Kresling origami-inspired electromagnetic energy harvester with reversible nonlinearity

This paper presents an electromagnetic energy harvester based on a unique nonlinear Kresling origami-inspired structure. By introducing the equilibrium shift phenomenon, reversible nonlinearity (i.e. mixed softening-hardening behavior) empowers the proposed harvester to work in a broad frequency band, confirmed by both simulation using a dynamic model and experimentation. The prototyped device can produce the open-circuit root mean square (RMS) voltage from 0.09 V to 0.20 V in the reversibly nonlinear response region in (6.19 Hz, 9.63 Hz) and a maximum output power of 0.4956 mW at an optimum load of 18.1 Ω under the excitation of 1.1 g. Moreover, detailed research further reveals that the design parameters of Kresling origami-inspired structure and electrical and mechanical loads influence reversible nonlinearity. Increasing the tip mass and γ 0 in the M2 region of the design map strengthens the softening behavior, and enlarging the electrical load enhances the hardening behavior. The findings from this work deepen the understanding of the nonlinear behavior of Kresling origami, unveils the great potential of origami structure in energy harvesting and offers a new method to realize broadband vibration energy harvesters.


Nomenclature
Name Definition Ue Induced electrical potential Ie Induced current Lc Inductance of the coil * Author to whom any correspondence should be addressed.
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Rc
Resistance Height of equilibrium point with tip mass p i , q i (i = 0, 1, …, 11) Parameters of fitting polynomial series
With advantages above, EMEHs are capable of generating milli-watt [21] to watt level [22] output power from ambient kinetic energy source, such as human motion [23], water currents [24] and wind flows [25] etc.For instance, Yang et al [19] proposed an EMEH based on a rotating motion rectifier, achieving a power output of 0.491 W with a 52.8% efficiency from a moving bicycle.The impressive performance of EMEHs has gained increasing acknowledgement from researchers as an alternative energy source for IoT devices [26].For example, Fan et al [27] proposed an H-shaped coupler energy harvester based on the electromagnetic generator in heavy railway energy harvesting.With 5.14 W average power and 56.46% energy transfer efficiency, the presented generator successfully powered railway sensors and charged supercapacitors in practical tests.Li et al [28] exhibited an electromagnetic-pendulum energy harvester to harness low-frequency water-wave energy.Under 1.5 Hz operation frequency and 0.5 g acceleration, the proposed design achieved an average power of 14.76 mW, providing enough energy for Bluetooth and Zigbee devices.Though the aforementioned linear resonator based designs possess the advantage of ease of design [18], narrow working bandwidth restricts their practical application [20].
To broaden the operation bandwidth of harvesters, researchers introduced nonlinearity into EMEHs [29] by various mechanisms, such as magnetic springs [30][31][32][33] empowering hardening [34,35], softening [36] and more complex multi-stability characteristics [37].For instance, based on the potential well escape phenomenon [38], Wang et al [39] introduced bistability into EMEH by the magnets arrangement, achieving ultra-wide bandwidth from 5.8 Hz to 22 Hz under the 0.5 g excitation, and Liu et al [40] combined the EMEH with four mechanical springs to achieve a tri-stable mechanism that provided broad bandwidth from 2 Hz to 11.5 Hz under 0.7 g excitation.However, the complexity introduced by mechanical and magnetic interactions [41,42] of nonlinear EMEHs hampers their practical application, especially in small-scale energy harvesting.
As compared to other origami structures, the following characteristics make Kresling origami highly promising for being utilized in nonlinear EMEH.Firstly, cylindrical structure of Kresling origami allows it to provide stable axial nonlinear stiffness to EMEHs [59].Besides, nonlinearity of Kresling origami is readily adjustable by small variations in geometric design and exhibits a rich diversity from monostable to multi-stable configurations [46,60].However, the nonlinearity feature and intriguing dynamic behavior of Kresling origami structures is yet to be exploited for designing vibration energy harvesters to broaden operation bandwidth.
This work proposes an origami-inspired electromagnetic energy harvester (OEMEH) with reversible nonlinearity.Key contributions of this work include: (1) adopting origamiinspired structure with reversible nonlinearity in the EMEH design to achieve broadband energy harvesting performance; (2) theoretical modeling of the harvester and a comprehensive analysis about reversible nonlinearity induced by equilibrium shift; (3) experimental revealing the relation between reversible nonlinearity and dynamic characteristics and the effects of electromotive damping and tip mass.The rest of this paper is organized as follows: section 2 provides the design and theoretical model of OEMEH.Section 3 discusses reversible nonlinearity induced by equilibrium shift with the established model.Section 4 experimentally examined and analyzed the reversible nonlinearity and output performance of OEMEH.In section 5, conclusive remarks are presented.

Design of OEMEH
As depicted in figure 1(a), the OEMEH is vertically established on the bottom plate, with two rods to support the top plate.The bottom of the Kresling origami is attached vertically to the center of the bottom plate, and the top of the origami supports the cover.A cylindrical permanent magnet at the top plate center and a toroidal coil at the cover center are arranged collinearly.The initial gap between the magnet and the coil can be adjusted by varying the length of the supporting rod.
The OMEH is excited by base vibration.As shown in figure 1(b), when the vertical base motion is applied to the bottom plate of OEMEH, the Kresling origami, as a spring, is stretched and compressed periodically, leading to the change of gap between magnet and coil, which induces the electromotive force (EMF) in the coil.Due to Faraday's law, the electrical potential between two output ends of the coil changes periodically, converting the vibratory kinetic energy into electrical energy.A finite element simulation preliminarily shows the energy conversion process (figure 1(c)), with the initial gap of 15 mm between the coil (24 mm diameter, 2 mm thickness, 10 mm height and 480 turns) and the cylindrical magnet (20 mm diameter, 5 mm height and the N45 grade).With the sinusoidal movement of the coil (2 mm and 8 Hz), the sinusoidal open-circuit voltage is in phase with excitation velocity and achieves an amplitude of 11 mV.

Dynamic model
To understand the dynamic behavior of the OEMEH, an electromechanically coupled dynamic model is established, as shown in figure 2. Since the Kresling origami is fabricated from light-weight material (i.e.paper), the weight of origami is negligible.In the electrical domain, U e , I e , L c , R c and R L represent the EMF, induced current, internal inductance of the coil, internal resistance of the coil and external resistive load, respectively.In the mechanical domain, M, F N , F D , F e , K N , C, C e , y 1 and y 2 are the tip mass of origami (cover and coil), nonlinear restoring force of the origami, structural damping force, electrical damping force, nonlinear stiffness of the origami, structural damping coefficient, electrical damping coefficient, base motion and displacement of the coil, respectively.
The governing equations of the system are: where u = y 2 − y 1 is the displacement of the tip mass relative to the base.Since the top plate that carries the magnet is fixed to the base and experiences the motion y 1 , u also represents the relative displacement between the coil and magnet.Based on Ampère's force law and Faraday's law, F e and U e can be expressed as ) where θ is the electromechanical coupling coefficient [24].Under the small excitation, θ can be treated as a constant value, in which the induced electrical potential and coil velocity are linearly correlated according to equation ( 4), and is confirmed by the finite element simulation results in figure 1(c).For the simplicity of showing the nonlinear feature of the proposed OEMEH, we use the constant θ in the simulations, irrespective of the excitation level.However, under the considerable excitation, θ, which can be expressed as θ = (N * dΦ /dt)/ u = NlB(u), actually varies with u, where N, Φ, l, B(u) are the number of the coil turns, the magnetic flux of the coil, the circumference of the coil and magnetic flux intensity.
Since the device works in the low-frequency regime, the inductive impedance due to L c is much smaller than the resistive impedance of R c and L c is ignored in the following deduction.According to equations ( 1)-( 4), the governing equation of the system can be simplified as where C e = θ 2 Rc+RL .

Structural analysis
The nonlinear restoring force and stiffness of the Kresling origami F N and K N need to be deduced through structural analysis to perform the dynamic simulation.The unfolded paper with a folding pattern for Kresling origami is shown in figure 3(a).The triangle facet is determined by a 0 , b 0 , c 0 , which are the lengths of the stress-free creases.γ 0 is the angle between creases a 0 and b 0 .After folding the paper, according to the creases, the 3D Kresling origami is obtained with top and bottom plates being regular n polygons, as shown in figure 3(b).In the proposed structure, a regular hexagon (n = 6) is designed as the shape of top and bottom plates (figure 3(c)).In our design, c 0 is equal to hexagon radius R.
Based on the geometric relation, b 0 can be expressed in terms of a 0 , γ 0 and R: In this work, the dynamic behavior of the Kresling origami is investigated by representing it as truss model [51,55], which By treating the creases as springs, the total potential energy of the Kresling origami, U N , can be written as where K a and K b are stiffnesses of equivalent springs of the creases a and b, separately.By substituting equations ( 7) and ( 8) into equation ( 9), the torque of the top plate T N = dU N /dϕ can be obtained as Considering that no external torque is exerted on the top plate during the motion of OEMEH, T N is equal to zero, i.e.
By substituting equations ( 6)-( 8) into equation (11), the relation between h and ϕ can be obtained.Since a, b and φ can be expressed as the function of h, U N can be witnessed as the function of h based on equation (9).Therefore, the vertical restoring force F N and stiffness K N of the Kresling origami can be obtained by

Equilibrium analysis
According to equation ( 9), the variation of designing parameters a 0 , R and γ 0 will influence the equilibrium of U N , leading to the variation of nonlinearity of the OEMEH (figure 4(a)).Hence, this section investigates the relationship between structural configuration and equilibrium.To simplify the analysis, the parameters are nondimensionalized as follows: The ratio between the spring stiffness of the creases b and a can be further assumed as µ K = a 0 /b 0 because Kresling origami is fabricated from homogeneous materials.According to equations ( 9) and (11), the design map of Kresling origami (figure 4(b)) can be qualitatively divided into five zones, monostable (M-type) and bistable (B-type) areas, which are based on numbers, energies and positions of equilibrium points.Notably, equilibrium points are defined as the local energy minimum points, where Kresling origami has no tendency to change its height.Detailed characterization of M-type and B-type are presented in table 1.
To reflect the characteristics of different Kresling designs, we selected five cases to represent different zones (M1: shown in figure 4(c).According to graphs, M3, B1, and B2 origamis are unsuitable for the OEMEH.The B1 and B2 Kresling origamis have two potential wells at the equilibrium points.If the Kresling origami is designed with bistable parameters, the gravity of the coil and the cover will force the origami stays in the lower state (compressed state).To achieve decent output, the length of the supporting rods should be shortened and the magnet and coil should be kept close to each other.However, under big excitation, though the OEMEH with B-type origamis has enough energy to overcome the potential well barrier, the shorter distance between the magnet and coil means that they will come into contact, impeding the coil's displacement to reach the other state of the origami (expanded state).In this scenario, the OEMEH cannot achieve the potential well escape phenomenon.If the supporting rods do not change to allow the snap-through behavior of the coil between two states, the large gap between coil and magnet and the consequent extremely small output are not meaningful in terms of energy harvesting, though this behavior might further broaden the operation bandwidth.Hence, B-type origamis are not preferred in the proposed OEMEH.For the M3 origami, the origami structure cannot support any load since its equilibrium point at h = 0. Nevertheless, the OEMEH with M1 or M2 origami can vibrate around the equilibrium points because of the single potential well.Compared to the M1, M2 origami has a wider well, allowing OEMEH to vibrate in a larger distance and achieve high outputs under the same condition, theoretically.Hence, M2 Kresling origami will be adopted as the nonlinear structure of OEMEH.

Foldability analysis
According to the design (figure 4(c)), h of Kresling origami decreasing to zero reflects the process from the 'deployed' status to the 'fully folded' status (figure 5(a)).However, with inappropriate design parameters may induce interference between facets, leading to the origami structure not being fully foldable.
To ensure the Kresling origami-based energy harvester to operate properly and achieve the intended nonlinear behavior,  the origami should be fully foldable.The foldability of origami thus needs to be analyzed.Figures 5(b) and (c) display the cases with facets interference.In the Kresling origami, triangle ABC is one of its facets.We assume that lengths of BC, AC and AB are creases of a, b and c 0 in figure 2(b), respectively.If a < b, the origami will rotate in counter-clockwise to fold the structure.Based on the geometry, the facet interference will occur when the rotation angle is −π, as shown in figure 5(b).If a > b, the origami rotates clockwise to fold the structure.Similarly, the maximum rotation angle is 2π/3, as shown in figure 5(c).Hence, the range of rotation angle φ is −π ⩽ ϕ ⩽ 2π/3.If the maximum ϕ is out of range, the origami cannot be fully folded due to the facet interference.Referring to equations ( 6)- (11), we calculate the maximum ϕ with different ā0 and γ 0 .Compared with the fully foldable range of ϕ, the design map, as shown in figure 5(d), indicates that the Kresling origami can be fully folded when ā0 ⩽ 1.73.

Reversible nonlinearity
In this section, a simulation is carried out to understand the nonlinear behavior of Kresling origami designed in M2 region in the design map (figure 4(b)).The parameters of the simulation are listed in table 2. The nonlinearity of the origami will be first probed from the tendency of F N and K N .Numerical results of F N (h) and K N (h) are obtained by referring to equations ( 9) and ( 12) and fitted by 11th-order polynomial series, as where p i and q i (i = 0, 1, …, 11) are parameters of fitting polynomial series.Equilibrium position of Kresling origami is calculated by Since the height at the equilibrium position (h 00 = 32.4mm) is assigned as the zero point of u, curves of F N (u) and K N (u) are obtained in figure 6(a) by introducing h = h 00 + u into equations ( 14) and (15).According to figure 6(a)(ii), increasing or decreasing u from the equilibrium position will increase K N (u), i.e. the hardening effect.However, considering the gravity of the tip mass (cover and coil), the origami is further compressed from the equilibrium position to generate external supporting force.Hence, the equilibrium will be shifted toward a lower height (figure 6(a)).New equilibrium of OEMEH is calculated by Assigning the new equilibrium (h 01 = 24.0mm) as the zero of u, the curves of F N (u) and K N (u) after the equilibrium shift are obtained in figure 6(b) by introducing h = h 01 + u into equations ( 14) and (15).According to figure 6(b)(ii), from the new equilibrium position, K N (u) first decreases and then increases with the increase of u, reflecting that both softening and hardening zones, i.e. mixed softening and hardening effect, exist in the K N (u) curve.
To confirm the conjecture about the reversible nonlinearity of the designed Kresling origami, equation ( 5) is written in the state space form and solved numerically using ODE 45 in Matlab.In the open-circuit condition of the OEMEH, a frequency sweep from 4.5 Hz to 14.5 Hz is performed and the displacements and velocities of the OEMEH are obtained under different base excitations (0.5 g, 0.7 g, 0.9 g and 1.1 g where g denotes gravitational acceleration).As shown in figures 6(c) and (d), displacement generally keeps the same pace with velocity during the frequency sweep.The OEMEH exhibits softening behavior under low excitation (0.5 g), i.e. the frequency response is extended from 8.89 Hz to 8.23 Hz during the backward sweep.With the increase of acceleration, the dynamic behavior of the OEMEH changes from softening to reversible nonlinearity, demonstrated by the appearance of multi-value regions during both backward and forward sweeps: due to softening behavior, multi-value region exists in low-frequency range (0.7 g: [7.40 Hz,8.21Hz], 0.9 g: [6.94 Hz, 7.59 Hz] The angle γ 0 is the key design parameter for the geometry (equation ( 6)) and closely related to the energy, force and stiffness curves of the origami (equation ( 9)).Hence, the influence of γ 0 on nonlinearity variation is investigated.Here, γ 0 is varied from 25.5 • to 26.01 • (figure 7 It is noted that the increase of γ 0 prolongs the range of the stiffness-decreasing zone, thus enhancing the softening effect.Figures 7(d

Prototyping and experimental setup
To investigate the electrical output characteristics of the OEMEH, a prototype is fabricated according to figures In OEMEH, cover, top and bottom plates are 3D printed with polylactic acid (PLA).For energy generation, a permanent magnet with grade of N45, diameter of 20 mm, and height of 5 mm, and a coil with 480 turns, diameter of 24 mm, thickness of 2 mm, height of 10 mm, and internal resistance of 18.3 Ω are used.
The nonlinear restoring force measurement and energy harvesting performance testing setups are shown in figures 9(a) and (b), respectively.Restoring force measurement setup is composed of a force sensor (Interface SMT1-5N), a controller and a universal testing machine (Instron 5567).As shown in figure 9(a), the bottom plate of OEMEH is fixed on the fixture and the cover is connected to the force sensor via the central bearing, which minimizes the influence of cover rotation on the vertical restoring force during the test.The controller manipulates the universal testing machine to change the height of OEMEH and acquires the real-time force signal to depict the force-displacement curve.The energy harvesting performance testing setup contains a power amplifier (APS Dynamics Inc. APS 126) and a shaker (APS Dynamics Inc. APS 113), a shaker controller (Vibration Research Corp. VR9500), an accelerometer (PCB 352A56), and an oscilloscope (Keysight DSOX2002A).As shown in figure 9(b), the bottom plate of OEMEH is bolted on the shaker table.The top plate with an embedded permanent magnet is fixed onto the shaker table via two threaded rods, keeping a vertical distance of 45 mm from the shaker table.With the tip mass of the cover and coil, in the static equilibrium, the height of the origami structure is 24 mm, and the vertical distance between the magnet and coil

Verification of reversible nonlinearity.
The reversible nonlinearity of the OEMEH is first confirmed by experiment from two aspects: analysis of the restoring force F N and frequency-sweep tests.First, the restoring force of Kresling origami is measured by the movement of the testing machine from the equilibrium state (restoring force equal to the gravity of cover and coil, height = 24.0mm), which is marked as the zero point of displacement.As shown in figure 10(a), the measured restoring force has the same tendency as the theoretical result.When u increases from 0, the slope of the measured force, which reflects the stiffness of the Kresling origami, first decreases and then increases, demonstrating the existence of reversible nonlinearity.
Then frequency-sweep tests are carried out to confirm the reversible nonlinearity further.The open-circuit voltage of the OEMEH is measured since it is positively proportional to the relative velocity between the coil and magnet according to equation (4).Besides, since one-side magnet arrangement results in the vertically asymmetric distribution of the magnetic field, the electromagnetic coupling coefficient θ varies spatially, leading to the asymmetry of the output voltage.Frequency-sweep results shows evident reversible nonlinearity occurring under high excitations, as shown in figures 10(b) to (f).For example, under the excitation of 1.1 g, one multivalue region of [6.17 Hz, 8.10 Hz] exists in the low-frequency range due to softening effect and the other multi-value region of [8.76 Hz,9.63 Hz] exists in the high-frequency range due to hardening effect.The prototyped device can produce the open-circuit RMS voltage from 0.09 V to 0.20 V in the With the increase of the excitation, the nonlinear behavior of the OEMEH transits from softening to reversible nonlinearity, showing a similar tendency observed in the simulations (figures 6(c) and (d)).Both softening and hardening behavior is enhanced with the increase in the excitation by widening the multi-value regions.When the excitation acceleration increases from 0.3 g to 0.9 g, the softening-related multi-value region is expanded from 0.99 Hz to 2.05 Hz.When the excitation acceleration increases from 0.9 g to 1.1 g, hardening related multi-value region exists and is expanded from 0.68 Hz to 1.53 Hz.The expansion of the multi-value regions shows the enhancement of reversible nonlinearity.In summary, the reversible nonlinearity is proven with the prototyped OEMEH from both restoring force curve and frequency-sweep tests.
Besides, a preliminary analysis of the influence of hysteresis in the restoring force on the output.As shown in figure 11  The high output zone is expanded toward both low and high frequencies due to the reversible nonlinearity.Firstly, softening behavior ensures that high output can be achieved during a backward sweep in the low-frequency range.According to the results, when the electrical load increases from 2.8 Ω to 1000 Ω, the RMS voltage at 7.2 Hz with backward sweep ranges from 13.23 mV to 93.37 mV, higher than that with forward sweep (from 3.50 mV to 15.11 mV), as shown in figure 12(b).High output power is consequently attained at the high energy orbit during the backward sweep.For instance, with an 18.1 Ω external load at 7.2 Hz, an output power of 0.1339 mW is achieved during the backward sweep, higher than that of the forward sweep (0.0071 mW), as shown in figure 12(d).Besides, hardening behavior ensures that high output can be achieved during a forward sweep in the highfrequency range.In figure 12(c), with the increase of electrical load (2.8 Ω-1000 Ω), the RMS voltage at 9.1 Hz with forward sweep ranges from 24.25 mV to 183.24 mW, higher than that with a backward sweep (from 14.65 mV to 92.51 mV).Besides, as shown in figure 12(e), the forward sweep achieves the higher optimal power (0.4808 mW) at the high energy orbit at 9.1 Hz as compared to the backward sweep (0.1143 mW).
To understand the overall performance of the proposed OEMEH, we obtain the maximum outputs in the swept frequency range (4.5 Hz to 14.5 Hz) with different load resistances (figure 12(f)).At each specified resistance, the maximum outputs (RMS voltage, average power) are the highest outputs in the reversibly nonlinear response region.It is noted in figure 12(f) that the increase of the load resistance from 2.8 Ω to 1000 Ω raises the maximum RMS voltage from 25.39 mV to 207.05 mV while the maximum average power of the OEMEH first increases and then declines, reaching a peak of 0.4956 mW at 18.1 Ω, which matches the internal resistance of coil.

Influence of electromotive damping.
According to figure 2 and equation ( 5), the electromotive damping C e , induced by energy-generating components (coil and magnet), has an important influence on the dynamic performance of OEMEH.Generally, under the same excitation, decreasing C e can enhance the displacement of the coil.Based on figure 13(a), increasing the tip displacement of Kresling origami will strengthen the hardening nonlinearity.Since the negative correlation between electrical damping C e and electrical load R L (C e = θ 2 /(R C + R L )), the increase of R L will result in the decrease of C e , leading to the enhancement of hardening nonlinearity.
We have analyzed the reversibly nonlinear response region in the frequency sweep responses at different load resistances to confirm our conjecture about electromotive damping.According to the results in figures 13  According to the analysis in section 3 , the variation of the tip mass (cover + coil) will influence the equilibrium shift, which is related to the transition of the reversible nonlinearity of the Kresling origami.As shown in figure 14(a)(i), equilibrium shift is positively correlated with restoring force of origami, which is equal to the gravity of tip mass.In other words, increasing tip mass will lead to a larger equilibrium shift.To confirm the relation between equilibrium shift and reversible nonlinearity, the variation of the softening and hardening zones with three different tip masses (13.5 grams, 16.1 grams and 18.9 grams) are depicted in figures 14(b)-(d).Results show that the softening zone of the origami stiffness will be expanded with the further equilibrium shift caused by the increase of the tip mass.Besides, the equilibrium height of the origami structure reduced from 25.65 mm to 24.57mm with the increase of tip mass from 13.5 grams to 18.9 grams.Hence, theoretically, the OEMEH with heavier tip mass, which shifts the equilibrium more to the left along u axis (figure 14(a)), will exhibit stronger softening nonlinearity.
To further show the influence of tip mass on the performance of the OEMEH and confirm our conjecture, frequency sweep tests are performed with tip masses of 13.5 grams, 16.1 grams and 18.9 grams, and the results are shown in figure 15.Firstly, reversibly nonlinear response region shifts to the left with the increase of tip mass.For example, under the  The findings from this work deepen the understanding of the nonlinear behavior of Kresling origami, unveil the great potential of Kresling origami in energy harvesting and offers design guidelines and a new method to realize broadband vibration energy harvesters.

Figure 1 .
Figure 1.Design and working mechanism of OEMEH: (a) schematic diagram and (b) working mechanism; (c) finite element simulation of electromagnetic induction due to coil movement.

Figure 4 .
Figure 4. Relation between design parameters and nonlinearity of Kresling origami: (a) equivalent nonlinear spring of Kresling origami in OEMEH; (b) dimensionless design map and (c) potential energy graphs in Kresling origami design.

Figure 6 .
Figure 6.Existence of reversible nonlinearity of OEMEH: vertical restoring force and stiffness (a) before and (b) after equilibrium shift; (c) displacements and (d) velocities of frequency-sweep simulations of OEMEH in open-circuit condition with different excitations.
(a)), and other parameters for simulation are the same as those in table 2. Considering the equilibrium shift, the restoring force F N (u) and stiffness K N (u) for different γ 0 are shown in figures 7(b) and (c), respectively.
) and (e) show the frequency-sweep simulation results with different γ 0 .It is noted that both hardening and softening behavior exist, reflected by the two multivalue regions.With the increase of γ 0 , the reversibly nonlinear response region shifts to the left from[7.03Hz, 11.17 Hz] of 25.5 • to [6.53 Hz, 9.99 Hz] of 26.01 • and the softening behavior is enhanced.

Figure 7 .
Figure 7. Influence of γ 0 on reversible nonlinearity: (a) variation of γ 0 on the design map and the corresponding variation of (b) vertical restoring force and (c) stiffness; (d) displacements and (e) velocities of frequency-sweep simulations of OEMEH in open-circuit condition with different γ 0 .

Figure 8 .
Figure 8. Fabrication process of the Kresling origami: (a) details of unfolded pattern: (i) geometric creases and (ii) Kirigami pattern; (b) unfolded pattern and (c) laser-cutted unfolded paper of the Kresling origami.

Figure 10 .
Figure 10.Verification of reversible nonlinearity: (a) comparison of measured and theoretical force-displacement curves; experimental results of voltages of frequency-sweep tests of OEMEH in open-circuit condition under different excitations: (b) 0.3 g, (c) 0.5 g, (d) 0.7 g, (e) 0.9 g and (f) 1.1 g.

Figure 11 .
Figure 11.Simulation with the hysteresis of the restoring force curve.(a) Restoring force curves.Simulated frequency-sweep results: (b) displacement, (c) velocity and (d) open-circuit voltage under the excitation of 1.5 g.
(a), two parallel dashed curves (namely, revised curve 1 and revised curve 2) are utilized to simulate the hysteresis of the restoring force.Based on the simulated magnetic flux density (B(u)) via COMSOL, the frequency-sweeping results under the 1.5 g excitation are simulated in figures 11(b)-(d).Compared to the simulation results in figures 6 and 7, the hysteresis of the restoring force results in an asymmetric of displacement and velocity, which will lead to the asymmetric tendency of the OEMEH output voltage according to U e = NlB(u) u, as shown in figure 11(d).

4. 2 . 2 .
Impedance matching.This section performs an impedance matching test to show the merits of reversible nonlinearity.With different electrical loads ranging from 2.8 Ω to 1000 Ω, the voltage across the loads under frequency sweep from 4.5 Hz to 14.5 Hz and excitation of 1.1 g is acquired, as shown in figure12(a).To demonstrate the output difference between softening and hardening nonlinear zones, we measured the outputs at two representative frequencies (7.2 Hz and 9.1 Hz) (figures 12(b)-(e)).

Figure 12 .
Figure 12.Impedance matching in multi-value regions and electrical outputs from experiment: (a) multi-value regions due to softening and hardening behaviors during backward and forward sweeps and two representative frequencies (7.2 Hz and 9.1 Hz); voltages with different load resistances at (b) 7.2 Hz and (c) 9.1 Hz; powers with different load resistances at (d) 7.2 Hz and (e) 9.1 Hz; (f) maximum outputs in the swept frequency range with different resistances.

Figure 13 .
Figure 13.Influence of load resistance on frequency band: (a) conceptual sketch of softening and hardening zones with varying displacement; experimental results of voltages of frequency-sweeps tests at different load resistances of (b) 2.8 Ω, (c) 10.0 Ω, (d) 18.1 Ω, (e) 27.0 Ω and (f) open circuit.

Figure 14 .
Figure 14.Equilibrium shift and reversible nonlinearity with different tip masses: (a) equilibrium shifts; stiffness softening and hardening zones with different tip masses of (b) 13.5 grams, (c) 16.1 grams and (d) 18.9 grams.