AMSC 660 Scientific Computing I. Term Project The Assignment

Transcription

1 AS 66 Scentfc omputn I Prof. O Leary Term Project The Assnment odeln of a Seat Suspenson System Greory J. Hemen Dec., 6

2 ure depcts a lumped parameter system that may be used to model the response of a seated human body n a seat suspenson system [,,]. In ths model, the seat, denoted by mass, s fxed to the floor throuh a damper or enery absorber that provdes a force, R, and throuh a sprn,. In addton, an end-stop buffer s mplemented, whch produces a nonlnear sprn reacton force, st, when the suspenson stroke exceeds ts free-suspenson travel. The soft seat cushon s smply represented as a stffness and dampn c and c, respectvely. Ths lumped parameter model assumes the human s seated and that 9% of the body weht s supported by the feet [, ]. The body s dvded nto four parts: pelvs, upper torso, vscera, and head, represented by mass, stffness, and dampn, where =,,, and, respectvely. The dsplacement of the floor s ven by, and throuh are the absolute dsplacements of masses -, respectvely. R ure echancal odel of Seat Suspenson System oupled wth a Human Body n Seated Posture

3 The equatons of moton for ths system are obtaned by summn the nertal, stffness, and dampn related vertcal forces on each mass. The nertal force actn on a ven mass s ven by the mass tmes absolute acceleraton - the dot denotn dervatve wth respect to tme. The stffness/sprn force s ven by the stffness tmes the relatve dsplacement between the two masses between whch the sprns are connected. Smlarly, the dampn force s ven by the dampn coeffcent tmes the relatve velocty between the two masses between whch the sprns are connected. Lastly, ravtatonal force on each mass must be ncluded,, where s the ravtatonal acceleraton 9.8 m/s. The moton of these masses s then ven by the follown system of dfferental equatons []: st R t t = t t = = = =, where, t = c c, and t = c c 6, 7. Problem : Arrane ths set of ODEs nto standard form. In don so, smplfy to matrx form and dentfy all matrces used.

4 Now, to complcate matters, the cushon and bodynamc stffnesses tend to be nonlnear. The cushon stffness s ven by: c e = The stffness of the pelvs,, s modeled by the nonlnear functon []: 8.7e7, f = 9, f < The stffness of the upper torso s also nonlnear []:.78e6.9e7.69e7,f. = 77, f <. The dampn coeffcent s ven by = ζ f =,,, where ζ s the dampn rato of each part of the human body. Because and are nonlnear functons, and are also nonlnear. Lastly, the nonlnear sprn reacton force, st, due to the end-stop buffer s ven by []:, f <. = st 8.e[ st sn ].e8[ st sn ], f. nally, the damper should be modeled usn a Bnham-Plastc force model, whch ncludes a vscous component and a frcton component: R =, f sn

5 where s the post-yeld vscous dampn coeffcent, f s the frcton force, and sn represents the Snum functon. Problem : a Usn the parameters lsted n Table [,, ], wrte a atlab functon xdot=seat_system_odet,x representn ths system of nonlnear ODEs. Pass other necessary parameters throuh as lobal varables. b Solve ths system usn ode for cycles of a. ampltude snusod floor acceleraton. Record the tme to complete ths soluton and plot the relatve dsplacement velocty between the seat and the floor and the absolute pelvs head acceleratons vs. tme. Note: The ntal veloctes are all ero. The ntal dsplacements are the statc dsplacements.e., x = /. Assume only the ntal cushon stffness c = 7.7e for t.e. x = / c. Also, for smplcty assume x =x =x =x. c Repeat part b usn ode, ode, odes, odes, odet, odetb. Use default optons for each case. Explan why some soluton methods fal and/or have loner soluton tmes than others. Table - Seat Suspenson odel Parameters Quantty Symbol Value Unts ass of seat. k ass of pelvs 9 k ass of upper torso.8 k ass of vscera 6.8 k ass of head. k Stffness of col sprn. kn/m Stffness of vscera.8 kn/m Stffness of head. kn/m Post-Yeld Dampn oeffcent 7 N s/m ushon Dampn c 9 N s/m Pelvs Dampn ζ. - Torso Dampn ζ. - Vscera Dampn ζ. - Head Dampn ζ. - Damper rcton orce f 7 N

6 The Bnham-plastc force model Eq. for the damper may be approxmated usn a hypertanent functon below as depcted n ure : R o = o f tanh, ε Damper orce N - Yeld Reon - Usn snum functon Usn tanh Relatve Velocty m/s ure - Approxmatn the Bnham-Plastc orce odel Problem : a Repeat Problem b c usn ths hypertanent model for the damper wth ε=.. How have the results chaned? Why? Do you recommend ths approxmaton? b Repeat wth a ampltude floor acceleraton. How have the tme results chaned? Why?

7 Problem : a Wrte a atlab functon [t,x]=rkf,t,x to perform a fxed step nteraton usn th order Rune-utta alorthm. Here, f s the ode functon, T s a tme vector, and X are the ntal condtons. b Rerun problem a usn rk nstead of ode. How do the resultn plots and soluton tme compare wth atlab s ODE solvers? Gve a beneft and a ptfall to usn ths routne over atlab s ODE solvers. References. ho, Y.T. and Wereley, N.., Bodynamc response mtaton to shock loads usn manetorheolocal helcopter crew seat suspensons, Journal of Arcraft, Vol., No.,, pp Zon, Z. and Lam,.Y., Bodynamc response of shpboard sttn subject to shp shock moton, Journal of Bomechancs, Vol.,, pp. -.. Lu, X.X., Sh, J., L, G.H., et al., Bodynamc response and njury estmaton of shp personnel to shp shock moton nduced by underwater exploson, Proceedns of the 69 th Shock and Vbraton Symposum, St. Paul, Vol. 8, 998, pp. -8.