Analysis of Natural Frequencies in the Universal Programs for Dynamic Processes Analysis
# 12, December 2016
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 sub-vector 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 sub-vector of the derivatives of the state variables was in the penultimate position, while a sub-vector 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 sub-matrix of coefficients that corresponds to the sub-vector 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.
- Timoshenko S.P., Iang D.Kh., Uiver U. Kolebaniia v inzhenernom dele. Mashinostroenie = Mechanical engineering. Moscow, 1985. 472 p.
- Modelirovanie sistem s sosredotochennymi parametrami (bazovyi kurs). The base and the generator of educational resources. Available at: http://bigor.bmstu.ru/?met/?doc=Mod/base.cou/?cou=Mod/base.cou, accessed 18.10.2016.
- Primenenie kompleksa PA9 dlia proektirovaniia ob"ektov mashinostroeniia. BMSTU Center for distance education. Available at: http://wwwcdl.bmstu.ru/Press/Press.html, accessed 18.10.2016.
- PRADIS — programmnyi kompleks dlia analiza dinamiki sistem razlichnoi fizicheskoi prirody. Laduga Engineering services. Available at: http://www.laduga.ru/pradis/pradis.shtml, accessed 18.10.2016.
- Resheniia LMS dlia modelirovaniia i provedeniia ispytanii. Siemens PLM Software. Available at: http://www.plm.automation.siemens.com/ru_ru/products/lms/index.shtml, accessed 18.10.2016.
- Petrenko A.I., Vlasov A.I., Timchenko A.P. Tablichnye metody modelirovaniia elektronnykh skhem na ETsVM. Vishcha shkola. Kiev, 1977. 190 p.
- Norenkov I.P., Trudonoshin V.A., Fedoruk V.G. Metod formirovaniia matematicheskikh modelei dlia adaptiruemykh programmnykh kompleksov analiza radioelektronnykh skhem. Radiotekhnika = Radiotechnics. 1986. № 9. P. 67-72.
- Norenkov I.P. Osnovy avtomatizirovannogo proektirovaniia. Bauman Moscow State Technical University. Moscow, 2009. 430 p.