Harmonic Base Excitation Research Articles

Seismic retrofit of reinforced concrete structures using magnetorheological mass driver: Evolving a design methodology

This paper proposes a design methodology for a damping enhancement-based seismic retrofitting of a damaged reinforced concrete (RC) model frame with a mass driver (MD) and magnetorheological (MR) damper. The MR-MD is applied with preset constant current inputs and the structure + MR-MD system is subjected to harmonic and seismic base excitations. Equivalent effective damping for a biviscous MR damper is derived both from energy and secant damping considerations. The effect of yielding force of the MR damper, mass value of the secondary system, and non-linearity associated with input accelerations are discussed. The paper proposes a suboptimum frequency tuning, that is, the frequency of the secondary MR-MD is marginally lower than the optimum value, anticipating a degrading stiffness and frequency drop of the target structure, typical of materials like masonry and concrete. Experimental investigations with undertuned, overtuned, and near-optimum tuned conditions of the secondary MR-MD system are performed to highlight that operation with near optimum damping plays a significant role, particularly when MR dampers are used. Higher effective damping associated with higher preset current inputs have a tendency to be counter-productive, which is proved experimentally and analytically. Four seismic performance parameters are calculated, which showed a substantial reduction of acceleration and displacement responses on the experimental structure when the damper is operated at lower input currents. From analytical and experimental investigations, a simplified retrofit design methodology for a degraded RC superstructure is proposed and experimentally validated through incorporating a mass driver coupled with a non-linear damper.

Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications

Energy harvesting at micro- and nanoscales has recently seen a renewed interest that the flexoelectric effect can counter the inability of piezoelectric energy harvesters to generate enough energy at small scales. Almost all small-scale energy harvesters use uniform rectangular geometries, whereas at the macroscale energy harvesters use a wide array of geometries including tapered rectangular geometries. The incorporation of non-uniform effects into a piezoelectric system considering the flexoelectric effect should give insight into how these systems can benefit from different geometries. A non-uniform flexoelectric Euler–Bernoulli cantilever energy harvester is modeled using classical continuum theories and is examined at the microscale. The non-uniformity of the energy harvester is governed by linear and nonlinear tapering effects, with the nonlinearities represented by high-order polynomials. The system is assumed to be linear, only undergoing harmonic base excitation. The varied tapering ratios and powers of the geometric tapering, considering that only the thickness and the width of the beam are tapered, are compared with uniform systems. The results show that non-uniform beams exhibit more harvested power than their uniform counterparts and also increase the range of resonant frequencies where significant power can be generated. Nonlinear tapering increases the amount of power that could be harvested compared to linear tapering; however, the nonlinearity of the tapering effects is limited to cubic and quadratic forms. It is demonstrated that higher-order tapering effects reduce the amount of harvested power compared to the linear taper counterpart. Non-uniform beams prove to be more effective than their rectangular counterparts within a linear system, whereas optimal resistive loads decrease as the tapering effects increase.

Analytical modelling of a nanoscale series-connected bimorph piezoelectric energy harvester incorporating the flexoelectric effect

This paper concerns the derivation of analytical solutions for a nanoscale series-connected bimorph piezoelectric energy harvester incorporating the flexoelectric and the non-local effects. The complexity involved in a parallel-connected bimorph piezoelectric energy harvester case is also briefly discussed here. Considering electric Gibbs free energy density and Euler Bernoulli beam theory, the governing equations and associated boundary condition are derived using the extended Hamilton principle. The convergent series of eigenfunctions and generalised modal coordinates related to mechanical deformation is utilised to obtain the closed-form analytical solutions for non-periodic and harmonic base excitations. The modelled harvester is validated separately from piezoelectric as well as flexoelectric point of views. Results for harmonic base excitations show that, at each excitation frequency, the consideration of flexoelectric effect in series connected bimorph piezoelectric energy harvester abruptly increases the voltage, current and power density outputs whereas it decreases the tip velocity. Here, the addition of flexoelectricity enhances the power density by ten times. The benchmark presented here may be useful for the verification of future experiments and the present study suggests that the flexoelectric effect is a significant aspect for the effective exploitation of bimorph piezoelectric energy harvester in advanced applications such as smart nanosensors.

Dynamic response of a tunable MEMS accelerometer based on repulsive force

This paper describes a tunable MEMS electrostatic accelerometer that uses a repulsive electrode configuration so that the design is not hampered by capacitive pull-in instability. The repulsive force configuration enables the increase of DC bias voltage without suffering from the pull-in failure mode. This flexibility in increasing voltage can be employed as a tuning parameter to widen the working frequency range and to improve the robustness of the accelerometer. A lumped parameter model is developed to simulate the response of the microstructure under a combination of electrostatic and dynamic mechanical loading. The electrostatic force is estimated using a finite element simulation. The nonlinear equations of motion are solved for harmonic base excitations and half-sine shock loads using the shooting and the long-time integration methods, respectively. To validate the model, a sensor is fabricated and characterized under harmonic base excitation and mechanical shocks. A mechanical sensitivity of 0.1 μm/g is achieved when the bias voltage is 40 V. The experimental data are in good agreement with the simulation results. The comprehensive dynamical characterization presented in this study contributes to the development of functional accelerometers with tunable capabilities to harmonic and shock accelerations.

Nonlinear Whirling Motion of Monopile Offshore Wind Turbines Subjected to Harmonic and Seismic Base Excitations

The triggering mechanism and the vibration patterns of the nonlinear whirling motion of monopile offshore wind turbines subjected to unidirectional base excitations are investigated both theoretically and experimentally via a 64:1 scaled model of the prototype NREL-5MW monopile offshore wind turbine. For motion, two nonlinear coupled integro-differential equations containing cubic nonlinearities due to curvature and inertia are solved by both analytical and numerical methods. Harmonic and random seismic base excitations with different amplitudes and frequencies are considered in the analysis to understand the instability mechanism. Extensive shake table tests show that the experimental results have good qualitative agreements with the theoretical results, and as observed in eight load cases, the nonlinear whirling motions of nacelle do exist and tend to be induced by large harmonic excitations with structural resonant frequency.

Multi-branch sandwich piezoelectric energy harvester: mathematical modeling and validation

Energy harvesting from ambient environment has attracted intensive attention over the years for powering low-power autonomous electronic devices. A conventional piezoelectric energy harvester (CPEH) can hardly meet the requirements of effective energy harvesting from wideband, low frequency and low amplitude vibration sources due to its simple substrate and single resonant peak response. Using sandwich structure comprised of a soft-core layer and two thin skins can drastically reduce the natural frequency and increase voltage output of the harvester. The stiffness of sandwich harvester and consequently its resonant frequency can be tuned more flexibly than the CPEH by choosing different materials and adjusting the geometric dimensions of the core and skins. In this paper, a novel multi-branch sandwich piezoelectric energy harvester (MSPEH) is proposed to harvest energy from wideband, low frequency and low amplitude vibration sources. The proposed harvester comprises of a main sandwich beam as substrate with a patch of piezoelectric layer bonded over it. Multiple inner single branches with tip masses are connected to the main substrate to generate multiple resonant peaks and tune the range of the interested frequency. Firstly, a mathematical model of the proposed harvester is presented, and the electromechanical coupling equations are obtained. Subsequently, a prototype of MSPEH with two inner single branches is fabricated and tested under harmonic base excitation to study its performance. When tested at a low harmonic excitation of 0.02 g, the harvester generates 2.48 V, 6.21 V and 1.55 V at three resonant frequencies of 18.18 Hz, 24.74 Hz and 28.12 Hz, respectively. Finally, the accuracy of derived mathematical model is verified by comparing with the predictions from finite element simulation and experimental results. The novel MSPEH offers excellent design flexibility to adjust resonant frequencies by choosing inner single branches and tip masses and thus it has potential to generate sufficient power output from wideband, low frequency and low amplitude vibration sources.

Theoretical and Experimental Study of Nonlinear and Electro‐Magneto‐Mechanical‐Based Piezoelectric Vibration Energy Harvester

This paper presents a folded nonlinear electro‐magneto‐mechanical (EMM) vibration‐based piezoelectric energy harvester system, which is built on the cantilevered beam structure and consists of one host beam and two substrate plates. The performance of the linearity and nonlinearity to the proposed EMM system is evaluated and compared. Moreover, the voltage response in time history and the phase portrait are studied under an external rectifier circuit with a resistor. The results show that the nonlinearity of the reported EMM system changes the coherent resonance vibration mode from single to double under a harmonic base excitation within the frequency range of 20 Hz–50 Hz. Meanwhile, the substrate plate D contributes more averaged voltage output at a lower frequency while the substrate plate A contributes the voltage output at the relatively higher frequency for the nonlinear EMM system. The experimental study indicates that the proposed nonlinear EMM vibration‐based piezoelectric energy harvester can yield a total voltage of 8.133 V@35.53 Hz while the baseline structure only produces 1.724 V@38.81 Hz. In addition, the bandwidth range of high‐power output is enlarged by the nonlinear EMM system, which makes this device more flexible and applicable to absorb the wasted vibration energy generated by industrial machines and public facilities.

An electromechanical finite element model for new CNTs-reinforced harvesters subjected to harmonic and random base excitations

Conventional piezoelectric energy harvesters are cantilever beams made of homogenous substructure and piezoelectric layers that produce alternating voltage due to their vibrations. Recently, a class of new emerging composite materials, the carbon nanotubes (CNTs)-reinforced composite, has been proposed making use of CNTs as the reinforcements in a functionally graded (FG) pattern. Harvesters made of functionally graded CNTs-reinforced substructure materials have not been studied in the energy harvesting literature, yet. Thus, in this study, functionally graded piezoelectric CNTs-reinforced harvesters as new energy harvesters subjected to harmonic and random vibrations are analyzed. The harvester is assumed to be comprised of piezoelectric and FG-CNTs-reinforced substructure layers. Five types of CNTs distributions through the substructure thickness direction are considered. The generalized Hamilton’s principle for electromechanical materials based on Euler–Bernoulli beam assumption is adopted in the derivation of governing equations. The finite element formulation of the equations is also presented. Time and frequency domain analyses of the finite element equations are performed. The output power from the CNTs-reinforced harvesters subjected to random and harmonic base excitations is calculated. The harvested energy of the CNTs-reinforced harvesters is compared, and it is concluded that the CNTs distribution has a significant effect on the deflection, produced voltage and harvested power.

Various Bifurcation Phenomena in a Nonlinear Curved Beam Subjected to Base Harmonic Excitation

Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation, which is governed by a coupled nonlinear equation with both quadratic and cubic nonlinearities, are investigated using the incremental harmonic balance (IHB) method. The nonlinear partial differential equation that governs the motion of the curved beam is given using Hamilton’s principle. A spatially discretized governing equation is derived using Galerkin’s method, yielding a set of second-order nonlinear ordinary different equations. A high-dimensional model that can take nonlinear model coupling into account is derived. Specific attention is paid to the different bifurcation phenomena of frequency responses and amplitude responses of the system without and with an anti-symmetric mode being excited. Numerical results reveal the rich and interesting diverse bifurcation phenomena that have not been presented in the existent literature on the nonlinear vibration of the curved beam system. Saddle-node, Hopf, and period-doubling bifurcations are observed without an anti-symmetric mode being excited. Besides, a symmetry-breaking bifurcation is observed with an anti-symmetric mode being excited. Furthermore, the phase portraits and bifurcation points obtained by the IHB method agree very well with those obtained by the numerical integration using the fourth-order Runge–Kutta method.

Complex modal analysis and coupled electromechanical simulation of energy harvesting piezoelectric laminated beams

In this paper, coupled electromechanical behavior of a vibrational energy harvesting system composed of a unimorph piezoelectric laminated beam with a large attached tip mass is investigated. To achieve this goal, first the electromechanically coupled partial differential equations governing the lateral displacement and output voltage of the harvester are extracted through exploiting the Hamilton’s principle. Considering vibration damping due to mechanical to electrical energy conversion, a complex modal analysis is performed to extract the complex eigenfrequencies and eigenfunctions of the system. Furthermore, an exact analytical solution is presented for the system response to the harmonic base excitations, including output voltage and harvested power. To validate the analytical results, at the next step a finite element simulation is conducted through ABAQUS software. To perform a fully-coupled analysis which brings into account the effect of harvesting circuit, user subroutine User-defined Amplitude (UAMP) is utilized to calculate the voltage–current relation and impose the correct electrical charge on the electrodes in each step by monitoring the output voltage of the system at previous time increments. Results of both analytical and numerical simulations are compared for a Micro-Electro-Mechanical Systems (MEMS) harvester as a case study, where a very good agreement is observed between them.

On the dynamics of a capacitive electret-based micro-cantilever for energy harvesting

In this paper, an electret-based capacitive energy harvesting device with out-of-plane gap closing scheme has been modeled and analyzed. The device is composed of a micro cantilever and a substrate which form a variable capacitor that is in series with a resistance. An electret material is used to provide the bias voltage which is needed in capacitive energy harvesters in order to scavenge energy from ambient vibration. The ambient vibration is applied to the system as a harmonic base excitation. The motion equations and the corresponding boundary conditions are derived using Hamilton's principle based on Euler-Bernoulli beam theory, and the Kirchhoff's voltage law is employed to couple the mechanical and electrical domains. The electro-mechanical governing equations are discretized using Galerkin procedure and integrated numerically over time. In order to verify the derived formulation, an energy conservation approach is employed in the free oscillation of the device. The stability of the system in the electrostatic field is examined in both static and dynamic cases. By investigating the behavior of the system in free oscillations, the resistance is found to affect the behavior of the micro-cantilever as a damper with a Lorentzian variation, with the maximum equivalent damping coefficient corresponding to the resistance of 100 MΩ. The performance of the energy harvester in the presence of ambient vibrations is evaluated based on the frequency response of the device, and the effect of various parameter as investigated on the device performance, among which the resistance appears to have the most dominant effect on the system. It is found that in low resistances, the micro-cantilever exhibit a softening behavior due to the large electrostatic pressure acting on it, however, by increasing the resistance, the frequency response of the system becomes linear. Besides, the resistance affects the amplitude of the micro-cantilever due to its damping effect. For the case studied here, a theoretical harvested power in the order of 1 μW is harvested by the device for a frequency of 2.5 MHz and the electret surface voltage of 180 V. Of course, by adjusting the physical properties, the device can be fitted for the needs of the target application.

Design and analytical modeling of magneto-electro-mechanical characteristics of a novel magneto-electro-elastic vibration-based energy harvesting system

In order to effectively design an energy harvesting system for any specific application, a model that accurately characterizes the energy harvesting parameters is needed. In the present paper a novel magneto-electro-elastic (MEE) cantilever beam has been proposed and modeled as an effective means to increase the harvested electrical power in a vibration-based energy harvesting system. The cantilever beam is composed of a linear homogeneous elastic substrate and two MEE layers with perfect bonds between their interfaces. Using the constitutive equations, Gauss's and Faraday's laws, based on the Euler–Bernoulli beam theory, the coupled magneto-electro-mechanical (MeM) differential equations are derived for a harmonic base excitation in the transversal direction with a superimposed small rotation. The resulting equations are then solved analytically to obtain the dynamic behavior as well as the harvested voltages and powers of the proposed energy harvesting system. Finally, parametric numerical studies are used to examine the effect of excitation frequency, external resistive loads, and material properties on the performance of the MEE energy harvester. The study reveals that the implementation of the coil circuit has resulted in an increase in the total useful harvested power. According to the numerical results, any increase in the Young's modulus and density of the substrate layer (across the ranges that have been studied and while the properties of the MEE layer are kept constant), increases the magnitude of the magnetoelectric harvested power in the unimorph MEE energy harvester system.

The Rayleigh–van der Pol oscillator on linear multibody systems

In nature many interactions of different oscillatory systems can be described by the phenomenon of synchronization. This phenomenon can be widely used for engineering applications. In this contribution, the synchronization between a self-sustained oscillator and linear multibody systems with harmonic force excitation is used successfully to damp unwanted vibrations of the host structure and additionally to harvest energy from the load. To this end, the parameters of the Rayleigh–van der Pol oscillator are adjusted to a given harmonic base excitation in such a way that the response is harmonic as well. Thereby the amplitude of the response and the phase-shift between the excitation and the oscillator response remain free. Afterwards the oscillator is attached to a forced multibody system and the amplitude and phase-shift are adjusted for the tasks of vibration absorption and energy harvesting.

Validation of Analytical Damping Ratio by Fatigue Stress Limit

The optimisation process of a vibration energy harvester is usually restricted to experimental approaches due to the lack of an analytical equation to describe the damping of a system. This study derives an analytical equation, which describes the first mode damping ratio of a clamp-free cantilever beam under harmonic base excitation by combining the transverse equation of motion of the beam with the damping-stress equation. This equation, as opposed to other common damping determination methods, is independent of experimental inputs or finite element simulations and can be solved using a simple iterative convergence method. The derived equation was determined to be correct for cases when the maximum bending stress in the beam is below the fatigue limit stress of the beam. However, an increasing trend in the error between the experiment and the analytical results were observed at high stress levels. Hence, the fatigue limit stress was used as a parameter to define the validity of the analytical equation.

Stick-slip vibration of a cantilever beam subjected to harmonic base excitation

Stick-slip vibration of a cantilever beam subjected to harmonic base excitation is analyzed in this paper. To model the stick-slip vibration, considering nonlinearity and discontinuity of the friction force, two kinds of equations of motion are used: one for motion in the stick state and the other for motion in the slip state. Using the finite element method, dynamic responses of stick-slip vibration are calculated by iteratively solving the equations and connecting the stick- and slip-state motions; such calculations are conducted for various excitation conditions. The dynamic responses show that the occurrence of stick-slip vibration depends on excitation amplitude and frequency, and an occurrence of stick-slip vibration is associated with the modal characteristic of a beam. Furthermore, to formulate an analytical expression for the criterion of stick-slip occurrence, the static friction force caused by the base excitation is determined analytically. Finally, excitation conditions for stick-slip occurrence are obtained and validated in a parametric plane of the excitation amplitude and frequency, and design guidelines for avoiding stick-slip occurrence are presented.

Experimental and Numerical Studies on Inelastic Dynamic Behavior of Stainless Steel Elbow Under Harmonic Base Excitation

In the present work, inelastic dynamic behavior of a pressurized stainless steel elbow is studied under harmonic base excitation with the emphasis on strain accumulation known as ratcheting. Initially, sine sweep test is carried out on the long radius stainless steel (SS 304L) elbow to evaluate the free vibration characteristics. Then, incremental harmonic base excitation with the first resonant frequency is applied to the elbow till failure and the resulting response is studied. The tested elbow is analyzed using a simplified method and the simulated ratcheting strain is compared with experimental results. The effect of thickness variation in the elbow on strain accumulation is also studied. Levels of base excitation corresponding to different failure criteria are evaluated and the details are provided in the paper.

Energy harvesting from vibration of Timoshenko nanobeam under base excitation considering flexoelectric and elastic strain gradient effects

The coupling between polarization and strain gradients is called flexoelectricity. This phenomenon exists in all dielectrics with any symmetry. In this paper, energy harvesting from a Timoshenko beam is studied by considering the flexoelectric and strain gradient effects. General governing equations and related boundary conditions are derived using Hamilton's principle. The flexoelectric effects are defined by gradients of normal and shear strains which lead to a more general model. The developed model also covers the classical Timoshenko beam theory by ignoring the flexoelectric effect. Based on the developed model, flexoelectricity effect on dielectric beams and energy harvesting from cantilever beam under harmonic base excitation is investigated. A parametric study was conducted to evaluate the effects of flexoelectric coefficients, strain gradient constants, base acceleration and the attaching tip mass on the energy harvested from a cantilever Timoshenko beam. Results show that the flexoelectricity has a significant effect on the energy harvester performance, especially in submicron and nano scales. In addition, this effect makes the beam to behave softer than before and also it changes the harvester first resonance frequency. The present study provides guidance for flexoelectric nano-beam analysis and a method to evaluate the performance of energy harvester in nano-dielectric devices.

Dynamics of a Upright Pole Coupled with Nonlinear Hysteretic Isolators under Harmonic Base Excitation

Leady isolator shows hysteretic nonlinearity. The isolation efficiency of leady isolator is an important problem in many engineering structures. In this paper, vibrations of a ceramic upright pole coupled with four leady isolators under harmonic base excitation are studied. A hysteretic force‐deformation model of leady isolator is derived from experimental results. With this model, the vibration of the pole in rotation is studied. The frequency response is obtained analytically by employing harmonic balance method. The analytical results are agreed well with numerical results. The vibration of the pole is decreased greatly by leady isolator, especially near resonant case. The influences of system parameters on vibration response and resonant peak are discussed in detail.

Modified pendular vibration absorber for structures under base excitation

The passive control of structures using a pendulum tuned mass damper has been extensively studied in the technical literature. As the frequency of the pendulum depends only on its length and the acceleration of gravity, to tune the frequency of the pendulum with that of the structure, the pendulum length is the only design variable. However, in many cases, the required length and the space necessary for its installation are not compatible with the design. In these cases, one can replace the classical pendulum by a virtual pendulum which consists of a mass moving over a curved surface, allowing thus for a greater flexibility in the absorber design, since the length of the pendulum becomes irrelevant and the shape of the curved surface can be optimized. A mathematical model for a building with a pendular tuned mass damper and a detailed parametric analysis is conducted to study the influence of this device on the nonlinear oscillations and stability of the main system under harmonic and seismic base excitation. In addition to the circular profiles, different curved surfaces with softening and hardening characteristics are analyzed. Also, the influence of impact on energy dissipation is considered. A detailed parametric analysis is presented showing that the proposed damper can not only reduce sharply the displacements, and consequently the internal forces in the main structure, but also the accelerations, increasing user comfort. A review of the relevant aspects is also presented.

Distributed parameter modeling to prevent charge cancellation for discrete thickness piezoelectric energy harvester

The charge cancellation due to the change of strain along single continuous piezoelectric layer can remarkably affect the performance of a cantilever based harvester. In this paper, analytical models using distributed parameters are developed with some extent of averting the charge cancellation in cantilever piezoelectric transducer where the piezoelectric layers are segmented at strain nodes of concerned vibration mode. The electrode of piezoelectric segments are parallelly connected with a single external resistive load in the 1st model (Model 1). While each bimorph piezoelectric layers are connected in parallel to a resistor to form an independent circuit in the 2nd model (Model 2). The analytical expressions of the closed-form electromechanical coupling responses in frequency domain under harmonic base excitation are derived based on the Euler–Bernoulli beam assumption for both models. The developed analytical models are validated by COMSOL and experimental results. The results demonstrate that the energy harvesting performance of the developed segmented piezoelectric layer models is better than the traditional model of continuous piezoelectric layer.