Discretized Equations Of Motion Research Articles

Envelope vector solitons in nonlinear flexible mechanical metamaterials

In this paper, we employ a combination of analytical and numerical techniques to investigate the dynamics of lattice envelope vector soliton solutions propagating within a one-dimensional chain of flexible mechanical metamaterial. To model the system, we formulate discrete equations that describe the longitudinal and rotational displacements of each individual rigid unit mass using a lump element approach. By applying the multiple-scales method in the context of a semi-discrete approximation, we derive an effective nonlinear Schrödinger equation that characterizes the evolution of rotational and slowly varying envelope waves from the aforementioned discrete motion equations. We thus show that this flexible mechanical metamaterial chain supports envelope vector solitons where the rotational component has the form of either a bright or a dark soliton. In addition, due to nonlinear coupling, the longitudinal displacement displays kink-like profiles thus forming the 2-components vector soliton. These findings, which include specific vector envelope solutions, enrich our knowledge on the nonlinear wave solutions supported by flexible mechanical metamaterials and open new possibilities for the control of nonlinear waves and vibrations.

Dynamical behavior of liquid-filled rotors with axial motion

In the present study, the dynamical behavior of liquid-filled rotors with axial motion is investigated. Based on the differential equations of motion for ideal fluids, the fluid forces exerted on the rotor are obtained. Then, the Euler–Bernoulli beam theory is utilized to establish the governing equations of motion for the rotor system. By applying Hamilton's principle, the governing equation and the corresponding boundary conditions are derived. Furthermore, in order to solve the eigenvalue problem of the system, the extended Galerkin method is applied to discrete equations of motion for the rotor system. As a result, the system stability, divergence, and flutter instability are defined. Moreover, a comparative study is presented to verify the accuracy of the proposed model. Finally, the effects of liquid parameters on the dynamic characteristics of the rotor system are investigated in detail. The results show that for liquid-filled rotors with axial motion, the stability of the system depends on the liquid parameters, axial velocity, and spinning velocity.

Envelope waves propagation in a one-dimensional tetratomic model of acoustic metamaterial

This paper explores the propagation of modulated waves in systems modelled as one-dimensional network whose elementary cell is constituted of four interacting subsystems. We derive the dispersion relation of its small amplitude plane waves as a quartic polynomial equation in the frequency squared; and numerically show the existence of four frequency bands. These include one acoustic band and three optical bands of which the middle one features the left-handedness. We also use the reductive perturbation method to reduce the discrete equations of motion of the system to the standard nonlinear Schrödinger, thereby predicting the possibility of the propagation modulated waves in the model. This prediction is firmly established through direct numerical simulations in which the analytical solutions of the nonlinear Schrödinger equation are used as input.

Extension of Galilean invariance to uniform motions for a relativistic equation of fluid flows

Galilean invariance and the Lorentz transformation are the two pillars of mechanics that an equation of motion must respect. The objective is to extend the Galilean invariance to uniform rotational motion and to expansion motion of which motion at constant translational velocity is only a special case. The second goal is to show that the discrete equation of motion is naturally relativistic without using the Lorentz transformation. The realization of these two aims leads to different concepts of special relativity: (i) space is homogeneous and isotropic and (ii) time, which flows regularly, is invariant by translation. These invariances, symmetric for translation and rotation, observed by the discrete equation of motion give it fundamental conservation properties to extend the scope of the equations of fluid mechanics to other fields of physics.

Nonlinear Thermo-Mechanical Response of Bi-Directional Functionally Graded Porous Beams with Initial Geometrical Imperfection

In this paper, the nonlinear vibration and buckling of bi-directional functionally graded (2D-FG) beam subjected to thermal loading are examined via the higher-order shear deformable beam theory incorporating the von Kármán geometric nonlinearity. Two types of initial defects including the internal porosity distribution and external geometrical imperfection shape are taken into consideration. The temperature-dependent material property varying along the length and thickness following the power-law function is employed, and the porosities inside the 2D-FG beam with even and uneven distributions are considered. The initial geometrical imperfection is described by the product of trigonometric and hyperbolic functions, which can capture both global and localized imperfection shapes. The discrete equations of motion for the 2D-FG beam are established by using the differential quadrature method (DQM), and the derived function is solved by an iterative procedure. The nonlinear thermo-mechanical vibration and buckling of 2D-FG beams under uniform, linear, and nonlinear temperature loads are systemically compared. Moreover, several key factors such as 2D-FG indexes, porosity distribution, geometrical imperfections, as well as boundary conditions are investigated in detail. The proposed model and obtained results can be used to guide the optimization design of multi-functional and multi-graded materials serviced under thermal environment.

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

In this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space mathbb{S}^{3}ltimes mathbb{R}^{3} with semi-direct product Lie group structure, where mathbb{S}^{3} is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

An alternative to the concept of continuous medium

Discrete mechanics proposes an alternative formulation of the equations of mechanics where the Navier-Stokes and Navier-Lam\'e equations become approximations of the equation of discrete motion. It unifies the fields of fluid and solid mechanics by extending the fields of application of these equations to all space and time scales. This article presents the essential differences induced by the abandonment of the notion of continuous medium and global frame of reference. The results of the mechanics of continuous medium validated by fluid and solid observations are not questioned. The concept of continuous medium is not invalidated, the discrete formulation proposed simply widens the spectrum of the applications of the classical equations. The discrete equation of motion introduces several important modifications, in particular the fundamental law of the dynamics on an element of volume becomes a law of conservation of the accelerations on an edge. The acceleration considered as an absolute quantity is written as a sum of two components, one soledoidal the other irrotational according to a local orthogonal Helmholtz-Hodge decomposition. The mass is abandoned and replaced by the compression and rotation energies represented by the scalar and vectorial potentials of the acceleration. The equation of motion and all the physical parameters are expressed only with two fundamental units, those of length and time. The essential differences between the two approaches are listed and some of them are discussed in depth. This is particularly the case with the known paradoxes of the Navier-Stokes equation or the importance of inertia for the Navier-Lam\'e equation.

Solitons in nerve axons

An axon or nerve fiber is a long, slender projection of a nerve cell, or neuron, in vertebrates that typically conducts electrical impulse known as action potentials away from the nerve cell body. Axons are the cable-like protrusions of neurons which wire up the nervous system. These delicate structures frequently need to survive for a lifetime. Parallel polar, bundles of microtubules run beside axons to form their structural backbones and are acting as highways for life-sustaining transportation. In this paper, we investigate the excitations of soliton along axons that are governed by the Nerve Soliton (NS) model. The NS model of nerve axon is governed by the discrete nonlinear equation of motion derived using the finite difference method. We done modulational instability (MI) analysis for the discrete equation of motion with effect of damping force by using Rotating wave approximation (RWA). The inherent intrinsic localized wave modes were generated which is responsible for energy localization in the nerve axons.

Nonlinear responses of suspended cable under phase-differed multiple support excitations

Cable structures may be subjected to multiple support excitations and exhibit complex nonlinear dynamic characteristics. To study the influence of phase-differed support excitations on the responses of suspended cable, a model with both ends subjected to 3-D non-uniform excitations is established, and the infinite-dimensional discrete motion equations are obtained. The in-plane primary resonance and out-of-plane primary resonance are studied using the multiple scales method. The general expressions of the equivalent excitation amplitude and the response phase of the multi-excitation system are obtained. The effects of different combinations of horizontal, vertical, and transverse excitations on the response amplitude and phase are analyzed theoretically, and the multi-scale solutions are verified by numerical integration. The results show that the multi-excited cable can be regarded as a nominal single excitation system (NSES). The effect of multiple support excitations on the cable response is mainly reflected in two aspects: First, these excitations make the response phase of the system shift based on the response phase of the NSES; second, these support excitations affect the equivalent excitation amplitude of the NSES. Both the response phase’s shift and the equivalent excitation amplitude depend on the combination of the excitations and the parity of the excited modes. When the phase difference between the two excitations is gradually changed from 0 to $$\pi $$ , the phase–frequency curve is also shifted by $$\pi $$ . However, the shifting process is not necessarily linear. The excitation phase difference does not change the soft and hard properties of the system. However, it can simultaneously change the response amplitude and the resonance region’s width so that the amplitude–frequency curve changes with a period of $$2\pi $$ in a specific region.

A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics

The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free and curl-free components. The derived equation of motion translates the conservation of acceleration over a segment, that of the intrinsic acceleration of the material medium and the sum of the accelerations applied to it. The scalar and vector potentials of the acceleration, which are the compression and shear energies, give the discrete equation of motion the role of conservation law for total mechanical energy. Velocity and displacement are obtained using an incremental time process from acceleration. After a description of the main stages of the derivation of the equation of motion, unique for the fluid and the solid, the cases of couplings in simple shear and uniaxial compression of two media, fluid and solid, make it possible to show the role of discrete operators and to find the theoretical results. The application of the formulation is then extended to a classical validation case in fluid–structure interaction.

Application of discrete mechanics model to jump conditions in two-phase flows

Discrete mechanics is presented as an alternative to the equations of fluid mechanics, in particular to the Navier-Stokes equation. The derivation of the discrete equation of motion is built from the intuitions of Galileo, the principles of Galilean equivalence and relativity. Other more recent concepts such as the equivalence between mass and energy and the Helmholtz-Hodge decomposition complete the formal framework used to write a fundamental law of motion such as the conservation of accelerations, the intrinsic acceleration of the material medium, and the sum of the accelerations applied to it. The two scalar and vector potentials of the acceleration resulting from the decomposition into two contributions, to curl-free and to divergence-free, represent the energies per unit of mass of compression and shear.The solutions obtained by the incompressible Navier-Stokes equation and the discrete equation of motion are the same, with constant physical properties. This new formulation of the equation of motion makes it possible to significantly modify the treatment of surface discontinuities, thanks to the intrinsic properties established from the outset for a discrete geometrical description directly linked to the decomposition of acceleration. The treatment of the jump conditions of density, viscosity and capillary pressure is explained in order to understand the two-phase flows. The choice of the examples retained, mainly of the exact solutions of the continuous equations, serves to show that the treatment of the conditions of jumps does not affect the precision of the method of resolution.

Investigation of bright and dark solitons in α, β-Fermi Pasta Ulam lattice

We consider the Hamiltonian of α, β-Fermi Pasta Ulam lattice and explore the Hamilton–Jacobi formalism to obtain the discrete equation of motion. By using the continuum limit approximations and incorporating some normalized parameters, the extended Korteweg–de Vries equation is obtained, with solutions that elucidate on the Fermi Pasta Ulam paradox. We further derive the nonlinear Schrödinger amplitude equation from the extended Korteweg–de Vries equation, by exploring the reductive perturbative technique. The dispersion and nonlinear coefficients of this amplitude equation are functions of the α and β parameters, with the β parameter playing a crucial role in the modulational instability analysis of the system. For β greater than or equal to zero, no modulational instability is observed and only dark solitons are identified in the lattice. However for β less than zero, bright solitons are traced in the lattice for some large values of the wavenumber. Results of numerical simulations of both the Korteweg–de Vries and nonlinear Schrödinger amplitude equations with periodic boundary conditions clearly show that the bright solitons conserve their amplitude and shape after collisions.

On Helmholtz–Hodge decomposition of inertia on a discrete local frame of reference

The notion of inertial reference frame is abandoned and I replaced it by a local reference frame on which the fundamental law of mechanics is expressed. The distant interactions of cause and effect are modeled by the propagation of waves from one local reference frame to another. The derivation of the equation of motion on a straight segment serves to express the proper acceleration as the sum of the accelerations imposed on it, in the form of an orthogonal local Helmholtz-Hodge decomposition, in one divergence-free and another curl-free contribution. I wrote the inertia term in the form of a gradient of a scalar potential and a dual curl of a vector potential. The adopted formalism opens the way to a reformulation of the material derivative in terms of potentials and allows me to remove the fictitious forces from continuum mechanics. The discrete equation of motion, invariant by rotation at constant angular velocity, is used to conserve the angular momentum per unit of mass, in addition to the conservation of energy per unit of mass and acceleration. All the variables in this equation are expressed only with two fundamental units, those of length and time.

On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity–potentials of acceleration

Discrete mechanics makes it possible to formulate any problem of fluid mechanics or fluid-structure interaction in velocity and potentials of acceleration; the equation system consists of a single vector equation and potentials updates. The scalar potential of the acceleration represents the pressure stress and the vector potential is related to the rotational-shear stress. The formulation of the equation of motion can be expressed in the form of a splitting which leads to an exact projection method; the application of the divergence operator to the discrete motion equation exhibits, without any approximation, a Poisson equation with constant coefficients on the scalar potential whatever the variations of the physical properties of the media. The a posteriori calculation of the pressure is made explicitly by introducing at this stage the local density. Two first examples show the interest of the formulation presented on classical solutions of Navier-Stokes equations; similarly as other results obtained with this formulation, the convergence is of order two in space and time for all the quantities, velocity and potentials. This formulation is then applied to a two-phase flow driven by surface tension and partial wettability. The last case corresponds to a fluid-structure interaction problem for which an analytical solution exists.

Vibration analysis of a free moving thin plate with fully covered active constrained layer damping treatment

A comprehensive dynamic model for a free moving thin plate with fully covered active constrained layer damping (ACLD) treatment is developed. The discrete equations of motion of the moving ACLD plate are derived by using Ritz method and Lagrange’s equations. Free vibration analysis of the composite plate undergoing large rotational or translational motions are performed by solving the complex modal eigenvalue problem of the system in open-loop and closed-loop cases. Interesting frequency curve veering, mode shape switching as well as damping ratio jumping phenomena are observed and discussed for the plate rotating around two different axes. It is demonstrated that the flexible plate undergoing rigid motions can not only generate dynamic stiffening effect but also dynamic softening effect. Effects of parameters such as angular velocity, acceleration, and control gain on the modal and damping characteristics of the ACLD plate are also investigated.

Derivation of equations of the model of the dynamic behaviour of the three-dimensional atmospheric cloud of electrically charged ice crystals under the influence of electrostatic forces

The purpose of this research is to create a theoretical model describing the dynamics of a cloud of electrically charged ice crystals in a three-dimensional space. The linear motion of the crystals was declined and considerations were limited to rotation of these crystals under the influence of electrostatic forces. The crystals were modeled as thin circular discs to simulate hexagonal plate ice crystals. This has reduced problem to the problem of two degrees of freedom (that allow for rotation around its center of gravity) per crystal. The basic assumption of the model is the crystals in the cloud are synchronized, which means that the differences in rotation of adjacent crystals are small. With this assumption, it was possible to derive continuous differential equations describing the dynamic behavior of such a medium on a macro scale from equilibrium of moments (from electrostatic forces) acting on the single crystal and discrete equations of motion of the cloud. Finally, the system of differential equations is second order with respect to time and space with trigonometric variable coefficients is derived. Examples of solutions of the system of equations are shown and compared with both solutions of similar theoretical problems available in the literature and optical effects in the clouds described by observers. It can be used in the future to better understand the dynamic optical phenomena occurring in the atmosphere.

A Dimensional Reduction Algorithm and Software for Acyclically Dependent Constraints

For discrete equations of motion with acyclic equality constraints and within the context of the null-space method, an original Algorithm is introduced. By first permuting and then topologically ordering the degrees-of-freedom in the constraint gradient matrix, the saddle point problem can be solved with a sparse triangular system for the constraint equations. In this work, we show that saddle problems resulting from constrained (nonlinear) mechanical problems can always be set in this form, with constraint pivots being selected a priori. Given n discrete motion equations and m equality constraints, the original square sparse system is replaced by a sparse system and a sparse triangular solve with m2 coefficients and n – m right-hand sides. This triangular solve, which involves three sparse matrices (in existing literature only two of the three matrices are sparse), is here discussed in detail. Seven sparse operations are addressed (five standard and two nonstandard) in addition to some specific ad-hoc operations. Algorithms, source code and examples are presented in this work.

Solitary waves in bistable lattices with stiffness grading: Augmenting propagation control

In this study we introduce small perturbations in the forms of graded intersite stiffnesses and graded on-site potentials to a lattice composed of bistable unit cells under elastic interactions. Based on a known soliton solution in the $\ensuremath{\phi}$-4 model, we use a perturbation approach to approximate the effects of the perturbations on the propagation speeds of transition waves. Numerical validations follow on the exact discrete equations of motion, from which we observe eventual stoppage of transition waves in the periodic lattice under physical damping, unidirectional propagation of the waves in the direction of softening properties, and boomerang-like reflection of the waves in the stiffening direction. Finally, we present three-dimensional-printed experimental lattices, confirming the theoretical and numerical results. The observed behaviors imply the extreme controllability of solitary waves through slight engineering manipulations in material-level structures. We further find that both kink (rarefaction) and antikink (compression) waves are allowed at any site in the lattice, extending the functionality of the lattice in engineering applications such as energy harvesting.

Free and forced whirling of a string with constant axial angular momentum using Cosserat Point Elements (CPE)

A chain of Cosserat Point Elements (CPEs) is used to model whirling of an extensible string. The discrete equations of motion are studied analytically and numerical results are presented. Particular emphasis is focused on steady-state solutions for which it is proved that the string must remain in a rotating plane. Free and forced non-steady solutions, together with their stability properties, are studied numerically.

Finite element modeling of a single-point multi-segment mooring in water waves

In this study, a finite element model is developed to simulate the problem of a single-point multi-segment mooring subjected to water waves. The governing equation for cable structure is derived from the principle of virtual work. A finite element method is then used to obtain the discrete equations for numerical computation. Due to high nonlinearity of the discrete equations of motion for cables, further manipulations, including incremental and iterative schemes, are used in the solution procedures, where the implicit Newmark's method is chosen for time integration. To describe the wave forces, the Morison equation is used to calculate the drag forces on cables, and the fluid motion generated by water wave is based on potential wave theory, in which the linear wave is applied. The initial equilibrium configuration of the cable structure, which is required before dynamic analysis, is calculated by the viscous relaxation technique in static analysis. Good agreement between numerical results is shown in calculation of a single-buoy relaxation. Comparisons of the present results of the four-segment mooring with an iterative model also confirm applicability of the present numerical model. Meanwhile, effects caused by nonlinear waves are also demonstrated.