si, t) between geometrical non-linearity relationship can indicate are as follows: mb1And mb2Respectively indicate the linear mass of horizontal beam and vertical beam；Under
Mark s and p respectively indicates structure sheaf and piezoelectric layer；The subscript 1,2 of s and p respectively indicates horizontal and vertical beam section structure sheaf and pressure
Electric layer；The density of ρ expression material；H and b respectively indicates the height and width of beam section；J1And J2Respectively indicate lumped mass M1And M2's
Rotary inertia；EI1And EI2Respectively indicate the bending stiffness of horizontal and vertical beam section；Symbol " ' " and " ", respectively indicate to siWith
T derivation；V (t) is piezoelectric patches due to deforming generated voltage；csAnd caRespectively the equivalent viscous strain of cantilever beam and air
Damped coefficient；IiIt is the cross sectional moment of inertia of beam；M1For first lumped mass；M2For second lumped mass.This uses Hamilton
Principle derives the detailed process of the governing equation of the energy collecting system of L type energy collecting device are as follows: wherein: Hamilton's equation are as follows:Wherein: T, V and WncRespectively indicate the virtual work that kinetic energy, potential energy and external force are done；Beam section section i (i
=1,2,3) the corresponding arc length s in placeiIt indicates, u is used in the axially and transversely displacement in each beam section respectivelyi(si, t) and vi(si,
T) it indicates；θiIndicate the corner displacement occurred before and after each beam section deforms at the i of section；In conjunction with the document mentioned in background technique
Paper 25, kinetic energy T and potential energy V can be respectively indicated are as follows:
Wherein, mb1And mb2The linear mass of horizontal beam and vertical beam is respectively indicated, expression formula is respectively as follows: mb1=
bs1ρs1hs1+2bp1ρphp, mb2=bs2ρs2hs2.Subscript s and p respectively indicate structure sheaf and piezoelectric layer, and subscript 1 and 2 respectively indicates
Horizontal and vertical beam section, ρ indicate the density of material, and h and b respectively indicate the height and width of beam section, J1And J2Respectively indicate concentration
Mass M1And M2Rotary inertia, EI1And EI2The bending stiffness of horizontal and vertical beam section is respectively indicated, expression formula is respectively as follows:Wherein EsAnd EpThe respectively Young springform of structure sheaf and piezoelectric layer
Amount.
Its, according to paper Z.Yan, H.Taha, T.Tan, Nonlinear characteristics of an
It is found that u in autoparametric vibration system, J.Sound V ib.390 (2017) 1-22.i(si,t)、vi
(si, t) and θi(si, t) between geometrical non-linearity relationship can indicate are as follows:
Therefore, the curvature θ of any positioni′(si, t), angular speedWith axial displacement ui(si, t) and it can be expressed as
In equation (4) and (5), v is had ignoredi′(si, t) 3 ranks and more
Higher order term.The virtual work that nonconservative force is done can indicate are as follows: Wne=Wele+Wdamp (7)
Wherein, WeleAnd WdampIt is expressed as the virtual work done by electricity and damping force.The several of piezoelectric patches are considered herein
What is non-linear, therefore the virtual work that electric power is done can indicate are as follows:
Wherein, MeleGenerated moment of flexure is influenced for charge, expression formula is as follows:
Wherein, V (t) is piezoelectric patches due to deforming generated voltage.H(s1) it is heaviside step function, e31=Epd31
For piezoelectric stress coefficient,It is piezoelectricity coupling terms, expression formula isThe virtual work done by damping force
WdampAre as follows:
Wherein,It is the moment of flexure as caused by strain rate, c in above formulasAnd caRespectively cantilever beam etc.
Imitate viscous strain and air damping coefficient, IiIt is the cross sectional moment of inertia of beam.
Equation (2), (3) and (7) is brought into the energy acquisition of available L-type energy collecting device in Hamilton's equation (1)
The governing equation of system.
Further, in step 4 piezoelectric patches load circuit circuit equation are as follows:
R is load resistor value；V (t) is piezoelectric patches due to deforming generated voltage；hpFor the thickness of piezoelectric patches；hs1For
The thickness of first beam section；It is the permittivity component under constant strain；e31=Epd31For piezoelectric stress coefficient； It is piezoelectricity coupling terms.
The circuit equation of the piezoelectric patches load circuit specifically derives step are as follows: by Gauss law, available circuit side
The expression formula of journey:
Wherein Gauss law can be detailed in paper IEEE 176-1987-IEEE, Standard on
Piezoelectricity.doi:10.1109/IEEESTD, 1988. wherein D be dielectric displacement vector, n is outer normal vector.Electricity
It is displaced D2Formula formula are as follows:
Consider piezoelectric patches geometrical non-linearity It is constant strain
Under permittivity component.It brings equation (17) into (16), obtains the circuit equation of electric piece load circuit.For analysing energy collection
The response of system, using the golden method of gal the Liao Dynasty by transverse vibrational displacement vi(si, t) and it is separated into space variable φij(si) and time change
Measure qj(t):
φij(si) and qj(t) be respectively system jth rank vibration shape and modal coordinate.The vibration shape of energy collecting system can be with
It indicates are as follows:
Coefficient Aij、Bij、 CijAnd DijFor the coefficient of system jth rank mode, these coefficients are obtained by boundary condition and orthogonality condition.
Border member after variables separation is as follows:
With orthogonal item:
In above formula, s and r represent the mode number of system, δrsIt is Kronecker δ function, and δrs=1 (r=s), δrs=0 (r
≠s)。
The specific derivation process for shape of wherein shaking are as follows: in order to derive the model function of vibration of L-type girder construction, in equation (11)-(13)
In remove damping term, piezoelectricity coupling terms and real number item, obtain the Linear Control equation of system:
By transverse vibrational displacement vI, jIt is separated into space variable and time variable, it may be assumed that
Then it (A.4) is brought into (A.1), obtains following expression:
Assuming thatSolution form are as follows:
K1It is real constant, will (A.6) be brought into (A.5) and obtain the expression formula of x:
The root of x in above formula are as follows: x1,2,3,4=± αj, ± i αj.Wherein,Therefore, the model function of vibration of beam section 1 can be with
It is expressed as:
It will (A.4) bring into and (A.2) obtain following expression:
Similarly, it is assumed thatExpression formula are as follows:
Wherein, K2It is real constant, will (A.10) be brought into (A.9) and obtain the expression formula about parameter y:
The root of above formula (A.11) can indicate are as follows:Its
In,The model function of vibration of second beam section can indicate are as follows:For the vibration shape for deriving beam section 3
Function introduces constant term in equation (A.3)So equation (A.3) can indicate are as follows:
V (s)=v3(s3)+v2(l2), with (A.1) similar, equation
(A.13) solution are as follows:
Further, combined circuit equation obtains the governing equation of the energy collecting system after depression of order in step S4 are as follows:
Indicate vertical acceleration；ζ1And ζ2Respectively indicate two rank mechanical damping ratios before system；ω1And ω2Point
Two rank circular frequency not before expression system；Indicate capacitor transverse vibrational displacement vi(si, t) and it is separated into space variable
φij(si) and time variable qj(t):φij(si) and qj(t) be respectively system jth rank vibration shape
And modal coordinate；
The vibration shape of energy collecting system are as follows:
Wherein,
Coefficient Aij、Bij、CijAnd DijFor the coefficient of system jth rank mode；S and r represents the mode number of system, δrsIt is
Kronecker δ function, and δrs=1 (r=s), δrs=0 (r ≠ s)；mk、nkAnd ηlFor dimensionless factor.
The governing equation of energy collecting system after depression of order be by by equation (19) be brought into (11), (12), (13) and
(18) in, and consider the preceding two ranks mode of system, obtained by orthogonality condition and boundary condition.
Wherein, dimensionless factor mk、nkAnd ηlDerivation process are as follows: it is convenient to write for coefficient expressions, assumes EI herein2
=EI3It sets up, therefore mk, nkAnd ηlExpression formula can indicate are as follows:
m13=m12 (B.13)
m19=m18 (B.19) m20=2m21 (B.20)
m23=2m22 (B.23)
n13=n12 (B.38)
n19=n18B.44； n20=2n21 B.45
n23=2n22 (B.48)
Further, when establishing L girder construction finite element model, Beam188 and Mss21 unit is respectively adopted and carrys out simulating beam
Structure and lumped mass, by the way that " NLGEOM, ON " order the geometrical non-linearity to consider L-type girder construction.
In the step s 7, the natural frequency of vibration and resistance of the analysis external load resistors to the L girder construction finite element model after verifying
When the influence of Buddhist nun's ratio, following vector relations are introduced:Based on the variable relation of (27), governing equation
It can be rewritten as formula (28):
The linear coefficient matrix of variable in formula (28) are as follows:
In L-type piezoelectric energy collector system, the frequency and resistance of system can be determined by the characteristic value of matrix B
Buddhist nun, to discuss external load resistors to the frequency of structure and the influence of damping.
Beneficial effects of the present invention: the present invention has derived consideration structure sheaf and piezoelectricity using Hamiton's principle and Gauss law
The mechanical-electric coupling distributed parameter model of layer geometrical non-linearity.It is tested using the model analysis and TRANSIENT DYNAMIC ANALYSIS of finite element
Demonstrate,prove the reasonability of the theoretical model proposed；The frequency of energy collecting system and damping are affected by load resistance.
Detailed description of the invention
Fig. 1 is flow chart of the method for the present invention；
Fig. 2 is L-type energy collecting device physical model schematic diagram of the present invention；
Fig. 3 is L-type energy collecting device coordinate distribution schematic diagram of the present invention；
Fig. 4 is L-type girder construction FEM model schematic diagram of the present invention；
Fig. 5 is theoretical prediction result of the present invention and finite element result comparison schematic diagram；
Fig. 6 be modal amplitudes of the present invention, end displacement unstable region time-history curves schematic diagram；
Fig. 7 be energy collecting system of the present invention frequency and damping ratio with resistance variation schematic diagram；
Fig. 8 is under second order analysis of main resonance different loads resistance of the present invention, and modal amplitudes, end displacement are with external excitation frequency
Variation schematic diagram；
Fig. 9 is under second order analysis of main resonance different loads resistance of the present invention, the energy of acquisition with external excitation frequency variation
Schematic diagram；
Figure 10 is modal amplitudes, end displacement and acquisition under second order analysis of main resonance difference external excitation power amplitude of the present invention
Energy with external excitation frequency variation schematic diagram；
Figure 11 is modal amplitudes, end displacement and collecting energy under second order analysis of main resonance different loads resistance of the present invention
With the variation schematic diagram of external excitation frequency；
Figure 12 is under single order analysis of main resonance different loads resistance of the present invention, and modal amplitudes, end displacement are with external excitation frequency
The variation schematic diagram of rate；
Figure 13 is under single order analysis of main resonance different loads resistance of the present invention, energy with external excitation frequency variation schematic diagram
Figure 14 is modal amplitudes, end displacement and acquisition under single order analysis of main resonance difference external excitation power amplitude of the present invention
Energy with external excitation frequency variation schematic diagram；
Figure 15 is modal amplitudes, end displacement and collecting energy under single order analysis of main resonance different loads resistance of the present invention
With the variation schematic diagram of external excitation frequency；
Specific embodiment
Specific embodiment and working principle of the present invention will be described in further detail with reference to the accompanying drawing.
A kind of Building Nonlinear Model of L-type piezoelectric energy collector and verifying analysis method, can be in conjunction with flow chart Fig. 1
Find out, including step S1-S7；Wherein:
S1: building L-type energy collecting device physical model and piezoelectric patches load circuit；
Fig. 2 is L-type energy collecting device physical model schematic diagram, and in Fig. 2, L-type piezoelectric energy collector includes L beam knot
Structure, first lumped mass M1, second lumped mass M2；L girder construction includes horizontal beam and vertical beam；First lumped mass
M1It is fixed on the corner of horizontal beam and vertical beam intersection；Second lumped mass M2In vertical beam and its position can more than
Lower slider；By sliding up and down come the natural frequency of vibration of adjustment structure to realize 2:1 internal resonance.It can also be seen that piezoelectric patches load
Circuit includes piezoelectric patches and load resistance R, and piezoelectric patches is respectively adhered on the upper and lower surface of horizontal beam, and be connected shape with load resistance R
At parallel circuit；
S2: establishing coordinate system, is based on L-type energy collecting device physical model, obtains and L-type energy collecting device physical model pair
The L-type energy collecting device coordinate distribution schematic diagram answered；L-type energy collecting device coordinate is distributed schematic diagram as can be seen from Figure 3.
In step s 2, L-type energy collecting device coordinate distribution schematic diagram introduces three rectangular coordinate systems: O1x1y1；O2x2y2；
O3x3y3, three coordinate systems are used to describe the movement of horizontal beam and vertical beam；
Horizontal beam and vertical beam are considered as three parts composition, wherein horizontal beam is the first beam section, vertical beam is considered as the second beam
Section and third beam section.The movement of horizontal beam and vertical beam is assumed using Euler-Bernoulli Jacob's beam.
S3: the governing equation of the energy collecting system of L-type energy collecting device is derived using Hamiton's principle；It is specific: to breathe out
Close equation is as follows:Wherein: T, V and WncKinetic energy, potential energy and external force is respectively indicated to be done
Virtual work；The corresponding arc length s in the place beam section section i (i=1,2,3)iIt indicates, the axially and transversely displacement in each beam section point
U is not usedi(si, t) and vi(si, t) and it indicates；θiIndicate the corner displacement occurred before and after each beam section deforms at the i of section.Kinetic energy T and
Potential energy V can be respectively indicated and is as follows:
Wherein, mb1And mb2The linear mass of horizontal beam and vertical beam is respectively indicated, expression formula is respectively as follows: mb1=
bs1ρs1hs1+2bp1ρphp, mb2=bs2ρs2hs2.Subscript s and p respectively indicate structure sheaf and piezoelectric layer, and subscript 1 and 2 respectively indicates
Horizontal and vertical beam section, ρ indicate the density of material, and h and b respectively indicate the height and width of beam section, J1And J2Respectively indicate concentration
Mass M1And M2Rotary inertia, EI1And EI2The bending stiffness of horizontal and vertical beam section is respectively indicated, expression formula is respectively as follows:Wherein EsAnd EpRespectively structure sheaf and piezoelectric layer
Young's modulus of elasticity.
ui(si,t)、vi(si, t) and θi(si, t) between geometrical non-linearity relationship can indicate are as follows:WithTherefore, the curvature θ of any positioni′(si, t), angular speedWith axial displacement ui(si, t) can be expressed as it is as follows:
Wherein, symbol " ' " and " ", respectively indicate to siWith t derivation.In equation (4) and (5), v is had ignoredi′(si,t)
3 ranks and higher order item.The virtual work that nonconservative force is done can indicate are as follows: Wne=Wele+Wdamp(7)；Wherein, WeleWith
WdampIt is expressed as the virtual work done by electricity and damping force.The geometrical non-linearity of piezoelectric patches, therefore electric power are considered herein
The virtual work done can indicate are as follows:
Wherein, MeleGenerated moment of flexure is influenced for charge, expression formula is as follows:
V (t) is piezoelectric patches due to deforming generated voltage.H(s1) it is heaviside step function, e31=Epd31For piezoelectricity
Stress coefficient,It is piezoelectricity coupling terms, expression formula isThe virtual work W done by damping forcedampAre as follows:
It is the moment of flexure as caused by strain rate, c in above formulasAnd caRespectively cantilever beam is equivalent
Viscous strain and air damping coefficient, IiIt is the cross sectional moment of inertia of beam.Bring equation (2), (3) and (7) into Hamilton's equation
(1) governing equation of energy collecting system is obtained in:
Wherein, N1、N2And N3The respectively xial feed of beam section 1,2 and 3, expression formula are as follows:
And linear barrier's condition (15) are as follows:
After S4: using Gauss law, obtain the circuit equation of piezoelectric patches load circuit, and combined circuit equation obtains depression of order
Energy collecting system governing equation；It is specific:
The expression formula of circuit equation is obtained by Gauss law:D is electricity
Motion vector, n are outer normal vectors.Dielectric displacement D2Expression formula be
Wherein, consider piezoelectric patches geometrical non-linearity It is permanent
Permittivity component under fixed strain.Equation (17) is brought into equation (16), the circuit equation of system can indicate are as follows:
For the response of analysing energy collection system, the present invention uses the golden method of gal the Liao Dynasty by transverse vibrational displacement vi(si, t) point
From for space variable φij(si) and time variable qj(t):
Wherein, φij(si) and qj(t) be respectively system jth rank vibration shape and modal coordinate.The vibration shape of energy collecting system
It can indicate are as follows:
Wherein,
The specific derivation of vibration shape is detailed in summary of the invention.
Coefficient A in above formulaij、Bij、CijAnd DijFor the coefficient of system jth rank mode, these coefficients are to pass through boundary condition
It is obtained with orthogonality condition.Border member after variables separation is as follows:
With orthogonal item:
In above formula, s and r represent the mode number of system, δrsIt is Kronecker δ function, and δrs=1 (r=s), δrs=0 (r
≠s)。
Equation (19) is brought into (11), (12), (13) and (18), and considers the preceding two ranks mode of system, by just
Governing equation after friendship condition and the available depression of order of boundary condition:
In above formula,Indicate vertical acceleration, ζ1And ζ2Two rank mechanical damping ratios before system are respectively indicated,
ω1And ω2Two rank circular frequency before system are respectively indicated,Indicate capacitor.Dimensionless factor mk、nkAnd ηlIt is specific in
Appearance is detailed in Summary, and details are not described herein.
S5: the physical parameter and geometric parameter of setting L-type energy collecting device physical model, from L-type energy collecting device physics
The physical parameter and geometric parameter of the parameter of L-type girder construction are obtained in the physical parameter and geometric parameter of model, and using large-scale
Universal Finite Element software ANSYS establishes L girder construction finite element model；
The physical parameter and geometric parameter of L-type energy collecting device physical model are as can be seen from Table 1.The ginseng of L-type girder construction
See Table 2 for details for several physical parameters and geometric parameter.
The physics and geometric parameter of table 1L type energy collecting device
The physical parameter and geometric parameter of the parameter of table 2L type girder construction
In step s 5, when establishing L girder construction finite element model, Beam188 and Mss21 unit is respectively adopted to simulate
Girder construction and lumped mass, by the way that " NLGEOM, ON " order the geometrical non-linearity to consider L-type girder construction.L girder construction is limited
Meta-model is detailed in Fig. 4.
In the present embodiment, it is respectively 8.15Hz and 16.49Hz that preceding two order frequency, which is respectively set,, FEM-software ANSYS
Preceding two order frequency calculated is respectively 8.3Hz and 16.48Hz, therefore theoretical model and finite element model frequency calculated
Worst error be 1.81%.
From fig. 5, it can be seen that modal amplitudes a1And a2Respectively indicate the first and second rank mode of structure third beam section end
Vibration displacement.The modal amplitudes a of finite element model1And a2It can be by carrying out TRANSIENT DYNAMIC ANALYSIS to model, to obtain third
The displacement time-history curves of beam section end carry out its available corresponding modal amplitudes of Fast Fourier Transform (FFT) to time-history curves,
From Fig. 5 it can also be seen that comparing the frequency response curve of theoretical model and finite element model under second order primary resonance.Reality in figure
Line and dotted line respectively indicate stable solution and unstable solution.Wherein, the unstable region that theoretical model is predicted is 16.4Hz-
16.435Hz, finite element model unstable region calculated are 16.4Hz-16.48Hz.
S6: the natural frequency of vibration, time-histories data and the internal resonance response of verifying L girder construction finite element model, the L after being verified
Girder construction finite element model；
In conjunction with Fig. 6 as can be seen that being counted in unstable region theoretical model modal coordinate calculated and finite element model
The comparison of the end displacement time-history curves of calculation.For theoretical model, the boundary that unstable region starts is that external excitation frequency is
16.4Hz, the response of structure is aperiodic motion at this time, (a) and (d) being detailed in Fig. 6.When external excitation frequency is gradually increased to
When 16.43Hz, the response of structure becomes chaotic motion, as shown in Fig. 6 (b) and (e).When external excitation frequency increases to
When 16.435Hz, the response of structure becomes aperiodic motion again, as shown in Fig. 6 (c) and (f).For finite element model, structure
Also there is similar nonlinear characteristic in corresponding unstable region.The displacement time-history curves of structure end are from external excitation frequency
The chaotic motion (Fig. 6 (h)) when external excitation frequency is 16.45Hz is arrived in aperiodic motion (Fig. 6 (g)) when 16.4Hz, then outside
Driving frequency is returned to aperiodic motion (Fig. 6 (i)) when being 16.48Hz.To sum up, Fig. 6 shows theoretical model and is predicted not
Stability region and finite element model unstable region calculated coincide good.
S7: analysis external load resistors are to the natural frequency of vibration of the L girder construction finite element model after verifying and the shadow of damping ratio
It rings, and analyzes load resistance, driving frequency and excitation amplitude to system capacity acquisition and position in the case of single order, second order primary resonance
Move the influence of response.External load resistors are analyzed to the natural frequency of vibration of the L girder construction finite element model after verifying and damping ratio
When influence, following variable relation is introduced:
Based on the variable relation of (27), governing equation can be rewritten are as follows:
Therefore, in (28) variable linear coefficient matrix are as follows:
In L-type energy collecting device system, the frequency and damping of system can be determined by the characteristic value of matrix B, with
Discuss external load resistors to the frequency of structure and the influence of damping.The frequency and damping ratio of system with load resistance variation
As shown in Figure 7.It can be seen that the ratio of the first second order frequency of system almost maintains 1:2 or so, from Fig. 7 from Fig. 6 (c)
(d) it can be seen that load resistor value influences obviously system damping, when load resistor value is R=4 × 10 in4System when ohm
First-order modal damping ratio reaches maximum value；When load resistor value is R=2.2 × 104The second-order modal damping ratio of system when ohm
Reach maximum value.The intrinsic frequency of coupled system is with damping to the dependence of load resistance for energy acquisition hereinafter and structure
The analysis of vibration displacement is of great significance.In the case of load resistance, driving frequency and excitation amplitude are analyzed to second order primary resonance
The influence of system capacity acquisition and dynamic respond.
Modal amplitudes and end portion vibration displacement under different loads resistance with the variation of external excitation frequency as shown in figure 8, its
Middle F=0.5m/s2.It can be seen from the figure that driving frequency corresponding to internal resonance occurs with the increase of load resistor value
Region is moving right, this second order frequency of system can change with the increase of resistance to explain from Fig. 7 (a), (b).
Fig. 8 shows in the end displacement of interior resonance zone system mainly by its single order modal amplitudes a1Determine, non-resonance region its
End displacement is mainly by second-order modal amplitude a2It determines.When load resistor value R=2.2 × 104Ohm and R=4 × 104When ohm,
Modal amplitudes a1Minimum can be controlled to.As can be seen that the load resistor value is the maximum second order of system respectively from Fig. 7 (d)
Resistance value corresponding to damping ratios.In addition, it can also be seen that working as load resistor value R=2.2 × 10 from Fig. 84Ohm and
R=4 × 104When ohm, the vibration displacement value of system end can control minimum.
Energy provided by the energy and each rank modal vibration that system acquisition arrives is under different loads resistance with external excitation frequency
The variation of rate is as shown in Figure 9.Wherein, F=0.5m/s2.Generated energy is vibrated by the first second-order modal of system and uses P respectively1
And P2It indicates, P1And P2It is to carry out Fast Fourier Transform (FFT) acquisition by the time-history curves of system gross energy collected.Fig. 9
Show energy that system acquisition arrives mainly by P2It provides, in addition when load resistor value R=2.2 × 104Ohm and R=4 ×
104System energy collected is larger when ohm, this is primarily due to the second-order modal resistance of the system under this load resistor value
Buddhist nun's ratio reaches maximum value, and the ceiling capacity that system acquisition arrives is corresponding with maximum damping.Importantly, working as resistance value R=4
×104System can not only collect maximum energy when ohm, and the vibration displacement of system end is also minimum.In Fig. 8 and Fig. 9
Dotted line indicates that system response of external excitation frequency field corresponding to the dotted line is unstable, unstable specific area
Between be 16.76Hz to 16.8Hz.
At different external excitation frequencies, modal amplitudes, end displacement and collected energy are with the variation of exciting force as schemed
Shown in 10.Wherein, R=105The influence of the width excited target power size in ohm internal resonance region is obvious, and exciting force is bigger, interior total
Vibration region is bigger.Simultaneously it can also be seen that exciting force is bigger, the bandwidth of collected energy is also bigger, collected energy value
Also bigger.In addition, the end displacement of system is also bigger.
Figure 11 be different loads resistance under, modal amplitudes, end displacement and collected energy with exciting force size change
Change.Wherein, f=16.55Hz.Modal amplitudes a1Increase with the increase of exciting force, however modal amplitudes a2With excitation amplitude
Increase first increases, when the amplitude of excitation is greater than 0.2m/s2, modal amplitudes a2Increase tendency be suppressed.End displacement and acquisition
To energy increase with the increase of exciting force, when load resistance R=2.2 × 104Ohm and R=4 × 104System acquisition when ohm
Also vibration displacement is controlled to minimum while to ceiling capacity.
It analyzes load resistance, driving frequency and excitation amplitude and system capacity in the case of single order primary resonance is acquired and is displaced and ring
The influence answered.
Under different loads resistance, modal amplitudes, end displacement are as shown in figure 12 with the variation of driving frequency.Wherein F=1m/
s2.Under Figure 12 (a) and Figure 12 (b) expression different loads resistance, the modal amplitudes a of endpoint1And a2With the variation of driving frequency.From
In figure as can be seen that in entire external excitation frequency range, modal amplitudes a1The contribution of opposite end point displacement all plays a major role.
It is also indicated that in Figure 13, in interior resonance zone, when load resistance R=4 × 104When ohm, system can collect ceiling capacity.Figure 12
It all indicates with the dotted line in Figure 13 as load resistance R=103The response of system is arrived in outer driving frequency for 8.23Hz when ohm
It is unstable within the scope of 8.33Hz；As load resistance R=106The response of system is arrived in outer driving frequency for 8.38Hz when ohm
It is unstable within the scope of 8.44Hz.
Under different exciting forces, the energy of end modal amplitudes, end displacement and system acquisition with driving frequency variation
As shown in figure 14.Wherein, R=105ohm.Similar with second order primary resonance, the resonance zone width of single order primary resonance is also with excitation
The increase of power and increase.Figure 14 (a) and Figure 14 (b) also indicate that under single order primary resonance, the vibration displacement of system end mainly by
The first-order modal amplitude a of system1Control, the energy that system acquisition arrives is mainly by P2It provides.In addition, Figure 14 is also indicated that, with outer
The continuous increase of exciting force, response at system fundamental frequency gradually become unstable from stabilization, this and paper
A.G.Haddow,A.D.S.Barr,D.T.Mook,Theoretical and experimental study of modal
interaction in a two-degree-of-freedom structure,J.Sound V ib.97(1984)451–
The rule found before 473. is similar.
In Figure 15, f=8.29Hz, modal amplitudes, end displacement and collected energy increase with the increase of external excitation power F
Greatly, similar with what is analyzed in Figure 11, when load resistance R=4 × 104Ohm and R=2.2 × 104When ohm, system can collect compared with
Big energy.In addition, it is noted that when load resistance is R=103When ohm, with being gradually increased for external excitation F, system
Response also gradually becomes unstable from stabilization, this phenomenon is equally similar with the rule found before Haddow [21].
It should be pointed out that the above description is not a limitation of the present invention, the present invention is also not limited to the example above,
Variation, modification, addition or the replacement that those skilled in the art are made within the essential scope of the present invention,
It should belong to protection scope of the present invention.
Claims (6)
1. a kind of Building Nonlinear Model of L-type piezoelectric energy collector and verifying analysis method, it is characterised in that:
S1: building L-type energy collecting device physical model and piezoelectric patches load circuit；
S2: establishing coordinate system, is based on L-type energy collecting device physical model, obtains corresponding with L-type energy collecting device physical model
L-type energy collecting device coordinate is distributed schematic diagram；
S3: the governing equation of the energy collecting system of L-type energy collecting device is derived using Hamiton's principle；
S4: Gauss law is used, obtains the circuit equation of piezoelectric patches load circuit, and combined circuit equation obtains the energy after depression of order
The governing equation of quantity collection system；
S5: the physical parameter and geometric parameter of setting L-type energy collecting device physical model, from L-type energy collecting device physical model
Physical parameter and geometric parameter in obtain L-type girder construction parameter physical parameter and geometric parameter, and using large-scale limited
First general software ANSYS establishes L girder construction finite element model；
S6: the natural frequency of vibration, time-histories data and the internal resonance response of verifying L girder construction finite element model, the L beam knot after being verified
Structure finite element model；
S7: influence of the analysis external load resistors to the natural frequency of vibration and damping ratio of the L girder construction finite element model after verifying, and
Load resistance, driving frequency and excitation amplitude are analyzed to system capacity acquisition and dynamic respond in the case of single order, second order primary resonance
Influence.
2. the Building Nonlinear Model of L-type piezoelectric energy collector according to claim 1 and verifying analysis method, special
When sign is to construct L-type energy collecting device physical model and piezoelectric patches load circuit in step S1, if:
L-type piezoelectric energy collector includes L girder construction, first lumped mass M1, second lumped mass M2；L girder construction includes
Horizontal beam and vertical beam；
First lumped mass M1It is fixed on the corner of horizontal beam and vertical beam intersection；
Second lumped mass M2In vertical beam and its position can slide up and down；
Piezoelectric patches load circuit includes piezoelectric patches and load resistance R, and piezoelectric patches is respectively adhered on the upper and lower surface of horizontal beam, and negative
It carries resistance R and is connected to form parallel circuit；
In step s 2, L-type energy collecting device coordinate distribution schematic diagram introduces three rectangular coordinate systems: O1x1y1；O2x2y2；
O3x3y3, three coordinate systems are used to describe the movement of horizontal beam and vertical beam；
Horizontal beam and vertical beam are considered as three parts composition, wherein horizontal beam be the first beam section, vertical beam be considered as the second beam section and
Third beam section.
3. the Building Nonlinear Model of L-type piezoelectric energy collector according to claim 2 and verifying analysis method, special
Sign is to derive the governing equation of the energy collecting system of L-type energy collecting device in step S3 using Hamiton's principle are as follows:
N1、N2And N3Respectively the first beam section, the second beam section, the xial feed of third beam section；
The corresponding arc length s in the place beam section section i (i=1,2,3)iIt indicates, u is used in the axially and transversely displacement in each beam section respectivelyi
(si, t) and vi(si, t) and it indicates；θi(si, t) and indicate the corner displacement occurred before and after each beam section deforms at the i of section；liIndicate the
The length of i beam section；
Wherein, ui(si,t)、vi(si, t) and θi(si, t) between geometrical non-linearity relationship can indicate are as follows:
mb1And mb2Respectively indicate the linear mass of horizontal beam and vertical beam；
Subscript s and p respectively indicate structure sheaf and piezoelectric layer；
The subscript 1,2 of s and p respectively indicates horizontal and vertical beam section structure sheaf and piezoelectric layer；
The density of ρ expression material；
H and b respectively indicates the height and width of beam section；
J1And J2Respectively indicate lumped mass M1And M2Rotary inertia；
EI1And EI2Respectively indicate the bending stiffness of horizontal and vertical beam section；
Symbol " ' " and " ", respectively indicate to siWith t derivation；
V (t) is piezoelectric patches due to deforming generated voltage；
csAnd caRespectively the equivalent viscous strain of cantilever beam and air damping coefficient；
IiIt is the cross sectional moment of inertia of beam；
M1For first lumped mass；M2For second lumped mass.
4. the Building Nonlinear Model of L-type piezoelectric energy collector according to claim 2 and verifying analysis method, special
Sign is the circuit equation of piezoelectric patches load circuit in step 4 are as follows:
R is load resistor value；
V (t) is piezoelectric patches due to deforming generated voltage；
hpFor the thickness of piezoelectric patches；
hs1For the thickness of the first beam section；
It is the permittivity component under constant strain；
e31=Epd31For piezoelectric stress coefficient；
It is piezoelectricity coupling terms.
5. the Building Nonlinear Model of L-type piezoelectric energy collector according to claim 4 and verifying analysis method, special
Sign is in step S4 that combined circuit equation obtains the governing equation of the energy collecting system after depression of order are as follows:
Indicate vertical acceleration；
ζ1And ζ2Respectively indicate two rank mechanical damping ratios before system；
ω1And ω2Respectively indicate two rank circular frequency before system；
Indicate capacitor；
Transverse vibrational displacement vi(si, t) and it is separated into space variable φij(si) and time variable qj(t):
Wherein, φij(si) and qj(t) be respectively system jth rank vibration shape and modal coordinate；
The vibration shape of energy collecting system can indicate are as follows:
Wherein,
Coefficient Aij、Bij、CijAnd DijFor the coefficient of system jth rank mode；
S and r represents the mode number of system, δrsIt is Kronecker δ function, and δrs=1 (r=s), δrs=0 (r ≠ s)；
mk、nkAnd ηlFor dimensionless factor.
6. the Building Nonlinear Model of L-type piezoelectric energy collector according to claim 1 and verifying analysis method, special
Sign is when establishing L girder construction finite element model, Beam188 and Mss21 unit is respectively adopted to simulate in girder construction sum aggregate
Quality, by the way that " NLGEOM, ON " order the geometrical non-linearity to consider L-type girder construction.
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