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Analysis of Natural Frequencies in the Universal Programs for Dynamic Processes Analysis
# 12, December 2016
DOI: 10.7463/1216.0852553
authors: V.A. Trudonoshin^{1,*}, V.G. Fedoruk^{1}
 ^{1} Bauman Moscow State Technical University, Moscow, Russia 
Finding the natural frequencies of complex technical objects is an important design procedure. This type of analysis allows us to determine the resonant frequencies and, as a consequence, to avoid their adverse impact on dynamics the projected object or that of under study. This applies to both the objects with distributed parameters, and the objects with lumped parameters. As to the first type of the objects, in almost every package that implements the finite element method, this type of analysis is available. The situation is different for the objects with lumped parameters. Methods to have the mathematical models for these objects look to implicit methods of numerical integration of ordinary differential equations. And the component equations of the reactive branches are sampled by numerical integration formulas, and the derivatives of state variables disappear from the vector of the unknowns of a mathematical model. In this case, talk about the implementation of the procedure for finding natural frequencies by finding eigenvalues is simply unnecessary. In cases where a mathematical model of the object is given in the normal Cauchy form, obtaining the natural frequencies is reduced to finding the eigenvalues of the coefficient matrix. There are methods to form the mathematical models in which the derivatives of the state variables make a subvector of the vector of unknowns. These are generalized, advanced nodal methods, and an advanced nodal one for mechanical systems. There can be a try for reduction of the mathematical models of objects, obtained by these methods, to the normal Cauchy form. The article discusses a similar procedure for the generalized and advanced nodal methods. As for the extended nodal method for mechanical systems there is specifics the article does not show. For the model obtained by generalized method the vector of unknown variables is permutated so that a subvector of the derivatives of the state variables was in the penultimate position, while a subvector of the state variables was in the last one. After that, through a direct Gauss course to the derivatives of the state variables and dividing into coefficients corresponding to the derivatives of the state variables, we obtain the normal form of Cauchy, but only in case there are no wrong distributions in the original model. For the advanced nodal method the unknowns are not permutated. The elimination approach is used to try to reset the submatrix of coefficients that corresponds to the subvector of node potentials for the equations. Here, the requirements for initial mathematical model are even more stringent than for the generalized method  each node of the circuit must be connected with the element C. This restriction is not fundamental, since taking the inertial elements in consideration is, usually, desirable. References  Timoshenko S.P., Iang D.Kh., Uiver U. Kolebaniia v inzhenernom dele. Mashinostroenie = Mechanical engineering. Moscow, 1985. 472 p.
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