Means of Enhancing the Power Output and Widening the Operational Frequency Band of Vibration-Based Electromagnetic Energy Harvester

Abstract

Purpose

This paper presents the design, development, and performance analysis of a Two-Degree-of-Freedom Vibration-Based Electromagnetic Energy Harvester (TDOF VBEEH). The purpose is to enhance power output and widen the operational frequency band of the harvester. In the literature, various techniques such as mechanical amplification, resonance tuning, and nonlinear oscillations etc., have been explored. The proposed TDOF VBEEH incorporates a mechanical amplifier in series with a traditional Single-Degree-of-Freedom (SDOF) VBEEH.

Methods

The paper derives an analytical expression for the average power output of the TDOF VBEEH based on the principles established by Tang and Zuo. The effects of mass ratio, electrical damping ratio, and tuning ratio on the power output and effective operational frequency band of TDOF VBEEH have been studied to provide design guidelines for TDOF VBEEH. Also, the experimental analysis is conducted to examine the effects of a purely resistive load and mass ratio on the average harvested power of the TDOF VBEEH. The study utilizes a specially designed setup for TDOF VBEEH. Furthermore, by employing the method of surface plots and contour diagrams, the global optimum values of power output for the TDOF VBEEH under various electrical damping ratios and normalized excitation frequencies have been determined.

Results

The results demonstrate that the appropriate selection of mass ratio, tuning ratio, and electrical damping ratio can enhance the power output and widen the effective operational frequency band of the TDOF VBEEH. Notably, the maximum harvested power is achieved when the electrical circuit connected to the harvester has a resistive load of approximately 1500 Ω, which aligns with the internal resistance of the copper coil. The results show that the increased power output is attainable over a widened operational excitation frequency band compared to the traditional Single-Degree-of-Freedom (SDOF) VBEEH. From the comparison of the experimental and analytical results, it is seen that the TDOF VBEEH performs better than the SDOF VBEEH when the parameters are properly chosen.

Conclusion

These findings highlight the superior performance and potential of TDOF VBEEHs over SDOF VBEEHs, emphasizing the importance of parameter selection in maximizing power output and widening the operational frequency band. Also, the findings offer valuable design guidelines for selecting system parameters when developing TDOF VBEEHs to power small electronic devices.

Introduction

In the last few decades, scientists and engineering researchers have been working on seeking solutions to the problem of the development of uninterrupted maintenance-free power sources for small electronic devices in general and remotely located wireless sensor nodes in particular. The development of a single degree of freedom (SDOF) vibration-based electromagnetic energy harvester (VBEEH) has come up as a satisfactory solution to this problem. SDOF VBEEHs are reliable, simple in construction and have huge design flexibility, and can be made available in small sizes and weights. However, this harvester gives maximum power output only at the resonant frequency and its effective operational frequency band is extremely narrow. Therefore, engineers—designers—researchers have proposed some innovative methods to enhance the power output and widen the operational frequency band. An excellent survey of electromagnetic electrostatic, piezoelectric and magnetoelectric [1,2,3], with vibration-based energy harvesting technology, hybrid mechanism [4] and architectural approach has been presented by various researchers [4,5,6]. Patil and Sakri [7] have shown that the maximum attainable power generated from a VBEEH is significantly influenced by the values of the mechanical damping ratio, during resonance and off-resonance conditions. From the results of theoretical and experimental investigation for optimal power output from VBEEH, the mechanical damping ratio should be as low as possible and the electrical damping ratio should be almost equal to the mechanical damping ratio. They have also studied experimentally the effect of shunted electrical loads on the performance of a VBEEH. Halim et al. [8, 9] have shown that the electrical damping ratio may be adjusted by varying the resistive load to achieve the maximum power output. The damping characteristics of the electromagnetic harvester system are determined by using an impulse response. Zhang et al. [10] investigated the effect of ohmic resistors, rectifiers, and capacitors on electromagnetic harvester performance. The main challenge to be addressed in adopting VBEEH is to increase its output voltage. VBEEHs in general have a very low voltage (piezoelectric, electromagnetic or electrostatic) only at one or a few excitation frequencies. Hence, the resonance generator produces maximum power when excited at its resonant frequency. Yildirim et al. [11] have reviewed the performance enhancement methods such as power amplification, resonance tuning, and means for increasing the effective operating bandwidth of a vibration energy harvester, and presented a summary of these methods with their merits and limitations, using different approaches such as the design of broadband frequency VBEEHs [12, 13], tunable frequency VBEEHs [14], nonlinear resonance [15] (dynamic magnifier [16, 17] and frequency up-conversion methods [18]). Malaji et al. [19, 20] presented the methods of enhancing the energy harvesting capability of coupled harvesters suitable for low-frequency applications. They have also proposed the multiple harvesters with intentional mistuning to obtain the wider bandwidth. Chaurha et al. [21] have reported a method of enhancement of bandwidth and vibration suppression of energy harvester using the nonlinear attachments. Chen and Wu [22] have presented a two degree of freedom electromagnetic energy harvester to increase the operating frequency bandwidth by incorporating a spiral diaphragm into a U-shaped cantilever. Tao et al. [23] have presented a novel two degree of freedom MEMS electromagnetic vibration energy harvester to address the fundamental issue of these energy harvesters that they give maximum power output in a narrow bandwidth around the resonance frequency of the ambient mechanical vibration sources which have distributed/multiple frequencies. Tang and Zuo [24] have proposed a dual-mass vibration energy harvester to obtain a higher energy harvesting rate and have shown that the local maximum power of the proposed harvester is larger than the global maximum power of the traditional single degree of freedom vibration energy harvester. Also, the parameters of the dual-mass energy harvester have been optimized, which maximize the energy harvesting capability of the dual-mass vibration energy harvester when subjected to harmonic excitation. Tai and Zuo [25] have obtained the condition for optimization for maximum power output from vibration-driven electromagnetic and piezoelectric energy harvester devices. These conditions have been derived by using resistive load, electrical damping ratio, and the normalized excitation frequency. Smilek and Hadas [26] have shown the performance enhancement of electromagnetic energy harvester is possible by maximizing electromechanical coupling by optimization of magnetic circuit design, placement of coil, its shape and size. Kim et al. [27] have proposed an approach for optimizing voltage and power from three different types of vibration-driven electromagnetic vibration energy harvesters having the same volume. Maroofiazar and Farzam [28] have carried out numerical modeling, statistical analysis and experimental investigation of a hybrid energy harvester. For the experimental investigation, the system is prepared by floating a magnet coiled fluid container subjected to vibrations peak to peak values of voltage and electrical power obtained are 740 mW and 58 µW, respectively. Wang et al. [29] have proposed a method of scavenging vibrational energy from ultra-low-frequency vibrations with low excitation levels. In the proposed electromagnetic energy harvester, a rolling magnet system has been used. An analytical solution of this harvester performs has been derived. The realistic ambient vibration accelerations from a bus and human body are applied to excite to this harvester. Researchers have also started to explore self-powered active vibration control technologies. Such dual degree systems can also be used to control the vibrations [21]. In recent years, active vibration control based on boundary control has been greatly developed, these type of harvesters can used in such cases such as Boundary Control of a Rotating and Length-Varying Flexible Robotic Manipulator System [30], Modeling and Observer-Based Vibration Control of a Flexible Spacecraft With External Disturbances and Asymmetric Input–Output Constraint Control of a Flexible Variable-Length Rotary Crane Arm [31].

The contribution of this paper lies in proposing a practical setup for a Two-Degree-of-Freedom (TDOF) Vibration-Based Electromagnetic Energy Harvester (VBEEH) and providing both theoretical and experimental analysis of its performance. While existing literature primarily focuses on theoretical analyses of TDOF systems, with only a few incorporating experimental studies, this paper bridges the gap by presenting tangible results. The paper begins by deriving an expression for the maximum average power generated by a TDOF VBEEH, employing the approach proposed by Tang and Zuo [24]. This theoretical analysis sets the foundation for understanding the power output capabilities of the TDOF system. To validate the theoretical findings, an experimental setup is designed and developed specifically for studying the effect of varying mass ratios and electrical resistive loads on the harvested power of the TDOF VBEEH. This experimental analysis provides practical insights into the performance of the TDOF system and allows for a comprehensive evaluation of its power output under different conditions. By combining theoretical derivation and experimental validation, this paper offers a comprehensive understanding of the TDOF VBEEH and its capabilities. The proposed practical setup enhances the applicability of the research, allowing for practical implementation and validating the theoretical predictions. This holistic approach contributes to the body of knowledge surrounding TDOF VBEEHs and facilitates further advancements in the field.

Analysis of Single Degree of Freedom (SDoF) and Two Degree of Freedom (TDoF) Harvesters

This section analyses the single degree of freedom (SDoF) and two degrees of freedom (TDoF) vibration energy harvesters subjected to base excitation as shown in Fig. 1. The SDoF harvester consists of mass m₂, spring k₂, damper c and energy transducer with load resistance R_h as shown in Fig. 1a. Figure 1b shows the TDoF harvester by connecting additional mass m₁ and spring k₁ between base and harvester setup.

Fig. 1

Schematic of harvesters a SDoF, b TDoF

The governing equation for SDoF harvester under base excitation ${x}_{0}=X\mathrm{sin}\omega t$, where, $\omega$ and ${X}_{0}$ are, respectively, the circular excitation frequency and amplitude of ambient vibration is given as.

$$\begin{aligned} m\ddot{z}+c\dot{z}+kz+{f}_{e}&=m\ddot{{x}_{0}},\\{f}_{e}&=\frac{{k}_{t}{k}_{e}\dot{z}}{{R}_{h}},\end{aligned}$$

(1)

where Z is the relative motion (x₂–x₀), f_e is the force provided by the energy transducer [24], k_t is thrust motor constant, k_e is back EMF, R_h is load resistance. The damping coefficient consists of both mechanical and electrical component c = c_m + c_e.

The steady-state amplitude Z and using ${\zeta }_{m}=\frac{c}{2\sqrt{mk}} ,{\zeta }_{e}=\frac{{c}_{e}}{2\sqrt{mk}}$, ${\omega }_{n}=\sqrt{\frac{k}{m}}$ will be

$$\left|Z\right|=\frac{{\left(\frac{\omega }{{\omega }_{n}}\right)}^{2}X}{\sqrt{{\left(1-{\left(\frac{\omega }{{\omega }_{n}}\right)}^{2}\right)}^{2}+{\left(2\left({\zeta }_{e}+{\zeta }_{m}\right)\left(\frac{\omega }{{\omega }_{n}}\right)\right)}^{2}}}.$$

(2)

The average power output from a single degree of freedom (SDOF) VBEEH is given as,

$${P}_{\mathrm{ave}}= \frac{1}{2}{c}_{e}{\dot{z}}^{2},$$

(3)

$${P}_{\mathrm{ave }= }\frac{{m}_{2}{\omega }^{3}{X}^{2}{\zeta }_{e}{\left(\frac{\omega }{{\omega }_{2}}\right)}^{3}}{{\left(1-{\left(\frac{\omega }{{\omega }_{2}}\right)}^{2}\right)}^{2}+{\left(2\left({\zeta }_{e}+{\zeta }_{m}\right)\frac{\omega }{{\omega }_{2}}\right)}^{2}}.$$

(4)

The non-dimensional form of ${P}_{\mathrm{ave}}$ is denoted as ${P}_{\mathrm{aven}}$ and is given as

$${P}_{\mathrm{aven }= }\frac{{P}_{\mathrm{ave}}}{{m}_{2}{\omega }^{3}{{\mathrm{X}}_{0}}^{2}}=\frac{{\zeta }_{e}{\left(\alpha \right)}^{3}}{{\left(1-{\left(\alpha \right)}^{2}\right)}^{2}+{\left(2\left({\zeta }_{e}+{\zeta }_{m}\right)\alpha \right)}^{2}}.$$

(5)

The equations of motion for TDoF system is given as,

$$\begin{aligned}{m}_{2 }\ddot{{x}_{2}} &=-{k}_{2}\left({x}_{2}-{x}_{1}\right)- c \left(\dot{{x}_{2}}-\dot{{x}_{1}}\right)+{f}_{e}\\ {m}_{1 }\ddot{{x}_{1}}+{f}_{e}&= {- k}_{2}\left({x}_{1}-{x}_{2}\right)- c\left(\dot{{x}_{1}}-\dot{{x}_{2}}\right)-{ k}_{1}\left({x}_{1}-{x}_{0}\right).\end{aligned}$$

(6)

The relative displacement z between mass ${m}_{2}$ and mass ${m}_{1}$ is $z={x}_{2}-{x}_{1}$

$$\begin{aligned}&{m}_{2 }\ddot{z}+{k}_{2}z+ c \dot{z}={- m}_{2 }\ddot{{x}_{1}}\\ &\quad {m}_{1 }\ddot{{x}_{1}}+{k}_{1}{x}_{1}{- k}_{2} z-c\dot{z}-{ k}_{1}{x}_{0}=0.\end{aligned}$$

(7)

The non-dimensional output power $\overline{{P }_{\mathrm{aven}}}$ of 2DOF VBEEH as

$$\overline{{P }_{\mathrm{aven}}}=\frac{{ \zeta }_{e }\mu f {\alpha }^{6}}{{\left[{{\alpha }^{4}+f}^{2}- \left(1+\left(1+\mu \right){f}^{2}\right){\alpha }^{2}\right]}^{2}+4{\left({ \zeta }_{e }+{ \zeta }_{m }\right)}^{2}{\left[f\alpha -\left(1+\mu \right)f{\alpha }^{3}\right]}^{2}},$$

(8)

where mass ratio µ $=\frac{{m}_{2}}{{m}_{1}} ,$ Tuning ratio f = $\frac{{\omega }_{2}}{{\omega }_{1}}$, and frequency ratio $\alpha$ =$\frac{\omega }{{\omega }_{1}}$. The detailed derivations are given in Appendix-1.

Figure 2a shows the variation of non-dimensional power Paven with excitation frequency α for various mass ratios µ of TDoF and SDoF. The tuning ratio f is kept at 1. The TDoF harvester can harvest power at two different peaks as compared to SDoF harvester indicating possibility of enhanced frequency bandwidth. The non-dimensional peak power of SDoF harvester is 2.4 it is further enhanced when a TDoF harvester is considered. Figure 2c shows the variation in non-dimensional peak power SDoF and TDoF at various mass ratios. An increase of 50–100% increase in peak power can be achieved when mass ratio µ is varied between 0.05 and 0.4. The frequency band width is measured at 20% of peak power of SDoF (P_max = 0.48) marked as BW in Fig. 2a. The variation of bandwidth is shown in Fig. 2b SDoF has a bandwidth of 0.43 and TDoF with mass ratio µ = 0.05 has a bandwidth of 0.38 indicating a small decrease in bandwidth. The bandwidth of TDoF increases with increase in mass ratio and attains a high value of 0.61 at µ = 0.2. The bandwidth is split in to two bands when µ is 0.3 and 0.4 around two different modes. It is also observed that with increase in mass ratio the distance between two peaks (P₁ and P₂) increases reducing the effective bandwidth.

Fig. 2

Comparison of SDoF and TDoF with different mass ratios a Power curve, b Frequency bandwidth, c Maximum power harvested (BW-frequency bandwidth)

To study the effect of mass ratio and tuning ratio on frequencies, the electrical sub-system from TDoF is removed and assume that the mechanical damping and electrical damping as zero i.e ${\zeta }_{m }= {\zeta }_{e }=0$. The expression for non-dimensional first mode frequency $\overline{{{\omega }_{n}}_{1}}$ and non-dimensional second mode frequency $\overline{{{\omega }_{n}}_{2}}$ have been obtained using well founded theory of vibration. These expressions are as follows; we get, non-dimensional first mode frequency $\overline{{{\omega }_{n}}_{1}}$, as

$$\overline{{{\omega }_{n}}_{1}}=\sqrt{\frac{\left[1+\left(1+\mu \right){f}^{2}\right]}{2}-\sqrt{{\left[\frac{1+\left(1+\mu \right){f}^{2}}{2}\right]}^{2}-{f}^{2 }}}.$$

(9)

Non-dimensional second mode frequency $\overline{{{\omega }_{n}}_{2}}$, as

$$\overline{{{\omega }_{n}}_{2}}=\sqrt{\frac{\left[1+\left(1+\mu \right){f}^{2}\right]}{2}+\sqrt{{\left[\frac{1+\left(1+\mu \right){f}^{2}}{2}\right]}^{2}-{f}^{2}}} .$$

(10)

The detailed expressions are given in Appendix II.

The variation of natural frequencies $\overline{{{\omega }_{n}}_{1}}$ and $\overline{{{\omega }_{n}}_{2}}$ vs. $\mu$ for cases f = 1.0, and 0, 8 are shown Fig. 3a and b. The distance between $\overline{{{\omega }_{n}}_{1}}$ and $\overline{{{\omega }_{n}}_{2}}$ increase with increase in mass ratio $\mu$ indicating increased distance between two peak powers P₁ and P₂ and decrease in bandwidth as shown in Fig. 2a. The variation in peak powers P₁ and P₂ with mass ratio $\mu$ is shown in Fig. 3c. The peak P₁ keeps on increasing and P₂ keeps on decreasing with increase in mass ratio $\mu$. Hence, with increase in µ there is decrease in peak P₂ and it occurs away from P₁, it leads to the reduction in bandwidth. Therefore, peaks should neither be too close or too far to obtain better band width. The optimum mass ratio seems to be 02–0.4 for better power output and bandwidth.

Fig. 3

Effect of µ on $\overline{{{\omega }_{n}}_{1}}$ and $\overline{{{\omega }_{n}}_{2}}$ a f = 1.0, b f = 0.8, c effect on P₁ and P₂ f = 1

The variation of $\overline{{{\omega }_{n}}_{1}}$ and $\overline{{{\omega }_{n}}_{2}}$ with tuning ratio f, for µ = 0.3 is shown in Fig. 4a. From Fig. 4a, b, it is seen that as the values of tuning ratio f increases, the first mode natural frequency $\overline{{{\omega }_{n}}_{1}}$ decreases initially and increases thereafter, reverse trend is followed by the second mode natural frequency $\overline{{{\omega }_{n}}_{2}}$. The values of P₁ and P₂ are equal when $f=\sqrt{\frac{1}{1+\mu }}$ [24], for $f<\sqrt{\frac{1}{1+\mu }}$ P₂ is greater than P₁ reverse happens when $f>\sqrt{\frac{1}{1+\mu }}$. To get maximum power output and better bandwidth, the tuning ratio f = 1.

Fig. 4

a Effect of f on $\overline{{{\omega }_{n}}_{1}}$ and $\overline{{{\omega }_{n}}_{2}}$ µ = 0.3, b effect on P₁ and P₂ µ = 0.3

The power output of 2DOF VBEEH is influenced by four major parameters, viz., the electrical damping ratio ξ_e, mass ratio µ, the tuning ratio f, and the excitation frequency ratio α. The power output of 2DOF VBEEH obtained by applying a single variable optimization technique will be an optimal value of P_ave with respect to that variable only and may not be of the same value for the optimal value of P_ave obtained for other variables such as mass ratio µ. As such, the problem here is to obtain global optimal value of output power P_ave of TDOF VBEEH when ξ_e and α are varied simultaneously for a set of values of tuning ratio f = 0.8, f = 1.0 and f = 1.2, and for a given value of µ or a set of values of mass ratio, µ = 0.1, and µ = 0.3 for a given value of tuning ratio f. To achieve this goal, the method of drawing surface plots and contour diagrams has been used to obtain global optimal value of power output of a TDOF VBEEH device. The details of drawing these plots are presented in the sections to follow. From these plots, the global optimal non-dimensional value of power output P_aven under a given set of conditions can be obtained.

It can be seen from Eq. 11, that the value of $\overline{{P }_{\mathrm{aven}}}$ is controlled by mass ratio μ, tuning ratio f, frequency ratio α, and electrical damping ratio ξ_e.

Using the theoretical Eq. 8 for non-dimensional average power output $\overline{{P }_{\mathrm{aven}}}$ from a 2DOF VBEEH and associated variables which control the value of $\overline{{P }_{\mathrm{aven}}}$, MATLAB has been used to draw surface plots and contour diagram.

• (i) The resulting surface plots showing the non-dimensional average power output $\overline{{P }_{\mathrm{aven}}}$ with respect to normalized excitation frequencyα, at various values of electrical dumping ratio ξ_e for given value of mass ratio µ are shown in Fig. 5 for tuning ratio f = 1, Fig. 6 for f = 0.8, and Fig. 7 for f = 1.2.

Fig. 5

The power output P_aven of 2DOF VBEEH at different values of excitation frequencies $\mathrm{\alpha }$ and electrical damping ratios ${\zeta }_{e}$, f = 1 and ${\zeta }_{m}$ = 0.046 when, a, c µ = 0.1, b, d µ = 0.3

Fig. 6

The power output P_aven of 2DOF VBEEH at different values of excitation frequencies $\mathrm{\alpha }$ and electrical damping ratios ${\zeta }_{e}$, f = 0.8 and ${\zeta }_{m}$ = 0.046 when, a, c µ = 0.1, b, d µ = 0.3

Fig. 7

The power output P_aven of 2DOF VBEEH at different values of excitation frequencies $\mathrm{\alpha }$ and electrical damping ratios ${\zeta }_{e}$, f = 1.2 and ${\zeta }_{m}$ = 0.046 when, a, c µ = 0.1, b, d µ = 0.3

Case I: f = 1.0. From Fig. 5, it is seen that $\overline{{P }_{\mathrm{aven}}}$ increases with the increase in value of ξ_e for μ = 0.1. The value of $\overline{{P }_{\mathrm{aven}}}$ is at maximum at first mode resonant frequency α₁ of TDOF VBEEH as shown in Fig. 5a and c. The case μ = 0.3 and ξ_e = 0.1 gives a better compromise between the maximum value of $\overline{{P }_{\mathrm{aven}}}$ and bandwidth as shown in Fig. 5b and d.

Case II: f = 0.8. From Fig. 6, it can be seen that for small values of μ and ξ_e the maximum value $\overline{{P }_{\mathrm{aven}}}$ occurs almost at the second normalized resonant frequency α₂ of TDOF VBEEH, however, the bandwidth is less than that for the case f = 1.0.

Case III: f = 1.2. From Fig. 7, it can be seen that for the values of μ = 0.1, and 0.3, for small values of ξ_e, almost all the non-dimensional power output $\overline{{P }_{\mathrm{aven}}}$ of the TDOF VBEEH occurs at the first mode resonant normalize excitation frequency α₁. The value of $\overline{{P }_{\mathrm{aven}}}$, in this case is the highest with extremely narrow bandwidth i.e. in the neighborhood of first mode natural frequency α₁.

Experimental Analysis

In this section, experimental analysis of SDoF and YdoF vibration energy is presented.

The Experimental setup has been designed in such a way that the SDOF VBEEH system can be converted to a 2DOF VBEEH with a little bit of a change in the setup.

The experimental setup designed and developed for 2 DOF VBEEH is shown in Fig. 8. The system (${m}_{2 ,}{k}_{2},{c}_{m},{c}_{e}$) is the energy harvester. The helical compression spring of stiffness ${k}_{2}$ is split into two springs of equal stiffness $\frac{{k}_{2}}{2}$ and these springs are placed in parallel to support the harvester mass ${m}_{2}$. The system (${m}_{1 ,}{k}_{1}$) is the amplifier system. The mass ${m}_{2}$ is supported by helical compression springs each of stiffness $\frac{{k}_{1}}{2}$ and are placed in parallel.

Fig. 8

The schematic and experimental setup of two DOF VBEEH

The rectangular plates of appropriate size are selected to represent harvester mass${m}_{2}$, amplifier mass ${m}_{1 ,}$ and the base plate. The base plate supports the amplifier system. All the plates are equipped with rolling contact bearings to ensure frictionless movement of these plates in the vertical direction in the two vertical parallel guide bars. The cam is driven by an electrical drive motor and is mounted at the end of the electrical drive motor shaft. The guide bars are mounted firmly in the foundation.

Electrical Copper Coil Magnet Sub-system

The coil-magnet system comprises of a ‘Magnet-in-line’ type configuration in which a cylindrical magnet moves in the vertical direction in an electrical coil of copper material. Figure 14 (appendix III) shows the magnet to be fixed to the harvester mass plate and the copper coil to be fixed to the amplifier mass plate with proper alignment in vertical direction. The relative movement between magnet and coil provides a magnetic field which when cut gives rise to an electromotive force.

Selection of Magnet and Coil Design

The method of selecting the magnet and design calculation for coil for magnet-coil sub-system are given in Appendix III.

Design Calculations

• (i) To increase the power output and to extend the effective operational frequency range of 2 DOF VBEEH device, and to keep its weight low, a reasonable choice for the mass ratio µ is in the range 0.3–0.4, hence for the analysis, the reasonable values of mass ratio µ are taken as 0.3 and 0.36.

• (ii) The VBEEH is designed essentially for low-frequency excitation. Hence, the excitation frequency ${f}_{w}$ is taken as 5.5 Hz, therefore, the circular excitation frequency is $\omega =2\pi {f}_{w}=34.54 \mathrm{rad}/\mathrm{s}$, for tuning ratio f = 1, hence, ${\omega }_{1}=\sqrt{\frac{{k}_{1}}{{m}_{1}}}={\omega }_{2}=\sqrt{\frac{{k}_{2}}{{m}_{2}}}=\omega =34.54 \mathrm{rad}/\mathrm{s}$

Sample Calculations for the Case µ = 0.36

With the harvester mass ${m}_{2}=2.4 \mathrm{kg}$, the amplifier mass ${m}_{1}$ is ${m}_{1}=\frac{{m}_{2}}{\mu }=\frac{2.4}{0.36}=6.67 \mathrm{kg}$. Using equations ${\omega }_{1}=\sqrt{\frac{{k}_{1}}{{m}_{1}}}$ and ${\omega }_{2}=\sqrt{\frac{{k}_{2}}{{m}_{2}}}$ and ${\omega }_{2}={\omega }_{1}$ for f = 1, the values of amplifier spring stiffness ${k}_{1}$ and harvester system spring stiffness ${k}_{1}$ and ${k}_{2}$ are calculated as ${k}_{1}=7957.38 \mathrm{N}/\mathrm{m}$ and ${k}_{2}=2863.22\mathrm{ N}/\mathrm{m}$. For harvester system, two helical compression springs of stiffness $\frac{{k}_{2}}{2}$ are used in parallel. Therefore each of these two springs is designed for stiffness $\frac{{\mathrm{k}}_{2}}{2}=1431.6\mathrm{ N}/\mathrm{m}$, using standard spring design procedure. Amplifier mass is supported by two springs of stiffness $\frac{{k}_{1}}{2}$ in parallel, hence each of these springs is designed for stiffness $\frac{{k}_{1}}{2}=3978.7 \mathrm{N}/\mathrm{m}$ using standard spring design procedure. Design specifications of these springs are given in Table 1

Table 1 Specifications of the amplifier and harvester springs

• (iii) Rectangular M.S. plate of size 325 × 75 × 15 mm is selected as the harvester mass ${m}_{2}$. Refer Fig. 9a and b. The amplifier mass ${m}_{1}$ is made up of 3 rectangular plates of sizes 325 × 75 × 12.

• (iv) A suitable M.S. plate is selected as a base plate to support amplifier spring-mass system. The base plate receives sinusoidal excitation by a cam driven by variable speed electric motor.

Fig. 9

The developed harvester and amplifier mass

Experimental Analysis

The entire experimental setup with attendant instrumentation for SdoF and TdoF VBEEH is shown in Fig. 10.

Fig. 10

Experimental setup and instrumentation for (a) SDOF (b) TdoF VBEEH

Experimental Analysis of SDOF VBEEH System (${{\varvec{m}}}_{2 ,}{{\varvec{k}}}_{2},{{\varvec{c}}}_{{\varvec{m}}},{{\varvec{c}}}_{{\varvec{e}}}$)

By removing amplifier (${m}_{1 ,}{k}_{1}$) system from the TDOF VBEEH set-up the SDOF (${m}_{2 ,}{k}_{2},{c}_{m},{c}_{e}$) VBEEH system is obtained. This SDOF VBEEH system is subjected to base harmonic excitation ${x}_{0}={X}_{0} \mathrm{sin}\omega t.$ Base excitation amplitude ${x}_{0}$ is provided by cam eccentricity and excitation frequency is varied by the change in the speed of the electrical drive motor.

The open circuit voltage V is measured across the copper coil terminals of SDOF VBEEH, at various excitation frequencies ω. The plot of V vs. ω is shown in Fig. 11a. The resistive electrical load R_L is connected across the copper coil terminals, and the voltage V_R across the load R_L is measured at different values of excitation frequency ω. R_L is varied from R_L = 1000 Ω to R_L = 2000 Ω using a resistive load bank developed for the same.

Fig. 11

a Voltage V_R vs. excitation frequency ω, b Harvested Power P_aveh vs. excitation frequency ω, for various values of R_L

The power P_aveh that can be harvested from the SDOF VBEEH is estimated using the formula ${P}_{\mathrm{aveh}}= {V}_{R} \times {i}_{R}$ where ${i}_{R}$ is the current in the electrical load circuit and ${i}_{R}=\frac{{V}_{R}}{{R}_{L}}$. Therefore average power harvested is ${P}_{\mathrm{aveh}}=\frac{{{V}_{R}}^{2}}{{R}_{L}}$, neglecting the internal resistance of the electrical copper coil. R_L is varied as R_L = 1000 Ω, R_L = 1500 Ω, and R_L = 2000 Ω. The plots of ${P}_{\mathrm{aveh}}$ vs. ω are obtained as shown in Fig. 11b. V_R, and ${P}_{\mathrm{aveh}}$ are maximum at resonant frequency ω = 33.5 rad/s. These values fall sharply at off-resonance frequencies. At the power magnitude of 10 mw, the frequency band is (32.6–34.6) 2 rad/s. As such, it is necessary to modify the SDOF VBEEH in such a way that increased harvested power can be made available over a wide operational frequency band.

Experimental Analysis of Developed 2DOF VBEEH Device

For this purpose, harvester system (${m}_{2 ,}{k}_{2},{c}_{m},{c}_{e}$) is mounted back in its place on the amplifier system (${m}_{1 ,}{k}_{1}$). Now, this set-up is ready for experimental analysis of the developed 2DOF VBEEH device.

The developed 2 DOF VBEEH device is subjected to the base harmonic excitation ${x}_{0}(t)={X}_{0} \mathrm{sin}\omega t$ by the cam driven by an electric motor. The responses ${x}_{2}(t)={X}_{2} \mathrm{sin}\omega t$ of the harvester mass ${m}_{2}$ and ${x}_{1}(t)={X}_{1} \mathrm{sin}\omega t$ of the amplifier mass ${m}_{1,}$ at different frequencies of excitation, have been recorded using the strip chart recorder.

The open circuit voltage V across the copper coil terminals of TDOF VBEEH is measured at various frequencies of base excitation. The curves of V vs. ω are plotted as shown in Fig. 12.

Fig. 12

Voltage V_R vs. excitation frequency ω for various values of R_L

The electrical load (pure resistive load R_L) is connected to the 2DOF VBEEH harvester and the voltage V_R across the resistance is measured at various values of frequencies of base excitation ω. R_L is varied as R_L = 1000 Ω, R_L = 1500 Ω, and R_L = 2000 Ω. The curves of V_R vs. ω at various values of R_L are plotted as shown in Fig. 12.

The power P_aveh that can be harvested across resistive load R_L is calculated as ${P}_{\mathrm{aveh}}=\frac{{{V}_{R}}^{2}}{{R}_{L}}$, at R_L = 1000 Ω, R_L = 1500 Ω, and R_L = 2000 Ω for various values of excitation frequency ω. The plots of ${P}_{\mathrm{aveh}}$ vs. ω are shown in Fig. 13.

Fig. 13

Harvested power P_aveh vs. excitation frequency ω at various values of R_L a µ = 0.36, b µ = 0.3

From Fig. 13, it is clearly seen that the band width of operating excitation frequency has been increased. The values of relative displacement amplitude Z is maximum at two resonant frequencies indicating that the open circuit voltage V and voltage across the electric resistive load V_R will be available over a wide band of excitation frequency. This result shows that the harvester power ${P}_{\mathrm{aveh}}$ of a 2 DOF VBEEH is increased as compared to that of a SDOF VBEEH device. The power ${P}_{\mathrm{aveh}}$ is available over the wide band of excitation and the peak values of ${P}_{\mathrm{aveh}}$ are at the two resonant frequencies. It should be noted that the maximum values of ${P}_{\mathrm{aveh}}$ is available at a resistive load of 1500 Ω which is approximately equal to internal resistance R_L of electrical copper coil.

From Fig. 13a and b, it is seen that the harvested power of 10 mw for mass ratio 0.36, the first mode band width = B_w1 = (34–23) = 11 rad/s and second mode band width = B_w2 = (44–42) = 3 rad/s. Total band width B_w = B_w1 + B_w2 = 14 rad/s compared to 2 rad/s of SdoF. For mass ratio 0.3, first mode band width = B_w1 = (37–23) = 14 rad/s. Second mode band width = B_w2 = (47–44) = 3 rad/s. Total band width B_w = B_w1 + B_w2 = 17 rad/s. The maximum power harvested is 45 mW, 70 mW and 60 mW, respectively, for SDoF, TDoF µ = 0.36 and 0.3, respectively. It is clearly seen that the TDOF VBEEH bandwidth of operating excitation frequency and magnitude of harvested power has been increased as discussed in “Analysis of Single Degree of Freedom (SDoF) and Two Degree of Freedom (TDoF) Harvesters”.

Conclusions

In conclusion, the analysis and experimental results presented in this paper provide important insights for the design of Two-Degree-of-Freedom Vibration-Based Electromagnetic Energy Harvesters (TDOF VBEEHs). The derived expression for the non-dimensional power output of the TDOF VBEEH highlights the dependence on the selection of mass ratio (µ), tuning ratio (f), electrical damping ratio (ξ_e), and normalized excitation frequency (α). The values of µ and f are also linked to the effective non-dimensional bandwidth (BW). Therefore, careful consideration must be given to the choice of these parameters when designing a TDOF VBEEH, including the harvester mass-spring system and amplifier mass-spring system. The utilization of surface plots and contour diagrams proves to be an invaluable method for determining the global optimal power output of a TDOF VBEEH during the design phase. By simultaneously varying the values of the electrical damping ratio (ξ_e) and normalized excitation frequency (α) for different values of the mass ratio (µ), the variation of non-dimensional power output (P̅_aven) of the TDOF VBEEH can be studied. The results show that increased power output is attainable over a widened operational excitation frequency band compared to the traditional Single-Degree-of-Freedom (SDOF) VBEEH. Comparing the experimental and analytical results, it is evident that the TDOF VBEEH outperforms the SDOF VBEEH when the parameters are properly chosen. The experimental results show a maximum power harvested of 45 mW, 70 mW, and 60 mW for the SDOF VBEEH, TDOF VBEEH with µ = 0.36, and TDOF VBEEH with µ = 0.3, respectively. Additionally, the bandwidth achieved is 2 rad/s, 14 rad/s, and 17 rad/s for the SDOF VBEEH, TDOF VBEEH with µ = 0.36, and TDOF VBEEH with µ = 0.3, respectively. These findings highlight the superior performance and potential of TDOF VBEEHs over SDOF VBEEHs, emphasizing the importance of parameter selection in maximizing power output and widening the operational frequency band. The research results presented in this paper serve as valuable design guidelines for developing TDOF VBEEHs capable of powering small electronic devices when the amplitude and frequency data of the ambient vibration source are known.

Appendices

Appendix 1

$${\mathrm{m}}_{2 }\ddot{{x}_{2}}\uparrow =-{k}_{2}\left({x}_{2}-{x}_{1}\right)\downarrow - c \left(\dot{{x}_{2}}-\dot{{x}_{1}}\right)\downarrow ,$$

(11)

$${m}_{1 }\ddot{{x}_{1}}\uparrow = {- k}_{2}\left({x}_{1}-{x}_{2}\right)\downarrow - c\left(\dot{{x}_{1}}-\dot{{x}_{2}}\right)\downarrow -{ k}_{1}\left({x}_{1}-{x}_{0}\right)\downarrow ,$$

(12)

where $c={c}_{m}+{c}_{e}$, the relative displacement z between mass ${m}_{2}$ and mass ${m}_{1}$ is $z={x}_{2}-{x}_{1},\mathrm{hence}, \dot{z} =\dot{{x}_{2}}-\dot{{x}_{1}}\mathrm{ and }\ddot{z}=\ddot{{x}_{2}}-{\ddot{x}}_{1}$ Substituting these values we get,

$${m}_{2 }\ddot{z}+{k}_{2}z+ c \dot{z}={- m}_{2 }\ddot{{x}_{1}} \mathrm{and},$$

(13)

$${m}_{1 }\ddot{{x}_{1}}+{k}_{1}{x}_{1}{- k}_{2} z-c\dot{z}-{ k}_{1}{x}_{0}=0.$$

(14)

Take complex forms of variables as $z=Z{e}^{j\omega t}$, ${x}_{1}={X}_{1}{e}^{j\omega t},\mathrm{ and }{x}_{0}={X}_{0}{e}^{j\omega t}$

then $\dot{z}=j\omega Z{e}^{j\omega t}$ and $\ddot{z}=-{\omega }^{2}Z{e}^{j\omega t}$, $\dot{{x}_{1}}=j\omega {X}_{1}{e}^{j\omega t}$ and $\ddot{{x}_{1}}=-{\upomega }^{2}{\mathrm{X}}_{1}{\mathrm{e}}^{\mathrm{j\omega t}}$.

Substituting these values in Eq. (13) we get Eq. (13) in the form

$${m}_{2 }{\upomega }^{2}Z+{k}_{2}Z+ c\left(j\mathrm{\omega Z}\right)={+ m}_{2 }{\upomega }^{2}{X}_{1} \mathrm{or }$$

$\left[\left({k}_{2}-{m}_{2 }{\upomega }^{2}\right)+jc\omega \right]Z={m}_{2 }{\upomega }^{2}{X}_{1}$ then

$${X}_{1}= \frac{\left[\left({k}_{2}-{m}_{2 }{\upomega }^{2}\right)+jc\omega \right]Z}{{m}_{2 }{\upomega }^{2}}.$$

(15)

Similarly, we get the Eq. (14) in the form

$$\begin{aligned}&{-m}_{1 }{\upomega }^{2}{X}_{1}+{k}_{1}{X}_{1}- {k}_{2}Z-cj\mathrm{\omega Z}-{k}_{1}{X}_{0}=0\\ & \quad \left({k}_{1}{-m}_{1 }{\upomega }^{2}\right){X}_{1}-\left({k}_{2}+cj\upomega \right)\mathrm{Z}={k}_{1}{X}_{0}.\end{aligned}$$

(16)

Substituting value of ${X}_{1}$, from Eq. (15) in Eq. (16) we get,

$\left\{\left({k}_{1}{-m}_{1 }{\upomega }^{2}\right)\left[\frac{\left({k}_{2}-{m}_{2 }{\upomega }^{2}\right)+jc\omega }{{m}_{2 }{\upomega }^{2}}\right]-\left({k}_{2}+cj\upomega \right)\right\}Z={k}_{1}{X}_{0},$ or

$$\left[\left({k}_{1}{-m}_{1 }{\upomega }^{2}\right)\left({k}_{2}-{m}_{2 }{\upomega }^{2}\right)+jc\omega \left({k}_{1}{-m}_{1 }{\upomega }^{2}\right)-\left({k}_{2}+cj\upomega \right){m}_{2 }{\upomega }^{2}\right]Z={m}_{2 }{\upomega }^{2}{k}_{1}{X}_{0},$$

(17)

$$\left[{k}_{1}{{k}_{2}-{k}_{1} m}_{2}{{\omega }^{2}-{k}_{2}m}_{1}{\omega }^{2}{+ m}_{1}{m}_{2 }{\omega }^{4}+jc\omega {k}_{1}-{jc\omega m}_{1}{\upomega }^{2}-{k}_{2}{m}_{2 }{\upomega }^{2}-cj{\omega }^{3}{m}_{2 }\right]Z={k}_{1}{{m}_{2 }{\omega }^{2}X}_{0}$$

$$\left\{\left[{k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+{k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right]+jc\left[\omega {k}_{1}-\left({m}_{1}+{m}_{2 }\right){\omega }^{3}\right]\right\}Z={k}_{1}{{m}_{2 }{\omega }^{2}X}_{0}.$$

(18)

Putting

$${a}_{1}=\left[{k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+{k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right]$$

${a}_{2}=c\left[\omega {k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{3}\right]={(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})c\omega ,$and

${{a}_{3}={k}_{1}m}_{2}{\omega }^{2},$

we can rewrite Eq. (18) in the form as,

$$Z=\left(\frac{{a}_{3}}{{a}_{1}+j{a}_{2}}\right) {X}_{0} \mathrm{or} { Z}^{2}=\left(\frac{{{a}_{3}}^{2}}{{{a}_{1}}^{2}+{{a}_{2}}^{2}}\right){{X}_{0}}^{2}.$$

(19)

Now, the average harvested power output P of 2DOF VBEEH is defined as

$$P =\mathrm{ Power output }={\int }_{0}^{\mathrm{v}}f\mathrm{ dv},$$

where f is electromagnetic force induced in copper coil and magnet due to their relative velocity. We have $\dot{z}=\dot{{(x}_{2}}-\dot{{x}_{1}})$ hence $f={c}_{e }\dot{z}={c}_{e }\dot{{(x}_{2}}-\dot{{x}_{1}})$

using, this relation, the equation of the average power output ${P}_{\mathrm{ave}}$ is obtained as

$${P}_{\mathrm{ave}} ={\int }_{0}^{z}\dot{z}\dot{\mathrm{d}z=\frac{1}{2}}{c}_{e }\lfloor {\dot{z}}^{2}\rfloor {\mathrm{ or }P}_{\mathrm{ave}}=\frac{1}{2}{c}_{e } {\dot{{(x}_{2}}-\dot{{x}_{1}})}^{2}.$$

(20)

Maximum Average Power output ${P}_{\mathrm{ave}}$ is given as,

$${P}_{\mathrm{ave}}=\frac{1}{2}{c}_{e }{({\omega X}_{2}-\omega {X}_{1})}^{2} \mathrm{or} {P}_{\mathrm{ave}}=\frac{1}{2}{c}_{e }{\omega }^{2}{({X}_{2}-{X}_{1})}^{2},$$

(21)

where ${X}_{2}$ and ${X}_{1}$ are amplitudes of ${x}_{2}(t)$ and ${x}_{1}(t)$ respectively.

Substituting $Z=\left({X}_{2}-{X}_{1}\right)$ in Eq. (19) and using the values of ${a}_{1}$,${a}_{2}$,$\mathrm{and} {a}_{3}$ just defined.

We can write,

$${\mathrm{Z}}^{2}={\left({\mathrm{X}}_{2}-{\mathrm{X}}_{1}\right)}^{2}=\left[\frac{\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right){{\mathrm{X}}_{0}}^{2}}{{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}^{2}+{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}{c}^{2}{\omega }^{2}}\right].$$

(22)

Substituting Eq. (22) in Eq. (21), we get maximum average power output ${\mathrm{P}}_{\mathrm{ave}}$ as

$${\mathrm{P}}_{\mathrm{ave}}=\left[\frac{\frac{1}{2}{\mathrm{c}}_{\mathrm{e }}{\omega }^{2}\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right){{\mathrm{X}}_{0}}^{2}}{{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}^{2}+{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}{c}^{2}{\omega }^{2}}\right].$$

(23)

We have $c={c}_{m}+{c}_{e}$, where ${c}_{m}={ 2\upzeta }_{\mathrm{m }}{m}_{2 }{\upomega }_{2}$ and ${c}_{e}={ 2\upzeta }_{\mathrm{e }}{m}_{2 }{\upomega }_{2}\mathrm{ hence}$

$c=2\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right){m}_{2 }{\upomega }_{2}$ and ${c}^{2}=4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}$

Using the value of ${c}^{2}$ and ${c}_{e}$ just defined, in Eq. (23), we can write,

$${\mathrm{P}}_{\mathrm{ave}}=\left[\frac{\frac{1}{2}{\times 2\upzeta }_{\mathrm{e }}{m}_{2 }{\upomega }_{2}{\omega }^{2}\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right){{\mathrm{X}}_{0}}^{2}}{{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}^{2}+{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}\right].$$

(24)

The non-dimensional average power output $\overline{{\mathrm{P}}_{\mathrm{aven}}}$ can be written as,

$$\overline{{\mathrm{P}}_{\mathrm{aven}}}=\frac{{\mathrm{P}}_{\mathrm{ave}}}{{{X}_{0}}^{2}{{{\omega }_{1}}^{3}m}_{1 }},$$

(25)

where the term ${{X}_{0}}^{2}{{{\omega }_{1}}^{3}m}_{1}$ has the dimensions of the power.

Substituting Eq. (13) in Eq. (15), we get,

$$\overline{{\mathrm{P}}_{\mathrm{aven}}}=\left[\frac{\frac{{{\upzeta }_{\mathrm{e }}m}_{2 }{\upomega }_{2}{\omega }^{2}{{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}}{{{{\omega }_{1}}^{3}m}_{1 }}}{{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}^{2}+{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}\right].$$

(26)

Consider the Numerator N of Eq. (26)

$$\begin{aligned}N&=\frac{{{\upzeta }_{\mathrm{e }}m}_{2 }{\upomega }_{2}{\omega }^{2}{{{{{k}_{1 }}^{2}m}_{2}}^{2}\omega }^{4}}{{{{\omega }_{1}}^{3}m}_{1}}\\ N& =\frac{{{\upzeta }_{\mathrm{e }}m}_{2 }}{{m}_{1 }}\times \left(\frac{{\upomega }_{2}}{{\omega }_{1}}\right)\left(\frac{{\upomega }^{2}}{{{\upomega }_{1}}^{2}}\right)\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right)\\ N&={\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{2}\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right)\\ N&={\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{2}\left({{k}_{1 }}^{2}{{{m}_{2}}^{2}\omega }^{4}\right)\left(\frac{{{\upomega }_{1}}^{4}}{{{\omega }_{1}}^{4}}\right)\\ N&={\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{2}\left({{k}_{1 }}^{2}{{m}_{2}}^{2}\right){\mathrm{\alpha }}^{4}{{\upomega }_{1}}^{4}\\ N&={\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}{{k}_{1 }}^{2}{{m}_{2}}^{2}{{\upomega }_{1}}^{4}\end{aligned}$$

where $\upmu = \frac{{m}_{2}}{{m}_{1}}$ is the mass ratio,$\mathrm{f}= \frac{{\omega }_{2}}{{\omega }_{1}}$ is the tuning ratio and $\mathrm{\alpha }= \frac{\omega }{{\omega }_{1}}$ is the frequency ratio. Using the above value of numerator N of the Eq. (26) we get,

$$\begin{aligned}\overline{{\mathrm{P} }_{\mathrm{aven}}}& =\left[\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}{{k}_{1 }}^{2}{{m}_{2}}^{2}{{\upomega }_{1}}^{4}}{{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}^{2}+{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}\right]\\\overline{{\mathrm{P} }_{\mathrm{aven}}}&=\left[\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{\frac{\left({k}_{1}{{k}_{2}-{(k}_{1} m}_{2}{+ {k}_{2}m}_{1}+{{k}_{2}{m}_{2}){\omega }^{2}+ m}_{1}{m}_{2 }{\omega }^{4}\right)}{{{k}_{1 }}^{2}{{m}_{2}}^{2}{{\upomega }_{1}}^{4}}}^{2}+\frac{{\left[{(k}_{1}-({m}_{1}+{m}_{2 }){\omega }^{2})\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}{{{k}_{1 }}^{2}{{m}_{2}}^{2}{{\upomega }_{1}}^{4}}}\right]\\ & =\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{\left[\frac{{k}_{1}{k}_{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}- \left(\frac{{k}_{1 }{m}_{2}{+ {k}_{2}m}_{1}+{k}_{2}{m}_{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right){\omega }^{2}+\left(\frac{{m}_{1}{m}_{2}{\omega }^{4}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right)\right]}^{2}+{\left[\frac{{k}_{1}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\frac{\left({m}_{1}+{m}_{2 }\right){\omega }^{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}\\ &=\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{D}_{1}+{\mathrm{D}}_{2}}.\end{aligned}$$

(27)

Putting the values of D₁ and D_2, as derived in Appendix I, 24, we get.

The non-dimensional output power $\overline{{\mathrm{P} }_{\mathrm{aven}}}$ of 2DOF VBEEH as

$$\overline{{\mathrm{P} }_{\mathrm{aven}}}=\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{\left[{{\mathrm{\alpha }}^{4}+\mathrm{f}}^{2}- \left(1+\left(1+\upmu \right){\mathrm{f}}^{2}\right){\mathrm{\alpha }}^{2}\right]}^{2}+4{\left({\upzeta }_{\mathrm{e }}+{\upzeta }_{\mathrm{m }}\right)}^{2}{\left[\mathrm{f\alpha }-\left(1+\upmu \right)\mathrm{f}{\mathrm{\alpha }}^{3}\right]}^{2}}.$$

(28)

Expression for D₁ and D_2, we have,

$$\begin{aligned}\overline{{\mathrm{P} }_{\mathrm{aven}}}&=\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{\left[\frac{{k}_{1}{k}_{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}- \left(\frac{{k}_{1 }{m}_{2}{+ {k}_{2}m}_{1}+{k}_{2}{m}_{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right){\omega }^{2}+\left(\frac{{m}_{1}{m}_{2}{\omega }^{4}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right)\right]}^{2}+{\left[\frac{{k}_{1}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\frac{\left({m}_{1}+{m}_{2 }\right){\omega }^{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}}\\ \overline{{\mathrm{P} }_{\mathrm{aven}}}&=\frac{{\upzeta }_{\mathrm{e }}\mathrm{\mu f }{\mathrm{\alpha }}^{6}}{{D}_{1}+{\mathrm{D}}_{2}}\end{aligned}$$

then

$$\begin{aligned}{D}_{1}&={\left[\frac{{k}_{1}{k}_{2}}{{{\omega }_{1}}^{2}{k}_{1 }{m}_{2}}- \left(\frac{{k}_{1 }{m}_{2}{+ {k}_{2}m}_{1}+{k}_{2}{m}_{2}}{{{\omega }_{1}}^{2}{k}_{1 }{m}_{2}}\right){\omega }^{2}+\left(\frac{{m}_{1}{m}_{2}{\omega }^{4}}{{{\omega }_{1}}^{2}{k}_{1 }{m}_{2}}\right)\right]}^{2}\\ & {=\left[{\left(\frac{{\omega }_{2}}{{\omega }_{1}}\right)}^{2}- \left(\frac{1}{{{\omega }_{1}}^{2}}+\frac{\upmu {\mathrm{f}}^{2}}{\upmu {{\omega }_{1}}^{2}}+\frac{\upmu {\mathrm{f}}^{2}}{{{\omega }_{1}}^{2}}\right){\omega }^{2}+\frac{1}{{{\omega }_{1}}^{2}}\frac{1}{{{\omega }_{1}}^{2}}{\left(\omega \right)}^{4}\right]}^{2}\\ &{=\left[{\mathrm{f}}^{2}- \left(1+{\mathrm{f}}^{2}+\upmu {\mathrm{f}}^{2}\right)\frac{{\omega }^{2}}{{{\omega }_{1}}^{2}}+ \frac{{\omega }^{4}}{{{\omega }_{1}}^{4}}\right]}^{2}\\ &{=\left[{\mathrm{f}}^{2}- \left(1+{\mathrm{f}}^{2}+\upmu {\mathrm{f}}^{2}\right){\mathrm{\alpha }}^{2}+{\mathrm{\alpha }}^{4}\right]}^{2},\end{aligned}$$

(29)

$$\begin{aligned} {D}_{2}&={\left[\frac{{k}_{1}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\frac{\left({m}_{1}+{m}_{2 }\right){\omega }^{2}}{{k}_{1 }{m}_{2{{\omega }_{1}}^{2}}}\right]}^{2}4{\left({\upzeta }_{\mathrm{m }}+{\upzeta }_{\mathrm{e }}\right)}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}\\&{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\frac{{k}_{1}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\frac{\left({m}_{1}+{m}_{2 }\right){\omega }^{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right]}^{2}{{m}_{2 }}^{2}{{\upomega }_{2}}^{2}{\omega }^{2}\\ &{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\frac{{k}_{1}{m}_{2}{\omega }_{2}\omega }{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\frac{\left({m}_{1}{m}_{2}{\omega }_{2}\omega +{m}_{2 }{m}_{2}{\omega }_{2}\omega \right){\omega }^{2}}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right]}^{2}\\&{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\frac{{k}_{1}{m}_{2}{\omega }_{2}\omega }{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}-\left[\frac{{m}_{1}{m}_{2}{\omega }_{2}\omega }{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}+\frac{\left({m}_{2 }{m}_{2}{\omega }_{2}\omega \right)}{{k}_{1 }{m}_{2}{{\omega }_{1}}^{2}}\right]{\omega }^{2}\right]}^{2}\\ & {=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\frac{{\omega }_{2}}{{\omega }_{1}}\frac{\omega }{{\omega }_{1}}-\left[\frac{1}{{{\omega }_{1}}^{2}}\frac{{\omega }_{2}{\omega }^{3}}{{{\omega }_{1}}^{2}}+\frac{{m}_{2 }}{{k}_{2 }}\frac{{k}_{2 }}{{k}_{1 }}\frac{\left({\omega }_{2}{\omega }^{3}\right)}{{{\omega }_{1}}^{2}}\right]\right]}^{2}\\ &{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\mathrm{f\alpha }-\left[\frac{{\omega }_{2}}{{\omega }_{1}}{\left(\frac{\omega }{{\omega }_{1}}\right)}^{3}+\frac{1}{{{\omega }_{2}}^{2}}\upmu {\mathrm{f}}^{2}\frac{\left({\omega }_{2}{\omega }^{3}\right)}{{{\omega }_{1}}^{2}}\right]\right] }^{2}\\ &{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\mathrm{f\alpha }-\left[\mathrm{f}{\mathrm{\alpha }}^{3}+\upmu {\mathrm{f}}^{2}\frac{\omega }{{\omega }_{2}}{\left(\frac{\omega }{{\omega }_{1}}\right)}^{2}\right]\right]}^{2}\\ & {=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\mathrm{f\alpha }-\left[\mathrm{f}{\mathrm{\alpha }}^{3}+\upmu {\mathrm{f}}^{2}\frac{\omega }{{\omega }_{1}}\frac{{\omega }_{1}}{{\omega }_{2}}{\mathrm{\alpha }}^{2}\right]\right]}^{2}\\ & {=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\mathrm{f\alpha }-\left[\mathrm{f}{\mathrm{\alpha }}^{3}+\upmu {\mathrm{f}}^{2}\mathrm{\alpha }\frac{1}{f}{\mathrm{\alpha }}^{2}\right]\right]}^{2}\\ &{=4{\left({\zeta }_{m }+{\upzeta }_{\mathrm{e }}\right)}^{2}\left[\mathrm{f\alpha }-\left[\mathrm{f}{\mathrm{\alpha }}^{3}+\mathrm{\mu f }{\mathrm{\alpha }}^{3}\right]\right]}^{2}\\ & \quad {4{\left({\zeta }_{m }+{\zeta }_{e }\right)}^{2}\left[\mathrm{f\alpha }-\left(1+\upmu \right)\mathrm{f}{\mathrm{\alpha }}^{3}\right]}^{2} .\end{aligned}$$

(30)

Appendix II

Calculation of First and Second Mode Natural Frequencies

If we remove electrical sub-system from 2DOF VBEEH and assume that the mechanical damping and electrical damping as zero i.e. ${\zeta }_{m }= {\zeta }_{e }=0$, an undamped 2 DOF vibrating system results as shown in Fig. 3

Using well founded theory of vibration, we can write the equation of motion of mass ${m}_{2}$ as,

${m}_{2}{\ddot{x}}_{2}=-{k}_{2}\left({x}_{2}-{x}_{1}\right)$ or

$${m}_{2}{\ddot{x}}_{2}+{k}_{2}\left({x}_{2}-{x}_{1}\right)=0.$$

(31)

Equation of motion of mass ${m}_{1}$ as

$$\begin{aligned}{m}_{1}{\ddot{x}}_{1}&=-{k}_{2}(({x}_{1}-{x}_{2})-{k}_{1}({x}_{1}-{x}_{0}) \\ &\quad {m}_{1}{\ddot{x}}_{1}+{k}_{1}({x}_{1}-{x}_{0})+{k}_{2}({x}_{1}-{x}_{2})=0.\end{aligned}$$

For free vibration, set ${x}_{0}=0$, then equation of mass ${m}_{1}$ is

$${m}_{1}{\ddot{x}}_{1}+{k}_{1}{x}_{1}+{k}_{2}({x}_{1}-{x}_{2})=0 .$$

(32)

Setting ${x}_{1}={X}_{1}\mathrm{ sin }{\omega }_{n}t$ and ${x}_{2}={X}_{2}\mathrm{ sin }{\omega }_{n}t$ where, ${X}_{1}$ and ${X}_{2}$ are, respectively, the amplitudes of free vibration of mass ${\mathrm{m}}_{1}$ and ${\mathrm{m}}_{2}$, ${\upomega }_{n}$ is the circular natural frequency. Substitute ${x}_{1}$ (t) and ${x}_{2}$ (t) in Eqs. (31) and (32). Solving the Eqs. (31) and (32) for ${X}_{1}$ and ${X}_{2}$. The frequency equation is obtained as

$${{\omega }_{n}}^{4}-\left[\frac{{k}_{2}}{{m}_{2}}+\frac{{k}_{1 }}{{m}_{1}}+\frac{{k}_{2}}{{m}_{1}}\right]{{\omega }_{n}}^{2}+\frac{{k}_{1 }{k}_{2}}{{m}_{1}{m}_{2}}=0.$$

(33)

Setting, Mass ratio$=\mu =\frac{{m}_{2}}{{m}_{1}}$, ${\omega }_{2}=\sqrt{\frac{{k}_{2}}{{m}_{2}}}$, ${\omega }_{1}=\sqrt{\frac{{k}_{1}}{{m}_{1}}}$, Tuning ratio = $f=\frac{{\omega }_{2}}{{\omega }_{1}}$ and substituting in Eq. (33), and after simplifying, we get

$$\frac{{{\upomega }_{\mathrm{n}}}^{4}}{{{\upomega }_{1}}^{4}}-\left[1+\left(1+\upmu \right){\mathrm{f}}^{2}\right]\frac{{{\upomega }_{\mathrm{n}}}^{2}}{{{\upomega }_{1}}^{2}}+{\mathrm{f}}^{2}=0.$$

(34)

Setting, $\frac{{{\omega }_{n}}^{2}}{{{\omega }_{1}}^{2}}={\overline{{\omega }_{n}}}^{2}$ in non-dimensional form and solving the Eq. (34) for ${\overline{{\omega }_{n}}}^{2}$, we get the dimensionless undamped natural frequency equation as

$${\overline{{\omega }_{n}}}^{2}=\frac{\left[1+\left(1+\mu \right){f}^{2}\right]}{2}\pm \sqrt{{\left[\frac{1+\left(1+\mu \right){f}^{2}}{2}\right]}^{2}-{f}^{2}}.$$

(35)

From the Eq. (35), we get, non-dimensional first mode frequency $\overline{{{\omega }_{n}}_{1}}$, as

$$\overline{{{\omega }_{n}}_{1}}=\sqrt{\frac{\left[1+\left(1+\mu \right){f}^{2}\right]}{2}-\sqrt{{\left[\frac{1+\left(1+\mu \right){f}^{2}}{2}\right]}^{2}-{f}^{2 }}}.$$

(36)

Non-dimensional second mode frequency $\overline{{{\omega }_{n}}_{2}}$, as

$$\overline{{{\omega }_{n}}_{2}}=\sqrt{\frac{\left[1+\left(1+\mu \right){f}^{2}\right]}{2}+\sqrt{{\left[\frac{1+\left(1+\mu \right){f}^{2}}{2}\right]}^{2}-{f}^{2}}} .$$

(37)

It is seen from Eqs. $(36)$ and $(37)$ operational frequencies $\overline{{{\omega }_{n}}_{2}}$–$\overline{{{\omega }_{n}}_{1}}$ depend on the mass ratio µ and tuning ratio f.

Appendix III

Selection of Magnet and Coil Design

• (i) Typically, four types of permanent magnets suitable for the coil-magnet system of a VBEEH are: Alnico, Ceramic, Samarium cobalt, and Neodymium Iron Boron (NdFeB). The properties such as high flux density and high magnetic field strength of NdFeB make it suitable for the coil-magnet system of VBEEH (Figs. 14, 15).

  The dimensions of cylindrical NdFeB magnet selected for magnet-coil system are: dimeter of magnet 10 mm and the height h_mag 30 mm and residual flux density B_r = 1.4 Tesla.

• (ii) The design of the coil is carried out according to the standard formulae [32]. These values are given in Table 2

Fig. 14

Magnet and coil in-line combination

Fig. 15

Developed coil

Table 2 Specifications of the designed coil