Showing papers in "International Journal of Non-linear Mechanics in 2019"
TL;DR: In this article, a new fractional derivative of the Caputo type is proposed and some basic properties are studied, such as adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered.
Abstract: In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grunwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition.
160 citations
TL;DR: In this paper, a unified formulation of full geometrically nonlinear refined plate theory in a total Lagrangian approach was developed to investigate the large-deflection and post-buckling response of isotropic rectangular plates.
Abstract: Accurate predictions of the in-service nonlinear response of highly flexible structures in the geometrically nonlinear regime are of paramount importance for their design and failure evaluation. This paper develops a unified formulation of full geometrically nonlinear refined plate theory in a total Lagrangian approach to investigate the large-deflection and post-buckling response of isotropic rectangular plates. Based on the Carrera Unified Formulation (CUF), various kinematics of two-dimensional plate structures are consistently implemented via an index notation and an arbitrary expansion function of the generalized variables in the thickness direction, resulting in lower- to higher-order plate models with only pure displacement variables via the Lagrange polynomial expansions. Furthermore, the principle of virtual work and a finite element approximation are exploited to straightforwardly and easily formulate the nonlinear governing equations. By taking into account the three-dimensional full Green–Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of unified plate elements are presented in terms of the fundamental nuclei, which are independent of the theory approximation order. The Newton–Raphson linearization scheme combined with a path-following method based on the arc-length constraint is utilized to solve the geometrically nonlinear problem. Numerical assessments, including the large-deflection response of square plates subjected to transverse uniform pressure and the post-buckling analysis of slender plates under compression loadings, are finally conducted to confirm the capabilities of the proposed CUF plate model to predict the large-deflection and post-buckling equilibrium curves as well as the stress distributions with high accuracy.
49 citations
TL;DR: By introducing a flux-controlled memristor with quadratic nonlinearity into the Liu-Chen system as a feedback term, a novel four-wing memristive chaotic system is derived in this article.
Abstract: By introducing a flux-controlled memristor with quadratic nonlinearity into the Liu–Chen system as a feedback term, a novel four-wing memristive chaotic system is derived in this paper. This memristive chaotic system with a line of equilibrium, which presented striking extreme multistability that has received extensive attention and research in recent years, can generate single-wing 1-periodic, single-wing 2-periodic, single-wing chaotic and other double-wing state rotational coexisting attractors depending on the memristor initial conditions. Furthermore, complex transient transition and sustained chaotic state behaviors can also be observed, the complex phenomenon of memristive chaotic system with infinite equilibrium is further revealed. The dynamical behaviors are numerically verified through investigating phase portraits, Lyapunov exponent spectra, bifurcation diagrams and basin of attraction. Finally, Multisim simulations and hardware experiments are carried out to validate the theoretical analysis.
49 citations
TL;DR: In this paper, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates.
Abstract: The multiple timescale evolution of polymers’ microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. Here, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.
45 citations
TL;DR: In this article, a simplified numerical technique is proposed to quantify the effects of non-linear stiffness in MEMS resonators and validate their approach on a clamped-clamped beam, a softening disk ring gyroscope (DRG) and a shallow arch showing internal resonance.
Abstract: We propose a simplified numerical technique to quantify the effects of non-linear stiffness in MEMS resonators and we validate our approach on a clamped–clamped beam, a softening Disk Ring Gyroscope (DRG) and a shallow arch showing internal resonance. We generate a Reduced Order Model (ROM) which is integrated with either a direct integration approach or a continuation technique with arc length control. Finally we compare and validate the results with a full FEM model.
44 citations
TL;DR: In this article, the nonlinear structural behavior of a non-parallel plates micro-actuator design was investigated via a reduced-order modeling, and it was shown that by increasing the actuation voltage the shallow arch behavior tend to reach a softening like behavior, mainly due to its dominant flexibility effect.
Abstract: In this research work, the nonlinear structural behavior of a non-parallel plates micro-actuator design is investigated via a reduced-order modeling. The micro-actuator is considered to be made of an initially flexible curved doubly-clamped microbeam and two evenly arranged stationary rectangular shaped out-of-plane electrodes of different lengths. The subsequent actuating attractive electrostatic force is mainly formed by the unevenness of the electric fringing-fields (non-parallel) electrodes arrangement. Results of this numerical investigation show that by increasing the actuation voltage the shallow arch behavior tend to reach a softening like behavior, mainly due to its dominant flexibility effect. With a further increase in the actuation voltage, the micro-arch starts to stretch and thus develops a hardening like behavior. Furthermore, and for certain values of the design parameters (mainly the shape and the length of the stationary electrodes), the shallow arch presented a snap-through like bi-stability behavior. Indeed, when the voltage approaches a certain critical value, the micro-arch static profile alters from a symmetrical shape to an asymmetric one, thus showing a symmetry breaking like behavior that depends a lot on the shape and length of the stationary non-parallel actuating electrodes. It is also demonstrated that by varying few of the micro-actuator design parameters, the variation of the normalized fundamental frequency showed rises and declines for certain ranges of the applied bias voltage. This approves that with such electrostatically actuated micro-actuator design, and with a smart tuning of its geometrical design parameters, one can obtain softening as well as hardening like behaviors. Modes frequency variations curves showed also that for certain actuation DC loads, it is possible to achieve a one-to-one internal resonance state involving both the first symmetric and the first antisymmetric modes of the shallow micro-arch.
42 citations
TL;DR: The rotary nonlinear energy sink (NES) reported in the literature is inertially coupled to an associated linear primary structure by a rigid rotating arm as discussed by the authors, which is now capable of oscillating in the radial direction along the coupling arm as well.
Abstract: The rotary nonlinear energy sink (NES) reported in the literature is inertially coupled to an associated linear primary structure by a rigid rotating arm. In this work, the rigid coupling arm is replaced by an elastic arm, with a linear coupling radial stiffness element used to provide the rotating NES with the added capacity for radial oscillation in order to achieve robust performance concerning passive nonlinear energy transfer and dissipation. Accordingly, the NES mass in addition to rotating about a fixed vertical axis, is now capable of oscillating in the radial direction along the coupling arm as well. In accordance to this structural modification, the resulting NES is referred to as rotary-oscillatory NES (RO NES), and as such, is capable of dissipating the transferred energy from the linear primary structure through its angular and radial damping elements during its combined angular rotation and radial oscillation. Moreover, this new NES configuration enables enhanced energy absorption and dissipation over a wide range of initial input energies. The optimized RO NES is compared to the corresponding optimized rotary NES, with the numerical results showing significant improvement in NES performance. In addition, the effectiveness of the RO NES to passively ‘redistribute’ the modal energies of the primary structure by means of nonlinear energy scattering of the input energy from low to high structural modes is studied.
39 citations
TL;DR: In this paper, the elastic structural stability analysis of the pressurized thin-walled functionally graded material (FGM) arches under temperature variation field was studied and the total potential energy function of the pinned-pinned arch was expressed explicitly by employing the classical thinwalled arch theories and admissible radial displacement functions.
Abstract: This paper focuses on the elastic structural stability analysis of the pressurized thin-walled functionally graded material (FGM) arches under temperature variation field. The material properties are temperature-dependent and thermo-elastic. The total potential energy function of the pinned–pinned arch was expressed explicitly by employing the classical thin-walled arch theories and admissible radial displacement functions. By means of the variational principle, the expressions of the critical buckling pressure were obtained analytically and verified numerically by developing a two-dimensional (2D) simulated model. The pre- and post-buckling equilibrium paths were depicted to explore the maximum pressure (buckling pressure). The comparison showed that the numerical results were in excellent agreement with the analytical solutions for different subtended angles, volume fraction exponents and temperature variations. In the end, the effects of volume fraction exponent and temperature variation were examined on the critical buckling pressure, the bending moment, the hoop force, the hoop strain and stress, the hoop and radial displacement components through the whole arch.
35 citations
TL;DR: In this article, the large amplitude free flexural vibration characteristics of fairly thick and thin functionally graded graphene platelets reinforced porous curved composite beams are investigated using finite element approach, which includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects.
Abstract: In this paper, the large amplitude free flexural vibration characteristics of fairly thick and thin functionally graded graphene platelets reinforced porous curved composite beams are investigated using finite element approach. The formulation includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects. The geometric non-linearity introducing von Karman’s assumptions is considered. The non-linear governing equations obtained based on Lagrange’s equations of motion are solved employing the direct iteration technique. The variation of non-linear frequency with amplitudes is brought out considering different parameters such as slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, graphene platelet geometry and boundary conditions. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and material parameters of the curved composite beam. Also, the degree of hardening behaviour increases with the weight fraction and aspect ratio of graphene platelet. The rate of change of nonlinear behaviour depends on the level of amplitude of vibration, shallowness and slenderness ratio of the curved beam.
34 citations
TL;DR: In this paper, a uniformly-valid plate theory based on series expansions about the bottom surface of a plate is derived. But this theory is not suitable for finite element implementation, and it cannot be used in finite element simulation.
Abstract: A uniformly-valid plate theory, independent of the magnitudes of applied loads, is derived based on the two-dimensional plate theory obtained from series expansions about the bottom surface of a plate. For five different magnitudes of surface loads, it is shown by using asymptotic expansions that this unified plate theory recovers five well-known plate models in the literature to leading-order. This demonstrates its uniform validity. More generally, it provides a uniformly-valid plate model provided that two asymptotic conditions are satisfied, which can be checked as a posteriori. The weak formulation of the uniformly-valid plate equations is furnished, which can be used for finite element implementation.
34 citations
TL;DR: In this article, the magnetic interaction is applied to a continuous structure as local negative-stiffness components to improve the vibration isolation performances and to extend the vibrational isolation band, and the relationship between structural parameters and effective isolation bandwidth is established by deriving the solution of steady state around zero equilibrium.
Abstract: In this study, magnetic interaction is applied to a continuous structure as local negative-stiffness components to improve the vibration isolation performances and to extend the vibration isolation band. According to the theory of electromagnetic fields, the mechanical model of interaction force is established. This model is used to discover that the magnetic interaction can induce the negative stiffness property locally using appropriate structural parameters. Because different structural parameters of the magnets can increase the negative stiffness strength with different effects, the parameters of the continuous beam and magnets are adjusted for the design of accurate zero stiffness property locally. Then, the relationship between structural parameters and effective isolation bandwidth is established by deriving the solution of steady state around zero equilibrium. As the nonlinearity and zero-stiffness property only occur locally around zero equilibrium and the system displays the positive-stiffness property for large vibration amplitudes, the effective isolation band is significantly extended, and the amplitude–frequency curve is similar to the linear system at resonance. The relevant experiments demonstrate the remarkable tuning characteristic of structural parameters of magnets on the isolation features, which realize the improvement of isolation/protection effectiveness for time-lasting excitation at a low frequency. This study not only demonstrates the advantages of nonlinearity induced by local magnetic interaction on the improvement of isolation performances but also realizes a continuous structure with local quasi-zero stiffness applied in fields of low-frequency engineering structures.
TL;DR: In this paper, a unified formulation of geometrically nonlinear refined beam theory based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach constitutes the basis of their analysis.
Abstract: Highly flexible laminated composite structures, prone to suffering large-deflection and post-buckling, have been successfully employed in a number of scenarios. Therefore, accurate predictions of their stress distributions in the geometrically nonlinear analysis are of paramount importance for their design and failure evaluation. In this paper, for composite beams subjected to large-deflection and post-buckling, we investigate the effectiveness of different geometrically nonlinear strain approximations for the description of their nonlinear static response and for the determination of stress distributions. For this purpose, a unified formulation of geometrically nonlinear refined beam theory based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach constitutes the basis of our analysis. Accordingly, various kinematics of one-dimensional structures are formulated via an appropriate index notation and an arbitrary cross-section expansion of the generalized variables, leading to lower- to higher-order beam models with only pure displacement variables for laminated composite beams. In view of the intrinsic scalable nature of CUF and by exploiting the principle of virtual work and a finite element approximation, nonlinear governing equations corresponding to various nonlinear strain assumptions can be straightforwardly and easily formulated in terms of fundamental nuclei, which are independent of the theory approximation order. Several numerical assessments are conducted, including large-deflection and post-buckling analyses of asymmetric and symmetric laminated beams under compression loadings. The numerical solutions are solved by using a Newton–Raphson linearization scheme along with a path-following method based on the arc-length constraint. Our numerical findings demonstrate the capabilities of the CUF model to calculate the large-deflection and post-buckling equilibrium curves as well as the stress distributions with high accuracy, which could be a basis to assess the validation ranges of various kinematics and different nonlinear strain approximations.
TL;DR: By adding a further variable in a three-dimensional chaotic system, a novel four-dimensional fractional-order chaotic system is developed that has no-equilibrium but it can also exhibit rich and complex hidden dynamics.
Abstract: The hidden attractor and extreme multistability are very important topics in nonlinear dynamics. In this paper, by adding a further variable in a three-dimensional chaotic system, a novel four-dimensional fractional-order chaotic system is developed. This system consists of eight terms including three different nonlinear terms and one constant term. What interests us is that this newly presented system has no-equilibrium but it can also exhibit rich and complex hidden dynamics. Furthermore, the offset boosting of a variable of the proposed chaotic system can be achieved by adjusting the constant term. The intricate hidden dynamic properties of the proposed chaotic system are investigated by employing conventional nonlinear dynamical analysis tools including equilibrium, phase planes, bifurcation diagrams and Lyapunov exponents, chaos diagrams, etc. Finally, Multisim simulations and the corresponding hardware experiments are implemented to validate the theoretical analysis.
TL;DR: In this article, the authors show that for nonlinear elastic materials satisfying Truesdell's so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear.
Abstract: In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell’s so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably.
TL;DR: In this paper, the nonlinear dynamic analysis of a parametrically base excited cantilever beam based piezoelectric energy harvester is carried out, where the attached mass is placed in such a way that the system exhibits 3:1 internal resonance.
Abstract: In this work, the nonlinear dynamic analysis of a parametrically base excited cantilever beam based piezoelectric energy harvester is carried. The system consists of a cantilever beam with piezoelectric patches in bimorph configuration and attached mass at an arbitrary position. The attached mass is placed in such a way that the system exhibits 3:1 internal resonance. The governing spatio-temporal equation of motion is discretized to its temporal form by using generalized Galerkin’s method. To obtain the steady state voltage response and stability of the system, Method of multiple scales is used to reduce the resulting equation of motion into a set of first-order differential equations. The response and stability of the system under principal parametric resonance conditions has been studied. The parametric instability regions are shown for variation in different system parameters such as excitation amplitude and frequency, damping and load resistance. Bifurcations such as turning point, pitch-fork and Hopf are observed in the multi-branched non-trivial response. By tuning the attached mass an attempt has been made to harvest the electrical energy for a wider range of frequency. Such kind of smart self-sufficient systems may find application in powering low power wireless sensor nodes or micro electromechanical systems.
TL;DR: In this paper, a displacement-based, geometrically nonlinear finite element model is developed for lattice core sandwich panels modeled as 2-D equivalent single-layer (ESL), first-order shear deformation theory (FSDT) micropolar plates.
Abstract: A displacement-based, geometrically nonlinear finite element model is developed for lattice core sandwich panels modeled as 2-D equivalent single-layer (ESL), first-order shear deformation theory (FSDT) micropolar plates. The nonlinearity is due to the moderate macrorotations of the plate which are modeled by including the von Karman nonlinear strains in the micropolar strain measures. Weak-form Galerkin formulation with linear Lagrange interpolations is used to develop the displacement finite element model. Selective reduced integration is used to eliminate shear locking and membrane locking. The novel finite element model is used to study the nonlinear bending and linear free vibrations of web-core and pyramid core sandwich panels. Clamped and free edge boundary conditions are considered for the first time for the 2-D micropolar ESL-FSDT plate theory. The present 2-D finite element results are in good agreement with the corresponding detailed 3-D FE results for the lattice core sandwich panels. The 2-D element provides computationally cost-effective solutions; in a nonlinear bending example, the number of elements required for the 2-D micropolar plate is of the order 1 0 3 , whereas for the corresponding 3-D model the order is 1 0 5 .
TL;DR: In this paper, the mass-damper dynamic absorber is modelled as a pendulum, hinged at the top of a rigid block, with the mass lumped at the end.
Abstract: In this paper the effectiveness of a mass-damper dynamic absorber in preventing rigid blocks from overturning is investigated. The mass-damper is modelled as a pendulum, hinged at the top of the block, with the mass lumped at the end. The equations of rocking motion, the uplift and the impact conditions are derived in a rigorous way, while the results are obtained from the equations’ numerical integration. Under an impulsive one-sine base excitation, an extensive parametric analysis is performed. In particular the frequency, the amplitude of the excitation along with the block and the mass damper’s geometrical characteristics have been taken as variable parameters. Overturning spectra, providing the excitation amplitude of block’s overturning versus the frequency of the one-sine pulse, are obtained. An experimental test is performed by using a linear electromagnetic motor that is used to realize a small mono-dimensional shaking table. A rigid block with fixed geometrical characteristics equipped with a pendulum is tested. Under a one-sine and a one-cosine impulsive base excitations, the overturning spectra are obtained. The results show that the presence of the mass damper leads to a general improvement of the dynamic response of the system. The experimental test confirms the validity of the analytical model as well as the effectiveness of the pendulum mass damper.
TL;DR: In this paper, an analytical solution capable of simulating the interfacial bond behavior between two structural materials subjected to thermal loading is presented. But, it is not yet well understood the influence of temperature variations on bonded joints, so more studies are needed to improve the current level of knowledge.
Abstract: Nowadays, adhesively bonded structures have received exhaustive attention mainly because, contrary to mechanical joints, they are able to avoid stress concentration. When a material is externally bonded to another structural member to improve the strength or stiffness of the latter, the adhesive joint is supposed to perform well for a long time, independently of the type of loading the bonded joint will be subjected to. However, studies dedicated to this topic are scarce when it comes to the influence of thermal action. The influence of temperature variations on bonded joints is not yet well understood, so more studies are needed to improve the current level of knowledge. The present study aims to develop an analytical solution capable of simulating the interfacial bond behaviour between two structural materials subjected to thermal loading. The complete debonding processes of such adhesively bonded joints are estimated based on a bi-linear bond–slip relationship. The proposed analytical model is validated by the numerical simulation of several examples, where some parameters previously identified as potentially affecting the bond behaviour are investigated. A commercial software based on the Finite Element Method (FEM) is used to support those examples in which either the analytical or the numerical simulations agreed very well.
TL;DR: In this article, the authors derived the Hessian criterion for determining the pull-in instability of dielectrics actuated by different loading methods: voltage control, charge control, fixed pre-stretch, and fixed prestretch by analyzing the free energy of the actuated systems.
Abstract: Pull-in (or electro-mechanical) instability occurs when a drastic decrease in the thickness of a dielectric elastomer results in electrical breakdown, which limits the applications of dielectric devices. Here we derive the criterions for determining the pull-in instability of dielectrics actuated by different loading methods: voltage-control, charge-control, fixed pre-stress and fixed pre-stretch, by analyzing the free energy of the actuated systems. The Hessian criterion identifies a maximum in the loading curve beyond which the elastomer will stretch rapidly and lose stability, and can be seen as a path to failure. We present numerical calculations for neo-Hookean ideal dielectrics, and obtain the maximum allowable actuation stretch of a dielectric before failure by electrical breakdown. We find that applying a fixed pre-stress or a fixed pre-stretch to a charge-driven dielectric may decrease the stretchability of the elastomer, a scenario which is the opposite of what happens in the case of a voltage-driven dielectric. Results show that a reversible large actuation of a dielectric elastomer, free of the pull-in instability, can be achieved by tuning the actuation method.
TL;DR: In this article, the effect of friction on the dynamic behavior of a crank-slider mechanism with clearance joint was investigated. But the authors only considered the acceleration and energy consumption of the mechanism, and they did not consider the clearance existence in the revolute joint.
Abstract: In this paper, we are interested in the effect of friction on dynamic behavior of a crank-slider mechanism with clearance joint. Due to the clearance existence in the revolute joint, it is important to choose an appropriate contact force model to analyze the dynamic response, the dynamic equations are established by combining Lagrange’s equation of the first kind with modified contact force model and LuGre friction model, the Baumgarte stabilization approach was used to improve the numerical stability. Simulation and experimental tests were carried out to verify this model. The system dynamic response hysteresis, and the more energy consumption when friction is considered in the mechanism. The influence of friction coefficient on the nonlinear dynamic characteristics of the mechanism with clearance joint is analyzed. The dynamic performance of the mechanism considering friction presents significant differences at the accelerations level. Friction plays a positive role in the stability of the system.
TL;DR: In this article, the authors deal with different classes of non-linear reaction-diffusion equations with variable coefficients c(x ) u t = [ a (x ) f ( u ) u x ] x + b ( x ) g( u ), that admit exact solutions.
Abstract: The paper deals with different classes of non-linear reaction–diffusion equations with variable coefficients c ( x ) u t = [ a ( x ) f ( u ) u x ] x + b ( x ) g ( u ) , that admit exact solutions. The direct method for constructing functional separable solutions to these and more complex non-linear equations of mathematical physics is described. The method is based on the representation of solutions in implicit form ∫ h ( u ) d u = ξ ( x ) ω ( t ) + η ( x ) , where the functions h ( u ) , ξ ( x ) , η ( x ) , and ω ( t ) are determined further by analyzing the resulting functional-differential equations. Examples of specific reaction–diffusion type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown u and /or the spatial variable x (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial–boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions are described.
TL;DR: In this article, a reduced order model is developed to simulate the dynamics of a bladed disk or blisk with nonlinear damping coatings adhered to its blades, where the nonlinear forces exerted by these coatings on the underlying linear blisk structure are a function of the local strain.
Abstract: In this paper, a reduced order model is developed to simulate the dynamics of a bladed disk or blisk with nonlinear damping coatings adhered to its blades. The nonlinear forces exerted by these coatings on the underlying linear blisk structure are a function of the local strain. It is known that coatings modify the stiffness and damping of each blade depending on its amplitude. Blisks, which are designed as perfectly cyclic symmetric structures with identical blades, never behave as such in practice due to various uncertainties encountered during their manufacturing. This asymmetry in the structure is also referred to as mistuning. Mistuning in the linear blisk structure, which causes different blades to respond with non-identical amplitudes, interacts with the coating nonlinearity to yield a mistuning pattern which depends on the blade amplitudes. Additional stiffness and damping parameters that are dependent on the blade amplitude are introduced into a reduced linear model to formulate the nonlinear reduced order model. It is found that this model captures the nonlinear amplitude dependent mistuning effect and predicts the nonlinear coated blisk responses accurately near isolated blisk mode families in blade-dominated frequency regions where these coating effects are likely to be dominant. Significant reductions in the computational effort are achieved through this reduction.
TL;DR: The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function.
Abstract: In this paper, a multiscroll snap oscillator with hyperbolic tangent function is proposed. There is no limitation in the number of scrolls and it can be increased by proper choice of a specific function. The Lyapunov exponents of the proposed system are obtained to testify the chaotic behavior of the system. Fractional order multiscroll system is derived from its integer order model by using the Adams–Bashforth–Moulton algorithm. A new scheme is applied in order to investigate the synchronization of the multiscroll systems. The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function. Such systems are less investigated in the literatures and has many real time applications like image and voice encryption, random number generators, chaos based communication systems and so on.
TL;DR: In this article, a numerical solution technique named as variational differential quadrature (VDQ) is adopted for the compressible nonlinear elasticity problems and the governing equations are obtained based on the virtual work principle by considering displacement as the unknown field.
Abstract: A numerical solution technique named as variational differential quadrature (VDQ) is adopted herein for the compressible nonlinear elasticity problems. The governing equations are obtained based on the virtual work principle by considering displacement as the unknown field. The neo-Hookean model is also considered for the hyperelastic behavior of material. In the solution method, an efficient vector–matrix formulation is developed from which the discretized governing equations are achieved from the weak form of equations in a direct approach. Simplicity in implementation and accuracy are among the features of the proposed approach. Moreover, it does not suffer from the locking problem and unphysical instabilities. Fast convergence rate and computational efficiency are other advantages of this method. A number of numerical examples are given to reveal the good performance of VDQ in the large deformation analysis of compressible and nearly-incompressible bodies.
TL;DR: In this paper, a multivariate version of the Newton-Raphson iterative scheme is used to reveal the structures of the basins of convergence associated with the coplanar libration points on various types of two-dimensional configuration planes.
Abstract: In the present work, the Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as numerical attractors), are unveiled in the axisymmetric five-body problem, where convex configuration is considered. In particular, the four primaries are set in axisymmetric central configuration, where the motion is governed only by mutual gravitational attractions. It is observed that the total number libration points are either eleven, thirteen or fifteen for different combinations of the angle parameters. Moreover, the stability analysis revealed that all the libration points are linearly stable for all the studied combination of angle parameters. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the structures of the basins of convergence, associated with the libration points, on various types of two-dimensional configuration planes. In addition, we present how the basins of convergence are related with the corresponding number of required iterations. It is unveiled that in almost every case, the basins of convergence corresponding to the collinear libration point L 2 have infinite extent. Moreover, for some combination of the angle parameters, the other collinear libration points L 1 , 2 also have infinite extent. In addition, it can be observed that the domains of convergence, associated with the collinear libration point L 1 , look like exotic bugs with many legs and antennas whereas the domains of convergence, associated with L 4 , 5 look like butterfly wings for some combinations of angle parameters. Particularly, our numerical investigation suggests that the evolution of the attracting domains in this dynamical system is very complicated, yet a worth studying problem.
TL;DR: In this article, a systematic and rigorous investigation is performed in an effort to unveil how the angle parameters affect the topology of the basins of convergence in axisymmetric restricted five-body problems with the concave configuration.
Abstract: The axisymmetric restricted five-body problem with the concave configuration has been studied numerically to reveal the basins of convergence, by exploring the Newton–Raphson iterative scheme, corresponding to the coplanar libration points (which act as attractors). In addition, four primaries are set in axisymmetric central configurations introduced by Erdi and Czirjak [13] and the motion is governed by mutual gravitational attraction only. The evolution of the positions of libration points is illustrated, as a function of the value of angle parameters. A systematic and rigorous investigation is performed in an effort to unveil how the angle parameters affect the topology of the basins of convergence. In addition, the relation of the domain of basins of convergence with required number of iterations and the corresponding probability distributions are illustrated.
TL;DR: In this paper, a compressible hyper-viscoelastic constitutive model for human brain tissue is proposed, and the parameters of the model are determined in a simultaneous calibration for tension, compression, shear, and compression relaxation tests data.
Abstract: In this paper, we have introduced a compressible hyper-viscoelastic constitutive model for human brain tissue. The model is calibrated with the reported experimental data from different regions of the brain. The parameters of the model are determined in a simultaneous calibration for tension, compression, shear, and compression–relaxation tests data. They are obtained in an iterative procedure in conjunction with a finite elements (FE) modeling of the tissue, as well as, with the Nelder–Mead Simplex optimization procedure. In the calibration procedure, the compressibility of the material is taken into account, and the respective time-dependent volumetric parameter is also determined. Additionally, the Drucker stability condition is enforced to assess the physical meaning of the extracted constitutive parameters. This proposed model provides an improved prediction of the experimental data and tissue response under various loading conditions. The results show that, under inhomogeneous deformation, the suggested approach will lead to a better material calibration of brain tissue compared to the simple mathematical model fitting.
TL;DR: In this article, the authors developed a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially independent random variables defined by probability density functions.
Abstract: When an elastic tube reinforced with helical fibres is inflated, its ends rotate. In large deformations, the amount and chirality of rotation is highly non-trivial, as it depends on the choice of strain–energy density and the arrangements of the fibres. For anisotropic hyperelastic tubes where the material parameters are single-valued constants, the problem has been satisfactorily addressed. However, in many systems, the material parameters are not precisely known, and it is therefore more appropriate to treat them as random variables. The problem is then to understand chirality in a probabilistic framework. Here, we develop a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially-independent random variables defined by probability density functions. The tube is subjected to uniform dead loading consisting of internal pressure, axial tension and torque. Assuming that the tube wall is thin and that the resulting deformation is the combined inflation, extension and torsion from the reference circular cylindrical configuration to a deformed circular cylindrical state, we derive the probabilities of radial expansion or contraction, and of right-handed or left-handed torsion. We refer to these stochastic behaviours as ‘likely inflation’ and ‘likely chirality’, respectively.
TL;DR: In this article, a rotative non-linear vibration absorber (NVA) is used as a passive suppressor for the vortex-induced vibrations phenomenon (VIV) in a structural model.
Abstract: This paper presents a numerical investigation on the use of a rotative non-linear vibration absorber (NVA) as a passive suppressor for the vortex-induced vibrations phenomenon (VIV) The structural model consists of rigid cylinders mounted on elastic supports and the hydrodynamic loads are calculated using phenomenological models The NVA is defined as a rigid bar, fitted with a tip-mass and hinged to the cylinder Energy is dissipated by means of a linear dashpot linked to the bar Two major groups are studied, the first one being that in which the cylinder is constrained to oscillate in the cross-wise direction (1-dof VIV) The second group, herein named 2-dof VIV, refers to the condition in which simultaneous oscillations in the cross-wise and in-line directions are allowed Characteristic oscillation amplitude curves are obtained as functions of reduced velocities covering the lock-in for different values of the control parameters that define the NVA (namely, its mass, radius and dashpot constant) In addition, quantitative and qualitative aspects of cylinder and suppressor responses are explored in the form of colormaps defined in the space of control parameters for three specific reduced velocities The systematic study shows that the mass parameter of the NVA has more influence on the VIV suppression for both 1-dof VIV and 2-dof VIV In general, the suppression has proved to be greater in the 1-dof VIV case
TL;DR: In this paper, the effects of pre-stretch, compressibility and material constitution on the period-doubling secondary bifurcation of a uni-axially compressed film/substrate bilayer structure were studied.
Abstract: We refine a previously proposed semi-analytical method, and use it to study the effects of pre-stretch, compressibility and material constitution on the period-doubling secondary bifurcation of a uni-axially compressed film/substrate bilayer structure. It is found that compared with the case of incompressible neo-Hookean materials for which the critical strain is approximately 0.17 when the thin layer is much stiffer than the substrate, the critical strain when the Gent materials are used is a monotonically increasing function of the constant J m that characterizes material extensibility, becoming as small as 0.12 when J m is equal to 1, whereas for compressible neo-Hookean materials the critical strain is a monotonically decreasing function of Poisson’s ratio; the period-doubling secondary bifurcation seems to become impossible when Poisson’s ratio is approximately equal to 0.307. The latter result may indicate that when Poisson’s ratio is small enough there are other preferred secondary bifurcations — an example is given where a secondary bifurcation mode with 7 ∕ 4 times the original period occurs at a lower strain value. The effect of a pre-stretch (compression or extension) in the substrate is not monotonic, giving rise to a critical strain that varies between 0.15 and 0.22.