Order Equations Of Motion Research Articles

Nonlinear analysis of sound transmission loss through cylindrical shell considering companion modes

This paper investigates the nonlinear vibro-acoustic performance of simply supported cylindrical shell under oblique incident plane sound wave including companion modes participation. The cylindrical shell equation of motion is obtained employing Donnell’s nonlinear shallow shell theory. The Galerkin method is applied to the equations of motion in order to obtain the system of nonlinear nonhomogeneous second order ordinary differential equations. The considered mode shapes are two asymmetric driven and companion modes and one axisymmetric mode. Multiple Scales Method is used to determine the response of resultant nonlinear differential equations. Next, the effects of incident sound characteristics on the transmission loss factor are studied. According to the results, effect of companion mode on the transmission loss (TL) through the shell can be seen only in high frequency of the incident sound wave. In addition, at high incident angle of acoustic wave, the companion mode effect on the TL of the system is very small.

Constraints on Yukawa gravity parameters from observations of bright stars

In this paper we investigate a Yukawa gravity modification of the Newtonian gravitational potential in a weak field approximation. For that purpose we derived the corresponding equations of motion and used them to perform two-body simulations of the stellar orbits. In 2020 the GRAVITY Collaboration detected the orbital precession of the S2 star around the supermassive black hole (SMBH) at the Galactic Center (GC) and showed that it is close to the general relativity (GR) prediction. Using this observational fact, we evaluated parameters of the Yukawa gravity (the range of Yukawa interaction Λ and universal constant δ) with the Schwarzschild precession of the S-stars assuming that the observed values as indicated by the GRAVITY Collaboration will have a small deviation from GR prediction [1]. GR provides the most natural way to fit observational data for S-star orbits, however, their precessions can be fitted by Yukawa gravity. Our main goal was to study the possible influence of the strength of Yukawa interaction, i.e. the universal constant δ, on the precessions of S-star orbits. We analyze S-star orbits assuming different strength of Yukawa interaction δ and find that this parameter has strong influence on range of Yukawa interaction Λ. For that purpose we use parameterized post-Newtonian (PPN) equations of motion in order to calculate the simulated orbits of S-stars in GR and Yukawa gravity. Using MCMC simulations we obtain the best-fit values and uncertainties of Yukawa gravity parameters for S-stars. Also, we introduce a new criterion which can be used for classification of gravitational systems in this type of gravity, according to their scales. We demonstrated that performed analysis of the observed S-stars orbits around the GC in the frame of the Yukawa gravity represent a tool for constraining the Yukawa gravity parameters and probing the predictions of gravity theories.

Well-posed formulation of Einstein-Maxwell effective field theory

We consider the well-posedness of the initial value problem for Einstein-Maxwell theory modified by higher derivative effective field theory corrections. Field redefinitions can be used to bring the leading parity-symmetric 4-derivative corrections to a form which gives second order equations of motion. We show that a recently introduced "modified harmonic" gauge condition can be used to obtain a formulation of these theories which admits a well-posed initial value problem when the higher derivative corrections to the equations of motion are small.

Effective action of string theory at order alpha ' in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order alpha ' in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order alpha ', we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.

Thermo-electro-mechanical buckling analysis of sandwich nanocomposite microplates reinforced with graphene platelets integrated with piezoelectric facesheets resting on elastic foundation

This paper investigates on the buckling treatment of a three layered rectangular nanocomposite microplate resting on elastic foundation. The core is a laminated nanocomposite layer reinforced with graphene platelets bonded with two piezoelectric facesheets. The microplate is subjected to thermo-electro-mechanical loads. The governing equations are developed in the framework of the first order shear deformation theory alongside with the modified couple stress theory. The effective laminated nanocomposite core layer thermo-mechanical properties are determined by the Halpin-Tsai micromechanical model. The Ritz method is employed to discretize the governing equations of motion in order to achieve the buckling loads for different boundary condition types. The outcomes are developed for both thermo-electrical and electro-mechanical buckling examinations. For the latter case, the microplate is studied when it is subjected to uniaxial, biaxial, and shear mechanical loads. The effects of boundary condition types, the external applied voltage, the graphene platelet distribution pattern, the elastic foundation parameters and the graphene platelet weight fraction as well as its geometrical features on the buckling loads are studied. The results reveal that the types of boundary condition, the graphene platelet dispersion pattern, the graphene platelet weight fraction, and the applied voltage affect noticeably the thermal/mechanical buckling load. Moreover, when the elastic foundation effect is considered the increment percent in the thermal and shear mechanical buckling loads is not affected by the graphene platelets dispersion pattern while for some certain boundary condition types, the graphene platelets distribution pattern affects the increment percent of the critical uniaxial and biaxial mechanical loads.

Causality in gravitational theories with second order equations of motion

This paper considers diffeomorphism invariant theories of gravity coupled to matter, with second order equations of motion. This includes Einstein-Maxwell and Einstein-scalar field theory with (after field redefinitions) the most general parity-symmetric four-derivative effective field theory corrections. A gauge-invariant approach is used to study the characteristics associated to the physical degrees of freedom in an arbitrary background solution. The symmetries of the principal symbol arising from diffeomorphism invariance and the action principle are determined. For gravity coupled to a single scalar field (i.e. a Horndeski theory) it is shown that causality is governed by a characteristic polynomial of degree $6$ which factorises into a product of quadratic and quartic polynomials. The former is defined in terms of an "effective metric" and is associated with a "purely gravitational" polarisation, whereas the latter generically involves a mixture of gravitational and scalar field polarisations. The "fastest" degrees of freedom are associated with the quartic polynomial, which defines a surface analogous to the Fresnel surface in crystal optics. In contrast with optics, this surface is generically non-singular except on certain surfaces in spacetime. It is shown that a Killing horizon is an example of such a surface. It is also shown that a Killing horizon satisfies the zeroth law of black hole mechanics. The characteristic polynomial defines a cone in the cotangent space and a dual cone in the tangent space. The latter is used to define basic notions of causality and to provide a definition of a dynamical black hole in these theories.

Higher order nonlocal viscoelastic strain gradient theory for dynamic buckling analysis of carbon nanocones

The dynamic buckling in viscoelastic carbon nanocones (CNCs) under the magnetic and thermal loads is considered in this article. This class of nano-structures has great application in nano-electro-mechanical system (NEMS) such as scanning probe microscopy. Structural damping influences are studied according to approach of Kelvin–Voigt. The viscoelastic medium around the CNC is assumed applying the model of visco-Pasternak. Utilizing the first order shear deformation theory (FSDT), Hamilton's principle, method of energy, the motion equations are determined where the influences of size are captured by higher order nonlocal strain gradient approach. Method of Bolotin and differential quadrature method (DQM) are utilized to solve the equations of motion in order to access the formation of dynamic instability region (DIR). The influences of different factors like, boundary conditions, nonlocal parameters, structural damping, magnetic field, viscoelastic medium, temperature changes along with CNCs semi vertex angle on the DIR are investigated. The outcomes show that with rising strain gradient parameter, the DIR will be occurred at high frequencies. It is furthermore found that the magnetic load has a positive influence on the DIR of the nano-structure.

On four-dimensional Einsteinian gravity, quasitopological gravity, cosmology and black holes

We show that the combination of cubic invariants defining five-dimensional quasitopological gravity, when written in four dimensions, reduce to the version of four-dimensional Einsteinian gravity recently proposed by Arciniega, Edelstein & Jaime, that produces second order equations of motion in a FLRW ansatz, with a purely geometrical inflationary period. We introduce a quartic version of the four-dimensional Einsteinian theory with similar properties, and study its consequences. In particular we found that there exists a region on the space of parameters which allows for thermodynamically stable black holes, as well as a well-defined cosmology with geometrically driven inflation. We briefly discuss the cosmological inhomogeneities in this setup. We also provide a combination of quintic invariants with those properties.

Dark sector evolution in Horndeski models

We use the Equation of State (EoS) approach to study the evolution of the dark sector in Horndeski models, the most general scalar-tensor theories with second order equations of motion. By including the effects of the dark sector into our code EoS_class, we demonstrate the numerical stability of the formalism and excellent agreement with results from other publicly available codes for a range of parameters describing the evolution of the function characterising the perturbations for Horndeski models, αx, with x = K, B, M, T. After demonstrating that on sub-horizon scales (k≳ 10−3 Mpc−1 at z=0) velocity perturbations in both the matter and the dark sector are typically subdominant with respect to density perturbations in the equation of state for perturbations, we find an attractor solution for the dark sector gauge-invariant density perturbation Δds by neglecting its time derivatives in the equation describing its time evolution, as commonly done in the well-known quasi-static approximation. Using this result, we provide simplified expressions for the equation-of-state functions: the dark sector entropy perturbations wdsΓds and anisotropic stress wdsΠds. From this we derive a growth factor-like equation for both matter and dark sector and are able to capture the relevant physics for several observables with great accuracy. We finally present new analytical expressions for the well-known modified gravity phenomenological functions μ, η and Σ for a generic Horndeski model as functions of αx. We show that on small scales they reproduce expressions presented in previous works, but on large scales, we find differences with respect to other works.

Asymptotic structure of electromagnetism in higher spacetime dimensions

We investigate the asymptotic structure of electromagnetism in Minkowski space in even and odd spacetime dimensions $\geq 4$. We focus on $d>4$ since the case $d=4$ has been studied previously at length. We first consider spatial infinity where we provide explicit boundary conditions that admit the known physical solutions and make the formalism well defined (finite symplectic structure and charges). Contrary to the situation found in $d=4$ dimensions, there is no need to impose parity conditions under the antipodal map on the leading order of the fields when $d>4$. There is, however, the same need to modify the standard bulk symplectic form by a boundary term at infinity involving a surface degree of freedom. This step makes the Lorentz boosts act canonically. Because of the absence of parity conditions, the theory is found to be invariant under two independent algebras of angle-dependent $u(1)$ transformations ($d>4$). We then integrate the equations of motion in order to find the behaviour of the fields near null infinity. We exhibit the radiative and Coulomb branches, characterized by different decays and parities. The analysis yields generalized matching conditions between the past of $\mathscr{I}^+$ and the future of $\mathscr{I} ^-$.

Friction-induced vibration in a two-mass damped system

A two degree of freedom mass-spring-damper model on a moving lubricated belt is considered in this paper. The motions of the two masses are coupled through spring and damper and each mass interacts with a moving lubricated belt. The friction model includes the boundary, mixed and hydrodynamic regimes of lubrication contact regimes. Friction coefficient is presented as a function of sliding velocity at each mass interface that includes a combination of linear, exponential and cubic functions to account for relatively large coefficient at low sliding speed, followed by Stribeck effect at low to moderate speed ranges and finally a full hydrodynamic lubrication regime. A set of dimensionless parameters is introduced and system's dynamic response behavior as well as stability are studied for various combination of parameters. The results show that the unstable system response can be quasiperiodic, or periodic in the form of limit cycles away from the origin when the sliding velocity is relatively small. Triple and double periodic behavior is observed in addition to other responses when mass suspended to the ground is larger than the second mass. All responses become asymptotically stable when dimensionless sliding velocity threshold is exceeded. Finally, for a qualitative analysis, the method of averaging is used to solve the nonlinear coupled equations of motion in order to find the amplitude and frequency of steady-state vibration. A comparison between the numerical results from direct time integration and analytical findings shows that approximate solution is acceptable.

Phenomenology of large scale structure in scalar-tensor theories: Joint prior covariance of wDE , Σ , and μ in Horndeski theories

Ongoing and upcoming cosmological surveys will significantly improve our ability to probe the equation of state of dark energy, $w_{\rm DE}$, and the phenomenology of Large Scale Structure. They will allow us to constrain deviations from the $\Lambda$CDM predictions for the relations between the matter density contrast and the weak lensing and the Newtonian potential, described by the functions $\Sigma$ and $\mu$, respectively. In this work, we derive the theoretical prior for the joint covariance of $w_{\rm DE}$, $\Sigma$ and $\mu$, expected in general scalar-tensor theories with second order equations of motion (Horndeski gravity), focusing on their time-dependence at certain representative scales. We employ Monte-Carlo methods to generate large ensembles of statistically independent Horndeski models, focusing on those that are physically viable and in broad agreement with local tests of gravity, the observed cosmic expansion history and the measurement of the speed of gravitational waves from a binary neutron star merger. We identify several interesting features and trends in the distribution functions of $w_{\rm DE}$, $\Sigma$ and $\mu$, as well as in their covariances; we confirm the high degree of correlation between $\Sigma$ and $\mu$ in scalar-tensor theories. The derived prior covariance matrices will allow us to reconstruct jointly $w_{\rm DE}$, $\Sigma$ and $\mu$ in a non-parametric way.

Scalar-vector-tensor gravity theories

We construct the consistent ghost-free covariant scalar-vector-tensor gravity theories with second order equations of motion with derivative interactions. We impose locality, unitarity, Lorentz invariance and pseudo-Riemannian geometry as the fundamental terms. In the tensor sector we require diffeomorphism invariance, whereas we allow the vector sector to be gauge invariant or not. The resulting Lagrangians consist of new genuine couplings among these fields with at most two derivatives per field. They propagate five physical degrees of freedom in the gauge invariant case and six degrees of freedom if the gauge invariance is broken. In the corresponding limit of the free general functions in the Lagrangian, one recovers the generalized Proca theories. These scalar-vector-tensor theories will have important implications for cosmological and astrophysical applications, among which we mention the application to inflation and generation of primordial magnetic fields, new black hole and neutron star solutions, dark matter and dark energy.

Large-scale structure phenomenology of viable Horndeski theories

Phenomenological functions $\Sigma$ and $\mu$, also known as $G_{\rm light}/G$ and $G_{\rm matter}/G$, are commonly used to parameterize modifications of the growth of large-scale structure in alternative theories of gravity. We study the values these functions can take in Horndeski theories, i.e. the class of scalar-tensor theories with second order equations of motion. We restrict our attention to models that are in a broad agreement with tests of gravity and the observed cosmic expansion history. In particular, we require the speed of gravity to be equal to the speed of light today, as required by the recent detection of gravitational waves and electromagnetic emission from a binary neutron star merger. We examine the correlations between the values of $\Sigma$ and $\mu$ analytically within the quasi-static approximation, and numerically, by sampling the space of allowed solutions. We confirm that the conjecture made in [Pogosian:2016pwr], that $(\Sigma-1)(\mu -1) \ge 0$ in viable Horndeski theories, holds very well. Along with that, we check the validity of the quasi-static approximation within different corners of Horndeski theory. Our results show that, even with the tight bound on the present day speed of gravitational waves, there is room within Horndeski theories for non-trivial signatures of modified gravity at the level of linear perturbations.

Vainshtein mechanism after GW170817

The almost simultaneous detection of gravitational waves and a short gamma-ray burst from a neutron star merger has put a tight constraint on the difference between the speed of gravity and light. In the four-dimensional scalar-tensor theory with second order equations of motion, the Horndeski theory, this translates into a significant reduction of the viable parameter space of the theory. Recently, extensions of Horndeski theory, which are free from Ostrogradsky ghosts despite the presence of higher order derivatives in the equations of motion, have been identified and classified exploiting the degeneracy criterium. In these new theories, the fifth force mediated by the scalar field must be suppressed in order to evade the stringent Solar System constraints. We study the Vainshtein mechanism in the most general degenerate higher order scalar-tensor theory in which light and gravity propagate at the same speed. We find that the Vainshtein mechanism generally works outside a matter source but it is broken inside matter, similarly to beyond Horndeski theories. This leaves interesting possibilities to test these theories that are compatible with gravitational wave observations using astrophysical objects.

Spacetime-bridge solutions in vacuum gravity

Spacetimes, which are representations of a bridge-like geometry in gravity theory, are constructed as vacuum solutions to the first order equations of motion. Each such configuration consists of two copies of an asymptotically flat sheet, connected by a bridge of finite extension where tetrad is noninvertible. These solutions can be classified into static and non-static spacetimes. The associated SO(3,1) invariant fields, namely the metric, affine connection and field-strength tensor, are all continuous across the hypersurfaces connecting the invertible and noninvertible phases of tetrad and are finite everywhere. These regular spacetime-bridge solutions do not have any analogue in Einsteinian gravity in vacuum.

Taming galileons in curved spacetime

Localising galileon symmetry along with Poincaré symmetry, we have found a version of the galileon model coupled with curved spacetime which retains the internal galileon symmetry in covariant form. Also, the model has second order equations of motion.

Priors on the effective dark energy equation of state in scalar-tensor theories

Constraining the dark energy (DE) equation of state, ${w}_{\mathrm{DE}}$, is one of the primary science goals of ongoing and future cosmological surveys. In practice, with imperfect data and incomplete redshift coverage, this requires making assumptions about the evolution of ${w}_{\mathrm{DE}}$ with redshift $z$. These assumptions can be manifested in a choice of a specific parametric form, which can potentially bias the outcome, or else one can reconstruct ${w}_{\mathrm{DE}}(z)$ nonparametrically, by specifying a prior covariance matrix that correlates values of ${w}_{\mathrm{DE}}$ at different redshifts. In this work, we derive the theoretical prior covariance for the effective DE equation of state predicted by general scalar-tensor theories with second order equations of motion (Horndeski theories). This is achieved by generating a large ensemble of possible scalar-tensor theories using a Monte Carlo methodology, including the application of physical viability conditions. We also separately consider the special subcase of the minimally coupled scalar field, or quintessence. The prior shows a preference for tracking behaviors in the most general case. Given the covariance matrix, theoretical priors on parameters of any specific parametrization of ${w}_{\mathrm{DE}}(z)$ can also be readily derived by projection.

Tensor Fields on Self-Dual Warped AdS3 Background

Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3-dimensional self-dual space-like warped AdS3 black hole background. We obtain a Klein–Gordon-like equation for tensor fields. By using gauge constraint and traceless condition, we will find the exact solutions of the equations of motion. Then, we will compute the quasi-normal modes by imposing appropriate boundary conditions at horizon and infinity.

Nondimensional analysis of a two-dimensional shear deformable beam in absolute nodal coordinate formulation

Absolute nodal-coordinate formulation is a technique that was developed in 1996 for expressing the large rotation and deformation of a flexible body. It utilizes global slopes without a finite rotation in order to define nodal coordinates. The method has a shortcoming in that the central processing unit time increases because of increases in the degrees of freedom. In particular, when considering the deformation of a cross section, the shortcoming due to the increase in the degrees of freedom becomes clear. Therefore, in the present research, the dimensional equation of motion concerning a two-dimensional shear deformable beam, developed by Omar and Shabana, is converted into a nondimensional equation of motion in order to reduce the central processing unit time. By utilizing an example of a cantilever beam, wherein an exact solution for the static deflection exists, the nondimensional equation of motion was verified. Moreover, by using an example of a free-falling flexible pendulum, the efficiency of the nondimensional equation of motion gained by increasing the number of elements was compared with that of the dimensional equation of motion.