Showing papers in "International Journal of Non-linear Mechanics in 2020"

Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances

Abstract: Micro and nanoelectromechanical systems M/NEMS have been extensively investigated and exploited in the past few decades for various applications and for probing fundamental physical phenomena. Understanding the linear and nonlinear dynamical behaviors of the movable structures in these systems is crucial for their successful implementation in various novel technologies and to meet the long list of new sophisticated requirements. This paper presents a review for some of the recent topics pertaining to the dynamical behaviors, linear and nonlinear, of M/NEMS resonating structures. First, an overview is presented of the various used dynamical approaches to enhance the sensitivity of resonators for sensing applications. Then a summary is presented of the recent works on the linear and nonlinear mode coupling in M/NEMS resonator. Next, recent research is reviewed on coupled M/NEMS resonators, mechanically and electrically, leading to collective behaviors like mode localization. The final part of the paper discusses analytical approaches that have been developed to better understand and investigate the dynamical behavior of M/NEMS resonators focusing on the perturbation method the multiple time scales.

A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory

Abstract: Among the most studied models in mathematical physics, Timoshenko beam is outstanding for its importance in technological applications. Therefore it has been extensively studied and many discretizations have been proposed to allow its use in the most disparate contexts. However, it seems to us that available discretization schemes present some drawbacks when considering large deformation regimes. We believe these drawbacks to be mainly related to the fact that they are formulated without keeping in mind the mechanical phenomena for describing which Timoshenko continuum model has been proposed. Therefore, aiming to analyze the deformation of complex plane frames and arches in elastic large displacements and deformation regimes, a novel intrinsically discrete Lagrangian model is here introduced whose phenomenological application range is similar to that for which Timoshenko beam has been conceived. While being largely inspired by the ideas outlined by Hencky in his renowned doctoral dissertation, the presented approach overcomes some specific limitations concerning the stretch and shear deformation effects. The proposed model is applied to get the solutions for some relevant benchmark tests, both in the case of arch and frame structures. It is proved that, also when shear deformation effects are of relevance, the enriched, yet simple, model and numerical computation scheme herein proposed can be profitably used for efficient structural analyses of non-linear mechanical systems in rather nonstandard situations.

On thermo-viscoelastic experimental characterization and numerical modelling of VHB polymer

Abstract: Recently, the so-called Very High Bond (VHB in short) tape has proven to be an ideal polymer for producing devices made of electric field-responsive functional materials, e.g., actuators in soft robotics, stretch sensors in wearable devices, and energy harvesters generating power from ambient motions. The VHB polymer is commercially available in several thicknesses. For this study, we have selected VHB 4905 due to its wide use as a common material for dielectric elastomers. The acrylic-based polymer is highly deformable, extremely viscoelastic, and highly sensitive to temperature fluctuations. Hence, in order to understand its mechanical and electro-mechanical behaviour, extensive experiments need to be conducted to unravel temperature dependencies in addition to strain-rate dependences. In this study, we present a wide variety of temperature experiments ranging from -30 ° C to 80 ° C at various strain rates and stretch levels under homogeneous deformation and temperature fields. The study demonstrates a pronounced influence of the temperature on the mechanical response of the VHB polymer. For VHB within the temperature range of our study, an increased temperature mechanically softens the material and vice-versa. After conducting a wide range of experiments, we propose a finite strain thermo-viscoelastic constitutive model that is an extension of a phenomenologically-motivated model where a non-linear evolution law is devised based on the classical concept of the multiplicative decomposition of the deformation gradient. Then, decoupled one-dimensional equations are derived and fitted to experimental data to identify relevant material parameters appearing in the model. The thermo-viscoelastic model validation shows its reasonably good capability to predict the experimental results.

Microstructurally-guided explicit continuum models for isotropic magnetorheological elastomers with iron particles

Abstract: This work provides a family of explicit phenomenological models both in the F − H and F − B variable space. These models are derived directly from an analytical implicit homogenization model for isotropic magnetorheological elastomers (MREs), which, in turn, is assessed via full-field numerical simulations. The proposed phenomenological models are constructed so that they recover the same purely mechanical, initial and saturation magnetization and initial magnetostriction response of the analytical homogenization model for all sets of material parameters, such as the particle volume fraction and the material properties of the constituents (e.g., the matrix shear modulus, the magnetic susceptibility and magnetization saturation of the particles). The functional form of the proposed phenomenological models is based on simple energy functions with small number of calibration parameters thus allowing for the description of magnetoelastic solids more generally such as anisotropic (with particle-chains) ones, polymers comprising ferrofluid particles or particle clusters. This, in turn, makes them suitable to probe a large set of experimental or numerical results. The models of the present study show that in isotropic MREs, the entire magnetization response is insensitive to the shear modulus of the matrix material even when the latter ranges between 0.003- 0 . 3 MPa , while the magnetostriction response is extremely sensitive to the mechanical properties of the matrix material.

3D printed elastomeric polyurethane: Viscoelastic experimental characterizations and constitutive modelling with nonlinear viscosity functions

Abstract: Digital Light Synthesis (DLS) technology creates ample opportunities for making 3D printed soft polymers for a wide range of grades and properties. In DLS, a 3D printer uses a continuous building technique in which the curing process is activated by an ultra-violet (UV) light. In this contribution, EUP40, a recently invented commercially available elastomeric polyurethane (EPU) printed by the DLS technology, is experimentally characterized. For characterizing the mechanical properties, an extensive viscoelastic experimental study on the digitally printed EPU taking the strain rate-dependence are conducted. The study reveals a significant time-dependency on its mechanical responses. Moreover, the material demonstrates noticeable nonlinear viscosities that depend on strain and strain rates. Based on the experimental findings for the printed elastomer, a large strain viscoelastic model is devised where evolution laws are enhanced by strain and strain rate-dependent nonlinear viscosities. Following identifications of relevant material parameters, we validate the model with the experimental data that show its good predictability. Such an extensive experimental study along with a constitutive model will help in designing and simulating more complex cellular and structured metamaterials using 3D printed elastomeric polyurethanes.

Active control for acoustic wave propagation in nonlinear diatomic acoustic metamaterials

Abstract: Wave propagation through nonlinear acoustic metamaterials has generated numerous scientific interests for their enormous potential in practical applications these years. This study focuses on the effects of nonlinearity on the band properties of diatomic mass-in-mass chain with active control. By applying the Lindestedt–Poincare (L–P) perturbation method, analytical dispersion relations of the linear and nonlinear diatomic mass-in-mass system have been established and investigated by numerical simulation. Different from the monatomic mass-in-mass chain, this two mass-in-mass units forming a unit cell of the periodic structure results in four branches of the dispersion relation. The effects of nonlinearity on the band gaps of the system have been exhaustively illustrated. By only tuning the nonlinear constitutive relation parameter of the spring, the fourth branch and the third gap are found to be more sensitive compared to the other branches and gaps. It is concluded that closing and re-opening of the band-folding-induced gap in this nonlinear system is still possible. Here, a piezoelectric spring model is applied to the diatomic mass-in-mass to make the system available for wider applications. With the negative proportional control, a new stop band is generated which can be also captured in the monatomic nonlinear system. The new results here will help better analyze the band gap properties in nonlinear mechanical metamaterials and emphasize the great potentials of the topological analysis of such a nonlinear local resonance system that induces band-folding-induced band gaps.

Multi-stable nonlinear energy sink for rotor system

Abstract: A multi-stable nonlinear energy sink (MNES) with piecewise linear stiffness and magnet negative stiffness is developed to suppress the vibration of unbalanced rotor system. The specific structure of the MNES is developed, and the working principles of the piecewise linear stiffness and negative stiffness are introduced. Then, the dynamic equations of the rotor-MNES system are established. Based on these, the transient and steady state responses of the rotor-MNES system are numerically studied. In addition, the vibration suppression ability of the MNES is analysed and compared with that of bi-stable nonlinear energy sink (BNES). Finally, experiments are carried out to verify the effectiveness of the MNES. The numerical and experimental results show that the designed MNES has a strong vibration suppress ability and can withstand a wide range of energy.

Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

Abstract: In this paper, an efficient numerical strategy is used to study the geometrically nonlinear static bending of functionally graded graphene platelet-reinforced composite (FG-GPLRC) porous plates with arbitrary shape. Porous nanocomposite plates including cutout with various shapes can be modeled by the present approach. Four types of porous distribution scheme and four GPL dispersion patterns are selected, and the material properties are calculated based on the closed-cell Gaussian random field scheme, the Halpin–Tsai micromechanical model together with the rule of mixture. First, the variational statement of governing equations based on the virtual work principle and higher-order shear deformation theory (HSDT) is derived and presented in vector–matrix form for computational aims. Then, using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a numerical approach called as VDQ-FEM is used to address the considered problem. In VDQ-FEM, the domain of problem is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to obtain the set of Studying effects of porosity and GPL distributions and porosity coefficient matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of the present work in implementing VDQ-FEM is proposing an efficient way based on mixed-formulation to guarantee the continuity condition of first-order derivatives in entire domain for the used HSDT model. A detailed parametric study is conducted to investigate the nonlinear bending of FG-GPLRC porous plates with different shapes. In the numerical results, the effects of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions on the nonlinear bending response of plates are analyzed.

Exploiting nonlinearity to enhance the sensitivity of mode-localized mass sensor based on electrostatically coupled MEMS resonators

Abstract: An ultrasensitive mass sensor is proposed by combining the benefits of mode localization and nonlinear dynamics in two clamped–clamped microbeams of different lengths. The coupling electrostatic stiffness between the two resonators can be tuned for modulating sensitivity, and the actuation voltage applied to the shorter beam can be adjusted in order to overcome mechanical defects such as geometric asymmetry. The analytical dynamic model considering the quadratic and cubic nonlinearities is established and solved by the asymptotic numerical method (ANM) combined with harmonic balance method (HBM), as well as validated by the long-time integration (LTI) method. A parametric study is performed in order to investigate the effects of the coupling voltage, gap ratio, position of added mass and length ratio on the device sensitivity. Beyond the critical Duffing amplitude and while taking advantage of mode localization, it is shown that the device sensitivity in terms of amplitude ratio is significantly enhanced with up to three orders of magnitude higher than the relative shift in resonance frequencies. The proposed model can be used as a design tool to tune the nonlinearity level enabling the performance improvement of multimodal MEMS mass sensors.

Geometrically nonlinear response of a fractional-order nonlocal model of elasticity

Abstract: This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler–Bernoulli beam. The finite nonlocal strains in the Euler–Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a mathematically well-posed formulation which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal Euler–Bernoulli beam are obtained using variational principles. Further, a nonlinear finite element model for the fractional-order system is developed in order to achieve the numerical solution of the integro-differential nonlinear governing equations. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the geometrically nonlinear response of a nonlocal beam subject to various loading and boundary conditions. Although presented in the context of a 1D beam, this nonlinear f-FEM formulation can be extended to higher dimensional fractional-order boundary value problems.

Nonlinear dynamic responses of layered skew sandwich composite structure and experimental validation

Abstract: The time-dependent deflection responses of the mechanically excited layered skew sandwich shell panels are computed numerically via a generic model developed mathematically using the higher-order shear deformation theory including the effects of the large displacement. The model includes the large displacements associated with the structural distortion under the small strain regime through Green–Lagrange nonlinear strain kinematics. The derived nonlinear system governing equation is converted to a set of algebraic form with the help of finite element steps. Subsequently, the time-dependent displacement values are computed numerically through the direct iterative technique including Newmark’s integration scheme. The dynamic deflections of the sandwich structural component under the influence of the externally excited mechanical loading are obtained through a generic computer code (developed in MATLAB) via the nonlinear higher-order finite element model. Before the implementation of the proposed model for the sandwich analysis, the solution stability and accuracy have been established by solving different kinds of numerical example from the published domain. Additionally, a few layered sandwich plates of different face sheet layers have been fabricated and the experimental dynamic data are recorded for the comparison purpose with the help of available modal test rig. Finally, the influences of the different structure-dependent design parameters on the nonlinear dynamic responses are investigated using the presently developed numerical model of skew sandwich shell panel, which also reveals that the present results give more accurate results.

Peridynamic correspondence model for finite elastic deformation and rupture in Neo-Hookean materials

Abstract: This study considers finite elastic deformation and rupture in rubber-like materials under quasi-static loading conditions by employing the bond-associated weak form of peridynamics with nonuniform horizon. The weak form of peridynamic equilibrium equation is derived based on the Neo-Hookean material model with slight compressibility. The nonlocal deformation gradient tensor is computed in a bond-associated domain of interaction using the PD differential operator. This approach is free of oscillations and spurious zero energy modes that are commonly observed in the PD correspondence models. Also, it permits the direct imposition of natural and essential boundary conditions. Its fidelity for predicting large deformation is established by comparison with those of finite element analysis of a rubber sheet with a hole under stretch. Also, its validity for predicting damage is demonstrated through simulations of experiments concerning progressive damage growth and final rupture in polymers undergoing large elastic deformation.

Electrostatic arch micro-tweezers

Abstract: This paper presents a novel electrostatic micro-tweezers to manipulate particles with diameters up to 14 μ m. The tweezers consist of two grip-arms mounted to an electrostatically actuated initially curved micro-beam. It exploits bistable equilibria, resulting from a snap-through instability, to close the separation distance between the two arms allowing them to grasp a large range of objects. The tweezers offer further control beyond the snap-through point, via electrostatic actuation, to increase pressure on larger objects or grasp smaller objects. The tweezers are fabricated in a p-type Silicon on Insulator (SOI) wafer. Euler-Bernoulli beam theory is utilized to derive the governing equation of motion taking into account the arms' rotary inertia and the electrostatic fringing field. A reduced-order model (ROM) is developed utilizing two, three and five symmetric modes in a Galerkin expansion. A finite element model (FEM) is also developed to validate the ROM and to study the arm tips' separation as a function of actuation voltage. The five-mode ROM is found to be convergent and accurate except in the vicinity of the snap-through saddle-node bifurcation. Our analysis shows that the tweezers can manipulate micro-particles with diameters ranging from 5 to 12 μ m with an operating voltage range limited by the snap-back voltage 100 . 2 V and the pull-in voltage 153 . 2 V.

Evaluation of geometrically nonlinear terms in the large-deflection and post-buckling analysis of isotropic rectangular plates

Abstract: The nonlinear mechanical response of highly flexible plates and shells has always been of primary importance due to the widespread applications of these structural elements in many advanced engineering fields. In this study, the Carrera Unified Formulation (CUF) is used in a total Lagrangian framework to analyze the large-deflection and post-buckling behavior of isotropic rectangular plates based on different nonlinear strain assumptions. The scalable nature of the CUF provides us with the ability to tune the structural theory approximation order and the strain–displacement assumptions opportunely. In this work, the Newton–Raphson linearization scheme with a path-following constraint is used in the framework of the 2-D CUF to solve the geometrically nonlinear problems to draw important conclusions about the consistency of many assumptions made in the literature on the kinematics of highly flexible plates. In this regard, the effectiveness of the well-known von Karman theory for nonlinear deformations of plates is investigated with different modifications such as the thickness stretching and shear deformations due to transverse deflection. The post-buckling curves and the related stress distributions for each case are presented and discussed. According to the results, the full Green–Lagrange nonlinear model could predict the nonlinear behavior of plates efficiently and accurately, whereas other approximations produce considerable inaccuracies in the case of thick plates subjected to large rotations and deflections.

Effect of large displacements on the linearized vibration of composite beams

Abstract: Natural frequencies and mode shapes are functions of the equilibrium state. In the large displacement regime, pre-stresses may modify significantly the modal behaviour of structures. In this work, a geometrical nonlinear total Lagrangian formulation that includes cross-sectional deformations is developed to analyse the vibration modes of composite beams structures in the nonlinear regime. Equations of motion are solved around nonlinear static equilibrium states, which are identified using a Newton–Raphson algorithm along with a path-following method of arc-length type. Different boundary conditions and stacking sequences are analysed. It is shown that vibration modes are strongly modified by nonlinear phenomena. Moreover, models that do not describe those effects accurately may results in misleading results, especially if compression is dominant. In fact, results show a crossing phenomenon in the post-buckling regime of an asymmetric cross-ply beam, whereas it is completely unforeseen by the linearized analysis.

Nonlinear aeroelastic behavior of a base-isolated beam under steady wind flow

Abstract: A homogeneous continuous visco-elastic shear-beam, describing the dynamics of base-isolated tall buildings exposed to a uniformly distributed steady wind flow, is studied. The shear beam is constrained at the bottom end by a nonlinear visco-elastic device and free at the top end. The aeroelastic effects, responsible for self-excitation, are evaluated via the quasi-static theory. The occurrence of Hopf bifurcation is detected. Critical and post-critical behavior is analyzed by applying a perturbation scheme. The critical wind velocity and the associated complex galloping mode are found. The steady value of the oscillation amplitude on the stable limit-cycle and its stability are evaluated as function of the mean wind velocity. The mechanical performances of the structure are investigated in according to the effectiveness of the visco-elastic isolation system. A passive controller is proposed to increase the galloping wind velocity and to reduce the amplitude of the limit-cycle.

Two families of explicit models constructed from a homogenization solution for the magnetoelastic response of MREs containing iron and ferrofluid particles

Abstract: This work puts forth two families of fully explicit continuum or phenomenological models that are constructed by approximating an analytical (but implicit) homogenization solution recently derived for the free-energy function describing the macroscopic magnetoelastic response of two classes of MREs comprised of an isotropic incompressible elastomer filled with a random isotropic distribution of: ( i ) spherical iron particles and ( i i ) spherical ferrofluid particles. Both families are given in terms of free-energy functions W H = W H ( F , H ) that depend on the deformation gradient F and the Lagrangian magnetic field H and are constructed so as to agree identically with the homogenization solution for small and large applied magnetic fields, this for arbitrary finite deformations and arbitrary volume fractions c of particles in the entire physical range c ∈ [ 0 , 1 ] . The accuracy of the proposed phenomenological models is assessed inter alia via the direct comparison of their predictions with that of the homogenization solution for a boundary-value problem of both fundamental and practical significance: the magnetostriction response of a spherical MRE specimen subject to a remotely applied uniform magnetic field.

New insights into the damping characteristics of a typical quasi-zero-stiffness vibration isolator

Abstract: A typical quasi-zero-stiffness (QZS) vibration isolator composed of two lateral springs and a vertical spring has been widely studied in previous literature, in which the dynamic equation is usually formulated with linear viscous damping. However, in practical applications, the damping may be generated from many sources and thus deserves special study. In this paper, a dynamic model with consideration of three main damping generation mechanisms is established to give further insights into the damping characteristics of this typical QZS isolator. The dynamic response and stability are analyzed based upon the newly formulated dynamic equation. The effects of the damping characteristics on the vibration isolation performance are investigated. It is shown that the QZS isolator can exhibit hardening or softening damping properties by tuning parameters of the three damping sources, and the hardening damping is beneficial for the overall isolation performance.

The Well Order Reconstruction Solution for Three Dimensional Wells in the Landau-de Gennes Theory

Abstract: We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by λ , and the well height denoted by ϵ . The WORS are distinctive equilibria reported in Kralj and Majumdar (2014) for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square center. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic surface energies on the top and bottom well surfaces, along with Dirichlet conditions on the lateral surfaces. Moreover, the WORS is globally stable for λ small enough in both cases and unstable as λ increases. We numerically compute novel mixed 3D solutions for large λ and ϵ followed by a numerical investigation of the effects of surface anchoring on the WORS, exemplifying the relevance of the WORS solution in a 3D context.

Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids

Abstract: We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner.

Dynamic buckling of classical/non-classical curved beams by nonlocal nonlinear finite element accounting for size dependent effect and using higher-order shear flexible model

Abstract: This paper investigates the dynamic snap-through buckling of classical and non-classical curved beams subjected to a suddenly applied step load The small scale effect prevalent in non-classical beams, viz, micro and nanobeam, is modeled using the nonlocal elasticity approach The formulation accounts for moderately large deflection and rotation The governing equilibrium equations are derived using the dynamic version of the principle of virtual work and are subsequently simplified in terms of the generalized displacements for the development of a nonlocal nonlinear finite element model The spatial domain comprises of 3-noded higher-order curved beam elements based on shear flexible theory associated with sine function The nonlinear governing equations are solved using the incremental stiffness matrices and by adopting direct time integration method The critical dynamic buckling load is identified by the smallest load at which there is a sudden rise in the amplitude of vibration The efficacy of model here is compared against the available analytical studies for the local and nonlocal beams A detailed study is made to highlight the effects of the geometric parameter, initial condition, nonlocal parameter, load duration, and boundary conditions on the dynamic stability of both classical and non-classical curved beams The nature and degree of participation of various eigen modes accountable for the dynamic snap-through behavior are examined a posteriori using the modal expansion approach Some interesting observations made here are valuable for the optimal design of such structural members against fatigue and instability

Buckling analysis of elastic–plastic nanoplates resting on a Winkler–Pasternak foundation based on nonlocal third-order plate theory

Abstract: The paper analyzes the elastic–plastic buckling behavior of thick, rectangular nanoplates embedded in a Winkler–Pasternak foundation, adopting the Reddy third-order plate theory in nonlocal elasticity. Elasto-plasticity is accounted for by considering two alternative plasticity theories, namely the J 2 flow incremental and the J 2 deformation theory, with material properties defined by a Ramberg–Osgood relation. An iterative procedure is proposed to obtain the critical load, and the corresponding critical mode, of plates simply supported on two opposite edges under applied uniaxial and biaxial loading conditions. Extensive analysis investigates the effects of geometrical, constitutive, and nonlocal parameters on the critical behavior of plates with different boundary conditions. To the best of the authors’ knowledge, there are no findings about elastoplastic buckling of nanoplates in the existing literature. It is therefore hoped that the results obtained may provide a helpful basis for comparison for future investigations.

Nonlinear dynamics of an inclined FG pipe conveying pulsatile hot fluid

Abstract: In this theoretical work, the nonlinear dynamics of a pinned–pinned inclined functionally graded (FG) pipe conveying pulsatile hot fluid is investigated. The equation of motion of the FG pipe is derived based on the Euler–Bernoulli beam theory and plug-flow model, and that is subsequently solved using Galerkin discretization in conjunction with the incremental harmonic balance/Runge–Kutta method. First, the divergence of the FG pipe is investigated where it is mainly revealed that the inclination of the pipe with the vertical axis yields buckling at a higher temperature while the type of the associated bifurcation changes from pitchfork to saddle–node bifurcation. On the basis of this static instability, the pre- and post-buckled equilibriums of the inclined FG pipe are identified, and its nonlinear dynamics associated with each of these equilibrium states is subsequently studied based on the variations of some system parameters namely inclination angle, temperature, graded exponent of FG material, mean flow velocity, amplitude of pulsatile flow velocity and material damping. The corresponding results reveal some notable nonlinear dynamic characteristics of the inclined FG pipe like the appearance of both the principal primary and secondary parametric resonances in the pre-buckled state, exchange between softening and hardening structural behavior through period doubling/period demultiplying/fold bifurcation, movement of saddle periodic orbit with temperature leading to the unequal domains of attraction over the post-buckled equilibriums and the appearance higher order parametric resonances at low material damping.

Novel two-parameter dynamics of impact oscillators near degenerate grazing points

Abstract: Following the previous work on the degenerate grazing bifurcations of impact oscillators, this paper aims to explore novel two-parameter dynamics near the degenerate grazing points using GPU parallel computing technology. By using the technology, a further understanding of the near-grazing dynamics can be developed for impact oscillators. Three main indicators, i.e., the largest Lyapunov exponent, number of excitation periods and number of impacts, are calculated for each grid of the two-parameter plane chosen. Based on these indicators, the dynamic response in the vicinity of degenerate grazing points can be characterized and more dynamic behaviors than the published results can be discovered. Phenomena of coexisting attractors and chaotic transitions including crisis are also discussed. The single and two degree-of-freedom impact oscillators are selected as illustrative examples to demonstrate the results.

Experiment study on nonlinear oscillator containing magnetic spring with negative stiffness

Abstract: The concept of negative stiffness may find extensive applications in wave manipulation and low frequency isolation of vibration. Many realizations use prestressed or buckled components, by which the negative stiffness possesses strong nonlinearity and should be taken into account. In this work, based on an air track platform, we propose a scheme to directly test the dynamics of lumped parameter model of a nonlinear vibration system containing negative stiffness spring. Repelling magnets are used to realize negative springs, and different nonlinearity and rigidity are achieved by varying the clearance in between the magnets. A mass–spring model of coupled oscillators linked by positive and negative springs in series is first investigated, the nonlinear behaviours of both the free vibration and the frequency response agree well with the theoretical analysis of Duffing’s equation adopting cubic approximation of the nonlinear stiffness. A model mimicking the high-static-low-dynamic stiffness isolator is also investigated under nonlinear regime. The proposed facility is flexible and useful for the proof-of-concept validation of relevant systems such as vibration isolators, energy sinkers involving nonlinearity.

Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators

Abstract: The nonlinear coupled vibrations widely exist in coupled resonant structures, which can lead to complex dynamic bifurcation behavior and expand the research scope of fundamental physics. A new micro-mass detection method is proposed by using bifurcation jumping phenomenon in nonlinear electrostatically coupled resonators in this article. Considering the fundamental frequency excitation, the one-to-one internal resonance equations to describe electrostatically coupled resonant sensor are obtained by using Hamilton’s principle and Galerkin method. Then, the perturbation analysis method is introduced to study the response and stability of the system for small amplitude vibration. Through bifurcation analysis, it is found that the isolated response branches appear in nonlinear electrostatically coupled resonators and present the physical conditions of this phenomenon. Typically, we demonstrate the exploitation of the bifurcation jump phenomena of two electrostatically coupled microbeam resonators to realize the mass quantitative detection and threshold detection, which overcomes the detection inaccuracy caused by frequency drift in the nonlinear vibration. Finally, the numerical experiments verify the validity of the method. The results of this paper can be potentially useful in micro-mass detection.

Analytical study on growth-induced bending deformations of multi-layered hyperelastic plates

Abstract: In this work, we study the growth-induced bending deformations of multi-layered hyperelastic plates. First, under the assumption of plane strain, we formulate the 3D governing system for a multi-layered hyperelastic (neo-Hookean) plate, which incorporates the growth parameters in the different layers. Based on the governing system and by adopting a series expansion–truncation approach, we derive a plate equation system. In the case of traction-free boundary conditions, we solve the plate equation system explicitly and obtain some simple analytical results, which can be used to describe the deformations of the multi-layered plates induced by growth. We verify the accuracy and efficiency of the analytical results through comparisons with some numerical and experimental results. We also provide a comparison between the analytical results obtained for single- and multi-layered hyperelastic plates. Furthermore, we analyze the properties of residual stresses in the multi-layered plates. The plate equation system and the analytical results obtained here have wide potential applications in the design of intelligent soft devices.

Non-linear mechanics in resonant inertial micro sensors

Abstract: Microsystems (or Micro Electro Mechanical Systems, MEMS) are important components of many popular products in the consumer market and are one of the enabling ingredients of incoming industrial revolutions like Industry 4.0 and Internet of Things (IoT). Behind many Microsystems there are important mechanical principles and coupling effects that must be completely mastered starting from the design phase. More sophisticated and smaller devices also imply to consider many non-linear effects that can be strictly related to the mechanical response or to coupled electro-thermo-mechanical phenomena. This paper contains an overview of key mechanical aspects in design and reliability of Microsystems with a particular focus on non-linear dynamics of oscillators in inertial sensors.

Nonlinear implementation of Finite Fracture Mechanics: A case study on notched Brazilian disk samples

Abstract: We formalize the coupled criterion (CC) in the framework of Finite Fracture Mechanics (FFM) to take into account the material nonlinear behavior. Once the true stress–strain curve is recorded, we estimate the Ramberg–Osgood (RO) parameters. We then exploit these values in the FFM numerical implementation, considering both an average stress requirement and the energy balance. As a case study, we consider experimental data on Brazilian disks containing a circular hole and involving two materials — PMMA and GPPS. The computational effort of the analysis increases with respect to the linear elastic case, but keeps reasonable. On the other hand, the nonlinear FFM investigation allows a significant improvement of the failure stress predictions, especially those related to smaller holes corresponding to a more pronounced nonlinear behavior detected during tests.

Nonlinear normal modes, resonances and energy exchange in single-walled carbon nanotubes

Abstract: The nonlinear resonance interaction and energy exchange between bending and circumferential flexure modes in single-walled carbon nanotubes is studied. First, the results of an analytical model of the resonance interaction between the considered nonlinear normal modes previously developed are reported. This approach was based on a reduced form of the Sanders–Koiter thin shell theory obtained by using simplifying hypotheses on the shell deformations. The analytical model predicted that the nonlinear resonance interaction leads to energy localization in a certain coherence domain over the carbon nanotube surface within a specific range of the initial oscillation amplitude. Then, a numerical model of the resonance interaction between the analysed nonlinear normal modes in the framework of the complete Sanders–Koiter thin shell theory is reported. Numerical simulations are performed to verify the energy localization phenomenon over the carbon nanotube surface and to compute the threshold values of the initial oscillation amplitude giving rise to energy localization. Finally, from the comparison between the two different approaches, it is obtained that the results of the numerical model for the threshold values of the nonlinear energy localization confirm with very good accuracy the predictions of the analytical model.