728. Mechanical and electrical elements in reduction of vibrations

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1 78. Mechancal and electrcal element n reducton of vbraton Katarzyna BIAŁAS The Slean Unverty of Technology, Faculty of Mechancal Engneerng Inttute of Engneerng Procee Automaton and Integrated Manufacturng Sytem Konarkego 8a, Glwce, Poland E-mal: katarzyna.bala@poll.pl (Receved 8 September 0; accepted 4 February 0) Abtract. The man am of th work the ntroducton of analy and ynthe of ytem ncludng mechancal and electrcal element reducng vbraton. In reult of ynthe were receved tructure and parameter of a dcrete model meetng the defned requrement concernng the dynamc feature of the ytem, n partcular, the frequency pectrum. The approach adopted make t poble to take acton amng at the reducton of phenomena reultng n the unwanted operaton of machnery or generaton of hazardou tuaton n the machnery envronment. Thank to the approach, the above mentoned preventve actvte can be conducted a early a durng the degnng of future functon of the ytem a well a durng the contructon of the ytem n queton. In th work alo ntroducton of thee two knd of element, mechancal and electrcal, of reducton of vbraton. Keyword: analy and modelng, proce ytem degn, ynthe, reducton of vbraton. Introducton Reducng the undered vbraton frequently dcued by centt becaue of harmful effect t ha on human organm and machnery operaton. Degner, manufacturer and uer alo have to face problem of preventng unwanted effect n the operaton of newly degned machnery or adaptng already manufactured and operatng machne to meet requrement reultng from current knowledge of hazard caued by machnery [, ]. Reducton of the advere effect of mechancal vbraton the ubject matter of numerou nvetgatng tude. Mnmzaton of undered vbraton can be acheved by varou method, where the ubdvon nto pave, actve and emactve technque the mot frequent approach [3 5, 7]. The pave method are not alway effectve, epecally n the cae of low-frequency vbraton, and for th reaon they are beng replaced by actve one. The am to perfect the ynthe een a modfcaton at the ub-aembly degn level n relaton to the requred pectrum of vbraton frequency of the ytem. Such defnton of the problem requre applcaton of the ynthe method, n categore approprate for the cla of ytem. The ynthe wll have two tage. The frt tage wll nclude a ynthe of the pave ytem, and then a ynthe of the actve ytem (dahed lne) or ytem wth dampng (contnuou lne) (Fg. ), [, ]. The model of the reearch The arrangement under conderaton a cacade dcrete vbraton ytem. To create the above-mentoned ytem t neceary to defne requrement for the reonance and antreonance frequence of the ytem (): ω ω 0,, ω 3, ω ω, ω n n reonant frequence ant - reonant frequence () VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

2 78. MECHANICAL AND ELECTRICAL ELEMENTS IN REDUCTION OF VIBRATIONS. n odd number. Synthe cont n dtrbutng the charactertc functon () onto the contnued fracton (3) [, ]: U U ( ) ( ) ( = ( )( c = + m + c 3 ) ( ) ( n ) n + m ) () (3) Sytem or requement chractertc Set of tructure of ytem contanng one or more actve element Determnaton of value of force or fore generated by actve element Synthe by mean of a elected method Determnaton of value of dampng element Checkng frequency and tme-related reult obtaned Sytem after dentfcaton Fg.. Idea of ynthe of mechancal ytem wth pave or actve element reducng vbraton The next tep requre the dentfcaton of external effect nfluence on the ytem. In th cae t wa etablhed that the ytem nfluenced by dynamc exctaton on nertal element (Fg. ). c c m m F( t) F( t) Fg.. Model of the ytem after ynthe wth dynamc exctaton 4 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

3 78. MECHANICAL AND ELECTRICAL ELEMENTS IN REDUCTION OF VIBRATIONS. The ymbol on the Fg. denote: m, m,, - nertal element, c, c,, - elatc element, F ( t), F ( t),, F ( t) - dynamc exctaton. k Pave element n reducton of vbraton Accordng to the flow chart that hown n Fg., decreae of vbraton can be acheved by applcaton of ether pave or actve element. The pave element can be mplemented, for ntance, a vcou damper. To fnd out the value of vcou damper one can ue the followng relatonhp (4, 6). The frt equaton ued for dampng element that are proportonal to the nertal element, the econd one applcable to elatc element []. The model of ytem where pave element are mplemented a damper proportonal to the nertal element hown n Fg. 3 and the model wth pave element proportonal to the elatc element hown n Fg. 4. b = h (4) m b value of the dampng element, m value of the nertal element calculated a a reult of the ynthe proce, h = dem contant of proportonalty. The value for the h contant elected from the range (5): 0< h < ω mn (5) ω lowet value of the reonance frequency dfferent from zero. mn b b b k c c m m F( t) F( t) Fg. 3. The model for a dcrete mechancal ytem wth pave element proportonal to nertal element b = λ (6) c c value of elatc element calculated a a reult of the ynthe proce, λ = dem contant of proportonalty. The value for the contant of proportonalty λ hould be elected from the range (7): VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

4 78. MECHANICAL AND ELECTRICAL ELEMENTS IN REDUCTION OF VIBRATIONS. 0 < λ< (7) ω n ω hghet value of the reonance frequency dfferent from zero. n c c b m m b F( t) F( t) b k Fg. 4. The model for a dcrete mechancal ytem wth pave element proportonal to elatc element Actve element n reducton of vbraton Applcaton of damper for decreae of vbraton make t poble to reduce dplacement of nertal element but complete elmnaton of uch dplacement not alway poble. To reduce dplacement of ndvdual nertal element of the ytem caued by the effect of dynamc and /or knematc exctaton one can apply actve element that are repreented by force actng from the outde (Fg. 5). Reolvng of the equaton ytem wrtten n the matrx form (8) lead to calculaton of value for ndvdual actve element that act onto pecfc nertal element: G = D A F (8) G matrx of exctaton generated by actve element, D matrx of dynamc tffne, A matrx of ampltude (approachng zero), F matrx of dynamc and/or exctaton. c c G m m G G k F( t) F( t) Fg. 5. The model of a dcrete mechancal ytem wth actve element Another mportant ue the way n whch the actve exctaton are mplemented. The one poble way to mplement uch exctaton applcaton of olely mechancal element n the form of knematc exctaton (Fg. 6) or electrcal element (Fg. 7). In cae of ung knematc exctaton, the only element that may change ther value are dplacement y whlt the elatc element c reman unaltered. To determne value for elatc element and ther dplacement one ha to beneft from the relatonhp (9): 6 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

5 78. MECHANICAL AND ELECTRICAL ELEMENTS IN REDUCTION OF VIBRATIONS. K c y = (9) K value of knematc exctaton, equvalent to the value of G, y dplacement that occur n the pecfnematc equaton, c value of the elatc element that occur n the pecfnematc equaton. Fgure 6 preent a model of the ytem that ncorporate knematc exctaton. c c K m m K K k F t ( ) F t ( ) Fg. 6. A model of the ytem wth knematc exctaton There are alo other oluton for actve component, uch a electrc or electromechancal applance. The force actng on a conductor kept n a magnetc feld gven by (0) [6]: F= BIL (0) F = force actng on the conductor, B = magnetc flux denty, I = current n the conductor, L = length of the conductor. Fgure 7 preent a model of the ytem wth electrc element a exctaton generated by actve element. Concluon Th paper decrbe degn example of ytem wth actve vbraton-reducng element. The ytem tructure and parameter were obtaned ung the method of decompong charactertc functon nto contnued fracton. One of the mportant ue condered n th paper how to create the actve element. Th paper cover actve ubaemble n the form of mechancal element ubjected to knematc exctaton and electrc element. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

6 78. MECHANICAL AND ELECTRICAL ELEMENTS IN REDUCTION OF VIBRATIONS. c c E m m E E k F t ( ) F t ( ) U I F Fg. 7. A model of the ytem wth electrc element Acknowledgement Th work ha been conducted a a part of the Reearch Project No. N N upported by Polh Mntry of Scentfc Reearch and Hgher Educaton n Reference [] Bała K., Buchacz A., Dztkowk T. Synthe of Vbratng Actve Mechancal Sytem wth Dumpng n Vew of Polar Graph and Structural Number. Monograph 30, Slean Unverty of Technology Pre, Glwce, 009, (n Polh). [] Buchacz A., Żurek K. Revere Tak of Actve Mechancal Sytem Depcted n Form of Graph and Structural Number. Monograph 8, Slean Unverty of Technology Pre, Glwce, 005, (n Polh). [3] Engel Z., Kowal J. Vbro-Accoutc Procee Control. AGH Pre, Kraków, 995, (n Polh). [4] Gudaa D., Nlvett F., Pappalardo C. M. Parameter dentfcaton of a full-car model for actve upenon degn. Journal of Achevement n Materal and Manufacturng Engneerng, Volume 40, Iue, Glwce, 00, p [5] Mchałowk S. Actve Sytem n Machne Contructon. Cracow Unverty of Technology Pre, Monograph 7, Cracow, 994, (n Polh). [6] Onwubolu G. Mechatronc: Prncple and Applcaton. Elever Butterworth-Henemann, Oxford, 005. [7] Śwtońk E., Mężyk A., Klen W. Applcaton of mart materal n vbraton control ytem. Journal of Achevement n Materal and Manufacturng Engneerng, Volume 4, Iue, Glwce, 00, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN