Defining parameters of differential equation of electric motor rotor mechanical rotation mathematical model while its' switching off
Mathematica modeling, numerical technique and program complexes
Аuthors*, **, ***
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
The presented work discussed the problems of modeling and mathematical methods for diagnosing the degradation wear of an electric motor in case of bearing unit destruction ‒ one of the most drasic failures. The investigations were carried out on the example of an induction motor with a «squirrel-cage» type rotor widely used in industry and for home appliances. For the degradation wear modeling, the mode of motor switch off from the network, the so-called «run-out» mode was sel ected. As a rule, the run-out mode is considered as the final one in the electric motor work cycle, and is not given due attention. It turned out that this mode presents a source of important information, which implementation makes possible to make judgement on the status of electric motor, its wear, operational safety and the residual resource.
This processes occurring in the electric motor is modeling as a second-order link has been carried out. The mechanical rotation of the rotor in the run-out mode is described by an ordinary homogeneous differential equation of the second order with constant coefficients. Parameters of the differential equation, i. e. its coefficients, reflect physical and mechanical properties of the electric motor: its geometry, material properties, etc.
The authors proposed to determine the state parameters of the electric motor differential equation fr om the known empirical function of its rotor rotational speed damping, registered in each cycle of its switching off. The paper presents the algorithm of mathematical approximation of the rotor damping rate empirical function by an analytic function with the required levels of accuracy and further determination of the corresponding differential equation parameters.
Moreover, the distortion of the rotor rotational speed damping function suggests mathematical methods for diagnosing the degradation wear of the electric motor elements, mainly wear of its bearing assembly. With the degradation of bearings, with embrittlement, dying of metal surfaces with small particles, the moment of resistance of their rotating parts increases substantially. In the run-out mode, the braking torque is increased in the bearings, the stopping time is shortened. Thus, if in the run-out mode the initial speed is less than the passport speed and the braking time is shortened ‒ the degradation processes begin in the bearings. The mechanical time constant TM of the rotor shaft braking function is the characteristic parameter of the degradation processes in bearings.
These features can be obtained with the simplest measuring instruments (voltmeter, stopwatch and tachometer) without planning and implementing special experiments, but directly during lectric motor operation.
Keywords:differential equation of natural damped vibrations of electric motor rotor, degradation changes modeling, mathematical methods for state diagnostics, service reliability
Gol’berg O.D., Khelemskaya S.P. Nadezhnost’ elektricheskikh mashin (Reliability of electric motors), Moscow, Akademiya, 2010, 288 p.
Gol’dberg O.D., Gurin Ya.S., Sviridenko I.S. Proektirovanie elektricheskikh mashin (Electric motors design), Moscow, Vysshaya shkola, 2001, 431 p.
Lisov A.A., Chernova T.A., Gorbunov M. Vestnik Moskovskogo aviatsionnogo instituta, 2017, vol. 24, no. 2, pp. 150-159.
Zakovryashin A.I., Koshel’kova L.V. Trudy MAI, 2016, no. 89, available at: http://www.mai.ru/science/trudy/published.php?ID=73384
Alekseev G.V. Vvedenie v chislennye metody resheniya differentsial’nykh uravnenii (Introduction to numerical methods for solving differential equations), Vladivostok, DVFU, 2010, 120 p.
Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniyam (Handbook of ordinary differential equations), St.-Petersburg, Lan’, 2003, 576 p.
Kopchenova N.V., Maron I.A. Vychislitel’naya matematika v primerakh i zadachakh (Computational mathematics in examples and problems), Moscow, St.-Petersburg, Lan’, 2009, 368 p.
Demidovich B.P., Maron I.A., Shuvalova E.Z. Chislennye metody analiza (Numerical methods of analysis), Moscow, Nauka, 1962, 368 p.
Lisov A.A. Izmeritel’naya tekhnika, 2001, no. 5, pp. 7-10.
Lisov A.A., Chernova T.A., Gorbunov M.S., Kubrin P.V. Kachestvo i zhizn’, 2016, no. 2, pp. 38-41.