Applied Mathematics and Mechanics (English Edition) ›› 2025, Vol. 46 ›› Issue (8): 1451-1474.doi: https://doi.org/10.1007/s10483-025-3285-8
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N. A. SAEED1,2,3, Y. Y. ELLABBAN4, Lei HOU2,†(
), Haiming YI2, Shun ZHONG5, F. Z. DURAIHEM6, O. M. OMARA1
Email Alert
RSSReceived:2025-04-10
Revised:2025-06-23
Published:2025-07-28
Contact:
Lei HOU, E-mail: houlei@hit.edu.cnSupported by:2010 MSC Number:
N. A. SAEED, Y. Y. ELLABBAN, Lei HOU, Haiming YI, Shun ZHONG, F. Z. DURAIHEM, O. M. OMARA. Nonlinear vibration of quasi-zero stiffness structure with piezoelectric harvester and RL-load: intra-well and inter-well oscillation modes under 1:1 internal resonance. Applied Mathematics and Mechanics (English Edition), 2025, 46(8): 1451-1474.
Fig. 1
QZS structure equipped with piezoelectric harvester and inductive-resistive load (color online)"
Fig. 2
(a) Bistable and monostable boundaries of considered quasi-zero mechanical oscillator in βγ-space. (b) Equilibrium point bifurcation of autonomous unperturbed oscillator. Phase space scenarios of autonomous unperturbed oscillator at different values of β when γ=1.0: (c) bistable oscillation scenario; (d) critical oscillation scenario; (e) monostable oscillation scenario (color online)"
Fig. 3
Monostable isolator response curves for base excitation amplitude F=0.015: (a) mass oscillation amplitude A versus Ω; (b) displacement transmissibility T versus Ω (color online)"
Fig. 4
Time-response curves of isolated mass Y(τ) and base excitation Z(τ)=0.015 cos(Ωτ) for F=0.015 corresponding to Fig. 3: (a) Ω=0.5; (b) Ω=2.0 (color online)"
Fig. 5
Response curves of monostable isolator: (a) A versus Ω and (b) T versus Ω when F=0.010, 0.015, 0.025; (c) A versus F and (d) T versus F when Ω=0.25, 0.75, 1.50; (e) vibration isolation region in ΩF-space (color online)"
Fig. 6
Influence of β and γ and dynamical behaviors of QZS system for F=0.015: (a) oscillation amplitude and (b) corresponding displacement transmissibility for γ=1.0; (c) vibration isolation region in Ωβ-space for γ=1.0; (d) oscillation amplitude and (e) corresponding displacement transmissibility for β=0.8; (f) vibration isolation region in Ωγ-space for β=0.8 (color online)"
Fig. 7
Response curves of monostable isolator for base excitation amplitude F=0.015 when coupled to resonant piezoelectric energy harvester: (a) mass oscillation amplitude A versus Ω; (b) harvested voltage amplitude B versus Ω; (c) displacement transmissibility T versus Ω (color online)"
Fig. 8
Time-response curves of isolator-harvester system under 1:1 internal resonance when Z(τ)=0.015 cos(Ωτ) corresponding to Fig. 7: (a)–(b) Ω=0.75; (c)–(d) Ω=2.00 (color online)"
Fig. 9
Response curves of isolator-harvester system for different base excitation amplitudes: (a) mass oscillation amplitude A versus Ω; (b) harvested voltage amplitude B versus Ω; (c) displacement transmissibility T versus Ω; (d) stability chart in FΩ-space (color online)"
Fig. 10
Vibration amplitudes of coupled system for strong base excitation F=0.025: (a) A versus Ω and (b) B versus Ω when β=0.7, 0.8, 0.9 for γ=1.0; (c) A versus Ω and (d) B versus Ω when γ=0.6, 1.2, 1.8 for β=0.8 (color online)"
Fig. 11
Vibration amplitudes of coupled system for strong base excitation F=0.025 when β=0.9 and γ=0.6 at different coupling parameters: (a) A versus Ω and (b) B versus Ω when δ=0.05, 0.15, 0.30 for σ=0.2; (c) A versus Ω and (d) B versus Ω when σ=0.05, 0.15, 0.25 for δ=0.2 (color online)"
Fig. 12
Vibration amplitudes of coupled system when β=0.9, γ=0.6, δ=0.01, σ=0.2, μ1=0.001, and μ2=0.01: (a) A versus F and (b) B versus F for Ω=0.75; (c) A versus F and (d) B versus F for Ω=1.00; (e) bifurcation diagram for Ω=1.00 (color online)"
Fig. 13
Response curves of bistable oscillator under base excitation amplitude F=0.015 when coupled to resonant piezoelectric energy harvester: (a) oscillation amplitude A versus Ω; (b) harvested voltage amplitude B versus Ω (color online)"
Fig. 14
Basins of attraction for intra-well and inter-well oscillations in bistable oscillator-harvester system, obtained based on Fig. 13 when F=0.015 for three different driving frequencies: (a) Ω=1.0; (b) Ω=1.5; (c) Ω=2.0 (color online)"
Fig. 15
Frequency-response curves of bistable oscillator under different values of base excitation amplitude F when coupled to resonant piezoelectric energy harvester: (a)–(b) F=0.010; (c)–(d) F=0.025 (color online)"
Fig. 16
Force-response curves of bistable oscillator-harvester system for base excitation frequency Ω=0.5: (a) oscillation amplitude A versus F; (b) harvested voltage amplitude B versus F (color online)"
Fig. 17
Time-response curves of oscillator-harvester system corresponding to Fig. 16 for F=0.025 and Ω=0.5: (a)–(b) intra-well and inter-well oscillation modes of mechanical oscillator, depending on initial conditions; (c)–(d) corresponding low- and high-amplitude harvested voltage (color online)"
Fig. 18
Steady-sate phase plane trajectories of oscillator-harvester system corresponding to Fig. 16 for Ω=0.5: (a) inter-well oscillation mode of mechanical oscillator using zero initial conditions; (b) corresponding high-amplitude harvested voltage (color online)"
Fig. 19
Force-response curves of bistable oscillator-harvester system for base excitation frequency Ω=1.0: (a) oscillation amplitude A versus F; (b) harvested voltage amplitude B versus F (color online)"
Fig. 20
Steady-sate phase plane trajectories of oscillator-harvester system corresponding to Fig. 19 for Ω=1.0: (a) intra-well oscillation mode of mechanical oscillator using zero initial conditions; (b) corresponding low-amplitude harvested voltage (color online)"
Fig. 21
Force-response curves of bistable oscillator-harvester system for base excitation frequency Ω=1.25: (a) oscillation amplitude A versus F; (b) harvested voltage amplitude B versus F (color online)"
Fig. 22
Bifurcation diagrams and steady-sate phase plane trajectories of oscillator-harvester system corresponding to Fig. 21 for Ω=1.25: (a) bifurcation diagram; (b)–(c) chaotic responses of oscillator-harvester system for Ω=1.25. Legends of (b) and (c) are as follows. Red lines: F=0.050; green lines: F=0.075; blue lines: F=0.100; purple lines: F=0.125; black lines: F=0.150 (color online)"
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