Non-classical method of modelling of vibrating mechatronic systems

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1 IOP Conference eries: Materials cience and Engineering PAPER OPEN ACCE Non-classical method of modelling of vibrating mechatronic systems To cite this article: K. Biaas and A. Buchacz 06 IOP Conf. er.: Mater. ci. Eng Related content - Hydraulic elements in reduction of vibrations in mechanical systems K Biaas and A Buchacz - On the conditional symmetries of Levi and Winternitz G Gaeta - imilarity solutions of the Kadomtsev- Petviashvili equation en-uye Lou View the article online for updates and enhancements. This content was downloaded from IP address on 06/09/08 at 9:

2 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 Non-classical method of modelling of vibrating mechatronic systems K. Białas A. Buchacz ilesian University of Technology University aculty of Mechanical Engineering Institute of Engineering Processes Automation and Integrated Manufacturing ystems Konarskiego 8A Gliwice Poland Abstract. This work presents non-classical method of modelling of mechatronic systems by using polar graphs. The use of such a method enables the analysis and synthesis of mechatronic systems irrespective of the type and number of the elements of such a system. The method id connected with algebra of structural numbers. The purpose of this paper is also introduces synthesis of mechatronic system which is the reverse task of dynamics. The result of synthesis is obtaining system meeting the defined requirements. This approach is understood as design of mechatronic systems. The synthesis may also be applied to modify the already eisting systems in order to achieve a desired result. The system was consisted from mechanical and electrical elements. Electrical elements were used as subsystem reducing unwanted vibration of mechanical system. The majority of vibration occurring in devices and machines is harmful and has a disadvantageous effect on their condition. Harmful impact of vibration is caused by the occurrence of increased stresses and the loss of energy which results in faster wear machinery. Vibration particularly low-frequency vibration also has a negative influence on the human organism. or this reason many scientists in various research centres conduct research aimed at the reduction or total elimination of vibration.. Introduction Vibration is a phenomenon which can be often observed in everyday life. ome types of vibration are used in the operation of machines and devices. However most of them are of a harmful nature. This fact is related to the influence they eert on real objects causing their malfunction. Another important issue is the negative impact vibration has on the human body. or this reason the issue of the reduction of unrequired vibration is of great significance. That is why many research institutes are investigating methods of preventing the improper application or eploitation/operation of newly built machinery or methods of adjusting already eisting and operating machines to specified requirements [-]. Methods of vibration prevention are many and varied. They have an effect on the machinery elements and components. One of the classifications is the division into passive semi-active and active methods of vibration reduction [5 6].. Non-classical method of modelling In order to present the modelling of a vibratory mechanical system with active elements by means of the polar method one should consider a system presented in igure. The system under consideration (ig. ) is composed of the following elements: Content from this work may be used under the terms of the Creative Commons Attribution.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work journal citation and DOI. Published under licence by IOP Publishing Ltd

3 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/ inert elements m m m n - elastic elements c c c n - ecitations acting on the system (t) (t) n (t) - active ecitations G (t) G (t) G(t) n. () t c () t c n() t c n m m m n G G G n igure. Model of mechanical system of cascade structure The description of the adopted discrete model utilises two basic sets of quantities and as well a set of elements Z set of polar quantities generalised linear coordinates set of flow quantities generalised forces Z set of the coefficients of the polar equations of vibratory mechanical system elements also referred to as dynamic rigidities. A polar relation refers to relations specified as follows []: s s s s s z s s s s z Z i k j j i k k i j k zk i j 0 n; i j; s z s polar equation of the element k in the system; s j i k k n n m n m n m w n m w g k () k n w n m - from to n inert elements - from (n+) to (n+m) elastic elements - from (n+m+) to (n+m+w) ecitations acting on the system - from (n+m+w+) to (n+m+w+g) ecitations generated by active elements. The polar relations is also designated as: s s s. sk i k j () The dynamic structure is the representation of the mechanical model of a discrete system in the form:. () The graph of the structure of a discrete mechanical system is the triplet of sets in the form: (4)

4 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 i f (i=) representation specified as follows: such that: in the following manner: s (i=0 n) i i : (5) f : : f (6) s i i f (ig..) (7) f: s s s n n : s0 : 0 igure. Representation of generalized coordinates in graph vertices and ' '' ''' '''' f f f f f (8) ' ' s f s (ig.) (9) sk k s k= n si sk s0 k k k k k f in the case of the inert elements k of the system in the case of the edges k of the graph k i 0 0 i (i= n) f : s s mp mp s n mp n n n : s0 0 igure. Representation of inertial elements into graph edges or the sake of the clarity and the legibility of the representations the grey-coloured lines in igure 4 and subsequent figures designate the individual elements of the system relations (ig. ) in the graph edges.

5 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 '' sk k f sk k '' f (ig. 4) (0) s k=n+ n+ n+m k k s s s s ; s s s ; s s s in the case of the elastic elements k of the k k k 0 k k system ; in the case of the edges k of the graph k 0 i 0 ; i (i= n) f : s s c c n s n n+ n n + m : s0 0 Ecitations acting on the system igure 4. Representation of elastic elements into graph edges s k k ''' f (ig. 5) () s k= n+m+ n+m+ n+m+w k k in the case of the edges k of the graph k i 0 0 i (i= n) f : : s s s0 n s n n+m+ n+m + 0 n n+m + w igure 5. Representation of ecitations into graph edges Active ecitations s k k sk k k= n+m+w+ n+m+w+ n+m+w+g '''' f (ig. 6) () 4

6 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 ; in the case of the edges k of the graph (i= n). k 0 i 0 ; i f : : s s G s0 s n G n n+m+w+ 0 n n+m + w+g igure 6. Representation of active ecitations into graph edges By way of mutual representations one obtains a graph (ig..7): () i f i= n. n+ n+m+w+ n n+m + G n+m + n + m n n+m +w n+m + w+g G n 0 igure 7. Polar graph as a model of system from figure 5

7 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 () t c () t c n() t c n m m m n G G G n U I G igure 8. A model of the system with electric elements In order to modelling a system including an active subsystem in the form of electric elements reducing vibration (ig.8) by means of a non-classical method of designing it is possible to use theory of graphs. Between vibratory mechanical systems and vibratory electric circuits (ig. 9) there are analogies in the mathematical description [67]. The analogy between longitudinally vibrating models and electric models is presented in Table.Modelling of electric systems using method of polar graph is analogical like modelling mechanical systems which are presented above. Relation between electrical and mechanical systems can be present by block graph (ig. 0). L L n- L n Ut () C C n- C n R R n- R n igure 9. Model of electric system Table. Analogies between longitudinally vibrating models and electric models mechanical system longitudinal vibration kg electric system m L H N c m Ns m t) N C b R ( U ( t) V ( t) m q ( t) C 6

8 IOP Conf. eries: Materials cience and Engineering 45 (06) 0400 doi:0.088/ /45/4/0400 s () s () s () s () mechanical part s () 0 electrical part s () 0 igure 0. Block graph of system with mechanical and electrical elements In figure 0 dashed line shows edge of two-arguments incidence between coordinates of electrical and mechanical parts.. Conclusions The presented manner of modelling mechatronic systems by means of polar graphs enables full automation and algorithmisation of calculations during the determination of the dynamic characteristics of the system as well as makes it possible to directly track implemented structural changes. In the case of systems composed of a great number of subsystems the determination of dynamic characteristics in a classical manner requires numerous labour-consuming activities. In situations when it becomes necessary to modify the structure of the system each time it is necessary to formulate and solve the system of differential equations of motion which is not required in the case of the system represented in the form of a polar graph. References [] Bialas K. 0 Electrical Elements in Reduction of Mechanical Vibrations. Applied Mechanics and Materials [] Bialas K. 0 Mechanical and electrical elements in reduction of vibrations Journal of Vibroengineering 4() -8. [] Białas K. Buchacz A. Dzitkowski T. 009 ynthesis of vibrating active mechanical systems with dumping in view of polar graphs and structural numbers. Gliwice Monograph 0 ilesian University of Technology Press (in Polish). [4] Engel Z. Kowal J. 995 Vibro-accoustic processes control Kraków AGH Press (in Polish). [5] Michałowski. 994 Active systems in machines construction Cracow Publication by Cracow University of Technology Monograph 7 (in Polish). [6] Onwubolu G. C. 005 Mechatronics Principles and Applications. Elsevier Butterworth- Heinemann. [7] Preumont A. 006 Mechatronics Dynamics of Electromechanical and Piezoelectric ystems. Published by pringer. The Netherlands. 7