linear vibration analysis using screw theory - helix - Georgia Institute ...

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4VFigure 1.3: Body Rotates and Translates about Vibration Axis Vvibration axis but also depends on non-geometric quantities. Similar results are obtained for a bodywhich is excited by the motion of its base.The motion of a body undergoing non-proportionally damped vibrations has always been dif-…cult to visualize. Chapter 6 shows that for planar motion, a mode is actually a rotation about apoint which is either stationary or traveling along a straight line depending on the type of damping(undamped, underdamped, critically damped or overdamped). In a similar manner, it is shown thatfor spatial motion, a mode is a screwing motion about a vibration axis which is either stationaryor traveling along a cylindroid. The analysis of forced damped vibrations also yields the conditionsunder which a mode will not respond to an excitation.Other results not directly related to this research are presented in Chapter 7. The …rst twosections detail some properties associated with the sti¤ness matrix. In addition, a decompositionof the damping matrix is presented; this decomposition is based on the same principles that Lipkinand Patterson used to decomposed the sti¤ness and inertia matrices.Finally, Chapter 8 concludes the thesis by summarizing the results and by proposing the applicationof this research to several problems. These include the design of mode shapes, the inverse
5mode shape problem where eigendata is used to reconstruct the sti¤ness and inertia properties of asystem, and also modal identi…cation problems.
Page 3 and 4: iiiDEDICATIONÀ mon …ls Matthieu
Page 6 and 7: vi5 FORCED VIBRATIONS : : : : : : :
Page 9 and 10: ixSUMMARYA novel analysis is propos
Page 11 and 12: CHAPTER IINTRODUCTIONThe purpose of
Page 13: 3properties have canonical forms: f
Page 17 and 18: 7where ~v O and ~ Á are respective
Page 19 and 20: 9yOA, I x , I y , Y, LMxm, m γFigu
Page 21: 1123·~T M =¸ T 1 T 2 T 3=640 0 00
Page 24 and 25: 14twist it induces the form T E = P
Page 26 and 27: where ~r S f and ~rC °are respecti
Page 28 and 29: 18m fi are all equal and represent
Page 30 and 31: 20Consider any wrench contained in
Page 32 and 33: 22on the subject is extensive [15],
Page 34 and 35: 24VEMyzxFigure 3.1: Elastically Sus
Page 36 and 37: 26! 2 =k ° + k x EVy 2 + k y EVx2m
Page 38 and 39: 28·where ^T M =~ ±T0¸Tis a trans
Page 40 and 41: 30yVMExFigure 3.2: Location of Vibr
Page 42 and 43: 32Theorem 10 When the center of mas
Page 44 and 45: 34where MV x1 , MV x2 , MV x3 are t
Page 46 and 47: 36At the center of mass,2M M =64m 0
Page 48 and 49: 38yV 2OMEV 1TranslationalDirectionx
Page 50 and 51: CHAPTER IVTRANSLATION AND COUPLE MO
Page 52 and 53: 42δw rFigure 4.2: Pure Translation
Page 54 and 55: 444.2.1 Translation ModeThe two equ
Page 56 and 57: 46sti¤ness and mass matrices have
Page 58 and 59: 484.3.1 Translation modeIt is known
Page 60 and 61: 50matrix and,23k tr =640 0 0a b cd
Page 62 and 63: 52rotations which intersect at M. T
Page 64 and 65:
54direct consequence of assuming eq
Page 66 and 67:
56be rewritten as,~ fTi~ ±1 = ~ ±
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58produced by a couple mode. The pr
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60should then be a pure translation
Page 72 and 73:
62be represented as,K M = X T EMK E
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64means that one principal sti¤nes
Page 76 and 77:
66Substituting (4.61) and (4.62) in
Page 78 and 79:
68δ 2δ 3τ 1E; MFigure 4.7: Coupl
Page 80 and 81:
70Equation (4.74) represents the ei
Page 82 and 83:
72² Design K 2 such that ^w is one
Page 84 and 85:
74Equation (4.83) shows that k fcan
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76make the common eigentwists paral
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78Note that ^w transforms according
Page 90 and 91:
80The proposed design places 13 con
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82construction is a problem of sti
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84where·¸2~ f2 ~ f3 p 6=66430 ¡2
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86fStiffnessAxisM 2Body 2Body 1τ M
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88the centers of mass and elasticit
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90where the vector from E 2 to M 2
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92where the arbitrary linear princi
Page 104 and 105:
94² The linear mass of the absorbe
Page 106 and 107:
96Pfk x kγ k yFigure 4.13: Possibl
Page 108 and 109:
98specialized to systems exhibiting
Page 110 and 111:
100where ! 2 j is the natural frequ
Page 112 and 113:
102V 1V 2ffτ MMV 3Figure 5.3: Mode
Page 114 and 115:
104γ iαMMV iMFV i FfFigure 5.4: L
Page 116 and 117:
106Equation (5.20) reveals that the
Page 118 and 119:
108Substituting (5.23) into (5.22)
Page 120 and 121:
110V 1EYMYMV 2V 3Figure 5.6: Mode 1
Page 122 and 123:
112δEMV 2V 3V 1Figure 5.8: Mode 1
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114a planar rotating unbalance for
Page 126 and 127:
116undamped, underdamped, criticall
Page 128 and 129:
118The location of point V is given
Page 130 and 131:
Note that ¡! AV can also be found
Page 132 and 133:
122For planar motion, the eigenvect
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124Aδ δxδByBxyB AδA By− δx
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126position. This observation means
Page 138 and 139:
Equation (6.53) reveals that the vi
Page 140 and 141:
130¯c ¯± B A¯Equation (6.64) re
Page 142 and 143:
13221.51Mode 1Mode 2Mode 3Imaginary
Page 144 and 145:
Y134-0.4-0.5-0.6-0.7Point APoint BM
Page 146 and 147:
13676δ ΑB543Mode 1Mode 2Mode 3210
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138where A and B are any points on
Page 150 and 151:
140where^W A = K^T A^W B = K^T BEqu
Page 152 and 153:
142where c = p ¤ =p. Equation (6.9
Page 154 and 155:
1442z 0-2420x-2-4420y-2-4Figure 6.1
Page 156 and 157:
146Equation (6.106) shows that the
Page 158 and 159:
1486.2.4 Overdamped ModesThe eigenv
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150where ¸j and X j are respective
Page 162 and 163:
152Equations (6.131) reveal that ^w
Page 164 and 165:
154Hence the orthogonality conditio
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156If ~ f R and ~ f I are linearly
Page 168 and 169:
CHAPTER VIIOTHER RESULTSThis chapte
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160(S) to coalesce, for the centers
Page 172 and 173:
162As before, all three ~r i = ~0 w
Page 174 and 175:
164The previous Theorem proved that
Page 176 and 177:
166Proof.From Theorem 49, the eigen
Page 178 and 179:
168the eigenwrenches with respect t
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170eigentwists of the damping matri
Page 182 and 183:
172Equation (7.53) shows that the e
Page 184 and 185:
174and inertia matrices to intersec
Page 186 and 187:
176In Chapter 4, the occurrence of
Page 188 and 189:
178APPENDIXA. Proof of Critically D
Page 190 and 191:
180where S is a non-singular matrix
Page 192 and 193:
182Equation (A.3) can be written as
Page 194 and 195:
184REFERENCES[1] Anton, H., 1984, E
Page 196 and 197:
186[35] Patterson, T. and Lipkin, H
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