Constraint Effects on Torque-Actuated Bistable Energy Harvesters

Design Considerations and Specifications of Fabricated Devices

The energy harvester of interest operates on the basic premise that proof masses mounted on cantilevered arms transfer torque to the mid-section of a central compressively buckled beam with piezoelectric PVDF strips adhered. A schematic of the device is shown in Figure 1. Upon compression of the center beam, quasi-pinned compliance arm supports induce “S”-shaped buckled deflection profiles of the center beam. Transverse motion (vibration) of the structure base introduces a moment at the center of the “S” shape and can induce switching between buckled states for sufficiently large accelerations. Snap-through (switching between buckled stable states) is desirable for potentially generating large strains within the central piezoelectric strips and maximizing power output.

Figure 1:

3×1 device in the (a) uncompressed, (b) compressed buckled up and (c) compressed buckled down. Main components of energy harvesting device, including the (d) torque arm PLA brace, the (e) steel shim stock center beam with PVDF strips, and the compliance arms constructed in the (f) pinned, (d) 3×1, and (g) 5×1 configurations.

Ideally, compliance arms that act as a perfect pin constraint would provide optimal torque transfer to the center and allow snap-through to be exhibited at the lowest possible acceleration levels. However, mimicking a perfectly pinned condition at the center beam adds fabrication cost/difficulty and is not feasible for planned MEMS-scale versions of the device. Using anything but a perfectly pinned method to constrain the deflection of the center beam will add both torsional and potential out-of-plane compliance. Torsional stiffness of the compliance arm will reduce the amount of force transferred to the center beam, increasing the energy threshold to be overcome to create stability state switching. Vertical compliance can change the buckling force needed to make the device bistable, increasing the acceleration needed to induce snap-through or pushing the center beam into first buckled mode deflection profile (dome shape) rather than the “S”-shaped second mode shown in Figure 1(c).

To test the effects of the compliance arm constraint and geometrical parameters, four device variations were constructed. The first device was a pinned beam that used a greased 20 gauge steel rod running through the entire center body width that allowed nearly free rotation at the compliance arm supports (Figure 1(f)). The second device was nearly identical to the first with the exception of the compliance arm constraint. In this case, the compliance arm rod was locked to the central beam using an industrial adhesive (JB-Weld Compound 8265-S), in addition to clamping at the base supports. The result was a clamped–clamped compliance arm made of steel with a round cross section. The third and fourth devices used 3×1 mm and 5×1 mm cross-sectional polylactic acid (PLA) compliance arms, respectively, with 2 mm fillets near the brace structure (Figure 1(g)–(h)). In all cases, 5/16″ steel nuts (4.62 g) were used as proof masses and were adhered 10 mm from the tip of the brace as shown in Figure 1.

Piezoelectric strips of 0.11 mm thickness (GoodFellow FV301960/3, d₃₁=19 pC N⁻¹, d₃₃=−20 pC N⁻¹) were cut to a width and length of 5 and 33 mm, respectively. Previous finite element analysis (FEA) results on the buckled center beam (Porter and Berfield, 2014) show that a strain sign change occurs at about 16% of the beam length from the end constraints and at the middle for these devices. Thus, to minimize charge neutralization, two separate strips of PVDF are adhered to the center beam on spans of one-sixth to one-half along the length of the beam, and again on a span from one-half to five-sixths of the beam (Figure 1(b)).

Shaker Table Setup

A custom-built shaker table setup was construed to test the devices under sinusoidal driving conditions. The shaker consisted of a 250 W_p−p (peak-to-peak) speaker, a 200 W_rms (root mean squared) amplifier, Arduino Nana v3.0 microcontroller, MPU-6050 accelerometer, and an op-amp signal buffer and conditioning circuit. The buffer and conditioning circuit are needed to accept a moderately high-voltage response and provide a high-input impedance so that the power generated can be accurately measured. To accommodate the large impedance loads that accompany single layer element PVDF strips (Jiang et al. 2010; Wen et al. 2014), a two-stage op-amp buffer circuit was used with an input impedance over 1 GΩ (OPA551PA). Driving specifications used in this work for the system were 10–500 Hz and 0.1–3 g_rms while the logging specifications for the Arduino utilized four voltage channels running 2,048 S/s.

A compression rig mounted atop the shaker plate allowed for each device to have variable length compliance arms and the ability to fine-tune a compression displacement by turning 8–32″ push rods and then locking down the rig by tightening M3 bolts. Compression displacement measurements for the system were performed via calipers. A render of the designed shaker table and the as-fabricated system with the 3×1 mm device and 8 mm compliance arms in the uncompressed state are shown in Figure 2. Great care was taken to buckle the device symmetrically as the shape of the potential energy function (specifically the maximum deflection point of the center beam which must be traversed to exhibit snap-through) must be within a certain range to provide energy harvesting benefits (Masana and Daqaq 2011). Near the critical buckling load, the resonant frequency of the center beam should theoretically approach zero and afterward stiffen up as more compression is applied (Masana and Daqaq 2013).

Figure 2:

(a) Render of designed shaker table and (b) the as-fabricated shaker table with 3×1 device mounted.

Optimum Load

To evaluate devices at peak power performance conditions, the optimum load was determined. PVDF strips with single electrodes tend to have high optimum resistance driving loads. This can introduce difficulties as the input impedance on standard laboratory testing equipment can be on the same scale as the devices being tested, which in turn can create power sharing that diminishes the signal. To reduce the optimum load of PVDF, multiple stacked PVDF and electrode layers can be used but the amount of layers needed would be very costly.

Four test cases were run at different impedance loads to determine the optimum load conditions for which all other tests would be performed. Three of the cases included the uncompressed pinned, 3×1 mm and 5×1 mm devices while the last test case was a compressed 3×1 mm device. All optimum load cases were run at 0.5 g_rms and near resonance.

A plot of normalized voltage response versus load impedance is shown in Figure 3. Optimum values ranged from around 20 to 40 MΩ. Since all tests showed minimal drop-off in power performance around 30 MΩ, 30 MΩ was chosen as the driving resistance load for all devices and testing cases.

Figure 3:

Normalized voltage response versus load impedance for the (a) back and (b) front electrode, all tested using 8 mm compliance arms.

Evaluation of Damping

Damping is a desirable parameter to quantify so that accurate simulations can be constructed and verified, in turn enabling optimization of device performance. Due to the large displacements that this device undergoes and the allowed range of motion for the displacement sensors, damping for low input accelerations is evaluated. Damping during very large motions would require a 3D digital image correlation (DIC) system. Viscous damping was evaluated for the device using three different methods: free, linear forced and nonlinear forced. Free vibration analysis works well for decaying responses, but can be a poor representation of damping in energy harvesting structures that have a constant input function exerted upon them. Despite this limitation, values for free damping were extracted using the voltage response to a mono-pulse chirp sent to the shaker table and using the logarithmic decrement method in eq. [1] Beards (1995):

where x0 and x1 are the amplitudes of the decaying sine wave at the beginning and end of each successive cycles, respectively.

A more realistic evaluation of linear viscous damping is to use the half-power bandwidth (HPB) method which evaluates the quality and damping ratio for a linear system (Beards 1995). This process requires the displacement measurement of the base shaker table and the tip of the torque arms, which was performed using two micro-epsilon laser displacement sensors (Micro-Epsilon NCDT1401). The gain, which is the ratio of the tip displacement to the base displacement, can be plotted against frequency to form a bode plot. From this plot, the linear quality factor can be determined using (Beards 1995)

where fr is the resonant frequency, f1 is the first −3 dB drop down point and f2 is the second. The damping ratio can then be determined from ζ=1/2Q. The buckled energy harvester device tested here was found to be nonlinear in both the uncompressed and compressed states, exhibiting a spring-stiffening or spring-softening effect characterized by a jump discontinuity. Physically, this means the frequency response depends on the direction of frequency sweeping. An even better estimate of the damping for a nonlinear system is to use a modified version of the HPB from Davis (2011)

where fp is the peak gain frequency, fjd1 is the frequency of the jump down point, fjd2 is the frequency of the jump down point that has the same gain as fjd1. The value n is the ratio of the gain of fp and fjd1. It is important to know which type of nonlinearity, stiffening or softening, the system falls under because one peak will be larger than the other during the directional sweep.

Parametric Evaluation of Compliance Arms

A parametric evaluation of the effect of compliance arm geometry on device behavior was performed using the length and the width of the arms as parameters. For clarity, a naming scheme was selected based on the arm width, with devices named as 3×1 and 5×1 referring to 3 mm×1 mm cross section and 5 mm×1 mm cross section, etc. The other two device variations were named for their free pinned or completely glued 20 gauge steel support. For the pinned, glued, 3×1, 5×1 the compliance length was varied between 8 and 12.5 mm. The 3×1 device was also tested in a 3 mm length setup as an additional run. Each device was frequency swept both forward and backward from 0.2 to 0.6 g_rms with a frequency range of 10–35 Hz.

ANSYS Modal Frequency Simulations

To get a rough estimate of the resonant frequencies for the fabricated devices, an ANSYS modal analysis was used. ANSYS linearizes all contact and nonlinear effects when estimating a modal frequency or harmonic sweep response. To capture more accurate, nonlinear responses for the devices, a transient analysis would be needed with a linearly varying frequency acceleration load. If the device exhibits snap-through under the simulated driving conditions, then the model would require a very small time step increment to capture this large displacement event. Due to the large computational times required to perform resonant frequency estimates under such snap-through conditions, these cases were not considered.

The model consisted of 8-node SHELL281 elements for the center beam and 20-node SOLID95 elements for the PLA brace and proof masses. Two cases were run for each device in which one had no compression and the other had a total of 0.1 mm of compression for the pre-stressed state. One item of great interest was the proximity of the second mode (each torque arm being 180° out of phase) to the first mode in terms of frequency, as the second mode tends to be detrimental to the power output of the device.

Promotion into HEOs

Nonlinear buckled devices such as the prototypes evaluated in this article generally exhibit chaotic behavior in response to an acceleration input. Normal, low-energy, snap-through behavior is governed by strange attractors that demonstrate stochastic voltage responses and provide little benefit in terms of peak rms power generation. However, if extra energy is put into the system, the device investigated herein can be pushed into a more periodic, HEO state that can be maintained. Associated with these HEOs are very large displacement amplitudes that may increase power performance and/or bandwidth for a given input acceleration when compared to the same device in the unbuckled state.

The buckled devices probed in this work were promoted into HEOs by generating a Gaussian input monopulse three times as high as the driving sinusoidal audio wave and superimposing it onto this signal in phase. The result created a rapidly decaying acceleration chirp or impulse that momentarily pushed the device at a higher acceleration. Based on previous works (Erturk and Inman 2011), it was theorized that this impulse would provide sufficient additional energy into the structure to enter an HEO state and increase device performance when compared with the same device in the unbuckled state.

Damping Effects

Free vibration tests were performed for the majority of the vibrational energy harvester (VEH) devices. After a monopulse was applied to the stationary device, the output voltage recorded typically appeared to be similar to a decaying sine wave. In the logarithmic decrement method, only peaks in the signal obtained after the base accelerometer had come to rest are used to determine damping behavior. Conversely, the nonlinear half-power method analysis is slightly more complicated due to deviations in devices performance based on sweep direction. Graphs for the logarithmic decrement and half-power method are shown in Figure 4 for the 3×1 mm device with 8 mm compliance arms. Sweeps for a few of the devices in their various configurations are shown in Table 1. An additional compressed case for the 5×1 mm device was also performed to investigate effects of damping on an over-compressed prototype. As seen below, over-compression tends to increase damping which is highly detrimental to device power performance. If the over-compressed case is neglected, the average dampening for all of the nonlinear devices combined is approximately 4.1%. This is a good starting estimate for FEA analysis. The results indicate that slight buckling on compliant structures should not change the damping ratio by much.

Figure 4:

(a) Voltage output and (b) gain response plots for the 3×1 mm device with 8 mm compliance arms.

Power Generation

Forward and reverse sweeps of all devices were performed while recording V_pp and V_rms voltages. Another interesting parameter is the full width at half max (FWHM) value of the sweep and is calculated by measuring the width of the peak (in Hz) at half of the maximum resonant value. This parameter provides an indication of device operating bandwidth. Data from the sweeps of the 3×1 mm device with the 8 mm compliance arm length configuration are shown in Figure 5. It should be noted that the V_rms amplitude increases greatly once snap-through behavior initiates for the buckled cases and that the response of the unbuckled devices looks inherently nonlinear. Such behavior can be caused by large out-of-plane displacements or imperfect boundary conditions (Spreemann, Folkmer, and Manoli 2011).

Figure 5:

Power results for the 3×1 mm device in the (a) unbuckled forward sweep, (b) buckled forward sweep, (c) unbuckled reverse sweep and (d) buckled reverse sweep case. Compliance arms are 8 mm long.

Power output is one of the critical, useful metrics for evaluating energy harvesting device performance. The rms power (Prms) from the PVDF strip voltage is

where R is the load resistance used for each piezoelectric strip (30 MΩ). General results for the peak resonant frequency, the linear modal frequencies and Prms are shown in Tables 2 and 3. Italic and superscript (a) elements show the acceleration and sweep cycles, where snap-through behavior was identified.

Snap-through action almost always tended to increase Prms values for the devices when compared with their unbuckled states. Improvements as high as 102.2% for the forward sweeps and 53.5% for the reverse sweeps are obtained using the data in Tables 2 and 3. With the exception of the glued pinned device (which buckling always increased power in both sweep directions) buckling was a detriment until intra-well actuation was achieved. Forward sweep performance without snap-through was almost always around 80% lower than their unbuckled counterparts.

There are many different ways to characterize the performance of an energy harvester (Pellegrini et al. 2013), some of which are quite complicated due to the many different variables that can be involved (Harne and Wang 2013). Since the proposed device has a broadening effect when promoted into a high-energy state, metrics that take into account broadening effects are used. The first metric is the figure of merit (FoM) which was created by Mitcheson et al. (2008) and is expressed as

where Y0 is the input amplitude, ρgold is the density of gold, V is the volume a device occupies and ω is the peak frequency in rad/s. To account for bandwidth the FoM_V is modified by

where BW1dB is the bandwidth measured at −1 dB from the maximum power generated. The creator of the metric wanted to favor flatter frequency curves as noted by Pellegrini et al., thus measuring down from the peak power by about ~80%.

Another interesting metric to use for this device is the performance index created by Pellegrini et al. (2013) and is expressed as

where m is a specified mass of the oscillator. The entire mass of the device was used in evaluation and not just the proof mass due to the rest of the system making up a considerable amount of the system mass. This would yield a conservative estimate of the performance index. To get the total performance index, all performance index values are integrated over a set range as

where a and b are frequency values picked by the user. A value of −3 dB (~50%) from the peak power generated was selected for the new bistable devices since that is where the FWHM is usually measured. Ia−b is typically thought of as a mean index value. The coefficient of variation for this metric which specifies a relative standard deviation is determined from

Plots showing the performance index and coefficient of variation swept in the forward direction are shown in Figure 6. The pure pinned device in the buckled configuration outperformed itself in the unbuckled configuration during HEO snap-through behavior at higher accelerations. Also, short compliance arms on the 3×1 device did give a wider spread of frequency content during HEO when compared to all other 3×1 device experiments. Reverse sweeps did not show any performance index metrics that outperformed their unbuckled case though most did approach an equivalent value to their unbuckled counterpart.

Figure 6:

Performance index and coefficient of variation for all devices swept in the forward direction.

FoMs modified by a −1 dB bandwidth for all experiments in the forward and reverse cases are shown in Figure 7. From the Mitcheson metric, once a device is kicked into HEO snap-through operation, the FoM_BW value does outperform the unbuckled case in both the forward and reverse direction. Also the 3×1 device shows an increased metric in the reverse direction at low acceleration inputs. This is primarily due to the wide bandwidth shown in Figure 5.

Figure 7:

Figures of merit for all devices swept in the (a) forward and (b) reverse directions.

Chirp Response

Determining what frequency to drive the devices at while imposing a monopulse was done by using the value found at peak performance in swept forward experiments from prior prototypes of the same geometry but swept at a faster rate. Data analysis on the power generation had not been done so it was not known whether the devices would exhibit spring stiffening or softening. To judge the stability and ease of promotion, experiments were performed at fp and ± 2 Hz. Acceleration for the experiments ranged from 0.1 to 0.8 g_rms. Chirp tests were not performed on the glued pin version of the device. A plot of the pinned devices undergoing a chirp excitation is shown in Figure 8.

Figure 8:

Chirp response of the pinned device in the unbuckled (above), compared to the maintained high-energy orbit in the buckled configuration (below).

All experiments for the chirp excitation are shown in Table 4. Values in italics and blue exhibited HEO operation after an in-phase chirp and either temporarily or constantly maintained that state. During HEO operation the proof mass displacement amplitude greatly increases as well as the voltage output. Some devices would exhibit HEO for 1–3 s until ultimately going back to their normal oscillation state which could be a non-snap-through or a continuous snap-through state.