1 Introduction

Energy harvesting of waste vibrations to power wireless sensors in remote environments has been widely investigated^[1-3]. Thus, a small size may offer additional flexibility in designing harvesters for practical applications. The maximum power harvesting of 1-degree-of-freedom mass-spring-damper harvesters can be achieved only at their natural frequency excitation, and it cannot be suitable for the random frequency-variant excitations in an ambient environment.

A promising approach involves the exploitation of stiffness nonlinearity^[4-9]. In this instant, these harvesters have a broader frequency bandwidth than the linear devices. Therefore, a tuning mechanism is not necessary for adaption to environmental excitation frequencies^[10-12]. Daqaq et al.^[6] demonstrated that the nonlinearity induced bending of the frequency response curves can be beneficial. Bi-stable energy harvesting has also received extensive attention^[13-19]. Harne and Wang^[20] described the various bi-stable energy harvesters. For non-resonance effects, this device exhibits coupling between the exterior excitation and the harvesting system over a wide range of frequencies. If carefully designed, considerable power over a broad frequency bandwidth can be produced using bi-stable harvesters^[19-21]. Jiang and Chen^[22] investigated piezoelectric harvesting from a random vibration source via the snap-through mechanism. McInnes et al.^[23] investigated the use of a stochastic resonance in a bi-stable harvester to enhance transduction in vibration-based energy harvesters. It has been shown that enhanced energy harvesting is realized by combining a periodic force with Gaussian noise. Wu et al.^[24] investigated the dynamics of a linear-bi-stable energy harvester. The results demonstrated the positive effects of the coupled system on energy transduction. Chen and Jiang^[25-26] concerned the use of coupled snap-through mechanism to enhance energy harvesting via internal resonance. The results indicated that the internal resonance design can produce more power than other designs.

Alternatively, nonlinearity can be realized using a snap-through mechanism. However, the stable equilibrium states for the conventional snap-through mechanism are hardly determined and easily bifurcate to the mono-stable state when mistuning of the weight of the mass^[27], and thus introducing a modified snap-through mechanism is necessary to overcome this. This article presents the results for an energy harvester using a modified snap-through mechanism configured by three springs. One of the springs supports the magnet mass which is capable of vibrating along the coils, and the other two springs, which are called auxiliary springs, act as a nonlinear stiffness. This spring configuration for energy harvesting is unique and compact, and thus can serve as a useful model to investigate the dynamic behavior of this type of nonlinear energy harvester. The multi-scale method (MSM) is applied to analyze such a system. The aim of this work is to analytically investigate the small vibration behavior of the snap-through harvester around one of the equilibrium points, and to compare the harvesting performance of the two types with the same distance between the two equilibriums. Analytical expressions are derived to plot the frequency response for small vibration constrained at one of the wells. The averaged power is then used to evaluate the performance by Gaussian white noise excitation.

The content is organized as follows. In Section 2, a mathematical model of the modified snap-through energy harvesting system is developed. In Section 3, approximation solutions for the amplitude-frequency response of the mono-stable configuration are derived, and the effect of nonlinearity on the performance of the harvester is studied. In Section 4, the characteristics of the two types of bi-stable configurations under low excitation are investigated, and the analytical results are compared with numerical data. In Section 5, the performance of two types of the bi-stable configuration is evaluated using the averaged power of Gaussian white noise excitation. Finally, Section 6 is an overview of the conclusions.

2 Description of the energy harvester based on a modified snap-through mechanism

Figure 1 shows a physical model of an energy harvester based on a modified snap-through mechanism for the vibration of a permanent magnet with mass m. The system has a vertical spring k_v in addition to two horizontal springs with stiffness k_h and damper c. The vibrating magnet mass producing magnetic flux intensity B, could move in the coils with the effective coil length L_coil, resulting in an electric current I. The electromagnetic part is characterized by the resistance R and the coil inductance L_ind.

The conversion of the mechanical energy into the electrical energy due to the excitatory motion of the base x_e is of particular interest in this model. The horizontal springs introduce geometric stiffness nonlinearity. The static force-deflection of the energy harvester is given by^[28]

(1)

which can be approximated by

in which

l is the length when the springs are in the lateral position, and l₀ is the initial length of the springs.

The dynamical equations of the system are given by

(2)

where z=x-x_e.

Equation (2) can be rewritten in a non-dimensional form as

(3)

where

with (·)'=d(·)/dτ. ^ represents the non-dimensional parameter.

If α >0, the system has only one equilibrium, that is,

If α < 0, the system has three equilibriums, which are represented as

(4)

These could be termed the mono-stable or bi-stable configuration.

3 Energy harvesting of the mono-stable configuration

If α >0, the system is mono-stable, and it is assumed α =Ω_mo². An appropriate rescaling of Eq. (3) is

in which ε is a small bookkeeping parameter.

Applying the MSM, a solution is obtained in the form of a second-order approximation^[29],

(5a)

(5b)

where T₀ =τ is the regular time, and T₁=ετ and T₂ =ε²τ are slow times. Note

(6a)

(6b)

where D_n (n=0, 1, 2) is the partial differential operator, defined as Therefore, the first equation in Eq. (3) can be written as

(7)

Equating the coefficient of each power of ε to zero gives an ordinary differential equation that could be solved to determine the unknown quantities in Eq. (7). To illustrate the main ideas, only the case of the main resonance Ω ≈ Ω₀ is considered. The detuning parameter σ is introduced to measure the difference between the nonlinear resonance and the linear resonance. Therefore,

(8)

Substituting Eq. (8) into Eq. (7) and equating the coefficient of each power of ε to zero yield the following equations.

ε⁰:

(9)

(10)

ε¹:

(11)

ε²:

(12)

The solution to Eq. (9) is sought in the complex form as

(13)

where A(T₁, T₂) is an undetermined function, and c.c. means complex conjugate. Substituting Eq. (13) into Eq. (10), we can obtain

(14)

Substituting Eq. (13) into Eq. (11) yields

(15)

exp is called the secular term, and must be eliminated from Eq. (15). This requires

(16)

which implies that A(T₁, T₂) does not depend on T₁. Substituting Eq. (16) into Eq. (15) yields

(17)

Since there are quadratic nonlinear terms in Eq. (3), it can be assumed that Substituting Eqs. (13), (14), and (16) into Eq. (12), we have

(18)

To eliminate the secular terms in Eq. (18), this requires

(19)

The equation above can be simplified, assuming

(20)

Substituting Eq. (20) into Eq. (19) and separating the real and imaginary parts yield

(21)

Assume

(22)

Substituting Eq. (22) into Eq. (21) yields

(23)

The stable-state solution is given by setting ϕ'=a'=0, which yields

(24)

Eliminating the harmonic terms cos φ and sin φ in Eq. (24), and considering Ω =Ω_mo+ε²σ, the implicit amplitude-frequency response is given by

(25)

where the term is produced by the stiffness of the electromagnetic generator (termed electrical stiffness), and the term is produced by the damping of the electromagnetic generator (termed electrical damping).

This is a quadratic equation in Ω² that can be solved to give

(26)

Combining Eq. (14) and Eq. (20) and separating the real and imaginary parts in the resulting equations give

(27)

where

For the closed electric circuit, the harvested power can be obtained as

(28)

The amplitude of the base motion that can be applied so that the peak frequency equals the natural frequency of the equivalent linear harvester can be predicted using an analytical method^[30], an overview of which is given. The maximum responses obtained by Eq. (26) are equal at this frequency, to give

(29)

where

which occurs at a frequency of

(30)

The excitation amplitude that can be applied so that the peak frequency equals the natural frequency of the equivalent linear harvester is given by

(31)

The frequency response curves of the mono-stable configuration are shown in Fig. 2 for various horizontal stiffness and the amplitude of the base motion is set to . For each value of the maximum occurs at a frequency Ω =1 as expected. A linear energy harvester has a harvesting bandwidth around the natural frequency. In Fig. 2, it is found that the frequency associated with harvesting is dependent on the extent of bending of the frequency response curve to the left. As is increased, the harvesting frequency band is extended to lower frequencies. However, the peak amplitude for the power is slightly reduced. If the system has a hardening behavior (mono-stable) instead of snap-through (bi-stable system). In that case, ≤ 2^[27] should be satisfied.

4 Energy harvesting of the bi-stable configuration

If α < 0, the system has bi-stability. Of particular interest is the case of low excitation, where the origin of the coordinate is shifted to the left or right stable equilibrium by introducing the new variable . Assuming -2α = Ω_bi², Eq. (3) can be rewritten as

(32)

which also has quadratic nonlinearity and a positive linear stiffness. The minus sign in Eq. (32) is for the left well solution and the plus sign applies the right well solution. In the following derivations, only the latter case is considered without loss of generality due to the symmetry of the system.

Since there is an additional quadratic nonlinear term compared with the mono-stable configuration, a solution is sought in the form of a second-order approximation as

(33a)

(33b)

Using the MSM in the case above, the amplitude-frequency response relationship yields

(34)

where the electrical stiffness is

and the electrical damping is given by

Comparing the amplitude-frequency relationship equation between mono-stable (see Eq. (25)) and bi-stable configurations (see Eq. (34)), it is found that it is hardening for the mono-stable design and softening for the bi-stable design. In addition, the degenerated linear natural frequency for the bi-stable configuration is twice that of the mono-stable configuration if electrical effects are neglected. It is also found that the electrical stiffness can strengthen the total linear stiffness in the case of the mono-stable design, and reduce this parameter in the case of the bi-stable design. However, the effects of the electrical damping on these two classes are the same. The harvested power can then be determined by combining Eqs. (26)-(28).

A bi-stable configuration is sometimes used because of its superior harvesting frequency bandwidth. However, this configuration introduces additional calculation complexity compared to the mono-stable energy harvester, which is a disadvantage. Nonetheless, the frequency bandwidth of the energy harvesting is extended compared with the mono-stable design.

Similar to the mono-stable energy harvesters, the performances of the bi-stable designs compared with the corresponding existing systems determine their effectiveness. Here, however, there are additional possibilities for achieving a bi-stable potential well. It is possible, for example, to have horizontal and vertical springs by inserting a vertical spring as in Fig. 1; or to make only horizontal stiffness by removing the vertical spring, however, in the latter case, to retain the symmetry of the two wells as in the former case, the weight of magnet must be offset by horizontal rotating. In this article, these are termed as follows: (ⅰ) vertical and horizontal springs (H-V), (ⅱ) only horizontal springs (H-O).

The performances of H-V and H-O configurations are compared for the same distance between two equilibriums. The length ratio for the H-O configuration can be applied so that the distances between two stable equilibriums for the H-V and H-O configurations are the same, and are given by^[31]

(35)

where is the length ratio of the H-V configuration.

To illustrate the harvesting behavior of the bi-stable system, the amplitude of the base motion is set to a value less than that is achieved numerically when the system just begins to vibrate across the two wells. For this condition, it is found that the system behaves intra-well vibration. Also, to illustrate the relative behavior between the H-V and H-O bi-stable energy harvesters, the non-dimensional length ratio and the horizontal stiffness for the H-O configuration are set to the value of In this case, the distances between the two stable equilibriums for the H-V and H-O configurations are the same.

To examine whether the MSM correctly captures the dynamic behavior, the frequency response curves for the displacement and power are plotted in Fig. 3, together with numerical results. These results are in agreement and the observations made in Fig. 4 can be considered to be valid.

Figures 4(a) and 4(b) show the displacement and power frequency response curves of the H-O configuration when nonlinearity is changed by varying It is found that as nonlinearity increases by changing Both the peak frequency and the magnitude of the response at this frequency increase. Figures 4(c) and 4(d) show the effects on the H-O configuration when increases. It is found that as nonlinearity increases, the harvesting range broadens and the frequency response curves for the displacement are almost the same as that at peak frequencies. Regardless of the amplitude-frequency responses for the displacement or power, the curve for the H-O configuration is bent further to higher frequencies compared with the H-V configuration. The reason for this is that the vertical spring can shorten the distance between the two equilibriums, but the compressed horizontal springs can enlarge the distance for the two equilibriums. To achieve the same distance for the two equilibriums, more compression of the horizontal springs for H-O is needed. Thus, nonlinearity for H-O is relatively large, and the frequency response curve for H-O exhibits more bending than that for the H-V configuration.

5 Energy harvesting from the random excitation

The modified snap-through mechanism can be used for a nonlinear electromagnet energy harvester to improve the performance when the device is subject to random excitation. The dynamic equation of the bi-stable energy harvester subject to a random base motion x_e = η_e (t), as shown in Fig. 1, can be approximated by

(36)

where

As previously discussed, the analytical results and the conclusions obtained using the MSM are only strictly applicable when the response is periodic. When the modified snap-through mechanism is subject to random excitation with a Gaussian random characteristic, Eqs. (25) and (34) cannot be used to quantify the harvesting power. Therefore, the harvesting electric power is defined in terms of the mean square (MS) of the electric current,

(37)

To calculate the harvesting power for random base motion, the fourth-order Runge-Kutta method is applied to directly deal with the dynamic equations. At each excitation level, the response can be obtained and the MS of the steady-state electric current response is employed to measure the effectiveness of the harvesting shown in Fig. 5 for H-V and H-O configurations. The area of the frequency response curve is used to evaluate the relative performance between the H-V and H-O configurations. The difference area of the energy harvesting process for these classes of configuration is shown as the shaded area. The averaged power based on the Gaussian white noise is used to measure the performance. It is evident that the averaged power of the H-O configuration is larger than that of the H-V configuration. Moreover, the difference harvesting power is increased as the intensity of the noise increases. The reason for this is that the shaded area for the H-O configuration is located at higher frequencies and larger amplitudes compared with the H-V configuration. The power is positive relative to the vibration frequency and vibration amplitude. The time responses of the magnet mass are shown in Fig. 6. For low excitation, it can be seen that the harvesting mass of both the H-V and H-O configurations exhibits intra-well vibration (see Figs. 6(a) and 6(b)). For moderate excitation levels, the harvesting mass exhibits inter-well vibration and the response shows some regularity (see Figs. 6(c) and 6(d)). For large excitation levels, the regular motion disappears and the response is once again stochastic (see Figs. 6(e) and 6(f)).

6 Conclusions

The modified snap-through mechanism is analyzed to determine the most efficacious utilization with regard to electromagnetic energy harvesting. The MSM is used for analysis of the intra-well vibration. The results are validated by some numerical studies. Then, the averaged power is numerically determined under the Gaussian white noise for the range of both intra-well and inter-well vibration. The main conclusions are listed as follows.

(ⅰ) The harvesting frequency band of the mono-stable configuration is extended to lower frequencies as the horizontal stiffness is increased.

(ⅱ) The displacement and power amplitude-frequency responses of both the H-V and H-O configurations exhibit more bending at high frequencies as increasing the horizontal stiffness.

(ⅲ) Under random excitation, the bi-stable energy harvester with only horizontal springs can outperform the harvester with both horizontal and vertical springs for the same distance between two equilibriums.