Linear Vibration Research Articles

Dynamic performance of the energy harvester with a fractional-order derivative circuit

Vibration energy harvesters are expected to replace chemical batteries and achieve self-power for low-power electronic devices, and their design has attracted significant attention. In previous studies, circuits consistent with the constitutive relationship with integer derivatives were frequently used, but fractional-order derivative circuits have received less attention. Because of the dissipation effects, such as internal friction and thermal memory, the fractional-order derivative is an effective mathematical approach for determining such nonconservative behaviors. In this study, the dynamic performance of a vibration energy harvester with a fractional-order derivative circuit in a random environment was investigated. Both linear and nonlinear vibration energy harvesters were evaluated using the linear random vibration theory and the equivalent linearization technique. The effects of the system parameters on the mean output power were analyzed, and the optimal system parameters, such as the oscillation frequency of the circuit and the coupling coefficient, were determined. Furthermore, the relationships between the optimal parameters and the order of the fractional derivative were established. Our study will potentially guide future designs of vibration energy harvesters.

Analysis of fiber optic gyroscope dynamic error based on CEEMDAN

The fiber optic gyroscope (FOG) is an angular rate sensor based on Sagnac effect. It has been widely used in aviation, unmanned, robot, and other fields due to its benefits of all solid state, high precision, and low cost. However, the photoelectric devices inside the FOG will produce a non-negligible error in the dynamic environment, resulting in zero bias drift and increased noise. Currently, research on the dynamic error of FOG focuses primarily on linear vibration, assuming that no input angular rate exists at this moment. However, the dynamic errors caused by angular vibration are more complicated in the practical application environment. The output of FOG includes both effective angular information and dynamic error at this moment, which means that the dynamic error should be removed while the effective angular information must be retained. In this paper, we focus on the effect of angular vibration on FOG, and propose a dynamic error compensation method of FOG based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) that properly separates the effective signal and dynamic error. We also consider the amplitude frequency characteristics of FOG under dynamic conditions and utilize the FOG bandwidth test experiment to verify the effectiveness of our method. The experimental results show that the compensated signal can correctly calculate the bandwidth of FOG. It indicates that our method can effectively compensate the dynamic error in angular vibration environment while improving the performance of FOG in dynamic environment.

Dynamic Characteristics of a Variable Damping Isolator with Translating Cam

A variable damping isolator based on a translating cam designed especially is proposed in this paper. The nonlinear variable damping isolator is mainly comprised of translating cam with a pair of rollers arranged symmetrically and horizontally and two linear dampers. The damping force of this isolator increases with the increase of its vertical displacement. The dynamic equation is established when the variable damping isolator is applied to the active isolation system under simple harmonic excitation. And the dynamic response of the equation is obtained by Harmonic Balance Method. After that, the numerical simulations are conducted to discuss the effects of the parameters of the isolation system on force transmissibility. The results show that the effects on attenuating the resonance and the force transmissibility in the region of high-frequency ratio are superior to that of the corresponding linear damping vibration isolator when appropriate design parameters (cam profile and damping coefficient of the horizontal linear damper) of the variable damping isolator are selected.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Influences of elastic foundations on the nonlinear free vibration of composite shells containing carbon nanotubes within shear deformation theory

In this work, the solution of nonlinear free vibration problem of composite shells structures containing carbon nanotubes (CNTs) resting on elastic soils within shear deformation theory (ST) is presented. After modeling the mechanical properties of nanocomposite shell structures containing CNTs and elastic soils, the basic relations, and governing equations of double curved shell structures within the ST are established considering the geometric nonlinearity. The frequencies of nonlinear and linear free vibrations and their ratios for inhomogeneous nanocomposite structures on the soils within the ST are obtained using perturbation method for the first time. After checking the methodology of the research, the effects of soils, nonlinearity, shear strains and patterns of CNT on the frequency-amplitude dependence of nanocomposite shell structures for various geometric parameters are carried out.

A FE-Based Macro-Element for the Assessment of Masonry Structures: Linear Static, Vibration, and Non-Linear Cyclic Analyses

A Finite Element (FE) based macro–element is described for the mechanical response of masonry structures within different ranges of analysis. The macro–element is composed of discrete rigid quadrilateral FE plates whose adjoining interfaces are connected through FE trusses. It allows representing both elasticity and strength orthotropy, full material nonlinearity and damage through a scalar–based model. The possibility of coupling with a so–called FE2 (multi–scale) strategy is also addressed. Validation of the macro–element is conducted within linear static, vibration, and cyclic (nonlinear) problems, in which both static and dynamic ranges are explored. Results are compared with those retrieved from traditional FE continuous models. Advantages are highlighted, as well as its robustness to cope with convergence issues and suitability to be applied within more general and larger–scale scenarios, such as the analysis of anisotropic materials subjected to static and dynamic loading. Formal details are given for its reproducibility by academics and practitioners—eventually within other FE platforms—as the improved running times may be of utmost importance in dynamic problems or highly nonlinear (material) quasi–static analysis.

Nonintrusive Nonlinear Reduced Order Models for Structures in Large Deformations: Validations to Atypical Structures and Basis Construction Aspects

The focus of this investigation is on reduced order models (ROMs) of the nonlinear geometric response of structures that are built nonintrusively, i.e., from standard outputs of commercial finite element codes. Several structures with atypical loading, boundary conditions, or geometry are considered to not only support the broad applicability of these ROMs but also to exemplify the different steps involved in determining an appropriate basis for the response. This basis is formed here as a combination of linear vibration modes and dual modes, and some of the steps involved follow prior work; others are novel aspects, all of which are covered in significant detail to minimize the expertise needed to develop these ROMs. The comparisons of the static and dynamic responses of these structures predicted by the ROMs and by the underlying finite element models demonstrate the high accuracy that can be achieved with the ROMs, even in the presence of significant nonlinearity.

How Intrusive Are Accelerometers for Measuring Nonlinear Vibrations? A Case Study on a Compressor Blade Subjected to Vibro-Impact Dynamics

Abstract A characteristic feature of nonlinear vibrations is the energy transfer among different parts or modes of a mechanical system. Moreover, nonlinear vibrations are often non-periodic, even at steady state. To analyze these phenomena experimentally, the vibration response must be measured at multiple locations in a time-synchronous way. For this task, piezoelectric accelerometers are by far the most popular technology. While the effect of attached sensors on linear vibration properties is well-known (mass loading in particular), the purpose of the present work is to assess their intrusiveness on nonlinear vibrations. To this end, we consider a compressor blade that undergoes impacts near the tip for sufficiently large vibrations. We consider two configurations, one in which five triaxial piezoelectric accelerometers are glued to the blade surface and one without sensors attached. In both configurations, the vibration response is measured using a multi-point laser Doppler vibrometer. In the linear case without impacts, the lowest-frequency bending mode merely sees the expected slight frequency shift due to mass loading. In the nonlinear vibro-impact case, unexpectedly, the near-resonant response to harmonic base excitation changes severely both quantitatively and qualitatively. In particular, pronounced strongly modulated responses and period doubling are observed only in the case without attached sensors. We conjecture that this is due to a considerable increase of damping, caused by the sensor cables, affecting mainly the higher-frequency modes.

The periodically extended stiffness nonlinear energy sink

Conventional nonlinear energy sinks (NES) are considered to be a more robust alternative to linear vibration absorbers such as the tuned-mass–damper (TMD). While the conventional NES has a larger efficient frequency bandwidth than the TMD, it is only really efficient for a small energy range. This implies a deterioration of the NES’s mitigation properties if the primary system’s amplitude varies. To overcome this issue, other researchers resort to increasing the complexity of the NES by adding degrees-of-freedom. Here, another line of thought is presented, by proposing an unconventional stiffness characteristic. To increase the energy bandwidth the NES in this paper features a non-smooth, periodically extended stiffness characteristic. This NES is attached to an uncertain primary system and its performance is compared with that of the conventional NES and of the TMD by deriving the slow invariant manifolds (SIMs) in transient 1:1 resonance. The SIMs are curves that relate the vibration amplitudes of the primary system and the NES, and serve as an easy and computationally efficient tool to analyze performance. The research in this paper will prove that the newly proposed NES can be both robust regarding energy and frequency uncertainty, by considering the novel periodically extended stiffness characteristic.

Vibration control of an unbalanced system using a quasi-zero stiffness vibration isolator with fluidic actuators and composite material: An experimental study

In this study, a new type of vibration isolator based on fluidic actuators and a composite slab was tested experimentally with an unbalanced disturbance. Quasi-zero stiffness vibration isolation techniques are advanced and provide effective isolation performance for non-nominal loads. The isolation performance of the proposed isolator was compared to that of a nonlinear vibration isolator equipped with fluidic actuators and a mechanical coil spring (NLVIFA). The NLVIFA system is better suited to non-nominal loads; however, the mechanical spring axial deflection leads to limited amplitude reduction in the system. To address this issue, a cross buckled slab was developed to replace a mechanical coil spring for absorbing vertical deflection by transverse bending, which is made of a specially developed composite material of Basalt fiber reinforced with epoxy resin and enhanced with graphene nano pellets. This current study was concerned with the theoretical analysis and experimental investigations of the proposed nonlinear vibration isolator with fluidic actuators and composite material (NLVIFA-CM), which performs under quasi-zero stiffness characteristics. Because of its reduced axial deflection, the theoretical and experimental results show that the NLVIFA-CM system outperforms the NLVIFA system and other linear type vibration isolators in terms of isolation performance. Furthermore, the proposed vibration isolator makes a significant contribution to low-frequency vibration.

A comprehensive study on the coupled multi-mode vibrations of cylindrical shells

As is discussed in the previous work (Dong et al. 2021), the functions of beam-like modes are not suitable for simulating the cylindrical shells with the clamped and free ends. Also, ignoring the influences of the in-plane inertias can lead to large relative differences of the solutions of vibration characteristics. This work aims to accurately calculate the vibration characteristics of the cylindrical shells under various classical boundary conditions comprised of movable simply-supported, immovable simply-supported, sliding, clamped and free ends. Meanwhile, this work deals with the nonlinear coupled multi-mode vibrations of the cylindrical shell under different boundary conditions with the in-plane inertias taken into consideration. Firstly, several modified Chebyshev polynomials in terms of the axial coordinate and new harmonic functions in terms of the circumferential coordinate are employed to establish the equations of natural frequency in the frame of Lagrange equations for all the single-modes, where the polynomials are independent of geometries and material properties of the shells. And then, the circumferential-wave-dependent mode functions of the cylindrical shells are proposed, based on these mode functions coming from the linear vibration, a unified nonlinear differential equation of motion of the cylindrical shells under different boundary conditions is established for the first endeavor, in which the coupled multi-mode vibrations containing various vibration waves are considered. Finally, the high dimensional differential equation with the square and cubic nonlinearities is investigated by employing the incremental harmonic balance method. Results shown that polynomials accurately satisfy various boundary conditions, and the contribution of the single-mode to the coupled multi-mode vibration is related to the loading patterns.

Linear vibration and displacement sensor with improved metrological charecteristics

A new design of the liner vibration and displacement sensor is considered. A mathematical model is developed, with the help of which analytical expressions for analyzing its metrological characteristics and compiling an engineering calculation procedure are obtained. The obtained data are experimentally confirmed within the acceptable limits. Keywords conical spiral, liner, copper, bronze, coils

Linear and nonlinear vibrations of isotropic rectangular plates resting on full or partial line supports

Linear and geometrically nonlinear free vibration analyses of isotropic rectangular plates with intermediate full or partial line supports in one or two directions are performed by a semi-analytical method. The Rayleigh-Ritz method is used first to calculate the plates examined linear frequency parameters and associated mode shapes which are then used as trial functions in the geometrically non-linear analysis. The line supports are modelled by distributions of translational springs contributing to the whole system strain energy with a stiffness tending to infinity at the limit case of rigid supports. The trial plate functions used in the RRM are obtained as products of x- and y- beam functions satisfying the appropriate boundary conditions only at the two ends and omitting the intermediate line supports. The calculation concept is classical, clear and requires no complicated mathematical tools for all aspect ratios, various edge conditions and any number of full or partial line supports in one or two directions. The mode shapes obtained from linear analysis are used in the non-linear theory to estimate the non-linear frequencies and mode shapes amplitude dependence at large vibration amplitudes of many multi-span plates. The problem of partial supports which are smaller than the size of the plate is also treated.

POD associated with modal projection-based reduced order model for prestress structural vibrations

The study deals with the computation of the linear vibrations of prestress elastic structures. The originality of the proposed approach is based on a Proper Orthogonal Decomposition (POD) associated with a modal projection-based Reduced Order Model (ROM) to evaluate the natural frequencies of an elastic structure in function of a static pressure load parameter. It is shown that a finite number of POD modes combined with linear prestress eigenmodes are able to evaluate efficiently the frequencies function of the evolution parameter. The generated reduced order model can be used in future works on parametric studies, sensitivity analyses, optimization procedure or feedback control loops systems.

Decrease of the air breakdown threshold by the subcritical streamer discharge when the initiator is placed near the metal screen

The influence of the screen size on the air breakdown by an induced subcritical streamer discharge is discussed based on the data of physical and computational experiments. A cylindrical metal vibrator mounted at a certain distance from the screen is used to initiate a streamer discharge. The initiator length is comparable with the half wavelength of electromagnetic field. Data on the effect of the distance from the emitting aperture to the screen and its width on the breakdown pressure are given. The dependence of the field amplification factor at the poles of a linear vibrator on the size of the reflector screen is obtained. Keywords: breakdown, streamer discharge, microwave discharge, plasma-assisted combustion.

Снижение порога пробоя воздуха подкритическим стримерным разрядом при размещении инициатора разряда вблизи металлического экрана

The influence of the screen size on the air breakdown by an induced subcritical streamer discharge is discussed based on the data of physical and computational experiments. A cylindrical metal vibrator mounted at a certain distance from the screen is used to initiate a streamer discharge. Data on the effect of the distance from the emitting aperture to the screen and its width on the breakdown pressure are given. The dependence of the field amplification factor at the poles of a linear vibrator on the size of the reflector screen is obtained.

Two-dimensional Coulomb plasmon-excitons, their spectrum and near-field excitation

A theory of two-dimensional non-radiative (Coulomb) plasmon-excitons is developed for atomically thin metal and semiconductor layers located nearby. An exactly solvable model of 2D Coulomb coupled excitations is proposed, and the equations of motion for their polarization are deduced within the classical electrodynamics. The spectra of plasmon-exciton waves are investigated, and the dispersion relations are shown to display the frequency anticrossing effect near the resonance between plasmons and excitons. The dynamics of 2D polarization waves is studied for the non-radiative plasmon-excitons under time-dependent near-field excitation by an external dipole. The theory reveals principal and practically valuable dynamical analogies of plasmon-excitons with coupled oscillations in other objects of linear vibration theory, such as mechanical oscillators, resonant electric chains, etc.

Vibrations of a Cylindrical Sandwich Shell with a Honeycomb Core Made Using FDM technology

Presented is a model of the dynamic deformation of a three-layer cylindrical shell with a honeycomb core, manufactured by fused deposition modeling (FDM), and skins reinforced with oriented carbon nano-tubes (CNT). A ULTEM 9085 thermoplastic-based honeycomb core is considered. To analyze the stress-strain state of the honeycomb core, a finite element homogenization procedure was used. As a result of this procedure, the dynamic response of the honeycomb core is modeled by a homogeneous orthotropic material, whose mechanical properties correspond to those of the core. The proposed model is based on the high-order theory, extended for the analysis of sandwich structures. The skin displacement projections are expanded along the transverse coordinate up to quadratic terms. The honeycomb core displacement projections are expanded along the transverse coordinate up to cubic terms. To ensure the integrity of the structure, shell displacement continuity conditions at the junction of the layers are used. The investigation of linear vibrations of the shell is carried out using the Rayleigh-Ritz method. For its application, the potential and kinetic energies of the structure are derived. Considered are the natural frequencies and modes of vibrations of a one-side clamped cylindrical sandwich shell. The dependence of the forms and frequencies of vibrations on the honeycomb core thickness and the direction of reinforcement of the shell skins have been investigated. It was found that the eigenforms of a sandwich shell are characterized by a smaller number of waves in the circumferential direction, as well as a much earlier appearance of axisymmetric forms. This means that when analyzing the resonant vibrations of a sandwich shell, it is necessary to take into account axisymmetric shapes. Changing the direction of reinforcement of the skins with CNTs makes it possible to significantly influence the frequencies of the natural vibrations of the shell, which are characterized by a nonzero number of waves in the circumferential direction. It was found that this parameter does not affect the frequencies of the axisymmetric shapes of the shell under consideration.

Protection of precision spacecraft equipment from internal sources of vibration

The work is devoted to the protection of a spacecraft from the influence of unacceptable internal vibration sources. The urgency of reducing the vibration activity on board the spacecraft to improve the accuracy of the target equipment is indicated. A particular problem of vibration protection of the spacecraft platform from a vibration source – an electric pump unit of a liquid thermal control system – is being solved. The basic requirements for electric pump unit vibration protection have been determined. Possible ways to reduce the level of vibration excited by the electric pump unit on the surface of the spacecraft fixation are considered. Particular attention is paid to such vibration protection methods as damping and vibration isolation, implemented by installing special vibration protection devices between the source (electric pump unit) and the object (spacecraft) – vibration isolators and vibration dampers. The principles of operation of vibration dampers and vibration isolators, the most common materials for vibration dampers are described. Examples of constructive solutions for linear single-axial vibration isolators are considered, recommendations for the use of promising products are developed. Particularemphasis is placed on the use of metal rubber as a material for vibration isolators. With regard to a specific design of electric pump unit, a diagram of the spatial structure of vibration isolation is proposed. Formulas for calculation are given in detail, a mathematical model of the vibration isolation system is developed. The procedure for calculating the parameters of the system has been formed. Based on the model, the maximum possible level of vibration suppression in the mid-frequency region was determined. Minimum required number of operable pixels was identified for monitoring the water surface with sufficient accuracy and reliability.

Lateral Parametric Vibration of Footbridge under Pedestrian Excitation considering the Time-Delay Effect

Pedestrian excitation may consequently cause large-scale lateral vibration of the long-span softness of footbridges. Considering the influence of structural geometric nonlinearity, a nonlinear lateral parametric vibration model is established based on the relationship between force and speed. Taking the London Millennium Footbridge as an example, the Galerkin method is applied to formulate parametric vibration equations. In addition, the multi-scale method is used to analyze the parametric vibration of footbridge system theoretically and numerically. The paper aims to find out the reasons for the large-scale vibration of the Millennium Footbridge by calculating the critical number of pedestrians, amplitude-frequency, and phase-frequency characteristics of the Millennium Footbridge during parametric vibration. On the other hand, the paper also studies the influence parameters of the vibration amplitude as well as simulates the dynamic response of the bridge during the whole process of pedestrians on the footbridge. Finally, the paper investigates influences of the time-delay effect on the system parameter vibration. Research shows that: the model established in the paper is reliable; the closer the walking frequency is to two times of the natural frequency, the fewer number of pedestrians are required to excite large vibrations; when the number of pedestrians exceeds the critical number in consideration of nonlinear vibration, the vibration amplitude tends to be stable constant-amplitude vibration, and the amplitude of vibration response is unstable constant-amplitude vibration when only linear vibration is considered; the following factors have an impact on the response amplitude, including the number of pedestrians on footbridge per unit time, damping, initial conditions, and the number of pedestrians in synchronized adjustment. At last, when considering the lag of the pedestrian’s force on the footbridge, the time-lag effect has no effect on the amplitude but has an effect on the time needed to reach a stable amplitude.