Showing papers in "International Journal of Non-linear Mechanics in 2018"
TL;DR: In this paper, the authors investigated some unrevealed features of a newly introduced megastable chaotic oscillator, which has a rich dynamical behavior including limit cycle, torus and strange attractor.
Abstract: In this paper, we investigate some unrevealed features of a newly introduced megastable chaotic oscillator. This oscillator has a rich dynamical behavior including limit cycle, torus and strange attractor. Also it has coexisting self excited and hidden attractors. Such multi-stable oscillator with coexisting self excited and hidden attractors is very rare in the literature. We have studied the oscillator dynamics using an analog circuit. Also a novel fuzzy-based robust and adaptive control method is designed to control this oscillator.
113 citations
TL;DR: In this article, a Euler-Bernoulli beam is coupled to a distributed array of nonlinear spring-mass subsystems acting as local resonators/vibration absorbers.
Abstract: In this work the multi-mode vibration absorption capability of a nonlinear metamaterial beam is investigated. A Euler–Bernoulli beam is coupled to a distributed array of nonlinear spring–mass subsystems acting as local resonators/vibration absorbers. The dynamic behavior of the metamaterial beam is first investigated via the classical approach employed for periodic structures by which the frequency stop bands of the single cell are determined. Subsequently, the frequency response is obtained for the metamaterial beam to study a multi-frequency stop band system by adding an array of embedded nonlinear local resonators. A path following technique coupled with a differential evolutionary optimization algorithm is adopted to obtain the optimal frequency-response curves of the metamaterial beam in the nonlinear regime. The use of the local absorbers, via a proper tuning of their constitutive parameters, allows a significant reduction of the metamaterial beam oscillations associated with the lowest three vibration modes.
80 citations
TL;DR: In this paper, a numerical procedure was proposed to predict nonlinear free and steady state forced vibrations of clamped-clamped curved beam in the vicinity of postbuckling configuration.
Abstract: This paper presents a novel numerical procedure to predict nonlinear free and steady state forced vibrations of clamped–clamped curved beam in the vicinity of postbuckling configuration. Nonlinear Euler–Bernoulli kinematics assumptions including mid-plane stretching are proposed to exhibit a large deformation but a small strain of von Karman. To simulate the interaction of beam with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The nonlinear integro-differential equation that governs the buckling of beam is discretized using the differential-integral quadrature method (DIQM) and then is solved using Newton’s method. The problem of linear vibration is discretized using DIQM and then is solved as a linear eigenvalue problem. Afterwards, a single-mode Galerkin discretization is used to reduce the nonlinear governing equation into a time-varying Duffing equation. The Spectral differentiation matrix operators are exploited to discretize the Duffing equation. The discretized Duffing equation is a nonlinear eigenvalue problem which is directly solved using pseudo arc length continuation method. Results obtained by the proposed numerical solution are compared with analytical solutions available in the literature and good agreement is obtained. Parametric studies are carried out to show the effects of applied axial load, imperfection and nonlinear elastic foundations on the natural frequency as well as forced damped vibration behavior of the beam. The above mention effects play very important role on the dynamic behavior of buckled curved beam.
63 citations
TL;DR: In this paper, a mathematical model was developed to predict the effective material properties of graphene nanoplatelets/fiber/polymer multiscale composites (GFPMC) through a theoretical study.
Abstract: In this paper, a mathematical model was developed to predict the effective material properties of graphene nanoplatelets/fiber/polymer multiscale composites (GFPMC). The large deflection, post-buckling and free nonlinear vibration of graphene nanoplatelets-reinforced multiscale composite beams were studied through a theoretical study. The governing equations of laminated nanocomposite beams were derived from the Euler–Bernoulli beam theory with von Karman geometric nonlinearity. Halpin–Tsai equations and fiber micromechanics were used in hierarchy to predict the bulk material properties of the multiscale composite. Graphene nanoplatelets (GNPs) were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A semi-analytical approach was used to calculate the large static deflection and critical buckling temperature of multiscale multifunctional nanocomposite beams. A perturbation scheme was also employed to determine the nonlinear dynamic response and the nonlinear natural frequencies of the beams with clamped–clamped, and hinged–hinged boundary conditions. The effects of weight percentage of graphene nanoplatelets, volume fraction of fibers, and boundary conditions on the static deflection, thermal buckling and post-buckling and linear and nonlinear natural frequencies of the GFPMC beams were investigated in detail. The numerical results showed that the central deflection and natural frequency were significantly improved by a small percentage of GNPs. However, addition of GNPs led to a lower critical buckling temperature.
48 citations
TL;DR: In this paper, the authors made use of GSRA grant GSRA2-1-0611-14034 from Qatar National Research Fund (a member of Qatar Foundation) to support their work.
Abstract: This publication was made possible by GSRA grant GSRA2-1-0611-14034 from Qatar National Research Fund (a member of Qatar Foundation). The finding achieved herein are solely the responsibility of author.
44 citations
TL;DR: In this paper, a tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested as a solution for preventing contact occurrence between disk and stator as result of undesirable vibrations produced by eccentricity of the disk.
Abstract: Contact occurrence between disk and stator as result of undesirable vibrations produced by eccentricity of the disk is one of the most destructive and common phenomena in rotor dynamics systems. In this work, utilizing tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested as a solution for preventing contact occurrence. The mass and angular position of absorbers determine their efficiency for resisting the eccentricity force produced by the disk, and their stiffness and damping coefficients determine the displacement scope of the absorber. In order to efficiently design absorbers, optimization is proposed. In this suggested optimization process, complex averaging method is used in order for deriving the equations of motion of the system in presence of dynamic absorbers at the steady state condition. Afterwards, for determining trustworthiness of each absorber’s performance, system’s behavior is studied for different values of its parameters such as rotational speed, stiffness, clearance and eccentricity in presence of each absorber. From the obtained results, it can be perceived that TMD and NES are as efficient as possible and they have exactly the same positive influence on the system’s vibrations. The reliability of the proposed optimization process can be determined by the results.
39 citations
TL;DR: In this article, a single-degree-of-freedom (SDOF) oscillator grounded through a linear spring in parallel with a linear viscous damper, and two inclined pairs of linear spring-damper elements forming an initial angle of inclination, ϕ 0, with the horizontal at equilibrium, is considered.
Abstract: A single-degree-of-freedom (SDOF) oscillator grounded through a linear spring in parallel with a linear viscous damper, and two inclined pairs of linear spring–damper elements forming an initial angle of inclination, ϕ 0 , with the horizontal at equilibrium, is considered. It is assumed that there is no pre-compression in any element. An impulsive excitation is applied to this system, and it is shown that, depending on the system parameters, the intensity of the applied impulse and the initial angle of inclination, there are strong stiffness and damping nonlinearities in the transient response induced solely due to geometric effects; these strong nonlinearities occur even though all elastic and dissipative elements of the system are governed by linear constitutive laws. Preliminary numerical simulations indicate that in different regimes of the dynamics the geometric nonlinearities are of hardening, hardening–softening or softening type. An analytical study is then performed to reveal two bifurcations in the dynamics with respect to the initial angle of inclination and detect the critical energy beyond which the nonlinearity changes from hardening to softening. Another effect of the initial angle of inclination is that it “slows” the decay rate of the transient response. To investigate this effect analytically, the complexification-averaging method is applied to an approximate (truncated) equation of motion, to show that, for non-zero initial angle of inclination, the time-scale of the slow dynamics of the system is directly related to the initial angle of inclination. An experimental study is then performed to verify the analytical and numerical predictions. The experimental system consists of a beam clamped at one of its ends and grounded by the inclined linear spring element at its other end. System identification is performed to identify the (linear) modal properties of the beam and detect the linear stiffness and viscous damping characteristics of the inclined spring. The experiments are performed for several different initial angles and initial conditions in order to obtain sufficient measured time series to be able to verify the theoretical predictions. The experimental results confirm the theoretical findings. This study highlights the strong hardening–softening stiffness and damping nonlinearities that may be induced by geometric (and/or kinematic) effects in oscillating systems composed of otherwise linear stiffness and damping elements.
38 citations
TL;DR: In this article, the two-to-one internal resonance between the first two symmetric vibrational modes of a micromachined arch resonator was investigated experimentally and theoretically.
Abstract: We investigate experimentally and theoretically the two-to-one internal resonance between the first two symmetric vibrational modes of a micromachined arch resonator, which is electrothermally tuned and electrostatically driven. An electrothermal voltage is applied across the beam anchors, which controls its stiffness, and hence, tunes the ratio between the first two symmetric frequencies close to two. The dynamic behavior of the arch beam is examined when excited using large harmonic AC voltages, which leads to direct simultaneous excitation of the first two symmetric vibrational modes, in addition to the activation of the internal resonance. Varieties of complex behaviors are demonstrated including quasi-periodic and aperiodic motion leading to potential chaotic motion. A reduced order model based on the Galerkin procedure and the Euler–Bernoulli beam theory is utilized to analyze the dynamic response of the structure. Poincare sections, power spectra, and time histories are used to analyze the dynamic responses. A good agreement is shown among the experimental and theoretical results.
38 citations
TL;DR: In this article, the authors investigated the global bifurcations and multi-pulse jumping chaotic dynamics of circular mesh antenna and employed an equivalent continuum circular cylindrical shell to represent the antenna.
Abstract: This paper investigates the global bifurcations and multi-pulse jumping chaotic dynamics of circular mesh antenna. An equivalent continuum circular cylindrical shell is employed to represent the circular mesh antenna. Based on the four-dimension non-autonomous nonlinear governing equations of motion for the equivalent continuum circular cylindrical shell derived by Zhang et al. (2016, 2017), the improved extended Melnikov theory of the non-autonomous nonlinear system is utilized to conduct a theoretical analysis of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The thermal excitation and damping coefficient are considered as the controlling parameters to analyze their effect on the nonlinear vibrations and bifurcations of the equivalent continuum circular cylindrical shell. Numerical simulations are also introduced to further verify the existence of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The results obtained from the numerical simulations are compared to those obtained from the Melnikov theoretical prediction.
37 citations
TL;DR: In this article, the authors apply the asymptotic homogenization technique to the equations describing the dynamics of a heterogeneous material with evolving micro-structure, thereby obtaining a set of upscaled, effective equations.
Abstract: In the present work, we apply the asymptotic homogenization technique to the equations describing the dynamics of a heterogeneous material with evolving micro-structure, thereby obtaining a set of upscaled, effective equations. We consider the case in which the heterogeneous body comprises two hyperelastic materials and we assume that the evolution of their micro-structure occurs through the development of plastic-like distortions, the latter ones being accounted for by means of the Bilby–Kroner–Lee (BKL) decomposition. The asymptotic homogenization approach is applied simultaneously to the linear momentum balance law of the body and to the evolution law for the plastic-like distortions. Such evolution law models a stress-driven production of inelastic distortions, and stems from phenomenological observations done on cellular aggregates. The whole study is also framed within the limit of small elastic distortions, and provides a robust framework that can be readily generalized to growth and remodeling of nonlinear composites. Finally, we complete our theoretical model by performing numerical simulations.
37 citations
TL;DR: In this article, a geometrically nonlinear energy sinks (NES) is employed for suppressing panel flutter and reducing the intensity of limit cycle oscillations (LCOs).
Abstract: High-speed flight can cause aircraft skin to exhibit an aeroelastic instability typically called panel flutter. Due to the risk of fatigue failure imposed by this undesired phenomenon, several techniques have been proposed over the years to passively or actively control such aeroelastic vibrations. One relatively new method that has been proven effective for controlling various aeroelastic phenomena is the use of nonlinear energy sinks (NES). Here, for the first time, this technique is employed for suppressing panel flutter and reducing the intensity of limit cycle oscillations (LCOs). The present work consists of a comprehensive study on the numerical modeling of a NES, its coupling to an aeroelastic finite element plate model, and several solutions attained through the resulting system. In order to simulate LCOs, a geometrically nonlinear plate model is employed. The supersonic aerodynamic loads are modeled by piston theory, and the final aeroelastic equations of motion are solved directly in time by an implicit integrator. The energy dissipated by the NES and the energy injected in the panel by the flow are obtained numerically as a function of time. This provides important insights on the mechanism through which a NES can either suppress flutter or mitigate LCOs by partially balancing the aerodynamic work. The performance of the NES is tested in three regimes: At the pre-flutter regime, the NES leads the panel to a faster return to equilibrium after a perturbation; at higher speeds, the NES is still able to suppress flutter, whereas an uncontrolled panel would exhibit LCOs; at even higher speeds, the NES is no longer able to completely suppress the motion but can pump enough energy to reduce the amplitude of the LCOs. In any of these scenarios, the practical outcome would be a longer lifespan for the skin structure. Furthermore, a parametric study is conducted to assess how different NES parameters such as damping coefficient and nonlinear stiffness affect the aeroelastic response. The results reveal that a NES can be used as a lightweight device for passively controlling panel flutter, and that such technique is suitable for optimization-driven design.
TL;DR: In this article, a mathematical framework and its numerical implementation for thermo-electro-viscoelasticity taking into account field-dependence of the relevant material parameters appearing in the constitutive model is proposed.
Abstract: In this contribution, we propose a mathematical framework and its numerical implementation for thermo-electro-viscoelasticity taking into account field-dependence of the relevant material parameters appearing in the constitutive model. Polymeric materials are typically viscoelastic and highly susceptible to thermal fluctuations. Several experimental studies suggest that major material parameters appearing in a constitutive model of a thermo-electro-mechanically coupled problem evolve with respect to temperature as well as the applied electric field. Hence we propose a framework for the realistic modeling of polymeric materials under coupled thermo-electro-mechanical loads in which the temperature and electric field are not only considered as independent fields but also show an effect on the material parameters. Furthermore we present the numerical discretization of the coupled balance laws within the context of the finite element method. To demonstrate the performance of the proposed thermo-electro-mechanically coupled framework, several boundary value problems are solved numerically.
TL;DR: In this paper, the Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions.
Abstract: The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided and consequently the unique α − solution is obtained within a convenient class of distributions including the Dirac delta measure. It is shown that our constructed α − solution is reasonable compared with the known results using other methods. Furthermore, the result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, which enables us to see that the α − solution is not self-similar any more. It is shown that the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term.
TL;DR: In this article, the authors investigated the discontinuous dynamical behaviors in a two-degree-of-freedom vibro-impact system with multiple constraints by using the flow switchability theory of discontinuous dynamic systems.
Abstract: In this paper, the discontinuous dynamical behaviors in a two-degree-of-freedom vibro-impact system with multiple constraints are investigated by using the flow switchability theory of discontinuous dynamical systems. Due to the interaction between the two masses in this discontinuous system, the following four cases are taken into account: both the masses are free-flight; one of the two masses is sticking; and both the masses are sticking. Different domains and boundaries are defined in absolute and relative coordinates based on the discontinuity caused by the impact between the masses and the constraints. From the above domains and boundaries, the analytical conditions of switching for stick motions and grazing motions in the vibro-impact system are obtained through the analysis of the corresponding vector fields and G-functions. The switching sets and four-dimensional mappings are introduced to describe different periodic motions and identify the corresponding mapping structures. Periodic motions with different mapping structures in the vibro-impact system are analytically predicted. The time histories of displacement, velocity, G-function and the corresponding trajectories in phase plane for stick, grazing and periodic motions are given to illustrate the dynamics mechanism of complex motions in such a vibro-impact system. A better understanding of the motion switching mechanism in mechanical systems with multiple constraints may be helpful for improving the efficiency of vibro-impact systems.
TL;DR: In this paper, an investigation on the nonlinear aeroelastic system of an airfoil with external store by incremental harmonic balance (IHB) method is presented, where the IHB method is implemented to obtain quasi-periodic (QP) solutions by introducing multiple irreducible time scales.
Abstract: This paper presents an investigation on the nonlinear aeroelastic system of an airfoil with external store by incremental harmonic balance (IHB) method. Besides solving limit cycle (LC) solutions, the IHB method is implemented to obtain quasi-periodic (QP) solutions by introducing multiple irreducible time scales. Steady state responses such as LC and QP oscillations obtained by the presented method are verified by numerical examples. One pair of Floquet multipliers for LC solutions first leave and then enter a unit cycle at complex conjugate values, indicating the existence of a secondary Hopf bifurcation and its reversal one. Along with the fundamental frequency of LC oscillation, an additional frequency arises at the secondary bifurcation, and finally disappears at the reversal bifurcation. The appearance and disappearance of the irreducible frequency cause the steady state responses changing from LC to QP and back to LC oscillation.
TL;DR: This paper investigates the effects of interval uncertain parameters on the dynamic behaviors of a rotor system with a transverse breathing crack in the shaft and develops a surrogate model for the uncertain problem to determine the bounds of the nonlinear dynamic responses.
Abstract: Parametric uncertainties are present in complex mechanical systems due to various reasons such as material dispersion and wear. This paper investigates the effects of interval uncertain parameters on the dynamic behaviors of a rotor system with a transverse breathing crack in the shaft. The uncertainties are modeled as uncertain-but-bounded interval inputs on the basis that no sufficient prior information is available to define their precise probabilistic distributions. A finite element rotor model is formulated and the harmonic balance method (HBM) is employed to solve the deterministic dynamic problem. Based on the Chebyshev approximation theory, a surrogate model for the uncertain problem is established and then used to determine the bounds of the nonlinear dynamic responses. The accuracy verification is performed by comparing with the scanning method. Simulations are carried out considering different uncertainties with several uncertain degrees. It will provide some references for early crack fault detection and condition monitoring in rotor systems with uncertainties included.
TL;DR: In this article, the authors derived a model of single-walled carbon nanotubes nonlinear vibrations based on shell theory, which is obtained using the Sanders-Koiter shell theory and nonlocal elasticity.
Abstract: The model of single-walled carbon nanotubes nonlinear vibrations, which is based on shell theory, is derived. The system of partial differential equations with respect to three displacements projections is obtained using the Sanders–Koiter shell theory and nonlocal elasticity. The system of nonlinear ordinary differential equations, which describes the nanotubes vibrations, is obtained by the Galerkin method. The harmonic balanced method is used to calculate free nonlinear vibrations. The bifurcation behavior of CNT vibrations owing to the Naimark–Sacker bifurcation is analyzed. Almost periodic vibrations of carbon nanotubes are investigated numerically. The nanotube periodic and almost periodic vibrations have three displacements projections u , v , w , which are commensurable. This is essential difference of the nanotube vibrations from macroshells.
TL;DR: In this paper, the Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton-Raphson basins of convergence, related to the out-of-plane equilibrium points.
Abstract: The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton–Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two-dimensional planes, are revealed by using the multivariate Newton–Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.
TL;DR: An important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence and its response to therapies in a patient-specific manner is represented.
Abstract: This work evaluates the predictive ability of a novel personalized computational tool for simulating the growth of brain tumours using the neuroimaging data collected during one clinical case study. The mathematical model consists of an evolutionary fourth-order partial differential equation with degenerate motility, in which the spreading dynamics of the multiphase tumour is coupled with a parabolic equation determining the diffusing oxygen within the brain. The model also includes a reaction term describing the effects of radiotherapy, that is simulated in accordance with the clinical schedule. We collect Magnetic Resonance (MRI) and Diffusion Tensor (DTI) imaging data for one patient at given times of key clinical interest, from the first diagnosis of a giant glioblastoma to its surgical removal and the subsequent radiation therapies. These neuroimaging data allow reconstructing the patient-specific brain geometry in a finite element virtual environment, that is used for simulating the tumour recurrence pattern after the surgical resection. In particular, we characterize the different brain tissues and the tumour location from MRI data, whilst we extrapolate the heterogeneous nutrient diffusion parameters and cellular motility from DTI data. The numerical results of the simulated tumour are found in good qualitative and quantitative agreement with the volume and the boundaries observed in MRI. Moreover, the simulations point out a consistent regression of the tumour mass in correspondence to the application of radiotherapy, with an average growth rate which is of the same order as the one calculated from the neuroimaging data. Remarkably, our results display the highest Jaccard index of the tumour region reported in the biomathematical literature. In conclusion, this work represents an important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence and its response to therapies in a patient-specific manner.
TL;DR: In this article, the authors discuss one-dimensional (1D) nonlinear problems of suspension-colloidal transport in porous media with two simultaneous particle capture mechanisms, and propose a filtration function that describes the breakthrough curves (BTCs) that monotonically increase with time and stabilise at some value lower than the injected concentration.
Abstract: We discuss one-dimensional (1D) non-linear problems of suspension-colloidal transport in porous media with two simultaneous particle capture mechanisms. The first mechanism corresponds to low retention concentration and constant filtration function. The second mechanism corresponds to large retention concentration with blocking (Langmuir) filtration function. The 1D flow problems are non-linear; however, they allow for exact solutions. The exact solutions are obtained for the general two-capture case, and also for piecewise-linear approximation of the filtration function. The proposed filtration function describes the breakthrough curves (BTCs) that monotonically increase with time and stabilise at some value lower than the injected concentration; those BTCs are observed for numerous suspension-colloidal flows. The tuning method for determining the model coefficients is developed. Close agreement between the laboratory and modelling data validates the proposed form of filtration function.
TL;DR: In this article, a model of coupled limit cycle oscillators is developed and analyzed to discover and map out the system's locking behavior, motivated by the physical problem of a pair of closely spaced doubly clamped, thin silicon beams coupled to each other through electrostatic fringing fields.
Abstract: A model of coupled limit cycle oscillators is developed and analyzed to discover and map out the system’s locking behavior. This sixth order model is motivated by the physical problem of a pair of closely spaced doubly clamped, thin silicon beams, coupled to each other through electrostatic fringing fields. The beams are assumed to be detuned with respect to each other. The beams are optically thin and are situated above a thick silicon substrate. When illuminated with continuous laser light a cavity interferometer is formed. Coupling of this optical interference with thermal stresses creates an inherent feedback loop that can drive the beams into limit cycle oscillation. Numerical analysis is used to study the range of coupling strengths and detunings over which 1:1, and other integer frequency ratio locking can be obtained. Results show that 1:1 locking can occur over a broad range of detuning even at relatively low levels of coupling. For coupling strengths just above the threshold for locking, both locked and drift states can exist, depending on the initial conditions. Locking at 2:1, 3:1, 3:2 and 5:2 frequency ratios are observed for detunings that are close but not exactly equal to these integer ratios.
TL;DR: In this paper, the effect of oblateness on the existence, locations and stability of the libration points in the restricted four-body problem with all the primary spheroids having equal masses is examined.
Abstract: The present manuscript deals with the restricted four-body problem when all the primaries are oblate spheroids. This paper unveils the effect of oblateness on the existence, locations and stability of the libration points. It is assumed that three oblate primaries having equal masses are set in Lagrangian equilateral triangle configuration. It is observed that there exist ten libration points in configuration ( x , y ) -plane out of which four are collinear and six are non-collinear for oblateness parameter A ∈ [ 0 , 0 . 681949 ) . Further, for the oblateness parameter A ∈ ( 0 . 681949 , 1 ) , there exist only four libration points out of which two are collinear and two are non-collinear. Out-of-plane libration points are also investigated and it is found that in-plane and out-of-plane libration points are unstable. It is further observed that oblateness parameter has substantial effect on the regions of possible motion. Further, we have numerically investigated the Newton–Raphson basins of convergence associated with the libration points to unveil that how the oblateness parameter influences the domain of convergence.
TL;DR: In this article, a multivariate version of the Newton-Raphson iterative scheme is used to reveal the attracting domain associated with the libration points on various type of two-dimensional configuration planes.
Abstract: The Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as attractors), are unveiled in the Copenhagen problem, where instead of the Newtonian potential and forces, a quasi-homogeneous potential created by two primaries is considered. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the attracting domain associated with the libration points on various type of two-dimensional configuration planes. The correlations between the basins of convergence and the corresponding required number of iterations are also presented and discussed in detail. The present numerical analysis reveals that the evolution of the attracting domains in this dynamical system is very complicated, however, it is a worth studying issue.
Abstract: We investigated interfacial instability of a thin liquid film flowing down an inclined plane, considering the linear variation of fluid properties such as density, dynamical viscosity, surface tension and thermal diffusivity, for the small variation of temperature. Using long wave expansion method and considering order analysis specially for very small Biot number ( B i ) we obtained a single surface equation in terms of the free surface h ( x , t ) . Considering sinusoidal perturbation method we carried out linear stability analysis and obtained the critical Reynolds number ( R e c ) and linear phase speed ( c r ) , both of which depend on K μ , K ρ but independent of K σ , K κ . Using the method of multiple scales, weakly nonlinear stability analysis is carried out. We demarcated subcritical, supercritical, unconditional and explosive zones and their variations for the variation of K μ , K ρ and K σ . Also we discussed the variations of threshold amplitude in the subcritical as well as in the supercritical zones for the variation of K μ , K ρ and K σ . Finally we discussed the variation of nonlinear wave speed N c r for the variation of K μ , K ρ and K σ .
TL;DR: In this article, the authors investigated the periodic motion bifurcations of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft and derived a mathematical model governing the cracked system lateral vibrations.
Abstract: This article investigates the periodic motion bifurcations of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft. The nonlinear spring characteristics due to Hertz contact force and bearing clearance, disc weight, disc eccentricity, breathing of the shaft crack, and angle between the crack and imbalance directions are included in the system model. A mathematical model governing the cracked system lateral vibrations is derived and then analyzed utilizing asymptotic analysis in the primary resonance case. Effects of disc eccentricity, creak depth, and angle between the crack and imbalance directions on the system response curves are studied. The analysis revealed that at a small crack depth, the system executes both forward and backward whirling motions at a specific range of the disc spinning speed, while the backward whirling orbits disappear as the crack depth increases. In addition, at zero disc eccentricity, the cracked system does not oscillate unless the system linear stiffness coefficient is reduced by about 11% as a result of shaft crack. Moreover, there is a spinning speed range of the rotating shaft at which two stable periodic solution attractors appear beside the trivial solution one when the linear stiffness coefficient of the system is reduced to 20% or more. The obtained analytical results are confirmed numerically that showed a very good agreement with the numerical ones. Finally, the acquired results are compared with the work published in the literature.
TL;DR: In this article, the authors developed the isogeometric model of size-dependent geometrically nonlinear shells subjected to large-amplitude vibrations, and the obtained ordinary differential equations from IGA were finally solved by the periodic grid approach which can be considered as a suitable solution strategy for the analysis of free and harmonically forced vibrations of different structures.
Abstract: Surface influences on the nonlinear vibrations of micro- and nano-shells are investigated by an efficient numerical approach. The seven-parameter geometrically nonlinear first-order shear deformation shell theory in Lagrangian description is formulated for the bulk part of structure. To consider surface stress effects, the Gurtin–Murdoch surface elasticity theory with considerations proposed by Ru (2016) and Shaat et al. (2013) is employed. In this regard, two thin inner and outer surface layers are considered, and the corresponding constitutive relations are incorporated into the shell formulations. The stress–strain and strain–displacement relations are represented in a novel matrix–vector form by which the governing equations of motion are derived based on Hamilton’s principle. The isogeometric analysis (IGA) is then utilized due to having the capability to construct exact geometries of shells and the associated powerful features. The obtained ordinary differential equations from IGA are finally solved by the periodic grid approach which can be considered as a suitable solution strategy for the analysis of free and harmonically forced vibrations of different structures. The present work contributes to the literature with developing the isogeometric model of size-dependent geometrically nonlinear shells subjected to large-amplitude vibrations.
TL;DR: In this paper, a model-less method for forecasting Hopf bifurcations and the complete post-bifurcation dynamics of nonlinear oscillatory systems is proposed.
Abstract: A unique method for forecasting Hopf bifurcations and the complete post-bifurcation dynamics of nonlinear oscillatory systems is proposed. The method is model-less and uses measurements of the system only in the pre-bifurcation regime. The forecasts are three-dimensional bifurcation diagrams using a few measurements of the system response to perturbations in the pre-bifurcation regime. To demonstrate the forecasting method and its advantages, it is employed to predict bifurcation diagrams of a 3-DOF fluid-structural system. Surrogate measurement data is obtained from simulations and used as input for the forecasting algorithm. Results show that the bifurcation diagrams are forecasted accurately in both supercritical and subcritical bifurcations. The method accurately forecasts the bifurcation point, the bifurcation type and the post-bifurcation limit cycle amplitudes despite the fact that only pre-bifurcation regime data are used for forecasting.
TL;DR: In this paper, the existence of solitary waves in a nonlinear square spring-mass lattice was investigated, where the masses interact with their neighbors through linear springs and are connected to the ground by a non-linear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement.
Abstract: We investigate the existence of solitary waves in a nonlinear square spring–mass lattice. In the lattice, the masses interact with their neighbors through linear springs, and are connected to the ground by a nonlinear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement. The low-order Taylor series expansions of the discrete equations lead to a continuum representation that holds in the long wavelength limit. Under this assumption, solitary wave solutions are sought within the long wavelength approximation, and the subsequent application of multiple scales to the resulting nonlinear continuum equations. The study focuses on weak nonlinearities of the ground stiffness and reveals the existence of 3 types of solitons, namely a ‘bright’, a ‘dark’, and a ‘vortex’ soliton. These solitons result from the balance of dispersive and nonlinear effects in the lattice, setting aside other relevant phenomena in 2D waves such as diffraction that may lead to a field that does not change during propagation in nonlinear media. For equal constants of the in-plane springs, the governing equation reduces to the Klein–Gordon type, for which bright and dark solitons replicate solutions for one-dimensional lattices. However, unequal constants of the in-plane springs aligned with the two principal lattice directions lead to conditions in which the soliton propagation direction, defined by the group velocity, differs from the wave vector direction, which is unique to two-dimensional assemblies. Furthermore, vortex solitons are obtained for isotropic lattices, which shows similarities with results previously found in optics, thermal media and quantum plasmas. The paper describes the main parameters defining the existence of these solitary waves, and verifies the analytical predictions through numerical simulations. Results show the validity of obtained solutions and illustrate the main characteristics of the solitary waves found in the considered nonlinear mechanical lattice. The study provides an analysis of the physics of waves in nonlinear systems, and may lead to novel designs of devices that can be used for high-performance waveguides.
TL;DR: In this paper, the influence of the internal debonding on the structural stiffness and the nonlinear modal characteristics of the layered structure are examined extensively in the context of two kinematic models and the desired responses are solved numerically with the help of robust (direct iterative method) technique and compared with available results to demonstrate the solution accuracy.
Abstract: The influence of the internal debonding on the structural stiffness and the nonlinear modal characteristics of the layered structure are examined extensively in the current research article. For the investigation purpose, the shell frequency responses are obtained numerically for both the linear and the nonlinear cases via a generic type of mathematical formulation using the Equivalent Single Layer (ESL) theory in the framework of two kinematic models. The current formulation not only includes the influence of the transverse shear deformations but also satisfies the parabolic variation of transverse shear stress through the thickness. Additionally, the geometrical nonlinear distortion modeled via Green–Lagrange strain–displacement relations. Further, the internal debonding between the adjacent layers are modeled using sub-laminate approach and the displacement continuity between segments (laminate and delaminate) have been established through the intermittent continuity conditions. The nonlinear system governing equation of the vibrated structure is obtained via Hamilton’s principle and converted to set of nonlinear algebraic equations through the isoparametric finite element (FE) steps. The desired responses are solved numerically with the help of robust (direct iterative method) technique and compared with available results to demonstrate the solution accuracy. Subsequently, an adequate number of examples are solved for the delaminated structure using the current higher-order nonlinear models and the influential parameters discussed in detail.
TL;DR: In this paper, a plasticity constitutive model is proposed to simulate the monotonic and cyclic behavior of granular soil-structure interfaces, and the model is built on two-surface plasticity models previously developed for interfaces between gravelly soils and structural materials.
Abstract: A plasticity constitutive model is proposed to simulate the monotonic and cyclic behavior of granular soil–structure interfaces. The model is built on two-surface plasticity models previously developed for interfaces between gravelly soils and structural materials (Saberi et al., 2016, 2017), which simulate strain hardening, stress degradation and phase transformation behavior. The proposed model in this study incorporates the softening behavior likely to occur in dense sandy soil–structure interfaces under monotonic and cyclic loading, and it provides a unified formulation for simulating the behavior of both sandy and gravelly soil–structure interfaces. The model accounts for the stress path dependency behavior of interfaces, and it requires a single set of nine calibration parameters, which can readily be obtained from standard interface shear tests. The interface model’s performance is evaluated for Constant Normal Load, Constant Normal Stiffness, and Constant Normal Height stress path conditions by comparing its predictions with experimental data.