The kinematic links motion is influenced mostly by the clearance size from the joint and the number of joints with clearance. The presented paper studies a particular kinematic chain as physical pendulum with clearance

Jan-Cristian GRIGORE

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  Fiabilitate si Durabilitate - Fiability & Durability    No 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     3 NUMERICAL CALCULUS REGARDING THE MOTION BEHAVIOUR ON THE KINEMATIC OPENED CHAIN WITH KINEMATIC LINKS WITH CLEARANCE  PhD. lecturer eng., Jan-Cristian GRIGORE, University of Piteşti, str.Tîrgu dinVale Nr.1   Abstract:  The kinematic links motion is influenced mostly by the clearance size from the joint and the number of joints with clearance. The presented paper studies a particular kinematic chain as physical pendulum with clearance.    Keywords: pendulum, angular, shocks, kinematic  1.INTRODUCTION  The  joints  from  the  kinematic  chain  structure  are  built  with  clearance,  necessary  from technological point of view. Without this clearance they are no relative motion between the kinematic chains elements. To find the kinematic parameters have to find the motion law of the  kinematic  chain,  which  in  the  most  cases  are  difficult  to  obtain  and  to  solve  this numerically, its need to use simplifying hypothesis. The most used numerical method to find the kinematics parameters and the impact force is Runge-Kutta, considering known the initial motion conditions and the kinematic chain characteristics.   2.THE PHYSICAL PENDULUM WITH BIG CLEARANCE WITHOUT FRICTION  Making the following notations can obtain the physical pendulum particularising the general model [1]:  0 ; ; 0 ;     y x A A F mg F y l x   (1) and on obtains the relations:  0 ; 0 ;     Q Q mg Q y x   (2) And the system   C CyxCCJBmBmBJQmQmQYX321    which is presented in the paper [1], becames CCCJBmBmBgYX32100     (3) The following initial conditions: 00 0 ; ; ; 0       C C C C Y Y X X t ;  00 0 ; ;        Cy C Cx C V Y V X   (4)   Fiabilitate si Durabilitate - Fiability & Durability    No 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     4 must to verify the restrictions means:           0 cos sin sin cossin cos00000000 0002 200 200               C C Cy C Cx CC CY X l V l Y V l Xr l Y l X  (5) Adding to restrictions presented above due to equality    sin sin l Y r C      (6) at the initial moment it exists the relations:   rl YC 000sinsin   (7) 00 00coscos     rl VCY    (8) If  on  analyse  the  cases  of  pendulum  with  big  clearance  (fig  1)  which  are  treated  by subchapters  (4.3)  from  [6],  for  the  small  clearance  then  on  obtains  the  following  initial conditions: 0 ; 0 ; ; 0       C C Y r l X t ;  0 0 ; ; 0              l r Y X C C   (9)   Fig. 1  For  the  pendulum  with  parameters  9 , 6 , 1    B kg m m l   is  analyzed  the  following  numerical  case  0 0 87,      lg  with  many  values  of  clearances  m r m r m r 1 . 0 02 . 0 ; 002 . 0     with diagrams drawings in figures 2 a), b), c), d); fig.3 a), b), c), d); fig.4 a), b), c), d).   Fiabilitate si Durabilitate - Fiability & Durability    No 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     5 0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.5timptetateta(t) 0 0.5 1 1.5 2 2.5 3-0.8-0.6-0.4-0.200.20.40.60.8timpalfaalfa(timp) a). Angular theta  versus time  b). Angular  versus time  -1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.200.20.40.60.8tetaalfaalfa(teta)  0 0.5 1 1.5 2 2.5 32030405060708090100110120timpNN(timp) c). Angular  versus angular  d). Reaction force N versus time Fig.2  0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.5timptetateta(t)  0 0.5 1 1.5 2 2.5 3-1-0.8-0.6-0.4-0.200.20.40.60.81timpalfaalfa(timp) a). Angular theta  versus time  b). Angular  versus time    Fiabilitate si Durabilitate - Fiability & Durability    No 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     6 -1.5 -1 -0.5 0 0.5 1 1.5-1-0.8-0.6-0.4-0.200.20.40.60.81tetaalfaalfa(teta)  0 0.5 1 1.5 2 2.5 320406080100120140timpNN(timp) c). Angular  versus angular  d). Reaction force N versus time Fig.3 0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.5timptetateta(t)  0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.5timpalfaalfa(timp) a). Angular theta  versus time  b). Angular  versus time  -1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5tetaalfaalfa(teta) 0 0.5 1 1.5 2 2.5 3050100150200250300timpNN(timp) c). Angular  versus angular  d). Reaction force N versus time Fig.4  3.CONCLUSIONS On observes that for the small clearances as  m r m r m r 1 . 0 02 . 0 ; 0001 . 0     the results are almost similar with those of the cases without clearances. In the case of big clearances m r 1 . 0   the results differ essentially from those obtained to the cases  of  pendulum  without  clearances.  On  observes  that  the  normal  reaction  goes  to  the negatives values, so lack of contact in some points are possible (shocks).     Fiabilitate si Durabilitate - Fiability & Durability    No 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     7  4.ACKNOWLEDGEMENTS  This work was supported by CNCSIS – UEFISCDI, project number PN-II-RU 683/2010   REFERENCES  [1] GRIGORE, J.-C., STĂNESCU, N.–D, Mathematical Models Of Motion’s Behaviour With And Without Friction In Plan For The Kinematical Bearings, The Annual Journal of the Faculty of Technical Sciences-MACHINE DESIGN 2011, ISSN 1821-1259; [2]AMIRONACHE, F., Fundamentals of multibody Dynamics. Theory and Applications,  Birkhäuser, Boston, Basel, Berlin, 2006; [3] Pandrea, N., Elements of the Solid’s Mechanics in Plückerian Coordinates, The Publishing House of the Romanian Academy, Bucharest, 2000; [4]STĂNESCU,  N.–D.,  MUNTEANU,  L.,  CHIROIU,  V.,  PANDREA,  N.,  Dynamical systems.  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[10]ERKAYA, S., UZMAY, I., Investigation on effect of joint clearance on dynamics of  four-bar mechanism, Nonlinear Dyn (2009) 58: 179–198,  [11]PANDREA, N., STĂNESCU, N.-D., OGARU, S., Multibody Dynamics, The Publishing House of the Romanian Academy, Bucharest, 2011 (in press). [12]STĂNESCU, N.-D., PANDREA, N., Theoretical Mechanics, The Publishing House of  the Romanian Academy, Bucharest, 2011 (in press). [13]. FLORES, P., A parametric study on the dynamic response of planar multibody systems  with multiple clearance joints, Nonlinear Dyn (2010) 61: 633–653. [14]FLORES, P., KOSHY, C.,S., AMBRÓSIO ,J., CLAHRO, J.C.P., LANKARANI M., Numerical and experimental investigation on multibody systems with revolute clearance  joints, Nonlinear Dyn  (in press) [15]GARCIA ORDEN, J.C., Analysis of Joint Clearances in Multibody Systems, Multibody System Dynamics (2005) 13: 401–420. 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NUMERICAL CALCULUS REGARDING THE MOTION BEHAVIOUR

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