Topological considerations and the method of averaging: a connection between local and global results

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

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Vibrations of Elastic Connecting Rod of a High-Speed Slider-Crank Mechanism

Journal of Engineering for Industry ◽

10.1115/1.3427974 ◽

1971 ◽

Vol 93 (2) ◽

pp. 636-644 ◽

Cited By ~ 37

Author(s):

Peter W. Jasinski ◽

Ho Chong Lee ◽

George N. Sandor

Keyword(s):

Small Parameter ◽

High Speed ◽

Elastic Stability ◽

Method Of Averaging ◽

Connecting Rod ◽

Transverse Vibrations ◽

Elastic Bar ◽

The Moment ◽

Steady State Solutions ◽

Crank Mechanism

The research involved in this paper falls into the area of analytical vibrations applied to planar mechanical linkages. Specifically, a study of the vibrations, associated with an elastic connecting-bar for a high-speed slider-crank mechanism, is made. To simplify the mathematical analysis, the vibrations of an externally viscously damped uniform elastic connecting bar is taken to be hinged at each end (i.e., the moment and displacement are assumed to vanish at each end). The equations governing the vibrations of the elastic bar are derived, a small parameter is found, and the solution is developed as an asymptotic expansion in terms of this small parameter with the aid of the Krylov-Bogoliubov method of averaging. The elastic stability is studied and the steady-state solutions for both the longitudinal and transverse vibrations are found.

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How to average logarithmic retrievals?

Atmospheric Measurement Techniques ◽

10.5194/amt-5-831-2012 ◽

2012 ◽

Vol 5 (4) ◽

pp. 831-841 ◽

Cited By ~ 11

Author(s):

B. Funke ◽

T. von Clarmann

Keyword(s):

Maximum Likelihood ◽

A Priori ◽

Mean Value ◽

Natural Variability ◽

High Signal ◽

Method Of Averaging ◽

Signal To Noise ◽

Mixing Ratios ◽

Priori Information ◽

The One

Abstract. Calculation of mean trace gas contributions from profiles obtained by retrievals of the logarithm of the abundance rather than retrievals of the abundance itself are prone to biases. By means of a system simulator, biases of linear versus logarithmic averaging were evaluated for both maximum likelihood and maximum a priori retrievals, for various signal to noise ratios and atmospheric variabilities. These biases can easily reach ten percent or more. As a rule of thumb we found for maximum likelihood retrievals that linear averaging better represents the true mean value in cases of large local natural variability and high signal to noise ratios, while for small local natural variability logarithmic averaging often is superior. In the case of maximum a posteriori retrievals, the mean is dominated by the a priori information used in the retrievals and the method of averaging is of minor concern. For larger natural variabilities, the appropriateness of the one or the other method of averaging depends on the particular case because the various biasing mechanisms partly compensate in an unpredictable manner. This complication arises mainly because of the fact that in logarithmic retrievals the weight of the prior information depends on abundance of the gas itself. No simple rule was found on which kind of averaging is superior, and instead of suggesting simple recipes we cannot do much more than to create awareness of the traps related with averaging of mixing ratios obtained from logarithmic retrievals.

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