Static and Modal Analysis and Optimization of Cross- Sectional Area of the Manipulator Arm

Transcription

1 Amercan Journal of Mechancal Engneerng, 2013, Vol. 1, No. 7, Avalable onlne at Scence and Educaton Publshng DOI: /ajme Statc and Modal Analyss and Optmzaton of Cross- Sectonal Area of the Manpulator Arm Ján Vavro jr., Ján Vavro * Unversty of Alexander Dubček n renčín, Faculty of Industres echnologes, Púchov, Slovaka *Correspondng author: jan.vavro@fpt.tnun.sk Receved November 01, 2013; Revsed November 11, 2013; Accepted November 25, 2013 Abstract he gven paper s focused on statc analyss, modal analyss and optmzaton of the cross-sectonal area of the loaded manpulator arm wth the help of the computatonal modellng based on fnte element method wth utlzaton of Pro-Engneer and Cosmos M programs. From the aspect of desgned object, the desgner's man task s to ncorporate such parameters whch wll be n harmony wth the requrements of ts practcal applcaton and whch wll also nclude ts mass, shape, geometry, and some dynamc propertes. he man am of ths paper s to enhance or optmze n-servce part of machne whle t s necessary to save the processed materal and t s also mportant to fnd the best solutons not only n terms of the machne operaton n the producton process, but also n terms of an adequate materal utlzaton and n terms of a sutable shape of the desgned structure. Keywords: statc analyss, modal analyss, optmzaton weght, fnte element method, Cosmos M, Pro/Engneer, expermental measurement, stran gauge Cte hs Artcle: Ján Vavro jr., and Ján Vavro, Statc and Modal Analyss and Optmzaton of Cross- Sectonal Area of the Manpulator Arm. Amercan Journal of Mechancal Engneerng 1, no. 7 (2013): do: /ajme Introducton Nowadays, the enhancement of the producton process, safety condtons at, cost effectveness, elmnaton of materal waste as well as relablty are necessary to be consdered because the optmsaton and safety durng the whole manufacturng process s closely connected wth requrements where all of these mentoned factors has to be taken n account. he ncreasng requrements n relaton to the mprovement of the workng approaches procedures can be observed n all felds of ndustres ncludng rubber ndustry. From the aspect of safety and n-servce effectveness, the attenton has to be pad to the devces and ther operatng parts and therefore, the man objectve of the gven paper s focused on dynamc and statc analyss as well as optmzaton of condtons for the cross-sectonal area of the manpulator K1 K5 (Fgure 1, 2) n the order to keep the requrements relatng to safety and effectveness. In relaton to the rubber ndustry, the gven devce s used as modfyng mould durng the vulcansaton procedure relatng to tyre manufacturng. he analyses and optmsaton was performed for manpulator arm LA 425 (Fgure 2) and t was done wth reference to operatng loadng. Manpulator was desgned by the help of the Pro/Engneer system. he geometry of the gven manpulator was transformed by the Igs fle to program for creaton of fnte element model. Fgure 1. Manpulator K1-K5 Fgure 2. Arm of manpulator 2. he Calculaton of Loadng for Manpulator Arm Based on the Statc Loadng Calculaton of arm LA 425 as a part of manpulator K1 - K5 was made on the bass of statc loadng [N] for the hghest extenson of the arm where the loadng s the hghest. Usng the volume elements, the fnte element model (Fgure 3) was made for manpulator arm LA 425 (Fgure 2).

2 Amercan Journal of Mechancal Engneerng he Calculaton of the Egenfrequences and Egenshapes for Manpulator arm Fgure 3. Fnte element model of the manpulator arm LA425 Arm s made of materal Stablty equatons are solved n ths way [1]: [ K]{ u} = { R} (1) Statc analyss of the manpulator arm ncludes soluton of the stablty equatons (1). System of equaton s solved by Gaussan method of the elmnaton. hs method ncludes postve defnte symetrc system of equaton. he algorthm performs mnmal operaton number. here are not performed L DL operatons wth zero elements. After the calculaton of the nodal pont dsplacements, calculaton of the loadng of elements s made. he relaton between deformaton vector { ε } and dsplacement are expressed by equaton: { ε } = [ B]{ u} (2) where [B] s transformaton of deformaton matrx of dsplacement. Loadng vector { σ } represents a conjuncton of the transformaton deformaton - matrx loadng [D ] wth the deformaton vector { ε } { σ} = [ D]{ ε} (3) Usage of (2) n (3), the relaton between loadng vector { σ} and vector of nodal pont dsplacements { u } can be obtaned { σ } = [ D][ B]{ u} (4) Dstrbuton of the stresses n arm of manpulator s gven n [MPa] and t can be seen n the Fgure 4 and detal of the dstrbuton of stresses s shown n the Fgure 5. Mathematcal model can be expressed n form (5) [ M] { u } + [ C] { u } + [ K] { u } = { f( t) } (5) where M s the mass matrx, C s the dampng matrx, f t s the tme varyng K s the stffness matrx, { ( )} loadng vector and { u }, { u }, { } u are the dsplacement, velocty and acceleraton vectors, respectvely. In relaton to the calculaton of the arm LA 425 of manpulator K1 - K5, the statc loadng was kg. he computatonal model (Fgure 2) representng the manpulator arm LA 425 (Fgure 3) was created by help of the eghtnodal volume elements. Modal and frequency analyss were done accordng to known equaton n the form: ( - 2 [ M] [ K] ) { } { 0 } ω + Φ = (6) where Φ s the egenvector representng the nodal shape of the -th egenfrequency. We calculated egenfrequences and from the egenfrequences and on the bass of the egenfrequences, the frequency characterstcs wth the modal dampng ξ = 0,001 was calculated he moton equaton for the -th egenshape s n accordance wth: ω s the -th egenfrequency, { } x 2 + 2ξ ω x + ω x = { Φ} { f } (7) hs relatonshp s generalzed problem of the egenvalues where soluton s made by the semautomatc-space teratve method. hs method s based on the dea of the nverse teraton converson wth several vectors at the same tme. In the Fgure 6, we can see the frst three shapes of the egenfrequences and n able 1, there are the frst 10 egenfrequences calculated by the help of the underspace method. NUM. able 1. he frst ten egenfrequences (RAD/SEC) (CYC./SEC) E E E E+03 Fgure 4. Dstrbuton of stresses n [MPa] E E E E E E E E E E E E E E E E+04 Fgure 5. Detal of dstrbuton of stresses n [MPa]

3 238 Amercan Journal of Mechancal Engneerng Fgure 6. he frst three shapes of the egenfrequences 4. Optmzaton of Cross-sectonal Area of the Manpulator Arm In relaton to the materal savng, long servce lfe, relable and safe operaton of mechancal equpment, nowadays strct requrements call for new approaches n puttng varous techncal deas nto practce. It can be fulflled only wth the usage of sutable computng systems for fast and effcent study of the parameters whch nfluence the statc or dynamc characterstcs of a machne n a greater or a lesser extent General Formulaton of the Optmzaton Problem In the frst step, t s necessary to create correct physcal machne model (statc or dynamc). Second step s focused on the nvestgaton and model descrpton by the proper mathematc means creaton of the computatonal model (equatons system, couplngs condtons, etc.) he thrd and often most dffcult procedure of optmum constructon desgnng s connected wth mathematcal formulaton of the optmzaton problem. It can be formulated as lookng for extreme condtons (t s mnmum value of extreme condtons) of the objectve functon: ({ }) z = F X (8) Furthermore, t s based on addtonal (lmtng) condtons whch are gven by rregulartes whch determnate admssble area: ({ }) { 0} where { } [,... ] Gj X (9) X = X1 X2 X n s vector of desgnng (desgn) varables. Equatons (8) and (9) contan the X. hey belong to problems of mathematc programmng whch s often based on non-lnearty. For research of global mnmum, Kuhn-ucker oblgatory condtons were used: general non-lnear functons of desgnng parameters { } m 1 F X + G X + Λ H X = j= 1 k= 1 ({ }) λ j j ({ }) m+ k k ({ }) { 0} (10) Mathematc programmng belongs to drect methods of optmzaton problems soluton. In consequence of general non-lnear character of objectve functon and lmtaton functons, equatons (8), (9), (10), there wll be nonlnearty. In majorty cases, ths state can not be solved analytcally. hen t s necessary to apply teratve methods, whch allow computng vectors chan { X } r (r = 1, 2...) whch converge to optmum { X } opt. Drect equaton applcaton (10) n optmzaton process mpacts on the problems wth the lmtatons n non-equatons form and therefore, there are other methods of non-lnear programmng used. he most common applcaton n optmzaton of problems relates to methods of zero place value (comparator) and methods of frst place value (gradent) but the second one mentoned methods are more effectve. her general dsadvantage s that there s the problem to perform the analyss of senstvty. Fundamentally, t deals wth varous modfed teratve methods of Newton type. General dea of these teratve approaches can be expressed by equaton: { X} K+ 1 = { X} K α F({ X} K) (11) Just lke n the equaton (10), the operator expresses gradent of objectve functon n the k-th teratve step for drecton of progress optmum fndng. It has been shown n practce that drect optmzng methods nvolve rather demandng computatons whch can be qute complcated from the techncal aspect. However, there are also ndrect methods that do not operate drectly wth the target functon and they are used only for evaluaton of the lmtatons (usually stresses). One of such methods s the Fully stress desgn, and we wll utlze ts basc prncples and procedures herenafter. he method s based on the dea of desgnng the structural varables n such a manner that the stresses n the entre structure wll be at the permssble lmt he method of Fully Stress Desgn hs method s used to fnd an optmum soluton utlzng the overall loadng capacty of the materal at the

4 Amercan Journal of Mechancal Engneerng 239 gven loadng. he lmtng condtons for the stresses may be wrtten as follows: σ σperm 0 (12) where σ sthe stress for the gven pont and σ perm s the maxmum permtted stress. As an example, an optmum desgn of beam structures wll be used wth respect to ts sectonal characterstcs Optmzaton of Sectonal Parameters by Beam Elements We consder duo node beam fnte element wth constant cross-secton. In relaton to ths element, there are slde forces, axal forces, torque constant and elastc moments changng n a lnear way between both endng nodal ponts. Maxmum stress wll be for one or other end of the beam, therefore ponts of calculated stresses wll be there. Except ths fact, we have defne also poston of ponts, for whch we compute stresses n cross-sectonal area. Fnal stresses wll be calculated accordng to equatons: - axal compound of normal stress: F M x y M. z σ x = Cz +. Cy (13) A Jy Jz - slde stress from slde forces: - slde stress from torque: Fy. Sy 1 = A Fz. Sz 2 = A ( Mx Fz ey Fy ez) (14) r 3 = + (15) Jk where F x, F y, F z are compounds of nternal force n gven cross-secton, M x, M y, M z are nternal moments, A s cross-sectonal surface, J y and J z are axal quadratc moments of persstence, J k s cross-sectonal torque moment, S y and S z are statc moments of cross-sectonal part dedcated by pont area where we calculate the stress, c y and c z are pont dstances n secton plane, n whch we calculate stresses from cross-sectonal centre of gravty, e y and e z are coordnates of eccentrcty (dstance between centre of gravty and slde centre), r s effectve radus of torson. Equvalent stress s calculated accordng to HMH hypothess (von Msses): 2 (( 1 2) 3 ) 2 2 ekv = x (16) σ σ Computatonal ponts for beam cross-secton have to be defned for the stes where the extreme values can be expected. Furthermore, we have to defne stffness matrx of three-dmensonal beam fnte element and calculaton of nternal force parameters, from whch the stresses n endpont nodes wll be assgned accordng to equatons (13), (14), (15). If axs y, z wll be central, then S y = 0, S z = 0, D yz = 0 and we obtan known, n lterature very often publshed, stffness matrx of beam element. here are many approaches n optmzaton of beam cross-secton. One of the approaches can be optmzaton of total heght, wdth or both cross-sectonal varables. he optmzaton modulus n Cosmos program can be used for ths work Optmzaton of Cross-sectonal Area of the Manpulator Arm Optmzaton of the manpulator arm was done n program COSMOS 2.85 and the optmum desgnng of the model (Fgure 2) led to optmzaton of the whole area of beam arm. he cross-sectonal area of arm for ndvdual beam elements was changed wth element wdth of arm and element heght was constant. Stress lmtaton value was specfed at σ dov = 150 [MPa]. he arm was loaded by own weght and maxmal (standard) force N at one and second sde of arm. Arm weght before optmzaton was 48.9 kg.. Stress dstrbuton n manpulator arm before optmzaton s n the Fgure 4. Stress dstrbuton n manpulator arm after optmzaton s at Fgure 7. Fgure 7. Stress dstrbuton after optmzaton n [MPa] Optmzaton results for ndvdual teratve steps are n the Fgure 8. Weght [kg] Convergence graph of manpulator arm objectve functon Iteratve steps Fgure 8. Convergence graph objectve functon Weght[kg] As we can see from Fgure 8, mnmal arm weght was reached n the tenth teratve step. We reduced arm weght by 3,975 kg. 5. Concluson In relaton to total manpulator arm loadng, there s the nfluence of the thrd shape of egenfrequency on manpulator arm vbraton whch causes vbraton n the vertcal drecton.. he frst and the second egenshape represent vbraton n horzontal drecton and ths vbraton does not have any nfluence on operaton of manpulator. he Fnte Element Method enables to solve

5 240 Amercan Journal of Mechancal Engneerng many complex practcal tasks that, n the past, were almost nsolvable. It cannot be generally concluded that some partcular method s the best and that t always leads to fndng the optmal soluton. Each problem s pecular n ts own way due to ts physcal or mathematcal model or due to the procedure used n the fndng of the best soluton. herefore, durng the lookng for the effectve procedure for the optmzaton of mechancally stressed structures, the followng prncples should be taken nto account: - specal approach to the problem under the soluton (statc or dynamc analyss, lnear or non-lnear behavour of the structure), - the way of stress and the number of loadng modes, - the number of desgned varables and ther character, - expected locaton of the optmum wth regard to the lmtatons. It s necessary to pont out that desgners cannot consder the statc optmzng analyss as the fnal desgn before the dynamc analyss s carred out. he mportant pont here s that the desgner should take n account the nfluence of the changes of rgdty and mass of the structure upon ts dynamc characterstcs. Acknowledgement he work presented n ths paper was supported by VEGA grant No. 1/0530/11. References [1] Azar, J., J.: Matrx Structural Analyss,Pergamon Press, New York, [2] Frankovský, P., Pástor, M.,Kenderová, M.: Automatzaton of Stran and Stress Felds Assessment Based on the Prncples of reflecton Photoelastcty.In: EAN 2012 Proceedngs of the 50 th annual conference on expermental stress analyss. Prague, [3] Frankovský, P., Ostertag, O., Ostertagová, E.: Automaton of experments n photoelastcty. In:Proceda Engneerng, 2012, No. 48, p ISSN [4] Frankovský, P., Kostelníková, A., Šarga, P.: he use of strengage method and PhotoStress method n determnng resdual stresses of steel console. In: Metalurga. Vol. 49, No 2 (2010), p , ISSN [5] Bocko, J., rebuňa, F., Pástor, M., Frankovský, P.,: Applcaton of Le groups n the study of beam and plate vbratons.in: EAN 2010,48 th Internatonal scentfc conference,velké Losny, Czech Republc, 2010, p.19-26, ISBN