Terms In Equations Of Motion Research Articles

Dynamic Stability of Sandwich Beams/Wide Plates Subjected to Axial Impulsive Loads

Abstract This paper presents an analysis for the dynamic stability of sandwich beams/wide plates subjected to axial impulsive loads. The formulation and solution of the problem is done by use of the extended high-order sandwich panel theory (EHSAPT). With the initial geometric imperfection included, the equations of motion in terms of seven generalized displacements are derived. The dynamic response of sandwich panels subjected to three different types of impulsive loads, namely, step, linear decay, and triangular impulse, is studied. Furthermore, the effects of the oscillation mode number, face/core materials, and geometries are investigated. It is observed that all measurements of the dynamic response, such as the maximum displacements, strains, and stresses, change at the same rate as the change of the impulse load magnitude and duration, for a specific impulse load profile. When the impulse load is lower than the static buckling load, the dynamic response is bounded no matter how long the load is applied. A step impulsive axial load with magnitude lower than the static buckling load can lead a sandwich panel to have a dynamic response as high as twice the static response. When the impulse load is higher than the static critical load, the dynamic response is unbounded with increasing load duration. However, it is possible that the dynamic response can be controlled at a low level if the duration of the impulse load is short enough, and thus, in this case, the load can safely exceed the static critical load.

Integrability of Differential Equations of Motion of an n-Dimensional Rigid Body in Nonconservative Fields for n = 5 and n = 6

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Electro-mechanical vibration characteristics of piezoelectric nano shells

Abstract This paper presents the application of higher-order shear deformation theory and nonlocal elasticity theory to electro-mechanical vibration analysis of a doubly curved piezoelectric nano shell resting on Pasternak's foundation. The piezoelectric doubly curved nanoshell is subjected to initial electro-mechanical loads. Effect of initial electro-mechanical loads is contributed in external works. Size effects are captured by nonlocal elasticity theory and Hamilton's principle is employed to derive the equation of motion in terms of three displacements of the middle surface, two rotational components and one electric potential. The main novelty of this paper is investigating concurrent effect of initial electro-mechanical loads, higher-order shear deformation theory and size dependent theory on the free vibration responses of doubly curved piezoelectric nano shell with various boundary conditions. The electro-mechanical vibration response of the doubly curved piezoelectric nano shell is investigated using an analytical method in terms of various parameters such as two opening angles, small scale parameter, spring and shear parameters of foundation and initial electric potential. It is concluded that increasing the nonlocal parameter leads to decrease of the natural frequencies of shell while increasing the applied electric potential increases the natural frequencies.

Waves over Curved Bottom: The Method of Composite Conformal Mapping

A compact and efficient numerical method is described for studying plane flows of an ideal fluid with a smooth free boundary over a curved and nonuniformly moving bottom. Exact equations of motion in terms of the so-called conformal variables are used. In addition to the previously known applications for shear flows with constant (including zero) vorticity, here a generalization is made to the case of potential flows in uniformly rotating coordinate systems, where centrifugal and Coriolis forces are added to the gravity force. A brief review is given of previous results obtained by this method in a number of physically interesting problems such as modeling of tsunami waves caused by the movement of nonuniform bottom, the dynamics of Bragg (gap) solitons over a spatially periodic bottom profile, the Fermi-Pasta-Ulam (FPU) recurrence phenomenon for waves in a finite pool, the formation of anomalous waves in an opposing nonuniform current, and the propagation of a solitary wave in a shear current and its runup on a depth difference. In addition, a number of new numerical results are presented concerning the nonlinear dynamics of a free boundary in closed rotating containers partially filled with a fluid -- centrifuges of complex shape. In this case, the equations of motion differ in some essential details from those of $x$-periodic systems.

Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model

This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

Active control of dynamic behaviors of graded graphene reinforced cylindrical shells with piezoelectric actuator/sensor layers

This work investigates the active vibration control and vibration characteristics of a sandwich thin cylindrical shell whose intermediate layer is made of the graphene reinforced composite that is bonded with integrated piezoelectric actuator and sensor layers at its outer and inner surfaces. The volume fraction of graphene platelets in the intermediate layer varies continuously in the shell's thickness direction, which generates position-dependent effective material properties. The constitutive relations of the graphene reinforced composite and piezoelectric materials are given by taking one-dimensional steady thermal field into account. Considering Donnell's shell theory, a final equation of motion in terms of the generalized radial displacement is derived by using Hamilton's principle and Galerkin method. Shell's natural frequencies are derived considering influences of the thermo-electro-elastic field. Introducing a constant velocity feedback control algorithm, active vibration control of the sandwich cylindrical shell is presented by employing the Runge-Kutta method. The feedback control gain has a pronounced effect on the damping, as well as the inertia of the system. Comparisons between the present results and those in other papers are done to validate the present solutions. Influences of weight fractions, distribution patterns and geometrical sizes of graphene platelets, temperature variations, thicknesses of layers and the feedback control gain on the vibration characteristics and active vibration control behaviors of the novel sandwich cylindrical shell are discussed.

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

This paper introduces a more efficient formulation for flexible multibody dynamics based on the floating frame of reference formulation (FFRF). A two-step condensation method is proposed. The first condensation involves coordinate reduction using a modal transformation, from nodal coordinates to modal coordinates, in the context of model order reduction (MOR). The second condensation involves re-fabrication of the equations of motion using a dynamic condensation of the deformation modal coordinates obtained from the first condensation. In the first condensation, constraint modes are corrected by condensation of the truncated modes that represent the effects of residual flexibility. As a result, the first condensation offers a more accurate deformation field in the reduced order model, which consequently leads to a more accurate flexible multibody simulation. In the second condensation, the equations of motion of a flexible multibody are once more reduced to the equations of motion in terms of absolute coordinates for the body reference frames. In this way, the proposed method deals with smaller sized equations of motion than in the conventional FFRF, where they are represented in terms of both absolute coordinates and deformation modal coordinates. The effect of the first condensation was verified using a benchmark example of a slider–crank mechanism with a flexible connecting rod. The efficiency of the proposed condensation method was also investigated using a computational complexity analysis with an $n$ -flexible body pendulum.

Exact holographic RG flows and the A1 \xd7 A1 Toda chain

We construct analytic solutions of Einstein gravity coupled to a dilaton field with a potential given by a sum of two exponentials, by rewriting the equations of motion in terms of an integrable Toda chain. These solutions can be interpreted as domain walls interpolating between different asymptotics, and as such they can have interesting applications in holography. In some cases, we can construct a solution which interpolates between an AdS fixed point in the UV limit and a hyperscaling violating boundary in the IR region. We also find analytic black brane solutions at finite temperature. We discuss the properties of the solutions and the interpretation in terms of RG flow.

Applications of Computer Algebra in the Study of the Two-Planet Problem of Three Bodies with Variable Masses

The classical two-planet problem of three bodies of variable masses is studied in the general case when the body masses vary anisotropically at different rates. Differential equations of motion in terms of osculating elements of aperiodic motion along quasi-conic sections are derived. An algorithm for computing the perturbation function in the form of power series in small parameters and the derivation of differential equations determining the secular perturbations of the orbital elements are discussed. All symbolic computations are performed using Mathematica.

Integrability and Dynamics of the n-Dimensional Symmetric Veselova Top

We consider the the n-dimensional generalisation of the nonholonomic Veselova problem. We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular we give a closed formula for the invariant measure, we indicate the existence of steady rotation solutions, and obtain some results on their stability. We then focus our attention on bodies whose mass tensor has a specific type of symmetry. We show that the phase space is foliated by invariant tori that carry quasi-periodic dynamics in the natural time variable. Our results enlarge the known cases of integrability of the multi-dimensional Veselova top. Moreover, they show that in some previously known instances of integrability, the flow is quasi-periodic without the need of a time reparametrisation.

Deformed Weitzenböck Connections and Double Field Theory

AbstractWe revisit the generalized connection of Double Field Theory. We implement a procedure that allow us to re‐write the Double Field Theory equations of motion in terms of geometric quantities (like generalized torsion and non‐metricity tensors) based on other connections rather than the usual generalized Levi‐Civita connection and the generalized Riemann curvature. We define a generalized contorsion tensor and obtain, as a particular case, the Teleparallel equivalent of Double Field Theory. To do this, we first need to revisit generic connections in standard geometry written in terms of first‐order derivatives of the vielbein in order to obtain equivalent theories to Einstein Gravity (like for instance the Teleparallel gravity case). The results are then easily extrapolated to DFT.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 2

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 1

In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Multibody model of railway vehicles with weakly coupled vertical and lateral dynamics

Abstract In the field of railroad dynamics, multibody models used for on-board state observation, parameter identification and track geometry measurement must ensure a high computational efficiency, particularly when real-time simulations are required. In this paper, a simplified model based on the assumption of weakly coupled lateral and vertical dynamics is proposed. Generalized forces included in the equations of motion are obtained using symbolic computations and the wheel-rail contact is modeled using knife-edge contact constraints and the concept of equivalent conicity. Resulting contact constraint equations can be solved analytically allowing the solution of ODE equations of motion in terms of independent coordinates. This general model considers arbitrary-geometry tracks including rail centerline irregularities. With the aim of showing the accuracy of the proposed model for the simulation of railways vehicle dynamics, a comparative study with a full 3D coupled dynamic model has been carried out for different vehicles and track geometries. The comparative study shows that the proposed simplified model in this work provides acceptable results for many applications. The model greatly improves the computational effort and it is applicable to real-life situations requiring a reduced set of parameters.

Swirling flow states of compressible single-phase supercritical fluids in a rotating finite-length straight circular pipe

Steady states of inviscid, compressible and axisymmetric swirling flows of a single-phase, inert, thermodynamically supercritical fluid in a rotating, finite-length, straight, long circular pipe are studied. The fluid thermodynamic behaviour is modelled by the van der Waals equation of state. A nonlinear partial differential equation for the solution of the flow streamfunction is derived from the fluid equations of motion in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. This equation reflects the complicated, nonlinear thermo-physical interactions in the flows, specifically when the inlet state temperature and density profiles vary around the critical thermodynamic point, flow compressibility is significant and the inlet swirl ratio is high. Several types of solutions of the resulting nonlinear ordinary differential equation for the axially independent case describe the flow outlet state when the pipe is sufficiently long. The approach is applied to an inlet flow described by a solid-body rotation with uniform profiles of the axial velocity and temperature. The solutions are used to form the bifurcation diagrams of steady compressible flows of real fluids as the inlet swirl level and the centreline inlet density are increased at a fixed inlet Mach number and temperature. Focus is on heavy-molecule fluids with low values of $R/C_{v}$. Computed results provide theoretical predictions of the critical swirl levels for the exchange of stability of the columnar state and for the appearance of non-columnar states and of vortex breakdown states as a function of inlet centreline density. The difference in the dynamical behaviour between that of a calorically perfect gas and of a real gas is explored. The analysis sheds new fundamental light on the complex dynamics of high-Reynolds-number, compressible, subsonic swirling flows of real gases.

Interpretation of f(R,T) gravity in terms of a conserved effective fluid

In the present work, we introduce a novel approach to study [Formula: see text] gravity theory from a different perspective. Here, [Formula: see text] denotes the trace of energy–momentum tensor ( EMT ) of matter fluids. The usual method (as discussed in the literature) is to choose an [Formula: see text] function and then solve for the resulted Friedman equations. Nevertheless, our aim here is, without loss of generality, to reformulate a particular class of [Formula: see text] gravity models in which the Einstein–Hilbert action is promoted by an arbitrary function of the trace of EMT . The strategy is the redefinition of the equation of motion in terms of the components of an effective fluid. We show that in this case the EMT is automatically conserved. As we shall see, adopting such a point of view (at least) in [Formula: see text] gravity is accompanied by two significant points. On one hand, [Formula: see text] function is chosen based upon a physical concept and on the other, we clearly understand the overall or effective behavior of matter in terms of a conserved effective fluid. To illustrate the idea, we study some models in which different physical properties for the effective fluid is attributed to each model. Particularly, we discuss models with constant effective density, constant effective pressure and constant effective equation of state ( EoS ) parameter. Moreover, two models with a relation between the effective density and the effective pressure will be considered. An elegant result is that in [Formula: see text] gravity, there is a possibility that a perfect fluid could effectively behave as a modified Chaplygin gas with four free parameters.

Element-free Galerkin model of nano-beams considering strain gradient elasticity

This paper deals with the investigation of the static bending response of nano-beams upon application of a mechanical load, considering the effect of gradient elasticity. The gradient elasticity is modelled using the strain gradient elasticity model with three independent material length constants. For the numerical modelling of the structural response, the higher-order displacement theory is used. The element-free Galerkin method is employed to interpolate the displacement field in the continuum in terms of the nodal displacements. Thereafter, using the variational principle, the governing equations of motion in terms of the nodal displacements are derived in the algebraic form. The model developed in this paper is validated by comparison with the exact solutions available in the literature, for the specific case of a simply supported beam under a sinusoidally distributed load. Following this, the numerical model is solved for various other boundary conditions and loading functions. The effect of the strain gradient elasticity is studied for different aspect ratios, and the results are compared. The reduction of the bending response due to the strain gradient effect is compared for each case. A comparison of the effects of each of the material length scales, corresponding to different forms of the higher-order gradients, over the structural response is performed. The present study reveals that the strain gradient elasticity must be considered for an accurate analysis of nanostructures.

Isospectral Flows for the Inhomogeneous String Density Problem

We derive isospectral flows of the mass density in the string boundary value problem corresponding to general boundary conditions. In particular, we show that certain class of rational flows produces in a suitable limit all flows generated by polynomials in negative powers of the spectral parameter. We illustrate the theory with concrete examples of isospectral flows of discrete mass densities which we prove to be Hamiltonian and for which we provide explicit solutions of equations of motion in terms of Stieltjes continued fractions and Hankel determinants.

The eccentric Kozai–Lidov effect as a resonance phenomenon

Exploring weakly perturbed Keplerian motion within the restricted three-body problem, Lidov (1962) and, independently, Kozai (1962) discovered coupled oscillations of eccentricity and inclination (the KL-cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the expression for the disturbing function, long-term modulation of the KL-cycles can established (Ford et al., 2000, Naoz et al., 2011, Katz et al., 2011). Specifically, flips between the prograde and retrograde orbits become possible. Since such flips are observed only when the perturber has a non-zero eccentricity, the term "Eccentric Kozai-Lidov Effect" (or EKL-effect) was proposed by Lithwick and Naoz (2011) to specify such behaviour. We demonstrate that the EKL-effect can be interpreted as a resonance phenomenon. To this end, we write down the equations of motion in terms of "action-angle" variables emerging in the integrable Kozai-Lidov model. It turns out that for some initial values the resonance is degenerate and the usual "pendulum" approximation is insufficient to describe the evolution of the resonance phase. Analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space.

Classical field theory on Lie algebroids: Multisymplectic formalism

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle we give a variational description of the equations of motion. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincare and Lagrange Poincare cases), variational problems for holomorphic maps, Sigma models or Chern-Simons theories. One of the advantages of our theory is that it is based in the existence of a multisymplectic form on a Lie algebroid.