###### Abstract
After deriving the distributed-parameter energy expressions, the extended Hamilton's principle is employed to obtain the discretized electromechanical Lagrange's equations. An axial displacement variable is kept in the formulation to account for its coupling with the transverse displacement (in the Euler-Bernoulli and Rayleigh models) or cross-section rotation (in the Timoshenko model) due to possible structural asymmetry. The steady-state electromechanical response expressions are obtained for harmonic base excitation. Experimental validations are given for the thin-beam case by comparing the assumed-mode predictions with the experimental and analytical results. The motivation of the present research is the fact that there exist many well-developed mathematical models of energy harvesters whose rigorous analytical treatment would be very desirable. Our main goal is to present rigorous mathematical analysis of a piezoelectric energy harvester with the Timoshenkobeam model as a substructure. We consider two cases: (i) the beam model with variable structural parameters; and (ii) the beam model with constant structural parameters. Our goal is to derive the asymptotic approximation for the infinite set of the natural frequencies of the model. To this end, we perform asymptotic analysis of the corresponding set of three partial differential equations subject to four boundary conditions, and homogeneous initial conditions. We would like to obtain the answer to the following question: _In which way do the parameters of piezoelectric layers and the electric circuit (such as the piezoelectric coupling constant \(\theta\), the capacitance of the layers \(C\), and the resistance of the external load \(R\)) enter the asymptotic formulas describing the distribution of the natural frequencies?_ The answer to this question is the following: _The asymptotic formula for the frequency distribution has the leading term, the second order term, and the remainder term. Based on our analysis, we obtain that the leading asymptotical term contains contribution only from the structural part of the model, the second order term contains contributions from piezoelectric and circuitry parts of the model (see Theorems 5 and 6 below)._ To prove such results, we have to derive refined asymptotic approximations for the vibrational modes. However, obtaining higher order asymptotics for the case of variable parameters structure is quite involved and extremely lengthy. That is why we consider two models of the harvester, one contains the structural part of the model with variable parameters, for which we derive the first order asymptotic approximation for the modes (that contains only the leading term and the remainder term), and the second model that contains constant parameter structural part of the model, for which we derive higher order asymptotic approximation for the modes (that contains the leading term, the second order term, and the remainder term). Now, we are in a position to describe the content of the present paper. In Section 2, the initial boundary-value problem is formulated in the form of a system of three coupled partial-differential equations (see Equations (1)-(5) below), in which Equation (2) contains the distributional terms. We represent the problem in an equivalent form so that there are no distributional terms in the equations but there is an additional term in the boundary conditions. In Section 3, the problem is reformulated and is given in the form of a first order in time evolution equation in the Hilbert state space corresponding to the model. The dynamics generator is the main object of interest for the rest of the paper. Its properties are presented in Theorem 1. The importance of the dynamics generator is that the set of its eigenvalues being multiplied by \"\(i\)\" (the imaginary unit) coincides with the set of the natural frequencies of the harvester model. In Section 4, we analyze asymptotic and spectral properties of the polynomial operator pencil whose eigenvalues coincide with the dynamics generator eigenvalues.