Embed Size (px)
Citation preview
FASTER, EASIER FINITE ELEMENT MODEL REDUCTION
NOMENCLATURE DOF degree of freedom FEA finite element analysis FEM finite element model FRF frequency response function
I identity matrix M mass matrix N arbitrary number
TAM test analysis model IRS improved reduced system
4 mode shapes partitioned to the measured DOF cp mode shapes partitioned to the omitted DOF Q, mode shapes normalized to unit generalized mass
n many industries the success of a product depends upon . an accurate knowledge of its dynamic characteristics. . Efficient methods and tools to correlate test/analysis dy- . namic results will have a significant effect on reducing .
time to market for those products. The most accurate way . for a modal test engineer to prove that a set of measured '
mode shapes is valid for correlation with a finite element : model is to show that they are orthogonal. The orthogonality . criterion is simply defined as:
[@.]&st [wb [@]Test (1) .
where,
Q, are the test modes, normalized to unit generalized mass M is the analytical mass matrix reduced to the measured
*
'
degrees of freedom
If the measured modes are orthogonal, then the product of . equation (1) will be the identity matrix. Equation (1) is also . useful for comparing test mode shapes with analytical mode . shapes, by replacing one of the mode shape matrices with . the analytical mode shapes.
Often test engineers will not compute orthogonality because it requires several time consuming and complex steps:
1. A finite element model of the test article needs to be created
2. The mass matrix of the finite element model must be . reduced to the measured degrees of freedom
3. The mass matrix and mode shapes must be brought . together so that the orthogonality product can be com- . puted.
Since often the purpose for conducting the modal test is to '
correlate a finite element model, then typically Step 1 is al- *
ready accomplished. Steps 2 and 3, however, have not come : easily in the past. And, because they are not easily accom- . plished they are sometimes skipped altogether, or a Modal .
'
T A Deiters is a senior project engineer with Structural Dynamics Research Cor- poration, San Diego, CA. K.S. Smith is associated with Alliance Spacesystems Inc., Pasadena, CA. This paper was presented at the 1 Fh International Modal Analysis Conference February 1999, Kissimmee, FL.
'
'
.
Assurance Criterion is substituted. The MAC is simply cal- culated by
[[@]Test (2)
after normalizing the shapes to unit length. Another way to think about a MAC is as the square of an orthogonality prod- uct, where the mass matrix is assumed to be an identity matrix, which is equivalent to each measured DOF having the same mass. For very simple test articles where the mass is evenly distributed across all DOF, this assumption may be valid (e.g., a flat plate). For more complex assemblies such as satellites or automobiles, this is not a valid assumption and the MAC can yield misleading results. Sometimes a MAC is a useful diagnostic, particularly when the analytical model is poor or unavailable, but orthogonality is generally the more accurate measurement of mode shape indepen- dence.
The reason Step 2 above is often skipped is that special so- lution sequences must be developed or learned by the ana- lyst in order to reduce the finite element mass matrix to the test DOF. Guyanl is the simplest reduction to implement in most commercial FEA codes, but Flanigan2 points out that the Guyan reduction loses accuracy if there is any significant mass at omitted DOF. A Guyan reduction should be avoided for test articles which do not permit measurement of impor- tant mass (i.e., solid rocket motors, where the interior core is massive and often inaccessible). Freed3 documented that the Hybrid reduction4 yields the most accurate mass reduc- tion if the original FEM model is accurate compared t o the physical test model. If the FEM has inaccuracies, the Hybrid reduction will produce poor test orthogonality and, like the MAC, yield misleading conclusions about the test mode shape independence. The IRS reduction5 has been shown by Freed to yield better test orthogonality even if the original FEM has errors; however, Flanigan' documented that if the measured DOF are poorly chosen, then the IRS reduction can encounter numerical problems and yield poor orthogon- ality results. The common conclusions of the authors re- ferred to above are:
Use Guyan reduction as a first start. I t is the easiest to implement into commercial codes. Beware of its limi- tations when massive DOF are omitted from the mea- sured set. Use more advanced methods (i.e., IRS, Hybrid) when Guyan reduction is inadequate. These methods are not generally available in commercial codes and must be implemented via special solution sequence^.^
Step 3 above is probably the largest barrier for most test engineers to calculating test orthogonality. Test mode shapes are estimated from measured frequency response functions (FRF) and reside on the test computer in the lab. Analysis mode shapes and reduced mass matrices reside with the FEM on the analysis computers, which can be physically lo- cated in another building or another part of the country from where the test is being conducted. Certain test software pro-
SeptemberlOctober 2000 EXPERIMENTAL TECHNIQUES 35
grams like SDRC I-DEAS’” have had the ability to calculate orthogonality for many years. However, the process of cre- ating the mass matrix and getting it into the proper format to be imported into the test software program or, alterna- tively, getting the test mode shapes exported into the proper format for orthogonality calculations elsewhere has never been an easy task. The method being introduced in this pa- per should go a long way toward eliminating the difficulty or impossibility of Step 3 above as an excuse for not com- puting test orthogonality. Another benefit of FEM reduction is that the process can be used in reverse to extrapolate the motion of the mode shape, thus allowing more efficient use of accelerometers to measure only critical DOF for mode shape independence. The back expansion of test modes is often not taken advantage of for the same reasons that or- thogonality is not calculated. When back expansion is not available, often triaxial measurements are made at every DOF to attain better test mode visualization at the cost of added test time and complexity. This paper will show that the back expansion matrix can be made as readily available as the reduced mass matrix, helping to reduce the test time and complexity and improve the results.
DESCRIPTION OF FEM REDUCTION By definition of any FEA modal solution, the following must be true:
(3)
where
@ are the FEM shapes at the full DOF M is the mass matrix for the full FEM DOF I is the identity matrix
In the case of a Guyan reduction solution method, the fol- lowing must be true:
have to be derived from a Guyan solution, however, to arrive at a useful reduced mass matrix. All that is required is that the mode shapes be mass normalized and the inverse in equation ( 5 ) replaced with a pseudo-inverse. In this case, equation ( 5 ) will not recover the original mass matrix, but i t will generate a matrix which satisfies equation (3) for the given mode shapes. This technique for generating a reduced mass matrix produces a Modal TAM [41. The matrix [+I should be well conditioned, which will be true if the selected measured DOF are sufficient to show that the modes in- cluded in [+] are linearly independent. It is not necessary that [+I be square (contain the same number of modes as selected DOF). In fact, in practice it is virtually impossible to use N DOF to identify N linear independent modes (except when using a Guyan technique), and best results are achieved when fewer modes than selected DOF are used. Since mode shapes are common output from most FEA codes, the input required for this reduction method is readily available and easily provided by the analyst.
The back expansion [Go,] from the measured DOF to the omitted DOF can be derived from the relationship:
[cpl = [G,,,l[+l
: where
(6)
[cp] is the shape vector partitioned to the omitted (‘‘0’’)
DOF , [Goal the transformation matrix that expands from the mea-
. [+] is the shape vector partitioned to the measured (‘‘a”) sured (“a”) to the omitted (“0”) DOF set
DOF
. Both [cp] and [+I can be partitioned from the FEM mode
. shape matrix. The transformation or back expansion matrix
. [Goal can be solved by rearranging (6)
where,
+ are the Guyan shapes at the reduced (measured) DOF M is the Guyan mass matrix for the reduced (measured)
. The same restrictions on the validity and accuracy of the
. mass matrix using this method apply to the back expansion * matrix. If the mode shape matrix [;@,I is a full set of Guyan
mode shapes, then equation (7) exactly recovers the Guyan ’ back expansion matrix. Otherwise, [;@I must be well condi- ’ tioned and a pseudo-inverse should be used in equation (7).
DOF I is the identity matrix
If Only the mode shapes are can be found by rearranging the terms in (3) or (4):
then the matrix : If local coordinates are used for displacement results, then . the mode shaDes used to calculate the mass and back ex-
( 5 ) . pansion matrices must be in local coordinates. [MI = “4lT1-“41-’ where, . PRACTICAL IMPLEMENTATION OF REDUCTION 4 are the mass-normalized analysis shapes partitioned to .
M is the reduced mass matrix
The matrix inversions in equation ( 5 ) are possible only if the . mode shape matrix [+I is square and nonsingular. If the . mode shapes 4 are derived from a Guyan reduction and all . of the Guyan modes are included, then the M that results ’
from ( 5 ) will be the Guyan reduced mass. This technique : computes the Guyan mass matrix using only the mass- . normalized Guyan mode shapes, which are easily computed . with most commercial FEA codes. The mode shapes do not .
the measured or reduced DOF A powerful commercial product, Matlab’” by Mathworks, was chosen for this application because of the simplicity with which the method could be implemented. Equations ( 5 ) and (7) above can be solved in nearly as few lines of Matlab com- mand language as it takes to write the equations them- selves.
A new interface called IMAT (I-DEAWMatlab interface) was a simple interface for getting mode shape information from analysis or test into Matlab. IMAT is essentially a toolbox which extends the capabilities inside Matlab to read test or analysis data directly from Universal File Format or, in the
36 EXPERIMENTAL TECHNIQUES September/October2000
case of test, function or time history data directly from I- DEAS associated data format (ADF).
The flow chart in Figure 1 shows the data process flow inside Matlab. The first step is to read in the full FEM mode shapes and define the measured DOF. Next a macro called m o d a l subst is executed by entering:
> >modal-subst(shape,aset,’subst.unv’,l)
The Matlab variables “shape” and “aset” are required for the macro. These are the shape vector and the measured DOF, respectively. The third argument, which is optional, is the name of the universal file to store the resulting reduced mass and back expansion matrix. If this argument is not provided, the program will prompt the user for a file name when it is ready to write it. The fourth argument is also optional and if equal to 0 will instruct the macro to not cal- culate the back expansion matrix. If it is l or not given, then the back expansion will be calculated and written to the uni- versal file.
Upon completion of the macro, a universal file exists which contains the reduced mass matrix and, if desired, the back expansion matrix. The back expansion matrix will expand from the measured DOF to the number of DOF included in the mode shape vector. Having this in a n ascii Universal file format means that it can be transported to any platform (Unix or PC) and easily imported into I-DEAS Test Corre- lation. The matrices are then available in the Correlation task of I-DEAS for computing:
Test vs. Test orthogonality Analysis vs. Analysis orthogonality Test vs. Analysis cross orthogonality Back expand test mode shapes to enough DOF to allow
It is important to understand the limitations of this method, however, when using mode shape vectors from other (non- Guyan) solution techniques. The most important considera- tion is that the measured DOF set is adequate to uniquely identify each mode shape included in the mode shape ma- trix. If the DOF set is not adequate, then the shape matrix will be ill conditioned and the resulting inverse will yield an
good mode shape understanding.
( [ M A T can be u.& to wnplify this step)
/ Do you want to hack eapand modes to hubset
ofthe full E M DOF”
Panitinn shape variable to include only DOF fur the display size model Using I M A T this i s done with >> rhdpe=shape(disp_nodes)
disp_node* i s a variahle which contain\ display nodes)
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
I Panition shape, to illedaured DOF. I M A T command foIliiw\ >> phia=shape(aset)
laset 15 a variable which contans meawred DOFI I I I h e n phia lUse pauedo inverse if phia is not square)
>whia inv=inv(Dhiill.
I I Calculate reduced m s s matria m Matlab with >>mrs=phia_inv’* phia-inv
- - I I I
1 1 I I
I I I I I I I I I I I I I I I I I I I I - -
I
I I I I I I
I I
>>phio=shape (met ) I I I
etermine omitted DOF set (difference between all DOF in
>xi\et=fullse-aset the mcluded mode shape and the measured WFI I r a back expansion
matria desued”
V I
Panition shape nutria to the ormtted W F set
0 Write mtrices to a universal file for i m p n into Test soh ware.
Calculate Back Expanwn matria >>hackeap=phin’phia~inv:
I
Fig. I: Flow chart depicting data processing steps for developing reduced mass matrix from FEM shapes using MATLAB and I- MAT
SeptemberlOctober 2000 EXPERIMENTAL TECHNIQUES 37
invalid mass matrix. A check has been put in m0daLsubst.m to ensure that the condition number of the shape matrix prior to inversion is less than 100. Experience has shown that if the shape matrix has a condition number less than 100 then the resulting mass matrix will be sufficiently ac- curate. If the condition number is higher than 100 then the user is warned that the mass matrix could be invalid and either more measured DOF or fewer mode shapes need to be included.
h 2
5 N 6
rn
e 9 r 1 0
Two descriptions which are often used to evaluate the good- ness of a reduction in the literature are accuracy and ro- bustness. One measure of accuracy is how close the reduced model's modal frequencies are to the full FEM. Another is how orthogonal the full FEM mode shapes appear when or- thogonality is calculated using the reduced mass matrix. For the method described here, if the Guyan mode shapes are used, then the accuracy of the resulting mass matrix will be equivalent to the accuracy of the Guyan reduced mass ma- trix. Studies have shown that this type of reduction works very well for the vast majority of test articles but will lose accuracy when important mass is omitted from the mea- sured DOF. If mode shapes from other modal solutions (i.e., Lanczos) are used, then the resulting mass matrix is similar to the accuracy of the Modal TAM method described ins4 The mass matrix when used to compute orthogonality with the FEM modes will yield perfect orthogonality since this was the definition upon which the mass matrix was derived.
Robustness describes how well the mass matrix continues to show good orthogonality when the test mode shapes deviate from the predicted model. This assumes that the test mode shapes are indeed valid and the deviation is due to errors in the FEM. Past research has shown3 that the Guyan reduc- tion is the most robust of the reduction methods. If the non- Guyan modes are used in the shape inverse method de- scribed above, then the resulting mass matrix will have the poor robustness characteristic of the Modal TAM methode3 This means that small errors in the model can introduce large off-diagonal terms in the orthogonality matrix for the test modes even though they may be valid and truly orthog- onal.
s 1 1 0 0 0 0 0 0 0 0 1 1 0 0100 0 0 0 0 1 0 0 0
a 3 0 0 1 0 0 0 0 0 1 0 0 0 ~ 4 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0
u 7 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 2 0 1 0 0 0 1 1 0 2100 0 0 0 0 0 0 0 0 0 0 1 0 0
Mainly for the reason of poor robustness it is advised that most test and analysis engineers only use this method as a convenient way to get a Guyan mass matrix for computing orthogonality. The Guyan reduction will work well for 90% of the test articles encountered. However, if Guyan is not an option or there is a high confidence in the FEA prior to or following the test, then using non-Guyan modes will produce a valid mass matrix which has similar accuracy and robust- ness to the Modal TAM method.
h 2
5 N 6
m
r 1 0
PLATE EXAMPLE To demonstrate the validity of the mass reduction, a flat plate in a free-free boundary condition was chosen for a test article. The finite element model of the plate is shown in Figure 2 with arrows representing the DOF chosen for mea- surement locations. A Guyan reduction was conducted for the DOF shown. Table 1 shows the orthogonality of the full FEM shapes using the Guyan reduced mass matrix as the FEA solver provided it. Only the first ten flexible modes are
0100 0 0 0 0 0 0 0 0 a 3 0 0 1 0 0 0 0 0 1 0 0 0 : 4 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0
u 7 0 0 1 0 0 0 1 0 0 0 0 0 b 8 1 0 0 0 0 1 0 1 0 0 1 0 e 9 0 0 0 0 1 1 0 1 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
. Fig. 2: Geometry of flat plate used for a simple FEM
. reduction example
~ Table I -Orthogonality of Lanczos Shapes for Flat , Plate Example Using Traditional Guyan Reduced . Mass Matrix Shows Acceptable Accuracy in the ' Reduction
' compared, and the mass reduction shows good accuracy with '
the highest off diagonal term equal to 2. Table 2 shows the , orthogonality result from using the mass matrix derived us- . ing the shape inverse method from the full Guyan mode . shape matrix. The condition number of the shape matrix . prior to inverse was 5.8. There are slight but not significant
: Table 2-Orthogonality of Lanczos Shapes for Flat . Plate Using Guyan Mass Matrix Derived from Guyan . Shapes in Matlab Shows Insignificant Differences
I
. I Shape Number 1 2 3 4 5 6 7 8 9 1 0
s 11100 0 0 0 0 0 0 1 0 01
38 EXPERIMENTAL TECHNIQUES SeptemberlOctober200D
differences in the orthogonality results shown in Tables 1 and 2. These differences can be attributed to numerical pre- cision.
s 2
P
5 N 6
m
e 9 r 1 0
. Table 5-Orthogonality of Modified Plate Modes are
. Still Acceptable when Using Guyan Mass Matrix
. Derived from the Un-Modified Plate Model
1 1 0 0 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0
a 3 0 0 1 0 0 0 0 0 0 0 1 0 ~ 4 0 0 0 1 0 0 0 1 0 0 0 1
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1
u 7 0 0 0 0 0 0 1 0 0 1 0 0 b 8 0 0 0 0 0 0 1 1 0 0 0 1
0 0 1 0 1 0 0 0100 0 0 0 0 1 0 1 0 1 0 1 0 0
Table 3 shows the full FEM orthogonality using a mass ma- trix, which was derived with the inverse shape method from the first ten flexible modes of a Lanczos solution. The con- dition number of the shape matrix prior to inversion was 4.1. The orthogonality is perfect within numerical precision due to the fact that the method “finds,” by definition, the mass matrix which will yield a n identity orthogonality matrix.
Table 4 shows what can happen to the orthogonality result if too many Lanczos modes a re included in the shape matrix prior to inversion. In this case 25 flexible modes were used. Since the DOF set is not adequate to identify in-plane bend- ing modes, the shape matrix which must be inverted be- comes poorly conditioned and the inversion yields erroneous results. The condition number of the shape matrix prior to inverse was 3e10.
2 3 4 5 6 7 8 9
10
Tables 5 and 6 compare the robustness of the Guyan mass reduction with the mass matrix derived from the first ten Lanczos modes. The mode shapes were changed by including a soft spring to ground along one edge of the plate model and
1 1 0 0 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0100 0 0 0 0 0 0 0 0 0 0100
Table 3-Orthogonality of Lanczos Shapes for Flat Plate is Perfect when Mass Matrix is Derived from Lanczos Shapes (Modal Reduction)
S h a P e
N
m b e r
U
Table 4-Errors Enter Mass Matrix Reduced from Lanczos Shapes when Measured DOF Set is not Adequate to Represent the Included Shapes
Shape Number 1 2 3 4 5 6 7 8 9 1 0
‘100 196 128 134 46 199 821 78 117 1 56 100 178 29 42 29 9 53 113 3 32 284 100 571 82 199 487 2 469 0 36 80 505 100 122 317 223 19 103 0 19 123 201 92 100 173 256 120 130 0 92 505 35 14 196 100 589 232 102 1
177 210 287 **** 118 105 100 212 968 9 74 153 180 17 35 206 250 100 214 0 78 421 212 238 19 812 **** 275 100 0
1 3 0 1 0 1 9 0 0 1 0 0
. Table 6-Orthogonality of Modified Plate Modes
. Degrades Significantly when Using Modal Reduced
. Mass Matrix Derived from Un-Modified Plate
S h a P e
N
m b e r
U
1 2 3 4 5 6 7 8 9
10
Shape Number 1 2 3 4 5 6 7 8 9 1 0
100 0 68 0 71 1 2 1 13 1 0 100 0 43 2 57 28 55 0 4
68 0 1 0 0 0 2 0 0 0 0 0 0 4 3 0 1 0 0 0 1 0 4 8 0 0
71 2 2 0 1 0 0 0 0 0 1 0 1 5 7 0 1 0 0100 1 2 0 0 2 28 0 4 0 1 1 0 0 0 0 0 1 5 5 0 8 0 2 0100 0 0
13 0 0 0 1 0 0 0100 0 1 4 0 0 0 0 0 0 0 1 0 0
. resolving. This simulates what might happen to orthogon-
. ality in a test if the test mode shapes do not look exactly like ’ the FEA predictions. The mass matrix is based upon a plate ’ without the change in boundary condition. The results show ’ tha t the Guyan based-mass matrix still shows decent ortho- ’ gonality for the modified shapes; however, the mass matrix ’
based upon the Lanczos modes (modal reduction) has de- , graded significantly.
. AEROSPACE EXAMPLE ‘ Finally, a typical aerospace example was chosen to show that ‘
this method works equally well with realistically compli- , cated test articles. The test article is a solid rocket motor as . shown by the geometry in Figure 3. The full FEM contained . 198,684 DOF and was reduced down to 231 DOF for the . reduced set. Table 7 shows the orthogonality of the first ten . measured test modes using a Guyan mass matrix. For com- . parison, Table 8 shows the orthogonality of the same ten ‘ modes using a mass matrix derived from the Lanczos modes ‘ using modal subst in Matlab. The orthogonality calculated ’ using the Guyan mass matrix shows the linear independence . of the test modes better than the mass matrix derived from , a modal reduction using the Lanczos modes. The errors in . the modal reduced mass matrix are due to inconsistencies
SeptemheriOctoher Pooo EXPERIMENTAL TECHNIQUES 39
I I .
Fig. 3: Geometry of the aerospace test article used to demonstrate FEM reduction using Matlab for a more realistic size model
:
a 3 P
5 N 6 ” 7 m b 8 e 9 r 1 0
I awe ~-unnogonaiiry 01 Aerospace cxampie I esr Modes Using a Guyan Reduced Mass Matrix
h 2 0 1 0 0 1 2 1 0 0 3 1 0 2 1100 1 1 1 1 1 1 1
~ 4 2 2 1 1 0 0 1 0 0 1 2 4 0 1 1 1100 2 4 1 2 1 4 0 1 0 2100 4 1 0 2 1 0 1 0 4 4100 1 1 0 2 3 1 1 1 1 1100 1 1 0 1 1 2 2 0 1 1 1 0 0 1 1 0 1 4 1 2 0 1 1100
Shape Number
2 a 3 P e 4
5 N 6
m 8
r 1 0 e 9
1 2 3 4 5 6 7 8 9 1 0 s 11100 0 2 2 0 4 1 2 0 11
0100 1 0 1 4 1 3 1 6 1 1100 0 2 1 4 0 6 1 0 0 0100 2 7 1 9 2 1 2 0 1 2 2100 1 8 1 0 1 4 4 1 7 1100 3 8 1 1 2
“ 7 7 1 4 1 8 3 1 0 0 0 1 1 2 3 0 9 1 8 0100 2 1
1 6 1 1 2 1 1 2 1 1 2100 3 1 6 2 0 1 1 2 1 0 0 2
Table 8-Orthogonality of Aerospace Example Test Modes Using a Modal Reduced Mass Matrix Based Upon the Lanczos Shapes
.
.
Shape Number 1 2 3 4 5 6 7 8 9 1 0
s 1)lOO 0 1 0 0 4 7 2 3 11
between the test and analysis modes. This example demon- strates the degradation in orthogonality results which can occur when using a modal reduction with shapes that are not consistent with those used to derive the mass matrix. The Guyan mass matrix is not as sensitive to these incon- sistencies.
CONCLUSIONS The best characteristic of the method described here for re- ducing a FEM is that it is very simple and as accurate as Guyan or Modal TAM methods. By virtue of its simplicity, more test engineers than ever before will have access to the valuable tools of orthogonality and back-expansion during a test. Orthogonality may save days of re-test time trying to improve mode shape independence, and back-expansion could save days of preparation and setup time spent install- ing extra measurement DOF in order to achieve decent mode shape visualization. It is recommended that this method be used primarily as an easy way to extract the Guyan reduced mass matrix from a full Guyan shape matrix due to the po- tential errors associated with deriving a modal reduced mass matrix. With understanding however, this method can be used with mode shapes from any solver provided they are mass normalized.
Using this method to extract the reduced mass matrix based on the Modal reduction method has an additional benefit. If during the test it becomes necessary to add extra DOF, then a new mass matrix can be easily computed and used on-site at the test. No additional analysis runs are required.
References 1. Guyan, R.J., “Reduction of Stiffness and Mass Matrices,”
AIAA Journal, Vl 3, Feb. 1965. 2. Flanigan, C.C., Hunt, D. “Integration of Pretest Analysis and
Model Correlation Methods for Modal Surveys,” 1 lth International Modal Analysis Conference, February 1993.
3. Freed, A.M., Flanigan, C.C., “A Comparison of Test-Analysis Model Reduction Methods,” 8th International Modal Analysis Con- ference, January 1990.
4. Kammer, D.C., “A Hybrid Approach to Test-Analysis-Model Development for Large Space Structures,” submitted for publication to Journal of Spacecraft and Rockets, November, 1989.
5. OCallahan, J., “A Procedure for an Improved Reduced System (IRS) Model,” 7’h International Modal Analysis Conference, Janu- ary, 1989.
6. Flanigan, C.C., “Modal Reduction Using Guyan, IRS, and Dy- namic Methods,” 16th International Modal Analysis Conference, February, 1998.
7. Flanigan, C.C., “Implementation of the IRS Dynamic Reduc- tion Method in MSC/Nastran,” MSC NASTRAN World Users Con- ference, March, 1990. lks
40 EXPERIMENTAL TECHNIQUES ~epternhr/0ctober2000
