For the piezoelectric cantilevers that Euler-Bernoulli assumptions are valid, the exact solution can be obtained from the mechanical equation of motion with electromechanical coupling. The exact solution utilizes a convergent series of eignefunctions and modal coordinates. For example see

"Erturk, A. and Inman, D. J., (2009) "An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations," Smart Materials and Structures"

In addition, there are approximate distributed parameter solutions for more complicated geometries and nonconforming beams with Euler-Bernoulli assumptions, such as Rayleigh-Ritz or assumed modes formulation. For example see

"Erturk, A. and Inman, D. J., (2010) "Assumed-modes Formulation of Piezoelectric Energy Harvesters: Euler-Bernoulli, Rayleigh and Timoshenko Models with Axial Deformations," in Proceedings of the ASME 2010 ESDA 10th Biennial Conference on Engineering Systems, Design and Analysis. pp. 405-414."

Dear Aldrin David Vargas Vargas

I Am interested in your work,so please don't hesitate to ask me.

Dear Aldrin David,

please, you may refer to the following paper and to related references there cited

Brach, S., Caruso, G., & Vairo, G. (2016). Optimization of a Tunable Piezoelectric Harvester Applied to Multimodal Structures. Journal of Wireless Communications, 1(1), 1-9.

I thank you very much, at the moment I am reading the papers that you have recommended me, you have been very helpful.

Dear Aldrin David Vargas Vargas

You can also use COMSOl software to model your results.

I suggest you to use equations from Erturk, A. and Inman based on first comment. There are some other modells that really do NOT work but you can find them have used in different papers. Just let you know Erturk and Inman model is working for the layer of piezoelectric material placed on a cantilever beam while we normally attach small piece of Piezo on the beam. However, you can also use lumped parameter modelling which is very simple and give you good results.

Single-Degree-of-Freedom (SDOF) electromagnetic generator model probably suitable for this purpose.

hi anyone can suggest a journal, article, or any related to piezoelectric cantilever beam for the mathematical model. I try to search the journal mostly dint show the formula they use to get the result ..help me