Modal effective mass is a very commonly used method for selecting target modes. Each mode has six effective mass values, corresponding to three translations and three rotations. The effective mass of the ith mode for the jth direction is based on the corresponding modal participation factor (MPF), calculated as follows:
(2.2.2-1)
where
- ith mode
shape
- unit translation or rotation
in the jth direction
- mass matrix
The unit translations and rotations are linear combinations of the rigid body modes, so the MPF is only non-zero for fixed base modes. This is because the flexible modes are orthogonal to the rigid body modes if the structure is not fixed. The modal effective mass (MEM) is the square of the MPF:
(2.2.2-2)
The MEM is often provided as a modal effective mass fraction (MEMF) by dividing by the mass in the associated direction as follows:
(2.2.2-3)
where:
- jth
diagonal element of the rigid body mass matrix
The modal effective mass fraction has the property
that if every mode of the structure were calculated, the sum . The MEMF, therefore, provides a very useful measure
of the relative MEM associated with each mode in each direction.
There are a number of interpretations of modal effective mass. One interpretation is that it is a measure of the amount of mass moving in direction j associated with a mode i. Another interpretation is that it is a measure of the contribution of the ith mode to the transfer function from a unit acceleration in the jth direction to the resultant force or moment in the jth direction. Using that interpretation, the modal effective mass is a special case of the frequency-response-based measures presented in the following section.
Typically only the translational components of MEM are considered since most structures are excited through translational motion of the base rather than rotational motion. However, there are cases where rotational motion is significant. In these cases the point about which the unit rotations are calculated is important.
The modal effective mass is calculated in Nastran using the MEFFMASS Case Control Card. There are a number of options for output, and the default will simply output the total MEM for all modes. The MEFFMASS(ALL)=YES will output all measures of the modal effective mass (MEM), including the modal effective mass fractions (MEMF) and the modal participation factors (MPF). Another useful option is the GRID option. This will calculate the rotation directions about a specified grid and should be used when rotational MEM is of importance. If this option is not used, the rotations are calculated about the origin of the basic coordinate system (CS0). In any case, the directions are defined by the basic coordinate system. A sample Nastran output is listed in Figure 2-1. The outputs are as follows:
1. Total Effective Mass Fraction This is the sum of the MEMF values for all calculated modes in each of six directions.
2. Effective Mass Matrix This is the rigid body mass matrix calculated by summing the effect of all the modes. The diagonals of this matrix are the sum of the MEM in that direction.
3. Rigid Body Mass Matrix This is the total rigid body mass matrix and should be identical the A-set mass matrix output using a WEIGHTCHECK Case Control Card. The diagonal elements of this matrix are used to calculate MEMF values.
4. Modal Effective Mass Fraction These are the MEMFij terms. The values in the first matrix are the translational terms and the values in the second matrix are the rotational terms. A column listing the cumulative effective mass fractions is also provided.
5. Modal Participation Factors These are the MPFij terms.
6. Modal Effective Mass These are the MEMij terms in units of mass.
7. Modal Effective Weight These are the MEMij terms in units of weight. If PARAM,WTMASS is not used the modal effective weight is the same as the modal effective mass.
M O D A L E F F E C T I V E M A S S S U M M A R Y
TOTAL EFFECTIVE MASS FRACTION
REFERENCE POINT AT ORIGIN OF BASIC COORDINATE SYSTEM
T1 T2 T3 R1 R2 R3
9.988360E-01 9.986904E-01 2.496829E-01 5.401772E-01 5.266303E-01 9.991691E-01
EFFECTIVE MASS MATRIX
*** ***
* 1.472220E+01 -1.521637E-10 -1.891113E-09 9.790985E-08 3.230237E+02 4.864870E-01 *
* -1.521637E-10 1.472005E+01 -2.791208E-02 -3.239704E+02 2.399714E-07 1.619823E-06 *
* -1.891113E-09 -2.791208E-02 3.680165E+00 -4.163848E+01 2.328655E-06 -4.264479E-07 *
* 9.790985E-08 -3.239704E+02 -4.163848E+01 1.590560E+04 -1.628817E-04 2.953477E-05 *
* 3.230237E+02 2.399714E-07 2.328655E-06 -1.628817E-04 1.316483E+04 8.364553E+02 *
* 4.864870E-01 1.619823E-06 -4.264479E-07 2.953477E-05 8.364553E+02 2.480995E+04 *
*** ***
A-SET RIGID BODY MASS MATRIX
*** ***
* 1.473936E+01 0.000000E+00 0.000000E+00 0.000000E+00 3.155251E+02 5.346715E-01 *
* 0.000000E+00 1.473936E+01 0.000000E+00 -3.155251E+02 0.000000E+00 1.612217E-06 *
* 0.000000E+00 0.000000E+00 1.473936E+01 -5.346715E-01 -1.612217E-06 0.000000E+00 *
* 0.000000E+00 -3.155251E+02 -5.346715E-01 2.944515E+04 -3.278980E-04 2.692480E-05 *
* 3.155251E+02 0.000000E+00 -1.612217E-06 -3.278980E-04 2.499824E+04 7.990648E+02 *
* 5.346715E-01 1.612217E-06 0.000000E+00 2.692480E-05 7.990648E+02 2.483058E+04 *
*** ***
1 GENERAL PURPOSE SPACECRAFT (GPSC) JUNE 21, 2004 NX NASTRAN 10/11/03 PAGE 16
FINITE ELEMENT MODEL (FEM)
0
MODAL EFFECTIVE MASS FRACTION
MODE FREQUENCY T1 T2 T3
NO. FRACTION SUM FRACTION SUM FRACTION SUM
1 1.359055E+01 2.341338E-21 2.341338E-21 2.078057E-06 2.078057E-06 1.799083E-01 1.799083E-01
2 1.603866E+01 1.746826E-18 1.749168E-18 7.902875E-07 2.868344E-06 4.679339E-03 1.845877E-01
3 1.765450E+01 1.696595E-18 3.445762E-18 3.045288E-05 3.332123E-05 4.558081E-02 2.301685E-01
4 2.026428E+01 7.555369E-02 7.555369E-02 4.202181E-17 3.332123E-05 6.086788E-19 2.301685E-01
5 2.524698E+01 3.357812E-18 7.555369E-02 6.731845E-02 6.735177E-02 4.956896E-07 2.301690E-01
6 2.526669E+01 5.599220E-02 1.315459E-01 1.471261E-18 6.735177E-02 2.060476E-20 2.301690E-01
7 3.626333E+01 8.595778E-20 1.315459E-01 1.549950E-04 6.750677E-02 1.945061E-02 2.496196E-01
8 3.734632E+01 1.110692E-16 1.315459E-01 6.245660E-01 6.920727E-01 1.151940E-05 2.496311E-01
9 4.006448E+01 5.522477E-01 6.837936E-01 3.324280E-17 6.920727E-01 2.234480E-20 2.496311E-01
10 4.270189E+01 6.752669E-18 6.837936E-01 1.373518E-02 7.058079E-01 3.082923E-06 2.496342E-01
11 4.634357E+01 2.889038E-01 9.726974E-01 5.414225E-18 7.058079E-01 3.513702E-19 2.496342E-01
12 4.910165E+01 1.971029E-17 9.726974E-01 2.878540E-01 9.936619E-01 3.377378E-05 2.496680E-01
13 5.453526E+01 1.671781E-02 9.894152E-01 7.863199E-19 9.936619E-01 1.010698E-19 2.496680E-01
14 5.724529E+01 2.863287E-19 9.894152E-01 4.791822E-03 9.984537E-01 8.394771E-08 2.496680E-01
15 6.022672E+01 9.420776E-03 9.988360E-01 4.421302E-16 9.984537E-01 1.279110E-17 2.496680E-01
16 6.089599E+01 9.582742E-16 9.988360E-01 2.366968E-04 9.986904E-01 1.484824E-05 2.496829E-01
17 6.946770E+01 5.084739E-08 9.988360E-01 7.268747E-20 9.986904E-01 6.309360E-19 2.496829E-01
1 GENERAL PURPOSE SPACECRAFT (GPSC) JUNE 21, 2004 NX NASTRAN 10/11/03 PAGE 17
FINITE ELEMENT MODEL (FEM)
0
MODAL EFFECTIVE MASS FRACTION
MODE FREQUENCY R1 R2 R3
NO. FRACTION SUM FRACTION SUM FRACTION SUM
1 1.359055E+01 1.237299E-05 1.237299E-05 1.406030E-21 1.406030E-21 3.480857E-17 3.480857E-17
2 1.603866E+01 7.932360E-07 1.316622E-05 7.472676E-19 7.486736E-19 3.384997E-20 3.484243E-17
3 1.765450E+01 9.265703E-02 9.267019E-02 3.400434E-16 3.407921E-16 1.966587E-18 3.680901E-17
4 2.026428E+01 2.112672E-17 9.267019E-02 3.814557E-02 3.814557E-02 1.319708E-01 1.319708E-01
5 2.524698E+01 1.317645E-01 2.244347E-01 6.967436E-18 3.814557E-02 9.688650E-20 1.319708E-01
6 2.526669E+01 5.273894E-18 2.244347E-01 1.440577E-01 1.822033E-01 5.161059E-04 1.324869E-01
7 3.626333E+01 5.087842E-05 2.244856E-01 2.381581E-20 1.822033E-01 3.347379E-19 1.324869E-01
8 3.734632E+01 2.094340E-01 4.339196E-01 3.615433E-17 1.822033E-01 4.676847E-16 1.324869E-01
9 4.006448E+01 2.595886E-17 4.339196E-01 3.362105E-01 5.184138E-01 1.424262E-02 1.467295E-01
10 4.270189E+01 1.061986E-01 5.401183E-01 3.715061E-18 5.184138E-01 9.610495E-16 1.467295E-01
11 4.634357E+01 8.846984E-19 5.401183E-01 7.221638E-03 5.256354E-01 2.123070E-02 1.679602E-01
12 4.910165E+01 5.729380E-05 5.401756E-01 8.785505E-18 5.256354E-01 3.155604E-15 1.679602E-01
13 5.453526E+01 1.765024E-21 5.401756E-01 9.343767E-06 5.256447E-01 1.255800E-02 1.805182E-01
14 5.724529E+01 1.777239E-07 5.401757E-01 7.824560E-21 5.256447E-01 7.992676E-17 1.805182E-01
15 6.022672E+01 1.486230E-18 5.401757E-01 9.855054E-04 5.266303E-01 8.186508E-01 9.991691E-01
16 6.089599E+01 1.427747E-06 5.401772E-01 1.064002E-16 5.266303E-01 8.722505E-14 9.991691E-01
17 6.946770E+01 1.383385E-18 5.401772E-01 2.060091E-08 5.266303E-01 4.331321E-08 9.991691E-01
1 GENERAL PURPOSE SPACECRAFT (GPSC) JUNE 21, 2004 NX NASTRAN 10/11/03 PAGE 18
FINITE ELEMENT MODEL (FEM)
0
MODAL PARTICIPATION FACTORS
MODE FREQUENCY T1 T2 T3 R1 R2 R3
NO.
1 1.359055E+01 1.857682E-10 -5.534367E-03 1.628414E+00 6.035929E-01 -5.928598E-09 9.296866E-07
2 1.603866E+01 -5.074159E-09 -3.412965E-03 -2.626222E-01 -1.528298E-01 -1.366762E-07 2.899163E-08
3 1.765450E+01 -5.000671E-09 2.118622E-02 8.196535E-01 -5.223314E+01 2.915560E-06 -2.209785E-07
4 2.026428E+01 -1.055278E+00 -2.488723E-08 2.995252E-09 7.887201E-07 -3.087996E+01 -5.724432E+01
5 2.524698E+01 7.035053E-09 -9.961077E-01 2.702988E-03 6.228826E+01 4.173412E-07 4.904843E-08
6 2.526669E+01 9.084541E-01 4.656763E-09 5.510907E-10 -3.940693E-07 6.000991E+01 -3.579834E+00
7 3.626333E+01 -1.125594E-09 4.779672E-02 -5.354339E-01 -1.223978E+00 -2.439987E-08 -9.116874E-08
8 3.734632E+01 -4.046095E-08 3.034090E+00 1.303029E-02 -7.852908E+01 -9.506812E-07 -3.407768E-06
9 4.006448E+01 -2.853029E+00 -2.213543E-08 -5.738884E-10 8.742783E-07 -9.167699E+01 1.880565E+01
10 4.270189E+01 9.976472E-09 4.499419E-01 -6.740942E-03 -5.591990E+01 -3.047458E-07 4.885020E-06
11 4.634357E+01 -2.063554E+00 -8.933207E-09 -2.275735E-09 -1.614004E-07 1.343608E+01 2.296020E+01
12 4.910165E+01 1.704456E-08 2.059802E+00 -2.231151E-02 1.298855E+00 4.686387E-07 8.851864E-06
13 5.453526E+01 -4.963967E-01 -3.404387E-09 -1.220534E-09 -7.209120E-09 -4.832988E-01 1.765849E+01
14 5.724529E+01 2.054337E-09 -2.657600E-01 1.112356E-03 7.234022E-02 1.398572E-08 1.408768E-06
15 6.022672E+01 3.726341E-01 -8.072617E-08 -1.373071E-08 2.091943E-07 4.963457E+00 1.425748E+02
16 6.089599E+01 1.188459E-07 5.906572E-02 1.479370E-02 2.050372E-01 1.630895E-06 4.653868E-05
17 6.946770E+01 8.657123E-04 1.035068E-09 3.049523E-09 2.018266E-07 2.269332E-02 -3.279470E-02
TOTAL -5.186305E+00 4.401066E+00 1.652598E+00 -1.235908E+02 -4.460811E+01 1.411423E+02
1 GENERAL PURPOSE SPACECRAFT (GPSC) JUNE 21, 2004 NX NASTRAN 10/11/03 PAGE 19
FINITE ELEMENT MODEL (FEM)
0
MODAL EFFECTIVE MASS
MODE FREQUENCY T1 T2 T3 R1 R2 R3
NO.
1 1.359055E+01 3.450981E-20 3.062922E-05 2.651733E+00 3.643244E-01 3.514827E-17 8.643172E-13
2 1.603866E+01 2.574709E-17 1.164833E-05 6.897043E-02 2.335695E-02 1.868038E-14 8.405145E-16
3 1.765450E+01 2.500672E-17 4.488559E-04 6.718318E-01 2.728300E+03 8.500489E-12 4.883150E-14
4 2.026428E+01 1.113613E+00 6.193744E-16 8.971534E-18 6.220795E-13 9.535722E+02 3.276913E+03
5 2.524698E+01 4.949198E-17 9.922305E-01 7.306145E-06 3.879827E+03 1.741737E-13 2.405748E-15
6 2.526669E+01 8.252889E-01 2.168544E-17 3.037009E-19 1.552906E-13 3.601190E+03 1.281521E+01
7 3.626333E+01 1.266962E-18 2.284526E-03 2.866894E-01 1.498123E+00 5.953535E-16 8.311738E-15
8 3.734632E+01 1.637088E-15 9.205699E+00 1.697886E-04 6.166816E+03 9.037948E-13 1.161288E-11
9 4.006448E+01 8.139775E+00 4.899774E-16 3.293480E-19 7.643626E-13 8.404672E+03 3.536526E+02
10 4.270189E+01 9.952999E-17 2.024477E-01 4.544030E-05 3.127035E+03 9.286999E-14 2.386342E-11
11 4.634357E+01 4.258256E+00 7.980219E-17 5.178971E-18 2.605008E-14 1.805283E+02 5.271706E+02
12 4.910165E+01 2.905169E-16 4.242782E+00 4.978037E-04 1.687025E+00 2.196222E-13 7.835549E-11
13 5.453526E+01 2.464097E-01 1.158985E-17 1.489703E-18 5.197141E-17 2.335778E-01 3.118224E+02
14 5.724529E+01 4.220300E-18 7.062837E-02 1.237335E-06 5.233108E-03 1.956002E-16 1.984628E-12
15 6.022672E+01 1.388562E-01 6.516714E-15 1.885325E-16 4.376227E-14 2.463590E+01 2.032758E+04
16 6.089599E+01 1.412434E-14 3.488759E-03 2.188535E-04 4.204024E-02 2.659819E-12 2.165849E-09
17 6.946770E+01 7.494578E-07 1.071366E-18 9.299590E-18 4.073398E-14 5.149866E-04 1.075492E-03
TOTAL 1.472220E+01 1.472005E+01 3.680165E+00 1.590560E+04 1.316483E+04 2.480995E+04
1 GENERAL PURPOSE SPACECRAFT (GPSC) JUNE 21, 2004 NX NASTRAN 10/11/03 PAGE 20
FINITE ELEMENT MODEL (FEM)
0
MODAL EFFECTIVE WEIGHT
MODE FREQUENCY T1 T2 T3 R1 R2 R3
NO.
1 1.359055E+01 1.332425E-17 1.182595E-02 1.023835E+03 1.406658E+02 1.357076E-14 3.337132E-10
2 1.603866E+01 9.940963E-15 4.497424E-03 2.662951E+01 9.018129E+00 7.212501E-12 3.245230E-13
3 1.765450E+01 9.655103E-15 1.733034E-01 2.593945E+02 1.053398E+06 3.282042E-09 1.885386E-11
4 2.026428E+01 4.299663E+02 2.391407E-13 3.463913E-15 2.401851E-10 3.681746E+05 1.265217E+06
5 2.524698E+01 1.910887E-14 3.831006E+02 2.820905E-03 1.498003E+06 6.724851E-11 9.288604E-13
6 2.526669E+01 3.186444E+02 8.372758E-15 1.172590E-16 5.995777E-11 1.390421E+06 4.947958E+03
7 3.626333E+01 4.891746E-16 8.820565E-01 1.106909E+02 5.784258E+02 2.298662E-13 3.209165E-12
8 3.734632E+01 6.320805E-13 3.554324E+03 6.555543E-02 2.381010E+06 3.489555E-10 4.483739E-09
9 4.006448E+01 3.142771E+03 1.891805E-13 1.271614E-16 2.951207E-10 3.245047E+06 1.365454E+05
10 4.270189E+01 3.842857E-14 7.816513E+01 1.754452E-02 1.207349E+06 3.585714E-11 9.213676E-09
11 4.634357E+01 1.644115E+03 3.081166E-14 1.999603E-15 1.005795E-11 6.970204E+04 2.035408E+05
12 4.910165E+01 1.121687E-13 1.638140E+03 1.922022E-01 6.513609E+02 8.479622E-11 3.025308E-08
13 5.453526E+01 9.513889E+01 4.474845E-15 5.751750E-16 2.006618E-14 9.018448E+01 1.203948E+05
14 5.724529E+01 1.629460E-15 2.726964E+01 4.777356E-04 2.020505E+00 7.552134E-14 7.662657E-10
15 6.022672E+01 5.361242E+01 2.516106E-12 7.279248E-14 1.689663E-11 9.511933E+03 7.848486E+06
16 6.089599E+01 5.453415E-12 1.347011E+00 8.449943E-02 1.623175E+01 1.026957E-09 8.362352E-07
17 6.946770E+01 2.893659E-04 4.136550E-16 3.590576E-15 1.572741E-11 1.988365E-01 4.152480E-01
TOTAL 5.684247E+03 5.683418E+03 1.420913E+03 6.141158E+06 5.082947E+06 9.579132E+06
Figure 2-1. Nastran Modal Effective Mass Output