Abstract This paper presents a design study of a miniaturized broadband nonlinear vibration energy harvester (VEH) with piecewise-linear restoring force based on a mechanically-sprung resonator with stoppers. It is commonly recognized that a VEH based on a nonlinearly-sprung resonator can show broadband frequency characteristics while keeping its maximum power performance due to its bent resonance peak. The resonator to be investigated in this study consists of a magnet composite as a mass moving through an induction coil, two planar springs, and mechanical stoppers. The magnet composite is comprised of two repelling cylindrical magnets and a steel disk between them, all encapsulated in a thin stainless-steel cylinder. The planar springs with spiral-like shape are respectively connected to the both ends of the magnet composite so that they provide soft linear stiffness in a compact size. The mechanical stoppers installed to constrain the deformation of the spring give the resonator piecewise-linear hardening characteristics which effectively broaden the resonance band. In this study, the prototype VEH developed in the previous study is presented, and the gaps between the springs and stoppers are adjusted so that the resultant piecewise-linear restoring force shows symmetric or asymmetric property with respect to the equilibrium point. Experimental studies and analyses are carried out to examine the performance of the presented VEH in terms of the frequency response. The comparison of three different configurations of the stopper illustrates how the asymmetry in the bilinear restoring force affects the shape of the resonance peak. It is also suggested that the asymmetry may help the VEH operate in broader band by exploiting its ability of tailoring the resonance characteristics, which still needs further investigation.

In this study, a dual-beam piezoelectric energy harvester is proposed. This harvester consists of a main beam and an auxiliary beam with a pair of magnets attached to couple their motions. The potential energy of the system is modeled to understand the influence of the potential wells on the dynamics of the harvester. It is noted that the alignment of the magnets significantly influences the potential wells. A theoretical model of the harvester is developed based on the Euler-Bernoulli beam theory. Frequency sweeps are conducted experimentally and numerically to study the dynamics of the harvester. It is shown that the dual-beam harvester can exhibit hardening effect with different configurations of magnet alignments in frequency sweeps. The performance of the harvester can be improved with proper placement of the magnets.

We explore the modeling and analysis of nonlinear non-conservative dynamics of macro-fiber composite (MFC) piezo-electric structures, guided by rigorous experiments, for resonant vibration-based energy harvesting, as well as other applications leveraging the direct piezoelectric effect, such as resonant sensing. The MFCs employ piezoelectric fibers of rectangular cross section embedded in kapton with interdigitated electrodes to exploit the 33-mode of piezoelectricity. Existing frameworks for resonant nonlinearities have so far considered conventional piezoceramics that use the 31-mode of piezoelectricity. In the present work, we develop a framework to represent and predict nonlinear electroelastic dynamics of MFC bimorph cantilevers under resonant base excitation. The interdigitated electrodes are shunted to a set of resistive electrical loads to quantify the electrical power output. Experiments are conducted on a set of MFC bimorphs over a broad range of mechanical excitation levels to identify the types of nonlinearities present and to compare the model predictions and experiments. The experimentally observed interaction of material softening and geometric hardening effects, as well as dissipative effects, is captured and demonstrated by the model.

Black box design is a constraint driven design approach that distills essential elements of a physical process into inputs and outputs. This paper details the black box design implementation and validation of shape memory alloy (SMA) coil actuators as active members in a Watt I six bar avian-inspired wearable morphing angel wing mechanism. SMA coil actuators leverage the unique characteristics of high energy density SMA wire by providing a compact structural platform for large actuation displacement applications. The moderate force and displacement performance of low spring index coil actuators paired with their virtually silent actuation performance made them an attractive actuator solution to an avian-inspired wearable morphing wing mechanism for the University of Minnesota Department of Theatre Arts and Dance production of ‘ Marisol ’. The wing design constraints (extended span of 7.5 ft, a closed span of 3 ft) required a tailorable actuator system with capacity to perform at particular target force and strain metrics cyclically. A low spring index parameter study was conducted to facilitate an accelerated phase of design prototyping. The parameter study featured six SMA coil actuator prototypes made with 0.012” diameter Dynalloy Flexinol ® wire of varying spring indexes (C = 2.5–4.9). The coil actuators were manufactured through a CNC winding process, shape set in a furnace at 450 °C for 10 minutes, and water quenched for hardening. A series of thermomechanical actuation tests were conducted to experimentally characterize the low spring index actuation performances. The coil actuation characterizations demonstrated increased force and decreased actuator displacement corresponding to decreased spring indexes. Scaling these results aided an accelerated design of an actuator system. The actuator system consisted of four C = 3.05 coil actuators wound with 0.02” diameter SMA that were integrated into each Watt I mechanism. The characterization of the force-displacement profiles for low index SMA coil actuators provides an effective empirical design strategy for scaling actuator performance to mechanical systems requiring moderate force, moderate displacement actuators.

Origami-inspired mechanical metamaterials could exhibit extraordinary properties that originate almost exclusively from the intrinsic geometry of the constituent folds. While most of current state of the art efforts have focused on the origami’s static and quasi-static scenarios, this research explores the dynamic characteristics of degree-4 vertex (4-vertex) origami folding. Here we characterize the mechanics and dynamics of two 4-vertex origami structures, one is a stacked Miura-ori (SMO) structure with structural bistability, and the other is a stacked single-collinear origami (SSCO) structure with locking-induced stiffness jump; they are the constituent units of the corresponding origami metamaterials. In this research, we theoretically model and numerically analyze their dynamic responses under harmonic base excitations. For the SMO structure, we use a third-order polynomial to approximate the bistable stiffness profile, and numerical simulations reveal rich phenomena including small-amplitude intrawell, large-amplitude interwell, and chaotic oscillations. Spectrum analyses reveal that the quadratic and cubic nonlinearities dominate the intrawell oscillations and interwell oscillations, respectively. For the SSCO structure, we use a piecewise constant function to describe the stiffness jump, which gives rise to a frequency-amplitude response with hardening nonlinearity characteristics. Mainly two types of oscillations are observed, one with small amplitude that coincides with the linear scenario because locking is not triggered, and the other with large amplitude and significant nonlinear characteristics. The method of averaging is adopted to analytically predict the piecewise stiffness dynamics. Overall, this research bridges the gap between the origami quasi-static mechanics and origami folding dynamics, and paves the way for further dynamic applications of origami-based structures and metamaterials.

Macro-fiber composite (MFC) piezoelectric materials are used in a variety of applications employing the converse piezo-electric effect, ranging from bioinspired actuation to vibration control. Most of the existing literature to date considered linear material behavior for geometrically linear oscillations. However, in many applications, such as bioinspired locomotion using MFCs, material and geometric nonlinearities are pronounced and linear models fail to represent and predict the governing dynamics. The predominant types of nonlinearities manifested in resonant actuation of MFC cantilevers are piezoelectric softening, geometric hardening, inertial softening, as well as internal and external dissipative effects. In the present work, we explore nonlinear actuation of MFC cantilevers and develop a mathematical framework for modeling and analysis. An in vacuo actuation scenario is considered for a broad range of voltage actuation levels to accurately identify the sources of dissipation. Several experiments are conducted for an MFC bimorph cantilever, and model simulations are compared with nonlinear experimental frequency response functions under resonant actuation. The resulting experimentally validated framework can be used for simulating the dynamics of MFCs under resonant actuation, as well as parameter identification and structural optimization for nonlinear operation regime.

Based on a recently developed shakedown theory for non-smooth nonlinear materials, we derive a criterion for high-cycle fatigue in shape memory alloys (SMAs). The fatigue criterion takes into account phase transformation as well as reorientation of martensite variants as the source of fatigue damage. The mathematical derivation of the criterion is based on the requirement of elastic shakedown for a given structure to achieve unlimited fatigue endurance. Elastic shakedown is defined as an asymptotic state in which damage due to time-varying load becomes confined at the mesoscopic scale, or the scale of the grain, with no discernable inelasticity at the macroscopic scale. From an energy standpoint, elastic shakedown corresponds to a situation where energy dissipation becomes bounded and the response elastic after a certain number of loading cycles. A sufficient condition to achieve this state was established by Melan (1936) [1] and Koiter (1960) [2] for elastoplastic materials and later generalized to hardening plasticity by Nguyen (2003) and to non-smooth non-linear materials by Peigney (2014). The latter formulation is applicable to SMAs obeying the ZM constitutive model (Zaki & Moumni, 2007) and is shown here to allow the derivation of a high-cycle fatigue criterion analogous to the one proposed by Dang Van (1973) for elastoplastic materials. The criterion allows establishing a safe domain in stress deviator space at the mesoscopic scale consisting of a hypercylinder with axis parallel to the direction of martensite orientation. The hypercylinder is delimited along its axis by two transverse hyperplanes representing bounds on admissible stress states consistent with the loading conditions for phase transformation. Safety with regard to high-cycle fatigue, upon elastic shakedown, is conditioned by the persistence of the macroscopic stress path, as the load varies and at every material point, strictly within the hypercylinder. The size of the hypercylinder is shown to strongly depend on the relative amount of martensite present in the SMA.

The paper presents a numerical implementation of the Large Time Increment (LaTIn) method for the integration of the ZM model [1] for SMAs in the pseudoelastic range. LaTIn was initially proposed as an alternative to the conventional incremental approach for the integration of nonlinear constitutive models [2]. It is adapted here for the simulation of pseudoelastic SMA behavior and is shown to be especially useful in situations where the phase transformation process presents little to no hardening. In these situations, a slight stress variation during a load increment can result in large variation of the volume fraction of martensite within a representative volume element of the SMA. This can lead to difficulty in numerical convergence if the incremental method is used. LaTIn involves two stages: in the first stage a solution satisfying the conditions of static equilibrium is obtained for each load increment without considering the consistency with the phase transformation conditions, then in the second stage consistent increments of the local state variables are determined for the entire loading path. The two stages take place sequentially, in contrast to the incremental method that requires satisfying the global equilibrium and local consistency conditions simultaneously at a given load increment before proceeding to the next. The numerical integration algorithm consists of the following steps: 1. Division of the loading path into a finite number of increments, 2. Solution for all the load increments of the static equilibrium problem in which the local consistency conditions are relaxed, 3. Update of the state variables in accordance with the consistency conditions for all the load increments. Steps 2 and 3 are repeated until a solution is reached that satisfies simultaneously the equilibrium and consistency requirements. An algorithm is presented for the implicit integration of the time-discrete equations. The algorithm is used for finite element simulations using Abaqus, in which the model is implemented by means of a user material subroutine. The simulation results are discussed in comparison with those obtained using conventional step-by-step incremental integration.

The paper presents a new constitutive model for iron-based shape memory alloys (Fe-SMAs) adapted from the ZM model initially proposed for Nitinol by Zaki and Moumni [JMPS2007]. The model introduces nonlinear hardening terms to account for interactions between the grains, martensite variants and slip systems that may exist within a volume element of the material. The expressions used for the hardening terms are similar to those in (Khalil et al. [JIMSS2012]). The equations of the model are derived from the expression of a Helmholtz free energy potential, with complementary loading conditions obtained within the framework of generalized standard materials with internal constraints. A detailed derivation of the implicit algorithm used for the integration of the model is provided and used for numerical simulations that are shown to agree with experimental data.

Impedance-based structural health monitoring (SHM) is a non-destructive, active technique for real-time structural damage assessment. Conventional impedance-based SHM practices apply a sinusoidal signal of fixed amplitude to excite the piezoceramic patch and obtain the impedance signature over a certain frequency range. Damage is then detected by comparing the measured impedance signature to a baseline measurement taken at the pristine state. In this work, the amplitude of the driving signal, which is directly related to the magnitude of the excitation force acting on the structure, is introduced as an additional variable, and sweeps over both frequencies and amplitudes are performed. Several structural defects, such as cracks and loose joints, are nonlinear in nature. Therefore, changing the excitation force will allow the detection of such damage induced nonlinearities and track their evolution. Numerical simulations are carried out to study the effects of nonlinearities on the impedance signature using a single mode model. Several types of structural nonlinearities, such as hardening, softening, and nonlinear damping are studied with the assumption that the piezoelectric actuator stays in its linear regime. Experiments are conducted on a single beam and a lap joint, and impedance signatures in the range of 12–15 KHz are measured at different levels of excitation. Nonlinear damping and softening behavior are detected experimentally by examining the measured impedance signatures. Numerical and experimental findings suggest the possibility of detecting and tracking structural nonlinearities using impedance measurements.

The use of nonlinear dynamic phenomena for frequency bandwidth enhancement in vibration-based energy harvesting has received growing attention over the last few years. Various designs have been studied to create Duffing oscillators usually by introducing magnetoelastic coupling. In such devices, magnetic forces are typically coupled with elastic beams involving ferromagnetic components to achieve a nonlinear restoring force of the monostable or bistable Duffing type. Other than the increased volume and structural complexity due to additional magnets and discrete components, these magnetoelastic piezoelectric energy harvesters are not suitable to use in various compact applications and in systems that are sensitive to magnetic fields. The M-shaped structural configuration studied in this work overcomes these issues due to the asymmetric stiffness nonlinearity created by a simple structural configuration composed of a bent spring steel beam with piezoelectric patches. The electroelastic dynamics of the M-shaped broadband piezoelectric energy harvester is governed by various interacting nonlinearities, such as the deliberately introduced stiffness nonlinearity of hardening type resulting from the substrate geometry, inherent elastic nonlinearities of softening type as well as hysteretic losses associated with piezoelectric patches, and other dissipative effects due to large velocities experienced in response to base excitation. A recently developed nonlinear non-conservative electroelastic modeling framework for the piezoelectric patches is combined with geometric nonlinearities of the M-shaped energy harvester to establish a nonlinear dissipative model of the electromechanically coupled system. Energy harvesting experiments for a set of resistors and base excitation levels are then performed to experimentally characterize the bandwidth enhancement using the M-shaped broadband piezoelectric energy harvester.

This ppppaper examines the performance of a galloping energy harvester possessing a nonlinear restoring force. To achieve this goal, a flow energy harvester consisting of a piezoelectric cantilever beam augmented with a square-sectioned bluff body at the free end is considered. Two magnets located near the tip of the bluff body are used to introduce the nonlinearity which strength and nature can be altered by changing the distance between the magnets. A lumped-parameter aero-electromechanical model adopting the quasi-steady assumption for aerodynamic loading is presented and utilized to numerically simulate the harvester’s response. Wind tunnel tests are also performed to validate the numerical simulations by conducting upward and downward wind velocity sweeps. Results comparing the relative performance of several harvesters with potential functions of different shapes demonstrate that a mono-stable potential function with a hardening restoring force can outperform all other configurations.

Harvesting energy from high-shock pulse events such as those experienced during high velocity impact and firing setback acceleration in munitions using piezoelectric-based harvesters and converting it to usable electrical energy is not new. However, efficient collection of the generated charges and its transfer to an electrical storage device such as a capacitor for pulse events with very short duration, such as those that last a small fraction of a millisecond such as 50–100 microseconds is challenging. Another challenge for such piezoelectric-based harvesters is their hardening to survive high levels of shock loading, sometimes in excess of 100–200,000 Gs. In this paper, the basic design and operation of two such piezoelectric-based energy harvesters and their charge collection and storage electronics are described. The novel methods described are shown to achieve this goal with very high efficiency. The paper also describes the development of test-beds to simulate electrical charge generation of the energy harvesting power sources during high velocity impact and munitions firing for use in the design and evaluation of the developed charge collection and storage electronics.

Analytical solutions are derived for the pseudoelastic response of a shape memory alloy (SMA) thick-walled cylinders subjected to internal pressure. A Tresca transformation criterion and linear hardening are used. Equations are given for the stress, total strain, transformation strain, and radial displacement.

Transforming aeroelastic vibrations into electricity for low-power generation has received growing attention over the past couple of years. The goal is to convert wind energy into electricity for powering small electronic components employed in wireless applications such as structural health monitoring. The potential applications of interest for aeroelastic energy harvesting range from lifting components in aircraft structures to several other engineering problems involving wireless electronic components located in high wind areas. This paper investigates linear and nonlinear aeroelastic energy harvesting using electromagnetic induction. A two-dimensional airfoil with plunge and pitch degrees of freedom (DOF) is considered. The electromagnetic induction is introduced to the plunge DOF by means of a coil-magnet combination and the nonlinearities are introduced through the pitch DOF. The governing dimensionless aeroelastic equations are given with electromagnetic coupling and a resistive load in the electrical domain. The effects of several dimensionless system parameters (electromechanical coupling, load resistance, and coil inductance) on the dimensionless electrical power as well as the dimensionless linear flutter speed are investigated. After considering the linear problem, combined nonlinearities are investigated to improve the electrical output. A cubic stiffness of the hardening type is combined with the free play nonlinearity to make the resulting nonlinear oscillations bounded with acceptable amplitude over a wide range of airflow speeds. The results and the dimensionless simulations presented in this work can be employed for designing and optimizing scalable aeroelastic energy harvesters for wind energy harvesting using electromagnetic induction.

Kiefer and Lagoudas proposed a thermodynamic model for predicting the magneto-mechanical behavior of magnetic shape memory alloys (MSMAs) and then confirmed their model experimentally [1]. The model was calibrated by placing the test specimen under a constant magnetic field and a varying compressive stress. Later, Feigenbaum and Ciocanel [2] used the model to predict behavior under a constant compressive stress and a varying magnetic field. Because the two experiments were done by different researchers on different specimens, the calibration gave different values for material paremeters. In this work, through experimental results from tests performed on the same specimen by the same researchers, the Kiefer and Lagoudas model, with any hardening function, will be shown to be unable to be calibrated so as to accurately predict the magneto-mechanical behavior of a specimen under both types of loading conditions.