Showing papers in "International Journal of Non-linear Mechanics in 2022"

Static and dynamic instability modeling of electro-magneto-active polymers with various entanglements and crosslinks

Abstract: Electro-magneto-active (EMA) polymers-based smart actuators suffer from electro-magneto-mechanical instability (EMMI) phenomena arising due to the positive feedback between the applied electric and magnetic fields and reduction in the polymer thickness. This work presents a theoretical investigation of EMMI phenomena of an EMA polymeric actuator in both static and DC dynamic modes of operation. A physics-based non-affine material model is adopted here for capturing the effect of the polymer chain entanglements and crosslinks on the instability phenomena. Simultaneously, a computationally efficient energy approach for obtaining the DC dynamic instability parameters is constructed, which depends on the energy balance at the position of maximal overshoot in an oscillation cycle. The findings from the current study illustrate the trends of variation in the deformation, electric field, and magnetic field at the onset of static and dynamic EMMI with the polymeric entanglements and crosslinks parameters. It is observed that large entanglements and crosslinks in polymer chains improve the stable operation travel range, which positively impacts the actuator performance.

Global dynamics and performance of vibration reduction for a new vibro-impact bistable nonlinear energy sink

Abstract: A new vibro-impact bistable nonlinear energy sink (VI-BNES) with relatively light weight attached to a damped linear oscillator (LO) is proposed to form a two-degree-of-freedom nonlinear mechanical system with bilateral rigid constraints. The state of the system after instantaneous impact is described by the simplified shock theory and the momentum conservation. When the LO subjected to impulse excitations with different levels is considered as the driver of the VI-BNES, a two-dimensional non-smooth reduced impact system with a harmonic perturbation and an irrational nonlinear restoring force is obtained to study the threshold for the homoclinic bifurcations and chaotic responses of the VI-BNES. Bistability of this mechanical model makes the reduced unperturbed system have a pair of symmetric homoclinic orbits as the pseudo-separatrix for in-well and cross-well oscillations. The Melnikov method developed by us for non-smooth planar systems with bilateral rigid constraints is firstly employed to detect the threshold of the amplitude of excitations of the LO for the onset of chaotic oscillations. Furthermore, cubic polynomial approximation of the restoring force on the premise of maintaining the equilibria of the system is carried out such that the analytical Melnikov analysis for the global dynamics becomes possible. The performance of the VI-BNES is also studied and compared with other NESs in detail through numerical analysis and experiments to show the high efficiency of vibration reduction of the VI-BNES due to the dual beneficial role of the bistability and the impacts with the bilateral rigid constraints. • The structure of a vibro-impact bistable nonlinear energy sink, abbreviated as VI-BNES, is novel to combine both the characters of bistability and vibro-impacts. • The Melnikov method for non-smooth systems developed by us is firstly introduced to analyze the global dynamics and detect the chaotic threshold and homoclinic bifurcations of the non-smooth two-degree-of-freedom coupled systems. • Numerical method for vibro-impacts coupled systems is improved to show the good performance in reducing vibration of the VI-BNES, and experiments are designed to verify the high efficiency of vibration reduction for the VI-BNES.

Geometrical nonlinear numerical frequency prediction of porous functionally graded shell panel under thermal environment

Abstract: The nonlinear eigenfrequency responses of a functionally graded material (FGM) panel in a thermal environment are numerically estimated in the present article using the finite element method (FEM). The constituents of the FGM are considered as the function of the temperature. For the evaluation of material properties of the FGM panel, Voigt’s micromechanical model is used in conjunction with three distinct types of material distribution patterns, namely power-law (PL), sigmoid (SM), and exponential (EN). Also, two kinds of porosity distributions, i.e. even (PT-I) and uneven (PT-II) through the panel thickness are considered in the present work. HSDT kinematics and Green–Lagrange nonlinear strain terms are employed to prepare a mathematical model of the FG panel. The governing equation is obtained using Hamilton’s principle, and a direct iterative technique is used to compute the final vibration responses. The convergence and validation are performed to check the stability and accuracy of the proposed model. Afterwards, several numerical examples are solved to demonstrate the proposed model efficacy in terms of currently adopted input parameters on the frequency (nonlinear) responses deliberated in detail.

Nonlinear and linearized vibration analysis of plates and shells subjected to compressive loading

Abstract: The present work provides a numerical model for carrying out virtual Vibration Correlation Technique (VCT) for computing the buckling load, identifying the natural frequencies variation with progressive higher applied load, and providing an efficient means for the verification of the experimental VCT results. The presented nonlinear approach is based on the Carrera Unified Formulation (CUF). Since far nonlinear regimes are investigated, the full Green–Lagrange strain tensor is adopted. Furthermore, geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method. For a robustness assessment of the virtual VCT, different flat panel and shell structures are studied and compared with results found in the available literature. The results prove that the proposed approach provides results with an excellent correlation with the experimental ones, allowing to predict the buckling load and the natural frequencies variation in the nonlinear regime with high reliability. • The paper deals with trivial linearized and full nonlinear vibration analysis of plates and shells. • Finite elements are proposed based on a unified approach. • Buckling and vibration results of structures subjected to compressive loadings are given. • Results are compared with experiments available from the literature. • Depending on the boundary conditions, curvatures and loading, the linearized approach may not be sufficiently representative.

A piezoelectric-electromagnetic hybrid flutter-based wind energy harvester: Modeling and nonlinear analysis

Abstract: This paper presents a piezoelectric–electromagnetic hybrid flutter-based energy harvester (HFEH), where the piezoelectric part and electromagnetic part are coupled with each other by magnetic forces. The working principle is explained in detail and the corresponding theoretical model of the HFEH is established based on Lagrange’s equations, Newton’s second law, Kirchhoff’s law, a semi-empirical nonlinear aerodynamic model, and a magnetic dipole model. The advantages of the HFEH, output performance, and nonlinear output characteristics under different magnet distances and load resistances are analyzed. Results show that the cut-in wind speed and the output power of the HFEH are, respectively, lower and higher than those of the typical flutter-based piezoelectric energy harvester (FPEH). When the distance between magnets A and B and that between magnets C and D is small, the amplitude jump phenomenon occurs, and the electromagnetic part has a satisfactory output power near the jump points. The output power of the piezoelectric part and the electromagnetic part of the HFEH, respectively, reaches 1.35 mW and 36.63 mW at a wind speed of 6.70 m/s. Overall, this study provides a theoretical framework for the design of high-efficiency wind energy harvesters.

Linear and nonlinear mechanical responses of FG-GPLRC plates using a novel strain-based formulation of modified FSDT theory

Abstract: In the present paper, a composite finite element is proposed using the assumed natural strain (ANS) method, which is based on the unified strain interpolations and the minimized potential energy. In the proposed formulation, the linearized weak form of compatibility and equilibrium equations is obtained for the geometrically nonlinear analysis. Additionally, the parabolic strain interpolation functions without any shear correction coefficient are imposed into the first-order shear deformation theory (FSDT) to verify the zero shear stresses condition at the top and bottom surfaces. The iterative arc-length method is applied to handle the post-buckling and free vibration responses. For the micromechanical part, the effective elasticity modulus based on the modified Halpin–Tsai micromechanical model is attained to analyze functionally graded graphene platelet-reinforced composite (FG-GPLRC) plates with and without hole. Furthermore, the effective mass density and the Poisson’s ratio are assessed via the rule of mixture. Therefore, the major novelty of this paper is in detecting the nonlinear thermomechanical behavior of GPLRC plates using robust strain-based interpolations. Hence, nonlinear equilibrium paths are plotted to illustrate the effects of different boundary conditions, weight fractions and distribution schemes on GPLRC plates. • A finite element is proposed ANS method. • Modified FSDT is used to establish the governing equations. • Halpin–Tsai micromechanical model is employed to measure the effective elasticity modulus. • Analysis of thermo-mechanically loaded GPLRC plates is performed.

Nonlinear dynamics and bifurcation analysis of journal bearings based on second order stiffness and damping coefficients

Abstract: Investigating dynamics and stability of rotors supported on journal bearings is a crucial step in the design of an efficient and reliable rotating machine. In the current work, a model for flexible rotor supported on two symmetric journal bearings is investigated. The nonlinear bearing forces are evaluated by either using direct solution of Reynolds equation or analyzing Reynolds equation to obtain linear and nonlinear bearing stiffness and damping coefficients using time dependent second order perturbation method. These coefficients are obtained for different operating conditions and bearing parameters such as length to diameter ratio, groove angle or applied groove pressure. The present results are validated with the previous literature and a perturbation analysis is used to investigate the validity range of the bearing linear and nonlinear coefficients. A novel technique based on polynomial fitting is used to present the bearing coefficient as a function of the bearing parameters. This enables the investigation of the dynamics of flexible rotor model using numerical continuation technique. Also, the effect of the bearing design parameters such as groove angle, length to diameter ratio and static pressure on the system stability is investigated. • Nonlinear journal bearing coefficients are investigated. • Time dependent perturbation method is used to obtain bearing coefficients. • Parametric analysis to the journal bearing coefficients is introduced. • Continuation analysis to the dynamics of rotor supported on Journal bearing is introduced.

Hyperelastic structures: A review on the mechanics and biomechanics

Abstract: Soft structures are capable of undergoing reversible large strains and deformations when facing different types of loadings. Due to the limitations of linear elastic models, researchers have developed and employed different nonlinear elastic models capable of accurately modelling large deformations and strains. These models are significantly different in formulation and application. As hyperelastic strain energy density models provide researchers with a good fit for the mechanical behaviour of biological tissues, research studies on using these constitutive models together with different continuum-mechanics-based formulations have reached notable outcomes. With the improvements in biomechanical devices, in-vivo and in-vitro studies have increased significantly in the past few years which emphasises the importance of reviewing the latest works in this field. Besides, since soft structures are used for different mechanical and biomechanical applications such as prosthetics, soft robots, packaging, and wearing devices, the application of a proper hyperelastic strain energy density law in modelling the structure is of high importance. Therefore, in this review, a detailed classified analysis of the mechanics of hyperelastic structures is presented by focusing on the application of different hyperelastic strain energy density models. Previous studies on biological soft parts of the body (brain, artery, cartilage, liver, skeletal muscle, ligament, skin, tongue, heel pad and adipose tissue) are presented in detail and the hyperelastic strain energy models used for each biological tissue is discussed. Besides, the mechanics (deformation, buckling, inflation, etc.) of polymeric structures in different mechanical conditions is presented using previous studies in this field and the strength of hyperelastic strain energy density models in analysing their mechanics is presented.

Nonstationary stochastic response determination of nonlinear oscillators endowed with fractional derivatives

Abstract: Fractional calculus has been broadly used in diverse engineering applications. In this regard, vibrations of fractional oscillators subject to stochastic loads have attracted considerable attention. This paper proposes a semi-analytical approach to determine the nonstationary response statistics of nonlinear oscillators endowed with fractional derivatives. Specifically, the fractional derivative term is represented by introducing a discretization scheme and associated first-order differential equations. This leads to an augmented dimension dynamic system. Further, in conjunction with the statistical linearization technique, the evolution of the system response is captured by a set of coupled ordinary differential equations with time-dependent coefficients. Furthermore, solving the associated Lyapunov equation for the randomly excited dynamic system yields the nonstationary statistics of the oscillator response. The reliability of the proposed method is demonstrated by Monte Carlo simulations pertaining to classical nonlinear oscillators. • A semi-analytical approach is proposed for nonlinear fractional oscillators. • The system augmentation strategy and statistical linearization technique are adopted. • Boole’s quadrature rule is implemented for enhanced computational efficiency. • Evolution of response displacement and the fractional derivative term is simulated. • The results are in satisfactory agreement with pertinent Monte Carlo simulations.

Random vibration analysis of vibro-impact systems: RBF neural network method

Abstract: The recent success of the radial basis function neural networks (RBFNN) method for random vibration analysis of smooth systems fuels in speculations that this approach may be extended to non-smooth problems. However, very little is known on the applicability of this approach to non-smooth systems. This work generalizes the RBFNN method to the randomly excited non-smooth vibro-impact system (VI-S). We first transform the non-smooth VI-S system to a continuous nonlinear system. Then, the solution of the reduced Fokker–Plank–Kolmogorov (FPK) equation for the transformed VI-S is expressed in terms of the RBFNN with Gaussian activation functions. The weights of the RBFNN are determined by solving an optimization problem to minimize the reduced FPK equation residual subjected to the constraint of the normalization condition. Three examples are presented to demonstrate the validity of the suggested scheme. Several remarks on the solution process also presented. All the results confirm the applicability and validity of the RBFNN method in dealing with the randomly excited non-smooth VI-S. • The RBFNN is extended to random vibration analysis of non-smooth systems for the first time. • Sampling technique is adopted to avoid intensive integration. • New guidelines on sampling are conducted to enhance computation accuracy. • Comprehensive numerical comparison shows the validity of proposed scheme.

Jeffery–Hamel flow of a shear-thinning fluid that mimics the reponse of viscoplastic materials

Abstract: The flow between two divergent plane walls with a source at the vertex (Jeffery–Hamel flow) of a shear-thinning fluid, that mimics the response of a class of seemingly viscoplastic materials, is studied. The semi-inverse approach is used to obtain the governing equations for the velocity profile. The third-order non-linear ordinary differential equation governing the flow is solved numerically, and the effects of the material moduli that characterize the fluid, the angle between plane walls, and the inertial term, on the velocity are reported and discussed. Results show that there is a critical angle beyond which flow reversal occurs, and the number and angular extent of inflow regions varies based on the material moduli and the inertial term. • The Jeffery–Hamel flow of a shear-thinning fluid that mimics viscoplastic materials is studied. • Governing equations for the velocity profile are obtained using the semi-inverse approach and solved numerically. • Effects of the material moduli, the angle between the plane walls, and the inertial term on the velocity are analyzed. • Variation of material moduli led to the appearance of flow reversal regions whose number and angular extent changed. • Variation of angle showed a critical angle at which flow reversal takes place.

Vibration analysis of a multi-disk bolted joint rotor-bearing system subjected to fixed-point rubbing fault

Abstract: This study aims to investigate the nonlinear vibration properties of a rotor-bearing system carrying a multi-disk bolted joint with a fixed-point rubbing fault. For this purpose, a mathematical model of a multi-disk bolted joint rotor system is explored to determine the effect of rotor-stator rub-impact and rub-impact positions on the nonlinear dynamics of a certain type of aero-engine compressor. First, the dynamics of the rotor system with a rubbing fault are examined numerically. Then, the effect of the rub-impact position in the multi-disk bolted joint on the nonlinear dynamics is studied. The shaft is modeled using a massless shaft, where the stiffness matrix of the shaft is determined based on the flexibility influence coefficient method. The governing equation of the multi-disk bolted joint is deduced considering both the lateral and bending stiffness between the adjacent disks. The dynamic model of the overall system can then be obtained by combining the shaft and multi-disk bolted joint. The resultant governing equations of motion are solved using direct numerical integration. The numerical results obtained from the established model are compared with those of a similar structure in other literature to verify the accuracy of the method used in this study. Then, the dynamic behavior of the rotor system and the mechanical properties of the multi-disk bolted joint under a rubbing fault are presented. Finally, the chaotic response with the rub-impact force acting on different positions of the multi-disk bolted joint is investigated through numerical simulations. • A unified approach for modeling of the multi-disk bolted joint rotor system is proposed. • Mechanical properties of multi-disk bolted joint under rubbing fault are evaluated. • Dynamic properties of the multi-disk bolted joint rotor system with rubbing fault are studied. • Effect of rubbing position in the multi-disk bolted joint on rotor dynamics was demonstrated.

Lever-type high-static-low-dynamic-stiffness vibration isolator with electromagnetic shunt damping

Abstract: Owing to the ultra-low frequency vibration isolation performance without compromising static stiffness, high-static-low-dynamic-stiffness (HSLDS) vibration isolators (VIs) have advantages over linear vibration isolators. This paper presents a lever-type HSLDS vibration isolator (L-HSLDS-VI) by employing negative resistance electromagnetic shunt damping (EMSD) together with eddy current damping to eliminate the jump phenomenon and thus to improve the stability of L-HSLDS-VIs. The lever inerter system can amplify the mass effect to broaden the isolation band. The theoretical model of L-HSLDS-VIs with EMSD (L-HSLDS-VI-EMSD) was established. The effects of negative resistance and lever ratio on the vibration isolation performance of L-HSLDS-VI-EMSD were investigated analytically and experimentally. The L-HSLDS-VI can provide significant nonlinear stiffness, which can realize the quasi-zero stiffness (QZS), and thus broaden the isolation band. EMSD produces a considerable damping effect to enhance the vibration mitigation performance of L-HSLDS-VI. The combination of the EMSD and nonlinear ECD damping is an efficient approach to improve the vibration isolation performance to overcome the jump phenomenon. This paper utilizes the lever effect to amplify damping effects, which could provide a guideline to modify the performance of HSLDS-VIs.

Taut domains in transversely isotropic electro-magneto-active thin membranes

Abstract: Actuation devices made of smart polymers typically show various instabilities, which can adversely affect their performance and lead to device failure. In general, smart polymers exhibit wrinkling instability when subjected to an electric or magnetic field. At the same time, wrinkles can be used constructively in certain applications demanding a controlled alternation of the surface morphology. Critical factors influencing thin films’ pull-in and wrinkling instabilities are discovered concerning the anisotropic taut domains with an applied electro-magneto-mechanical field control. A continuum mechanics-based electro-magneto-mechanical model is developed for predicting the thresholds on the taut domains in the plane of principal stretches. Also, the concept of natural width under simple tension is implemented to derive the coupled nonlinear equation that evaluates the associated taut domains. The findings of the model solution indicate that the extent of taut domains can be controlled by modifying the level and the principal direction of the transverse isotropy. Additionally, the taut domain for a particular level of applied electromagnetic field increases with an increase in the anisotropy parameter, while it depleted with an increase in the fiber orientations from 0° to 90° for an applied level of electromagnetic loading.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

Abstract: In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found. • A novel nonlocal anisotropic elastic shell model for the vibrations of double-walled carbon nanotubes is developed. • Molecular dynamics simulations are performed to obtain the proper value of nonlocal parameter to be adopted. • The estimate of the natural frequencies from the nonlocal model is globally more accurate than that from the local one. • A nonlinear distribution of van der Waals interaction forces between single-walled carbon nanotubes is considered. • The nonlinear response is significantly more hardening than that given by the corresponding linear distribution.

Higher order discontinuity mapping for double grazing bifurcations in a slender rigid block confined between side-walls

Abstract: We investigate the near grazing dynamics in a slender rigid block confined between two side-walls. We obtain conditions of the existence of double grazing orbits of the system, namely, periodic orbits that have two grazing points. By extending the method of discontinuity mapping for grazing orbit with a single grazing point, we compute the lower and the higher order approximations of the Poincaré map respectively near the double grazing orbit. The results of computing Monte Carlo bifurcation diagrams obtained by the lower and the higher order maps respectively are compared with those from direct simulations of the original system. We find that there are large disagreements between the lower order map and the original system. Thus the lower order map is not accurate enough to study double grazing bifurcations. On the other hand, the higher order map can effectively reduce such disagreements and we expect that it plays an important role in the study of the near grazing dynamics of the system.

Stall-induced fatigue damage in nonlinear aeroelastic systems under stochastic inflow: Numerical and experimental analyses

Abstract: This study focuses on characterizing the fatigue damage accumulated in nonlinear aeroelastic systems subjected to stochastic inflows through both numerical simulations and wind tunnel experiments. In the mathematical model, nonlinearities are assumed to exist either in the structure (via a cubic hardening nonlinearity in the pitch stiffness), or in the flow (via dynamic stall condition), or simultaneously in both the structural and aerodynamic counterparts. The aerodynamic loads in the attached flow and dynamic stall conditions are estimated using Wagner’s formulation and semi-empirical Leishman–Beddoes model, respectively. To augment the findings to in-field flow conditions, the oncoming wind flow is considered to be randomly time-varying in nature. The stochastic input flow fluctuations are modeled using a Karhunen–Loeve Expansion formulation. The response dynamics and the associated fatigue damage of the aeroelastic system, possessing different sources of nonlinearities, are systematically investigated under isolated cases of deterministic and stochastic input flows. Specifically, the pertinent role of stochasticity in the input flow is brought out by presenting the response dynamics and the associated fatigue damage accumulation for different values of noise intensity and time scale of the input flow fluctuation. It is demonstrated that under fluctuating flow conditions, the dynamics intermittently switch between attached flow and the dynamic stall regimes even at low mean flow speeds. The intermittent nature of the response varies as the time scale and intensity of the oncoming flow are varied. The role of torsional stresses as the predominant component dictating the fatigue damage accumulation irrespective of the source of nonlinearity is illustrated. Using the rainflow counting method and Miner’s linear damage accumulation theory, it is shown that the accumulated fatigue damage is substantially higher under stochastic flow conditions as compared to deterministic input flows. Importantly, it is observed that different time scales and intensities of the oncoming flow fluctuation play a pivotal role in dictating the fatigue damage in aeroelastic systems. Finally, fatigue damage is observed to be significantly higher for torsionally dominant oscillations in the dynamical stall regime compared to the oscillations at the attached flow regime. The numerical findings are strengthened by drawing comparisons with the preliminary results obtained from wind tunnel experiments performed on a NACA 0012 airfoil undergoing dynamic stall. To the best of our knowledge, this is the first study that systematically bridges the dichotomy between the stall-induced dynamical signatures in stochastic aeroelastic systems and maps the same to the corresponding structural damage. • Role of coupled nonlinearities and stochastic inflows on the aeroelastic dynamics is studied. • Coupled nonlinearities is via a cubic hardening stiffness and dynamic stall behavior. • Effect of time scales and intensity of stochastic inflow on aeroelastic responses is analyzed. • Accumulated fatigue damage is computed via rainflow counting algorithm. • Fatigue damage is higher for dynamic stall cases than in attached flow conditions.

Nonlinear dynamics of mode-localized MEMS accelerometer with two electrostatically coupled microbeam sensing elements

Abstract: In the presented work, a model of a microelectromechanical accelerometer with two microbeam sensing elements located between two fixed electrodes is proposed. The action of the inertia forces in the longitudinal direction affects the spectral properties of the system. The dynamics of the system in the presence of a weak electrostatic coupling between the sensitive elements is characterized by the phenomenon of modal localization - a significant change in the amplitude ratios for the forms of in-phase and anti-phase oscillations with small changes in the applied acceleration. Diagrams of equilibrium positions are obtained varying the potential difference between a fixed electrode and a movable element and between two movable elements. The dependences of the frequencies and the ratio of the components of the eigenvectors on the magnitude of the inertial action are investigated. It is concluded that the amplitude sensitivity of the proposed sensor is orders of magnitude higher than frequency shift sensitivity. Peculiarities of the sensor nonlinear dynamics is investigated in case of external harmonic electrostatic actuation. Principal difference in the characteristics of mode localization phenomenon is revealed between the linearized model describing the modal characteristics of the system and the model describing the real dynamic mode of operation taking into account nonlinear effects.

Simultaneous identification of structural damage and nonlinear hysteresis parameters by an evolutionary algorithm-based artificial neural network

Abstract: An evolutionary algorithm-based artificial neural network (ANN) to identify structural damage and nonlinear hysteresis parameters simultaneously is presented. To avoid the ‘dimensional disaster’, the principal component analysis technique is applied to eliminate the redundant dimensionality of the acceleration data. The acquired principal components, which covers over 95% of the variability in data, are then employed as the input to the ANN, while the system parameters to be identified are defined as the output. ANN is an effective tool to tackle complex problems in numerous fields. However, if using the gradient-descend algorithm to train the ANN model, with vanishing or exploding gradients, ANN may suffer the local minimal during the training process. To address this drawback, a new evolutionary algorithm, termed the K-means Jaya, is employed to train the ANN model to obtain optimal weights and biases by minimizing the discrepancies between real outputs and desired ones. The optimal weights and biases are then used to configure the ANN. The proposed method is applied to single and multiple degree-of-freedom nonlinear systems subjected to different external loadings. Results demonstrate the structural damage and nonlinear hysteresis parameters can be accurately identified.

Nonlinear stochastic dynamics of detuned bladed-disks with uncertain mistuning and detuning optimization using a probabilistic machine learning tool

Abstract: The paper deals with the nonlinear stochastic dynamics concerning the detuning optimization in presence of random mistuning of bladed-disks with geometrical nonlinearities. We present an efficient computational methodology for reducing the computational cost, an analysis of the detuning, and the detuning optimization, based on the use of a high-fidelity computational model. A deep computational analysis is presented for a 12-bladed-disk structure that is representative of industrial turbomachines in order to understand the role played by the geometrical nonlinearities on the dynamical behavior and to exhibit the consequences on the detuning effects. For the detuning optimization with a very large number of possible detuned configurations, we propose a reformulation of the combinatorial optimization problem in a probabilistic framework, which is adapted to a probabilistic machine learning tool in order to limit the number of evaluations of the cost function with the high-fidelity computational model. The methodology proposed is validated for the 12-bladed-disk structure for which the exact optimal detuned configuration has been identified. A very good prediction is obtained.

Intelligent modeling of nonlinear dynamical systems by machine learning

Abstract: An intelligent data-driven method of modeling nonlinear dynamical systems named as LSTM network with output recurrence (OR-LSTM) is proposed, which can learn the inherent characteristics of the dynamical systems from data and predict the states of the systems given external excitation and initial conditions. For the proposed OR-LSTM architecture, the most important key point is output recurrence connections. In order to evaluate it, a teacher-forcing LSTM network without output recurrence connections (TF-LSTM) is designed for comparison. The differences between OR-LSTM and TF-LSTM from the perspective of gradient propagation and parameter updating are analyzed firstly. To verify the capability of the LSTM networks in dynamical systems modeling, the oscillation of a 42-story reinforced concrete frame-core tube building under severe and extreme seismic excitations are modeled. The results show that the OR-LSTM network performs well and maintains the long-term robustness in earthquake responses prediction, even only known displacements of 9 stories. However, the TF-LSTM network fails. In order to evaluate the extensive practicability of the OR-LSTM network, the famous Van der Pol dynamic system and Lorenz dynamic system with strong nonlinearity are also modeled by the proposed method. The results show that OR-LSTM network is competent in modeling most of nonlinear dynamic systems, while has limited in fully chaotic systems, but the predictable length of time sequence and prediction accuracy are improved.

Seismic analysis of liquid storage tank using oblate spheroid base isolation system based on rolling friction

Abstract: The seismic isolation of ground-supported cylindrical liquid storage tank using the newly developed oblate spheroid base isolation (OSBI) system is investigated. The OSBI is an ellipsoidal-shaped isolator in both of its horizontal axes and dissipates seismic input energy in rolling friction apart from period lengthening achieved through the oblateness. The detailed mathematical formulation and governing dynamical equations of motion for a tank mounted on the non-linear OSBI are derived and analyzed using the numerical integration technique to unravel the underlying mechanics behind its dynamic behavior. An idealized mechanical model of the tank consists of three-lumped masses: convective, impulsive, and rigid. The dynamic responses are investigated for the tank subjected to uni-directional sinusoidal harmonic motion as well as to five time-varying, bi-directional horizontal components of earthquakes. The seismic response of the liquid storage tanks with the OSBI system is compared with that of the same tank either isolated using pure-friction (P-F) system or kept non-isolated. In order to understand the dynamic behavior of the tank mounted on the OSBI system, the influence of different isolator parameters such as eccentricity, coefficient of rolling friction, and aspect ratio of the tank are studied. The energy responses are also evaluated to assess the performance of the OSBI system. From the present study, it is found that the OSBI system is effective in mitigating the seismic response in the liquid storage tanks.

Effect of odd-viscosity on the dynamics and stability of a thin liquid film flowing down on a vertical moving plate

Abstract: In this study, we focus on the problem of hydrodynamic instability of a thin, viscous, Newtonian liquid film with broken time-reversal-symmetry flowing down along the surface of a vertical moving plate. The presence of odd viscosity gives rise to new terms in the pressure gradient of the flow. Utilizing the classical long-wave perturbation method, we obtain the analytical solutions as well as derive the nonlinear evolution equation of Benney-type in terms of film thickness h ( x , t ) which is significantly modified due to the presence of odd viscosity in the liquid. We solve the linear model by using the normal mode approach and for three different conditions, namely, the quiescent, up-moving and down-moving plate velocity. The linear study shows that the effect of the down-moving motion of the vertical plate is to enhance the stability of the film flow whereas the up-moving motion of the vertical plate tends to reduce it. Further, the study shows that odd viscosity always has a stabilizing effect on the flow field. In addition, the Orr–Sommerfeld equation is also derived and solved analytically to obtain the critical Reynolds number. Finally, we show the numerical solution of the evolution equation in a periodic domain which clearly demonstrates the role of odd-viscosity on the dynamics of the plate motions of thin film flows coating in isothermal environments. Our study clearly shows how odd viscosity influences the stability of the flow. • Odd-viscosity plays a vital stabilizing role on the flow. • The moving style of the plate significantly affects the stability of the flow. • Nonlinear stability analysis by means of numerical simulations.

Nonlinear supersonic flutter of sandwich truncated conical shell with flexible honeycomb core manufactured by fused deposition modelling

Abstract: The self-sustained vibrations of sandwich conical shell with a flexible honeycomb core manufactured by fused deposition modeling are studied. The interaction of the sandwich thin-walled structure with supersonic gas flow is analyzed. The geometrical nonlinearity of the shell structure is taken into account to predict the self-sustained vibrations. Vibrations of the structure layer are described by three displacements of the middle surface of each layer and two rotations of the normal to the middle surface. The higher-order shear deformation theory is used to describe the strain–displacement relationships. The self-sustained vibrations of the sandwich conical shell are described by a system of nonlinear ordinary differential equations with respect to the generalized coordinates. The assumed mode method is used to derive the equations. The bifurcations of the nonlinear self-sustained vibrations are analyzed numerically using a continuation technique. Quasi-periodic and chaotic self-sustained vibrations are numerically studied for the shell subjected to different boundary conditions. Numerical results show that the amplitudes of quasi-periodic and chaotic vibrations are significantly larger than the amplitudes of periodic vibrations for this shell.

Nonlinear interaction of the damped large dynamic deformations of the Mooney–Rivlin hyperelastic plates with the viscoelastic and shear reactions of the supporting substrate

Abstract: In this research, the interaction between the damped large deformation dynamic responses of the Mooney–Rivlin hyperelastic plates and the viscoelastic and shear characteristics of the supporting substrate (that constitute a visco-hyperelastic system) is investigated for various distributions and various time variations of the transverse loads and different boundary conditions. A viscoelastic Winkler–Pasternak model is chosen for the supporting substrate to account for the dissipative, shearing, and supporting features of the foundation. The governing equations of motion are obtained by using Hamilton’s principle, von Karman assumptions, left Cauchy–Green deformation tensor, and a modified classical plate theory whose accuracy is enhanced through the incorporation of an appropriate high-order incompressibility condition. The combination of the nonlinear and coupled motion equations and boundary conditions is solved by an iterative 2D differential quadrature (DQ) spatial-discretization and Newmark’s time-marching methods. The effects of the constitutive parameters of the hyperelastic material, magnitude and type of the distributed loads, the thickness and aspect ratios of the plate, parameters of the Winkler–Pasternak viscoelastic foundation, and boundary conditions on the dynamic lateral deflections of the plate are studied examined during the parametric studies. Results emphasize the role of the visco-hyperelasticity features interaction of the combined plate-substrate system on the vibration suppression and dynamic performance of the structure in different spatial and time-variation patterns of the distributed loads and various boundary conditions. • Interaction between the nonlinear dynamic responses of the hyperelastic plate and viscoelasticity and shear of the support is investigated. • Mooney–Rivlin constitutive model, various load distributions, and different boundary conditions are considered. • The incompressibility condition is employed to propose a fourth-order expression for the strain energy density function. • The motion equations and boundary conditions combination is solved by an iterative 2D-DQ and Newmark methods.

Stochastic bifurcations and transient dynamics of probability responses with radial basis function neural networks

Abstract: Stochastic noise is common in many engineering systems. Stochastic noise and nonlinearity of dynamical systems will lead to the occurrence of complex dynamic phenomena, such as stochastic bifurcations and transitions. In this paper, a radial basis function neural networks (RBF-NN) method is applied to study stochastic bifurcations and transient dynamics of probability response. A Duffing oscillator under additive and multiplicative random noise is presented to show the effectiveness of the proposed solution method. A new type of stochastic bifurcations is found with an increase of system control parameter, which is called a stochastic double P-bifurcation. It occurs when a stationary probability density function (PDF) changes from a single peak to another single one through two double peaks with their maximum exchange. It is revealed that such a double P-bifurcation is linked to two catastrophic bifurcations of its deterministic counterpart. Moreover, these different types of transient evolutions of the PDFs are observed with transitions of probability maximum peak to one, and to the other, as well as to the both of stable invariant sets from different initial PDFs. The evolutionary orientation of transient PDFs aligns with unstable invariant manifolds leading to stable invariant sets. The previous conjecture is confirmed that the noise-induced transient dynamics and bifurcations are dominated by the global topology and its sudden changes of corresponding deterministic systems without noise. • The method of radial basis function neural networks with Gaussian activation functions is extended to study complex bifurcations of transient dynamics of stochastic systems. • Discovered complex stochastic bifurcation scenarios involving two attractors and a saddle. • Discussed the general guidelines for implementing the method of radial basis function neural networks with Gaussian activation functions for nonlinear stochastic systems.

Density inversion phenomenon in porous penetrative convection

Abstract: Penetrative convection occurs in many natural phenomena where an unstable stratified fluid moves into a stable one. This topic is of interest in many research fields like, for example, in geophysics and astrophysics. In the present paper, on taking into account for quadratic density law, the onset of penetrative convection in a horizontal porous layer is investigated. For the problem at stake, since the principle of exchange of stabilities has been proved, the convection can occur only through a secondary stationary motion. The critical Rayleigh numbers for the onset of stationary convection have been found via the linear instability analysis of the conduction solution. Moreover, the nonlinear stability of the motionless state has been investigated via the energy method. Numerical simulations have been performed through Chebyshev-τ spectral method in order to analyse the behaviour of stability and instability thresholds with respect to the upper plane temperature.

On the role of friction and remodelling in cell–matrix interactions: A continuum mechanical model

Abstract: We discuss a mathematical model addressing the mechanical interactions exchanged between living cells and the extra-cellular environment, by specialising our study to focal adhesions. Many biological functions, such as cell locomotion, proliferation or orientation, require living cells to establish stable connections with the extra-cellular matrix. In this respect, focal adhesions anchor cells to the extra-cellular matrix and regulate the transmission of signals in response to internal or external stimuli. Within the adopted mechanical setting, both the focal adhesion and the extra-cellular matrix are described as rectified elastic fibres, subjected to a system of elastic forces. Moreover, the proposed model relies upon two peculiar features. The former aspect concerns the characterisation of a friction-like interaction between the focal adhesion complex and the extra-cellular matrix. Friction does, indeed, play a role in determining stability and growth of focal adhesion. The latter phenomenon considered in the present model is remodelling. Here, remodelling is understood as the occurrence and development of irreversible, plastic-like distortions related to the internal structure of both the focal adhesion complex and the extra-cellular matrix. The obtained formulation encompasses and generalises a class of models formerly proposed in the literature to the case where friction-like interactions and remodelling of both focal adhesion and substratum are taken into account. The reported numerical results illustrate the influence of both friction and remodelling on the distribution of tractions, displacements and effective stiffness of the ensemble comprising focal adhesion, extra-cellular-matrix and interacting cells.

Nonlinear behavior and instabilities of a hyperelastic Von Mises truss

Abstract: Recent decades have witnessed a renewed interest in the field of structural stability due to new applications involving smart and deployable structures, micro- and nanocomponents and mechanical metamaterials, among others. In many of these structures multistable behavior is desirable, which can be accomplished by traditional and new materials capable of undergoing large elastic deformations. In this paper the nonlinear behavior, bifurcations and instabilities of a hyperelastic von Mises truss exhibiting multistable behavior is investigated. Most papers dealing with the von Mises truss are restricted to linear elastic materials. Here, the nonlinear equilibrium equations are derived considering elasticity in the fully non-linear range and the incompressible Mooney–Rivlin constitutive law is adopted to model the hyperelastic material. The nonlinear equations are solved by using the Newton–Raphson method and continuation techniques. Then, all equilibrium paths and bifurcation points are obtained and their stability is investigated using the energy criterion. A detailed parametric analysis of shallow and nonshallow trusses under horizontal and vertical loads is conducted. Load and geometric imperfections are considered and their influence on the bifurcation scenario and the truss load carrying capacity is clarified. The influence of the material parameters on the nonlinear response is also examined. The results show that the simultaneous presence of geometric and material nonlinearities leads to several equilibrium paths, some of which are not expected for linear elastic materials or found in the existing literature on nonlinear materials, resulting in several coexisting stable and unstable solutions and a complex potential energy landscape, thus clarifying the influence of the constitutive hyperelastic model on the results. Analytical expressions for the normalized snap-through and pitchfork bifurcation loads are derived as a function of the material parameters, truss geometry and imperfections for practical applications. The influence of Eulerian buckling on the truss load carrying capacity is also investigated and formulas to evaluate the buckling load under both vertical and horizontal loads are derived. The present results may help in the development of new engineering applications where multistability and large deformations are desired.