Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

•Higher-order perturbation analysis for piezoelectric energy harvesting systems with multiple aspects of nonlinearity.•Geometric nonlinearity, damping nonlinearity and inertia nonlinearity are considered.•Considering multiple nonlinear factors provides a more accurate evaluation of the energy harvesting effects.•Higher-order computational methods improve the solution accuracy of piezoelectric energy harvesting systems.•Higher-order method shows stronger nonlinear softening effects and it is more suitable for low frequency environments. A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power. [Display omitted]