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118 http://vestnik.donstu.ru studied bifurcations, periodic and chaotic dynamics of atetrahedral compositemultilayer piezoelectric rectangular plate others and Zhanga Y.F. [10], In frequency [9]. in resonance the and adjust collectingabilityanalyzed the improve to was cantilever composite gradient using functionally a collectors energy model ofmagnetic A theoretical modes. magnetoelectricfrequencies resonantforc the at response Guo [7,8], In generated. is machine, the layerwith then is piezomagnetic along piezoelectricDue the deformed element. this, to an current electric under themechanical studied. of loadsare action https://doi.org/10.23947/1992- energy are widely used to power this kind of apparatus. In [1 kind apparatus. 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The study is objective to resonance dependence the of antiresonancefrequencies, and electromechanical and piezomagnetic, Introduction. Don State Technical University Solov'ev N. A. , layers piezomagnetic and piezoelectric with bimorph circular a of vibrations Transverse UDC entre layer dead made finite of The steel. method implemented element methodANSYS the in as used afor packageis - . 539.3 active layers for a given working frequency with the highest electromechanical coupling factor. coupling electromechanical highest the with frequency working given a for layers active - A finite element model of a piezoelement ofa model element A finite Ifsystem the an in field is magnetic alternatingmounted bymagnets created permanent on rotating the parts of Introduction. Introduction. electro energy storageenergy piezoelectrics, device, piezomagnetics, finitemethod, natural element oscillation frequencies Vestnik Transverse axisymmetric oscillations of a bimorph withtwo piezo bimorph Transverse a oscillations axisymmetric of A. N. Solov'ev , are studied. This element can be applied in an energy storage device which is in an alternating magnetic which alternating an in is an in device storage applied energy be studied. can element This are - Do Thanh Binh, O. Binh, Do Thanh on the geometr the on elasticity. The elementelasticity. The consists of three layers: twopiezo of Don State Technical Univers In themeasuring of development sensor and systems,modern small the research done is ( A mathematical model of the piezoelement action is a boundary value problem of linear value of problem aboundary is action piezoelement mathematical the model of A - Rostov - Liang and Yu others discussed theoreticalmodels multilayer of magnetoelectric composites - active elements that directly convert the energy of mechanical vibrations into electrical electrical into vibrations mechanical of the energy convert that directly elements active 5980- , The resultsThe obtained can used be designing under working the element of energy the ic parameters the of element. Do Thanh Binh, O. Binh, Thanh Do on - MECHANICS Don, Don, 2020- N. Lesn N. ) and the Government of the Russian Federation (contract no. 075 (contract andПЧ) RussianFederation the of Government Russian Federation Russian yers. yers. 20- with the support from the RF Ministry of Education and Science (project (project Science and ofEducation Ministry RF from the support the with y 2 ity. 2020.Vol. no. 20, ak - obility and nonvolatility of the above devicesmandatory. are Energy 118 Vestnik of DSTU, 2020, vol. 20, no. 2, pp. 11 - in the ANSYSthein Problems developed. packageis of determiningthe 124 N. Lesnyak. N. ) orrespondingvibrations to of the bending and extensional - 3], ener3], 2 https:/ , pp. , pp. Transverse vibrationsTransverse of acircularwith bimorph 11 /doi.org/10.23947/1992- gy storage devices using piezoelectric generators using piezoelectricgy devices storage 8 – 12 4 . . 1992 ISSN - active layers ( - - active layers, piezoelectric and active layers, piezoelectric and 5980 eISSN 1992 eISSN 5980 - PZT sized householdsized appliances, 5980- - 4 and CoFe 2020- - - 6006 active layers, are active layers, are 20- 2 2 O - 15- - 118 4 ) 8 2019 – and a and - 124 12 4 . - . is generated. In [7,8], Guo [7,8], In generated. is machine, the layerwith then is piezomagnetic along piezoelectricDue the deformed element. this, to an current electric under themechanical studied. of loadsare action studied bifurcations, periodic and chaotic dynamics of atetrahedral compositemultilayer piezoelectric rectangular plate others and Zhanga Y.F. [10], In frequency [9]. in resonance the and adjust collectingabilityanalyzed the improve to was cantilever composite gradient using functionally a collectors energy model ofmagnetic A theoretical modes. magnetoelectricfrequencies resonantforc the at response 1928) part of state order no. 9.1001.2017/ energy are widely used to power this kind of apparatus. In [1 kind apparatus. In of this usedwidelyenergy power to are storage deviceswith piezo m but objects, various of condition technical cellphoneswireless and sensor systems, powerful sources of energy are not required for monitoring and diagnosing the https://doi.org/10.23947/1992- Discussion andConclusions. constructed. of piezo the geometrics,electromechanical ondius ra device the factor thickness the and coupling the frequenciesof and these Graphic solved. dependences are antiresonance frequencies and natural resonance of Results. solving a boundary value problem. c Materials andMethods. Funding information: Funding piezoelectric and piezomagnetic la piezo the sizes of selecting geometricthe provide of parameters the piezoelement the on of antiresonance resonance the and storage device due the to action of an alternatingmagneticfield. constructed The dependences of the eigenfrequencies magneto coupling factor workfield. The study is objective to resonance dependence the of antiresonancefrequencies, and electromechanical and piezomagnetic, Introduction. Don State Technical University Solov'ev N. A. , layers piezomagnetic and piezoelectric with bimorph circular a of vibrations Transverse UDC For citation: For Keywords: entre layer dead made finite of The steel. method implemented element methodANSYS the in as used afor is package - . 539.3 active layers for a given working frequency with the highest electromechanical coupling factor. coupling electromechanical highest the with frequency working given a for layers active - A finite element model of a piezoelement ofa model element A finite Ifsystem the an in field is magnetic alternatingmounted bymagnets created permanent on rotating the parts of Introduction. Introduction. electro energy storageenergy piezoelectrics, device, piezomagnetics, finitemethod, natural element oscillation frequencies Vestnik Transverse axisymmetric oscillations of a bimorph withtwo piezo bimorph Transverse a oscillations axisymmetric of A. N. Solov'ev , are studied. This element can be applied in an energy storage device which is in an alternating magnetic which alternating an in is an in device storage applied energy be studied. can element This are - Do Thanh Binh, O. Binh, Do Thanh on the geometr the on elasticity. The elementelasticity. The consists of three layers: twopiezo of Don State Technical Univers In themeasuring of development sensor and systems,modern small the research done is ( A mathematical model of the piezoelement action is a boundary value problem of linear value of problem aboundary is action piezoelement mathematical the model of A - Rostov - Liang and Yu others discussed theoreticalmodels multilayer of magnetoelectric composites - active elements that directly convert the energy of mechanical vibrations into electrical electrical into vibrations mechanical of the energy convert that directly elements active 5980- , The resultsThe obtained can used be designing under working the element of energy the ic parameters the of element. Do Thanh Binh, O. Binh, Thanh Do on - MECHANICS Don, Don, 2020- N. Lesn N. ) and the Government of the Russian Federation (contract no. 075 (contract andПЧ) RussianFederation the of Government Russian Federation Russian yers. yers. 20- with the support from the RF Ministry of Education and Science (project (project Science and ofEducation Ministry RF from the support the with y 2 ity. 2020.Vol. no. 20, ak - obility and nonvolatility of the above devicesmandatory. are Energy 118 Vestnik of DSTU, 2020, vol. 20, no. 2, pp. 11 - in the ANSYSthein Problems developed. packageis of determiningthe 124 N. Lesnyak. N. ) orrespondingvibrations to of the bending and extensional - 3], ener3], 2 https:/ , pp. , pp. Transverse vibrationsTransverse of acircularwith bimorph 11 /doi.org/10.23947/1992- gy storage devices using piezoelectric generators using piezoelectricgy devices storage 8 – 12 4 . . 1992 ISSN - active layers ( - - active layers, piezoelectric and active layers, piezoelectric and 5980 eISSN 1992 eISSN 5980 - PZT sized householdsized appliances, 5980- - 4 and CoFe 2020- - - 6006 active layers, are active layers, are 20- 2 2 O - 15- - 118 4 ) 8 2019 – and a and - 124 12 4 . - . conditions [4]: tool. solution a as is selected model, piezoelectric problem of the lineartheory piezo of should be added: tensor of elastic modules; modules; elastic of tensor moreover moreover of massof forces; dielectric permeabilities; permeabilities; dielectric where where studied. with surfa plate magnetoelectroelastic ofa vibrations free and bending In[11], supports. with simple potentials. potentials. strength; strength; fixed. 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The are device of radius the and thickness of Mechanical boundary condition s Boundary conditions specified are formechanical, themagnetic and electrical, of the components fields. 0 and and Γ Γ B is surface 12 34 ∩ ∩ and and Vestnik of Don State Technical Univers ε Γ Γ σ are tensors of mechanical tensorsare strain; of stress and Ω H ∅ = ∅ = is volume density electric of charges; - are vectors of magnetic induction and magnetic field strength; strength; field magnetic and induction magnetic of vectors are charge In density. addition, if connected theare an to electrodes external circuit, twoconditions . . α e ismagnetoelectric ofmodules; tensor is tensor of piezoelectric modules;tensoris piezoelectric of Fig. 1. Scheme of half axial section of piezoelectric generator piezoelectric of section axial half of Scheme 1. Fig. τ u z rz r - =0 =0 e piezoelectric layer are covered with covered e piezoelectric layer are thatconnected are electrodes an to external axis of symmetry of axis uu ε ⋅+ ρ∇ σ∇ = ∇⋅ =σ ∇⋅ =ρ +ρ ∇⋅ ∇+ = - . Let. surface the ϕ magnetoelectric ela sticity ANSYS The which [4]. package, a implements 1 2 . Let. surface the ,DB D u, f σ = according to the harmonic law; the outer radius of the substrate is pivotally is pivotally ofthe substrate radius outer the law; harmonic the to according ( uU ϕ ity. 2020.Vol. no. 20, x ) (x, = : H h E e c:ε σ h: B e: D t Metal plate Metal Piezomagnetic plate platePiezoelectric magnetoelectric body consistssystem of a of equations and boundary ( −⋅ ⋅− − = = +⋅+⋅ = ∇ on - on statevibrations of the described construction is the boundary  ) Γ T EαH κE ε μH E α ε Γ 1 ) +⋅ 3 ,E , u , TT n T D nD S is displacement vector; vector; is displacement ⋅ ⋅ S ⋅ σp = and and 2 consist of two parts = , pp. , pp. + − consist of two parts −σ = h ∇ϕ Ω μ E , is tensor of piezomagnetic modules; modules; piezomagnetic of is tensor on 11 ,B is permeability tensor; tensor; is permeability 0 8 are vectorsare offield and electric induction – Γ on 12 2 − = 4 . . . 1992 ISSN Γ 4 ∇φ , 0 , ϕ ρ - Γ 5980 eISSN 1992 eISSN 5980 Γ and and 3 is density of the material; material; ofthe is density r 1 and and φ f Γ are electric and magnetic electricare and Γ 4 is vector of the density 2 , so that so , , so that so , - 6006 ce effects were were ce effects κ S S is tensor of = = Γ Γ 34 1 - ∪ ∪ value value c Γ Γ (4) (1) (5) (3) (2) er er 2 is is , , 119 Mechanics 120 http://vestnik.donstu.ru piezomagnetic inwhich plates polarized are thickness, have thicknessh h thickness has (steel) metal substrate The vector where corresponding constants. corresponding element grid of the energy storage device is shown in Fig. 2. 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Material properties of CoFe of properties Material Transverse vibrationsof acircular bimorph withpiezoelectric andpiezomagnetic layers 5 , - 0.29 The computer model computerThe ofis device the ANSYS built finite the in element package. 4 piezoceramic4 С CoFe2O4 [5, 6], adhesive layers a Material properties ofPZT properties Material 33 E 45.3 Hpa С 115 , 44 M , density density , Hpa φ , and radius r radius and = φ Q x ) (x, ϕ 580 31 С S in the system (1) system the in ρ= m 44 , t S E E 25 N density . , = 3 ρ= . on / Hpa consist of two parts 6 7860 A SI dS v nD , . 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When analyzing the natural vibrations of a piezo a of vibrations natural the analyzing When ntiresonance f requency 56. 11. kHz Vestnik of Don State Technical Univers 748 675 Surface Surface Surface Surface surfaces 2, 3 are contact surfaces between piezoelectric layers piezoelectric between surfaces contact are 3 2, surfaces 4 1 2 3 The natural vibrations of a piezoelectric element are considered, the radius of whichradius vibrations the natural element The ofis considered, are apiezoelectric of The finite element grid of the energy storage device. storage energy the of grid element Fig. finite The 2. Surfaces 1, 4 are outer surfaces of piezoelectric layers; piezoelectric of surfaces outer are 4 1, Surfaces distribution of electric and magnetic potential on on potential magnetic and electric of distribution – 4. However, the electric potential on surface 1 is unknown and is found from found unknown is is surfaceelectric and potential 1 However, the 4. on A Distribution of vertical displacement on deformed state and and state deformed on displacement vertical of Distribution ntiresonance ity. 2020.Vol. no. 20, sents first the natural three frequencies of antiresonance. - n p electromagneti atural frequencies =0.5 mm, the radius of the substrate is r=10 mm, and the mm, the and substrate mm, r=10 is =0.5 the of radius the 2 , pp. , pp. 11 8 – 12 c energy storage device, it is assumed that the it that the device,is storage c energy assumed 4 . . 1992 ISSN and metal layer. metal and - flux density set is on surface In 4. the - 5980 eISSN 1992 eISSN 5980 undeformed sample device, electric potentials - 6006 culating the the culating Table 3 Table 121 Mechanics 122 http://vestnik.donstu.ru all equations all possible use to based approach on an the of the effective piezo properties and antiresonance, the and electromechanicalon geometricfactor coupling parameters investigated. is the bending of plates. the whichfor lower showsmodes, that hypotheses bending dis can there accepted about be their in the range of 6.8 ÷ 9.8 mm. 9.8 ÷ 6.8 of range the in piezoelectric layers r the piezoelectric layers. shows Fig. 4 that value the of natural the frequencywithincreases increasing radius the of the

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fr1 3 x 10 - 4 A The thicknessThe value piezoelectric the layersh of number alternatingWhen sufficiently large, layers of is becomes piezoelectric the and it piezomagnetic of eigenfrequencies the dependence numerical the resonance In of calculations below, of results the presented Fig. 3 shows the natural frequency dependences on the thickness h thickness the on dependences frequency natural the shows 3 Fig. while are the latter potential magnetic and electric displacement, ofvertical the distribution 3 presents Table Fig. shows 4 theof the dependence electromechanical factor coupling on thickness the h ntiresonance f requency 115. kHz of the problem - (1) hp Solov'ev 572 p , but decreases with but, increasing decreases thickness h A. N., et al. а ) Transverse vibrationsof acircular bimorph withpiezoelectric andpiezomagnetic layers (3) are(3) used. Fig. 3. Natural frequencies: a) resonance frequency, resonance a) frequencies: Natural 3. Fig. rp distribution of electric and magnetic potential on on potential magnetic and electric of distribution Distribution of vertical displacement on deformed state and and state deformed on displacement vertical of Distribution b ) antiresonance frequency antiresonance x 10 - 3 p varies in the range of 0.3 ÷ 0.7 mm, and the radius value r value radius the and mm, 0.7 ÷ 0.3 of range the in varies

fr1 3 x 10 - 4 A The thicknessThe value piezoelectric the layersh of When the number alternatingWhen sufficiently large, layers of is becomes piezoelectric the and it piezomagnetic of eigenfrequencies the numericaldependence the resonance In of calculations of below, results the presented Fig. 3 shows the natural frequency dependences on the thickness h thickness the on dependences frequency natural the shows 3 Fig. Table 3 presents the distribution of vertical displacement, electric and magnetic potential while the latter are while are the latter potential magnetic and electric displacement, ofvertical the distribution 3 presents Table Fig. shows 4 theof the dependence electromechanical factor coupling on thickness the h ntiresonance f requency 115. kHz of the problem - (1) hp Solov'ev 572 p , but decreases with but, increasing decreases thickness h A. N., et al. а ) Transverse vibrationsof acircular bimorph withpiezoelectric andpiezomagnetic layers (3) are(3) used. Fig. 3. Natural frequencies: a) resonance frequency, resonance a) frequencies: Natural 3. Fig. rp distribution of electric and magnetic potential on on potential magnetic and electric of distribution Distribution of vertical displacement on deformed state and and state deformed on displacement vertical of Distribution b ) antiresonance frequency antiresonance x 10 - 3 p varies in the range of 0.3 ÷ 0.7 mm, and the radius value r value radius the and mm, 0.7 ÷ 0.3 of range the in varies

fa1 p x 10 of theof piezoelectric layers. - 4 - magnetoelectric In compositethis 6]. [5, case, hp - strainmagnetic electric state, and fields, p and the radius r radius the and undeformed sample b ) tribution corresponding to p p of the piezoelectric of piezoelectric the rp and the radius r radius the and x 10 - 3 p p of of – magneticfield. Journal ofAlloys and Compounds. 2017;693:989– https://doi.org/10.1016/j.engstruct.2018.04.100. laminated piezoelectric rectangular plate. EngineeringStructures Journal. 476. https://doi.org/10.1016/j.compstruct.2015.02.001. piezoelectric multilayer https://doi.org/10.1016/j.compstruct.2014.09.049. . 2015;119:738 Journal – Structures Composite composites. magnetoelectric multilayer materials with applications to multilayered structures. International Journal of Engineering Science. 2011; Science. ofEngineering Journal International structures. multilayered to with applications materials multilayer composites. International Journal of Engineering Science. 2011; Science. ofEngineering Journal International composites. multilayer Springer. Smart Systems. Series “Advanced Structured Materials”. Altenbach H, Carrera E, Kulikov G. Altenbach Kulikov Carrera E, “Advanced StructuredH, SmartSeries Materials”. Systems. piezoelectric materials modeling ACELANin - 10.1142/S1758825117500843 Type Energy Piezoelectric Harvester. International Journal of Applied 2017;9(6): Mechanics. doi:1750084. 2014. 214 p. energy storage devices withmechanicalenergy physical devices storage complicated Cand.Sci. properties: and uslozhnennymi fiziko uslozhnennymi Heidelberg: Springer; 2018. limits. the considered within respectively, values, smallest and the largest take layers valuemaximumfactor ofcouplingachievedwhen radius elecis the tromechanical thickness piezoelectric the the and of piezoelectric elements operating at acertain frequency having and the greatest efficiency. This analysis shows that the the energy certain restrictions dimensions, on the on the device which electromechanical factor, coupling shows the efficiency of metalpiezomagnetic a fixed plateson effect plate.geometric The of the the characteristics of piezoelectric layers under piezoelectric the and are storage device energy active the The of elements considered. ANSYS package is 9. Yang Shi, HongYao, Yuan Analysis on nonlinear vibrations near internal r internal near vibrations nonlinear Zhang10. on YF, Zhang YaoZG. W, Analysis 8. Guo 8. 7. Guo 7. 6. Challagulla KS, Challagulla 6. 5. Jin 5. 4. Kurbatova NV, Nadolin DK, Nasedkin Nadolin DK, Kurbatova NV, 4. 3. DuongLV, Pham Chebanenko MT, VA, et al. Finite Element Modeling and 2. DuongLV. Konechno - Shevtsov1. AN, Parinov SolovievIA, SN, et al.Actuators Piezoelectric Generators Energy and for Harvesting. References Conclusion. Conclusion. 2018; storage device,studied. is calculation The presented results allow in paper the selecting sizesrational of (In Russ.) (In Vestnik of Don State Technical Univers - 81(5):69- - - Yeon Kim. Micromechanical analysis of effective properties of magneto of analysisproperties effective Yeon of Kim.Micromechanical Liang Yu, Huai Yu, Liang Liang Yu, Huai Yu, Liang - An axisymmetric finite element model of an energy storage device based on round plates in the on round the in plates energy based model device storage finitean axisymmetric element An of mekhanicheskimi svoistvami: diS…. kand. tekhn. nauk tekhn. kand. diS…. svoistvami: mekhanicheskimi - magnetostrictive composite cantilever structures. Composite Structures Journal. 2015;125:367– Journal. Structures Composite structures. cantilever composite magnetostrictive 88. Georgiades AV. Micromechanical analysis of magnetoAV. of MicromechanicalGeorgiades analysis - -

Wu Zhang, Fei Zhang, Wu k x 10 Wu Zhang, Yuan Zhang, Wu ehlementnoe modelirovanie p'ezoehlektricheskikh ustroistv nakopleniya ehnergii s s ehnergii nakopleniya ustroistv p'ezoehlektricheskikh modelirovanie ehlementnoe - 4 - wen Gao. Afu Gao. wen Fig. 4. ity. 2020.Vol. no. 20, COMPOS package. Analysis and Modelling of Advanced Structures and and Structures Advanced of Modelling and Analysis package. COMPOS hp Electromechanical - Ming Bai, et al. et Bai, Ming - Xun Li, et al. Equivalent Li, al. et Xun AV, et al. Finite for approach element magnetocomposite nctionally graded composite cantilever to harvest energy from compositegraded from nctionallyharvest energy to cantilever 2 , pp. , pp. coupling 999. https://doi.org/10.1016/j.jallcom.2016.09.242. Theoretical investigation of magnetoelectric effect in in effect magnetoelectric of investigation Theoretical 11 8 – 12 factor 4 49:1001 rp . . 1992 ISSN [Finite circuit method for resonant analysis of of analysis resonant for method circuit – 1018. - element modeling piezoelectric of element - - electro 5980 eISSN 1992 eISSN 5980 Experimental Studies of Stack Studies Experimental (Eng.), x 10 - esonances of acomposite thermo - 3 diss. - electro (Eds.). - - ]. Rostov- 2018;173:89 6006 elastic composite composite elastic - 49:85– thermo Singapore: on- 104. - elastic elastic – Don; 106. 748. 748. - - 123 Mechanics 124 http://vestnik.donstu.ru University ( University issue the in Scheduled Submitted https://doi.org/10.1016/j.ijmecsci.2019.05.029. [email protected] [email protected] 2016 effects. International Journal of Mechanical 2019;157 Sciences. - results. O.N. Lesnyak: discussion of the results. ofthe results. discussion Lesnyak: O.N. results. ofthe discussion analysis; model; computational computer and mathematical a constructing for method solution Technical University ( University Technical Technical University ( University Technical , S copusID Solov'ev, Arkadii N.,the of Head theAbout authors 11. Ying Yang, Xian Yang, Ying 11. All authors have read and approved the final manuscript. A.N. Solov'ev: task formulation; discussion of the results. Do Thanh Binh: Thanh Do results. the of discussion formulation; task Solov'ev: A.N. Claimed contributorship Lesnyak, OlgaN , Binh Thanh Do 19.02.2020 1, Gagarin sq., Rostov- Solov'ev 55389991900 09.04.2020 1, Gagarin sq.,Rostov 1, Gagarin sq.,Rostov A. N., : ., associate ., professor postgraduate student of et al. - , Fang Li. BendingFang free and vibration acircular of magnetoelectroelast ORCID: ORCID: Transverse vibrationsof acircular bimorph withpiezoelectric andpiezomagnetic layers on- Don, 344000, RF http://orcid.org/0000 - - Theoretical andTheoretical Applied Mechanics State Department, Don Technical on on - - Don, 344000, RF Don, 344000, RF) ofthe andTheoretical Applied MechanicsDon State Department, the Theoretical and Theoretical the Applied State MechanicsDepartment, Don ), Dr.Sci. (Phys. - 0001- ), ), 8465- , ORCID , ORCID 5554 - Math.), professor, : : , http://orcid.org/0000 http://orcid.org/0000 S [email protected] survey conducting; selection of a a of selection conducting; survey ResearcherID ic plate with surface with surface plate ic - - 0001- 0003- 158:858– 7410- 1002- H - 7906- 0061 2468 871. , ,