Recently, i have read some papers with similar ideas which one piezoelectric layer is laminated to a cantilevered beam with a tip magnet. sometimes they consider an electric enthalpy into the equations of using Hamilton's principle or Euler-Lagrangian to governing equations , sometimes they did not consider it.
example 1: Ts+Tp+Tm-Us-Hp-Um
Here, Ts,Tp,Tm are the kinetic energy of the substrate, piezoelectric and magnet tip mass, respectively; Us and Um are the potential enegry of subtrate and magnet tip mass.
(Ref:Samuel C. Stanton, Clark C. McGehee, Brian P. Mann, Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator, Physica D: Nonlinear Phenomena, Volume 239, Issue 10, 15 May 2010, Pages 640-653, ISSN 0167-2789, )
example 2: T-U-UAB-UAC+W
Here, T and U represent the kinetic energy and potential energy for total system, UAB and UAC means magnetic potential energy; W means the work done by the moment produced by voltage V , across the piezoelectric layer.
(Ref:Peyman Firoozy, Siamak E Khadem, S Mehrdad Pourkiaee, Broadband energy harvesting using nonlinear vibrations of a magnetopiezoelastic cantilever beam, International Journal of Engineering Science, Volume 111, February 2017, Pages 113-133, ISSN 0020-7225, )
my questions:
1: The W in example 2 is equal or similar to Hp in example 1? or some kind of transformation?
2. if the W has nothing with the Hp, then how to decide when considering the electric enthalpy with this similar kind of laminated cantilevered structure?
Dear Mr. Sunny,
They both play similar role in order to describe the mechanical and electrical behaviors along with a coupling parameter that yields electromechnanical effects in the energy harvesters.
I am encouraging you to read other published works of the Dr. Stanton in this regard which I am citing it in the following message.
" Samuel C Stanton, Alper Erturk, BrianPMann,,EarlHDowell, Daniel J Inman Nonlinear nonconservative behavior and modeling of piezoelectric energy harvesters including proof mass effects"
I hope this helps.
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