The Averaging Method

The eigenvalue analysis does not reveal much information regarding the behavior of the nonlinear system once instability occurs. The existence of periodic solutions (limit cycles), region of attraction of the stable trivial solution, and the effects of system parameters on these features as well as the size of the limit cycles (amplitude of steady-state vibrations) are important problems that cannot be solved using the linearized system s equations. 

Wherever applicable, we use the method of averaging [55, 56] to study the behavior of the equations of motion as a weakly nonlinear system. There are a few variations on the basic theorem for the first-order periodic averaging [56-58]. A slightly modified version of the theorem proven in [57] - suitable for otur subsequent analyses - is presented below which establishes the error estimate of the solution of the averaged system, with respect to the that of the original differential equation. 

Theorem 3.1. (First-Order Averaging) Consider the following system in standard form 

Setting r = 0, the amplitude equation has two equilibrium points f = 0 (trivial solution) and r = 2 (periodic solution). The stability of these solutions can be established by analyzing the eigenvalues of the Jacobian of the amplitude equation. 

We find that the trivial solution is unstable whereas the periodic solution i.e., the limit cycle) is exponentially stable. 

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Each sample collected In all of the methods was transferred to a clean stainless-steel pan for blending as described In the averaging method ... 

The electron microscope offers a unique approach for measuring individual nano-sized volumes which may be catalytically active as opposed to the averaging method employed by spectroscopic techniques. It is just this ability of being able to observe and measure directly small crystallites or nano-volumes of a catalyst support that sets the microscope apart from other analyses. There have been many studies reported in the literature over the past fifteen years which emphasize the use of analytical and transmission electron microscopy in the characterization of catalysts. Reviews (1-5) of these studies emphasize the relationship between the structure of the site and catalytic activity and selectivity. Most commercial catalysts do not readily permit such clear distinction of physical properties with performance. The importance of establishing the proximity of elements, elemental distribution and component particle size is often overlooked as vital information in the design and evaluation of catalysts. For example, this interactive approach was successfully used in the development of a Fischer-Tropsch catalyst (6). Although some measurements on commercial catalysts can be made routinely with a STEM, there are complex catalysts which require... 

As described above, the flow within the canopy is heterogeneous at the scales d and AS. While this flow structure must be resolved to understand processes that occur at the scale of individual stems, such resolution is not needed, and in fact is quite cumbersome, when describing the flow over larger spatial scales. An averaging scheme is applied to reduce the complexity to tractable equations. The averaging method is nicely described in [522, 523], Following their notation, the velocity (a, v, w) and pressure p) field is first decomposed into a time average (overbar) and deviations from the... 

A more appropriate approach to the construction of the mechanics of concentrated disperse systems based on the cell model is developed in [77], This approach, based on the averaging methods over the ensemble of randomly situated particles, allows one, by using a unique methodology, to obtain both the equations of continuum mechanics of disperse systems and the closing rheological relations theoretically rather than phenomenologically. In particular, the... 

Not all of the fragment constants and factors of the Leo method were incorporated in this program. Their inclusion would have reduced the error for a few compounds and thus slightly lowered the average method error. 

Estimated areas were normalized by the averaging method. The values presented in this table are geometric means. 

From all the averaging methods, it is customary to use the number-average, Mn, and weight average, M , which are both well defined in a mathematical way. 

Some estimates of the averaging method for multi-frequency systems of ordinary differential equations. Dif. Uravn., 4, (1968), 459-473. 

On the problem ofJustification of the averaging method for equations of the second order under pulse irfluence. Ukrain. Mat Zhum., 29, (1977), 750-762. 

The averaging method for a turbulent flow divides the time series into an averaged value and a fluctuation with respect to that average [6]. Or,... 

The advantage of the above form is that it can be fit into conventional non-rela-tivistic codes since the two-component spinor projections have been eliminated and we obtain simple Im > projections involving ordinary spherical harmonics. However, the averaging method eliminates the spin-orbit operator. Fortunately, the spin-orbit operator itself can be expressed in terms of RECPs as shown by Hafner and Schwarz (1978,1979) and Ermler et al. (1981). This form is shown below. 

Similarly to Karplus-Strong synthesis, the cellular automata lookup table also employs a recirculating wavetable to simulate the behaviour of a vibratory medium. The main difference is that the new sample values of the delay line are computed here by means of cellular automata, rather than by the averaging method. 

The averaging method of Ahmed and Fahien utilizes three-parameter radial profiles (1980) given by ... 

In the stick-slip vibrations, the mass moves with the belt for a part of one period. This intermittent motion results in a dynamical system with varying degrees-of-freedom (DOFs). Obviously, during the stick phase, the number of DOFs is zero and it is one otherwise [see (4.2) and (4.3)]. Due to this discontinuity, the averaging method used in previous section is no longer applicable. A common but intricate approach to construct an approximate solution for the stick-slip motion is to treat the stick phase and slip phase separately and then stitch the two results together [57, 62]. Here, however, we take a different approximation approach smoothing ... 

A numerical example is given next to demonstrate the utility of the averaging method described above. 

In this section, the averaging method is applied to (6.30). To obtain the first-order averaged equations, the right-hand side of (6.30) must be averaged over a period (i.e., T = 2n) while keeping a as constant . This gives... 

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