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Nonlinear dynamics and performance evaluation of an asymmetric bistable energy harvester with unilateral piecewise nonlinearity

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Abstract

Due to the snap-through characteristics and broadband response, bistable energy harvesters (BEHs) have attracted significant attention. Previous studies revealed that asymmetry in potentials has an adverse impact on performance. To broaden the response bandwidth, a stopper is introduced and positioned at the side with a deeper potential well to realize unilateral piecewise nonlinearity, and this paper emphasizes the system’s nonlinear dynamics and output performance. Bifurcation diagrams and basins of attraction are utilized to study the system’s local and global dynamics under harmonic excitations. Numerical results illustrate that a proper collision gap between the stopper and beam could enable the system to realize large-amplitude oscillation within a wider frequency range, and this phenomenon is closely related to excitation amplitude and collision stiffness. Under stochastic excitation, numerical output reveals that there is an optimal collision gap for low to moderate excitation intensities, while the introduction of a stopper degrades the performance when the noise intensity is large enough. Regarding the collision stiffness, the collision gap and noise intensity should be clarified to examine its influence on performance. In general, this numerical study provides an effective strategy, but still less be explored, to enhance the performance of asymmetric BEHs.

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Data availability

Data will be made available on reasonable request.

Abbreviations

C :

Damping matrix

C p :

Capacitive coupling matrix

C p :

Equivalent capacitive, F

c :

Modulus of elasticity, Pa

c 1 :

Equivalent damping, N/(m/s)

c 2 :

Additional damping during the collision, N/(m/s)

D :

Vector of electric displacement

d :

Dimensionless collision distance/gap

d 0 :

Collision distance/gap, m

d x :

Distance between one point on the cantilever to the neutral layer, m

E :

Vector of electric field

E a :

External energy, J

E k :

Kinetic energy, J

E p :

Potential energy, J

e :

Piezoelectric coupling coefficient

F :

Dimensionless nonlinear restoring force

\(\overline{F}\) :

Amplitude of the external mechanical force, N

\(\overline{f}\) :

Applied external force, N

f :

Dimensionless excitation amplitude

G :

Collision force, N

g :

Dimensionless collision force

H :

Hamiltonian function of the undisturbed system

K :

Stiffness matrix

K :

Dimensionless collision stiffness

k :

Equivalent stiffness, N/m

\(k_{1} ,k_{2} ,k_{3}\) :

Coefficients for nonlinear restoring

k s :

Collision stiffness, N/m

l c :

Length scale, m

M :

Mass matrix

m :

Equivalent mass, kg

m tip :

Tip magnets’ mass, kg

N :

Numbers of modes

\(N_{{\overline{f}}}\) :

The numbers of forces applied to the cantilever

N q :

The numbers of charges applied to the cantilever

P :

Dimensionless average power

q :

Charge, C

R :

Load resistance, \(\Omega \)

\({\varvec{r}}(\overline{t})\) :

Total vibration mode function matrix

\(r_{i} (\overline{t})\) :

Coefficient matrix of the ith mode function

S :

Vector of strain

T :

Vector of stress

\(\overline{t}\) :

Time, s

t p :

Thickness of piezoelectric layers, m

t s :

Thickness of the substrate, m

U :

Dimensionless potential function

u :

Deflection vector along z direction

V I :

Variational indicator

V p :

Volume of the piezoelectric layers, m3

V s :

Volume of the substrate, m3

\(\overline{V}\) :

Output voltage, V

v :

Applied voltage, V

x :

Dimensionless displacement

\(\overline{x}\) :

Tip displacement of the cantilever beam, m

y :

\(y = \dot{x}\)

z :

Position along the beam

α :

Ratio between the period of the mechanical system and the time constant of the harvesting circuit

δ :

Variational symbol

\(\hat{\delta }\) :

Dirac-delta function

\(\overline{\beta },\overline{\delta }\) :

Coefficients for dimensionless nonlinear restoring

\(\lambda ,\beta\) :

Weighting coefficients for Rayleigh damping

η :

Gaussian white noise process

\({\varvec{\theta}}\) :

Electromechanical coupling matrix

θ :

Equivalent electromechanical coupling coefficient

\(\omega_{n}\) :

Natural frequency, rad/s

\(\omega\) :

Dimensionless excitation frequency

ρ p :

Material density of the piezoelectric layers, kg/m3

ρ s :

Material density of the substrate, kg/m3

ε :

Dielectric constant

\(\xi_{{1}} ,\xi_{{2}}\) :

Dimensionless damping ratio

κ 2 :

Dimensionless electromechanical coupling coefficient

\(\overline{\Omega }\) :

Excitation frequency

\(\phi_{i} (z)\) :

The ith mode function

\({\varvec{\varPhi}}(z)\) :

Total mode function matrix

φ :

Electric field matrix

\([]_{p}\) :

Subscript for piezoelectric layers

\([]_{s}\) :

Subscript for substrate

\([]_{t}\) :

Matrix transpose

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Funding

This study was supported by National Natural Science Foundation of China (No. 12202400, 52171193), High-level Foreign Expert Introduction Plan of Henan Province (HNGD2023001), Key Research Development and Promotion Project in Henan Province (Grant No. 242102221044, 242102241026,222102320337, 232102240037, 222102240028), Scientific Research Team Plan of Zhengzhou University of Aeronautics (23ZHTD01010), and Engineering Technology Research Center of Henan Province for General Aviation.

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Authors and Affiliations

  1. Engineering Technology Research Center of Henan Province for MEMS Manufacturing and Applications, School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China

    Wei Wang, Jianhui Wang & Ronghan Wei

  2. Institute of Intelligent Sensing, Zhengzhou University, Zhengzhou, 450001, China

    Wei Wang & Ronghan Wei

  3. School of Aero Engine, Zhengzhou University of Aeronautics, Zhengzhou, 450046, China

    Shuangyan Liu

  4. School of Cyber Science and Engineering, Hanwei Institute of Internet of Things, Zhengzhou, 450001, China

    Ronghan Wei

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WW: Conceptualization, Methodology, Investigation, Funding acquisition, Writing–original draft. JW: Investigation, Methodology. SL: Conceptualization, Investigation, Funding acquisition. RW: Writing–review & editing, Funding acquisition.

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Wang, W., Wang, J., Liu, S. et al. Nonlinear dynamics and performance evaluation of an asymmetric bistable energy harvester with unilateral piecewise nonlinearity. Nonlinear Dyn 112, 8043–8069 (2024). https://doi.org/10.1007/s11071-024-09491-1

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