Elastic Degrees Of Freedom Research Articles

Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part I—Element Level Equations

Equations of motion are derived using Lagrange’s equation for elastic mechanism systems. The elastic links are modeled using the finite element method. Both rigid body degrees of freedom and the elastic degrees of freedom are considered as generalized coordinates in the derivation. Previous work in the area of analysis of general elastic mechanisms usually involve the assumption that the rigid body motion or the nominal motion of the system is unaffected by the elastic motion. The nonlinear differential equations of motion derived in this work do not make this assumption and thus allow for the rigid body motion and the elastic motion to influence each other. Also the equations obtained are in closed form for the entire mechanism system, in terms of a minimum number of variables, which are the rigid body and the elastic degrees of freedom. These equations represent a more realistic model of light-weight high-speed mechanisms, having closed and open loop multi degree of freedom chains, and geometrically complex elastic links.

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.

On the Use of the Momentum Balance in the Impact Analysis of Constrained Elastic Systems

In elastic systems, impulsive forces that act at a point on a deformable body produce stress waves that travel with finite speeds. This paper examines, both theoretically and numerically, the validity of using the generalized impulse momentum approach in modeling impact or collisions in the constrained motion of deformable bodies. The generalized impulse momentum equations that involve the coefficient of restitution and the kinematic constraint Jacobian matrix are used to predict the jump discontinuity in the velocity vector as well as the joint reaction forces. The series solutions obtained by solving these algebraic equations are used to establish a closed form relationship between the jump discontinuity in velocities and joint reactions due to impact and the number of elastic degrees of freedom. It is shown that by increasing the number of elastic coordinates these series converge to their limits. The convergence of these series is used to prove that the generalized impulse momentum equations with the coefficient of restitution can be used with confidence to study impact problems in constrained multibody systems consisting of interconnected rigid and deformable bodies. The results obtained are compared with the classical treatment of the impact problems in the theory of elasticity wherein the case of perfectly plastic impact is assumed.

Experimental Validation of a Plane Flexible Robot Modelling

We present in this paper the first results concerning the experimental validation of a plane flexible robots modelling. This last one is related to robots which move in the plane with rotative joints and which links are supposed to be elastic. The spatial discretization of the links is performed using a finite elements method.We firstly describe the experimental device of a plane robot with two links and two rotational joints. The structure has been defined in order to get a large elasticity in the horizontal plane and very rigid out of this plane.The measurements are performed with the angular position and the| angular velocity of the origin cross section of each link. Otherwise the elastic degrees of freedom are determined using strains gauges. By the end we compare the experimental and the simulation results. The developments of these studies concern the control of flexible robots. Spatial robotics is a well known application of a such modelisation; the problem of interactions between control and structural vibrations is another one in industrial robotics.

Some developments in the theory of modulated order. II. Deformable-lattice models and the axial next-nearest-neighbor Ising model as a random magnet.

This paper presents a new class of models capable of producing modulated order with solely nearest-neighbor forces. These deformable-lattice models create modulated phases out of the interaction between spin and elastic degrees of freedom through a polarization, or feedback, mechanism---as opposed to the purely spin or purely elastic models that employ either competing force or competing periodicity mechanisms. We show that one of the deformable-lattice models, in particular, can be formally reduced to the axial-next-nearest-neighbor Ising (ANNNI) model. This observation turns out to imply, first, that the ANNNI model can be regarded as an ordinary Ising model with a distribution of coupling constants (a random magnet), and second, that other spin models might be profitably thought of not as Hamiltonians, but as potentials of mean force resulting from integrating out elastic degrees of freedom. The possible implications are considered for both the range and nature of the interaction between chemisorbed species as well as that between intercalates in graphite.

Orientational order in liquids: A possible scenario of freezing.

A new model of liquid-solid freezing, based on the orientational order in a three-dimensional liquid, is proposed. The novel aspect of the model is the inclusion of the coupling between the orientational and elastic degrees of freedom in a dense liquid. This allows for the explicit computation of several thermodynamic quantities of interest near the freezing transition.

A brief study of the classical statistical mechanics of spin–Peierls chains

We present and discuss the statistical mechanics of the static properties associated with a one-dimensional classical spin–Peierls Heisenberg chain. Thermodynamic properties and correlation functions are calculated numerically from the eigenvalues of the transfer matrix. All fluctuations from the magnetic and elastic degrees of freedom are considered. Among other results, it is shown that magnetization and dimerization do not tend to occur simultaneously.

The Anderson lattice with elastic degrees of freedom

Abstract Kondo lattice systems with elastic degrees of freedom are investigated via perturbation theory. The results lead to an understanding of the Kondo volume collaps effect for Ce-metal and of a possible mechanism for Heavy Fermion Superconductivity as observed in CeCu 2 Si 2 . It is pointed out how a microscopic calculation may lead to a description of these phenomena in a universal way, i.e. in terms of the influence of lattice deformations on the Kondo temperature.

Unified theory of rearrangement scattering

We present a unified microscopic theory of rearrangement scattering which satisfies the Pauli principle. The equations for the transition amplitudes and wave functions are reduced to effective two-body equations with the complexity put into the distortion potentials and the transition potentials. The central theme in the development of this theory is the importance of elastic scattering in initial and final channel, although this is not a prerequisite for its validity as the theory is exact. The elastic degrees of freedom are absorbed into the distorted waves using projection techniques in combination with the Faddeev formulation. The resulting transition amplitude can be expanded in the transition potential, leading to a modified coupled channel series. The transition potential, which has a constructive nature (i.e., no counter terms), can also be expanded. We provide arguments for the convergence of both expansions. Various contributions to the transition potential are discussed, and we put special emphasis on a rescattering term which has not been considered before in the context of coupled channel equations. Various inconsistence which may hamper standard coupled channel calculations are discussed as well.

On the observability of critical fluctuations in uniaxial ferroelectrics

Abstract In order to examine the observability of critical fluctuations in uniaxial ferroelectrics of the triglycine group we started from the mear field solution of an Ising model coupled to elastic degrees of freedom. To exhibit size and temperature range of logarithmic correction factors modifying the mean-field theory due to these fluctuations, non-universal parameters of our model have to be determined from experimental data. While logarithmic factors should be observable in various static quantities due to our quantitative analysis of TGS and TGSe, their fractional powers are not expected to be detectable in contrast to uniaxial dipolar ferromagnets.

Dynamic damping in magnetic Sine-Gordon systems

We study the influence of the elastic degrees of freedom of a one-dimensional spin-chain with ferromagnetic Heisenberg coupling in situations, where the magnetic dynamics can be described by a Sine-Gordon model. Compatibility of the Sine-Gordon approach with spin-lattice coupling is shown, and CsNiF3 and domain wall motion are discussed as concrete physical realizations. As a consequence of the spin-lattice interaction a soliton-phonon coupling arises, which destroys the formal Lorentz invariance of the Sine-Gordon system. The influence of this coupling on the soliton dynamics is investigated. It results in a dynamic damping of the soliton motion, which in general is not of viscous type. Moreover, the influences of the soliton-phonon coupling on the phonon dispersion are estimated and possible experimental consequences are discussed.

Non-linear excitations in a compressible chain of dipoles

A linear chain of XY dipoles coupled to the elastic degrees of freedom of the lattice is studied. For second-order coupling, in the presence of an external electric field, the dipolar kink-soliton excitations lead to soliton-like deformation of the lattice. First-order coupling mediates dipolar kink-solitons excitations which themselves induce kink-soliton deformation of the lattice.

Magnetic solitons and elastic kink-like excitations in compressible Heisenberg chain

Shows that the compressible Heisenberg chain remains, in the continuum limit, an example of a completely integrable system. The soliton excitations of the magnetic degrees of freedom lead to the kink-like excitations of the elastic degrees of freedom.

Recent developments in critical magnetic behaviour

Competition between various types of interaction leads to complex phase diagrams, exhibiting multicritical points. Bicritical points appear due to uniaxial assymmetry, resulting e.g. from uniaxial stress. These may become tetracritical when cubic symmetry exists. Tricritical and bicritical points appear in metamagnets. Tetracritical points exist for randomly mixed ferromagnets and antiferromagnets. Competing interactions also lead to new types of critical behaviour, e.g. “dipolar”, “cubic”, etc. All of these may be affected by interactions with elastic degrees of freedom, random impurities, etc. Present theoretical understanding of these phenomena, based mainly on renormalization group arguments, is reviewed, and relevant experiments are mentioned.

On the critical behaviour of spin-reorientation phase transitions

The authors show that, as a consequence of the linear coupling between the magnetic and elastic degrees of freedom, rotational spin-reorientation type phase transitions in three dimensions will be described by classical critical exponents.

Phase transitions of ann-vector model on an elastic anisotropic lattice

Wegner's model for magnetic phase transitions on elastic isotropic lattices is generalized to elastic anisotropy and the general spin-lattice coupling linear in lattice distortion. Cyclic boundary conditions are employed. The elimination of the elastic degrees of freedom gives rise to an effective spin Hamiltonian which consists of a shortrange and a long-range part. Renormalization group analysis yields the following result: If the lattice couples to a relevant operator with an exponenty >d/2, the long-range part contains a relevant contribution with exponent 2y—d leading away from any fixed pointH* associated with the short-range interactions. In particular,H* is unstable for positive specific heat exponentα, and, in most cases, if a spin-anisotropy is involved.

Coupling to anisotropic elastic media: Magnetic and liquid-crystal phase transitions

A generalization of the Larkin-Pikin-Sak model in which an $n$-component order parameter is coupled to a general anisotropic elastic continuum is studied using the $\ensuremath{\epsilon}$ expansion. It is found that the fixed-point structure is the same as the isotropic model but that all fixed points are unstable with respect to anisotropic perturbations, independent of external boundary conditions. The compressible smectic-$A$ to smectic-$C$ liquid-crystal transition is also studied. It is found to be unaffected by elastic degrees of freedom and is, therefore, expected to have helium exponents, as previously predicted by de Gennes.

Planar motion of a rotating cable-connected space station in orbit.

The planar motion of a rotating space station composed of a manned compartment and a counterweight linked by a long flexible cable is analyzed. It is assumed that initially the system is spinning in the orbital plane. The effects of cable elasticity and vibrational motion are included in the derivation of equations, which reveal that the orbital, rotational, and elastic degrees of freedom are coupled through the gravitational gradient terms. The re- sponses of the coupled nonlinear differential equations to the influence of gravitational gradi- ents are solved by the fourth-order Runge-Kutta numerical integration method. Viscous damping is included in the investigation and found to be effective to the gravitational excita- tion. From the results of the analysis, the feasibility of the configuration is evaluated, and the critical problem areas are discussed. Nomenclature a = sublength of cable d, = position vector from center of mass to a point on cable EA = extensional rigidity Gri, G'>, dGc — gravity forces on mi, m2, and elementary podrjo of cable members, cables, or chains are of considerable interest as structural elements for connecting the compartments or counterweights to form a space station. However, the exact dynamic behavior of a cable-linked configuration is difficult to predict. The deployment, spin-up, artificial gravity, docking capacity, control system, and wobble damping are all formidable problems of a cable-linked configuration. An- other critical question to be answered is whether the cable- connected system, oscillating under the influence of gravity gradient and other perturbing forces, is a stable configuration. Published information concerning dynamic problems of a cable-connected space station is limited. A majority of the authors confined their treatment of the libration of the orbit- ing vehicle to the assumption of small oscillation, and they neglected perturbations from the orbital elements.14 Past experience in the analysis of orbiting satellites indicated that, for a configuration like the cable-connected space sta- tion, the rotational and vibrational motion cannot be treated independently from orbital motion without violating the laws of conservation of energy and conservation of angular momentum.5 In this paper, the planar motion of a rotating space station composed of a manned compartment and a counterweight linked by a long flexible cable is analyzed. This space sta- tion is assumed to be in orbit around a spherical earth with an inverse-squar e-law gravitational field and to be spinning in its orbital plane. (The spin-up phase is not included in the analysis.) The stability and control characteristics may be significantly influenced by the distortions of the cable under transient loading conditions. The effects of cable elasticity and vibrational motion are included in the formu- lation. The time dependence of the elastic motion is based upon the deformation of the cable expressed in terms of its normal modes of free vibration. The equations of planar motion in the earth-fixed coordi- nates are derived from the total kinetic energy of the system, the strain energy of the cable, and the generalized force re- sulting from the work done by the gravitational gradient over the extended body. The resulting equations reveal that the orbital, rotational, and elastic degrees of freedom