###### Abstract  
In this paper, we extend the optimization analysis found in the current literature for single-degree-of-freedom vibrational energy harvesters. We numerically derive and analyze the optimization conditions based on unified expressions for piezoelectric and electromagnetic energy harvesters. Our contribution lies in the detailed analysis and comparison of both resonant and anti-resonant states while fully including the effect of intrinsic resistance. We include both the case of excitation by inertial load and prescribed displacement, as the latter has not been elaborated on in the previous literature and provides new insights. We perform a general analysis but also consider typical values of applied piezoelectric and electromagnetic energy harvesters. Our results improve upon previous similar comparative studies by providing new and useful insights regarding optimal load, load power and power input to output efficiency. Our analysis shows an exponential increase in the critical mechanical quality factor due to the resistive loss coefficient. We find that the ratio of mechanical quality factor to resistive loss coefficient, at resonance, increases drastically close to the theoretical maximum for load power. Under the same optimization conditions, an equivalent conclusion can be drawn regarding efficiency. We find that the efficiency at anti-resonance behaves differently and is equal to or larger than the efficiency at resonance. We also show that the optimal load coefficient at resonance has a significant dependence on the mechanical quality factor only when the resistive loss coefficient is large. Our comparison of excitation types supports the previous literature, in a simple and intuitive way, regarding optimal load by impedance matching and power output efficiency. Our modeling and exploration of new parameter spaces provide an improved tool to aid the development of new harvester prototypes. v 30 August 2026 September 2022 29 September 2922 29 September 2922 29 September 2922 292929292929292929292929292929292929292929292929292929292929292929292929292929292929292929229292929229292929  
Essential parameters in these models are mechanical quality factor, ratio of excitation frequency to natural frequency, intrinsic impedance and resistance, load resistance and the electromechanical coupling coefficient. The latter is defined by the ratio of electrostatic (PEH) or electromagnetic (EMEH) energy to mechanical energy. There is a critical value of the effective electromechanical coupling coefficient, which is determined by the mechanical quality factor, at which the output power reaches a theoretical maximum, and an anti-resonant peak starts to emerge, as derived by [PERSON] et al. [3]. The lumped mechanics of a PEH or EMEH can be modeled in a unified way, assuming they are both systems with linear stiffness and damping. This has been exploited by [7; 8; 9] to derive a general model applicable for both systems. [PERSON] et al. [8] performs a thorough derivation of a general expression for generated power, normalized by the theoretical power limit. The analysis focuses on the generated power for varying load, mechanical quality factor and effective electromechanical coupling. Optimal load coefficients are described, but the optimal frequency, normalized by the natural frequency, is simply taken to be \(1\), thus not addressing the case of anti-resonance. In their paper, they note that the effective electromechanical coupling and resistive loss coefficient are both large for a typical EMEH and small for a typical PEH. From their survey, they also find a few existing PEHs achieving power performances close to the theoretical limit, while the typical EMEH has room for improvement. [PERSON] et al. has modeled the power output for VEHs in several papers and included the harmonic case [7], cases with different types of loads, reactive and passive [10; 11], and a derivation method which allows both harmonic and stochastic input to be used [12; 13]. In [7], [PERSON] et al.