A comparative study on vibration suppression and energy harvesting via mono-, bi-, and tri-stable piezoelectric nonlinear energy sinks

Abstract

This paper investigates the dynamical responses and performances of mono-, bi-, and tri-stable nonlinear energy sinks in simultaneous vibration suppression and energy harvesting as well as solely vibration mitigation. To this end, a realizable multi-stable nonlinear energy sink composed of a bimorph cantilever beam and arrays of magnets is considered for vibration mitigation. The target vibrating structure is a simply-supported beam under a harmonic excitation. The proposed absorber can exhibit multi-stability based on the magnet gaps. First, using extended Hamilton’s principle, the coupled continuous magneto-electromechanical equations governing the system are obtained. Next, the bifurcations of the absorber fixed points that lead to different stability states are analyzed. Then, the time- and frequency-responses of the coupled system are studied using time integration. The results show that the bi- and tri-stable absorbers perform well in case of strongly modulated responses. Furthermore, the response of the coupled system is verified using the harmonic balance method and pseudo-arclength continuation. Next, the potential well escape method based on the harmonic balance solution is exploited to investigate the strongly modulated response emergence and disappearance as well as its variations with the system parameters. The coupling mechanism of the nonlinear absorber to the host beam is also studied using linearization and compared with the harmonic balance solution. In addition, in order to compare the performances of the bi-stable and tri-stable absorbers, energy-based analyses are performed. The results reveal the average harvested power per average kinetic energy of the main system for a bi-stable absorber is higher than a tri-stable absorber. Furthermore, the bi-stable absorber can mitigate the host structure vibration more than the tri-stable absorber. Finally, it is observed that the bi-stable absorber suppresses the vibration more than its corresponding mono-stable absorber and harvests more energy over a broader frequency region.

Appendices

Appendix

Appendix A1

The mass per unit lengths of the main beam and MDVA are:

$${m}_{m}={\rho }_{m}{W}_{m}{t}_{m}$$

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$${m}_{a}={\rho }_{s}{W}_{s}{t}_{s}+2({\rho }_{p}{W}_{p}{t}_{p})$$

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where $\rho$, W, and t denote the density, width, and thickness, respectively.

Bending stiffnesses of the main beam and absorber are:

$${Y}_{m}{I}_{m}={Y}_{m}\left(\frac{{W}_{m}{t}_{m}^{3}}{12}\right)$$

(32)

$${Y}_{a}{I}_{a}={Y}_{s}\left(\frac{{W}_{s}{t}_{s}^{3}}{12} \right)+ {Y}_{p}\left(\frac{2}{3}{W}_{p}\left\{{\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right\}\right)$$

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Electromechanical coupling and piezoelectric layer capacitance are:

$$\theta =-\frac{{Y}_{p}{d}_{31}}{2{t}_{p}}{W}_{p}\left\{{\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right\}\left[H\left(s\right)-H\left(s-{L}_{p}\right)\right]$$

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$${C}_{p}=\frac{{e}_{33}{W}_{p}{L}_{p}}{{t}_{p}}$$

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where $H\left(s\right)$ is the Heaviside function and ${L}_{p}$, ${d}_{31}$, and ${e}_{33}$ are the PZT layers’ length, strain constant, and permittivity, respectively.

The boundary conditions of the main and absorber beams are:

$$S=0: \left\{\begin{array}{c}{w}_{m}=0\\ {w}_{m}^{{\prime}{\prime}}=0\end{array}\quad,\quad\right.S={L}_{m}: \left\{\begin{array}{c}{w}_{m}=0\\ {w}_{m}^{{\prime}{\prime}}=0\end{array}\right.$$

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$$s=0: \left\{\begin{array}{c}{w}_{a}=0\\ {w}_{a}^{\prime}=0\end{array}\quad,\quad\right.s={L}_{a}: \left\{\begin{array}{c}{Y}_{a}{I}_{a}{w}_{a}^{{\prime}{\prime}}=-\frac{{M}_{t}D}{2}{\ddot{w}}_{m}-({I}_{t}+\frac{{M}_{t}{D}^{2}}{4}) {\ddot{w}^{\prime}}_{m} \\ {{Y}_{a}{I}_{a}w}_{a}^{{\prime}{\prime}{\prime}}={M}_{t}{\ddot{w}}_{a}\end{array}\right.$$

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Appendix A2

The I-th mode shapes and natural frequencies of the simply-supported beam are:

$${\psi }_{I}\left(S\right)=\sqrt{\frac{2}{{m}_{m}{L}_{m}}} \text{sin} \left(\frac{I\pi }{{L}_{m}}S\right)$$

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$${\overline{\omega }}_{I}=(I\pi )\sqrt{\frac{{Y}_{m}{I}_{m}}{{m}_{m}{L}_{m}^{4}}}$$

The i-th mode shapes and natural frequencies of the absorber beam are:

$${\phi }_{i}\left(s\right)={C}_{i}(\text{cos}\frac{{\lambda }_{i}}{{L}_{a}}s-\text{cosh}\frac{{\lambda }_{i}}{{L}_{a}}s+{\mathcal{H}}_{i}\left(\text{sin}\frac{{\lambda }_{i}}{{L}_{a}}s-\text{sinh}\frac{{\lambda }_{i}}{{L}_{a}}s\right))$$

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$${\omega }_{i}={\lambda }_{i}^{2}\sqrt{\frac{{Y}_{a}{I}_{a}}{{m}_{a}{L}_{a}^{4}}}$$

where ${C}_{i}$, ${\lambda }_{i}$, and ${\mathcal{H}}_{i}$ are the i-th modal amplitude, eigenvalue, and modal constant, and they are respectively calculated by:

$${\int }_{0}^{{L}_{b}}{\phi }_{i}\left(s\right){(m}_{a}){\phi }_{j}\left(s\right)ds+{[{\phi }_{i}\left(s\right){M}_{t}{\phi }_{j}\left(s\right)+{\left[{\phi }_{i}\left(s\right){M}_{t}\frac{D}{2}{\phi }_{j}\left(s\right)\right]}^{\prime}+{\phi }_{i}^{{\prime}}\left(s\right)({{I}_{t}+M}_{t}{\frac{{D}^{2}}{4}){\phi }_{j}^{{\prime}}}\left(s\right)]}_{{L}_{a}}={\delta }_{ij}$$

$$\frac{{M}_{t}{I}_{t}}{{m}_{a}^{2}{{L}_{a}}^{4}}\left(1-\text{cos}{\lambda }_{i}\text{cosh}{\lambda }_{i}\right){{\lambda }_{i}}^{4}-\left(\frac{{M}_{t}{D}^{2}/4}{{m}_{a}{{L}_{a}}^{3}}+\frac{{I}_{t}}{{m}_{a}{{L}_{a}}^{3}}\right)\left(\text{sin}{\lambda }_{i}\text{cosh}{\lambda }_{i}+\text{sinh}{\lambda }_{i}\text{cos}{\lambda }_{i}\right){{\lambda }_{i}}^{3}-2\frac{{M}_{t}D/2}{{m}_{a}{{L}_{a}}^{2}}\left(\text{sin}{\lambda }_{i}\text{sinh}{\lambda }_{i}\right){{\lambda }_{i}}^{2}+\frac{{M}_{t}}{{m}_{a}{L}_{a}}\left(\text{sinh}{\lambda }_{i}\text{cos}{\lambda }_{i}-\text{cosh}{\lambda }_{i}\text{sin}{\lambda }_{i}\right){\lambda }_{i}+\left(1+\text{cos}{\lambda }_{i}\text{cosh}{\lambda }_{i}\right)-\left(\frac{{M}_{t}{D}^{2}/4}{{m}_{a}{{L}_{a}}^{3}}+\frac{{I}_{t}}{{m}_{a}{{L}_{a}}^{3}}\right){{\lambda }_{i}}^{3}\left(1-\text{cos}{\lambda }_{i}\text{cosh}{\lambda }_{i}\right)=0$$

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$${\mathcal{H}}_{i}=\frac{\left(\text{sin}{\lambda }_{i}-\text{sinh}{\lambda }_{i}\right)+\frac{{M}_{t}}{{m}_{a}{L}_{a}}{\lambda }_{i}[\left(\text{cos}{\lambda }_{i}-\text{cosh}{\lambda }_{i}\right)-\frac{D}{2}{\lambda }_{i}\left(\text{sin}{\lambda }_{i}+\text{sinh}{\lambda }_{i}\right)]}{\left(\text{cos}{\lambda }_{i}+\text{cosh}{\lambda }_{i}\right)-\frac{{M}_{t}}{{m}_{a}{L}_{a}}{\lambda }_{i}[\left(\text{sin}{\lambda }_{i}-\text{sinh}{\lambda }_{i}\right)-\frac{D}{2}{\lambda }_{i}\left(\text{cos}{\lambda }_{i}-\text{cosh}{\lambda }_{i}\right)]}$$

Appendix A3

The coefficients of the ODEs defined in Eq. (13) are defined as:

$$A{M}_{I}={\psi }_{N}\left({S}_{0}\right) ({m}_{a}{L}_{a}+{M}_{t}){\psi }_{I}\left({S}_{0}\right)$$

$$B{M}_{i}={\psi }_{N}\left({S}_{0}\right)({m}_{a}{\int }_{0}^{{L}_{a}}{\phi }_{i}\left(s\right)ds+{M}_{t}{\phi }_{i}\left({L}_{a}\right))$$

$$C{M}_{N}=2{\overline{\zeta }}_{N}{\overline{\omega }}_{N}$$

$$D{M}_{N}={\overline{\omega }}_{N}^{2}$$

$$E{M}_{NI}=2{\psi }_{N}\left({S}_{0}\right) ({M}_{t}){\psi }_{I}\left({S}_{0}\right)$$

$$F{M}_{N}={\psi }_{N}\left({S}_{0}\right)$$

$$G{M}_{I}={\psi }_{I}\left({S}_{0}\right)({m}_{a}{\int }_{0}^{{L}_{a}}{\phi }_{n}\left(s\right)ds+{M}_{t}{\phi }_{n}\left({L}_{a}\right))$$

$$H{M}_{n}=2{\zeta }_{n}{\omega }_{n}$$

$$I{M}_{n}={\omega }_{n}^{2}$$

$$J{M}_{nijk}={\int }_{0}^{{L}_{a}}{Y}_{a}{I}_{a}{\phi }_{n}{\left({{\phi }^{{\prime}}}_{i}{\left({{\phi }^{{\prime}}}_{j}{{\phi }^{{\prime}{\prime}}}_{k}\right)}^{{\prime}}\right)}^{{\prime}}ds$$

$$K{M}_{n}=\frac{{Y}_{p}{d}_{31}}{2{t}_{p}}{W}_{p}\left({\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right) { \phi ^{\prime}}_{n}\left({L}_{p}\right)$$

$$L{M}_{nij}=\frac{{Y}_{p}{d}_{31}}{2{t}_{p}}{W}_{p}\left({\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right) {\left(\frac{1}{2}{\left({\phi }_{n}{\phi }_{i}^{{\prime}}{\phi }_{j}^{{\prime}}\right)}^{\prime}-{\phi }_{n}{\phi }_{i}^{{\prime}}{\phi }_{j}^{{\prime}{\prime}}\right)}_{s={L}_{a}}$$

$$M{M}_{nijk}={\int }_{0}^{{L}_{a}}{\phi }_{n}({\phi }_{i}^{{\prime}}{\int }_{{L}_{a}}^{s}{m}_{a}{\int }_{0}^{s}({\phi }_{j}^{{\prime}}{\phi }_{k}^{{\prime}})dsds)^{\prime}ds$$

$$N{M}_{i}=\frac{{Y}_{p}{d}_{31}}{2{t}_{p}}{W}_{p}\left({\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right) {\phi }_{i}^{{\prime}}\left({L}_{p}\right)$$

$$O{M}_{ijk}=\frac{{Y}_{p}{d}_{31}}{2{t}_{p}}{W}_{p}\left({\left(\frac{{t}_{s}}{2}+{t}_{p}\right)}^{2}-{\left(\frac{{t}_{s}}{2}\right)}^{2}\right) {\int }_{0}^{{L}_{a}}{\phi }_{i}^{{\prime}}{\phi }_{j}^{{\prime}}{\phi }_{k}^{{\prime}{\prime}}ds$$

$$MF{C}_{n}={\phi }_{n}\left({L}_{a}\right)$$

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