A Practical Computational Method for Reducing a Dynamical System with Constraints to an Equivalent System with Independent Coordinates
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION HAMPTON VA LANGLEY RESEARCH CENTER
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The purpose of this paper is to present a method by which equations of motion of a linear mechanical system can be derived in terms of independent coordinates when basic information about the system is available in terms of coordinates which are not independent but, instead, are governed by linear homogeneous equations of constraint. Necessity for this derivation arises frequently in practical vibration analysis. The method is believed to be new, and experience in analyzing the vibrations of shells indicates that it very often offers decided advantages over methods previously used. In the method, a real symmetric matrix is constructed by an operation that involves only the coefficients in the equations of constraint. The eigenvectors and corresponding eigenvalues of the symmetric matrix are computed. Then, a transformation matrix leading to independent coordinates is assembled from the eigenvectors corresponding to eigenvalues having the value zero. The new method is demonstrated to have the following advantages 1 for the most general constraint equations, solution of the equations is reduced to computing the eigenvalues and eigenvectors of a symmetric matrix and 2 the method is applicable when there are redundancies in the equations of constraint. Two examples of application of the new method are presented i.e., spring-mass systems and cylindrical elastic shells, and the paper closes with a discussion of numerical considerations involved in practical computing with the method.
- Theoretical Mathematics