Order Equations Of Motion Research Articles

Differential transform method and Adomian decomposition method for free vibration analysis of fluid conveying Timoshenko pipeline

The free vibration analysis of fluid conveying Timoshenko pipeline with different boundary conditions using Differential Transform Method (DTM) and Adomian Decomposition Method (ADM) has not been investigated by any of the studies in open literature so far. Natural frequencies, modes and critical fluid velocity of the pipelines on different supports are analyzed based on Timoshenko model by using DTM and ADM in this study. At first, the governing differential equations of motion of fluid conveying Timoshenko pipeline in free vibration are derived. Parameter for the nondimensionalized multiplication factor for the fluid velocity is incorporated into the equations of motion in order to investigate its effects on the natural frequencies. For solution, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Timoshenko beam theory. After the analytical solution, the efficient and easy mathematical techniques called DTM and ADM are used to solve the governing differential equations of the motion, respectively. The calculated natural frequencies of fluid conveying Timoshenko pipelines with various combinations of boundary conditions using DTM and ADM are tabulated in several tables and figures and are compared with the results of Analytical Method (ANM) where a very good agreement is observed. Finally, the critical fluid velocities are calculated for different boundary conditions and the first five mode shapes are presented in graphs.

Effect of flow pulsatility and time-dependent release kinetics on stent-based delivery through atherosclerotic plaque

The present work is concerned with an investigation of the effect of flow pulsatility and time-dependent release kinetics on the transport and retention of drug in a stent-based delivery through atherosclerotic plaque. Keeping the relevance of the physiological situation in view, an axi-symmetric two-dimensional model of drug delivery from a strut has been developed in cylindrical polar coordinate system. Here, the luminal drug transport is considered as an unsteady convection–diffusion process, while the drug transport within the arterial tissue is considered as a two-species model (both free and bound), capable of predicting the delivery of drug and its retention in the arterial tissue, in which the specific binding site action is modelled using a reversible chemical reaction. The transport of free drug is modelled as an unsteady convection–diffusion–reaction process, while the bound drug, assuming completely immobilised in the tissue, as an unsteady reaction process. An explicit finite-difference scheme in staggered grids has been developed to tackle numerically the governing equations of motion in order to compute the physiologically significant quantities with desired degree of accuracy. Model results predicted that, unlike within the tissue, the averaged free drug concentration increases with the increase of Re in the lumen and also the averaged concentrations of both drug forms decrease with increasing plaque thickness. Moreover, higher Peclet number in the tissue and coating results in lowering down the averaged concentrations of free as well as bound drug in the tissue and lumen. The present results predict that the presence of plaque dramatically affects the local pharmacokinetics and provides higher resistance to drug transfer in the arterial wall. The averaged concentration of free drug increases in the lumen due to flow pulsatility while a reverse trend is observed in the tissue. Additionally, modelling results also indicate that time-dependent release of drug in stent-based delivery diminishes the averaged concentration of both drug forms as compared to constant release kinetics. All the above observations are consistent with several existing results in the literature.

What can cosmology tell us about gravity? Constraining Horndeski gravity with Σ and μ

Phenomenological functions $\Sigma$ and $\mu$ (also known as $G_{\rm light}/G$ and $G_{\rm matter}/G$) are commonly used to parameterize possible modifications of the Poisson equation relating the matter density contrast to the lensing and the Newtonian potentials, respectively. They will be well constrained by future surveys of large scale structure. But what would the implications of measuring particular values of these functions be for modified gravity theories? We ask this question in the context of general Horndeski class of single field scalar-tensor theories with second order equations of motion. We find several consistency conditions that make it possible to rule out broad classes of theories based on measurements of $\Sigma$ and $\mu$ that are independent of their parametric forms. For instance, a measurement of $\Sigma \ne 1$ would rule out all models with a canonical form of kinetic energy, while finding $\Sigma-1$ and $\mu-1$ to be of opposite sign would strongly disfavour the entire class of Horndeski models. We separately examine the large and the small scale limits, the possibility of scale-dependence, and the consistency with bounds on the speed of gravitational waves. We identify sub-classes of Horndeski theories that can be ruled out based on the measured difference between $\Sigma$ and $\mu$.

Intensification of nonlinearity by dispersed events in media with structure and generation of various dominant frequencies in P- and S-waves

A model of a continuum with a structure described by infinite order equations of motion is proposed. In case that a wave is very long as compared with the size of the structure, equations are reduced to the fourth-order equations. A closed equation of motion, including nonlinear, dispersed and wave members, is derived. It is shown that solutions in the form of soliton waves exist only in media where wave velocity grows with pressure. In the media, where soliton waves do not exist, quasi-stationary solutions with multiple frequencies prevail. It is found that the nonlinear effect of multiple frequencies is unexpectedly high even for small deformation as dispersion violently intensifies nonlinear events. Moreover, in the domain of small deformation, there exist solutions for longitudinal and transversal waves with the same length and different frequencies. The solutions for the same length waves with different frequencies most often occur in seismology and seismic explorations.

Efficiency of non-dimensional analysis for absolute nodal coordinate formulation

Absolute nodal coordinate formulation was developed in the mid-1990s. The adoption of the continuum mechanics concept has allowed large displacements and large deformations to be expressed in flexible body analysis. However, the analysis time increases due to the increased number of degrees of freedom at nodal points. Therefore, we aimed to reduce the analysis time by converting a dimensional equation of motion (EOM) to a non-dimensional EOM by using non-dimensional variables of time, length, and force. A non-dimensional mass matrix, a non-dimensional longitudinal stiffness matrix, and a non-dimensional conservative force vector are derived and applied to the non-dimensional EOM. To verify the non-dimensional EOM, a cantilever beam with static deflection, for which an exact solution exists, is considered. As the number of elements is increased, the mean value by the non-dimensional EOM converges to the static deflection. Revolute and spherical joints are used to propose two- and three-dimensional numerical solutions based on the non-dimensional EOM of a free-falling pendulum. These solutions are compared with the numerical solutions from using a dimensional EOM in order to verify the non-dimensional EOM. The analysis results for simple pendulum motion using dimensional and non-dimensional EOMs are in good agreement.

Squeeze film characteristics of circular contacts at impact and rebound motion with micropolar lubricants

In this paper, pure squeeze elastohydrodynamic lubrication (EHL) motion of circular contacts with micropolar lubricant is explored at the impact and rebound processes from a lubricated surface. On the basis of micro-continuum theory, the transient modified Reynolds equation is derived. Then it is solved simultaneously with the elasticity deformation, rheology, and ball motion equations in order to obtain the transient pressure profiles, film shapes, normal squeeze velocities, and accelerations. The simulation results reveal that the effect of the micropolar lubricant is equivalent to enhancing the lubricant viscosity, which would also enlarge the damper effect. Therefore, as the characteristic length (L) and coupling number (N) of the micropolar lubricants increase, the pressure spike and the dimple form earlier, the maximum pressure and the film thickness increase, the diameter of the dimple decreases, the rebounding velocity and the maximum acceleration decrease, and the maximum relative impact force decreases. Furthermore, the secondary peak of central pressure occurred at the rebound end is greater than the first peak of central pressure occurred at maximum impact force. As the effects of micropolar lubricants increase, the first and second peaks form earlier, the total impact time decreases, and the first and the second peaks increase.

Towards the most general scalar-tensor theories of gravity: A unified approach in the language of differential forms

We use a description based on differential forms to systematically explore the space of scalar-tensor theories of gravity. Within this formalism, we propose a basis for the scalar sector at the lowest order in derivatives of the field and in any number of dimensions. This minimal basis is used to construct a finite and closed set of Lagrangians describing general scalar-tensor theories invariant under Local Lorentz Transformations in a pseudo-Riemannian manifold, which contains ten physically distinct elements in four spacetime dimensions. Subsequently, we compute their corresponding equations of motion and find which combinations are at most second order in derivatives in four as well as arbitrary number of dimensions. By studying the possible exact forms (total derivatives) and algebraic relations between the basis components, we discover that there are only four Lagrangian combinations producing second order equations, which can be associated with Horndeski's theory. In this process, we identify a new second order Lagrangian, named kinetic Gauss-Bonnet, that was not previously considered in the literature. However, we show that its dynamics is already contained in Horndeski's theory. Finally, we provide a full classification of the relations between different second order theories. This allows us to clarify, for instance, the connection between different covariantizations of Galileons theory. In conclusion, our formulation affords great computational simplicity with a systematic structure. As a first step we focus on theories with second order equations of motion. However, this new formalism aims to facilitate advances towards unveiling the most general scalar-tensor theories.

Investigation of non-Hermitian Hamiltonians in the Heisenberg picture

The Heisenberg picture for non-Hermitian but η-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but η-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two new non-Hermitian Hamiltonians are constructed and investigated, where one is non-Hermitian and non-PT-symmetric and the other is non-Hermitian but PT-symmetric. Moreover, the complementarity and compatibility between our treatment and the PT symmetry are discussed.

Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation

In this present study, Chebyshev collocation method is utilized to solve buckling and vibration problems of functionally graded (FG) sandwich beams resting on two-parameter elastic foundation including Winkler and shear layer springs. The faces of FG sandwich beam are assumed to be made by functionally graded materials (FGMs) composing of ceramic and metal phases and the core of the beam is made from homogenous material. Timoshenko beam theory is employed to construct the governing equations of motion in order to cover the significant effects of shear deformation and rotary inertia. The beams with various boundary conditions are considered to find out their critical loadings and natural frequencies. An accuracy of the present solutions is confirmed by comparing with some available results in the literature. Moreover, many important parametric studies of layer and beam thickness ratios, material volume fraction index, spring constants, etc. are taken into investigation. According to numerical exercises, it is revealed that the spring constants of elastic foundation have significant impact on buckling and vibration results of such beams.

New Formulations on Motion Equations in Analytical Dynamics

Using the main author's researches on the energies of acceleration and higher order equations of motion, this paper is devoted to new formulations in analytical dynamics of mechanical multibody systems (MBS). Integral parts of these systems are the mechanical robot structures, serial, parallel or mobile on which an application will be presented in order to highlight the importance of the differential motion equations in dynamics behavior. When the components of multibody mechanical systems or in its entirety presents rapid movements or is in transitory motion, are developed higher order variations in respect to time of linear and angular accelerations. According to research of the main author, they are integrated into higher order energies and these in differential equations of motion in higher order, which will lead to variations in time of generalized forces which dominate these types of mechanical systems. The establishing of these differential equations of motion, it is based on a generalization of a principle of analytical differential mechanics, known as the D`Alembert – Lagrange Principle.

Hyperscaling violating black holes in scalar-torsion theories

We study a gravity theory where a scalar field with potential, beyond its minimal coupling, is also coupled through a non-minimal derivative coupling with the torsion scalar which is the teleparallel equivalent of Einstein gravity. This theory provides second order equations of motion and we find large-distance non-perturbative static spherically symmetric four-dimensional solutions. Among them a general class of black hole solutions is found for some range of the parameters/integration constants with asymptotics of the form of hyperscaling violating Lifshitz spacetime with spherical horizon topology. Although the scalar field diverges at the horizon, its energy density and pressures are finite there. From the astrophysical point of view, this solution provides extra deflection of light compared to the Newtonian deflection.

Modified Gravity Models Admitting Second Order Equations of Motion

The aim of this paper is to find higher order geometrical corrections to the Einstein-Hilbert action that can lead to only second order equations of motion. The metric formalism is used, and static spherically symmetric and Friedmann-Lema\^itre space-times are considered, in four dimensions. The FKWC-basis are introduced in order to consider all the possible invariant scalars, and both polynomial and non-polynomial gravities are investigated.

Non-extremal black hole solutions from the c-map

We construct new static, spherically symmetric non-extremal black hole solutions of four-dimensional ${\cal N}=2$ supergravity, using a systematic technique based on dimensional reduction over time (the c-map) and the real formulation of special geometry. For a certain class of models we actually obtain the general solution to the full second order equations of motion, whilst for other classes of models, such as those obtainable by dimensional reduction from five dimensions, heterotic tree-level models, and type-II Calabi-Yau compactifications in the large volume limit a partial set of solutions are found. When considering specifically non-extremal black hole solutions we find that regularity conditions reduce the number of integration constants by one half. Such solutions satisfy a unique set of first order equations, which we identify. Several models are investigated in detail, including examples of non-homogeneous spaces such as the quantum deformed $STU$ model. Though we focus on static, spherically symmetric solutions of ungauged supergravity, the method is adaptable to other types of solutions and to gauged supergravity.

Third order equations of motion and the Ostrogradsky instability

It is known that any nondegenerate Lagrangian containing time derivative terms higher than first order suffers from the Ostrogradsky instability, pathological excitation of positive and negative energy degrees of freedom. We show that, within the framework of analytical mechanics of point particles, any Lagrangian for third order equations of motion, which evades the nondegeneracy condition, still leads to the Ostrogradsky instability. Extension to the case of higher odd order equations of motion is also considered.

Nonlocal damage propagation in the dynamics of masonry elements

In this work, a nonlocal damage-plasticity model for dynamic finite element analyses of cohesive structural elements is presented. The proposed cohesive model is able to reproduce the main relevant behaviors of quasi-brittle materials despite being quite simple, i.e. governed by only a few parameters which can be determined by standard laboratory tests. In particular, the model is able to reproduce the mechanisms of cohesive materials under static or dynamic loads: degradation of the mechanical properties (damage) and accumulation of irreversible strains (plasticity). Moreover, the model also simulates the cyclic macroscopic behavior of quasi-brittle materials, taking into account the loss and recovery of stiffness due to crack closure and reopening. The latter effect represents a particularly important characteristic in the case of dynamic loads. The proposed formulation is implemented as a constitutive model for two-dimensional plane stress four-node quadrilateral elements. The second order equations of motion are solved adopting the implicit Newmark time integration scheme. The proposed model is validated and its dynamic performance is numerically demonstrated through the analysis of a large-scale structural element.

Hot attractors

The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions.

Post-Newtonian effects in N-body dynamics: conserved quantities in hierarchical triple systems

Conventional approaches to incorporating general relativistic effects into the dynamics of N-body systems containing central black holes, or of hierarchical triple systems with a relativistic inner binary, may not be adequate when the goal is to study the evolution of the system over a timescale related to relativistic secular effects, such as the precession of the pericenter. For such problems, it may be necessary to include post-Newtonian (PN) ‘cross terms’ in the equations of motion in order to capture relativistic effects consistently over the long timescales. Cross terms are PN terms that explicitly couple the two-body relativistic perturbations with the Newtonian perturbations due to other bodies in the system. In this paper, we show that the total energy of a hierarchical triple system is manifestly conserved to Newtonian order over the relativistic pericenter precession timescale of the inner binary if and only if PN cross-term effects in the equations of motion are taken carefully into account.

Relative Orbital Elements Formulation Based upon the Clohessy-Wiltshire Equations

For on-orbit applications where two or more spacecraft are flying in close proximity, it is often convenient to apply the Clohessy-Wiltshire differential relative motion equations in order to calculate the relative motion of a deputy spacecraft about a chief spacecraft that is assumed to be in a circular orbit. Under these assumptions, the solutions to the Clohessy-Wiltshire equations can be re-parameterized as a set of relative orbital elements that fully characterize the relative motion of the deputy about the chief. In contrast to the Cartesian relative position and velocity states, relative orbital elements provide a clear geometric interpretation of the relative motion and yield an intuitive understanding of how the unforced relative motion will evolve with time. In this paper, the derivation of relative orbital elements is given, and the transformation between relative orbital elements and Cartesian state elements expressed in the local-vertical, local-horizontal frame is provided. The evolution of relative orbital elements with time is evaluated, and characteristics of the unforced motion in terms of relative orbital elements are described.

NONLINEAR STRAIN GRADIENT THEORY BASED VIBRATION AND INSTABILITY OF BORON NITRIDE MICRO-TUBES CONVEYING FERROFLUID

Nonlinear vibration and instability of a boron nitride micro-tube (BNMT) conveying ferrofluid under the combined magnetic and electric fields are investigated. Based on Euler–Bernoulli beam (EBB), piezoelasticity strain gradient theory and Hamilton's principle, high order equations of motion are derived for three boundary conditions namely as clamped–clamped (C–C), simply–simply (S–S) and clamped–simply (C–S). The differential quadrature method (DQM) is applied to discretize the motion equations in order to obtain the nonlinear frequency and critical fluid velocity using a direct iterative method. A detailed parametric study is conducted to elucidate the influences of the various boundary conditions, size diameter and magnetic field on vibrational characteristic of BNMT. Numerical results indicate that the effect of magnetic field appears in higher speed of ferrofluid and increases the critical velocity or enlarges the stability region. The results are in good agreement with the previous researches. The results of this study can be used to manufacture smart micro/nano electromechanical systems in advanced biomechanics applications with magnetic and electric fields as parametric controllers.

Generalization of the Proca Action

We consider the Lagrangian of a vector field with derivative self-interactions with a priori arbitrary coefficients. Starting with a flat space-time we show that for a specialchoice of the coefficients of the self-interactions the ghost-like pathologies disappear. This constitutes the Galileon-type generalization of the Proca action with only three propagating physical degrees of freedom. The longitudinal mode of the vector field is associated to the usual Galileon interactions for a specific choice of the overall functions. In difference to a scalar Galileon theory, the generalized Proca field has more free parameters. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order equations of motion on curved space-times.