Dissipative properties of composite structures. 1. Statement of problem

Object and purpose of research. The object under study is a sandwich plate with two rigid anisotropic layers and a filler of soft isotropic viscoelastic polymer. Each rigid layer is an anisotropic structure formed by a finite number of orthotropic viscoelastic composite plies of arbitrary orientation. The purpose is to develop a mathematical model of sandwich plate. Materials and methods. The mathematical model of sandwich plate decaying oscillations is based on Hamilton variational principle, Bolotin’s theory of multilayer structures, improved theory of the first order plates (Reissner-Mindlin theory), complex modulus model and principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. In description of physical relations for rigid layers the effects of oscillation frequencies and ambient temperature are considered as negligible, while for the soft viscoelastic polymer layer the temperaturefrequency relation of elastic-dissipative characteristics are taken into account based on experimentally obtained generalized curves. Main results. Minimization of the Hamilton functional makes it possible to reduce the problem of decaying oscillations of anisotropic sandwich plate to the algebraic problem of complex eigenvalues. As a specific case of the general problem, the equations of decaying longitudinal and transversal oscillations are obtained for the globally orthotropic sandwich rod by neglecting deformations of middle surfaces of rigid layers in one of the sandwich plate rigid layer axes directions. Conclusions. The paper will be followed by description of a numerical method used to solve the problem of decaying oscillations of anisotropic sandwich plate, estimations of its convergence and reliability are given, as well as the results of numerical experiments are presented.

Features of complex vibrations of flexible micropolar mesh panels

In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent. An additional independent material parameter of length associated with a symmetric tensor by a rotation gradient is introduced into consideration. The equations of motion of a panel element, the boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Kirchhoff – Love’s kinematic hypotheses. It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges. The material is isotropic, elastic and obeys Hooke’s law. To homogenize the rib system over the panel surface, the G. I. Pshenichnov continuous model is used. The dissipative mechanical system is considered. The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations. The Cauchy problem is solved by the Runge – Kutta method of the 4th order of accuracy. Using the establishment method, a study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted.

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.

NEWMARK ALGORITHM FOR DYNAMIC ANALYSIS WITH MAXWELL CHAIN MODEL

This paper investigates a time-stepping procedure of the Newmark type for dynamic analyses of viscoelastic structures characterized by a generalized Maxwell model. We depart from a scheme developed for a three-parameter model by Hatada et al. [1], which we extend to a generic Maxwell chain and demonstrate that the resulting algorithm can be derived from a suitably discretized Hamilton variational principle. This variational structure manifests itself in an excellent stability and a low artificial damping of the integrator, as we confirm with a mass-spring-dashpot example. After a straightforward generalization to distributed systems, the integrator may find use in, e.g., fracture simulations of laminated glass units, once combined with variationally-based fracture models.

Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.

Theoretical modeling and vibration control for pre-twisted composite blade based on LLI controller

Theoretical modeling and vibration control for divergent motion of thin-walled pre-twisted wind turbine blade have been investigated based on “linear quadratic Gaussian (LQG) controller using loop transfer recovery (LTR) at plant input” (LLI). The blade section is a single-celled composite structure with symmetric layup configuration of circumferentially uniform stiffness (CUS), exhibiting displacements of vertical/lateral bending coupling. Flutter suppression for divergent instability is investigated, with blade driven by nonlinear aerodynamic forces. Theoretical modeling of CUS-based structure is implemented based on Hamilton variational principle of elasticity theory. The discretization of aeroelastic equations is solved by Galerkin method, with blade tip responses demonstrated. The LLI controller is characterized by LTR at the plant input. The effects of LLI controller are achieved and illustrated by displacement responses, controller responses and frequency spectrum analysis, respectively.

A Nonlinear Mathematical Model of Thermoelastic Thin Plates with Voids and Its Applications

Following the linear theory of thermoelastic materials with voids, a generalized Hamilton variational principle is extended to the thermoelastic plates with voids under the case of large deflection, and a 3D nonlinear mathematical model is presented. In this process, the balance equation of the entropy is converted to an equivalent form without the first-order time-derivative by integral, and the concept of the moments for the volume fraction of voids and temperature field is introduced. As application, the nonlinear dynamic and aerodynamic characteristics of simply-supported rectangular thermoelastic plates with voids for four different materials are studied and compared by using a Galerkin approach. The effects of the initial deflections and material parameters are considered in detail. In addition to providing a generalized Hamilton variational principle and a 3D nonlinear mathematical model, it is also provided a valuable numerical method to solve the dynamic problem directly in the paper. The theory and the method can be applied to solving various problems of the thermoelastic plates with voids easily.

An Analytical Symplecticity Method for Axial Compression Plastic Buckling of Cylindrical Shells

This study is mainly concerned with the analytical solutions of plastic bifurcation buckling of cylindrical shells under compressive load. The analysis is based on the J2 deformation theory with a linear hardening and proportional loading is adopted in the calculation. A symplectic solution system is established and Hamilton's governing equations are derived from the Hamilton variational principle. The basic problem in plastic buckling is converted into solving for the symplectic eigenvalues and eigensolutions, respectively. The obtained results reveal that boundary conditions have a very limited influence on bucking loads but its influence on buckling modes and plastic borders cannot be neglected. Meanwhile, it is demonstrated that the shell material properties significantly affect the plastic buckling behavior. This proposed symplectic method is shown to be a rigorous approach. It also provides a uniform and systematic way to any other similar problems.

SYMMETRY PROPERTIES OF THE EXACT EM RADIATION-REACTION EQUATION FOR CLASSICAL EXTENDED PARTICLES AND ANTIPARTICLES

Based on recent theoretical developments (Cremaschini and Tessarotto, 2011–2013), in this paper the issue is addressed of the first-principle construction of the nonlocal relativistic radiation-reaction (RR) equation for classical spherical-shell, finite-size particles and antiparticles. This is achieved invoking the axioms of Classical Electrodynamics by means of the Hamilton variational principle. In connection with this, the Lagrangian conservation laws, together with the possible existence of adiabatic invariants, and the transformation laws of the RR equation with respect to CPT and time-reversal transformations are investigated. The latter properties make possible the parametrization of the RR equations, holding respectively for particles and antiparticles of this type, in terms of the same coordinate time t and the investigation of the qualitative properties of their solutions. In particular, in both cases the RR self-force is found to have the same signature, which implies that the dynamics of classical finite-size antiparticles is equivalent to that of classical extended particles of opposite charge. Therefore, in the framework of Classical Mechanics, a distinction between particles and antiparticles cannot be made based solely on the electromagnetic interactions associated with electromagnetic RR phenomena. As a basic application of the theory, the Lagrangian conservation laws and symmetry properties for the Hamiltonian asymptotic approximations of the exact RR equation are also addressed.