Dynamic instability of a non-uniform beam modeled in a two-degree-of-freedom system

The behaviour of a non-uniform beam loaded by a parallel or tangential compressive force respectively is analyzed by exploiting a two-degree-of-freedom dynamic model. The same system has been analyzed previously by Lee and Reissner from a static point of view and by Neer and Baruch by a dynamic approach using a one-degree-of-freedom model. The previous analyses revealed only part of the phenomena by the present approach.Here, for the tangential force a classical flutter instability is obtained and for the parallel force only static instability is possible.     

Note on flow instability in heated ducts

Zusammenfassung Das Problem der Stabilität der Strömung durch beheizte horizontale Kanäle gegen zweidimensionale Störungen wird in linearisierter Form als Variationsaufgabe behandelt. Die Abhängigkeit der Stabilitätsparameter von der Gestalt des Kanals wird untersucht und verglichen mit dem Fall der Stabilität der von unten beheizten Flüssigkeitsschicht.     

Torsion of a circular cone with static and dynamic loading

Integral transformation methods—the Mellin transform for statics and the Lebedev-Kontorovich transform for dynamics—are used to construct analytic solutions of the problem of the torsion of an elastic circular cone. Assuming that external forces are concentrated in the neighbourhood of the vertex of the cone, the asymptotic behaviour of the far field is investigated. It is shown that the leading term of the asymptotic expansion is governed by the magnitude of the moment of the external forces, so that the St Venant principle is satisfied in the cases under consideration.     

Analysis and simulations of a nonlinear elastic dynamic beam

A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions.     

An optimization of a Mindlin-Timoshenko beam with a dynamic contact on the boundary

We deal with the optimal control problem governed by a hyperbolic variational inequality describing the perpendicular vibrations of a Mindlin-Timoshenko beam clamped on the left end with a rigid obstacle at the right end. A variable thickness of a beam plays the role of a control parameter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A survey of static and dynamic potential games

Potential games are noncooperative games for which there exist auxiliary functions, called potentials, such that the maximizers of the potential are also Nash equilibria of the corresponding game. Some properties of Nash equilibria, such as existence or stability, can be derived from the potential, whenever it exists. We survey different classes of potential games in the static and dynamic cases, with a finite number of players, as well as in population games where a continuum of players is allowed. Likewise, theoretical concepts and applications are discussed by means of illustrative examples.     

Maximum thick paths in static and dynamic environments

We consider the problem of finding a large number of disjoint paths for unit disks moving amidst static or dynamic obstacles. The problem is motivated by the capacity estimation problem in air traffic management, in which one must determine how many aircraft can safely move through a domain while avoiding each other and avoiding “no-fly zones” and predicted weather hazards. For the static case we give efficient exact algorithms, based on adapting the “continuous uppermost path” paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem.Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. Deciding the existence of just one path, even for a 0-radius disk, moving with bounded speed is NP-hard, as shown by Canny and Reif [J. Canny, J.H. Reif, New lower bound techniques for robot motion planning problems, in: Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987, pp. 49–60]. Moreover, we observe that determining the existence of a given number of paths is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying “dual” approximations, compromising on the radius of the disks and on the maximum speed of motion.Our main result is a pseudopolynomial-time dual-approximation algorithm. If K unit disks, each moving with speed at most 1, can be routed through an environment, our algorithm finds (at least) K paths for disks of radius somewhat smaller than 1 moving with speed somewhat larger than 1.     

Discrete and dynamic versus continuous and static loading policy for a multi-compartment vehicle

In this paper, we address the problem of loading non-intermixable products in a vehicle consisting of compartments of different sizes. The demands of the products are different but uniform over time. The objective is to meet product demands and minimize setup rate (that is, the number of deliveries per unit time). Two approaches, namely, dynamic and static, are investigated and their performances are compared with each other. In the dynamic approach, deliveries are made in several discrete periods and, then, repeated in a cyclic fashion. In each of these deliveries, the allocation of products to compartments can be different. The static approach, on the other hand, assumes a continuous time scale and determines a single assignment of products to compartments that maximizes the time in which the product demands are fully satisfied by this single delivery. The comparison between the two approaches shows that the dynamic approach is superior to the static approach when a discrete time scale is considered. However, even when the discrete time scale constraint is relaxed, the dynamic approach still provides better results for relatively long cycle times.     

A dynamic analysis of a coupled beam/slider system

The paper deals with transverse vibration of a beam with moving boundary conditions. In order to examine the transfer of energy between a moving support and a vibrating beam under assumption of zero slope of the beam elastic line at the moving support, dynamic analysis of a coupled beam/slider system is carried out. The moving support is modelled as a slider attached to a spring which realizes definite boundary conditions. Equations of motion are derived using Hamilton’s principle. Because length of the beam varies appropriate transformations of time and position variables are made to convert the solution space into a rectangle and, subsequently, to solve the partial differential equation of transverse vibration of the beam using the FDM. The phenomenon of the energy flow between the slider and the beam is a subject of the detailed analysis. The beam vibration with a fixed formula of length is examined, too. The dynamic characteristics of the system is brought forward from spectral analysis of numerical solutions.     

Self-reinforcement of liquid-crystal polymers at static and dynamic loading

The self-reinforcement effect of a solid uniaxially oriented SVM-K liquid-crystal polyamide and a copolyester of hydroxybenzoic and hydroxynaphthoic acids has been investigated by tensile-strength, stress-relaxation, and dynamic methods. The samples were prepared by spinning from lyotropic solution (SVM-K) and from a thermotropic melt (polyester). The tensile-strength and stress-relaxation tests were performed on complex fibers and the dynamic test on single fibers. The set of stress-strain curves, changing from a convex shape with two linear sections (at room temperature) to a concave shape (at high temperatures) is shown for both materials in Fig. 1. There is a pronounced difference between the deformation mechanisms at low and high strains in the stability of rigidity. At high temperatures the rigidity becomes less than the initial one during deformation and the current modulus at high strains has the same value within large ranges of temperatures and strains (Fig. 2). A low-deformation transition of another physical parameter than the yield-stress has been found. The stress-strain diagram for both investigated polymers has been generalized by using the constant value of the current modulus for the normalization of the stress value (Fig. 3). The stress-relaxation phenomena are shown to be anomalous. At high temperatures the stress-relaxation intensity decreases with increasing deformation, i.e., after deformation the polymer is characterized by a stability of rigidity which is higher than the initial value (Fig. 4). The dynamic modulus appears to increase with increasing deformation rate (Fig. 5). Due to these peculiarities the liquid-crystal polymers must be considered not only as normal high-modulus reinforcements for composite materials but also as materials, self-reinforcing under loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 4, pp. 435–441, July–August, 1994.     

Stabilizing the XFEM for static and dynamic crack simulations

For many crack geometries the standard XFEM may lead to badly conditioned equation systems. Recently a number of remedies to that problem have been published. In this contribution the stabilization technique presented in [1] is extended to dynamic and possibly nonlinear fracture mechanics and additionally combined with the SGFEM / SXFEM [2]. The performance of the different strategies is demonstrated in a linear elastic 3D mode 1 crack problem for which the analytical solution exists. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multipolar routing: where dynamic and static routing meet

Assuming that the traffic matrix belongs to a polytope, we describe a new routing paradigm where each traffic matrix is routed a combination of a number of extreme routings. This combination depends on the current traffic matrix. Multipolar routing can be seen as a generalization of both routing and robust static routing. Moreover, the time complexity of multipolar routing is under control since it depends on the number of poles (i.e. the number of extreme routings) which can be defined by the network planner     

Resonant response of a strong electromagnetic wave beam to a gravitational wave in a static magnetic field

Concrete forms of resonant response (ER) for a strong electromagnetic (EM) wave beam (photon flux) propagating in a static magnetic field to a standing gravitational wave (gravitons) are given, and the corresponding perturbation solutions and resonant conditions are obtained. It is found that perturbed EM fields (PEMFs) contain three new components with frequencies Io,* w,l and ωP_g respectively. In the case of ω_e⋙ω_g, the PEMFs are manifested as the EM wave beams with frequency ω_e and a standing EM wave with ω_g. The former and the background EM wave beam (BE-MWB) have the same propagating direction, while in the case of ω_g⋙ω_e, all PEMFs are expressed as the standing EM waves with frequency ω_g. The resonant response occurs in two cases of ω_e = 1/2 ω_g andω_e, = ω_g only. Then not only the first order perturbed energy fluxes (PEFs) propagating in the same and opposite directions of the BEMWB can be generated, but also radial and tangential PEFs which are perpendicular to the above directions can be produced. This effect might provide a new way for the EM detection of the gravitational waves (GWs). Moreover, the possible schemes of displaying perturbed effects induced by the standing GW withh = 10^-33 - 10^-35 and λ_g = 0.1 m at the level of the single photon avalanche and in a typicla laboratory dimension are reviewed.     

Solutions of dynamic problems for incompressible and slightly compressible helically orthotropic non-uniform thick-walled cylinders

Exact analytic solutions are obtained of the one-dimensional dynamic problem for an incompressible elastic radially non-uniform helically orthotropic thick-walled cylinder under plane-strain conditions, loaded with a time-dependent pressure from inside and/or outside. The necessary and sufficient conditions for solutions to exist and to be unique are established. The convergence of the wave solutions for slightly compressible cylinders to the analytic relations obtained for incompressible cylinders is investigated.     

The dynamics of a non-uniform spheroid on a horizontal plane

The dynamics of a spheroid with a displaced mass centre on a horizontal plane with friction is considered. It is assumed that the mass centre of the spheroid lies on its dynamic symmetry axis. The steady motions of the spheroid are investigated using the general theory of invariant sets of mechanical systems with symmetry, and a geometrical interpretation of the results is given using generalized Smale diagrams.