Mathematical Modeling and Nonlinear Dynamic Analysis of Flexible Manipulators using Finite Element Method

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1 Unvesa Jouna of Non-nea Mechancs (), Mathematca Modeng and Nonnea Dynamc Anayss of Fexbe Manpuatos usng Fnte Eement Method J. Jafa, M. Mzae, M. Zandbaf Depatment of Mechancs, Damavand Banch, Isamc Azad Unvesty, Damavand, Ian. Abstact By the cuent eseach, mathematca modeng and dynamc anayss of fexbe manpuatos ae pesented based on fnte eement method. In ode ths to be effected, each n of fexbe manpuato s modeed by fnte numbe of eements and the dspacement of eement s fomuated based on noda coodnates and shape functons of beam eement. Then, the netc and potenta enegy of the system s deveoped usng the dspacement n the efeence coodnate systems. In addton, by empoyng the Lagange pncpe, the nonnea dynamc mode of the system s deved. Besdes, smuaton esuts ae pesented to vadate the poposed method. Key Wod and Phases Fexbe Manpuato, Mathematca Modeng, Nonnea Dynamc Anayss, Fnte Eement Method.. Intoducton Fexbe manpuatos exhbt many advantages ove the tadtona gd ones: they have ght weght, the motos ae smae, they consume ess enegy, and the poducton s fuga. Because of these mpotant featues, the appcaton of fexbe manpuatos ae exceedngy deveoped dung ast decades, and they have been acheved an mpotant oe n many feds of scence such as sugca opeaton [, ], nucea appcaton [, ], and aeo space stuctues [5, 6]. Thus, the mathematca modeng and dynamc anayss of such system s mpotant and teated by some authos: Boo [7] anayzed the dynamc behavo of fexbe manpuatos based on ecusve agangan method. Moeove, a Newton-Eue appoach s pesented n [8] to mode the dynamc of a fexbe obot. Meghda and Fahm [9] used an anaytca method to decoupe the dynamc equatons of eastc manpuatos. Futhemoe, a umped mode of a pane fexbe manpuato s pesented n []. Sngh [] used an extended Hamton s pncpe to deve the equaton of moton of the fexbe manpuato. Besdes, Koayem et a. [, ] pesented the dynamc modeng of fexbe manpuato systems, based on assumed mode method. In the method, the fexbe behavo of the system s modeed va egenvaue functons mutped by moda coodnate of the system. In ths pape, the mathematca anayss and dynamc modeng of fexbe manpuato s pesented based on fnte eement method. Each n of the system s modeed by fnte numbe of eements, and the dspacement vecto of each pont of the obot s fomuated n the efeence coodnate by means of fnte fomuaton of beam eement. Then, the netc and potenta eneges of the system ae pesented, and the dynamc mode of the system s deved usng Lagange pncpe. Fnay, smuaton esuts ae pesented.. Fnte eement fomuaton fo mathematca mode of the system To pesent the mathematca and dynamc mode of the fexbe manpuatos, the system wth m numbe of ns, each n s dvded to n eements wth ength of. As the tota dspacement of each pont of the fexbe manpuato can be pesented as. Accodng to Fgue, the efeence coodnate system s shown by OXY, and the oca coodnate system attached to th n s assumed as O X Y. 56

2 J. Jafa, M. Mzae & M.Zandbaf Fg. The fexbe manpuato The paametes of the fexbe manpuato ae shown n Tabe. Tabe. Paametes of fexbe manpuato Paametes Jth eement of th n Dspacement vecto of eement Dspacement vecto of th ont Angua dspacement of th ont Numbe of eements of th n Length of th n Mass pe ength of th n Gavtatona constant of eath Length of th eement of th n Eastcty moduus of th n Moment of neta of th n Rotaton matx between oca and goba coodnate system Nomencatues o θ n L m g E I T To pesent the tota dspacement vecto of th eement of the system n the goba coodnate system, ths vecto s assumed as a summaton of dspacement of O, and the defecton of the n n the oca coodnate O X Y : ( ) + x =, +, f = O + T (.) y whee y s the defecton of eement due to fexbty of system n the oca coodnates. By mpementaton of fnte eement method, ths dspacement s pesumed a summaton of Hemtan shape functon mutped to noda coodnate of the eement []: 57

3 J. Jafa, M. Mzae & M.Zandbaf ( x,t) = y φ ( x ) u (.) = + whee φ shows the shape functon and u + ae the noda coodnate of the systems, and ae gven as []: x x φ ( x ) = + (.) ( ) = x x x φ x + (.) ( x ) = x x φ (.5) x x φ ( x ) = + (.6) As the dspacement vecto of eement s fomuated, the netc enegy of the eement s stated as foows: T T = m. dx < x < (.7) t t If, the vectos = [ θ u u u u ] T and ψ = [ u u u ] T ae defned, then Eq. (.7) z + + can be ewtten as: T T = z& M z& T M (, p) = m. dx z z p Besdes, the potenta enegy of the eement s shown as V g potenta enegy, and eastc potenta enegy : V = V + V g e u + + V e V (.8), and s a summaton of gavtatona (.8a) The gavtatona potenta enegy s gven as: Vg = m g[ ] dx (.9) And the eastc enegy of the system s: y T Ve = EI ( )dx ψ Kψ = (.) x whee the stffness matx K s pesented as: 6 6 EI 6 6 K = (.) Beyond the above, the geneazed coodnate vecto s defned as q, and the tota netc and potenta enegy of the system can be wtten as: m n m n T( q,q & ) = T V( q ) = V (.) = = = = Then, the Lagange functon s ntoduced as L( q, q &) = T V, and the Lagange pncpe s deveoped. The pncpe of Lagange fo dynamc systems s expessed as: 58

4 J. Jafa, M. Mzae & M.Zandbaf d L L = Q dt q (.) & q whee q epesents the geneazed coodnates, Q s the geneazed extena foce. Thus, by mpementaton of Lagange pncpe, the nonnea dynamc equatons of the system ae summazed as foows: ( q q& ) B τ M q& + f, = (.) As n Eq. (.) s pesented, the nonnea dynamc mode of the system s deveoped, and no neazaton s done. Consequenty, the nonnea tems affect the dynamc of the system. It must be futhe notced that fo each n of the fexbe manpuato, the fst node s concded on the ont of the n. Thus, these noda coodnates ae zeo: u ( t ) =, u ( t ) = (.5). Dynamc mode of a snge n manpuato Fo a snge-n fexbe manpuato, as the n modeed by one eement, the geneazed coodnate vecto of the system s q = [ θ u u ], whee θ s angua dspacement of the obot ont, u and u ae the eastc defecton and sope of the end pont of the fexbe manpuato. Moeove, the otaton matx of the system s: cos( θ) sn( θ ) T = (.) sn( θ) cos( θ) and the tota dspacement of any pont of the obot s: x = x x cos( θ ) sn( θ) ( L x sn( θ + ) cos( θ) ( L x ) u L x ) u L x + ( L x + ( L x + ) u L x + ) u L (.) So, the netc and potenta eneges of the snge-n fexbe manpuato ae stated as: ( L & θ u + 7L & θ u& L & θ u& + 78& θ u + L u& L & θ u u + 78u& + 7L & L u& u& ) T = m L θ V g = m gl ( 6u cos( θ) Lu cos( θ) + 6L sn( θ) ) (.) EI V e = ( u Lu u + L u ) L (.5) Thus, the dynamc equaton of the system can be deved, usng Lagange pncpe. To smuate the dynamc behavo of the system, the paametes ae gven as: m=5 g, L=m, I=5e-9, E=e9 pa, g=9.8. The smuaton esuts ae as foows: (.) 59

5 J. Jafa, M. Mzae & M.Zandbaf Fg. Angua dspacement of fexbe manpuato Fg. Dspacement mode shape of the fexbe manpuato. Concusons In ths pape, the nonnea dynamc anayss of the fexbe manpuatos has been studed usng fnte eement method. The tota dspacement vecto of the system has been fomuated usng Hemtan shape functon. Hence, the tota dspacement of the eastc am n efeence coodnate system has been pesented, the Lagange pncpe has been used to deve the nonnea dynamc moton of the eastc manpuato. Fnay, the poposed method has been empoyed to deve the dynamc equatons of a snge-n manpuato, and some smuatons ae done. 6

6 J. Jafa, M. Mzae & M.Zandbaf Refeences. Lao H., Suzu H., Matsumya K., Masamune K., Doh T., Chba T., Fetus-suppotng Fexbe Manpuato wth Baoon-type Stabze fo Endoscopc Intautene Sugey, Int J of Medca Robotcs and Compute Asssted Sugey, (8),.. Kuma R., Beeman P., Gupta P., Banes A., Jensen P. S., Whtcomb L. L., Tayo R., Pemnay Expements n Coopeatve Human/obot Foce Conto fo Robot Asssted Mco-sugca Manpuaton, Poc of IEEE Int Conf on Robotcs and Automaton, 6 (), -.. Jansen J. F., Bus B. L., Babcoc S. M., Kess R. L., Hame W. R., Long-each manpuaton fo waste stoage tan emedaton, ASME Dyn. Syst. Conto. Dv., DSC, (99), Peot Y., Code J. J., Fconneau J. P., Masonne D., Matn E., Pame J. D., Deveopment of a ong each atcuated manpuato fo ITER n vesse nspecton unde vacuum and tempeatue, Fuson Eng. Desgn, 69 (), Satoo A., Kazuya Y., Adaptve Reacton Conto fo Space Robotc Appcatons wth Dynamc Mode Uncetanty, Advanced Robotcs, (), Zhong J. L., Dynamc Modeng and Vbaton Conto of a Space Fexbe Manpuato Usng Pezoeectc Actuatos and Sensos, Advanced Mateas Reseach, 5 (), Boo W. J., Recusve agangan dynamcs of fexbe manpuato ams, Int. J. of Robotcs Reseach, (999), Rahsha F., Godenbeg A. A., Dynamc Modeng of a Snge Ln Fexbe Robot, Poc of the IEEE Int Conf on Robotcs and Automaton, (985) Meghda A., Fahm F., On the Fst-ode Decoupng of Dynamca Equatons of Moton fo Eastc Mutbody Systems as Apped to a Two-n Fexbe Manpuato, Mutbody Systems Dynamcs, 5 (), -.. Megahed S. M., Hamza K. T., Modeng and Smuaton of Pana Fexbe Ln Manpuatos wth Rgd Tp Connectons to Revoute Jonts, Robotca, (), Koayem M. H., Rahm H. N., Nonnea dynamc anayss fo eastc obotc ams, Fontes of Mechanca Engneeng, 6 (8), Koayem M. H., Rahm H. N., Noobn A., Mathematca modeng and taectoy pannng of mobe manpuatos wth fexbe ns and onts, Apped Mathematca Modeng, 6 (), 9-.. Zenewcz O. C., Tayo R. L., Zhu J. Z., The fnte eement method, ts bass and fundamentas, Eseve Buttewoth Henemann, Oxfod, Unted Kngdom, 5. 6