VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G

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1 B Tom Irvine Emil: Jnur 8, 3 VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G Introdution An vionis omponent m e mounted with isoltor grommets, whih t s soft springs. The gol of the isoltor design is to provide ttenution of sho nd virtion energ. This is hieved lowering the nturl frequen of the omponent sstem. Consider omponent with omple geometr tht is to e mounted vi four isoltors, s shown in Figures nd. Assume tht the omponent s hrdmounted nturl frequen is t lest one otve greter thn n of its isoltion frequenies. The ojetive is to derive the equtions of motion for this sstem, ounting for si degrees-of-freedom. Bground The motivtion for this report me from se histor where the omponent s enter of grvit ws offset suh tht the dimension in Figure ws negtive. Thus, the omponent s C.G. ws overhnging the footprint. The detiled finite element model of the isolted omponent hd some numeril stilit prolems due to the omintion of the C.G. offset nd the soft springs. As result, the model inorretl gve some of the nturl frequenies s ero, or pproimtel ero. Hnd lultions should e used to he finite element results regrdless of numerill stilit, ut this verifition is prtiulrl ritil when the model is potentill unstle. The hnd lultion pproh in this report represents the omponent point mss with inerti. The isoltors re represented springs. The nturl frequenies for relworld emple re lulted using softwre progrm tht implements the hnd lultion model. The progrm is given in Appendi A. To some etent, the numeril instilit m represent mehnil instilit. The finite element model ws eventull rendered numerill stle dding rottionl dof springs for eh isoltor. This ws rther rtifiil euse the true rottionl stiffness of eh isoltor ws unnown.

2 Derivtion m, J Figure. Isolted Avionis Component Model The mss nd inerti re represented t point with the irle smol. Eh isoltor is modeled three orthogonl DOF springs. The springs re mounted t eh orner. The springs re shown with n offset from the orners for lrit. The tringles indite fied onstrints. indites the origin.

3 C. G. Figure. Isolted Avionis Component Model with Dimensions All dimensions re positive s long s the C.G. is inside the o. At lest one dimension will e negtive otherwise. The vriles α, β, nd θ represent rottions out the X, Y, nd Z es, respetivel, using the right-hnd rule onvention. 3

4 4 The totl ineti energ is J J J m T () The totl potentil energ is V ()

5 The energ is E m J J J (3) 5

6 E m J J J (4) 6

7 E m J J J (5) 7

8 8 4 3 J J J m E (6)

9 E m J J J (7) 9

10 E m J J J (8)

11 The energ method is sed on onservtion of energ. d dt E (9) Appl the method. m J J J () Eqution () n e seprted into si individul equtions.

12 m () m () m J (3) (4) J

13 (5) J (6) Tpill, 3 4 (7) 3 4 (8) 3 4 (9) 3

14 Thus m 4 4 () m 4 () m 4 4 () J 4 4 J J 4 4 (3) (4) (5) 4

15 The equtions n e simplified s m 4 4 (6) m 4 (7) m 4 4 (8) J 4 4 (9) J J 4 4 (3) (3) The equtions n e rrnged in mtri formt. M K (3) 5

16 6. The mss nd stiffness mtries re shown in upper tringulr form due to smmetr. J J J m m m M (33) K = (34)

17 Aelertion Bse Eittion in the X-is β u m Figure 3. Isolted Avionis Component Model, Bse Eittion in X-is The se mss is onneted to the springs vi rigid lins. 7

18 8 The totl ineti energ is J J m m m u m T (35) Let r = u (36) J J m m r u m u m T (37) J J m m mr mu r mu u m T (38) J J m m mr mu r u m m T (39)

19 9 The totl potentil energ is r r r r V (4)

20 The energ is r r r r J J m m m r m u r u m m E (4)

21 E m mu mu r mr m m J J r r r (4)

22 The energ method is sed on onservtion of energ. d dt E (43) Appl the method. m m uu m u r m u r m r r m m J J J 3 4 rr r r 3 4 r r (44)

23 Eqution () n e seprted into seven individul equtions. muu mu r m (45) mur m r r 3 4 r r 3 4 r 3 4 r (46) m (47) m J (48) (49) 3

24 J r 4 (5) J r (5) Tpill, 3 4 (5) 3 4 (53) 3 4 (54) 4

25 Thus muu mu r m (55) r m ur m r r 4 r r 4 r (56) m 4 (57) m 4 4 (58) J 4 4 J J 4 4 r r (59) (6) (6) 5

26 The equtions n e simplified s mu m r m (6) mu mr 4 r 4 (63) m 4 (64) m 4 4 (65) J 4 4 (66) J J 4 4 r r (67) (68) The equtions n e rrnged in mtri formt. M r r m u K (69) The mss nd stiffness mtries re the sme s those given in equtions (33) nd (34) respetivel. 6

27 Aelertion Bse Eittion in the Y-is v m Figure 4. Isolted Avionis Component Model, Bse Eittion in Y-is The se mss is onneted to the springs vi rigid lins. 7

28 8 Let r = v (7) The equtions n e rrnged in mtri formt. m v r K r M (7) The mss nd stiffness mtries re the sme s those given in equtions (33) nd (34)

29 Aelertion Bse Eittion in the Z-is β w m Figure 5. Isolted Avionis Component Model, Bse Eittion in Z-is The se mss is onneted to the springs vi rigid lins. 9

30 Let r 3 = w (7) The equtions n e rrnged in mtri formt. M r3 r3 m w K (73) The elertion trnsmissiilit funtions of equtions (69), (7) nd (73) n then e determined vi Referene. The modl trnsient response to se input time histor n e lulted vi Referenes nd 3. Referenes. T. Irvine, Frequen Response Funtion Anlsis of Multi-degree-of-freedom Sstem with Enfored Motion, Virtiondt,.. T. Irvine, The Generlied Coordinte Method for Disrete Sstems, Sujeted to Bse Eittion, Revision B, Virtiondt, T. Irvine, Sho Response of Multi-degree-of-freedom Sstems, Revision F, Virtiondt,. 3

31 APPENDIX A Emple A mss is mounted to surfe with four isoltors. The sstem hs the following properties. M J J J K K K = 4.8 lm = 44.9 lm in^ = 39.9 lm in^ = 8.8 lm in^ = 8 lf/in = 8 lf/in = 8 lf/in = 6.8 in = -.68 in = 3.85 in = 3. in = 3. in Agin, is negtive due to overhng. Assume tht the dmping is 8% for ll modes. 3 onverting ineti energ to het energ. Note tht the isoltors provide dmping The nturl frequen results re lulted using the Mtl sript: si_dof_four_iso.m The output is given on the net pges. 3 Other dmping vlues ould e used to perform trde stud if the tul vlues were unnown. Dmping nnot e predited nltill. It must e mesured. 3

32 3

33 33

34 34

35 Geometr Dimensions (inh) = 6.8 = -.68 = 3.85 = 3 = 3 Mss (lm) Mss= 4.8 Polr MOI (lm in^) J= 44.9 J= 39.9 J= 8.8 Stiffness per Isoltor (lf/in) K= 8 K= 8 K= 8 The mss mtri is m = The stiffness mtri is =.e+4 * Eigenvlues lmd =.e+5 *

36 Nturl Frequenies = H.. H H H H H Modes Shpes (rows represent modes) lph et thet Prtiiption Ftors (rows represent modes) lph et thet Effetive Modl Mss (rows represent modes) lph et thet Totl Modl Mss Visous Dmping Rtios for Si Modes Mode

37 Frequen Response Funtions 37

38 Figure A-. The Z-is response ws elow the lower mplitude plot limit. 38

39 Figure A-. The Z-is response ws elow the lower mplitude plot limit. 39

40 Figure A-3. The Z-is response ws elow the lower mplitude plot limit. 4

41 Figure A-4. The Z-is response ws elow the lower mplitude plot limit. 4

42 Figure A-5. Both the X nd Y-es responses were elow the lower mplitude plot limit. 4

43 Figure A-6. Both the X nd Y-es responses were elow the lower mplitude plot limit. 43

44 Sted-stte Sine Bse Input Appl sine input s follows: Y-is input G 7.3 H C.G. Aelertion Response X-is:.765 G Y-is: 3.44 G Z-is: G C.G. Reltive Displement Response X-is:.573 in Y-is:.5853 in Z-is: in 44

45 Rndom Virtion Bse Input Appl rndom se input using the NAVMAT P-949 level s follows: 45

46 Figure A-7. The Z-is response is elow the lower plot limit. 46

47 Figure A-8. The Z-is response is elow the lower plot limit. Y-is input C.G. Aelertion Response X-is:.9383 GRMS Y-is:.846 GRMS Z-is: GRMS C.G. Reltive Displement Response X-is:.5 in RMS Y-is:.3 in RMS Z-is: in RMS 47

48 Hlf-sine Pulse Bse Input Appl 5 G, mse hlf-sine pulse s follows: Y-is input 5 G. se Hlf-Sine Pulse Aelertion Response (G) m min X-is: Y-is: Z-is: Reltive Displement Response (inh) m min X-is: Y-is: Z-is: 48

49 Figure A-9. 49

50 Figure A-. The resulting displement is rther high, nerl.4 inhes mimum. The isoltors must e le to withstnd this displement without ottoming out. But this displement requirement is too high for most isoltor models. Sw spe nd lerne re lso onerns. Thus, stiffer isoltors m e neessr in order to redue the pe displement. 5

51 Aritrr Bse Input Appl simulted protehni sho pulse s follows. Figure A-. Y-is input Aelertion Response (G) m min X-is: Y-is: Z-is: Reltive Displement Response (inh) m min X-is: Y-is: Z-is: 5

52 Figure A-. The sho dt is from: pro_sho_smple.tt 5

53 Figure A-3. 53

54 Figure A-4. 54

55 Figure A-5. Agin, the reltive displement m eond the isoltors llowle limit. 55

56 Figure A-6. Note tht the Z-is response plots re omitted euse the response in this is ws effetivel ero. 56

57 Rigid-od Aelertion Appl rigid-od elertion s follows. Rigid-Bod Aelertion (G) 6 C.G. Displement (in) (in) (in) C.G. Rottion thet- thet- thet- (rd) (rd) (rd).6 57

58 APPENDIX B Perpendiulr Ais Theorem B the perpendiulr is theorem, the following eqution reltes J to the re moments of inerti out the other two mutull perpendiulr es: J = I + I Referene: 58