Dynamic Frictional Problems Research Articles

Numerical algorithms for two-dimensional dynamic frictional problems

Conventional integration schemes for the equations of motion encounter numerical problems near transitions between stick and slip in two-dimensional frictional contact problems. Here we explore the dynamical behaviour close to these transitions and develop analytical patches that can be used to develop a state-variable algorithm, where the system toggles between states of stick and slip.

The test for suppressed dynamical friction in a constant density core of dwarf galaxies

The dynamical friction problem is a long-standing dilemma about globular clusters (hereafter,GCs) belonging to dwarf galaxies. GCs are strongly affected by dynamical friction in dwarf galaxies, and are presumed to fall into the galactic center. But, GCs do exist in dwarf galaxies generally. A solution of the problem has been proposed. If dwarf galaxies have a core dark matter halo which has constant density distribution in its center, the effect of dynamical friction will be weakened considerably, and GCs should be able to survive beyond the age of the universe. Then, the solution argued that, in a cored dark halo, the suppression of dynamical friction is caused by a new equilibrium state constructed by the interaction between the halo and the GC, in which a part of the halo rotates along with the GC (co-rotating state). In this study, I tested whether the solution is reasonable and reconsidered why a constant density, core halo suppresses dynamical friction, by means of N-body simulations. As a result, I conclude that the true mechanism of suppressed dynamical friction is not the co-rotating state, although a core halo can actually suppress dynamical friction on GCs significantly.

On the use of variational inequalities to model impact problems of elasto–plastic media

A new variational inequality-based formulation is presented for treating dynamic elasto–plastic contact problems. The incremental variational inequality representing this class of problems is developed in an updated Lagrangian framework. A new technique for representing the kinematic contact conditions, based on C 1-continuous spline interpolation and an intermediate surface with uniquely defined normal, is developed. The solution algorithm is based upon the iterative use of mathematical programming and Lagrange multipliers to identify the candidate contact surface and contact stresses. This approach guarantees the accurate imposition of the active kinematic contact constraints and avoids the use of special contact elements. The dynamic variational inequalities formulations for the two sub-problems are solved using the generalized- α time integration scheme. The solution strategy accounts for the effect of friction through the use of an appropriate regularization technique in the virtual work expressions. This newly developed approach leads to a significant reduction in numerical oscillations in impact and dynamic frictional problems, and is less sensitive to variations in the time increment. It also reduces the number of iterations needed to achieve convergence. The robustness and accuracy of the proposed FE algorithm are demonstrated by application to a number of case studies.

Analysis of dynamic frictional contact problems using variational inequalities

This investigation is concerned with the development, implementation and application of variational inequalities (VI) to treat the general elastodynamic contact problem. Two VI algorithms are used to treat the general frictional contact formulation and to identify the candidate contact surface and contact stresses. The first algorithm, which decomposes the proposed into two sub-problems, is based upon the iterative use of mathematical programming and Lagrange multipliers. The second algorithm treats the VI formulation in a single step, using nondifferentiable optimization algorithm. In order to ensure that the dynamic contact problem is modeled accurately, a generalized- α method is adopted for time marching. The selected time integration parameters significantly reduce the spurious high-frequency modes, which are present in the traditional Newmark method in impact and dynamic frictional problems. In order to demonstrate the versatility and accuracy of the proposed FE algorithms, a number of numerical examples are examined.

Dynamical friction in dwarf galaxies

We present a simplified analytic approach to the problem of the spiraling of a massive body orbiting within the dark halo of a dwarf galaxy. This dark halo is treated as the core region of a King distribution of dark matter particles, in consistency with the observational result of dwarf galaxies having solid body rotation curves. Thus we derive a simple formula which provides a reliable and general first order solution to the problem, totally analogous to the one corresponding to the dynamical friction problem in an isothermal halo. This analytic approach allows a clear handling and a transparent understanding of the physics and the scaling of the problem. A comparison with the isothermal case shows that in the core regions of a King sphere, dynamical friction proceeds at a different rate, and is sensitive to the total core radius. Thus, in principle, observable consequences may result. In order to illustrate the possible effects, we apply this formula to the spiraling of globular cluster orbits in dwarf galaxies, and show how present day globular cluster systems could in principle be used to derive better limits on the structure of dark halos around dwarf galaxies, when the observational situation improves. As a second application, we study the way a massive black hole population forming a fraction of these dark halos would gradually concentrate towards the centre, with the consequent deformation of an originally solid body rotation curve. This effect allows us to set limits on the fraction/mass of any massive black hole minority component of the dark halos of dwarf galaxies. In essence, we take advantage of the way the global matter distribution fixes the local distribution function for the dark matter particles, which in turn determines the dynamical friction problem.

Dark clusters in galactic halos?

It is proposed that the invisible mass in galactic halos may consist of one million solar masses dark clusters. Such clusters would be able to heat the stellar disks in galaxies, just as in the Lacey and Ostriker (1985) supermassive black hole scenario, but dynamical friction would not necessarily drag too many of them into the galactic nucleus and would avoid accretion of interstellar gas, making them excessively luminous as they traverse the disk. The dynamical friction problem can be circumvented because the clusters may be disrupted by encounters before the drag can be effective, providing the halo core radius is less than 2-4 kpc, the clusters have a size of about 1 pc, and the components of the clusters have masses less than about 10 solar masses. A variety of ways is suggested in which the clusters required in this model could arise.