The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

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For SOC (state of charge) assessment techniques based on electrical circuit models, the parameters of the model are strongly biased by: battery aging, temperature, causing some errors in the estimation of the SOC. One approach to solve this problem is to update the model parameters constantly. We suggest a new algorithm VRLS (Variable recursive least squares) to update the parameters of a 2-resistor-capacitor (RC) network and to estimate the output battery voltage. VRLS is compared to the recursive least squares (RLS) and the adaptive forgetting factor recursive least squares (AFFRLS) algorithms. For algorithm assessment, we utilized real experimental data conducted on the Samsung 18650-20R lithium-ion cell. The tests indicate that compared to RLS and AFFRLS methods, VRLS recorded a low distribution in the high error range, in addition to small predictive performance indicators (RMSE, MAE, and MAPE) in all tests, which implies that VRLS has a good parameter identification ability.

The main objective of this work is to study the geometrically non-linear free vibration of stepped beams carrying multiple masses. These beams are studied on the basis of the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. The discrete expressions for the beam total strain and kinetic energies are derived. By applying Hamilton's principle, the problem is reduced to a non-linear algebraic system solved by an approximate method (the so-called second formulation). A parametric study is performed to explore the effect of non-linearity on the dynamic behaviour of stepped beams with several added masses. The free vibration case is discussed by considering three types of stepped beams which differ in the cross-section type.

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