Dielectric elastomers (DEs) are a class of highly deformable electroactive polymers (EAPs) employed for electromechanical transduction technology. When electrostatically actuated dielectric elastomer actuators (DEAs) are subjected to an input signal comprising multiple Heaviside voltage steps, the emerging inherent residual vibrations may limit their motion accuracy in practical applications. In this paper, the systematic development of a command-shaping scheme is proposed for controlling residual vibrations in an electrically driven planar DEA. The proposed scheme relies on invoking the force balance at the point of maximum lateral stretch in an oscillation cycle to bring the actuator to a stagnation state followed by the application of an additional electric input signal of predetermined magnitude at a specific time. The underlying concept of the proposed control scheme is articulated for a single Heaviside step input-driven actuator and further extended to the actuator subjected to the multistep input signal. The equation governing the dynamic motion of the actuator is derived using the principle of virtual work. The devised dynamic model of the actuator incorporates the effects of strain stiffening of elastomer and viscous energy dissipation. The nonlinear dynamic governing equation is solved using matlab ode solver for extracting the dynamic response of the actuator. The applicability of the devised command-shaping control scheme is illustrated by taking a wide range of parameters including variations in the extent of equilibrium state sequences, damping, and polymer chain extensibility. The proposed scheme is found to be adaptable in controlling the vibrations of the actuator for any desired equilibrium state. The results presented in this paper can find its potential application in the design of an open-loop control system for DEAs.

This paper presents complete nonlinear electromechanical models for energy harvesting devices consisting of multiple piezoelectric bimorphs connected in parallel and series, for the first time. The proposed model is verified against available experimental results for a specific case. The piezoelectric and beam constitutive equations and different circuit equations are utilised to derive the complete nonlinear models for series and parallel connections of the piezoelectric bimorphs (PBs) as well as those of piezoelectric layers in each bimorph, i.e. four nonlinear models in total. A multi-modal Galerkin approach is used to discretise these nonlinear electromechanical models. The resultant high-dimensional set of equations is solved utilising a highly optimised and efficient numerical continuation code. Examining the system behaviour shows that the optimum load resistance for an energy harvester array of 4 PBs connected in parallel is almost 4% of that for the case with PBs connected in series. It is shown an energy harvesting array of 8 PBs could reach a bandwidth of 14 Hz in low frequency range, i.e. 20-34 Hz. Compared to an energy harvester with 1 PB, it is shown that the bandwidth can be increased by more than 300% using 4 PBs and by more than 500% using 8 PBs. Additionally, the drawbacks of a multi-PB energy harvesting device are identified and design enhancements are proposed to improve the efficiency of the device.

Comparing to linear vibration absorbers, nonlinear energy sinks (NESs) have attracted worldwide attention for their intrinsic characteristics of targeted energy transfer or energy pumping in a relatively wide frequency range. Unfortunately, they are highly dependent on the vibration amplitude to be attenuated and will play its role only if the external load exceeds a specific threshold value. Different from the passive bistable NES, a novel piezoelectric nonlinear energy sink (PNES) is designed by introducing in-phase actuation to compensate or enhance the external vibration loads, thus triggering the NES operating in high attenuation efficiency. The nonlinear mathematic model of the PNES is established for investigating the dynamic response and determining the threshold compensation strategy. And the results show that the maximum attenuation efficiency can be improved by 58.16% compared to the traditional passive NES. Also, the amplitude-dependent coefficient (ADC) can be significantly reduced to 0.33 from 1.0, which means that the PNES can effectively mitigate vibrations even when the excitation amplitude is 67% smaller than the original threshold value. Finally, the feasibility of the in-phase compensation method is experimentally validated, which can further expand the application range of NES.

Maintaining preload in bolted joints is critical for the safe and efficient operation of nearly all built-up structures. Dynamic loss of preload during operation occurs when sufficient shear force is applied to the joint such that slip is induced in at least the threads if not the entire bolt. Such shear forces are often realized when the joint is subjected to sustained vibrations, resulting in loosening over relatively long periods of time, or extreme shock loading where loosening occurs over fractions of a second. Modeling of joint loosening often focuses on complex analytical approaches or high-fidelity simulations using finite element models. While such approaches may succeed for a single bolt, they are unfeasible for use in simulations of entire built-up structures, which may possess dozens to thousands of joints. Thus, there is a need for reduced-order models (ROMs) that capture the dominant governing physics, but at drastically lower computational costs. This research introduces a phenomenological ROM for loosening in bolted joints subjected to axial shock excitation. The model introduces a mathematical relationship between the stiffness of the joint and torque of the fastener and treats the torque as a dynamic internal variable governed by a first-order, ordinary differential equation. The proposed ROM is presented then applied to an experimental study of a split-Hopkinson pressure bar with a threaded joint subjected to extreme shock loading. The results demonstrate that the proposed ROM is able to reproduce the dominant effects of loosening in bolted joints subjected to axial shock excitation.

Band gaps in metamaterials and phononic crystals provide a way to engineer vibration mitigation into a material’s geometry. Here, we present a comprehensive experimental characterization of band gaps in lattice-resonator metastructures, which have been previously analyzed with finite element simulations, to better understand this phenomenon in 3D-printed materials. We fabricate the metastructures with a new approach to obtain multimaterial structures using stereolithography. We experimentally characterize the material’s frequency-dependent storage and loss modulus over the band gap frequencies to confirm that the measured band gaps are due to geometry and not due to material properties. Experimental results using both frequency sweep and impulse excitations show that band gaps and attenuation efficiencies strongly depend on the lattice geometry as well as loading direction, and a comparison between axial and bending excitation responses reveals frequency ranges of “fluid-like” and “optical-like” behaviors. Comparison between finite element simulations and experimental results demonstrate the robustness of the metastructure design. While the experiments used here are well established, their combination allows us to gain additional insights into band gaps measurements. Specifically, we show that the coherence function, a common concept in signal processing, is a strong predictor of band gaps in linear materials and that the attenuation efficiency inside the measured band gap can be physically limited by fluid–structure interactions.

This paper investigates the nonlinear static response as well as nonlinear forced dynamics of a clamped–clamped beam actuated by piezoelectric patches partially covering the beam from both sides. This study is the first to develop a high-dimensional nonlinear model for such a piezoelectric-beam configuration. The nonlinear dynamical resonance characteristics of the electromechanical system are examined under simultaneous DC and AC piezoelectric actuations, while highlighting the effects of modal energy transfer and internal resonances. A multiphysics coupled model of the beam-piezoelectric system is proposed based on the nonlinear beam theory of Bernoulli–Euler and the piezoelectric constitutive equations. The discretized model of the system is obtained with the help of the Galerkin weighted residual technique while retaining 32 degrees-of-freedom. Three-dimensional finite element analysis is conducted as well in the static regime to validate the developed model and numerical simulation. It is shown that the response of the system in the nonlinear resonant region is strongly affected by a three-to-one internal resonance.

Shaft vibration caused by rotor dynamic (RD) fluid force generated by the seal clearance flow has caused several problems. Because such vibration is a coupled phenomenon of clearance flow and shaft vibration, a coupling analysis is essential to solve these problems. In this study, a two-way coupling fluid–structure interaction (FSI) analysis of the seal clearance flow and shaft vibration of a rotor system was conducted and verified through experiments. The rotor system used was a vertical, flexible rotor system with a plain annular seal. In the numerical analysis of the seal clearance flow, the continuity equation and momentum equations, which were averaged across the film thickness, were numerically solved. To suppress the numerical instability, which is unique to the coupling analysis, and improve its numerical stability, a method of successively correcting pressure and shaft acceleration values was adopted so that the continuity equation and rotor equations of motion could be satisfied at every time step. By performing the coupling simulation, the frequency response characteristics of whirling amplitude and leakage flow were investigated. In regard to the stability of the system, the rotational speeds at which self-excited vibration occurs (onset speed of instability: OSI) in its increasing condition and ceases (onset speed of dropdown: OSD) in its decreasing condition were investigated. The coupling analysis results reasonably agree with the experimental results, which demonstrate the validity of the analysis method. In addition, the influence of static eccentricity and whirling amplitude on stability (OSI and OSD) was clarified, which are useful in the design stage of turbomachinery.

The deployment dynamics of a simplified solar sail quadrant consisting of two Euler–Bernoulli beams and a flexible membrane are studied. Upon prescribing the in-plane motion and modeling the tension field based on linearly increasing stresses assumed on the attached boundaries, the coupled equations of motion that describe the system's transverse deflections are obtained. Based on these equations and their boundary conditions (BCs), deployment stability is studied by deriving simplified analytic expressions for the rate of change of system energy. It is shown that uniform extension and retraction result in decreasing and increasing energy, respectively. The motion equations are discretized using expansions in terms of “time-varying quasi-modes” (snapshots of the modes of a cantilevered beam and a clamped membrane), and the integrals needed for the resulting system matrices are rendered time-invariant via a coordinate transformation. Numerical simulation results are provided to illustrate a sample deployment and validate the analytic energy rate expressions.

A new kind of nonlinear energy sink (NES) is proposed to control the vibration of a flexible structure with simply supported boundaries in the present work. The new kind of absorber is assembled at the end of structures and absorbs energy through the rotation angle at the end of the structure. It is easy to design and attached to the support of flexible structures. The structure and the absorber are coupled just with a nonlinear restoring moment and the damper in the absorber acts on the structure indirectly. In this way, all the linear characters of the flexible structure will not be changed. The system is investigated by a special perturbation method and verified by simulation. Parameters of the absorber are fully discussed to optimize the efficiency of it. For the resonance, the maximum motion is restrained up to 90% by the optimized absorber. For the impulse, the vibration of the structure could attenuate rapidly. In addition to the high efficiency, energy transmits to the absorber uniaxially. For the high efficiency, convenience of installation and the immutability of linear characters, the new kind of rotating absorber provides a very good strategy for the vibration control.

This paper presents results related to the stability of gyroscopic systems in the presence of circulatory forces. It is shown that when the potential, gyroscopic, and circulatory matrices commute, the system is unstable. This central result is shown to be a generalization of that obtained by Lakhadanov, which was restricted to potential systems all of whose frequencies of vibration are identical. The generalization is useful in stability analysis of large scale multidegree-of-freedom real life systems, which rarely have all their frequencies identical, thereby expanding the compass of applicability of stability results for such systems. Comparisons with results in the literature on the stability of such systems are made, and the weakness of results that deal with only general statements about stability is exposed. It is shown that the commutation conditions given herein provide definitive stability results in situations where the well-known Bottema–Karapetyan–Lakhadanov result is inapplicable.

Vibrational microplatforms that exploit complex three-dimensional (3D) architectures assembled via the controlled compressive buckling technique represent promising candidates in 3D micro-electromechanical systems (MEMS), with a wide range of applications such as oscillators, actuators, energy harvesters, etc. However, the accuracy and efficiency of such 3D MEMS might be significantly reduced by the viscoelastic damping effect that arises from material viscosity. Therefore, a clear understanding and characterization of such effects are essential to progress in this area. Here, we present a study on the viscoelastic damping effect in complex 3D structures via an analytical model and finite element analysis (FEA). By adopting the Kelvin–Voigt model to characterize the material viscoelasticity, an analytical solution is derived for the vibration of a buckled ribbon. This solution then yields a scaling law for the half-band width or the quality factor of vibration that can be extended to other classes of complex 3D structures, as validated by FEA. The scaling law reveals the dependence of the half-band width on the geometries of 3D structures and the compressive strain. The results could serve as guidelines to design novel 3D vibrational microplatforms for applications in MEMS and other areas of technology.

Two novel nonparametric identification approaches are proposed for piezoelectric mechanical systems. The novelty of the approaches is using not only mechanical signals but also electric signals. The expressions for unknown mechanical and electric terms are given based on the Hilbert transform. The signals are decomposed and re-assembled to obtain smooth stiffness and damping curves. The current mapping approach is developed to identify accurately a piezoelectric mechanical system with strongly nonlinear electric terms. The developed identification approaches are successfully implemented to simulate signals obtained from different nonlinear piezoelectric mechanical systems, including Duffing nonlinearity, softening and hardening nonlinearity, and Duffing nonlinearity with strong nonlinear electric terms. The proposed approaches are successfully applied to experimental signals of a circular laminated plate device in order to identify the nonlinear stiffness functions, damping functions, electromechanical coupling functions, and equivalent capacitance functions. The results show both softening and hardening nonlinearity in the stiffness characteristic and weak nonlinearity in electric characteristics. The results of the Hilbert transform based approach and the current mapping approach are compared, and the outcomes show good agreements.

Nonlinear dynamics and mode aberration of rotating plates and shells are discussed in this work. The mathematical formalism is based on the one-dimensional (1D) Carrera unified formulation (CUF), which enables to express the governing equations and related finite element arrays as independent of the theory approximation order. As a consequence, three-dimensional (3D) solutions accounting for couplings due to geometry, material, and inertia can be included with ease and with low computational costs. Geometric nonlinearities are incorporated in a total Lagrangian scenario and the full Green-Lagrange strains are employed to outline accurately the equilibrium path of structures subjected to inertia, centrifugal forces, and Coriolis effect. A number of representative numerical examples are discussed, including multisection blades and shells with different radii of curvature. Particular attention is focused on the capabilities of the present formulation to deal with nonlinear effects, and comparison with s simpler linearized approach shows evident differences, particularly in the case of deep shells.

In many applications, coupling between thermal and mechanical domains can significantly influence structural dynamics. Analytical approaches to study such problems have previously used assumptions such as a proscribed temperature distribution or one-way coupling to enable assessments. In contrast, time-stepping numerical simulations have captured more detailed aspects of multiphysics interactions at the expense of high computational demands and lack of insight of the underlying physics. To provide a new tool that closes the knowledge gap and broadens potential for analytical techniques, this research formulates and analytically solves a thermomechanical beam model considering a combination of thermal and mechanical excitations that result in extreme nonlinear behaviors. Validated by experimental evidence, the analytical framework facilitates the prediction of the nonlinear dynamics of multi-degree-of-freedom structures exhibiting two-way thermomechanical coupling. The analysis enables the investigation of mechanical and thermomechanical impedance metrics as a means to forecast future nonlinear dynamic behaviors such as extreme bifurcations. For the first time, characteristics of mechanical impedance previously reported to predict the onset of dynamic bifurcations in discrete systems are translated to illuminate the nearness of distributed parameter structures to bifurcations. In addition, fundamental connections are discovered in the thermomechanical evaluations between nonlinear low amplitude dynamics of the postbuckled beam and the energetic snap-through vibration that are otherwise hidden by studying displacement amplitudes alone.

In this paper, inerter-based dynamic vibration absorbers (IDVAs) are applied in elastic metamaterials to broaden low-frequency band gaps. A discrete mass-spring lattice system and a distributed metamaterial beam carrying a periodic array of IDVAs are, respectively, considered. The IDVA consists of a spring and an inerter connected to a traditional mass-spring resonator. Compared to the traditional resonators, the special designed IDVAs generate two local-resonance (LR) band gaps for the discrete lattice system, a narrow low-frequency band gap and a wider high-frequency one. For the distributed IDVA-based metamaterial beam, in addition to the generated two separated LR band gaps, the Bragg band gap can also be significantly broadened and the three band gaps are very close to each other. Being able to amplify inertia, the IDVAs can be relatively light even operated for opening up low-frequency band gaps. When further introducing a dissipative damping mechanism into the IDVA-based metamaterials, the two close-split LR band gaps in the lattice system are merged into one wide band gap. As for the metamaterial beam with the dissipative IDVAs, an even wider band gap can be acquired due to the overlap of the adjacent LR and Bragg-scattering band gaps.

Structural damping, that is the presence of a velocity dependent dissipative term in the equation of motion, is rationalized as a thermalization process between a structure (here a beam) and an outside bath (understood in a broad sense as a system property). This is achieved via the introduction of the kinetic temperature of structures and formalized by means of an extended Lagrangian formulation of a structure in contact with an outside bath at a given temperature. Using the Nosé–Hoover thermostat, the heat exchange rate between structure and bath is identified as a mass damping coefficient, which evolves in time in function of the kinetic energy/temperature history exhibited by the structure. By way of application to a simple beam structure subjected to eigen-vibrations and dynamic buckling, commonality and differences of the Nosé–Hoover beam theory with constant mass damping models are shown, which permit a handshake between classical damping models and statistical mechanics–based thermalization models. The solid foundation of these thermalization models in statistical physics provides new insights into stability and instability for engineering structures. Specifically, since two systems are considered in (thermodynamic) equilibrium when they have the same temperature, we show in the case of dynamic buckling that a persistent steady-state difference in kinetic temperature between structure and bath is but indicative of the instability of the system. This shows that the kinetic temperature can serve as a structural order parameter to identify and comprehend failure of structures, possibly well beyond the elastic stability considered here.

Artificial periodic structures are used to control spatial and spectral properties of acoustic or elastic waves. The ability to exploit band gap structure creatively develops a new route to achieve excellently manipulated wave properties. In this study, we introduce a paradigm for a type of real-time band gap modulation technique based on parametric excitations. The longitudinal wave of one-dimensional (1D) spring-mass systems that undergo transverse periodic vibrations is investigated, in which the high-frequency vibration modes are considered as parametric excitation to provide pseudo-stiffness to the longitudinal elastic wave in the propagating direction. Both analytical and numerical methods are used to elucidate the versatility and efficiency of the proposed real-time dynamic modulating technique.

The membrane structure has been applied throughout different fields such as civil engineering, biology, and aeronautics, among others. In many applications, large deflections negate linearizing assumptions, and linear modes begin to interact due to the nonlinearity. This paper considers the coupling effect between vibration modes and develops the theoretical analysis of the free vibration problem for orthotropic rectangular membrane structures. Von Kármán theory is applied to model the nonlinear dynamics of these membrane structures with sufficiently large deformation. The transverse displacement fields are expanded with both symmetric and asymmetric modes, and the stress function form is built with these coupled modes. Then, a reduced model with a set of coupled equations may be obtained by the Galerkin technique, which is then solved numerically by the fourth-order Runge–Kutta method. The model is validated by means of an experimental study. The proposed model can be used to quantitatively predict the softening behavior of amplitude–frequency, confirm the asymmetric characters of mode space distribution, and reveal the influence of various geometric and material parameters on the nonlinear dynamics.

This investigation considers the dynamic stability of the steady-state frictional sliding of a finite-thickness elastic layer pressed against a moving rigid and flat surface of infinite extent. The elastic layer is fixed on its bottom surface; on its entire top surface, the rigid surface slides with constant speed and with a constant friction coefficient. The plane-strain equations of motion for a linear isotropic elastic solid are solved analytically for small dynamic disturbances. The analysis shows that even with a constant (speed-independent) friction coefficient, the steady solution is dynamically unstable for any finite friction coefficient. Eigenvalues with positive real parts lead to self-excited vibrations which occur for any sliding speed and which increase with increasing coefficient of friction. This is in contrast to the behavior of an elastic half-space sliding against a rigid surface in which the instability only occurs if the coefficient of friction is greater than unity. This work and its extensions are expected to be relevant in the theoretical aspects of sliding friction as well as in a variety of areas such as earthquake motion and brake dynamics.

A majority of dielectric elastomers (DE) developed so far have more or less viscoelastic properties. Understanding the dynamic behaviors of DE is crucial for devices where inertial effects cannot be neglected. Through construction of a dissipation function, we applied the Lagrange's method and theory of nonequilibrium thermodynamics of DE and formulated a physics-based approach for dynamics of viscoelastic DE. We revisited the nonlinear oscillation of DE balloons and proposed a combined shooting and arc-length continuation method to solve the highly nonlinear equations. Both stable and unstable periodic solutions, along with bifurcation and jump phenomenon, were captured successfully when the excitation frequency was tuned over a wide range of variation. The calculated frequency–amplitude curve indicates existence of both harmonic and superharmonic resonances, soft-spring behavior, and hysteresis. The underlying physics and nonlinear dynamics of viscoelastic DE would aid the design and control of DE enabled soft machines.