Substitute Model of Deep-groove Ball Bearings in Numeric Analysis of Complex Constructions Like Manipulators

Transcription

1 Mechanics and Mechanica Engineering Vo. 12, No. 4 (2008) c Technica University of Lodz Substitute Mode of Deep-groove Ba Bearings in Numeric Anaysis of Compex Constructions Like Manipuators Leszek Podsȩdkowski Piotr Zawiasa Technica University of Lódź, Institute of Machine Toos and Production Engineering Stefanowskiego 1/15, Lódź, Poand Received (18 November 2008) Revised (25 November 2008) Accepted (15 December 2008) It is very important, whie modeing a robot, to obtain proper accuracy and repeatabiity of movements thanks to the proper stiffness of the whoe construction. Robot configuration and parameters depend on the constructor s knowedge and experience. CAD FEM programs aow to create optima robot construction as they verify the functionaity and durabiity of the three dimensiona construction. The anaysis of stiffness of robot singe eements under externa oad may be reaized in FEM programs thanks to the eement s inear characteristics. However, a robot needs to treated as an assemby of eements connected in a fexibe way. There are a few ways of modeing the connections of robot eements. That is why in this research a numeric method of composed eements is shown using inear joints with regard to compiance of joints at a point of robot work. Keywords: Deep-groove ba bearing, CAD FEM program 1. Introduction Let us anayze a kinematic scheme of a robot with two degrees of freedom and rotationa joint robot (Tab. 1). Since in robotics a construction with an open kinematic chain happens quite often, every eement of the construction makes a contribution to the fina position of the effectors. Thus, the anaysis of manipuator stiffness shoud incude a eements that have an infuence on manipuator deformations. FEM (Finite Eement Method) programs, open to the pubic due to a fast deveopment of computer technoogy, are abe to define stiffness of an eement. However, joints in compex machines cause frictiona probem, which must be soved in an iterative way because of its non inearity.

2 350 Podsȩdkowski, L. and Zawiasa, P. Figure 1 Infuence of bearing fexibiity on arms deformation Roing bearings are commony used because preoad may be induced in them. Thanks to the preoad, bearing sackness, which is inadmissibe because of its ambiguous position whie steering, is reduced. A mobie joint made by deep groove ba bearings is a specific kind of a joint. These bearings, with radia and axia stiffness, are used in robot constructions because of their sma size. To define the stiffness, a vaue of an contact ange of deep groove ba bearing is necessary. It can be determined basing on Fig. 2. The ange of deep groove ba bearings that takes into account the radia cearance may be determined by the foowing equation according to [1] : ( α o = arccos 1 P ) d (1) 2A A = r b.zew + r b.wew 2r k P d radia cearance of a bearing It can be proved that in deep- groove ba bearings the contact ange is consideraby variabe. For exampe, in 6202 bearing for radia cearance that is contained in norms [2] for the particuar vaues of the ba race radius r b.zew = r b.wew = 1, 04 r k, the ange changes by seven degrees between the owest and the highest vaue of cearance. According to[3], it is possibe to determine the stiffness of roing bearings in axia and radia directions. Due to a fact that manipuators are compex, specia and open, the anaysis of deformations with the friction taken into account is quite difficut. After introducing preoad into bearing system, system inearity is obtained. A mode of a bearing with inear characteristics and stiffness that corresponds to bearing work was proposed. Few three dimensiona modes showing a deep groove ba bearing were created. Its fina version is shown in Fig. 4.

3 Substitute Mode of Deep-groove Ba Bearings Figure 2 Dependence of the contact ange of deep - groove ba bearing on radia cearance Figure 3 The infuence of radia cearance on the contact ange 2. Mathematica mode Foowing assumptions were taken for cacuations: The outer dimensions correspond to dimensions of the bearing The mode consists of one part with two outer rings which are connected with each other by beams Young s moduus E is a constant vaue of bearing stee Data that mode the stiffness are foowing dimensions of a beam: h height, b - width, ength and the number of beams in a mode, are

4 352 Podsȩdkowski, L. and Zawiasa, P. Figure 4 Mode 3D bearings It may be written where: E Young s moduus I inertia moduus Mu fixing moment EI d2 y dx = P ( x) + Mu (2) f Mg(x=) = P 3 12EI (3) Figure 5 Digitizing of a bearing mode This equation for beam deformation takes into account ony bending force. This deformation iustrates the defection for beams with high proportion of ength and height. In the anayzed case the proportion is cose to one. Therefore, shearing forces need to be taken into account. According to [4], tota deformation of the beam is: f(x) = f Mg(x) + f T (x) (4)

5 Substitute Mode of Deep-groove Ba Bearings where: f Mg(x) defection caused by bending moment f T (x) defection caused by shearing moment Therefore: T (x) = dmg(x) dx The differentia equation of f T (x) function is: f dt (x) k (x) = dx GA where : k dimensioness coefficient depending on the shape of rectanguar section k = 1.2 [4] G shear moduus of easticity A section exposed to shearing (5) (6) Figure 6 Scheme of radia force distribution in a mode After doube integrating, an equation for maxima deformation as a consequence of shearing force is obtained: f (x) = P k (7) GA After taking into consideration that G = E 2(1 + ν) and that section of the sheared beam is rectanguar (its coefficient of section shape k = 1.2; section inertia moduus I = bh 3 /12 and Poisson s ratio ν = 0.3) f (x) = P ( ) 2 3 h Ebth (9) (8)

6 354 Podsȩdkowski, L. and Zawiasa, P. the tota deformation of the beam is g f (x=) = P 3 Ebh 3 [ 1 + 3, 12 ( ) ] 2 h (10) Radia force P r acting on the bearing mode may be expressed: P r = P r(i) i = w r E bh 3 [ , 12 ( ) h 2 ] i sin2 (α(i)) + bh i cos2 (α(i)) (11) Axia force Po acting on the bearing, decomposes uniformy on a beams (Fig. 7). Therefore: P o = Zw o E 3 hb 3 Ebh 3 [ , 12 ( ) b 2 ] + hb (12) where: Z the number of beams in a mode w o dispacement of the bearing Figure 7 Scheme of axia force oading of the bearing mode Basing on equations (8)(9) parameters of a beam b and h were determined and program Mode 3D in Visua Basic cooperating with program CAD was created. It uses methods of cacuating the stiffness of bearings according to [3] and generates 3D soid that modes the bearing of a given axia and radia stiffness.

7 Substitute Mode of Deep-groove Ba Bearings Resuts of cacuations To compare the infuence of the way of modeing the deep groove ba bearings on deformations, foowing soid modes used in robot construction were used: 1. ring shaped mode with outer dimensions corresponding to bearing dimensions 2. mode taken from the internet data base [5] consisting of outer and inner rings and nodues; r b.zew = r b.wew = r k. 3. anaytica mode 4. mode generated with Mode 3D program of a given stiffness. An exampe of a shaft ( Fig. 8) supported on two identica deep groove ba bearings, which is oaded with forces F x = 200 N and F y = 1000 N appied on the arm R = 100 mm was anayzed. Figure 8 Scheme of oading of bearing system Anayzing the bearing system with preoad, it may be stated that the system characteristics is quasi inear [3]. That is why, joints between eements eading to inear soution may be used in numeric cacuations. The contact ange was obtained by

8 356 Podsȩdkowski, L. and Zawiasa, P. formua 1, assuming that axia cearance in 6202 bearing was P e = 0, 06 mm, the diameter of a nodue r k = 6 mm and the race radius r b.zew = r b.wew = 1, 04r k. Resuts of deformations are shown in tabet (1) Tabe 1 Infuence of bearing fexibiity on arms deformation. Mode Mode Mode Mode Fixed beam Deformations 0,018 0,019 0,047 0,053 0,022 in X axis [mm] Deformations 0,004 0,005 0,013 0,016 0,008 in Y axis [mm] It is seen that whie modeing a shaft being supported by bearings with modes 1 and 2, the stiffness of a node is very high and actuay takes into account the deformation of the beam itsef as if it were fixed. When mode 4 was used, the whoe bearing node was defected. Mode 4, in contrast to mode 3, takes into consideration overturning moments of the bearing. Therefore, there are differences of defections between the modes. 4. Concusions Thanks to the usage of numerica methods and the Mode 3D program in anayzing the deformations of manipuators it became very easy to quicky generate the modes of deep groove ba bearings of given stiffness and to anayze the infuence of this stiffness on the deformations of a whoe construction. According to the numerica cacuations shown in the tabet, modeing of a bearing is of a great importance as it consideraby infuences the deformations of a whoe construction. References [1] Harris, T.A.: Roing bearing anaysis, New York: John Wiey & Sons, [2] Norma PN ISO 5753: Lożyska toczne Luz promieniowy [3] Krzemiński Freda, H.: Lożyska toczne, Warszawa, PWN, [4] Gawȩcki, A. Mechanika materia ów i konstrukcji prȩtowych. Poznań: WPP, [5] [Onine]