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Range of Validity of the Method of Averaging


Prelewicz, Daniel Adam (1970) Range of Validity of the Method of Averaging. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4S5M-N506.


Sufficient conditions are derived for the validity of approximate periodic solutions of a class of second order ordinary nonlinear differential equations. An approximate solution is defined to be valid if an exact solution exists in a neighborhood of the approximation.

Two classes of validity criteria are developed. Existence is obtained using the contraction mapping principle in one case, and the Schauder-Leray fixed point theorem in the other. Both classes of validity criteria make use of symmetry properties of periodic functions, and both classes yield an upper bound on a norm of the difference between the approximate and exact solution. This bound is used in a procedure which establishes sufficient stability conditions for the approximated solution.

Application to a system with piecewise linear restoring force (bilinear system) reveals that the approximate solution obtained by the method of averaging is valid away from regions where the response exhibits vertical tangents. A narrow instability region is obtained near one-half the natural frequency of the equivalent linear system. Sufficient conditions for the validity of resonant solutions are also derived, and two term harmonic balance approximate solutions which exhibit ultraharmonic and subharmonic resonances are studied.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Applied Mechanics)
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Caughey, Thomas Kirk
Thesis Committee:
  • Unknown, Unknown
Defense Date:5 May 1970
Funding AgencyGrant Number
Department of HealthUNSPECIFIED
State of CaliforniaUNSPECIFIED
Record Number:CaltechTHESIS:09012015-094327595
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9128
Deposited On:01 Sep 2015 17:27
Last Modified:20 May 2024 23:15

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