TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW. NUMERICAL INVESTIGATIONS OF THE NATURAL FREQUENCIES AND MODE SHAPES

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1 TRIESKOVÉ A BEZTRIESKOVÉ OBRÁBANIE DREVA, 8(): 7 78, Zvolen, Tehniká univezita vo Zvolene, ISBN TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW NUMERICAL INVESTIGATIONS OF THE NATURAL FREQUENCIES AND MODE SHAPES Geogi Vukov Zhivko Gohev Valentin Slavov Abstat A numeial investigation of the natual feuenies and mode shapes of the iula s saw unit is pesented in this study The eseah is done on the base of an adeuate mehani-mathematial model fo investigation of fee tosional vibations of a iula saw developed by the authos The model pesents featues in the onstution of a kind of iula saws As a esult this study allows the detemination of the esonant wok egimes The detemination of these egimes is impotant fo intodution of adeuate measues whih an guaantee thei using The esults of the investigation an be used as a base fo making some eommendations onening the inease of eliability of the mahine as well as the auay and uality of the podution Key wods: Ciula saws, modeling, tosional vibations INTRODUCTION The deease of the level of the vibations and the noise duing the wok of moden iula mahines is one of the main poblems, whih is imposed of high euiements to the paametes of the tehnial euipment in the wood industy It is neessay to study the essene of poesses that ae typial fo the mahine and its elements to find out easons fo the appeaane and inease of the vibations and the noise It is also neessay to make some investigations whih onside the mahine as a mehanial vibation system with some haateistis [6], [] The estition of the vibations and the noise level demands to be fomulated onete ways and methods fo influene on the vibation system It leads to intodution of auate euiements onneted with the onstution and the wok of its elements Theefoe, fomulation and analysis of the euations whih desibe the vibations of the elements of the woodwoking mahines ae vey impotant These euations an be used as a base fo investigations and giving some eommendations to the onstution and the way of its wok [4] The kind of iula mahines whih ae wide used in patie ae investigated in this study [5], [] Figue shows a iula mahine and a sheme of this iula mahine Univesity of Foesty, Sofia givukov@ltubg, zhivkog@yahooom Univesity of Chemial Tehnology and Metallugy, Sofia valslav@abvbg

2 7 TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW The eleti moto is pesented by, is the belt gea, the wok table, 4 the main shaft, 5 the mahine s body, 6 the aiage, 7 the teated detail, 8 the iula saw with the flanges and the nut of the main shaft Fig Ciula Mahine Eleti Moto, Belt Gea, Wok Table, 4 Main Shaft, 5 Mahine s Body, 6 Caiage, 7 Teated Detail, 8 Ciula Saw Figue shows the unit saw of the iula mahine whih is investigated in this study 8 Fig Unit saw of the iula mahine As evey mehanial vibation system, the vibation haateistis of the iula saw have thei natual vibation feuenies When the feuenies of the extenal influenes whih ause vibations ae eual to a feueny of thei natual feuenies appeas the phenomenon esonane Resonane egimes an lead to signifiant inease of vibation amplitudes Signifiant vibation amplitudes hange the nomal wok egimes of the mahine and damage the auay and uality of the podution [7], [] Exta stess, whih is aused by inease of vibation amplitudes, sometimes an eah suh values that an damage o even destoy some mahine s elements [8] Pinipally esonant effets ae unwilling They an be avoided by a suitable seletion of paametes of the iula saw and othe details and units of the mahine, as well as the wok egimes [], [9] It is neessay to make in advane an evaluation of the esonant dange when the iula mahine is designed and dimensioned It does not allow dange wok egimes duing the opeation exploitation To solve this poblem it is a must to study

3 GEORGI VUKOV ZHIVKO GOCHEV VALENTIN SLAVOV 7 the natual feuenies The hanges in the onstution o in the wok egimes ae advisable if the esonant dange is available The aim of this study is to make a numeial investigation of the natual feuenies and mode shapes of the iula s saw unit The investigation is done on the base of an adeuate mehani-mathematial model fo investigation of fee tosional vibations of a iula saw developed by the authos [] The model pesents featues in the onstution of a kind of iula saws Some eommendations onening the pevention fom the esonant wok egimes an be made on the base of this study It is onneted with the inease of eliability of the mahine as well as with the auay and uality of the podution Mehani-Mathematial Model An oiginal mehani-mathematial model fo investigation of the dynamial poesses and vibations in the saw unit of a kind of iula saws is built It is shown on the fig This model inludes fou disete mass onneted with thee massless elasti elements φ i, i =,,, 4 ae the angles of the otation of the oesponding oto The elastiity oeffiients of the eleti moto s shaft, the belt and the main shaft ae taken into aount The elastiity angula oeffiient of the eleti moto s shaft is maked with, and this one of the main shaft with (Nm/ad) The elastiity linea oeffiients of the two pats of the belt between the belt pulle ae с and с (N/m) l 4 φ 4 J 4 d φ φ J φ J d J Fig Dynamial model l The edued mass inetia moments (kg m ) ende in aount: J the mass inetia moment of the eleti moto s oto; J the mass inetia moment of the belt pulle on the eleti moto s shaft; J the mass inetia moment of the belt pulle on the main shaft; J 4 the mass inetia moment of the iula saw Hee ae some symbols: d, d diametes of the eleti moto s shaft and main shaft (m); l, l omputing length of the eleti moto s shaft and main shaft (m);, adius of the belt pulles on the eleti moto s shaft and main shaft (m); G modulus of sheaing

4 74 TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW The investigation of the vibations of the iula s unit saw euies fomulation and solution of the diffeential euations whih desibe these poesses Theefoe, it is used the matix mehanis [], [] The mehani-mathematial model is done by using the applied enginee pogam (Mathematia) It is developed an algoithm fo fomulation of the matixes whih desibe the mass-inetial popeties and the elasti popeties of the mehanial system The diffeential euations whih desibe the fee vibations ae dedued by using the Lagange s method d dt T T L, () whee i ae the genealized oodinates, T and L ae espetively the kineti and the potential enegy of the multibody systems The veto of the genealized oodinates is 4 T () The kineti enegy of the mehanial system is obtained as a sum of the kineti enegy of the fou basi bodies (the eleti moto s oto, the belt pulle on the eleti moto s shaft, the belt pulle on the main shaft, iula saw) T J J J J 4 4 () The potential enegy of the mehanial system is obtained as a sum of the potential enegies eeived fom the defomations of the eleti moto s shaft, the belt and the main shaft L (4) The system of paameti diffeential euations whih desibe the fee tosional vibations of the iula s saw unit ae M C (5) 4 is The matix, whih haateizes the mass-inetial popeties of the mehanial system,

5 GEORGI VUKOV ZHIVKO GOCHEV VALENTIN SLAVOV 75,5,,8,8,, 4 J J J J T a a j i ij ij M M (6) The matix, whih haateizes the elasti popeties of the mehanial system, is 9,6 9,6 9,6 94, ,7 4,7 4,7 4,7,, L j i ij ij C C (7) Patiula solutions to the system of the diffeential euations (5) ae seahed as: t h sin, (8) whee h is the amplitude of the small vibation on the genealized oodinate with natual feueny, and is the initial phase Afte diffeentiation of (8) and substituting in (5) it obtains a system of linea algebai euations In the matix desiption they ae V M C (9) To detemine the natual feuenies and the mode shapes, it is neessay to solve the task about finding the natual values and the natual vetos of the euations (9) The satisfation of the euations (9) euies the following M C det ()

6 76 TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW The oots of the haateistis euation detemine the natual feuenies The natual feuenies fom the matix of the natual values They ae, i, 4 diag,,, () The natual feuenies [Hz] ae deteminated by () ω, f Hz () The natual values of the system () detemine the natual vetos of the mehanial system A natual veto v, whih gives oelation between amplitudes of the vibations, oesponds to evey natual feueny The veto s omponents define the matix of the natual vetos (modal matix) of the system (9) that is whee v j 4 4 V,, () v v, v, v, v, 4, is the natual mode veto on the genealized oodinate fo -th natual feueny RESULTS The elements of the iula s saw unit ae modeled by using the applied enginee pogam Solid Woks [4] The mass, elasti and geometial haateistis ae shown in the table J inetia moment of the eleti moto s oto (kg m ), 8 J inetia moment of the belt pulle (kg m ), 8 J inetia moment of the belt pulle (kg m ), J 4 inetia moment of the iula saw (kg m ), 5 stiffness of the eleti moto s shaft (Nm/ad) 4, 7 stiffness of the main shaft (Nm/ad) 9,6 stiffness of the belt (N/m) 5 5 stiffness of the belt (N/m) 5 5 d diamete of the eleti moto s shaft (m),7 d diamete of the main shaft (m),7 adius of the belt pulle (m),6 adius of the belt pulle (m),9 l distane between the belt pulle and the eleti moto (m),5 l distane between the iula saw and the belt pulle (m),75 Table

7 GEORGI VUKOV ZHIVKO GOCHEV VALENTIN SLAVOV 77 The alulations ae done with help of the applied enginee pogam Mathematia [5] The natual feuenies [s ] (and in [min ]) ae 55699,; 746,; 556,54; ; ( 5884; 657; 475; ) The alulated natual feuenies [Hz] and mode shapes of the mehanism s tosional vibations ae illustated on fig 4 mode shapes,6,,,7,7,5,5,99, DOF's,6,,4,, 4,6 feuenes Hz 79,6 Hz 455,95 Hz 8864,74 Hz 8864,74 Hz 455,95 Hz 79,6 Hz Hz Fig4 Mode shapes CONCLUSION A numeial investigation of the natual feuenies and mode shapes of the iula s saw unit is pesented in this study The investigation is done on the base of an adeuate mehani-mathematial model fo investigation of fee tosional vibations of a iula saw developed by the authos The model pesents featues in the onstution of a kind of iula saws The esults of this study allow making some eommendations to avoid the esonant wok egimes Thus the esults an be used as a base fo inease of eliability of the mahine as well as the auay and uality of the podution REFERENCES Amiouhe, F, Fundamentals of Multibody Dynamis Theoy and Appliations, Bikhäuse, Boston, 6

8 78 TORSIONAL VIBRATIONS IN THE SAW UNIT OF A KIND OF CIRCULAR SAW Angelov, I, Matix mehanis - Dynamis, Avangad Pima, Sofia, Angelov, I, Slavov, V R, Book of poblems Matix mehanis - Dynamis, Avangad Pima, Sofia, 4 Coutinho, M, Dynami Simulations of Multibody Systems, Spinge-Velag, New- Yok, 5 Filipov, G, Woodwoking mahines Sofia (977) 6 Geogieva, D, P Dihev Eonomi Effetiveness of Cuent Tehnial Systems fo Monitoing of Euipment Wok in the Logging and Woodwoking Podution XX th Intenational Confeene fo Young Sientists, Univesity of Foesty, Sofia,, pp Mainov, B, Kineti Enegy Loss in Some Classes Ciula Mahines as a Result of the Ciula Saws Unbalane, th National Congess on Theoetial and Applied Mehanis, -5 Sept 9, Boovets, Bulgaia 8 Mainov B, Dynami Stesses in Conseuene of Impulse Impats in the Links of Cetain Classes Band Saw Mahines with a Feeding Meshanism Mehanis of mahines, (96) Vana 9 Mainov, K, Analysis of the motion of ganulifeous patiles in the sew mehanisms Woodwoking and Funitue Manufatuing Univesity of Foesty, Sofia, 8, pp 6 Obeshkov P, Woodwoking mahines Sofia, 996 Veits V, А Kohua, А Matinenko, Dynamial investigations of dive mahines - Mosow: 97 Vukov, G, J Gohev, V Slavov, Tosional Vibations in the Saw Unit of a Kind of Ciula Saws Mehani-Mathematial Model Poeedings Sientifi Papes, Thid Sientifi and Tehnial Confeene Innovation in Foest Industy and Engineeing Design, Sofia,, pp Vukov, G Y, Tosional Vibations of the Diving Mehanism of a Caved Venee Mahine Amplitude-Feueny Spetum Mehanis of mahines (75) Vana, 8, pp 6 4 wwwsolidwoksom 5 wwwwolfamom/mathematia