# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 4 Estimation of the Effective Model Parameters  
Table 1 lists the model physical parameters divided into four categories,  
* Those parameters whose values may be empirically measured.
* Unknown parameters whose values were guessed based upon expertise and approximate empirical measurement.
* Parameters whose values were determined by a particle swarm optimization which fit the model spectrum to the empirical spectrum, that minimized the difference between the two spectra. A particle swarm optimizer was selected because it was hypothesized that the nonlinear nature of the problem resulted in an error measure with multiple local minima.
* The device static parameters which cannot be solved analytically due to the nonlinear nature of the problem and had to be solved via an optimization method.  
Figure 3: The counterposed magnetic poles cause the magnets to deflect and separate. The static values are determined by numerical optimization.  
Identifying the unknown parameters and verifying and validating the lumped parameter nonlinear ordinary differential equation (ODE) model involved utilizing experimental measurements obtained from the vibration harvesting device prototype. The measurements served as a basis to fine-tune and validate the predictions made by the model. By comparing the experimental results with the numerical simulations, adjustments and optimizations can be made to improve the accuracy of the model's predictions.  
To accomplish the verification, a particle swarm optimization (PSO) algorithm [23; 24] was employed. The PSO algorithm was utilized to optimize and fit the parameters of the theoretical model to align with the experimental results. Through an iterative process, the algorithm searches for the best parameter values that minimize the discrepancy between the model predictions and the experimental data. The cost function was the root mean square error between the empirical and modeled output voltage spectra:  
\[f(\mathbf{p})=\sqrt{\frac{1}{N}\ \sum_{n=1}^{N}\left(V_{e}(n)-\alpha V_{m}(n; \mathbf{p})\right)^{2}}, \tag{35}\]  
where \(V_{e}(n)\) is the empirical spectrum, \(V_{m}(n;\mathbf{p})\) is the model spectrum, \(N\) is the number of sample points and \(\mathbf{p}\) is the vector of parameters to be optimized:  
\[\mathbf{p}=\left[\begin{array}{c}k_{11}\\ k_{12}\\ k_{21}\\ k_{22}\\ J_{1}\\ J_{2}\\ \alpha\end{array}\right]. \tag{36}\]  
\(\alpha\) is a positive amplitude used to account for unknown scaling between the empirical and model spectra.  
The effectiveness of this calibration approach hinges upon the assumption that the theoretical model adequately captures all the relevant physics associated with the system under investigation. It is crucial for the model to approximate the behavior of the system with an acceptable degree of accuracy. By validating the model against experimental data, an assessment of whether this assumption holds true and determination of the reliability of the calibrated model can be made. The results obtained from the PSO optimization indicate that the fitted parameters of the model are not significantly different from the independently measured physical parameters, such as the inertial masses and spring stiffness. This suggests that the calibrated model successfully captures the essential characteristics of the system. The agreement between the model parameters and the independently measured values further reinforces the confidence in the accuracy and reliability of the numerical model.