Terms In Equations Of Motion Research Articles

Dynamic modeling, input-shaped maneuvering and vibration suppression of flexible body using quasi-coordinates and euler parameters

In dealing with the dynamics of a flexible body, the rigid-body motions and elastic vibrations are analyzed separately. However, rigidbody motions cause vibrations, and elastic vibrations affect rigid-body motions, indicating the inherent coupling between rigid-body motions and elastic vibrations. The coupled equations of motion for a flexible body can be derived by means of Lagrange’s equations in terms of quasi-coordinates. The resulting equations of motion are hybrid and nonlinear. This paper proposes a unified approach for the equations of motion for a flexible body based on the perturbation method and the Lagrange’s equations of motion in terms of quasicoordinates and Euler parameters to analyze a more general case maneuvering. The resulting equations consist of zero-order nonlinear equations of motion which depict rigid-body motions and first-order time-varying linear equations of motion which depict perturbed rigid-body motions and elastic vibration. Hence, the input-shaped maneuvering can be applied to the zero-order equation considering the induced vibrations. Since the input-shaped maneuvering alone cannot achieve vibration suppression, the vibration suppression controller combined with the input-shaped maneuvering is proposed in this study. As a numerical example, a hub with elastic appendages is considered. Numerical results show that the unified modeling approach proposed in this paper is effective in numerical simulation and control design.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

A first-order secular theory for the post-Newtonian two-body problem with spin – I. The restricted case

We revisit the relativistic restricted two-body problem with spin employing a perturbation scheme based on Lie series. Starting from a post-Newtonian expansion of the field equations, we develop a first-order secular theory that reproduces well-known relativistic effects such as the precession of the pericentre and the Lense-Thirring and geodetic effects. Additionally, our theory takes into full account the complex interplay between the various relativistic effects, and provides a new explicit solution of the averaged equations of motion in terms of elliptic functions. Our analysis reveals the presence of particular configurations for which non-periodical behaviour can arise. The application of our results to real astrodynamical systems (such as Mercury-like and pulsar planets) highlights the contribution of relativistic effects to the long-term evolution of the spin and orbit of the secondary body.

‘Stringy’ Newton–Cartan gravity

We construct a ‘stringy’ version of Newton–Cartan gravity in which the concept of a Galilean observer plays a central role. We present both the geodesic equations of motion for a fundamental string and the bulk equations of motion in terms of a gravitational potential which is a symmetric tensor with respect to the longitudinal directions of the string. The extension to include a nonzero cosmological constant is given. We stress the symmetries and (partial) gaugings underlying our construction. Our results provide a convenient starting point to investigate applications of the AdS/CFT correspondence based on the non-relativistic ‘stringy’ Galilei algebra.

Fractional differential equations of motion in terms of combined Riemann—Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann—Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

Nonlocal wave propagation in an embedded DWBNNT conveying fluid via strain gradient theory

Abstract Based on the strain gradient and Eringen’s piezoelasticity theories, wave propagation of an embedded double-walled boron nitride nanotube (DWBNNT) conveying fluid is investigated using Euler–Bernoulli beam model. The elastic medium is simulated by the Pasternak foundation. The van der Waals (vdW) forces between the inner and outer nanotubes are taken into account. Since, considering electro-mechanical coupling made the nonlinear motion equations, a numerical procedure is proposed to evaluate the upstream and downstream phase velocities. The results indicate that the effect of nonlinear terms in motion equations on the phase velocity cannot be neglected at lower wave numbers. Furthermore, the effect of fluid-conveying on wave propagation of the DWBNNT is significant at lower wave numbers.

Dynamic modelling and actuation of the adaptive torsion wing

This article presents the dynamical modelling of a novel active aeroelastic structure. The adaptive torsion wing concept is a thin-wall, two-spar wingbox whose torsional stiffness can be adjusted by translating the spar webs in the chordwise direction inward and towards each other using internal actuators. The reduction in torsional stiffness allows external aerodynamic loads to induce twist on the structure and maintain its deformed shape. Here, the adaptive torsion wing system is considered as integrated within the wing of a representative unmanned aerial vehicle to replace conventional ailerons and provide roll control. The adaptive torsion wing is modelled as a two-dimensional equivalent aerofoil using bending and torsion shape functions to express the equations of motion in terms of the twist angle and plunge displacement at the wingtip. The full equations of motion for the adaptive torsion wing equivalent aerofoil were derived using Lagrangian mechanics. The aerodynamic lift and moment acting on the aerofoil were modelled using Theodorsen’s unsteady aerodynamic theory. A low-dimensional, state-space representation of an empirical Theodorsen’s transfer function was adopted to allow time-domain analyses. Four actuation strategies were investigated. Figures of merit, including plunge displacement, twist angle, actuation forces and actuation powers, were quantified and discussed for each of the scenarios. This study allows the conceptual design and sizing of the internal actuators that are required to drive the webs.

Hydroelasticity of marine vessels advancing in a seaway

An important branch of hydroelasticity deals with the dynamic interactions between elastic structures and ocean waves. While the modal superposition method has been the primary approach in hydroelastic analysis of marine vessels, we present a directly coupled approach in the frequency domain with a rigorous treatment of the vessel forward speed. The formulation adopts a translating coordinate system with the free surface boundary conditions accounting for the double-body flow around the vessel and the radiation condition taking into account the Doppler shift of the scattered waves. A boundary element model, based on the Rankine source formulation, describes the potential flow and the hydrodynamic pressure on the vessel. A finite element model relates the vessel response to the hydrodynamic pressure through a kinematic and a dynamic boundary condition on the wetted hull surface. This direct coupling of the structural and hydrodynamic systems leads to an equation of motion in terms of the nodal displacement of the finite elements. The results are evaluated against predictions from a seakeeping model with forward speed and the modal superposition method at zero speed. A parametric study of a Wigley hull shows that forward speed introduces new resonance modes, which amplify the elastic response of the vessel in a seaway.

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.

A distributed-mass approach for dynamic analysis of Timoshenko plane frames

The dynamic stiffness method is the exact method for the dynamic analysis of plane frames using the continuous-coordinate system to consider the effect of mass distribution in beam elements. The dynamic stiffness method may create some null modes where the joints of beam element have null deformation. Unlike the Bernoulli–Euler frames, adding an interior node at the middle of the beam elements cannot normalize all the null modes of flexural vibration in the Timoshenko frames. The floating interior-node scheme is proposed to eliminate the null modes of flexural vibration in the Timoshenko frames. Orthogonal properties of vibration modes in Timoshenko plane frames are theoretically derived, through which the equations of motion in beam elements can be transformed into the decoupled equations of motion in terms of mode amplitudes.

LRS Bianchi type I universes exhibiting Noether symmetry in the scalar–tensor Brans-Dicke theory

Following up on hints of anisotropy in the cosmic microwave background radiation (CMB) data, we investigate locally rotational symmetric (LRS) Bianchi type I spacetimes with non-minimally coupled scalar fields. To single out potentially more interesting solutions, we search for Noether symmetry in this system. We then specialize to the Brans-Dicke (BD) field in such a way that the Lagrangian becomes degenerate (nontrivial) and solve the equations for Noether symmetry and the potential that allows it. Then we find the exact solutions of the equations of motion in terms of three parameters and an arbitrary function. We illustrate with families of examples designed to be generalizations of the well-known power-expansion, exponential expansion and Big Rip models in the Friedmann-Robertson-Walker (FRW) framework. The solutions display surprising variation, a large subset of which features late-time acceleration as is usually ascribed to dark energy (phantom or quintensence), and is consistent with observational data.

Electric charge in the field of a magnetic event in three-dimensional spacetime

We analyze the motion of an electric charge in the field of a magnetically charged event in three-dimensional spacetime. We start by exhibiting a first integral of the equations of motion in terms of the three conserved components of the spacetime angular momentum, and then proceed numerically. After crossing the light cone of the event, an electric charge initially at rest starts rotating and slowing down. There are two lengths appearing in the problem: (i) the characteristic length $\frac{qg}{2\ensuremath{\pi}m}$, where $q$ and $m$ are the electric charge and mass of the particle, and $g$ is the magnetic charge of the event; and (ii) the spacetime impact parameter ${r}_{0}$. For ${r}_{0}\ensuremath{\gg}\frac{qg}{2\ensuremath{\pi}m}$, after a time of order ${r}_{0}$, the particle makes sharply a quarter of a turn and comes to rest at the same spatial position at which the event happened in the past. This jump is the main signature of the presence of the magnetic event as felt by an electric charge. A derivation of the expression for the angular momentum that uses Noether's theorem in the magnetic representation is given in the Appendix.

Expansion of the Glauber equation of motion in terms of cumulants powers

We present an expansion of the Glauber equation of motion in terms of cumulants of the effective fields. We show that we recover a part of the fluctuations with a simple third order expansion. We illustrate the method with an application to the 2D square lattice Ising model with nearest neighbors. For the special case of perturbed non gaussian effective fields, we show that only the first cumulants are significative. Furthermore it can be shown that the resulting distribution is closer to the normal distribution than the unperturbed fields one. It leads to the conclusion that we can consistently truncate the expansion of the exact Glauber equation of motion.

N Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies

This paper presents the explicit dynamic equations of a mechanical system. The equations are presented so that they can easily be implemented in a simulation software or controller environment and are also well suited for system and controller analysis. The dynamics of a general mechanical system consisting of one or more rigid bodies can be derived from the Lagrangian. We can then use several well known properties of Lie groups to guarantee that these equations are well defined. This will, however, often lead to rather abstract formulation of the dynamic equations that cannot be implemented in a simulation software directly. In this paper we close this gap and show what the explicit dynamic equations look like. These equations can then be implemented directly in a simulation software and no background knowledge on Lie theory and differential geometry on the practitioner’s side is required. This is the first of two papers on this topic. In this paper we derive the dynamics for single rigid bodies, while in the second part we study multibody systems. In addition to making the equations more accessible to practitioners, a motivation behind the papers is to correct a few errors commonly found in literature. For the first time, we show the detailed derivations and how to arrive at the correct set of equations. We also show through some simple examples that these correspond with the classical formulations found from Lagrange’s equations. The dynamics is derived from the Boltzmann–Hamel equations of motion in terms of local position and velocity variables and the mapping to the corresponding quasi-velocities. Finally we present a new theorem which states that the Boltzmann–Hamel formulation of the dynamics is valid for all transformations with a Lie group topology. This has previously only been indicated through examples, but here we also present the formal proof.

Dynamic Responses of Ground Vibration due to a Moving Load by using Cole–Cole Model

The dynamic responses of viscoelastic ground vibration due to moving loads by using Cole-Cole model is investigated in this study. First, the paper gives the equation of motion in terms of the displacements for the homogeneous isotropic viscoelastic medium in the frequency domain. Then, the governing equations can be reduced to two Helmholtz equations and the displacement of the viscoelastic medium can be expressed via the potentials. By applying the double Fourier transform with respect to the horizontal coordinates, the general solution for the potentials of the visoelastic medium in the frequency wavenumber domain can be derived.

The inertial terms in equations of motion for bubbles in tubular vessels or between plates

Equations resembling the Rayleigh-Plesset and Keller-Miksis equations are frequently used to model bubble dynamics in confined spaces, using the standard inertial term RR+3R([middle dot]) (2)/2, where R is the bubble radius. This practice has been widely assumed to be defensible if the bubble is much smaller than the radius of the confining vessel. This paper questions this assumption, and provides a simple rigid wall model for worst-case quantification of the effect on the inertial term of the specific confinement geometry. The relevance to a range of scenarios (including bubbles confined in microfluidic devices; or contained in test chambers for insonification or imaging; or in blood vessels) is discussed.

On solutions to the “Faddeev-Niemi” equations

Recently it has been pointed out that the so-called Faddeev-Niemi equations that describe the Yang-Mills equations of motion in terms of a decomposed gauge field, can have solutions that obey the standard Yang-Mills equations with a source term. Here we argue that the source term is covariantly constant. Furthermore, we find that there are solutions of the Yang-Mills equation with a covariantly constant source term that are not solutions to the Faddeev-Niemi equations. We also present a general class of gauge field configurations that obey the Faddeev-Niemi equation but do not solve the Yang-Mills equation. We propose that these configurations might have physical relevance in a strongly coupled phase, where spin-charge separation takes place and the Yang-Mills theory cannot be described in terms of a Landau liquid of asymptotically free gluons.

The importance of kinematics accuracy in modelling the dynamics of rail vehicle moving in a curved track with variable velocity

The paper is devoted to accurate description of kinematics of rail vehicle in its motion along curved track with variable velocity. Failing to do this description properly means inaccurate determination of inertia terms in the dynamical equations of motion. Determining how important are these terms is the paper’s main objective. Simulation results from complete model that includes all inertia terms arising from motion of the track centre-line based reference frames and from incomplete models with such terms selectively omitted are compared. The comparison proved particular significance of the omitted terms for vehicle moving along transition curves with variable velocity.

Reliable analysis for the nonlinear fractional calculus model of the semilunar heart valve vibrations

AbstractThe aim of this paper is to solve the equation of motion of semilunar heart valve vibrations using the homotopy perturbation method. The vibrations of the closed semilunar valves were modeled with fractional derivatives. The fractional derivatives are described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Analytical solution is obtained for the equation of motion in terms of Mittag–Leffler function with the help of Laplace transformation. These solutions can be interesting for a better fit of experimental data. Copyright © 2010 John Wiley & Sons, Ltd.

A Simple Phenomenological Model for the Effective Kinematic Viscosity of Helium Superfluids

A number of experiments where quantum turbulence in helium superfluids has been generated by various means (such as towed/oscillating grids, thermal counterflow, pure superflow, spin-down, ion/vortex rings emission) displays a temporal decay of the observed vortex line density, of the power law form L=Γ t −3/2 at late times. The prefactor, Γ, in analogy with classical homogeneous isotropic turbulence, allows deducing the temperature dependent effective kinematic viscosity, νeff, for turbulent helium superfluids. It appears to be a robust quantity, independent of methods of generating quantum turbulence and detecting the decaying vortex line density. We present a simple phenomenological model to estimate νeff based on comparison of dissipation terms in equations of motion for classical viscous flow and vortex flow of a superfluid in a stationary normal fluid. This model leads to νeff≈κ q, where q=α/(1−α′); α and α′ being dimensionless mutual friction parameters. Within the temperature range where mutual friction dissipation mechanism is dominant this simple model prediction agrees well with the experimental data and with the recent theoretical estimate of Roche, Barenghi and Leveque (Europhys. Lett. 87:54006, 2009).