Wuhan University of Technology12.3 Uncoupled equations of motion-viscousdampingmv(t) +cv(t) +kv(t) =p(t)mY(t) +cY(t) + TY(t) =Tp(t)Φmmon=0mtnΦmkpn =0Tcon=0mtn12-11
12-11 Wuhan University of Technology 12.3 Uncoupled equations of motion - viscous damping
Wuhan UniversityofTechnologyL12.3 Uncoupled equations of motion- viscous dampingMn Yn(t) +Cn Yn(t) +Kn Yn(t) = Pn(t)wherethedefinitionsofmodal coordinatemass,stiffness,andloadhavebeenintroducedfromEq.(1212)and wherethemodal coordinateviscousdampingcoefficienthas been defined similarlyCn=OTconPn(t)Yn(t)+2SnwnYn(t)+w,Yn(t) =Mn12-12
12-12 Wuhan University of Technology 12.3 Uncoupled equations of motion - viscous damping where the definitions of modal coordinate mass, stiffness, and load have been introduced from Eq. (1212) and where the modal coordinate viscous damping coefficient has been defined similarly
Wuhan Universityof Technology-12.3 Uncoupled equations of motion-viscousdampingwhereEg.(1212d)hasbeenusedtorewritethestiffnesstermandwherethesecondtermonthelefthand siderepresentsadefinitionofthemodal viscousdamping ratioCn52wnMnAswasnotedearlier,itgenerallyismoreconvenientandphysicallyreasonabletodefinethedampingofaMDOFsystemusingthedampingratioforeachmodeinthiswayratherthantoevaluatethecoefficientsof thedampingmatrixcbecause the modal damping ratios E, can be determined experimentally orestimatedwithadequateprecisioninmanycases12-13
12-13 Wuhan University of Technology 12.3 Uncoupled equations of motion - viscous damping where Eq. (1212d) has been used to rewrite the stiffness term and where the second term on the left hand side represents a definition of the modal viscous damping ratio As was noted earlier, it generally is more convenient and physically reasonable to define the damping of a MDOF system using the damping ratio for each mode in this way rather than to evaluate the coefficients of the damping matrix c because the modal damping ratios can be determined experimentally or estimated with adequate precision in many cases
Wuhan University of Technology12.4 Response analysis by mode disp. superposition- viscous dampingmv(t) +cv(t) +kv(t) =p(tPn(t)Yn(t) +25n wn Yn(t) +w2 Yn(t)n=1,2,...,N3MnMn =oT m onPn(t) = ΦT p(t)[k-w?m] = 012-14
12-14 Wuhan University of Technology 12.4 Response analysis by mode disp. superposition - viscous damping