Effective modal masses - Promoptica

1 5 me Congr s National de M canique Th orique et Appliqu eLouvain-la-Neuve, 23 & 24 mai 2000p 1/4 Effective modal , , de Li ge - Centre Spatial de Li geAvenue du Pr -Aily, B-4031, Angleur (Li ge)1 AbstractUsual dynamic analysis techniques express structuredeformation in a new base of rigid modes and elasticmodes. Each mode is characterised by a modal massand Effective masses associated to differentdirections. The normalisation of these modes isarbitrary, this imply that they have no physicalmeaning when considered alone. On the other hand,the Effective mass has a physical meaning and allow,in a lot of cases, simplification of the computation ofdeformations and stresses. They allow a trade-off ofuseful modes and simple computations on , each mode can be interpreted as a mass-spring-dashpot system oriented in the rigid modesspace along a specific direction.

2 The mass in thiscase is the Effective mass. An elastic mode will beexcited by a junction degree of freedom (dof)acceleration {a> if the projection of {a> along themode specific direction is non-null. When a force isapplied on the internal dof's, each elastic mode willbe excited by the projection of the force on thismode. Moreover, the reaction at the junction dofimplied by a mode will have a direction parallel tothe mode specific paper outlines the real representation that can beattributed to the Effective masses and to thesedirections. A new normalisation of these modes isproposed in order to give them a physicalsignificance. We introduce also modal stresses toallow easy computation of dynamic IntroductionThe use of modal analysis for structure analysis leadsto very interesting and useful results.}}

3 Nevertheless,for large structures or to obtain important accuracy,the necessary computation time is very better understanding of the representation bymodes can improve the analysis by a better selectionof the useful modes and a better estimation of theresults expected [1], [2], [3], [5].In this paper, a new representation of matrices andvectors is proposed. Explanations are proposed inappendix ( )3 Reminder of modal analysisThe equation we want to solve is equation (1).(1) []]{[{[]]{[{[]]{[{[{FFqq}K{K}K[Kqq}C{C}C [Cqq}M{M}M[M=++&&&&&&Commonly, we define a base of modes. Some ofthem are considered as rigid modes as they imply nostructure deformation (in isostatic or rigidly mountedcases). As a normalisation, the displacement of thedegrees of freedom (dof) corresponding to junctionnodes are set to one in these modes. We considerhere applications where the support is infinitely rigidor isostatic.]]]]}}}}

4 In those cases, the rigid modes (m = 6)will be defined as [ } = - [K]-1[K} if [K]-1 exists. Inthis case, we can also observe that {K}-{K][K]-1[K}is null. The rigid modes can be written as (2),(2) } = }I{}[with {I} = {diag (.. )}.The elastic modes [ >> can be extracted fromequation (3):(3) [ [ diag( k )][M] + [K] ] [ = 0k = 1,2,..nThe normalisation of these modes is arbitrary. Foreach mode we associate a frequency (3), a modalmass (4), a modal stiffness (5), a modal structuraldamping ( k = ck/2mk k) and a vector linking thismode to the rigid modes (6).(4) mk = < k] [M] [ k>(5) kk = < k] [K] [ k>(6) {Lk > = { | M | k >The resolution of the equation using theseparameters gives the following results.(7) [q0> = [ [ }+[ diag(.. ) L} }{q0> + [ diag(.. ) ] [F0>and(8) {R0> = {-{L diag(.. ) ] { ]][F0> + { - {L diag(.)]]]]]]]]}}

5 L} - { M } + { K } } {q0>with the dynamic amplification factor (here writtenHk),(9) kk2k2ki211)(H + = In (7), the first term [ } {q0> represents the rigiddisplacement; the second term represents thedisplacement due to junction dof displacement and5 me Congr s National de M canique Th orique et Appliqu eLouvain-la-Neuve, 23 & 24 mai 2000p 2/4the third term represents the displacement due to theforces applied to the internal dof. In (8), the firstterm represents the reaction forces due to the forceapplied to the internal dof, the second termrepresents the reaction forces due to the junction the parameters, the frequency and thedamping are independent of the mode normalisationand thus are the only parameters that have a combining them, we can define one effectivemass matrix for each mode (10).(10) {Mk} = {Lk> 1/mk <Lk}k = 1,2.]

6 NThese matrices are now independent of modalnormalisation and have a physical significance. Bydefinition, only 6 elements of this matrix areindependent, the elements outside main diagonalbeing a combination of the diagonal ones. There isalso only one non null eigenvalue with onecorresponding eigenmode. This eigenvalue is thetrace of the matrix, the mode is parallel to vector{Lk>.The main difficulty of this approach is the fact thatthe mass matrix combines different kinds ofelements: mass ones (expressed in kg) and inertiaones (expressed in ). As a result, the modalmass, the Effective mass, .. are also combinations ofmass and order to avoid this problem, we will assume thatthe modes have units: displacements in meters androtations in radians. By this way, the modal massesas well as the Effective mass matrices componentsare all expressed in.}

7 4 Effective parametersLet's define the Effective mass meff,k as theeigenvalue of the Effective mass matrix. We proposeto modify the modes normalisation in order that thenorm of {Lk> equals meff, solution is to multiply all modes by the squareroot of the Effective mass divided by the modal mass.(11) [ eff,k> = [ e,k> . kk,effmmThis will result in a new set of modes with modalmasses equal to Effective masses , {Lk> vector as anorm of meff,k and a direction parallel to the directionof the eigenmode of the kth Effective mass matrix anda new modal stiffness appears that will be calledeffective stiffness (12).(12) keff,k = meff,k . k Those new modes are called Effective Resolution of equations with thisnew Imposed acceleration along junction dofWhen the {a0> acceleration is imposed at thejunction dof, from equation (8), we can extract theunknown junction reaction {R0>:(13) {{{}{>+> => 0kk,eff0alk, )(TRkwhere a0 is the acceleration vector norm, {leff,k> isthe unitary vector in direction {Leff,k>, alk is thecosine of the angle between acceleration vector andeach {leff,k> vector and Tk is given by:(14) )(H1)(Tk2kk += In (13), we can see that for each mode, theacceleration is projected on corresponding {leff,k>.]]}}}}}}}}}}}

8 This projected acceleration (acc) excites the mass-spring-dashpot system that gives a forceTk( ).meff, (the response of a 1-dof system in the{leff,k> acceleration can also be projected along theeigenmodes of {MB}. Along each of these direction,a mass, whose value is the corresponding eigenvalue,is accelerated and also transmit a force to the sum of all these forces is the resulting identical computation can show that thedisplacement of the structure.(15) [[[} > +> => 20kk,eff0al2kk0aa.)(HqkOnce again, equation (15) shows that theacceleration is projected along {leff,k> and thedisplacement of the mass defines now the coefficientto apply to the Effective Force imposed on internal dofIf we apply forces on the internal dof, we can extractthe junction reaction from (8).(16) {][{]{[> >> < => 0k,eff0kk, ).(HRSo to obtain the solution, we can again separate inforces (< eff,k][F0>) applied at each mass.)]]]}}}}

9 Thereaction force is applied along {leff,k>. The sum of allforces is the reaction at the junction displacement are expressed by (17).(17) [][[> > < => k,eff0kk, )(HqAs in the previous case, the displacement of themasses is the coefficient to apply to the me Congr s National de M canique Th orique et Appliqu eLouvain-la-Neuve, 23 & 24 mai 2000p 3/46 Physical Effective modesThe kth Effective mode is associated to a specificdirection defined by {Lk> in the rigid modes space ofthe structure. The Effective mode [ eff,k> is theresulting deformation, multiplied by a factor k ,when a uniform unitary acceleration is imposedalong direction {Lk> at the junction we compute the deformation energy in this casewe have:(18) ][][2kk,eff4kk,eff2kk,eff2kk,effm21k21K2 1 = = > < Effective massesThe Effective mass meff,k of mode k is the fraction ofthe total static mass that can be attributed to thismode (static inertia for rotation modes).]]]}}}

10 As shownby (18), this also represents the energy absorbed bythis mode multiplied by 2. k when nominallyexcited by a static unitary acceleration at the junctionnodes. The meff,k units are in any case . Effective participation factorsThe {Leff,k> vectors define a set of directions in therigid mode space (in general a 6-dimension space ).Each Effective mode can be represented by a mass-spring-dashpot system oriented along {leff,k> in therigid modes space . The mass value is meff,k, thespring constant is keff,k and the damping coefficient isunchanged. The elastic behaviour of the structure canbe represented in the rigid modes space by this set of1-D mass-spring-dashpot systems (Fig ).meff,3keff,3 eff,3meff,1keff,1 eff,1meff,2keff,2 eff,2meff,4keff,4 eff,4XY zAccelerationFigure 6-1: Representation of elastic modes in rigidmodes space (X, Y, z)To obtain the mass associated to a direction, we canproject the Effective mass along this vector.}}