Energy dissipation prediction in a line of colliding oscillators

In this paper the impact of a line of adjacent structures, or oscillators, is studied using an energy formulation. The energy exchange and dissipation from a collision of a pair of oscillators is studied by creating an equivalent oscillator pair, one has the energy of the in-phase motion and the other has the out-of-phase energy. It is found that the energy exchange between colliding oscillators is proportional to the initial kinetic energy difference of the oscillators and that work in the collision is proportional to the out-of-phase energy or difference energy. The kinetic energy at contact is then related to the mean oscillator energy, permitting a power balance equation to be written for each oscillator in line. The power balance equations have three independent variables for each pair of oscillators: the oscillator time averaged energies and the phase difference. This equation is run in a time-stepping procedure, with steps at the mean collision rate. The work in the collisions and internal oscillator dissipation is output as a function of time. A parameter study is conducted to see how the work changes with oscillator: separation, contact stiffness and contact damping.     

Sustained high-frequency energy harvesting through a strongly nonlinear electromechanical system under single and repeated impulsive excitations

This work investigates a vibration-based energy harvesting system composed of two oscillators coupled with essential (nonlinearizable) stiffness nonlinearity and subject to impulsive loading of the mechanical component. The oscillators in the system consist of one grounded, weakly damped linear oscillator mass (primary system), which is coupled to a second light-weight, weakly damped oscillating mass attachment (the harvesting element) through a piezoelastic cable. Due to geometric/kinematic mechanical effects the piezoelastic cable generates a nonlinearizable cubic stiffness nonlinearity, whereas electromechanical coupling simply sees a resistive load. Under single and repeated impulsive inputs the transient damped dynamics of this system exhibit transient resonance captures (TRCs) causing high-frequency ‘bursts’ or instabilities in the response of the harvesting element. In turn, these high-frequency dynamic instabilities result in strong and sustained energy transfers from the directly excited primary system to the lightweight harvester, which, through the piezoelastic element, are harvested by the electrical component of the system or, in the present case, dissipated across a resistive element in the circuit. The primary goal of this work is to demonstrate the efficacy of employing this type of high-frequency dynamic instability to achieve enhanced nonlinear vibration energy harvesting under impulsive excitations.     

M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation

Over the past few years, nonlinear oscillators have been given growing attention due to their ability to enhance the performance of energy harvesting devices by increasing the frequency bandwidth. Duffing oscillators are a type of nonlinear oscillator characterized by a symmetric hardening or softening cubic restoring force. In order to realize the cubic nonlinearity in a cantilever at reasonable excitation levels, often an external magnetic field or mechanical load is imposed, since the inherent geometric nonlinearity would otherwise require impractically high excitation levels to be pronounced. As an alternative to magnetoelastic structures and other complex forms of symmetric Duffing oscillators, an M-shaped nonlinear bent beam with clamped end conditions is presented and investigated for bandwidth enhancement under base excitation. The proposed M-shaped oscillator made of spring steel is very easy to fabricate as it does not require extra discrete components to assemble, and furthermore, its asymmetric nonlinear behavior can be pronounced yielding broadband behavior under low excitation levels. For a prototype configuration, linear and nonlinear system parameters extracted from experiments are used to develop a lumped-parameter mathematical model. Quadratic damping is included in the model to account for nonlinear dissipative effects. A multi-term harmonic balance solution is obtained to study the effects of higher harmonics and a constant term. A single-term closed-form frequency response equation is also extracted and compared with the multi-term harmonic balance solution. It is observed that the single-term solution overestimates the frequency of upper saddle-node bifurcation point and underestimates the response magnitude in the large response branch. Multi-term solutions can be as accurate as time-domain solutions, with the advantage of significantly reduced computation time. Overall, substantial bandwidth enhancement with increasing base excitation is validated experimentally, analytically, and numerically. As compared to the 3 dB bandwidth of the corresponding linear system with the same linear damping ratio, the M-shaped oscillator offers 3200, 5600, and 8900 percent bandwidth enhancement at the root-mean-square base excitation levels of 0.03g, 0.05g, and 0.07g, respectively. The M-shaped configuration can easily be exploited in piezoelectric and electromagnetic energy harvesting as well as their hybrid combinations due to the existence of both large strain and kinetic energy regions. A demonstrative case study is given for electromagnetic energy harvesting, revealing the importance of higher harmonics and the need for multi-term harmonic balance analysis for predicting the electrical power output accurately.     

Nonlinear vibration control and energy harvesting of a beam using a nonlinear energy sink and a piezoelectric device

This paper presents an optimal design for a system comprising a nonlinear energy sink (NES) and a piezoelectric-based vibration energy harvester attached to a free–free beam under shock excitation. The energy harvester is used for scavenging vibration energy dissipated by the NES. Grounded and ungrounded configurations are examined and the systems parameters are optimized globally to both maximize the dissipated energy by the NES and increase the harvested energy by piezoelectric element. A satisfactory amount of energy has been harvested as electric power in both configurations. The realization of nonlinear vibration control through one-way irreversible nonlinear energy pumping and optimizing the system parameters result in acquiring up to 78 percent dissipation of the grounded system energy.     

Improvement of efficiency of piezoelectric element attached to beam based on mechanical impedance matching

This paper describes new methods that improve the efficiency of a piezoelectric element attached to a beam based on mechanical impedance matching. Piezoelectric elements are often used to suppress bending vibration. They are also used as sensors or energy-harvesting sources. In such cases, the piezoelectric element is usually bonded onto the host structure by an adhesive bond. The efficiency of the piezoelectric element depends on the bonding location. When the efficiency is insufficient despite a good location, the size or number of piezoelectric elements is increased. However, the efficiency of the piezoelectric element is usually insufficient even if these methods are applied. In order to enhance the efficiency of the piezoelectric elements without using active methods, this paper proposes a mechanical impedance matching method that uses spacers or tuning for the size of the piezoelectric element. Because the attached piezoelectric element and host structure in this region behave as springs in parallel to the bending deformation, the stored strain energy in the piezoelectric element is maximized under the condition that their spring constants match. The proposed methods were theoretically investigated with consideration for the effects of the bonding layer, spacers, and host structure. The optimum conditions for the proposed methods were theoretically formulated, and the effectiveness of the proposed methods and theoretical analysis was verified through simulations and experiments.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

Harmonic balance analysis of the bistable piezoelectric inertial generator

This paper applies the method of Harmonic Balance to analytically predict the existence, stability, and influence of parameter variations on the intrawell and interwell oscillations of bistable piezoelectric inertial generator. Existing work on the bistable piezoelectric harvester in the presence of varying harmonic environmental loading has been relegated to simulation and experimental analyses. Furthermore, linear piezoelectric behavior and linear damping has always been presumed. This paper improves upon an existing model for the bistable piezoelectric harvester by incorporating nonlinear dissipation and cubic softening influences in the electroelastic laminates before applying analytical methods. A framework for theoretically predicting empirical observations, such as optimal impedance loads for steady-state motions, is provided as well as other dynamic considerations such as potential well escape phenomena.     

Approximations for motion of the oscillators with a non-negative real-power restoring force

Oscillators with a non-negative real-power restoring force are considered in this paper. This type of restoring force is related to systems with a quasi-zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature. Examples of these types of restoring force are grounded in real physical and engineering systems. Periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and Hamilton's variational principle. Then, the approach is extended to non-conservative oscillators by adjusting the elliptic Krylov-Bogoliubov method. The methods proposed for the conservative and non-conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force. Several examples, the majority of which are so far unsolved, are given to illustrate the methods proposed and to demonstrate their generality, which permits unforeseen solutions for motion, containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force. The results obtained are compared with numerical results and have excellent accuracy.     

Asymmetries on a three-coupled-oscillator system: Their use in a gravitational-radiation detector

Summary A system composed of three coupled oscillators having the same resonant frequency is analysed. This system, which is the representation of the real physical system composed of a resonant gravitational-wave antenna equipped with a capacitive transducer and connected to a superconducting current amplifier, presents a series of asymmetries in the normal modes. An analytical evaluation of the resonant frequencies of the three normal modes of this system is given. These three normal modes are not equivalent, and their behaviour is the object of the present paper. The quality factors of the normal modes are calculated, showing that the central-mode quality factor is very weakly affected by the lowest one of the three oscillators, namely the electricalQ. A second asymmetry is then found by calculating the spread of a pulse of energy deposited into the first oscillator: it is shown that most of this energy is detected in the central mode. These two asymmetries permit to increase the sensitivity of the system calculated for the central mode. A third asymmetry is then analysed concerning the different contribution of each oscillator to the thermal noise of each mode. Supposing, as it is in the real case, that the three oscillators are in equilibrium, but at different thermodynamic temperatures, an equivalent thermodynamic temperature is calculated for each mode. This last feature can then be used to lower the equivalent temperature of the central mode without the need of cooling all the three oscillators at the same low temperature.     

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.     

Hamiltonian systems with indefinite kinetic energy

Some interesting features of a class of two-dimensional Hamiltonians with indefinite kinetic energy are considered. It is shown that such Hamiltonians cannot be reduced, in general, to an equivalent dynamical Hamiltonian with positive definite kinetic energy quadratic in velocities. Complex nonlinear evolution equations like the K-dV, the MK-dV and the sine-Gordon equations possess such Hamiltonians. The case of complex K-dV equation has been considered in detail to demonstrate the generic features. The two-dimensional real systems obtained by analytic continuation to complex plane of one-dimensional dynamical systems are also discussed. The evolution equations for nonlinear, amplitude-modulated Langmuir waves as well as circularly polarized electromagnetic waves in plasmas, are considered as illustrative examples.     

Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems

A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude.     

Enhancing energy harvesting by coupling monostable oscillators

The performance of a ring of linearly coupled, monostable nonlinear oscillators is optimized towards its goal of acting as energy harvester – through piezoelectric transduction – of mesoscopic fluctuations, which are modeled as Ornstein-Uhlenbeck noises. For a single oscillator, the maximum output voltage and overall efficiency are attained for a soft piecewise-linear potential (providing a weak attractive constant force) but they are still fairly large for a harmonic potential. When several harmonic springs are linearly and bidirectionally coupled to form a ring, it is found that counter-phase coupling can largely improve the performance while in-phase coupling worsens it. Moreover, it turns out that few (two or three) coupled units perform better than more.     

Nonlinear dynamics of the heartbeat: I. The AV junction: Passive conduit or active oscillator?

Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V₀) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3:2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1:1 (SA:AV) phase-locking of the oscillators occurs over a critical range of V₀. Externally driving the SA oscillator at increasing rates results in 3:2 AV Wenckebach periodicity and a 2:1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1:1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.     

Strong coupling of nonlinear electronic and biological oscillators: reaching the "amplitude death" regime

Interaction between an electronic and a biological circuit has been investigated for a pair of electrically connected nonlinear oscillators, with a spontaneously oscillating olivary neuron as the single-cell biological element. By varying the coupling strength between the oscillators, we observe a range of behaviors predicted by model calculations, including a reversible low-energy dissipation "amplitude death" where the oscillations in the coupled system cease entirely.     

Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian's method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator's drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.     

A generic double-curvature piezoelectric shell energy harvester: Linear/nonlinear theory and applications

Converting vibration energy to useful electric energy has attracted much attention in recent years. Based on the electromechanical coupling of piezoelectricity, distributed piezoelectric zero-curvature type (e.g., beams and plates) energy harvesters have been proposed and evaluated. The objective of this study is to develop a generic linear and nonlinear piezoelectric shell energy harvesting theory based on a double-curvature shell. The generic piezoelectric shell energy harvester consists of an elastic double-curvature shell and piezoelectric patches laminated on its surface(s). With a current model in the closed-circuit condition, output voltages and energies across a resistive load are evaluated when the shell is subjected to harmonic excitations. Steady-state voltage and power outputs across the resistive load are calculated at resonance for each shell mode. The piezoelectric shell energy harvesting mechanism can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and non-shell (beam, plate, ring, arch, etc.) distributed harvesters using two Lamé parameters and two curvature radii of the selected harvester geometry. To demonstrate the utility and simplification procedures, the generic linear/nonlinear shell energy harvester mechanism is simplified to three specific structures, i.e., a cantilever beam case, a circular ring case and a conical shell case. Results show the versatility of the generic linear/nonlinear shell energy harvesting mechanism and the validity of the simplification procedures.     

Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis methods we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.     

Factorization Method for a Class of Quantum Nonlinear Harmonic Oscillators

In this work, we study a more general class of quantum nonlinear harmonic oscillators by the factorization method. The energy spectrum and the associated ground state are also determined.