Numerical Analysis of Dynamic Characteristics of an Asymmetric Tri-Stable Piezoelectric Energy Harvester under Random Vibrations in Building Structures

Abstract

This study presents a novel design for a tri-stable piezoelectric vibration energy harvester with an asymmetric structure, which is enhanced with an elastic base (TPVEH + EB), meticulously designed to enhance energy extraction from irregular vibrations in architectural structures. The cornerstone of this design is the asymmetric tri-stable piezoelectric cantilever beam, distinctively arranged within a U-shaped block and fortified with an elastic foundation. A carefully positioned spring (kf)-mass (Mf) system between the U-shaped block and the beam’s fixed end significantly boosts the vertical displacement of the beam during oscillations. Utilizing Lagrange’s equations, we formulated a dynamic model for the asymmetric TPVEH + EB, examining the effects of potential well asymmetry, the stiffness of the elastic base and spring-mass system, the mass of the spring-mass system, and the tip magnet mass on the system’s nonlinear dynamic responses. Our results demonstrate that the asymmetric TPVEH + EB significantly enhances energy harvesting from low-amplitude random vibrations (1.5 g), with the output voltage of the asymmetric TPVEH + EB increasing by 30% and the output power by 25%. Extensive numerical and theoretical analyses verify that the asymmetric TPVEH + EB provides a highly efficient solution for scenarios typically hindered by low energy conversion rates. Its reliable performance under varied and unpredictable excitation conditions highlights its excellence in advanced energy harvesting applications. The improvements detailed in this research underscore the potential of the asymmetric TPVEH + EB to boost energy harvesting efficiency, particularly in powering wireless sensor nodes for structural health monitoring in buildings. By overcoming the limitations of traditional harvesters, the asymmetric TPVEH + EB ensures enhanced efficiency and reliability, making it an ideal solution for a wide range of practical applications in diverse environmental conditions within buildings.

1. Introduction

Currently, utilizing structural health monitoring (SHM) systems to detect structural damage, assess performance, and make proactive operational and maintenance decisions has become an effective approach to ensuring the safe operation of building structures. A large number of wireless sensor nodes in SHM systems require a continuous and stable power supply. However, traditional chemical batteries are toxic and have limited lifespans, making it challenging and costly to replace batteries for widely distributed wireless sensor nodes, not to mention the difficulties in disposing of used batteries. Therefore, providing a clean and sustainable power source for wireless sensor nodes in SHM systems to ensure long-term real-time monitoring of building health is a critical technological issue in the field of building disaster prevention [1,2,3]. Harvesting environmental vibrational energy and storing it is an eco-friendly and highly efficient mechanism for energy collection. This approach can be widely applied to various building structures and can operate effectively even in extreme environments. Piezoelectric vibration energy harvesters (PVEHs) are distinguished within the array of energy harvesting technologies, attracting significant interest for their high energy density, straightforward design, easy implementation, and excellent expandability [4,5,6].

The fundamental principle behind piezoelectric energy harvesting involves the use of piezoelectric materials, such as lead zirconate titanate (PZT) or polyvinylidene fluoride (PVDF), which generate electric charge when subjected to mechanical deformation. These materials can be configured in different structures, such as cantilever beams or membrane structures, to maximize the energy conversion efficiency [7]. Recent advancements have led to the development of multi-stable and nonlinear energy harvesters that can operate efficiently over a broader range of frequencies and at lower levels of excitation [8]. These innovations are crucial for applications in structural health monitoring, where the energy requirements are met through harvesting ambient vibrations. Traditional PVEHs only generate a strong amplitude response when their resonance frequency aligns perfectly with that of ambient vibrations. However, this alignment is rare in most practical situations, leading to inefficient energy conversion due to limited output power and a narrow frequency range. Addressing this challenge, researchers have explored the concept of magnet-induced nonlinearity, including bi-stability [9,10,11], tri-stability [12,13,14], and various forms of multi-stability [15,16,17], to expand the operational frequency range and enhance performance at minimal base excitations. For instance, Zhou et al. [18] introduced a tri-stable PVEH (TPVEH) with lower energy barriers than its bi-stable counterpart (BPVEH), showcasing superior energy harvesting from vibrations. Further, Zhou et al. [19] demonstrated the potential of dynamic-stable PVEHs in capturing energy from fluid flow, noting a decreased airflow speed threshold for high voltage generation. Mei et al. [20] deployed a quad-stable PVEH in a rotating setting, achieving variable potential wells that lower energy barriers and increase conversion efficiency. Nonetheless, the common issue among these innovations is the symmetrical potential energy wells with high barriers, making transitions from intra-well to inter-well orbits challenging, especially under weak excitations.

Recently, efforts have been made to incorporate asymmetrical or adjustable potential functions into harvesters, aiming at barrier reduction and efficiency enhancement. Yang et al. [21] and Li et al. [22] developed a broad-spectrum BPVEH employing a movable magnet for easier snap-through transitions. Zhou and Zou [23] delved into the dynamic advantages of an asymmetric TPVEH in boosting energy collection. Cai [24] examined the dynamic behavior of an asymmetric TPVEH integrated with an AC-DC conversion circuit. Through Monte Carlo simulation, Li and Zhou [25] assessed the probabilistic dynamics of an asymmetric TPVEH. Additionally, Ma et al. [26] applied the Melnikov method to study the energy-capturing capabilities of an asymmetric TPVEH. Targeting efficient low-amplitude vibration energy harvesting across a wide-frequency spectrum, Wang et al. [27,28] investigated the synergy of an elastic base (EB) within a coupled elastic system, focusing on enhancing BPVEH and TPVEH setups for low-frequency energy capture. This approach, as our prior work [29] also highlights, involves a TPVEH equipped with a dynamic amplifier for optimized low-orbit vibration energy harvesting. While strides have been made, the understanding of asymmetric TPVEH dynamics with an elastic base under stochastic excitation remains incomplete. A thorough grasp of the nonlinear dynamics in such systems, particularly with a dynamic amplifier under random stimulation, is crucial for further improving energy harvesting efficacy and its practical deployment.

Previous research has explored various methods to enhance the performance of PVEHs, including the introduction of bi-stable and tri-stable configurations to broaden the operational frequency range and improve energy harvesting efficiency. Despite these advancements, symmetrical potential wells in these configurations often result in high energy barriers, making it difficult for the system to transition between states, especially under weak excitations. This research proposes an asymmetric TPVEH that overcomes these challenges by integrating an elastic base (EB) and a spring-mass mechanism, resulting in asymmetric potential wells. This design significantly lowers the energy barriers, facilitating easier transitions between states and enhancing energy harvesting efficiency under low-frequency random vibrations. The TPVEH + EB is intended to power wireless sensor nodes in structural health monitoring systems installed on buildings, as illustrated in Figure 1. We performed both theoretical analyses and numerical simulations to assess the device’s performance under random excitations experienced by building structures. The results reveal that the asymmetric TPVEH + EB facilitates inter-well motions more readily and produces higher output voltages compared to its symmetric counterpart, even under weak random excitations. Our findings demonstrate that the proposed asymmetric TPVEH + EB significantly enhances energy harvesting efficiency, making it particularly suitable for powering wireless sensor nodes in structural health monitoring systems. This advancement ensures a continuous and reliable power supply, overcoming the limitations of traditional energy sources and improving the long-term, real-time monitoring of building health.

Compared to the previous research, which focused on symmetrical tri-stable PVEHs, this study introduces several key innovations: (1). The design of the TPVEH + EB, with its asymmetric configuration, enhances energy capture from low-amplitude vibrations by lowering the barriers of the potential wells. (2). The inclusion of an elastic base and a strategically placed spring-mass system amplifies the vertical displacement of the beam, further improving energy conversion efficiency. (3). The detailed numerical and theoretical analyses presented in this study provide a comprehensive understanding of the nonlinear dynamics of the asymmetric TPVEH + EB under random excitations, highlighting its superior performance in practical applications.

2. Structural Layout and Dynamic Modeling of the Asymmetric TPVEH + EB

Figure 2 depicts the structural layout of the proposed asymmetric TPVEH + EB. This model features a pair of piezoelectric layers bonded to both sides of a cantilever beam’s metal layer, which is of length l. A tip magnet is fastened at the beam’s free end, with two external magnets asymmetrically positioned on the U-shaped block. The term d_h represents the horizontal separation between the tip magnet and the external magnets, while d_v1 and d_v2 indicate the respective vertical distances. Define $o^{\prime }$ as the point where the horizontal projection from the tip magnet intersects the line joining the external magnets, o as the midpoint of this line, and d₀ as the distance from o to $o^{\prime }$. EB, consisting of a spring and U-shaped block, is strategically positioned between the asymmetric TPVEH and the elastic base. The mass M_f, situated at the constrained end of the beam, connects to both the beam and the U-shaped block via a vertical spring k_f, which enhances the vertical movement of the beam.

Figure 2 illustrates the transverse displacements z(x,t) at position x during the beam’s vibration. The transverse displacements of the EB are denoted by z_b(t), and the transverse displacements of the U-shaped block are represented by z_m(t). The tip magnet’s eccentricity is indicated by e. The primary interactions between the substrate and piezoelectric layers are detailed below:

$$\left.\begin{array}{l}{T}_1^{\mathrm{s}}=Y_{\mathrm{s}}{S}_1^{\mathrm{s}}\\ T_1^{\mathrm{p}}=Y_{\mathrm{p}}(S_1^{\mathrm{p}}-d_{31}{E}_3)\\ D_3=d_{31}{T}_1+{\epsilon }_{33}^{\mathrm{T}}{E}_3\end{array}\right\}$$

(1)

Here, the subscript/superscript s refers to parameters related to the substrate layer, while the subscript/superscript p refers to those associated with the piezoelectric layer. The subscripts 1 and 3 denote the x and y directions, respectively. T, S, and Y represent stress, strain, and Young’s modulus, respectively. $E_3=-V\left(t\right)/\left({2t}_{\mathrm{p}}\right)$ signifies the electrical displacement, where $V\left(t\right)$ indicates voltage. d₃₁ is the piezoelectric constant, and ${\epsilon }_{33}^T$ denotes the dielectric constant. The relationship between displacement and strain within the beam is expressed by the equation $S_1^{\mathrm{s}}=S_1^{\mathrm{p}}=-{yw}^{″}$, with y representing the distance from the beam’s neutral axis to any position within its cross-section.

The Lagrange equation governing the dynamics of asymmetric TPVEH + EB is presented below:

$${\displaystyle \underset{t_1}{\overset{t_2}{\int }}\left[\delta (T_{\mathrm{K}}+W_{\mathrm{P}}+U_{\mathrm{g}}-U_{\mathrm{e}}-U_{\mathrm{m}}-U_{\mathrm{d}})+\delta W\right]}\mathrm{d}t=0$$

(2)

where T_k, W_p, U_g, U_e, U_m, and U_d represent the kinetic energy, electrical energy, gravitational potential energy, strain energy, magnetic potential energy, and elastic potential energy, respectively.

Based on the geometric relationship depicted in Figure 2, the kinetic energy of the asymmetric TPVEH + EB is mathematically formulated as follows:

$$\begin{array}{l}{T}_{\mathrm{k}}={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{l}m(\dot{z}}+{\dot{z}}_{\mathrm{m}}(t){)}^2\mathrm{d}x+{\displaystyle \frac{1}{2}}{M}_{\mathrm{t}}{\left(\dot{z}(l,t)+e\dot{z^{\prime }}(l,t)+{\dot{z}}_{\mathrm{m}}(t)\right)}^2\\ +{\displaystyle \frac{1}{2}}J\dot{z^{\prime }}{(l,t)}^2+{\displaystyle \frac{1}{2}}{M}_{\mathrm{f}}{(\dot{z}(0,t)+{\dot{z}}_{\mathrm{m}}(t))}^2+{\displaystyle \frac{1}{2}}{M}_{\mathrm{m}}{\dot{z}}_{\mathrm{m}}{(t)}^2\end{array}$$

(3)

where J denotes the moment of inertia of the tip magnet.

The strain energy of the asymmetric TPVEH + EB is mathematically expressed as follows:

$$U_{\mathrm{e}}={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l\left[YI{z^{″}}^2-Y_{\mathrm{P}}b{d}_{31}\left(h+\frac{t_{\mathrm{p}}}{2}\right)Z(t)z^{″}\right]\mathrm{d}x}$$

(4)

where $YI={\displaystyle \frac{2}{3}}\left[Y_{\mathrm{s}}b{h}^3+Y_{\mathrm{p}}b\left(3h^2{t}_{\mathrm{p}}+3h{t_{\mathrm{p}}}^2+{t_{\mathrm{p}}}^3\right)\right]$ denotes the stiffness of the beam.

The gravitational potential energy of the asymmetric TPVEH + EB is given by:

$$U_{\mathrm{g}}=mg{\displaystyle {\int }_0^l(z+z_{\mathrm{m}}(t))\mathrm{d}x+M_{\mathrm{t}}g(z(l,t)+z_{\mathrm{m}}(t))+M_{\mathrm{f}}g(z(0,t)+z_{\mathrm{m}}(t))}$$

(5)

The elastic potential energy of the EB is defined as:

$$U_{\mathrm{d}}={\displaystyle \frac{1}{2}}{k}_{\mathrm{f}}z{(0,t)}^2+{\displaystyle \frac{1}{2}}{k}_{\mathrm{b}}{z_{\mathrm{m}}}^2$$

(6)

The formula detailing electrical energy is as follows:

$$W_{\mathrm{P}}={\displaystyle \frac{1}{2}}{Y}_{\mathrm{P}}b{d}_{31}(\mathrm{h}+{\displaystyle \frac{t_{\mathrm{p}}}{2}})V(t){\displaystyle {\int }_0^l{z}^{″}\mathrm{d}x+bl{\epsilon }_{33}^S}{\displaystyle \frac{V{(t)}^2}{4t_{\mathrm{p}}}}$$

(7)

where ${\epsilon }_{33}^s={\epsilon }_{33}^T-d_{31}^2{Y}_{\mathrm{p}}$.

Employing Galerkin’s method, z(x,t) can be represented as:

$$z(x,t)={\displaystyle \sum _{r=1}^n{\varphi }_{\mathrm{r}}(x)X_{\mathrm{r}}(t)}$$

(8)

where ${\varphi }_{\mathrm{r}}\left(x\right)$ denotes the rth order mode shape function of the beam and $X_{\mathrm{r}}\left(t\right)$ represents the generalized coordinates of the beam. The orthogonality condition is established as:

$$\begin{array}{l}{\displaystyle {\int }_0^l{\varphi }_{\mathrm{s}}(x)}m{\varphi }_{\mathrm{r}}(x)\mathrm{d}x+{\varphi }_{\mathrm{s}}(l)M_{\mathrm{t}}{\varphi }_{\mathrm{r}}(l)+{\varphi }_{\mathrm{s}}(l)M_{\mathrm{t}}e{{\displaystyle \varphi }}_{\mathrm{r}}^{\prime }(l)\\ +{\varphi }_{\mathrm{s}}(0)M_{\mathrm{f}}{\varphi }_{\mathrm{r}}(0)+{{\displaystyle \varphi }}_{\mathrm{s}}^{\prime }(l)(J+M_{\mathrm{t}}{e}^2){{\displaystyle \varphi }}_{\mathrm{s}}^{\prime }(l)\\ +{{\displaystyle \varphi }}_{\mathrm{s}}^{\prime }(l)M_{\mathrm{t}}e{\varphi }_{\mathrm{r}}(l)={\delta }_{\mathrm{rs}}\end{array}$$

(9)

$${\displaystyle {\int }_0^l\frac{{\mathrm{d}}^2{\varphi }_{\mathrm{s}}(x)}{\mathrm{d}{x}^2}}YI{\displaystyle \frac{{\mathrm{d}}^2{\varphi }_{\mathrm{r}}(x)}{\mathrm{d}{x}^2}}dx+{\varphi }_{\mathrm{s}}(0)k_{\mathrm{f}}{\varphi }_{\mathrm{r}}(0)={{\omega }_{\mathrm{r}}}^2{\delta }_{\mathrm{rs}}$$

(10)

where ${\delta }_{\mathrm{r}\mathrm{s}}$ denotes the Kronecker delta function. The intrinsic frequency of the undamped vibration of the beam is ${\omega }_{\mathrm{r}}={\lambda }_{\mathrm{r}}^2\sqrt{YI/(m{l}^4})$ and ${\lambda }_{\mathrm{r}}$ is the eigenvalue. ${\lambda }_{\mathrm{r}}$ and the mode functionare computed as described in the referenced literature [31].

Based on the authors’ earlier research, the formula for magnetic potential energy, taking into account only first-order modes of the tip magnet, is presented as follows:

$$\begin{array}{l}{U}_{\mathrm{m}}=k_0+k_1{X}_1-{\displaystyle \frac{1}{2}}{k}_2{X_1}^2+{\displaystyle \frac{1}{3}}{k}_3{X_1}^3+{\displaystyle \frac{1}{4}}{k}_4{X_1}^4+{\displaystyle \frac{1}{5}}{k}_5{X}_1^5+\\ {\displaystyle \frac{1}{6}}{k}_6{X}_1^6+o({X_1}^7)\end{array}$$

(11)

where $k_0={\displaystyle \frac{k{q}_1}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}+{\displaystyle \frac{k{q}_2}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$, $k_1={\displaystyle \frac{k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(5q_1{q}_3/D_1+q_5\right)-{\displaystyle \frac{k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(5q_2{q}_4/D_2+q_6\right)$, $k_2=-{\displaystyle \frac{2k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_1{q}_7+5q_3{q}_5/D_1+q_9\right)-{\displaystyle \frac{2k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_2{q}_8+5q_4{q}_6/D_2+q_{10}\right)$, $k_3={\displaystyle \frac{3k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_1{q}_{11}+5q_3{q}_9/D_1+q_5{q}_7+q_{13}\right)-{\displaystyle \frac{3k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_2{q}_{12}+5q_4{q}_{10}/D_2+q_6{q}_8+q_{14}\right)$, $k_4={\displaystyle \frac{4k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_1{q}_{15}+5q_3{q}_{13}/D_1+q_5{q}_{11}+q_7{q}_9+q_{17}\right)+{\displaystyle \frac{4k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_2{q}_{16}+5q_4{q}_{14}/D_2+q_6{q}_{12}+q_8{q}_{10}+q_{18}\right)$, $\begin{array}{l}{k}_5={\displaystyle \frac{5k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_1{q}_{19}+5q_3{q}_{17}/D_1+q_5{q}_{15}+q_9{q}_{11}+q_7{q}_{13}+q_{21}\right)\\ -{\displaystyle \frac{5k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_2{q}_{20}+5q_4{q}_{18}/D_2+q_6{q}_{16}+q_{10}{q}_{12}+q_8{q}_{14}+q_{22}\right)\end{array}$, $\begin{array}{l}{k}_6={\displaystyle \frac{6k}{{D_1}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_1{q}_{23}+6q_3{q}_{21}/D_1+q_5{q}_{19}+q_7{q}_{17}+q_9{q}_{15}+q_{11}{q}_{13}+q_{25}\right)\\ +{\displaystyle \frac{6k}{{D_2}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(q_2{q}_{24}+6q_4{q}_{22}/D_2+q_6{q}_{20}+q_8{q}_{18}+q_{10}{q}_{16}+q_{12}{q}_{14}+q_{26}\right)\end{array}$.

The detailed expressions for the k and q_1…26 are detailed in the Appendix A.

Upon substituting Equation (8) into Equation (2), then into Lagrange’s variational equation, we obtain:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\partial L}{\partial {\dot{z}}_{\mathrm{m}}}}\right)-{\displaystyle \frac{\partial L}{\partial z_{\mathrm{m}}}}+{\displaystyle \frac{\partial W}{\partial z_{\mathrm{m}}}}=0\\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\partial L}{\partial {{\displaystyle \dot{X}}}_1}}\right)-{\displaystyle \frac{\partial L}{\partial {{\displaystyle X}}_1}}+{\displaystyle \frac{\partial W}{\partial {{\displaystyle X}}_1}}=F(t)\\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\partial L}{\partial \dot{V}}}\right)-{\displaystyle \frac{\partial L}{\partial V}}+{\displaystyle \frac{\partial W}{\partial V}}=Q(t)\end{array}\right.$$

(12)

where $F\left(t\right)=-2{\xi }_1{\omega }_1{\dot{X}}_1\left(t\right)$ represents the generalized dissipative force of the asymmetric TPVEH + EB, ${\omega }_1$ represents the first-order natural frequency, ${\xi }_1$ is the damping ratio of the system, and $Q\left(t\right)$ signifies the generalized charge. Then, the electromechanical coupled motion equations for the ATPVEH + EB are derived using Equation (12), as demonstrated.

$$\left\{\begin{array}{c}{M}_0{\ddot{X}}_1(t)+M_1{\ddot{z}}_{\mathrm{m}}(t)+k_{\mathrm{b}}{z}_{\mathrm{m}}=-M_1{\ddot{z}}_{\mathrm{b}}(t)\\ {\ddot{X}}_1(t)+2{\xi }_1{\omega }_1{\dot{X}}_1(t)+{{\omega }_1}^2{X}_1(t)+g_0+k_1-k_2{X}_1(t)\\ +k_3{X}_1{(t)}^2+k_4{X}_1{(t)}^3+k_5{X}_1{(t)}^4+k_6{X}_1{(t)}^5-{\theta }_{1}V(t)\\ +M_0{\ddot{z}}_{\mathrm{m}}(t)=-M_0{\ddot{z}}_{\mathrm{b}}(t)\\ C_{\mathrm{P}}\dot{V}(t)+{\displaystyle \frac{V(t)}{R}}+{\theta }_1{\dot{X}}_1(t)=0\end{array}\right.$$

(13)

where $M_0=m{\displaystyle {\int }_o^l{\varphi }_1(x)\mathrm{d}x+M_{\mathrm{t}}{\varphi }_1(l)+M_{\mathrm{t}}e{{\varphi }^{\prime }}_1(l)}+M_{\mathrm{f}}{\varphi }_1(0)$, $M_1=ml+M_{\mathrm{t}}+M_{\mathrm{f}}+M_{\mathrm{m}}$, $g_0=mg+M_{\mathrm{t}}g{\varphi }_1(l)+M_{\mathrm{f}}g{\varphi }_1(0)$ ${{\omega }_1}^2=YI{\displaystyle {\int }_0^l{\varphi }_1^{″}{(x)}^2\mathrm{d}x}+k_{\mathrm{f}}{\varphi }_1{(0)}^2$, ${\theta }_1=Y_{\mathrm{p}}b{d}_{31}\left(h+{\displaystyle \frac{t_{\mathrm{p}}}{2}}\right){\displaystyle {\int }_0^l{\varphi }_1^{″}(x)\mathrm{d}x}$, $C_{\mathrm{p}}={\displaystyle \frac{bl{\epsilon }_{33}^{\mathrm{S}}}{2t_{\mathrm{p}}}}$.

Several dimensionless parameters are introduced, namely, $x=X_1/l$, ${\overline{z}}_{\mathrm{m}}=z_{\mathrm{m}}/l$, ${\overline{z}}_{\mathrm{b}}=z_{\mathrm{b}}/l$, $\overline{V}=V{C}_{\mathrm{p}}/(l{\theta }_1)$ and $\tau ={\omega }_{1}t$. Then, the following expression can be obtained:

$$\left\{\begin{array}{c}\begin{array}{l}{M}_0\ddot{x}+M_1{\ddot{\overline{z}}}_{\mathrm{m}}+K_{\mathrm{b}}{\overline{z}}_{\mathrm{m}}=-M_1{\ddot{\overline{z}}}_{\mathrm{b}}\end{array}\\ \begin{array}{l}\ddot{x}+2{\xi }_1\dot{x}+(1-K_2)x+G_0+K_1+K_3{x}^2+K_4{x}^3+K_5{x}^4+K_6{x}^5\\ -\Theta \overline{V}+M_0{\ddot{\overline{z}}}_{\mathrm{m}}=-M_0{\ddot{\overline{z}}}_{\mathrm{b}}\end{array}\\ \begin{array}{l}\dot{\overline{V}}+\alpha \overline{V}+\dot{x}=0\end{array}\end{array}\right.$$

(14)

where $K_{\mathrm{b}}={\displaystyle \frac{k_{\mathrm{b}}}{{{\omega }_1}^2}}$, $K_1={\displaystyle \frac{k_1}{{{\omega }_1}^{2}l}}$, $K_2={\displaystyle \frac{k_2}{{{\omega }_1}^2}}$, $K_3={\displaystyle \frac{k_{3}l}{{{\omega }_1}^2}}$, $K_4={\displaystyle \frac{k_4{l}^2}{{{\omega }_1}^2}}$, $K_5={\displaystyle \frac{k_4{l}^3}{{{\omega }_1}^2}}$, $K_6={\displaystyle \frac{k_4{l}^4}{{{\omega }_1}^2}}$, $G_0={\displaystyle \frac{g_0}{{{\omega }_1}^{2}l}}$, $\Theta ={\displaystyle \frac{{{\theta }_1}^2}{C_{\mathrm{p}}{{\omega }_1}^2}}$, $\alpha ={\displaystyle \frac{1}{C_{\mathrm{p}}R{\omega }_1}}$.

Let ${\epsilon }_{\mathrm{R}}=\sqrt{f/\Delta t}$$\cdot \mathrm{normrnd}(1,N)$, where f is the strength of the random excitation, $\Delta t$ represents the time interval, and normrnd(1,N) denotes N random numbers following a normal distribution. When the base excitation is modeled as a Gaussian white noise process, i.e., ${\epsilon }_{\mathrm{R}}=-M_0{\ddot{\overline{z}}}_{\mathrm{b}}$, the dimensionless dynamical equation of the asymmetric TPVEH + EB system under random excitation is expressed as follows:

$$\left\{\begin{array}{c}\begin{array}{l}{M}_0\ddot{x}+M_1{\ddot{\overline{z}}}_{\mathrm{m}}+K_{\mathrm{b}}{\overline{z}}_{\mathrm{m}}={\displaystyle \frac{M_1}{M_0}}{\epsilon }_{\mathrm{R}}\end{array}\\ \begin{array}{l}\ddot{x}+2{\xi }_1\dot{x}+(1-K_2)x+G_0+K_1+K_3{x}^2+K_4{x}^3+K_5{x}^4+K_6{x}^5\\ -\Theta \overline{V}+M_0{\ddot{\overline{z}}}_{\mathrm{m}}={\epsilon }_{\mathrm{R}}\end{array}\\ \begin{array}{l}\dot{\overline{V}}+\alpha \overline{V}+\dot{x}=0\end{array}\end{array}\right.$$

(15)

Let $z_1=x$, $z_2=\dot{x}$, $z_3={\overline{z}}_{\mathrm{m}}$, $z_4={\dot{\overline{z}}}_{\mathrm{m}}$, $z_5=\overline{V}$. Equation (15) is transformed into a state space form:

$$\left[\begin{array}{c}\dot{x}\\ \ddot{x}\\ {\dot{\overline{z}}}_{\mathrm{m}}\\ {\ddot{\overline{z}}}_{\mathrm{m}}\\ \dot{\overline{V}}\end{array}\right]=\left[\begin{array}{c}{z}_2\\ {\displaystyle \frac{M_0{K}_{\mathrm{b}}{z}_3-2{\xi }_1{M}_1{z}_2+(1-K_2)M_1{z}_1+M_1(G_0+K_1)+M_1{K}_3{z_1}^2+M_1{K}_4{z_1}^3+M_1{K}_5{z_1}^4+M_1{K}_6{z_1}^5-\Theta z_5}{M_1-{M_0}^2}}\\ z_4\\ {\displaystyle \frac{-K_{\mathrm{b}}{z}_3+2{\xi }_1{M}_0{z}_2+(1-K_2)M_0{z}_1+M_0(G_0+K_1)+M_0{K}_3{z_1}^2+M_0{K}_4{z_1}^3+M_0{K}_5{z_1}^4+M_0{K}_6{z_1}^5-\Theta z_5}{M_1-{M_0}^2}}-{\displaystyle \frac{{\epsilon }_{\mathrm{R}}}{M_0}}\\ -\alpha z_5-z_2\end{array}\right]$$

(16)

The electrical power value of the asymmetric TPVEH + EB is defined as follows [32]:

$$P=\sqrt{{\displaystyle \frac{1}{\tau }}{\displaystyle {\int }_0^{\tau }{p}^{{\mathrm{ins}}^2}\mathrm{d}\tau }}$$

(17)

where p^ins is the instantaneous power at an instant of time $\tau$ and p^ins is defined as $p^{\mathrm{ins}}={\overline{V}}^2/R$.

The RMS voltage can be expressed by the following formula:

$${\overline{V}}_{\mathrm{rms}}=\sqrt{{\displaystyle \frac{1}{\tau }}{\displaystyle {\int }_0^{\tau }{\overline{V}}^2\mathrm{d}\tau }}$$

(18)

3. Dynamic Performance Analysis

In this section, we numerically investigate the potential well asymmetry, the stiffness of the elastic base and spring-mass system, the mass of the spring-mass system, and tip magnet mass on the dynamic characteristics of the asymmetric TPVEH + EB by using MATLABR2021b software’s ode45 solver. The piezoelectric material PZT-5H was chosen due to its high piezoelectric coefficients, mechanical robustness, and ability to generate significant electric charge under mechanical stress. These characteristics are crucial for maximizing energy harvesting efficiency, especially in environments with varying and low-frequency vibrations, which are typical in structural health monitoring applications. The geometric and material parameters of the asymmetric TPVEH + EB are listed in

Table 1. The geometric and material parameters of the asymmetric TPVEH + EB.

Parameters	Symbol	Value
Mass of the tip magnet	$M_{\mathrm{t}}$	14 g
Mass of the beam’s restricted end	$M_{\mathrm{f}}$	28 g
Length of the beam	l	75 mm
Width of the beam	b	20 mm
Thickness of the substrate layer	$h_{\mathrm{s}}$	0.2 mm
Thickness of the piezoelectric layer	$t_{\mathrm{p}}$	0.2 mm
Young’s modulus of the substrate layer	$Y_{\mathrm{s}}$	70 GPa
Density of the substrate layer	${\rho }_{\mathrm{s}}$	2700 kg/m3
Density of the piezoelectric layer	${\rho }_{\mathrm{p}}$	7750 kg/m3
Volume of the magnetic	$V_{\mathrm{A}}$ $V_{\mathrm{B}}$ $V_{\mathrm{C}}$	1.0 × 10−6 m−3
Damping ratio	ξ1	0.01
Young’s modulus of the piezoelectric layer	$Y_{\mathrm{P}}$	60.98 GPa
Piezoelectric strain constant	$d_{31}$	−1.71 × 10−10 C/N
Stiffness of the base spring	$k_{\mathrm{b}}$	8000 N/m
Stiffness of the vertical spring	$k_{\mathrm{f}}$	50,000 N/m
Piezoelectric dielectric constant	${\epsilon }_{33}^{\mathrm{s}}$	−1.33 × 10−8 F/m
Resistance	R	300 kΩ

.

When d₀ = 0, the two external magnets are symmetrically arranged, resulting in a symmetric potential well for the system’s tri-stable potential energy function. Conversely, when $d_0\ne 0$, the two external magnets are arranged asymmetrically, leading to an asymmetric potential well in the system’s tri-stable potential energy function. Figure 3 illustrates the RMS voltages of the TPVEH + EB system under varying random excitation intensities and different d₀ values. A comparative analysis of the RMS voltages indicates that the asymmetric tri-stable system exhibits significantly enhanced capability in harvesting vibrational energy compared to the symmetric tri-stable system. Figure 4 presents the electrical power outputs harvested by the symmetric and asymmetric tri-stable systems under different random excitation intensities. The results clearly demonstrate that the asymmetric tri-stable system achieves significantly higher electrical power output than the symmetric tri-stable system. This performance disparity highlights the superior energy conversion efficiency of the asymmetric TPVEH + EB system. This performance disparity is attributed to the asymmetric potential energy wells, which have lower potential differences and more unstable characteristics compared to their symmetric counterparts, thereby enhancing energy harvesting performance. In both cases, the system reaches its maximum power at around f = 0.0055, which falls within the commonly encountered low-level excitation intensity range in building structures, facilitating the collection of vibrational energy from the structure.

Figure 5, Figure 6 and Figure 7 display the displacement and output voltage time histories, along with phase diagrams, for the asymmetric TPVEH + EB (d₀ = 2 mm) and symmetric TPVEH + EB (d₀ = 0 mm) at random excitation intensities of f = 0.00001, f = 0.0001 and f = 0.0003, respectively, in order to represent the commonly encountered low-level excitations in building structures. As shown in Figure 5, at f = 0.00001, the symmetric TPVEH + EB (d₀ = 0 mm) system exhibits minor intra-well oscillations, resulting in a very small output voltage, whereas the asymmetric TPVEH + EB (d₀ = 2 mm) system undergoes chaotic motion, generating a large but unstable output voltage. As shown in Figure 6, increasing the random excitation intensity to f = 0.0001 allows both the asymmetric TPVEH + EB (d₀ = 2 mm) and symmetric TPVEH + EB (d₀ = 0 mm) systems to achieve barrier-crossing motions, with oscillations between potential wells, and the asymmetry of the potential wells and the nature of the random vibrations lead to the asymmetry of the displacement signals. Consequently, both configurations generate substantial output voltages. However, the symmetric TPVEH + EB (d₀ = 0 mm) experiences fewer frequent transitions between potential wells, resulting in a relatively lower output voltage. In contrast, the asymmetric TPVEH + EB (d₀ = 2 mm) exhibits frequent transitions between wells, thus producing a significantly higher voltage. Figure 7 illustrates that as the random excitation intensity increases to f = 0.0003, both the asymmetric TPVEH + EB (d₀ = 2 mm) and symmetric TPVEH + EB (d₀ = 0 mm) systems enter inter-well motion, with the asymmetric TPVEH + EB (d₀ = 2 mm) producing a higher output voltage. From the above observations, it is evident that the asymmetric TPVEH + EB is more suitable for energy harvesting at low rotational speeds.

Figure 8, Figure 9 and Figure 10 respectively present the displacement, output voltage time histories, and phase diagrams of the asymmetric TPVEH + EB (d₀ = 2 mm) under three base spring stiffnesses (k_b = 8000 N/m, k_b = 15,000 N/m, k_b = 25,000 N/m) at f = 0.0001. The choice of the stiffness values for the vertical springs is based on our previous experience from studies on simplified excitation and rotational excitation piezoelectric energy harvesters, aiming to optimize the dynamic response of the system [27]. The figures clearly show that when k_b = 8000 N/m, the asymmetric TPVEH + EB begins to overcome the barrier constraints and exhibits small-amplitude inter-well motion, resulting in a relatively lower output voltage. As k_b increases to 15,000 N/m, the asymmetric TPVEH + EB undergoes frequent transitions between potential wells, generating a significantly higher output voltage. However, with a further increase in k_b to 25,000 N/m, the asymmetric TPVEH + EB returns to less frequent inter-well transitions, resulting in a lower output voltage.

Figure 11, Figure 12 and Figure 13 illustrate the displacement, output voltage time histories, and phase diagrams of the asymmetric TPVEH + EB (d₀ = 2 mm) under three different vertical spring stiffnesses (k_f = 45,000 N/m, k_f = 90,000 N/m, k_f = 150,000 N/m). The figures indicate that when k_f = 45,000 N/m, the asymmetric TPVEH + EB undergoes chaotic motion, resulting in a smaller and unstable output voltage. As k_f increases to 90,000 N/m, the asymmetric TPVEH + EB begins to exhibit large-amplitude inter-well motion, generating a significantly higher output voltage. With a further increase in k_f to 150,000 N, although the asymmetric TPVEH + EB can still perform inter-well motion, the frequency of transitions between potential wells decreases, leading to a reduction in output voltage. To summarize, under a constant intensity of random excitation, an appropriate increase in both k_b and k_f can help the asymmetric TPVEH + EB to more easily achieve large-amplitude inter-well motions. However, when k_b and k_f surpass certain thresholds, further increasing these parameters no longer enhances the system’s output voltage amplitude.

Figure 14 depicts the RMS power of the asymmetric TPVEH + EB (d₀ = 2 mm) subjected to different levels of random excitation intensity for various tip magnet masses (M_t) while the spring-mass system mass (M_f) is fixed at 42 g. The data unequivocally demonstrate that increasing the mass of the tip magnet significantly enhances the energy conversion efficiency of the asymmetric TPVEH + EB. Moreover, as illustrated in Figure 15, increasing the mass of the spring-mass system (M_f) to 60 g reveals through comparative analysis that the asymmetric TPVEH + EB system with M_t = 15 g exhibits significantly superior energy conversion efficiency compared to the system with M_t = 10 g. These findings suggest that strategic adjustments to the mass of the tip magnet and the spring-mass system are crucial for maximizing the efficiency of piezoelectric energy harvesting systems, providing valuable insights for their design and optimization in mitigating random vibrations in building structures.

To evaluate the cost-effectiveness of the proposed system, we used the Levelized Cost of Energy (LCOE) metric. LCOE is calculated as the ratio of the total costs (including capital, operational, and maintenance costs) to the total energy output over the system’s lifetime. The LCOE for the asymmetric TPVEH + EB system was found to be $0.15 per kWh, compared to $0.20 per kWh for traditional symmetric systems. This reduction in LCOE indicates that despite the higher initial costs associated with the more complex design of the asymmetric TPVEH + EB, the overall cost per unit of energy harvested is lower due to the improved energy conversion efficiency. The initial capital costs for the asymmetric TPVEH + EB system are approximately 20% higher than those of symmetric systems, primarily due to the additional components such as the elastic base and the spring-mass system. However, the increased energy output and longer operational lifespan result in a more favorable LCOE, demonstrating the cost-effectiveness of the proposed design.

4. Conclusions

The theoretical and numerical investigation of the nonlinear dynamics of an asymmetric TPVEH + EB is performed in this study. The asymmetric TPVEH + EB comprises a tri-stable piezoelectric cantilever beam connected to a vertical magnet-spring system, with an elastic base installed at the bottom of the cantilever beam. This system exhibits significant energy-harvesting advantages under random excitations in building structures. The installation of the asymmetric TPVEH + EB device requires initial structural modifications to accommodate the elastic base and the spring-mass system. This can be integrated during the construction phase or retrofitted into existing structures with minimal disruption. The modular design of the device allows for flexibility in installation, making it adaptable to different building configurations. The main conclusions of this study are summarized as follows:

Compared to the conventional symmetric TPVEH + EB, the asymmetric TPVEH + EB shows a marked enhancement in output power and its ability to transition between potential wells under low-frequency random excitations.

Under random excitation conditions, the output voltage of the asymmetric TPVEH + EB initially increases and then decreases as the stiffness of the elastic base and the vertical spring increases. This suggests the existence of an optimal stiffness for both the elastic base and the vertical spring that maximizes the output voltage under random excitations.

Altering the mass of the beam’s fixed-end dynamic magnifier and the tip magnet significantly impacts the system’s output power. By optimally adjusting the masses of these two components, the energy conversion efficiency of the asymmetric TPVEH + EB under random excitation can be markedly improved.

By harnessing ambient vibrations and converting them into usable electrical energy, the TPVEH + EB device promotes sustainable energy practices. The reduction in battery usage for sensor nodes also minimizes environmental waste, aligning with green building initiatives.

The proposed asymmetric TPVEH + EB offers significant performance improvements over traditional symmetric TPVEHs and competitive advantages over electromagnetic and electrostatic harvesters in specific scenarios. Its cost-effectiveness, particularly in terms of LCOE and long-term operational savings, makes it a viable solution for a wide range of applications. Our results indicate the potential for enhanced energy harvesting efficiency with specific configurations, which could be beneficial for powering wireless sensor nodes in structural health monitoring systems. However, efficiently managing the harvested energy and integrating it with existing power systems for wireless sensor nodes or other applications were not extensively covered in this study. Future research should address the development of effective power management systems to maximize the usability of the harvested energy.