I have two equations for nonlinear piezoelectric energy harvester.
Mechanical domain:
πΜ(π‘)^2+πππΜ(π‘) + π^2π(π‘)+XΞΈ Μ - ππ£(π‘) + πππβππ(π‘) = βπ€π§Μ(π‘)
Electrical domain:
πΆππ£Μ(π‘) + π£(π‘)/π = ππΜ(π‘)
harmonic excitation terms
π(π‘) = Qe^jπt, z(t) = Ze^jπt, π£(π‘) = Ve^jπt, these terms and its derivative are substituted in the above two equations.
(-π^2Q+πππjπ + π^2QβπV) e^jπt +XΞΈ Μ + πππβππ(π‘) = βπ€(-π^2Z)e^jπt
After substitution, the nonlinear term (πππβππ(π‘) - nonlinear magnetic force) and centrifugal force term (XΞΈ Μ ) are there. We cannot able to cancel the harmonic term e^jπt because of the presence of the nonlinear term and centrifugal term.
(-π^2Q+πππjπ + π^2QβπV) e^jπt = βπ€(-π^2Z)e^jπt
(-π^2Q+πππjπ + π^2QβπV) = βπ€(-π^2Z) (1)
Without that nonlinear term, we can able to cancel the harmonic term e^jπt. Then get one equation to solve.
Similarly,
(πΆπjπV + V/π )e^jπt = πjπQe^jπt
(πΆπjπV + V/π ) = πjπQ (2)
solving (1) and (2) we able to get Voltage expression and displacement expression.
Similarly, how to get voltage and displacement expression for the nonlinear piezoelectric energy harvester with that nonlinear term?
Tell me suggestions regarding this.
I guess you can search for approximate solution as just a sum of first and third harmonic.
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