Reconstructing the transient, dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Zhang, Chunlin; Harne, Ryan L.; Li, Bing; Wang, K.W.

doi:10.1016/j.ijnonlinmec.2015.11.002

Cited by 15 publications

(20 citation statements)

References 24 publications

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“…It is well demonstrated that the snap-through and intrawell vibrations of conservative bistable harvesters can be exactly described using the Jacobian elliptic functions of which the argument and modulus are explicitly derived as functions of the displacement amplitude [41]. While in the technique frame of averaging method for the dissipative system with γ , θ > 0, the relations between the argument and modulus of elliptic functions and the vibration amplitude are assumed to satisfy the solution form derived from the conservative case [36] x…”

Section: Averaging the Transient Dynamicsmentioning

confidence: 99%

“…Via substituting the first-and second-order derivatives of the displacement in (8) into the 7, the first order derivative of the amplitude and argument are derived. For the snap-through vibration, dC dt and dψ c dt can accordingly be solved as [36]…”

Section: Averaging the Transient Dynamicsmentioning

confidence: 99%

“…Polynomial fit method is employed to derive explicit and integrable approximations of (13) and (14). The aforementioned limitations of the EA method are resulted from two aspects: (a) fitting f k 2 c and g k 2 d using one integrable polynomial in EA method possesses poor accuracy near the dynamics boundary α βC 2 = α βD 2 = 0.5; and (b) the phase approximation asymptotically increases prior to the boundary α βC 2 = 0.5 which leads to undesired jump phenomenon on the analytically predicted displacement and velocity at the final time in the snap-through regime [36]. To overcome these limitations, a strategy that adopting two-piecewise polynomials approximating f k 2 c and g k 2 d is proposed to derive accurate estimations of the vibration amplitude and the critical time that demarcating the intrawell from the snapthrough oscillation regimes.…”

Section: Analytical Approximation Of the Dissipative Transient Dmentioning

confidence: 99%

“…Since the transient dynamics under successive impulses can be sequentially estimated under each impulsive excitation, the prediction accuracy of single-impulse excited harvester is especially important which determines the initial state under next impulsive excitation. In recent years, the transient dynamics and converted electric power of impulsively excited bistable harvesters was analytically predicted via an enhanced averaging method (EA) with Jacobian elliptic functions [32], [36]. The EA method reconstructs the transient dynamics via separately predicting the responses in the snapthrough and intrawell regimes using different Jacobian elliptic functions, and is validated to be efficient and accurate in predicting the dissipative vibration amplitude and harvested power.…”

Section: Introductionmentioning

confidence: 99%

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Transient Dynamics and Electric Power Estimation of Bistable Energy Harvesters Under Impulsive Excitations

2019

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Bistable vibration energy harvesters are sensitive to impulsive excitations widely existed in environments, and the favorable large-orbit snap-through oscillation possesses high energy conversion efficiency and output voltage. The enhanced averaging method (EA) with Jacobian elliptic functions provides an analytical, efficient tool of guiding the structural design and parameters optimization of bistable harvesters excited in impulsive environments. However, the EA method still suffers some limitations: the predicted transient dynamics undergo undesired sudden jump at the end moment of the snap-through regime, and; phase error occurs during the intrawell regime when large initial velocity is applied. To address these limitations, an improved EA method is proposed in this paper. The transient dynamics in the snap-through and intrawell regimes are separately approximated via two-piecewise expressions, leading to high-fidelity estimation of the vibration amplitude and critical moment dividing the snap-through from intrawell dynamics which occur later in the time. A novel, more accurate estimation of the vibration phase is derived. The prediction performance of the improved EA method is validated via numerous comparisons with the EA method and numerical simulations. It shows that the improved EA method effectively eliminates the aforementioned limitations induced by EA method, providing high-fidelity reconstruction of the transient dynamics of the bistable harvesters excited in impulsive environments. INDEX TERMS Bistable harvester, electric power, impulsive excitation, improved EA method.

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Section: Averaging the Transient Dynamicsmentioning

confidence: 99%

Section: Averaging the Transient Dynamicsmentioning

confidence: 99%

Section: Analytical Approximation Of the Dissipative Transient Dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transient Dynamics and Electric Power Estimation of Bistable Energy Harvesters Under Impulsive Excitations

2019

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“…The traditional quantitative analysis methods for nonlinear system mainly include Lindstedt-Poincaré method [9], multi-scale method [10], average method [11] and KBM method [12]. The common point of these methods is to treat the nonlinear factors as a small perturbation attached to linear system [13].…”

Section: Introductionmentioning

confidence: 99%

Analytical solution for nonlinear vertical vibration model of mill roll system based on improved complexification averaging method

Zhu¹,

Jiang²,

Kong³

et al. 2016

J. vibroeng.

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Rolling mill is the core equipment in modern iron and steel industry, and the reliability of its mill roll system (MRS) is the key to ensure the rolling process with high precision, high speed, continuity and stability. However, the MRS possesses some features such as high nonlinearity, time variability and strong coupling. The vertical vibration easily happens in its working process. Nevertheless, the mathematical model of MRS is difficult to be established and hard to be solved. In this paper, the nonlinear dynamics theory and modern signal processing method were introduced to solve this difficult problem. A two degree of freedom (DOF) nonlinear vertical vibration model of MRS was established. And the model was analytically solved by using complexification averaging (CA) method. The solution error of CA method was thoroughly analyzed. Moreover, the CA method was improved by combining the fast empirical mode decomposition (FEMD) method. Research results indicate that the improved CA method presents a significant advantage in improving the solution precision, and can be used to solve strong nonlinear vibration system (NVS) with two DOF.

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Vibration Energy Harvesting

2017

Harnessing Bistable Structural Dynamics

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We have developed an electromagnetic generator to bury in subcutaneous area or abdominal cavity of the birds. As we can't use a solar battery, it is extremely difficult to supply a power for subcutaneous implantation such as biosensors under the skin due to the darkness environment. We are aiming to test the antigen-antibody reaction to confirm an avian influenza. One solution is a very small generator with the electromagnetic induction coil. We attached the developed coil to chickens and pheasants and recorded the electric potential generated as the chicken walked and the pheasant flew. The electric potential generated with physical simulator is equal to or exceeds the 7 V peakto-peak at maximum by 560/min of flapping of wings. Even if we account for the junction voltage of the diode (200 mV), efficient charging of the double-layer capacitor is possible with the voltage doubler rectifier. If we increase the voltage, other problems arise, including the high-voltage insulation of the double-layer capacitor. For this reason, we believe the power generated to be sufficient for subcutaneous area of birds. The efficiency, magnetic 2 mm in length and coil 15mm in length, if axial direction is rectified, the magnetic flux density given to the coil could calculated to 7.1 % and generated power average 0.47mW. The improvements in size and wire insulation are expected in the future.

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Reconstructing the transient, dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Cited by 15 publications

References 24 publications

Transient Dynamics and Electric Power Estimation of Bistable Energy Harvesters Under Impulsive Excitations

Transient Dynamics and Electric Power Estimation of Bistable Energy Harvesters Under Impulsive Excitations

Analytical solution for nonlinear vertical vibration model of mill roll system based on improved complexification averaging method

Vibration Energy Harvesting

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