In this paper an algorithm based on [1] [2] are numerical simulations, achieving generalized coordinates of motion, positions, speeds of a rigid rotating kinematic coupling with big clearance in joint, case without frictio

Jan-Cristian GRIGORE

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                                    Fiabilitate si Durabilitate - Fiability & Durability    no 2(6)/ 2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  29 NUMERICAL SIMULATIONS FOR THE CASE OF RIGID ROTATING KINEMATIC COUPLING WITH BIG CLEARANCE  Lecturer dr. eng., Jan-Cristian GRIGORE, University of Piteşti  Abstract: In this paper an algorithm based on [1] [2] are numerical simulations, achieving generalized coordinates of motion, positions, speeds of a rigid rotating kinematic coupling with big clearance in joint, case without friction   Keywords: algorithms, energy, angular velocities.  1.Numerical application  Based on algorithms developed in [1] and [2] will be solved for a revolute motion rectangular table with  kg m 15 =  (fig. 1) and dimentions  m h m b 6 . 0 ; 4 . 0 = =  with big clearance joint  mm L mm r 10 2 ; 2 = = , wherea the moment  m N M ⋅ − = 80 , moments of inertia are z x y z x J J J m kgmbJ m kgmhJ + = ⋅ = = ⋅ = = ; 8 . 03; 45 . 0122222   Fig.1  Potential energy given weight mg is ϕ θ sin cos2⋅ ⋅ ⋅ =bZ mg V C   (1) and so the components are generalized forces, ϕ θϕϕ θθϕ θ cos sin2; sin cos2' ' ⋅ ⋅ − =∂∂− = ⋅ ⋅ − =∂∂− =bmgVQbmgVQ   (2)                                         Fiabilitate si Durabilitate - Fiability & Durability    no 2(6)/ 2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  30 The moments are: []⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅ =⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡0cos sinsin sin00ϕ θϕ θMMRMMMTzyx  (3) components of generalized forces given the moment M is  ϕωϕωϕωθωθωθωψωψωψωϕθψ& & && & && & &∂∂+∂∂+∂∂=∂∂+∂∂+∂∂=∂∂+∂∂+∂∂=zzyyxxzzyyxxzzyyxxM M M QM M M QM M M Q"""  (4) Into account expressions  ϕ θ ψ ω ϕ θ ϕ θ ψ ω ϕ θ ϕ θ ψ ω & & & & & & + = − = + = cos ; sin cos sin ; cos sin sin z y x   (5) can writer: θ ϕ θ ψ cos ; 0 ;" " " M Q Q M Q = = =   (6) and obtain expressions for generalized forces total θ ϕ θϕ θϕ ϕ ϕθ θ θψ ψ ψcos cos sin2sin cos2" '" '" 'Mbmg Q Q Qbmg Q Q QM Q Q Q+ − = + =− = + == + =  (7)   a). If the point O is fixed If the point O is fixed then Lrarctg ct = = θ .  and obtain the equations of motion  ()⎥⎥⎦⎤⎢⎢⎣⎡⋅ ⋅− = ⋅⎥⎥⎦⎤⎢⎢⎣⎡−⋅ − ++ ⎥⎦⎤⎢⎣⎡⋅ ⎥⎦⎤⎢⎣⎡ + +ϕ θ θ ϕ θψϕ ψθψθθ θ ϕ θ ϕ θcos sin2cos 2 sin sin21coscos cos cos sin sin sin222 2 2 2 2g b mMMJ JJ JJ J J Jy xz zz z y x&& && && &  (8) The initial conditions 0 ; / 200 ; 0 ;2; 0 = = = = = ϕ ψ ϕπψ & & s rad t   (9) and using the calculation of annexes are obtained table 1 and diagrams of variation of parameters  ψ ϕ,  from fig. 2 a), b) and angular velocities  ψ ϕ & &,  from fig. 3 a), b) so.                                              Fiabilitate si Durabilitate - Fiability & Durability    no 2(6)/ 2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  31 Tab.1. t  ψ & &   ϕ & &   ψ   ϕ  ψ &   ϕ &  c z−  0 -5.14  -101.74  1.5708  0  200  0  0 0.001 -6.3449  -100.34  1.7708 -5.0752e-005 199.99 -0.10127 -3.7698e-006 0.002 -9.921  -96.187  1.9708  -0.00020161 199.99 -0.19977 -1.4975e-005 0.003 -15.761  -89.403  2.1708 -0.00044839 199.97 -0.2928  -3.3305e-005 0.004 -23.701  -80.179  2.3707 -0.00078442 199.95 -0.37778 -5.8265e-005 0.005 -33.516  -68.773  2.5707  -0.0012005 199.93 -0.4524 -8.9174e-005 0.006 -44.93  -55.506  2.7706 -0.0016853 199.89 -0.5147 -0.00012518 0.007 -57.628  -40.746  2.9704  -0.0022254 199.84  -0.56301 -0.0001653 0.008 -71.257  -24.903  3.1702  -0.0028061 199.77  -0.59593 -0.00020843 0.009 -85.448  -8.4085 3.37  -0.0034116 199.69  -0.61256 -0.00025341 0.01 -99.814  8.2868  3.5696 -0.0040257  199.6 -0.61256 -0.00029902 0.011 -113.97  24.731  3.7692  -0.0046317 199.49  -0.59604 -0.00034403 0.012 -127.53  40.476  3.9686  -0.0052131 199.37  -0.56344 -0.00038722 0.013 -140.11 55.08  4.1679  -0.0057538 199.24  -0.51558 -0.00042738 0.014 -151.38  68.151  4.3671  -0.0062392 199.09  -0.45378 -0.00046343 0.015 -161.04  79.348  4.5661  -0.0066566 198.94  -0.37977 -0.00049444 0.016 -168.85  88.387  4.765  -0.0069954 198.77  -0.29569 -0.0005196 0.017 -174.61  95.032  4.9636  -0.0072467  198.6 -0.20376 -0.00053826 0.018 -178.13  99.081  5.1621  -0.0074027 198.42  -0.10652 -0.00054985 0.019 -179.32  100.42  5.3605  -0.0074591 198.24  -0.0065789  -0.00055404 0.02 -178.16  99.028  5.5586 -0.0074146 198.06  0.093383  -0.00055074 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5-0.45-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050timpfifi 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2050100150200250timppsipsi a) b) Fig. 2  0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4-3.5-3-2.5-2-1.5-1-0.500.51timpfi1dfi1d 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2020406080100120140160180200timppsi1dpsi1d a) b) Fig. 3                                          Fiabilitate si Durabilitate - Fiability & Durability    no 2(6)/ 2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  32 References  [1] Grigore, J., Algoritm de calcul în vederea integrării pentru rigid în mişcare de rotaţie cu joc mare în cupla cinematică, Simpozion Târgu Jiu, iunie, 2010. [2] Grigore, J., Rigid în mişcare de rotaţie cu joc mare în cupla cinematică., UTCB Sesiunea de comunicări ştiinţifice SIMEC 2010, Bucureşti 2010. [3] Amironche, F., Fundamentals of multibody dynamics. , Birkhäser, Boston, Berlin, 2004 [4] Alexandru, C., Mecanisme articulate-analiză- sinteză-simulare cinematică. Editura Universităţii "Transilvania" Braşov, 2000 [4] Grigore, J., Pandrea, N., Aspecte privind jocurile din cuplele cinematice de rotaţie. Simpozion Târgu Jiu, iunie, 2008. [5] Grigore, J., Contribuţii la studiul dinamic al mecanismelor cu jocuri. Teză de doctorat, Piteşti, 2008. [6] Pandrea, N., Popa, D., Mecanisme. Teorie şi aplicaţii CAD, Editura Tehnică, Bucureşti, 2000. [5] Pandrea, N., Mecanica, vol. I, II, Litografia Universităţii din Piteşti, 1994                                   

NUMERICAL SIMULATIONS FOR THE CASE OF RIGID ROTATING KINEMATIC COUPLING WITH BIG CLEARANCE

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