Integro-differential Equations Of Motion Research Articles

A Stress-State based Peridynamics model for elastio-plastic material modeling

A Stress-State based PD (SSPD) model using a well-known yield criteria is proposed in this paper and tested on the modeling of two dimensional bars under different loading levels as a first step for further development. SSPD is based upon peridynamics (PD) which utilize temporal spatial integro-differential equation of motion and formulates continuum problems in terms of integral equations, which are capable of modeling discontinuities such as cracks. The proposed bond strength not only depends on the bond stretch, but on the current state of all bonds connected to a particle as well. Thus, a stress-based peridynamics model is obtained. The tensile simulation compared to conventional FEM shows promising performance with an error of 5%. Compression simulations, however, need more investigation to include the effect of contact forces.

Continuous point symmetries in group field theories

We discuss the notion of symmetries in non-local field theories characterized by integro-differential equations of motion, from a geometric perspective. We then focus on group field theory (GFT) models of quantum gravity and provide a general analysis of their continuous point symmetry transformations, including the generalized conservation laws following from them.

Dynamic response of mechanical systems to impulse process stochastic excitations: Markov approach

Methods for determination of the response of mechanical dynamic systems to Poisson and non-Poisson impulse process stochastic excitations are presented. Stochastic differential and integro-differential equations of motion are introduced. For systems driven by Poisson impulse process the tools of the theory of non-diffusive Markov processes are used. These are: the generalized Itô’s differential rule which allows to derive the differential equations for response moments and the forward integro-differential Chapman-Kolmogorov equation from which the equation governing the probability density of the response is obtained. The relation of Poisson impulse process problems to the theory of diffusive Markov processes is given. For systems driven by a class of non-Poisson (Erlang renewal) impulse processes an exact conversion of the original non-Markov problem into a Markov one is based on the appended Markov chain corresponding to the introduced auxiliary pure jump stochastic process. The derivation of the set of integro-differential equations for response probability density and also a moment equations technique are based on the forward integro-differential Chapman-Kolmogorov equation. An illustrating numerical example is also included.

Satellite motion in the gravitational field of a viscoelastic planet with a core

The motion of a “planet-satellite” system in a gravitational field of mutual attraction forces is investigated. The planet is modeled by a body consisting of a solid core and a viscoelastic shell made of Kelvin-Voigt material. The satellite is modeled by a material point. A system of integro-differential equations of motion of a mechanical system is derived from the variational d’Alembert-Lagrange principle within the linear model of the theory of elasticity. Using the asymptotic method of separation of motions, an approximate system of equations of motion is constructed in vector form. This system describes the dynamics of a system with allowance for disturbances caused by elasticity and dissipation. The solution of the quasistatic problem of elasticity theory for the deformable shell of a planet is obtained in the explicit form. An averaged system of differential equations describing the evolution of satellite’s orbital parameters is derived. For partial cases phase trajectories are constructed, stationary solutions are found, and their stability is investigated. As examples, some planets of the Solar system and their satellites are considered. This problem is a model for studying the tidal theory of planetary motion.

Dynamic Stability of Viscoelastic Flexible Plates of Variable Stiffness Under Axial Compression

The dynamic stability of viscoelastic plates of variable stiffness is analyzed. The deflections are described by partial integro-differential equations of motion. The Bubnov–Galerkin method based on monomial and polynomial approximation of deflections is used to reduce the problem to ordinary integro-differential equations with time as an independent variable. These equations are solved numerically using a singular kernel. A numerical problem-solving algorithm based on this method is described. A number of new mechanical effects are revealed

Non-Linear Vibrations of a Microbeam Resting on an Elastic Foundation

In this study, response of a microbeam bonded to a non-linear elastic foundation is investigated. The model accounts for mid-plane stretching, an applied axial load, and an AC harmonic force. The microbeam is resting on a non-linear elastic foundation which introduces a cubic non-linear term to the equations of motion. Immovable end conditions introduce integral type nonlinearity. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. Frequency-response curves and non-linear frequencies are analyzed with respect to the effective physical parameters. Influence of elastic foundation coefficients, coefficients related to dielectric constants, and mid-plane stretching on the vibrations of the microbeam are investigated.

BIFURCATION ANALYSIS OF DAMPED VISCO-ELASTIC PLANAR BEAMS UNDER SIMULTANEOUS GRAVITATIONAL AND FOLLOWER FORCES

The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.

Compatibility conditions and equations of motion in the linear micropolar theory of elasticity

An analog of Cesaro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.

Time Domain Response Analysis of Barge Floater Supporting an Offshore Wind Turbine

Wind energy is a reliable source of sustainable power generation and has been an active area of research globally to economically harness the energy for human use. Reliable source of wind energy pushed the engineers to install wind turbines near and far off the coasts. In shallow water upto 100 m, fixed structures like tripods, jackets, monopiles and gravity base are functionally and economically feasible. In deep waters, a floating substructure can be more economical for offshore wind turbine. In this study a barge type floater of different aspect ratios from 0.4 to 1.0 is investigated for its performance under wave and wind loading. All these floaters were designed with a defined transverse metacentric height (GM) equal to 1.0 m and the hydrodynamic analysis is carried out using WAMIT. The barge with aspect ratio B/L = 1.0 is found to have lowest pitch RAO. The time domain surge, heave and pitch response for this barge has been obtained using Integro-differential equation of motion and the statistical response characteristics are compared for two different cases of excitation namely, wave excitation alone and combined wave and wind excitation. Statistics of surge, heave and pitch responses are obtained for three different seas states and for two different wave heading angles.

Stochastic stability of a viscoelastic column axially loaded by a white noise force

This paper deals with the analysis of stability of a hinged-hinged viscoelastic column subjected to a non-zero mean stochastic axial force. The randomly variable part of this is described by a stationary Gaussian white noise process. The viscosity affects the curvature of the column, for which the classic Euler–Bernoulli's model is adopted. The viscosity is described by the linear Kelvin–Voigt's model. A dynamic stability analysis is performed. Normal modes are introduced in the integro-differential equation of motion so that uncoupled modal equations are retrieved. With reference to the first mode, by using an additional state variable, three Itô’s ODE are obtained, from which the differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. The zero solution, that is undeformed straight column, corresponds to zero moments. If the column is perturbed, it is stable when the response moments tend to zero. A necessary and sufficient condition of stability in the moments of order r is that the matrix A r of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. If the mean axial force μ P or the white noise intensity w P are increased, there exist critical values μ Pcr , w P c r for which almost an eigenvalue is positive. The critical mean axial force is found to be inversely proportional to the parameter φ ∞ , which measures the amount of viscous deformation. The search for the critical values of w P is made numerically, and several graphs are presented for a representative column.

Derivation of ODEs and Bifurcation Analysis of a Two-DOF Airfoil Subjected to Unsteady Incompressible Flow

An airfoil subjected to two-dimensional incompressible inviscid flow is considered. The airfoil is supported via a translational and a torsional springs. The aeroelastic integro-differential equations of motion for the airfoil are reformulated into a system of six first-order autonomous ordinary differential equations. These are the simplest and least number of ODEs that can present this aeroelastic system. The differential equations are then used for the bifurcation analysis of an airfoil with a structural nonlinearity in the pitch direction. Sample bifurcation diagrams showing both stable and unstable limit cycle oscillation are presented. The types of bifurcations are assessed by evaluating the Floquet multipliers. For a specific case, a period doubling route to chaos was detected, and mildly chaotic behavior in a narrow range of velocity was confirmed via the calculation of the Lyapunov exponents.

Parametric Resonance of Hopf Bifurcation in a Generalized Beck’s Column

A generalized damped Beck’s column under pulsating actions is considered. The nonlinear partial integrodifferential equations of motion and the associated boundary conditions, expanded up to cubic terms, are tackled through a perturbation approach. The multiple scales method is applied to the continuous model in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. This codimension-2 bifurcation entails two control variables, namely, the amplitude of the static and dynamic components of the follower force, playing the role of detuning and bifurcation parameters, respectively. In the postcritical analysis bifurcation diagrams and relevant phase portraits are examined. Two bifurcation paths associated with specific values of the follower force static component are discussed and the birth of new stable period-2 subharmonic motion is observed.

Monotonically convergent algorithms for solving quantum optimal control problems described by an integrodifferential equation of motion

A family of monotonically convergent algorithms is presented for solving a wide class of quantum optimal control problems satisfying an inhomogeneous integrodifferential equation of motion. The convergence behavior is examined using a four-level model system under the influence of non-Markovian relaxation. The results show that high quality solutions can be obtained over a wide range of parameters that characterize the algorithms, independent of the presence or absence of relaxation.

Divergence, Hopf and double-zero bifurcations of a nonlinear planar beam

The multiple scale method is directly applied to a one-dimensional continuous model to derive the equations governing the system asymptotic dynamic around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped visco-elastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded upto cubic terms in the transversal displacements and velocities of the beam. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence and Hopf and double-zero bifurcations. The spectral properties of the linear operator are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system posses an incomplete system of eigenvectors at the double-zero point (i.e. it is defective or nilpotent), thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Finally, they are numerically studied to furnish a comprehensive scenario of the postcritical behaviour around the bifurcations.

Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam

The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.

충격하중을 받는 점탄성 균열의 응력확대계수 계산

In this paper, A new finite element method for the time domain analysis of the dynamic stress intensity factor of two-dimensional viscoelastic body with a stationary central crack under the transient dynamic load is presented, which is based on the integrodifferential equations of motion in the isotropic linear viscoelasticity and the Galerkin's method. The viscoelastic material is assumed to be elastic in dilatation and behaves like a standard linear solid in shear. As a numerical example, the Chen's problem in viscoelastodynamic version is solved for the parametric study about the effect of viscosity and relaxation time on the dynamic stress intensity factor.

NON-LINEAR VIBRATIONS OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR ELASTIC FOUNDATION

In this study, non-linear vibrations of slightly curved beams are investigated. The curvature is taken as an arbitrary function of the spatial variable. The initial displacement is not due to buckling of the beam, but is due to the geometry of the beam itself. The ends of the curved beam are on immovable simple supports and the beam is resting on a non-linear elastic foundation. The immovable end supports result in the extension of the beam during the vibration and hence introduces further non-linear terms to the equations of motion. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. Both free and forced vibrations with damping are investigated. Effect of non-linear elastic foundation as well as the effect of curvature on the vibrations of the beam are examined. It is found that the effect of curvature is of softening type. For sufficiently high values of the coefficients, the elastic foundation may suppress the softening behaviour resulting in a hardening behaviour of the non-linearity.

Effect of capillary wave radiation on the oscillations of a vibrator

The oscillations of a rigid body on an elastic tie (vibrator) in an ideal incompressible fluid with a free boundary, on which surface tension forces act, are considered. The linearized problem of hydrodynamics is solved approximately in the self-consistent formulation, the reaction forces exerted on the body by the fluid are calculated, and an integrodifferential equation of motion is obtained. Using asymptotic methods, the average characteristics determining the damping coefficient and the frequency shift of the oscillations of the vibrator are obtained with allowance for the effect of the capillary waves radiated by the vibrator. Qualitative effects depending on the parameters of the system are revealed. The authors' numerical simulation of the motion of the vibrator completely confirms the qualitative conclusions concerning the nature of the oscillations of a body in a fluid having surface tension.

Time-dependent wave-packet dynamics with memory: the electron-HCl collision complex

Vibrational excitation by electron impact and dissociative electron attachment in HCl are described in a time-dependent picture. The calculations are based on the nonlocal resonance model of low-energy electron-HCl scattering proposed earlier by Domcke and Mundel. The space-time integrodifferential equation of motion for the time-dependent wave packet representing the electron-HCl collision complex has been solved using standard numerical techniques. The results provide insight into the nature of the pronounced threshold phenomena observed in the vibrational excitation and dissociative attachment cross sections of HCl.

Stability Properties of a Viscoelastic Column Under a Periodic Force

The dynamic stability of a viscoelastic column subjected to a periodic longitudinal load is investigated. The viscoelastic behavior is given in terms of the Boltzmann superposition principle which yields an integro-differential equation of motion. The stability boundaries of this equation are determined analytically by using the multiplescales method. It is shown that due to the viscoelasticity the stability regions are expanded, relative to the elastic ones, and the time for which a stable system becomes unstable is given. In addition, the stability properties of the viscoelastic column are time dependent and an initially stable system can turn unstable after a finite time, unlike columns that are described by the elastic model.