**1. Introduction**

With the ongoing development of microelectronic technologies, multiple low-power consuming wireless sensor devices are being embedded within hand-held and wearable consumer electronics. These devices are mainly powered by an external power source (e.g., electrochemical batteries) of the consumer electronics and their continuous use allows it to run out of power quickly. Compared to device technologies, the development of the power sources (i.e., batteries) are still slower, even though the devices require less power to operate. Electrochemical batteries have a limited lifespan, and require periodic charging that is inconvenient or sometimes impossible. Moreover, since most batteries contain toxic chemicals, disposal of the expired batteries produces hazardous waste that enhances environmental pollution and poses threats to human and animal health. Therefore, there is

grea<sup>t</sup> interest in developing self-powered electronics for uninterruptible and long-lasting operation by eliminating the need for recharging or replacing the power source. In recent years, energy harvesting from surrounding energy sources (e.g., light, heat, sound, vibration, etc.) has drawn much attraction to address these circumstances [1–3]. Among these sources, vibration is the most attractive physical energy source due to its versatility, incorruptibility, and abundance in nature [4]. However, di fferent vibration sources (e.g., human and machine motion, water and wind flow, rotary motion etc.) generate vibrations of di fferent frequencies and amplitudes, and mostly exhibit low-frequency, large-amplitude characteristics with various cyclic movements in di fferent directions [5–7]. These vibrations, in the form of kinetic energy, can e ffectively be converted into electrical energy by employing compatible electromechanical transduction mechanisms that include piezoelectric [8], electromagnetic [9], electrostatic [10], magnetostrictive/magnetoelectric [11], and triboelectric [12] mechanisms.

The performance of a vibration energy harvester greatly depends on the characteristics of vibration, the type of transducer, and how the transducer is coupled to the mechanical system. Generally, vibration energy harvesters utilize an inertial mechanism employed by a cantilevered spring-mass system, having a specific resonant frequency. Harvested energy (power) is at its maximum when the harvester's resonant frequency matches the applied vibration frequency. Unfortunately, the power output decreases dramatically as the frequency of excitation (i.e., the resonant frequency of the harvester) decreases [13]. Moreover, employing a cantilevered spring-mass system for low-frequency (<10 Hz) energy harvesting is quite challenging due to the size constraints for specific application. Human-body-induced motion (e.g., walking, running, shaking limbs, etc.) also generates low-frequency (<6 Hz) vibrations, which do not allow the cantilever structure to be employed conveniently [14]. Hence, efficient energy harvesting from human-body-induced motion for hand-held and wearable smart devices requires clever design choices. Micro/nano-structured triboelectric nanogenerators [15,16], flexible piezo-composite based piezoelectric nanogenerators [17,18] etc. have shown grea<sup>t</sup> application potential in wearable biomechanical energy harvesting and motion sensing. However, they require huge e fforts in material development, which was not of our interest. Our primary interest was to design and develop inertial based, low-frequency (e.g., human-body-induced motion) energy harvesters.

The mechanical frequency up-conversion mechanism [19], among numerous design approaches over the past few years, has become the mainstream approach for human-motion based energy harvesting. It allows the transducer element (in the form of a spring-mass system) to actuate at its own resonant frequency (considerably high) by a low-frequency oscillatory or rotary system that responds to the external low-frequency vibration generated by human-motion. Commonly used methods of mechanical frequency up-conversion include mechanical impact and plucking [20–23]. Impact excitation transfers an instantaneous momentum into the transducer element whereas plucking excitation implies a slow deflection of the transducer element followed by its sudden release. In general, these methods exert direct force straight to the transducer element that could potentially lead to damage, especially in the case of piezoelectric devices. In order to overcome these issues with piezoelectric energy harvesters, we introduced the transverse impact-based frequency up-conversion mechanism in a human handy-motion driven electromagnetic energy harvester by employing a double-clamped FR4 cantilever beam as a high-frequency oscillator and a freely movable sphere as a low-frequency oscillator [24]. The transverse impact mechanism meets the reliability challenge and the freely-movable sphere allows the device to operate e fficiently at extremely low frequencies (with su fficiently large amplitudes) of handy-motion vibration, meaning its non-resonant behavior [25]. However, the device generates low power and its average power density is poor.

In order to improve its performance, we attempted to hybridize our previous work by incorporating a piezoelectric transducer without cost to the harvester volume. A hybrid energy harvesting technology combines two or more types of transducers that simultaneously capture energy from the same excitation [26–28]. In this paper, we present the theoretical modeling and experimental characterization of a piezoelectric (PE) and electromagnetic (EM) hybrid energy harvester for human-limb motion by utilizing the transverse mechanical impact-based frequency up-conversion strategy. Transverse impact, created by a sliding sphere over the parabolic tip of a mass attached to a clamped–clamped piezoelectric beam, eliminates the reliability issue from rapid damage of the piezoelectric cantilever due to direct impact. Moreover, simultaneous power generation from both PE and EM transducers offers a higher power density. A theoretical model for the hybrid generator under transverse impact was developed and experimentally validated with a prototype device. The proposed approach has the potential of reliable operation under low-frequency and high-amplitude excitation of human-body-induced motion toward the development of self-powered portable and wearable smart devices.

#### **2. Design and Modeling**

#### *2.1. Harvester Structure and Its Operation*

Figure 1a shows the schematic structure of the proposed transverse-impact driven hybrid energy harvester for human-limb motion. A PE transducer in the form of a clamped–clamped lead zirconate titanate (PZT) bimorph beam and an EM transducer consisting of one cylindrical magne<sup>t</sup> attached to the center of the piezo-beam and a hollow cylindrical multi-turn copper coil fixed to the housing constituted the hybrid generator structure. An additional mass with a parabolic-top was attached to the piezo-beam, opposite to the magnet. A hollow rectangular channel that contained a freely-movable spherical ball was placed on top of the piezo-beam in parallel. The channel had an opening at the center of its bottom wall that allowed the parabolic-top of the mass to be positioned through it, so that the ball was able to slide over the parabolic-top while the device was operated.

**Figure 1.** Schematics of the proposed transverse-impact driven frequency up-converted hybrid energy harvester (**a**) and its operation principle (**b**).

The principle of frequency up-conversion by transverse impact in the proposed hybrid energy harvester is illustrated in Figure 1b. When the device is excited by a low-frequency vibration with a sufficiently large amplitude (i.e., human-limb motion), the ball moves back and forth along the length of the channel. In its back and forth motion, the ball slides over the parabolic-top of the mass and produces a transverse impact on it that pushes mass as well as the piezo-beam downward, allowing it to vibrate freely at its own resonant frequency in a direction perpendicular to the direction of the ball movement. As a result, stresses are generated on the surfaces of the piezo-beam that generate voltage by virtue of the piezoelectric effect. Simultaneously, the magne<sup>t</sup> attached to the piezo-beam vibrates with respect to the adjacent coil and an electromotive force (e.m.f) voltage is generated by electromagnetic induction between them. The frequency (resonant) at which the beam vibrates is much higher than that of the excitation applied by human-limb motion and can be determined by the material and structural parameters of the beam. As shown in Figure 1b, the ball exerts transverse impact twice in one cycle of its back and forth motion; each time the system undergoes an impulse excitation, resulting in an exponentially decayed oscillatory motion between two consecutive impacts, so the output responses from both piezoelectric and electromagnetic transducers will be.

## *2.2. Electromechanical Modeling*

The system is considered as a single-degree-of-freedom (SDOF) forced spring-mass-damper system excited by a periodic force *<sup>F</sup>*(*t*). When the ball slides over the parabolic-top of the cantilevered proof-mass, the collision between them is stated as the low-velocity transverse impact of a rigid body on a flexible element [20]. Accordingly, when the bodies (ball and proof-mass) come into contact, they tend to interpenetrate each other, and a local compression force develops in their interface, which increases as the ball slides over the parabolic-top of the proof-mass, resulting in bending of the beam. When the compression force is large enough, the ball slides over the parabolic-top before the force becomes large enough. However, the curvatures of both the ball and parabolic-top as well as the overlap between them plays crucial roles on the transverse impact mechanism. Finally, the bodies are separated and each vibrates independently until the next collision occurs. According to the force diagram of the transverse impact mechanism [20], the force experienced on the proof-mass in the transverse direction is *<sup>F</sup>*(*<sup>L</sup>*, *t*) = μ*k F*(*t*) *sin*θ and the governing equation of motion of the proposed system can be expressed as

$$m\ddot{y}(t) + c\dot{y}(t) + ky(t) = \mu\_k F(t) \sin \theta \int\_0^{l./2} \varphi(\mathbf{x})d\mathbf{x} \tag{1}$$

where *m* is the mass (including the masses of the attached proof-mass and magnet); *y*(*t*) is the mass displacement; *c* is the equivalent damping coefficient; *k* is the stiffness of the beam; *L* is the length of the beam; μ*k* is the coefficient of kinetic friction while the ball slides over the parabolic-top; and ϕ(*x*) is the mass normalized eigenfunction of the first vibration mode for the boundary condition *x* = *L*/2, which is [29]

$$\varphi(\mathbf{x}) = \sqrt{\frac{2}{m\_l L}} \left[ \cos h \frac{2\lambda}{L} \mathbf{x} - \cos \frac{2\lambda}{L} \mathbf{x} - \varepsilon \left( \sin h \frac{2\lambda}{L} \mathbf{x} - \sin \frac{2\lambda}{L} \mathbf{x} \right) \right] \tag{2}$$

where *ml* is the mass per unit length; λ is the dimensionless frequency parameter for the first mode; and ς = (sin *h*λ − sin <sup>λ</sup>)/(cos *h*λ + cos <sup>λ</sup>).

The beam is a piezoelectric bimorph and generates voltage when lateral stress is generated on the surfaces of the piezoelectric material due to the transverse impact. The cross-section of the beam with a metallic shim sandwiched between two piezoelectric layers is shown in Figure 2. Each piezo-material is poled along its thickness direction and are connected in parallel. During bending of the beam, the stresses in the top and bottom piezoelectric layers will be in opposite directions: one is in tension and the other is in compression. An equivalent moment of inertia of the bimorph beam is defined as [30]

$$I\_{c\eta} = 2\left(\frac{wh\_p^3}{12} + wh\_ph\_{c\eta}^2\right) + \frac{E\_s wh\_s^3}{12E\_p} \tag{3}$$

where *Es* and *Ep* are the Young's modulus of the shim and piezoelectric materials; *hs* and *hp* are the thickness of the shim and piezoelectric layers; and *w* and *heq* = *hs* + *hp*/2 are the width and the equivalent thickness of the beam, respectively. The maximum stress on the piezoelectric surface due to the transverse impact at the center of the beam (*x* = *L*/2) can be calculated as [31]

$$\sigma\_{\text{max}} = \frac{M(x)\left(h\_{\text{p}} + \frac{h\_{\text{p}}}{2}\right)}{I\_{eq}} = \frac{\mu\_k F\_{\text{max}} L\left(h\_{\text{p}} + \frac{h\_{\text{p}}}{2}\right)}{8I\_{eq}}\tag{4}$$

where *<sup>M</sup>*(*x*) is the bending moment and *Fmax*(= *kymax*) is the magnitude of the transverse force determined by Hook's law where *k* is the stiffness and *ymax* is the maximum displacement of the bimorph beam. Now, the generated peak open circuit voltage can be determined as

$$V\_{oc} = \frac{-d\_{31}h\_p \sigma\_{\text{max}}}{\varepsilon\_0 \varepsilon\_r} \tag{5}$$

where −*d*<sup>31</sup> and ε0 are the piezoelectric charge constant and dielectric constant of the piezoelectric material, respectively and ε*r* is the permittivity of free space. According to the dynamics of the transverse mechanical impact by the freely movable spherical ball described earlier, the output voltage from the piezoelectric transducer can be written as a function of time *t* as

$$V\_{PE}(t) = V\_{\alpha} e^{-\zeta\_{m^{0}t}t} \sin(\omega\_{d}t); \; n \frac{2\pi}{\omega\_{d}} < t < (n+1) \frac{2\pi}{\omega\_{d}}, \; (n = 0, 1, 2, 3, \dots) \tag{6}$$

where *n* is the number of impacts; ω*r*, ω*d*, and ζ*m* are the resonant frequency, damped resonant frequency, and mechanical damping ratio, respectively, which are defined as [31,32]

$$
\omega\_r = \sqrt{\frac{k}{m}} = \frac{\lambda}{L^2} \sqrt{\frac{E\_p I\_{eq}}{m}};
\
\omega\_d = \omega\_r \sqrt{\left(1 - \zeta\_m^2\right)};
\
\zeta\_m = \frac{\varepsilon\_m}{2m\omega\_r} \tag{7}
$$

**Figure 2.** Cross-section of the piezoelectric bimorph beam.

As the magne<sup>t</sup> attached to the piezoelectric beam also vibrates simultaneously, it induces voltage in the coil due to relative motion between them. According to Faraday's law of electromagnetic induction, the induced open circuit e.m.f voltage generated by the electromagnetic transducer is [33]

$$V\_{EM}(t) = -N\frac{d}{dt} \left[ \int \stackrel{\rightarrow}{B}\_{} d\vec{A} \right] = -NBl\dot{y}(t) \tag{8}$$

where N is the number of coil turns and : →*<sup>B</sup>*.*d*<sup>→</sup>*<sup>A</sup>* indicates the net magnetic flux through the differential element area *dA* of the magnet-coil assembly. *B* is the magnetic flux density; *l* is the coil length across the magnetic flux lines; and .*y*(*t*) is the relative velocity between the magne<sup>t</sup> and coil, which is determined by solving Equation (1) as

$$\dot{y}(t) = -\frac{\mu\_k F\_{\text{max}} \omega\_r \Big[ \int\_0^{L/2} q(\mathbf{x}) d\mathbf{x} \Big]}{k \sqrt{1 - \zeta\_m^2}} e^{-\zeta\_m \omega\_r t} \sin(\omega\_d t) \tag{9}$$

In the case of both transducers, the instantaneous power delivered to corresponding load resistance *Rl* can be expressed as

$$P(t) = \frac{1}{T} \int\_0^T \frac{V(t)^2}{R\_I} dt\tag{10}$$

It is to be noted that the damping (ζ*T*) for each standalone (either piezoelectric or electromagnetic) transducer includes the mechanical damping ζ*m* and the electrical damping ζ*e* of the corresponding transducer: ζ*T* = ζ*m* + ζ*e* (*PE*) for the piezoelectric transducer and ζ*T* = ζ*m* + ζ*e* (*EM*) for the electro-magnetic transducer. However, for coupled transducers (when both transducers are terminated to corresponding loads simultaneously), the damping values are the same: ζ*T* = ζ*m* + ζ*e* (*PE*) + ζ*e* (*EM*) for both transducers.
