Non-resonant energy harvesting via an adaptive bistable potential

To find the logic, we consider a model of a single-degree-of-freedom ideal energy harvester characterized by the mass m and displacement x(t) that is subject to the energy harvesting force f (t) and exogenous excitation force $F(t).$ Then, the equation of motion will simply be

$$\begin{eqnarray}&&m\ddot{x}(t)=F(t)+f(t).\end{eqnarray} \tag{ 1 }$$

Here we assume that the ideal harvesting force can harvest all the energy that flows to the system (there is no long-term accumulation of energy in the system). Hence, maximizing the harvested energy will be equivalent to maximizing the energy flow to the system. In other words, we want to maximize

$$\begin{eqnarray}&&{E}_{\mathrm{max}}=\underset{x(t)}{\mathrm{max}}\int {\rm{d}}t\;F(t)\dot{x}(t)\end{eqnarray} \tag{ 2 }$$

over admissible trajectories of $x(t).$ It is easy to show that this integral is unbounded if x(t) is unconstrained. Indeed, the trajectory defined by a simple relation $\dot{x}(t)=\lambda F(t)$ that can be realized with the harvesting force $f=m\lambda \dot{F}-F$ results in a harvesting rate of $\lambda {F}^{2}$, which can be made arbitrarily large by increasing the mobility constant λ. This trivial observation illustrates that the question of fundamental limits is only well-posed for models that incorporate some technological or physical constraints. This is a general observation that applies to most of the known fundamental limits. For example, the Carnot cycle limits the efficiency of cycles with bounded working fluid temperature, and Shannon capacity defines the limits for signals with bounded amplitudes and bandwidth.

As a common constraint to vibratory energy harvesters, we constrain the harvester displacement in a symmetric fashion i.e. $| x(t)| \leqslant {x}_{\mathrm{max}},$ where ${x}_{\mathrm{max}}$ is the displacement limit. Rewriting equation (2) as $-\int {\rm{d}}t\;\dot{F}(t)x(t),$ it can be seen by inspection that the integral is maximized by the optimal trajectory:

$$\begin{eqnarray}&&{x}_{*}(t)=-{x}_{\mathrm{max}}\;\mathrm{sign}\left[\dot{F}(t)\right].\end{eqnarray} \tag{ 3 }$$

This optimal trajectory is indeed realizable by the ideal harvesting force of $f(t)=m{\ddot{x}}_{*}(t)-F(t).$ The interpretation of equation (3) is easy; it says that when F(t) is increasing, x(t) should be kept at its lower limit, and vice versa, when F(t) is decreasing, x(t) should be kept at its upper limit. Thus, the transitions between displacement limits occur when the sign of $\dot{F}(t)$ is changing, i.e. at the extrema of $F(t).$ In other words, in this logic, the harvester mass is kept at its lowest position ($-{x}_{\mathrm{max}}$) until the excitation force F(t) reaches its maximum, when the mass should be pushed to its highest position (${x}_{\mathrm{max}}$) (either by the excitation force, or by the harvesting force if the local maximum of the excitation force is still negative or not big enough to push the mass to the highest position limit¹ ). Similar dynamics occur in the reverse direction, and this strategy continues in the same fashion at every extremum of the excitation force $F(t).$

If the harvester is incapable of injecting energy into the system (a passive-only harvester), the harvested mass should traverse between the limits ($\pm {x}_{\mathrm{max}}$) by the excitation force F(t) only. In this case, the logic is slightly modified; the harvester mass should be kept at its lowest (highest) displacement limit until the largest maximum (most-negative minimum) of the excitation force is reached. Only then is the harvester mass pushed from one displacement limit to the other. This logic is very similar to the well-known buy-low-sell-high strategy in the stock market; hence, we call this logic a buy-low-sell-high (BLSH) strategy hereafter.

Now the question is how to implement this logic. The BLSH strategy could be realized by an adaptive bistable potential. In essence, the passive BLSH strategy keeps the harvester mass at one end ($\pm {x}_{\mathrm{max}}$) before letting it go to the other end according to its logic. A bistable potential with stable points at $\pm {x}_{\mathrm{max}}$ and an adaptive potential barrier could do this. To realize the BLSH logic, the potential barrier should be large enough to confine the harvester mass in one well (${x}_{\mathrm{max}}$ or $-{x}_{\mathrm{max}}$). Then, when, according to the logic, the harvester mass should traverse to the other end, the potential barrier should vanish. This logic is schematically shown in figure 1. Note that a harvester equipped with this nonlinear logic is essentially a non-resonant harvester.

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The harvester is modeled as a lumped-parameter mechanical oscillator coupled to a simple electrical circuit via an electromechanical coupling mechanism. The formulation here is generic and could be applied to both capacitive (e.g. piezoelectric) and inductive (e.g. electromagnetic) transduction mechanisms. The nondimensionalized governing dynamic equations could be written as [8]

$$\begin{eqnarray}\ddot{x} & + & 2\zeta \dot{x}+\displaystyle \frac{\partial U(x,t)}{\partial x}+{\kappa }^{2}y=-{\ddot{x}}_{b}\\ \dot{y} & + & \alpha y=\dot{x}.\end{eqnarray} \tag{ 4 }$$

In the above equations, x is the oscillator's displacement relative to base displacement (x_b). Linear mechanical damping is characterized by the damping ratio ζ, and κ denotes the linear electromechanical coupling coefficient. y represents the electric quantity that would be voltage or current in capacitive or inductive transduction mechanisms, respectively, and α is the ratio of the mechanical to the electrical time constants. The adaptive bistable potential is denoted by $U(x,t)$, and the overdot denotes differentiation with respect to dimensionless time. All parameters and variables are dimensionless.

Two common techniques to realize bistability are the buckling phenomenon and magnets (to create negative stiffness) in addition to the positive mechanical stiffness. When using a magnetic field to realize bistability, if permanent magnets are replaced by electromagnets [28] (giving a controllable magnetic field) one can change the potential shape and, hence, create an adaptive bistability. A passive bistable potential admits a quartic form [29], and when made adaptive, we model it as

$$\begin{eqnarray}&&U(x,t)=\displaystyle \frac{1}{2}(1+\sigma (x,F,\dot{F}){r}_{k}){x}^{2}\\ &&\quad -\displaystyle \frac{1}{4}\sigma (x,F,\dot{F})(1+{r}_{k})\displaystyle \frac{{x}^{4}}{{x}_{s}^{2}},\end{eqnarray} \tag{ 5 }$$

where ${r}_{k}\lt -1$ is the strength of the negative stiffness of the magnetic field relative to the linear mechanical one. x_s denotes the dimensionless stable position of the bistable potential, and $\sigma (x,F,\dot{F})$ is a signal function which repeatedly switches between 1 and 0 according to the BLSH logic. The signal function depends on the system states and excitation statistics. $\sigma (x,F,\dot{F})$ is always equal to unity except when we want the harvester mass to traverse from one end to the other (according to the BLSH strategy), when it is set to zero. Based on the BLSH logic, $\sigma (x,F,\dot{F})$ could be formulated as follows

$$\begin{eqnarray}\sigma (x,F,\dot{F})=\left\{\begin{array}{ll}0; & \dot{F}(t)=0\;\mathrm{and}\;F(t)x(t)\lt 0\\ 1; & | x(t)| \approx {x}_{\mathrm{max}}\;\mathrm{and}\;F(t)x(t)\gt 0.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

In equation (6), the signal function is set to unity when $| x(t)|$ is approximately and not exactly equal to x_max. The reason is twofold: first, once the potential is activated the mass still oscillates in that well even though by a small amount; hence, to make sure it does not exceed the displacement limits, the potential is activated slightly before it reaches $\pm {x}_{\mathrm{max}}.$ Second, once the mass reaches one well, we want to keep it trapped in that well until the condition for the release of the mass arises, i.e. the first condition in equation (6). However, before this condition has arisen, the mass oscillates slightly in that well, so its displacement will be approximately and not exactly equal to $\pm {x}_{\mathrm{max}}.$ In simple words, the second condition in equation (6) says that the mass should be trapped and kept at one end once it reaches the displacement limits before the first condition arises and it is released. It is also worth mentioning that, in the limit where the potential barrier height goes to infinity, the approximation changes to equality.

Figure 2(a) depicts an energy harvester with a piezoelectric (capacitive) transduction mechanism equipped with adaptive bistable potential. The adaptive bistability is realized by an electromagnet and a permanent magnet (the proof mass). An on/off controller is used to implement the BLSH logic. The controller senses the excitation and the harvester states and then according to the BLSH strategy sends a signal to the current supplier to supply an appropriate current ($\sigma (t)=1$) or to shut down the current supply ($\sigma (t)=0$).

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It should be noted that using electro- and permanent magnets is not the only way to realize adaptive bistability. Although this technique is easy to implement, care should be taken to design the electromagnets with minimal losses. Since the harvester mass is at the displacement limits for a substantial fraction of the time, ohmic losses could be larger than the harvested energy if the electromagnets are poorly designed. Another possible way to realize adaptive bistability, as mentioned earlier, is via adaptive buckling. Buckling as a means to create bistability in the context of energy harvesting is well studied (see e.g. [30]). Making it adaptive could solve the issue of ohmic losses although it entails its own practical difficulties, e.g. adaptively changing the axial force to switch between the buckled and normal states of the beam.

Figure 2(b) shows how the potential shape changes based on the controller signal $\sigma (t),$ and graphically depicts the sequence of the harvester mass trajectory following BLSH logic on admissible potential curves² . It should be noted with this type of implementation (equation (5) and figure 2(a)) the adaptive bistable system following BLSH logic will not always be passive. For instance, when the harvester mass is moved $\unicode{9312} \to \unicode{9313}$ ($\unicode{9315} \to \unicode{9314}$) a positive amount of energy is added to the system because of the way the potential shape is changed. However, in the transition right before the one that adds energy, i.e. in $\unicode{9313} \to \unicode{9312}$ ($\unicode{9314} \to \unicode{9315}$), the same amount of energy is taken out of the system; hence, the net energy injected into the system by this type of implementation is zero in half a cycle (if not at all times) when the cycle is referred to transitions from $-{x}_{\mathrm{max}}$ to $+{x}_{\mathrm{max}}$ and then back to $-{x}_{\mathrm{max}}.$ In order to have a passive system at all times, one should come up with a bistable mechanism whose potential barrier could be deepened without changing the potential energy level of its stable points, like a latching mechanism. This is not the case with the current techniques for realizing bistability (buckling and magnetic fields).

The potential function considered here for the bistable system is the same as the one used for the adaptive bistable harvester with a small change in the parameter notation ($1+{r}_{k}\to -a$). The potential used is of the form $U(x)=-\frac{1}{2}{{ax}}^{2}+\frac{1}{4}a\displaystyle \frac{{x}^{4}}{{x}_{s}^{2}}$ where $a\gt 0.$ Figure 3 shows the average power and displacement amplitude of the bistable system when subjected to harmonic excitation of the form $-{\ddot{x}}_{b}={F}_{0}\mathrm{sin}(\omega t).$ This paper intends to mainly cover the low-frequency excitation where the linear harvesters fail to work efficiently; hence, the dimensionless excitation frequency is set to $\omega =0.05.$ The average power is calculated by $\frac{1}{T}{\displaystyle \int }_{0}^{T}{y}^{2}(t){\rm{d}}t$ for a long simulation time T. One should note that this expression gives the normalized dimensionless average power. The dimensional instantaneous power is equal to $(m{\omega }_{n}^{3}{l}_{c}^{2})\alpha {\kappa }^{2}{y}^{2}$ where m, ${\omega }_{n},$ and l_c are the harvester mass, time-scaling frequency, and length scale, respectively. Hence, the average power used here is nondimensionalized by $m{\omega }_{n}^{3}{l}_{c}^{2},$ and further normalized by $\alpha {\kappa }^{2}$.³

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It can be seen from figure 3 that the average power increases monotonically with a and x_s up to a maximum and then drops sharply. This is where the interwell oscillation turns into intrawell oscillation (the potential barrier increases linearly with a and x_s²). A drastic decrease in the amplitude of the oscillations verifies this. It should be noted that for values below the optimum value of a (for a given x_s), the system is still in interwell motion; however, the power monotonically decreases as a is decreased from its optimum value. This can be seen more clearly in figure 4. This suggests the robustness issues with the conventional bistable system, that is, the harvester works efficiently only when the potential barrier is slightly below its critical value, when it triggers the interwell oscillations, which agrees with Zhao and Etrurk's claim [20].

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Next, we compare the performance of the adaptive bistable harvester with that of optimized conventional bistable and linear harvesters when they are subjected to harmonic excitation. To this end, we first optimize the parameters of the bistable system for given excitation inputs and displacement limits. The same harmonic excitation used in figures 3 and 4 is considered here (${F}_{0}=10$ and $\omega =0.05$). According to figures 3 and 4 the optimal parameters corresponding to maximum displacement of 3.4 are x_s = 2 and a = 12. For a fair comparison, the parameters of the adaptive bistable and linear harvesters are set such that their maximum displacements do not exceed this value (${r}_{k}=-300$ and x_s = 2.8 for the adaptive bistable harvester and the natural frequency of $\sqrt{3}$ for the linear harvester).

Figures 5 and 6 show time histories of the displacement and electrical domain state (voltage or current for capacitive or inductive transduction mechanisms, respectively) for the three adaptive bistable, conventional bistable, and linear harvesters. According to the figures, although they all have the same maximum displacement, the maximum induced voltage (current) in them is quite different with the adaptive bistable harvester having the largest and the linear having the smallest induced voltage (current). One could notice the BLSH logic in the adaptive bistable harvester by comparing the moments of the transition from one end to the other and the excitation force extrema. It should also be noted that the conventional bistable harvester is trying to mimic the BLSH strategy in a less effective way.

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Another way to compare the harvesters' performances is via their phase portraits. Figure 7(a) shows these phase portraits. As seen in the figure, the transition of the oscillator's mass between the two displacement limits occurs at a higher velocity for the adaptive bistable harvester than for the other two. The force–displacement diagram in figure 7(b) illustrates even better how the adaptive bistable harvester outperforms the other two. This diagram shows the force capable of doing positive work versus displacement. An ideal harvester, i.e. a harvester with BLSH strategy and ideal harvesting force, will have a perfect rectangle on this diagram, given the displacement limits. This rectangle represents the maximum amount of energy that could be pumped into the harvester (which will consequently be harvested by the ideal harvesting force) in one cycle. The ideal harvester with the perfect rectangle in the force–displacement diagram is analogous to the Carnot cycle with its perfect rectangle in the temperature–entropy diagram given the temperature limits of the hot and cold reservoirs. In both cases, all the other systems (harvesters and heat engines) fall within this perfect rectangle, enclosing a smaller area. Time histories of the harvested energy via the three harvesters depicted in figure 8 prove the greater effectiveness of the adaptive bistable system compared to the other two.

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As mentioned earlier, most real-world excitations are random and non-stationary rather than harmonic, and the linear and bistable harvesters do not work effectively when subjected to these types of excitations. To examine and compare the performance of the three harvesters with random excitations, we subject all the harvesters to experimental and relatively low-frequency walking motion. This data is experimentally recorded at the hip level while walking [31]. The time history and spectral representation of the walking excitation used here are depicted in figure 9.

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For simulations the experimental data are first non-dimensionalized with a scaling frequency of 500 Hz and scaling length of $20\;\mu {\rm{m}}.$ Again, first the conventional bistable potential parameters (a, and x_s) are optimized for maximum harvested energy for a displacement constraint of 1.5; then the parameters of the adaptive bistable and linear harvesters are set such that they do not exceed this displacement limit. The harvested energy is computed in the same way as in the case of the harmonic excitation, with the only difference being that it is multiplied by the constant $\alpha {\kappa }^{2}$ for the sake of easier numerical comparison between different harvesters.

Figure 10(a) illustrates the displacement time history of the harvester with adaptive bistability following a BLSH logic. Energy harvested by the different harvesters is compared in figure 10(b). In addition to the optimal conventional bistable system (${x}_{s}=0.9,$ and a = 1.6), two other bistable systems with detuned a parameters are simulated. According to the figure, the BLSH adaptive bistable harvester outperforms the optimal conventional bistable and the linear harvesters. It can also be seen that changes in the bistable system parameters could significantly diminish the harvester's effectiveness. Despite the the differences in the governing dynamic equations and proposed harvester mechanisms, the results in figure 10 look similar to those in [22]; the reason is that both the latch-assisted mechanism in [22] and the adaptive bistable system in this study try to mimic the same BLSH logic, and given the same transduction mechanism and excitation input these two mechanisms will ideally harvest the same energy.

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