SIMULATION MODELING OF AN ESSENTIALLY NON- LINEAR DYNAMIC SYSTEM

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1 7. medzinárodná vedecká konferencia Riešenie krízovýc situácií v špecifickom prostredí, Fakulta špeciálneo inžinierstva ŽU, Žilina, máj 0 SIMULATION MODELING OF AN ESSENTIALLY NON- LINEAR DYNAMIC SYSTEM Detelin Vasilev *) ABSTRACT Te forced vibrations of an essentially non-linear dynamic system are examined. Te pysical nature of te essential non-linearity is due to te existence of dry friction forces wit a canging magnitude. A comparative researc of te forced vibrations of te two-mass essentially non-linear system by te presence of dry friction of armonics type is done. A simulation model of a system of differential equations wit different parameters is completed from a view of building up an effective vibroprotection system. A board spectrum of a disturbing frequency is searced so tat te system completes its functions as a protector against vibrations. Conclusions of te decisions caracterizing te influence of te model parameters on te movement of te mecanical system are drawn. Key words: simulation modeling, Vibroprotection, Non-linear dynamic system ABSTRACT В работе рассматриваются вынужденные колебания существенно нелинейные динамические системы. Физическая природа существенные нелинейности из-за наличия сил сухого трения переменчивая величина. Сделаны сравнительные исследования вынужденных колебаний двух масс существенно нелинейная система наличие сухого трения "гармонического" типа. Имитационная модель системы дифференциальных уравнений с различными параметрами завершена с целью построения эффективной системы виброзащиты. Сделаны выводы решений, характеризующие влияние параметров модели на движение механической системы. *) Detelin Vasilev, doc. Pd, Todor Kableskov University of Transport, 58Geo Milev Str., Sofia, Bulgaria, tel.: , mobile tel: , fax: , dvasilev@vtu.bg 667

2 Key words: Компьютерное моделирование, Виброзащиты, Нелинейная динамическая система. DYNAMIC MODELS Te dynamic dampening of vibrations is a metod were additional devices called dynamic dampers are introduced into a vibrating system. Tey realize te dynamic dampening of vibrations using te principle of redistributing te vibration energy and directing in from te object protected to te damper and te principle of increasing te quantity of scattered energy in te system. Te paper examines te vibrations of te simplest inertia dynamic damper (te so called Fram s damper) in te presence of dry friction of armonic type. Five dynamic models given below are studied. Te differential equations describing te oscillations of te corresponding models ave te kind of: X X Model-I & x k ν( x x) ( x x ) = 0. = sin () X X Model-II & x kν( x x) ( x x ) = 0. + f cos pt sign x& = sin () X X Model-III & x kν( x x) + f cos pt sign x& = ( x x ) + f cos pt sign x& = sin pt. sin (3) 668

3 X Model-IV X & x kν( x x) + f cos pt sign x& = ( x x ) + f cos pt sign x& = sin pt. sin (4) X X Model-V & x k ν( x x) = sin ( x x ) + f sign x& = 0. (5) Here x and x are te generalized co-ordinates corresponding to te dynamic models; k and k are te corresponding natural frequencies; p is te frequency of te disturbing m force; ν = is te ratio between inertia features of te two masses. Coefficient m caracterizes te amplitude of te armonic disturbing force and coefficient f caracterizes te amplitude of te dry friction force, wic is of armonic type as it can be seen.. RESULTS OF SIMULATIONS Using MATLAB software package, simulation modeling of te differential equations written above as been done. Te integration is based on an explicit Runge- Kutta (4,5) formula, te Domain-Prince pair. Te following parameters of te system ave been accepted: p=; =; f=0,; ν=0,5. Natural frequencies k and k cange wile carrying out te simulation modeling. A frequency analysis of te oscillations along te two generalized coordinates is made for corresponding combination. Te distribution of te maximal 669

4 values of te amplitude special density wit different values of k and k (k and k vary in te range of 0 40) is given in next ten figures. Amplitude a Amplitude a Model - I Figure Figure Model - II Figure 3 Figure 4 Model - III Figure 5 Figure 6 Model - IV 670

5 Amplitude a Amplitude a Figure 7 Figure 8 Model - V Figure 9 Figure 0 3. CONCLUSIONS Te general analysis of te results sows te following peculiarities:. Te amplitude of te basic mass a decreases insignificantly in comparison to te basic model-i wit switcing to te damper wit dry friction of armonics type (model-v). Here it is caracteristic tat te system locking as not been watced of te bot models as it is of systems wit dry friction and a constant magnitude.. Zones of system locking ave been watced in te presence of disturbing and resistance forces acting togeter on bot of te masses. Tese zones are in te area of ig natural frequencies wit anti-pase disturbances (model-iv) and mainly around te resonance areas wit syncronic disturbances (model-iii). 3. Witout looking for te coefficient of vibroprotection efficiency, we can estimate tat a wide area around te resonance is caracteristic for model-ii: low values of basic mass a amplitude ave been watced in tat area. REFERENCES [] Tcerneva-Popova, Z., & Vasilev, D. (975). Forced vibrations of a system wit two degree of freedom by te presence of dry friction of "armonics" type. Announcements of HMEI, XXXIV(book 6). [] Te Mat Works. (005). Matlab v.7 Users Manual. Mat Works Publising. [3] Tomson, W. (998). Teory of Vibration wit Applications (5t ed.). Prentice- Hall International Inc. Článok recenzoval: doc. Ing. B. Leitner, PD. 67

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