# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 6 Empirical and Model Analysis of the VEH Sensor
### Swept Sine Measurements of VEH Sensor and Optimization of Model  
A range of sinusoidal input voltages are generated using the internal frequency generator of the lock-in amplifier (Stanford Research, SRS856, Stanford Research Systems, Sunnyvale, CA USA). The signal is swept across the estimated operational bandwidth of the prototype. The drive voltage to the shaker is adjusted using a reference accelerometer (PCB Piezotronics, model 352A24, Depew, NY, USA). The accelerometer is mounted on the end of the beam and provides a measurement of the input acceleration experienced by the harvesting sensor. Using the calibrated accelerometer, the drive voltage from the power amplifier was adjusted to create an approximate input acceleration from the shaker of \(\dot{y}(t)=10^{-2}\) g across the range of 1 Hz to 200 Hz. The controlled response of the shaker (BK vibration exciter 4809, Marlborough MA, USA) across this frequency range and drive level is flat. A laser vibrometer is used to verify the shaker response. Additional vibrometer measurements detected no variation in acceleration magnitude across the length of the aluminum bar either, thus verifying the shaker and the attached aluminum bar are reproducing the drive signal characteristics from either the lock-in amplifier or the signal generator. The goal is to determine an estimate of the harvester's coupling sensitivity (V/g) and power conversion efficiency for low-level acceleration input. Figure 5 shows the empirical voltage spectrum from the VEH prototype measured by the SRS 856. The load resistance is 50 \(\Omega\) and the input drive acceleration from the shaker is \(8\times 10^{-3}\) g. The spectrum shows one large peak at 138 Hz and a secondary smaller one at near 140 Hz. The larger peak at 138 Hz has a usable bandwidth of approximately 20 Hz. Outside this range, the sensor coupling sensitivity of the prototype is insufficient for harvesting; this will be elaborated upon in the next section. This measurement can now be utilized to improve the analytical model. Since the input acceleration is constant, the root mean square (rms) sensitivity of the sensor is estimated as (rms(Vout)/\(10^{-2}\) g); so, between 130 Hz and 150 Hz, the rms coupling sensitivity is 0.58 mV/g, with a peak sensitivity of 0.87 mV/g at 138 Hz. The optimization of the model requires tuning the parameters of the model to fit a given measured output. The parameters utilized in the model described in Equations (19)-(22) are listed in Table 1. Some of these parameters can be easily measured while others cannot. The initial fabrication data of the components (i.e., mass, magnetic strength, dimensions, estimated spring stiffness, etc.) are measured using standard laboratory techniques. The spring stiffness is estimated using a measurement of the first resonances of each of the magnet spring sub-assemblies using the laser vibrometer. The magnetic field strength of each magnet was measured using a EXTECH AC/DC Magnetic Meter, Nashua, NH, USA. These techniques will introduce errors when coupled with the complexity of the equations of motion. It is found through trial and error that the stiffness estimates for the spherical springs and the rotational inertial terms have the strongest effect on the response of the model for the low-level input acceleration. The optimization step is utilized in conjunction with the parameters that have the least uncertainty to estimate the parameters that have higher uncertainty. With this approach, the model is \"tuned\" to fit the empirical results. For each optimization iteration, the model of Equation (19) is run for 30 s of model time using MATLAB's ODE45 [25] solver (this is also used to solve the static case of Equation (34)). The model output is the state vector of Equation (18). The time series of interest is the current, \(i(t)\), which when multiplied by the 50 \(\Omega\) load resistance yields the harvested voltage, \(v(t)\).