System degrees of freedom

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

An instability of the impulse MTS method for At slightly less than half the period of a normal mode is confirmed by an analytical study of a linear model problem [7]. For another analysis, see [2]. A special case of this model problem, which gives a more transparent description of the phenomenon, is as follows Consider a two-degree-of-freedom system with Hamiltonian p + 5P2 + + 4( 2 This models a system of two springs con-... 

Figure 5-5. Singie degree of freedom system (spring mass system).

A quick review of system torsional response may help explain why a resilient coupling works. Figure 9-14 is a torsional single degree of freedom system with a disk having a torsional moment of inertia J connected to a massless torsional spring K. 

L h degree of freedom system will have (n - 1) natural fr... 

The theory for a one-degree-of-freedom system is useful for determining resonant or natural frequencies that occur in all machine-trains and process systems. However, few machines have only one degree of freedom. Practically, most machines will have two or more degrees of freedom. This section provides a brief overview of the theories associated with two degrees of freedom. An undamped two-degree-of-freedom system is illustrated in Figure 43.16. 

Figure 43.16 Undamped two-degree-of-freedom system with a spring couple...

Nesting of KAM-tori For two-degree-of-freedom system, the KAM theorem states that for sufficiently weak perturbations, all sufficiently irrational KAM-tori are pre-... 

For a simple system, such as a rod under compression, one can define the stiffness, or the spring constant. If we examine Hookes law, we get 6L = L SFjEA, where A is the cross sectional area, and 6F is the applied load. The spring constant k is defined as dF/dL, or k = EA/L. The basic physics equation F = kx is just a statement of this. For many degree of freedom systems there will be multiple spring constants, each connected to a modal shape. 

General transformation factor equations for distributed mass systems and multi-degree of freedom systems are given by Higgs 1964 (Chapter 5), and Clough 1993 (Chapter 2). These general methods can be used in determining transformation factors for nonprismatic members or members which have nonuniform mass distributions. 

For practical purposes, manual solutions of this equation can be obtained for only two or possible three degrees of freedom. An example of an elastic-plastic, two degree of freedom system analysis is given by Biggs 1964 (pp. 237-242). Even this... 

The shear wall is effectively a single degree of freedom system,... 

Some studies on dynamic interaction effects for two degree of freedom systems have been done by Baker 1983 (pp. 415-418). Although these studies were made using a limited range of variables, results indicate that conservative responses can be obtained using uncoupled SDOF system approximations versus a coupled approach. 

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schrodinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. 

That this condition is a generalization of the standard WKB (Wentzel-Kramers-Brillouin) condition can be seen by considering a one-degree-of-freedom system, where we have... 

In doing so it proves convenient to carry out the computations in the Wigner representation. That is, as in quantum mechanics based upon the wave function, it is necessary to deal with a representation of the density operator p. The (convenient) Wigner representation pw of p is defined, for an N degree of freedom system, by... 

Another possibility is the representation in a two-dimensional diagram, as in Figure A.4 (right). The component C being chosen as the reference, the relation (A.3) gives the transformed co-ordinates XA = (xA + xc)/(l + xc) and X, = (xB + xc) / (1 + xc). The residue curves run from the reactive azeotrope to the vertex of component I. This situation is denoted by two degrees of freedom systems . 

In multidimensional systems, these intersections would exhibit much more variety than they do in the lower-degrees-of-freedom systems that have been traditionally studied in nonlinear physics. One of the new aspects is tangency , which was found in the predissociation of a van der Waals complex of three bodies [11,12]. The tangency gives birth to transition in chaos [12], which is called a crisis [13]. Then, the extention of the concept of reaction rates to multidimensional chaos, which was first proposed in Ref. 9, breaks down [11,14]. 

From these observations, we learn that even simple double-well potential systems demonstrate a variety of dynamics, depending on the shape of potential functions. Note that this sort of question—that is, what information in the potential function is necessary and sufficient conditions which make the system chaotic—dates back to just the beginning of the study of chaos in few-degrees-of-freedom systems. 

In this case the system is called integrable. From the first equation we have /, = const, for all i that is, /, are constants of motion. Thus / , (/) = 0//o/8/,- are also constants, and we have

phase space of the system is on a regular orbit (torus). 

Before undertaking the formal analysis of vibrations in solids, it is worthwhile to see the machinery described above in action. For concreteness and utter simplicity, we elect to begin with a two-degrees-of-freedom system such as that depicted in fig. 5.3. Displacements in all but the x-direction are forbidden, and we assume that both masses m are equal and that all three springs are characterized by a stiffness k. By inspection, the total potential energy in this case may be written as... 

The results of this section show that semlclasslcal estimates of resonance energies may be obtained — using the Solov ev—Johnson ansatz In regions of phase space where Invariant tori do not exist. This technique may well turn out to be the semlclasslcal method of choice for many-degrees-of-freedom systems even If tori exist, visualization of the caustic structure Is difficult. 

Local Trajectory Instability That is, the local (in phase space) exponential sensitivity of trajectories to changes in initial conditions. This property is the primary characteristic of a C system. Specifically, consider, in an N degree of freedom system, two trajectories with initial coordinates and momenta q0, p0 and qo, Po- For convenience we denote the column vector [q(f), p(t)] associated with p0, q0 as x(t) and the trajectory [q ( ), p (t)] associated with Po, q 0 as x (t). The phase space separation d(t) between trajectories is given by... 

This result makes clear that considerably greater insight is necessary before a set of adequate diagnostics are available to define a multi-degree-of-freedom system as sufficiently chaotic to apply statistical models of bound relaxation. 

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