Multi-Degree-of-Freedom System Response to Multipoint Base Excitation
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1 lti-degree-of-freeo Syste Response to ltipoint Base Ecitation By o Irvine Eail: to@vibrationata.co October 6, Introction, J Figre. ( - ( + Figre. he free-boy iagra is given in Figre.
2 he syste has a CG oset if. he syste is statically cople if. he rotation is positive in the clocise irection. he variables are y J i z i is the base isplaceent is the translation of the CG is the rotation abot the CG is the ass is the polar ass oent of inertia is the stiness for spring i is the relative isplaceent for spring i Sign Convention: ranslation: par in vertical ais is positive. Rotation: clocise is positive. - ( - - ( + Figre. Figre 4.
3 S the forces in the vertical irection for ass. F ( ( ( ( ( (4 S the forces in the vertical irection for ass. F (5 ( (6 ( (7 (8 S the forces in the vertical irection for ass. F (9 ( ( ( ( ( ( ( ( ( S the oents abot the center of ass.
4 4 J (4 ( ( J (5 ( ( J (6 J (7 J (8 he eqations of otion are ( ( J (9 Variables ass atri Stiness atri I Ientity atri ransforation atri U Displaceent vector Displaceents at riven noes f Displaceents at free noes
5 5 he eqation of otion for a lti-egree-of-freeo syste is [][] ][] [ ( f ] [ ( Partition the atrices an vectors as follos ( ( J ( f f f f ( here (4 (5 f (6 J (7
6 6 (8 (9 f ( ( ( ( Create a transforation atri sch that f ( I ( f f (4 Preltiply by, f f (5 Again, the partitione eqation of otion is f f (6
7 ransfor the eqation of otion to ncople the stiness atri so that the reslting stiness atri is ˆ ˆ (7 I I (8 f I (9 f f f (4 f (4 f f (4 f 7
8 et I (4 f f ˆ ˆ (44 (45 (46 f (47 I I (48 ˆ f (49 ˆ (5 I I (5 I f I 8
9 By siilarity, the transfore ass atri is ˆ ˆ ˆ ˆ f f (5 ˆ ˆ ˆ ˆ ˆ ˆ (5 ˆ ˆ ˆ (54 he eqation of otion is ths ˆ ˆ ˆ (55 he final isplaceents are fon via f (56 I f I (57 9
10 APPENDIX A Eaple, J Figre A-. Consier the syste in Figre A-. Assign the folloing vales. he vales are base on a slener ro, alin, iaeter = inch, total length=4 inch. able A-. Paraeters Variable J Vale lb lb 8.9 lb 97 lb in^, lbf/in, lbf/in 8 in 6 in et t & sin 5t sin here the aplite is in nits of G an tie t is in secons. Note that the ass vales for an are actally arbitrary since these egrees-of-freeo are riven. he folloing paraeters ere calclate for the saple syste via a atlab script.
11 >> enforce_acceleration enforce_acceleration. ver. October 6, by o Irvine Enter the nits syste =English =etric Asse syetric ass an stiness atrices. Select inpt ass nit =lb =lbf sec^/in stiness nit = lbf/in Select file inpt etho =file preloae into atlab =Ecel file ass atri Enter the atri nae: Stiness atri Enter the atri nae: Inpt atrices ass = sti = Select oal aping inpt etho =nifor aping for all oes =aping vector Enter aping ratio.5
12 nber of ofs =4 Enter the starting tie (sec Enter the en tie (sec Enter the saple rate (saples/sec 4 Enter the nber of ofs ith enforce acceleration. (ai = 4 Each inpt file st have to colns: tie & acceleration Enter the first of Enter the applie acceleration inpt atri nae for this of. a Enter the secon of Enter the applie acceleration inpt atri nae for this of. a5 = =.e+6 * Natral Freqencies No. f(hz..8e e
13 oes Shapes (coln forat oeshapes = = = = Natral Freqencies No. f(hz oes Shapes (coln forat oeshapes = Participation Factors part = Otpt arrays: ea_isp - isplaceent ea_vel - velocity ea_acc - acceleration
14 ACCE (G ACCEERAION 8 6 ass ass ass IE (SEC Figre A-. 4
15 RE DISP (inch REAIVE DISPACEEN.6 ass - ass ass - ass IE (SEC Figre A-. 5
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