Order Equations Of Motion Research Articles

Self-dual hopfions

We construct static and time-dependent exact soliton solutions with non-trivial Hopf topological charge for a field theory in 3+1 dimensions with the target space being the two dimensional sphere S**2. The model considered is a reduction of the so-called extended Skyrme-Faddeev theory by the removal of the quadratic term in derivatives of the fields. The solutions are constructed using an ansatz based on the conformal and target space symmetries. The solutions are said self-dual because they solve first order differential equations which together with some conditions on the coupling constants, imply the second order equations of motion. The solutions belong to a sub-sector of the theory with an infinite number of local conserved currents. The equation for the profile function of the ansatz corresponds to the Bogomolny equation for the sine-Gordon model.

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.

Periodic response of nonlinear dynamical system with large number of degrees of freedom

In this paper, a methodology based on shooting technique and Newmark’s time integration scheme is proposed for predicting the periodic responses of nonlinear systems directly from solution of second order equations of motion without transforming to double first order equations. The proposed methodology is quite suitable for systems with large number of degrees of freedom such as the banded system of equations from finite element discretization.

Exact null tachyons from renormalization group flows

We construct exact two-dimensional conformal field theories, corresponding to closed string tachyon and metric profiles invariant under shifts in a null coordinate, which can be constructed from any two-dimensional renormalization group flow. These solutions satisfy first order equations of motion in the conjugate null coordinate. The direction along which the tachyon varies is identified precisely with the world sheet scale, and the tachyon equations of motion are the renormalization group flow equations.

Boundary terms and junction conditions for the DGP π-Lagrangian and galileon

In the decoupling limit of DGP, π describes the brane-bending degree of freedom. It obeys second order equations of motion, yet it is governed by a higher derivative Lagrangian. We show that, analogously to the Einstein-Hilbert action for GR, the π-Lagrangian requires Gibbons-Hawking-York type boundary terms to render the variational principle well-posed. These terms are important if there are other boundaries present besides the DGP brane, such as in higher dimensional cascading DGP models. We derive the necessary boundary terms in two ways. First, we derive them directly from the brane-localized π-Lagrangian by demanding well-posedness of the action. Second, we calculate them directly from the bulk, taking into account the Gibbons-Hawking-York terms in the bulk Einstein-Hilbert action. As an application, we use the new boundary terms to derive Israel junction conditions for π across a sheet-like source. In addition, we calculate boundary terms and junction conditions for the galileons which generalize the DGP π-Lagrangian, showing that the boundary term for the n-th order galileon is the (n−1)-th order galileon.

Instantons, black holes, and harmonic functions

We find a class of five-dimensional Einstein-Maxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterised by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of four-dimensional Euclidean sigma models, which includes those occuring for four-dimensional Euclidean N = 2 vector multiplets as a subclass. For solutions taking values in a completely isotropic submanifold of the target space, we show that the solution can be expressed in terms of harmonic functions if an integrability condition is met. This condition can either be solved by imposing that the solution depends on a single coordinate, or by imposing that the target space is a para-Kähler manifold which can be obtained from a real Hessian manifold by a generalized r-map. In the latter case one obtains multi-centered solutions. Moreover, if the integrability condition is met, the second order equations of motion can always be reduced to first order equations, which become gradient flow equations if the solution is further required to depend on one coordinate only. The dualization of axions into tensor fields and the lifting of four-dimensional instantons to five-dimensional solitons are used to motivate the addition of a boundary term to the action, which accounts for the instanton action. If the sigma model is coupled to gravity, and if the Hesse potential is of a suitable form which we specify, then the four-dimensional Euclidean Lagrangian can be lifted consistently to a five-dimensional Einstein-Maxwell type Lagrangian. Instanton solutions lift to extremal black hole solutions, and the instanton action equals the ADM mass.

Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

Constraining modified gravity with large non-Gaussianities

In writing a covariant effective action for single-field inflation, one is allowed to add a Gauss-Bonnet and axion-type curvature couplings. These couplings represent modifications of gravity, and are the unique higher-curvature terms that lead to second order equations of motion in four dimensions. In this paper we study the observational consequences of such couplings for models with large non-Gaussianities. Our focus is on the Gauss-Bonnet term. In particular, we study an effective action where the scalar Lagrangian is a general function of the inflaton and its first derivative. We show that, for large non-Gaussianities, one can write ${f}_{\mathrm{NL}}$ in terms of only three parameters. The shape of ${f}_{\mathrm{NL}}$ is also studied, and we find that it is very similar to that of $k$-inflation. We show that the Gauss-Bonnet term enhances the production of gravitational waves, and allows a smaller speed of sound for scalar perturbations. This, in turn, can lead to larger non-Gaussianities which can be constrained by observations. Using current Wilkinson microwave anisotropy probe limits on ${f}_{\mathrm{NL}}$ and the tensor/scalar ratio, we put constraints on all parameters. As an example, we show that for Dirac-Born-Infeld inflation, the Gauss-Bonnet coupling leads to an interesting observational window with both large ${f}_{\mathrm{NL}}$ and a large amplitude of gravitational waves. Finally, we show that the Gauss-Bonnet coupling admits a de Sitter phase with a relativistic dispersion relation for scalar perturbations.

Bulk emission of scalars by a rotating black hole

We study in detail the scalar-field Hawking radiation emitted into the bulk by a higher-dimensional, rotating black hole. We numerically compute the angular eigenvalues, and solve the radial equation of motion in order to find transmission factors. The latter are found to be enhanced by the angular momentum of the black hole, and to exhibit the well-known effect of superradiance. The corresponding power spectra for scalar fields show an enhancement with the number of dimensions, as in the non-rotating case. On the other hand, the proportion of the total (i.e., bulk+brane) power that is emitted into the bulk decreases monotonically with the angular momentum. We compute the total mass loss rate of the black hole for a variety of black-hole angular momenta and bulk dimensions, and find that, in all cases, the bulk emission remains significantly smaller than the brane emission. The angular-momentum loss rate is also computed and found to have a smaller value in the bulk than on the brane.

Reconstruction of a distributed force applied on a thin cylindrical shell by an inverse method and spatial filtering

The localization and the magnitude of acting steady-states forces resulting from the dynamic response of a cylindrical shell are investigated. The derivatives present in the equation of motion are expressed in terms of displacements using finite-difference schemes. Displacements of the structure are introduced into the equation of motion in order to calculate the distribution of these forces. Numerical simulations show that when the force distribution is calculated using well-defined displacements, the exact position of the force is easily determined. However, when the displacements are mixed with uncertainties, noise appears and the real force is relatively inaccurate. In order to handle this instability in the inverse problem, a regularization method based on signal processing techniques such as windowing and filtering in the spatial wavenumber domain should be performed. Numerical examples that illustrate the regularization process show that when the local filtering is applied to the force distribution, calculated from noisy data, reasonable results are obtained. Experiments are performed on a cylindrical shell in order to measure its velocities. Unfortunately, using the displacements derived from measured velocities, high sensitivity to the noise in the measurements is observed. Overall, the experimental identification gives also good results when the same technique of filtering is applied to the force distribution.

Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method

In this study, the out-of-plane free vibration analysis of a double tapered Euler–Bernoulli beam, mounted on the periphery of a rotating rigid hub is performed. An efficient and easy mathematical technique called the Differential Transform Method (DTM) is used to solve the governing differential equation of motion. Parameters for the hub radius, rotational speed and taper ratios are incorporated into the equation of motion in order to investigate their effects on the natural frequencies. Calculated results are tabulated in several tables and figures and are compared with the results of the studies in open literature where a very good agreement is observed.

Kinematic optimization of ball-screw transmission mechanisms

Abstract The paper proposes a method for the kinematic optimization of transmission mechanisms, where non-circular (NC) gears are used to perform a mechanical control on the output motion. The investigation presented here deals with the motion control of a ball-screw transmission mechanism. The objective is lowering the peak acceleration value of the screw, by designing a pair of variable radius gears as a driving mechanism. The kinematic characteristics of the ball-screw mechanism are analyzed by means of non-dimensional motion equations in order to formulate the optimization problem. A genetic algorithm (GA) is then implemented to optimize the objective function, and a penalty method is used to fulfil the design rules. The kinematic analysis of the optimal mechanism revealed a 37% reduction of the peak acceleration of the screw in comparison with a constant pitch screw, operated at a constant speed. A kinematic simulation is used to validate the method.

Hybrid Constraint Space Dynamics and Control for Robot Manipulators

In this paper we present the development of a hybrid constraint space dynamics modeling technique and position/force controller for robotic manipulator control in constrained environments. The method utilizes a constraint space dynamic model in which the model coordinates are displacement along the constraint trajectory and the normal force between the manipulator end-effector and the environment. The dynamic model is constructed by transforming the conventional joint space manipulator dynamics equations into their constraint space equivalents through the application of mapping functions, which relate differential displacements and velocities in the constraint space coordinate system to the joint space coordinate system. Control algorithms may then be applied to the simplified dynamic structure of the constraint space equations of motion in order to produce a vector of manipulator joint torques which will satisfy both position and force requirements along the environmental constraint. Actuator constraints and momentum compensating techniques are also used to ensure that the position and force control problems are completely decoupled from one another. A computer torque control algorithm is then applied to a two-degrees-of-freedom prismatic robot and simulations are carried out with two different constraint surfaces, i.e. a planar, and a concave circular environment. The results of these simulations show that the controller, implemented in hybrid constraint space provides good position and force control.

An effective two-dimensional model for MHD flows with transverse magnetic field

This paper presents a model for quasi-two-dimensional MHD flows between two planes with small magnetic Reynolds number and constant transverse magnetic field orthogonal to the planes. A method is presented that allows three-dimensional effects to be taken into account in a two-dimensional equation of motion thanks to a model for the transverse velocity profile. This model is obtained by using a double perturbation asymptotic development both in the core flow and in the Hartmann layers arising along the planes. A new model is thus constructed that describes inertial effects in these two regions. Two separate classes of phenomena are found: one related to inertial effects in the Hartmann layer gives a model for recirculating flows and the other introduces the possibility of having a transverse dependence of the velocity profile in the core flow. The ‘recirculating’ velocity profile is then introduced in the transversally averaged equation of motion in order to provide an effective two-dimensional equation of motion. Analytical solutions of this model are obtained for two experimental configurations: isolated vortices generated by a point electrode and axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence) experiment. The theory is found to give a satisfactory agreement with the experiment so that it can be concluded that recirculating flows are actually responsible for both vortex core spreading and excessive dissipative behaviour of the axisymmetric sidewall layers.

Constrained systems and statistical distribution

Theoretical advances in the statistical description of non-Hamiltonian systems [Tuckerman, Mundy, and Martyna, Europhys. Lett. 45, 149 (1999)] have recently led us to rethink the definition of the phase-space probability by using the proper invariant measure. Starting from this point of view, we will derive the statistical distribution of constrained systems, considered as non-Hamiltonian, in Cartesian coordinates, in a way independent from the standard Lagrangian treatment. Furthermore, we will analyze the statistical distribution of the Nose-Hoover isothermal (canonical or NVT) dynamics, considering with care the conservation laws hidden in the evolution equations. Consequently, we will correct the equations of motion in order to obtain the proper statistical ensemble. The isobaric-isothermal (constant pressure or NpT) form of the Nose-Hoover dynamics is then considered as a nontrivial extension of the described procedure and, similarly to the NVT case, we will correct the equations of motion with respect to previous versions. The case of a constrained system coupled to a thermostat and a piston is easily handled by means of the same formalism, whereas the Lagrangian treatment becomes involved.

Nonlinear Vibration of Composite Shell Subjected to Resonant Excitations

Analysis of the vibration of a shallow, simply supported, nonsymmetric unbalanced cross-ply laminated, circular cylindrical composite shell is presented. The subject is particularly relevant considering the widespread use of cylindrical shell structures in engineering applications. This research applies the discretized Lagrangian/method of multiple scales solution technique. The Donnell shallow shell strain-displacement relations and the single-mode displacement field from the linear eigenvalue problem are applied. The system Lagrangian is developed and integrated over the spatial domain and then substituted into Lagrange’s equation. The resulting equation of motion is a second-order temporally nonlinear ordinary differential equation in the form of the Duffing oscillator. The natural frequency, the coefficient of the cubic nonlinearity, and the strength of the nonlinearity are investigated. The method of multiple scales is applied to the nonlinear equation of motion in order to analyze the frequency response. Primary resonance, subharmonic resonance, and superharmonic resonance are analyzed.

Two derivations of the Feynman formulae for the fields of a moving charge

The Feynman expressions for the electric and magnetic fields of a charge moving on an arbitrary trajectory are found by two simple methods. The first involves a direct integration of Maxwell's time-dependent equations. The second is based on the formalism of quantum optics which uses the first order equations of motion for creation and annihilation operators. As an illustration of the simplicity of these methods, the fields of an oscillating electric dipole are obtained.

Size Limitations in Semicircular Duct Systems

The present article discusses mechanical requirements and limitations which are applicable to the construction of the system of semicircular ducts, especially to its size. The simplified case of a single, uniform duct system has been considered which can be described by a second order equation of motion. The principal functional quantities for this rotation-sensor are: (1) response speed; (2) sensitivity; and (3) regular flow. The response speed of a single, uniform semicircular duct is characterized by the short time constant (T2) which is dependent on the duct radius (r). Its estimated range is from 0.04 ms in the smallest to 140 ms in the largest known labyrinth. The sensitivity is characterized by the maximal endolymph displacement after a step stimulus (xmax). Its estimated range is from 0.0016 μm to 5.97 mm (6.56 decades!), assuming an input angular velocity of ω=1 rad s−1. The Reynolds number is a measure for an undisturbed laminar flow. Its estimated range varies from 7.38·10−4to 45.1 for ω=1 rad s−1. The above data follow from graphs in which, for a single uniform duct, circuit radius (R) is plotted against duct radius (r) for labyrinths of 233 species belonging to different vertebrate-groups. A relation R =38.9·r1.60was determined. The smallest labyrinth was found in a carp larva (Cyprinus), the largest in a whale shark (Rhincodon). Large whales possess labyrinths of average mammalian size. It is revealed that semicircular duct size is bound by requirements concerning regular flow and by a too low response speed for large labyrinths, and by a too low sensitivity for small labyrinths. Other important quantities are mechanical amplification factors which are a consequence of more complex vestibular constructions than a single uniform duct circuit. Allometric relationships are interpreted as compromises between the quantities mentioned. A hypothesis for the relatively large semicircular duct sizes of fishes, especially Elasmobranchii , compared with mammals and birds is presented.

Numerical computation of differential-algebraic equations for nonlinear dynamics of multibody android systems in automobile crash simulation.

The principle of virtual work is used to derive the Euler-Lagrange equations of motion in order to describe the dynamics of multibody android systems. The constrained variational equations are in fact differential-algebraic equations of high index and are cast as ordinary differential equations through differentiation of the constraint equations. The integration routine LSODAR and the fourth-order Runge-Kutta method are used to compute the generalized coordinates, their time derivatives and the body forces of two android models. The graphs of the constraint forces reveal the whiplash effect on the neck and that the stiffness of both multibody systems is due to large magnitude impulsive forces experienced by many bodies simultaneously.

Contribution to parallel and vector computation in multibody dynamics

In spite of the continuous and sustained increase in CPU-time performances of modern computer processors, one notes a real eagerness in the scientific community to investigate both the vector and parallel computational approaches. The goal is obviously to attain high-speed simulation whatever the envisaged numerical treatment may be. The aim of the present contribution is, firstly, to point out the advantage of splitting a multibody system into distinct parts in a `coarse grain level' parallelization context, and secondly, to stress the advisability of vectorizing multibody equations of motion in order to compute them in an adequate computer environment.