NASA/ TM-2003-212343 

Hybrid Residual Flexibility/Mass-Additive 

Method for Structural Dynamic Testing 

M.L. Tinker 

Marshall Space Flight Center, Marshall Space Flight Center, Alabama 

March 2003 

https://ntrs.nasa.gov/search.jsp?R=20030032438 2018-05-15T15:56:26+00:00Z
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NASA / TM-2003-212343 

Hybrid Residual Flexibility/Mass-Additive 

Method for Structural Dynamic Testing 

M.L. Tinker 

Marshall Space Flight Center, Marshall Space Flight Center, Alabama 

National Aeronautics and 

Space Administration 

Marshall Space Flight Center MSFC, Alabama 35812 

March 2003 
I 

Acknowledgments 

Kathy Kappus, Tim Driskill, and Blane Anderson, of the Structural and Dynamics Testing Group at Marshall Space Flight 

Center, are acknowledged for their support of this project. They have participated extensively in studies of the residual 

flexibility, mass additive, and hybrid test techniques by providing modal and frequency response data for the payload 

simulator structure described in this Technical Memorandum (TM), as well as other structures. Kathy Kappus 

frequency response functions used in section 4. 

Eric Blades, formally of Structural Dynamics Research Corporation (SDRC) in San Diego, California, and currently 

at Mississippi State University, developed the FOKRAN computer code implementation of the hybrid equations developed 

in this TM. He also performed the initial computer simulations upon which many of the results in section 3 are based. 

I arranged for fabrication of the rectangular boundary mass discussed in this TM and provided the experimental 

Available from: 

NASA Center for Aerospace Information 

7121 Standard Drive 

Hanover, MD 21076-1320 

(301) 621-0390 

National Technical Information Service 

5285 Port Royal Road 

Springfield, VA 22 16 1 

(703) 487-1650 

.. 

II 
TABLE OF CONTENTS 

1 . INTRODUCTION AND BACKGROUND ................................................................................... 1 

1.1 Mass-Additive Modal Test Method ...................................................................................... 1 

1.2 Residual Flexibility Test Method .......................................................................................... 4 

2 . DEVELOPMENT OF THE HYBRID TEST METHOD .............................................................. 6 

2.1 Derivation of Governing Equations for Hybrid Method ....................................................... 7 

2.2 Frequency Response Formulation of Hybrid Method .......................................................... 17 

2.3 Statistical Least-Squares Curve-Fitting Procedure for Identification of Experimental 

Residual Flexibility Values ................................................................................................... 18 

3 . ANALYTICAL APPLICATION OF HYBRID METHOD TO SHUTTLE PAYLOAD 

SIMULATOR ................................................................................................................................ 21 

3.1 Correlation of Free-Boundary Mode Shapes and Frequencies ............................................. 21 

3.2 Model Updating for Boundary Frequency Response ........................................................... 24 

3.3 Comparison of Residual Flexibility Values for Boundary Degrees of Freedom .................. 25 

3.4 Use of Payload Simulator Model for Hybrid Method Parameter Studies ............................ 27 

4 . EXPERIMENTAL IMPLEMENTATION OF HYBRID METHOD FOR PAY LOAD 

SIMULATOR WITH MASS-LOADED BOUNDARIES ............................................................. 38 

4.1 

4.2 

4.3 

4.4 

Translational Frequency Response Function and Residual Flexibility 

Measurements Compared to Analysis .................................................................................. 40 

Approximate Rotational Frequency Response Functions Derived From 

Translational Test Data and Comparison With Analysis ...................................................... 47 

Experimental Rotational Residual Flexibility Results and Comparison 

With Analysis ....................................................................................................................... 50 

Using Models Correlated to Free-Suspension Modes and Residual Flexibility ................... 55 

5 . SUMMARY ................................................................................................................................... 59 

REFERENCES .................................................................................................................................... 61 

... 

111 
LIST OF FIGURES 

2 

3 

5 

5 

22 

23 

23 

25 

38 

39 

39 

1. Shuttle payload carrier with orbiter connections ................................................................ 

2. Materials Science Laboratory payload in mass-additive test configuration .. .. .. ......... .... ... . . 

3. Space Station common module prototype in free-suspension test configuration ............... 

4. Residual functions for Space Station common module prototype . ... . .. .. ..... .. .. .. .... .. ... .. . . .... . 

5. Space Shuttle payload simulator structure .......................................................................... 

6. Shuttle payload simulator in free-suspension modal test configuration ............................. 

7. Trunnion interface region of payload simulator .................................................................. 

8 Testlanalysis trunnion response functions, after model updates (zero mass loading) ......... 

9. Payload simulator with rectangular mass attached to one trunnion .................................... 

10. Detailed view of added mass and trunnion region of payload simulator ............................ 

11. Rectangular mass with excitation and response points ....................................................... 

Initial tesdanalysis comparison of Y-bending responses for attached mass, excitation 

at mass corner point 5, response at mass center point 3 (5 Y, 3 Y) ........................................ 

12. 

41 

13. Comparison of test/analysis Y-bending responses (5 Y, 3 Y) for attached mass 

after model updates ............................................................................................................. 41 

Comparison of test and updated analytical Z-bending responses (5Z, 32) 

for attached mass ................................................................................................................. 

14. 

42 

15. 42 

16. 

Measured and updated analytical axial responses (6X, 3X) for attached mass ................... 

Analytical residual function as difference between full and modal response 

functions (5Y, 3Y) ................................................................................................................ 43 

Test residual function and smooth curve-fitted function, shown as difference between 

full and synthesized response functions (5Y, 3Y) for attached mass ................................... 

17. 

44 

V 
LIST OF FIGURES (Continued) 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 

26. 

27, 

28. 

29. 

30. 

31. 

32. 

Comparison of test and updated model residual functions, Y-bending (5Y, 3Y) 

for attached mass ................................................................................................................ 

Measured and updated analytical residual functions for trunnion Z-bending at mass 

attachment location (52, 32) .............................................................................................. 

Residual functions from test and updated model for mass-loaded trunnion 

in axial direction (6X, 3X) .................................................................................................. 

Residual functions from test and analysis for X direction (6X, 3X), showing 

an example of breakdown of second-order curve fit for noisy test data ............................ 

Rotational responses of mass (3RX, 3Rx), direct and calculated from translational 

responses . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Comparison of direct and calculated rotational responses (3RY, 3RY) for attached 

mass .............. ......... ... .. ....................................... .. .. .......... .. . .. ...... . . ...... ..... ... ....... ..... ....... ... .. 

Direct and calculated rotational response functions (3RZ, 3R2) for attached mass .......... 

Comparison of test/analysis rotational responses about X axis at mass 

center (3RXIMX) ................................................................................................................ 

Test/analysis 2-bending responses (5Z, 22) at attached mass before model updates ........ 

Test and analytical trunnion Z-bending responses (5Z, 22) after model correlation ......... 

Comparison of test and updated analytical rotations about Y axis (3RYIMY) at mass 

center . .... .. ......... . ...... . . .. .. ..... .. ... .... ........ ........ ........ ..... .. . .. .... ... ..... . . . ... . ... .. ... ... . ... ..... .. . ........ .. . 

Test and updated analytical trunnion rotations about Z axis (3RZIMZ) at mass center ..... 

Test and updated analytical residual functions about X axis (3RXIMX) for attached 

mass ..... . . . ..... .. .. .... ... ........ . ....... ............... .... .... .. ........ .... .. .. .. .. .. .. ..... . ... ... .. ... ... ..... .. .. ... ... ..... .. . 

Test and updated analytical rotational residuals about Y axis (3RYIMY) at mass center ... 

Test and updated analytical rotational residuals about 2 axis (3RZIMz) at mass center ... 

44 

45 

45 

46 

48 

49 

49 

51 

52 

52 

53 

54 

56 

57 

58 

vi 
LIST OF TABLES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

IO. 

11. 

12. 

Comparison of test and analytical free-free modes for payload simulator, after model 

correlation (zero mass loading) ........................................................................................... 24 

Residual flexibility values for Shuttle payload simulator, before and after model 

updates to trunnions (zero mass loading, 14 free-free modes) ........................................... 26 

Effect of different boundary mass loadings on derived constrained frequencies 

and mode orthogonality for hybrid Rubin method (24 boundary DOFs, 20 modes) .......... 28 

Derived constrained modes for hybrid Rubin method (0.5-lb mass, 12 boundary DOFs, 

20 modes) ............................................................................................................................ 29 

Comparison of derived constrained modes for hybrid MacNeal method with different 

mass loadings (20 mass-additive modes, full residual flexibility matrix, 24 boundary 

DOFs) .................................................................................................................................. 30 

Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (0.5-lb loading, full residual flexibility, 24 boundary DOFs) ................. 3 1 

Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (5-lb loading, full residual flexibility, 24 boundary DOFs) .................... 3 1 

Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (zero mass loading, full residual flexibility, 24 boundary DOFs) .......... 32 

Effect of retained residual terms on derived constrained modes in hybrid MacNeal 

method (0.5-lb loading, 20 mass-added modes, 24 boundary DOFs) ................................. 33 

Derived constrained modes for different numbers of mass-added modes in hybrid 

MacNeal method, with only diagonal Grbb residual terms (0.5-lb loading, 

12 boundary DOFs) ............................................................................................................. 34 [I 

Derived constrained modes for different numbers of mass-added modes in hybrid 

MacNeal method, with only diagonal Grbb residual terms (zero mass loading, 

12 boundary DOFs) ............................................................................................................. 35 [I 

Variation of residual flexibility values for different mass loadings and number 

of retained mass-additive modes ......................................................................................... 36 

vii 
13. 

14. 

15. 

I 

LIST OF TABLES (Continued) 

Derived constrained modes for the basic mass-additive method with no residual terms 

(0.5-lb loading, 12 boundary DOFs) .................................................................................. 36 

Derived constrained modes for basic rnass-additive method (24 boundary DOFs) ........... 37 

Residual flexibility values for test and correlated model at trunnion mass attachment 

location ............................................................................................................................... 46 

... 

Vlll 
TECHNICAL MEMORANDUM 

HYBRID RESIDUAL FLEXIBILITYAMASS-ADDITIVE METHOD 

FOR STRUCTURAL DYNAMIC TESTING 

1. INTRODUCTION AND BACKGROUND 

A large fixture was designed and constructed for modal vibration testing of International Space 

Station elements. This fixed-base test fixture, which weighs thousands of pounds and is anchored to a 

massive concrete floor, initially utilized spherical bearings and pendulum mechanisms to simulate 

Shuttle orbiter boundary constraints for launch of the hardware. Many difficulties were encountered 

during a checkout test of the common module prototype structure, mainly due to undesirable friction and 

excessive clearances in the test-article-to-fixture interface bearings. Measured mode shapes and frequencies were not representative of orbiter-constrained modes due to the friction and clearance effects in the 

bearings. As a result, a major redesign effort for the interface mechanisms was undertaken, The total 

cost of the fixture design, construction and checkout, and redesign was over $2 million. 

Because of the problems experienced with fixed-base testing, alternative free-suspension methods were studied, including the residual flexibility and mass-additive approaches. Free-suspension 

structural dynamics test methods utilize soft elastic “bungee” cords and overhead frame suspension 

systems that are less complex and much less expensive than fixed-base systems. The cost of freesuspension fixturing is on the order of tens of thousands of dollars, as opposed to millions, for large 

fixed-base fixturing. In addition, free-suspension test configurations are portable, allowing modal tests to 

be done at sites without modal test facilities. For example, a mass-additive modal test of the ASTRO-I 

Shuttle payload was done at the Kennedy Space Center launch site.* In the following sections, the massadditive and residual flexibility test methods are described in detail. A discussion of a hybrid approach 

that combines the best characteristics of each method follows and is the focus of study described in this 

Technical Memorandum (TM). 

1.1 Mass-Additive Modal Test Method 

It is well known that, for coupled substructures, the interfaces must be adequately characterized 

to allow accurate transient response and operational loads analyses. This is especially true for Shuttle 

payloads that are constrained at a number of discrete points (fig. 1). The dynamic response of the payload is largely controlled by the stiffness of the interfaces (trunnions and keel). For this reason, one 

of the primary objectives of modal testing of Shuttle payloads is to measure interface characteristics 

so that the mathematical model can be refined at the interfaces. 
Trunnions 

Trunnions 

Figure 1. Shuttle payload carrier with orbiter connections. 

Obviously, a constrained-boundary test allows direct measurement of the modes that are controlled by the interfaces. However, in a free-boundary support configuration, the interface modes occur 

at higher frequency than the structure global modes. Due to the large frequency bandwidth involved, this 

increases the time required to measure both the local interface modes and the global modes in one test. 

Further, it is difficult to develop a mathematical model with fidelity over a large frequency range. 

In the mass-additive the structure boundaries are mass loaded to lower the frequencies of the interface modes and bring the modes into the frequency bandwidth of the global modes. This 

allows global free-free modes and the local interface modes to be easily measured in a single test. The 

idea of attaching masses to structure boundary degrees of freedom (DOFs) in modal testing is similar to 

the concept presented by Benfield and Hruda9 for modal synthesis. A Shuttle payload with mass-loaded 

boundaries is shown in figure 2. In addition to narrowing the frequency bandwidth, the added masses 

allow the interfaces to be exercised to a greater degree than is possible in a usual free-boundary configuration. Interface modes are obtained due to extensive excitation of the structure in the interface regions. 

From a modeling standpoint, the added masses are easily included as rigid bodies having known 

mass and inertia attached to the boundary DOFs. Constrained-boundary modes and frequencies can be 

obtained from a finite element model that has been updated to agree with measured free-free massadditive modal data. First the masses are analytically removed from the updated model, then the boundary DOFs are constrained. Alternatively, the test-verified model with boundary masses removed can be 

coupled to a model of the launch vehicle and used for liftoff and/or landing loads analysis. 

2 
Air Bag Suspension (Four Places), Note: Air Bags attached to 

modal node Doint at trunnion 

Trunnion Mas 

and Trunnion 

Simulator 

/ 

;ses 

CPSS Mass 

Simulator. 

EAClMass 

ilator 

Figure 2. Materials Science Laboratory payload in mass-additive test configuration. 

A disadvantage of the mass-additive technique is that, when properly applied in a component 

mode synthesis approach, many mass-added free-boundary modes are required to derive accurate constrained modes and frequencies, In reference 4, it is shown that for some structures the number of modes 

required is impractical. Advantages of the method are general ease of performing the measurements 

in the context of the global modal test and ease of understanding the methodology. Further, it is possible 

to obtain rotational measurements for the locations where masses are attached to the structure. As shown 

in figure 2, rectangular masses are typically used in the mass-additive technique. Accelerometers can be 

attached at the corners and outer edges of the masses to obtain rotational data indirectly, or rotational 

sensors can be attached at the mass center for direct measurements. If rotational accelerometers are used, 

more accurate measurements are possible in comparison to non-mass-loaded cases due to the increased 

amplitudes of the rotational motion. This can be of particular importance for torsional measurements. 

3 
1.2 Residual Flexibility Test Method 

The concept behind the residual flexibility approachlo is that a subset of the structure’s free-free 

(unconstrained) mode shapes, along with the residual flexibility of the boundary DOFs, can be used 

to derive the constrained-boundary modes and frequencies. This can be easily understood for the case 

of Shuttle payloads, where the interface or boundary DOFs are the translational coordinates of the 

payload trunnions and keel that are constrained by the orbiter (fig. 1). Residual flexibility of the 

interfaces provides an approximation of the global free-free modes not measured. Further, it has been 

shown for simple structures that residual flexibility for a given boundary DOF is simply the flexibility 

of that boundary. This technique of using an approximation of the effects of neglected higher-order 

modes, or residual modes, to improve the accuracy of reduced-basis mathematical models was first 

presented by MacNeal. l1 An improvement of the basic residual approach by including second-order 

effects, or residual inertia, was presented by Rubin. l2 References 13-27 also present investigations 

of the residual flexibility approach and related techniques. 

Reference 10 describes the application of the residual flexibility approach to the Space Station 

common module prototype (fig. 3). The general test procedure is as follows: (1) Free-boundary mode 

shapes and frequencies are obtained in the usual manner by using shakers to excite the structure at the 

boundary DOFs; (2) modal parameters are then obtained by curve-fitting the resulting responses as 

functions of frequency over the desired bandwidth; and (3) residual functions, or residual flexibilities, 

are obtained by subtracting a curve-fitted frequency response function (FRF), which only covers a 

frequency band of interest, from the full measured FRF, which covers a larger bandwidth. Figure 4 

shows two residual functions in acceleration/force format for the common module prototype. For most 

Shuttle payloads, there are seven of these residual functions to be measured-one for each trunnion/keel 

translational DOF that is constrained in the orbiter. After the required test data are obtained, the mathematical model of the structure is modified to obtain the best possible agreement with test mode shapes, 

frequencies, and residual functions. The model is then used in either of two ways: (1) With boundary 

DOFs analytically constrained to obtain fixed-base modes or (2) with boundary DOFs connected to the 

orbiter interfaces for coupled Shuttle/payload loads analysis. This general procedure applies for any kind 

of structure that could be constrained or coupled to other structures in service. 

The main drawback using the residual flexibility method is the difficulty in obtaining accurate 

measurements for the boundary or interface residual functions. Although the functions in figure 4 are 

clean, they represent average values generated using modal testing software. Most residual measurements are very noisy, making it difficult to identify the residual function. Residual flexibility values 

are also very small numbers, typically ranging from 1.OE-3 to 1.0E-6 in/lb for translational terms. 

In addition, the interface FRF must have a well-defined stiffness line (a general upward-sloping trend 

in acceleration/force format, as evident for higher frequencies in fig. 4) for accurate residual values 

to be obtainable. The major advantage of this approach is the quick convergence of constrained modes 

which are derived from free-free modes and residuals, with a reasonably small number of modes. 

I 4 
Figure 3. Space Station common module prototype in free-suspension test configuration. 

- Test 

--- Model 

Fwd Trunnion, y 

104 

105 I I I Ill 

10’ 102 1 03 

I 

I I I I I I1111 

1 00 

10’ 

102 

1 03 

104 

105 

- Test 

Model 

Aft Trunnion, x 

--- 

- 

E I 

lI I I I II 1061 I I I I I I Ill 

10’ 1 02 103 

Frequency (Hz) Frequency (Hz) 

Figure 4. Residual functions for Space Station common module prototype. l7 

5 
2. DEVELOPMENT OF THE HYBRID TEST METHOD 

Comparative analysis of the mass-additive and residual flexibility modal test methods described 

in the previous section shows the following, in summary: 

Residual flexibility approach 

I - Major advantage: More accurate in terms of deriving constrained modes from freesuspension data; that is, the convergence rate is higher in the component mode synthesis 

formulation. 

- Major disadvantages: Difficult to accurately measure residual flexibility values, 

and rotational data for structure boundaries can only be obtained using expensive 

rotational sensors. 

Mass-additive technique 

- Major advantages: Very attractive in terms of simplicity of the methodology and general 

ease of performing measurements; rotational boundary responses can be measured rather 

easily using translational accelerometers. 

- Major disadvantages: Large number of mass-added modes are required in deriving fixedboundary modes, making the method impractical for some structures. 

I Consideration of the advantages and disadvantages of each technique led to the development 

of a hybrid free-suspension test/analysis approach with more general applicability and flexibility than 

either of the methods alone. The hybrid formulation is essentially the residual flexibility approach 

modified to allow mass loading of the structure boundary DOFs for which residual values are to be 

obtained. Such a formulation overcomes or helps in solving at least three problem areas identified 

in the comparative analysis: 

Large number of mass-additive modes required for deriving constrained-boundary modes: 

Overcome because the inclusion of residual flexibility terms with mass-additive modes 

results in a high rate of convergence for derivation of constrained modes. 

Inability in the residual approach to measure rotations at boundary DOFs: Overcomedue 

to the possibility of attaching translational accelerometers to corners and edges of boundary 

masses. Viewed a different way, rotational motion is enhanced allowing the possibility of 

more accurate measurements with rotational sensors if such instrumentation is available. 

6 
Large bandwidth required for measuring residual terms: Large bandwidth required 

for measuring residual terms is reduced due to presence of boundary masses, which lowers 

the frequency of local boundary or interface modes. 

A hybrid approach of this type should make free-suspension modal test methods applicable 

to a wider variety of payloads and structures. For example, a Shuttle payload could be considered 

for testing using the residual flexibility method even though it has interface FRFs exhibiting a large 

number of modes that effectively hide the stiffness lines. In such a case, the interfaces could be mass 

loaded to bring each fundamental interface mode to a lower frequency-below some of the modes that 

mask the stiffness lines. As discussed in the previous section, a characteristic of the mass-additive 

approach is that interface bending frequencies can be lowered into the frequency bandwidth of the test. 

The use of mass loading in residual flexibility testing should also make the technique feasible for industrial applications where rotational residual flexibilities may be required. Application of the method for 

general boundary conditions has been hindered due to difficulties in performing rotational measurements. Reference 7 describes the use of rectangular masses and translational accelerometers to estimate 

the rotational FRF of a structure. Such an approach, or use of rotational sensors with boundary masses, 

should allow rotational residual functions to be measured with greater accuracy than in the past. 

2.1 Derivation of Governing Equations for Hybrid Method 

2.1.1 Development of Hybrid Transformation Matrix 

In a hybrid residual flexibility/mass-additive modal test, rectangular masses are attached 

to boundary or interface DOFs, as shown in figure 2. For this configuration, the undamped equations 

of motion are written as 

or 

where the mass matrix for the structure with boundary mass loading is [a] = [MI + [AM], and [AM] 

represents the added mass. Free-free elastic and rigid-body modes [ an] and [ aR] are obtained from an 

eigensolution of equation (1) for the case {F} = 0 and are used in subsequent equations discussed in this 

section. These eigenvectors are referred to as the mass-additive or mass-loaded mode shapes. 

Rubin12 showed that flexible-body displacements for a structure can be written as a first-order 

approximation of residual effects, 

where [GI and [G, I are the free-free and constrained flexibility matrices, respectively, and for configurations discussed in this TM, apply to the mass-loaded structure. Generally, terms in the equations 

I 

7 
presented in this section refer to quantities for the structure with mass-loaded boundaries unless noted 

otherwise. The flexibility transformation matrix is [A] = [I] - [a] [@R] [MR]-'[@R]~. It is noted that 

the mode shape matrices in the transformation matrix [A] are the rigid-body modes of the mass-loaded 

structure (from eigensolution of eq. (1) with {F}= 0), and the generalized mass [MR] is generated using 

the rigid-body modes. If the contribution of modes to be retained or measured is removed from the 

deflection for the flexible structure, the residual flexibility matrix is obtained, as shown in equation (4): 

Note that [G,] = [ @,][K,]-l[@,]Tis the flexibility corresponding to the retained or measured elastic 

mass-additive modes [Qn], and that the generalized stiffness in this expression is based on the retained 

modes. Equation (4) demonstrates the physical meaning of the residual flexibility [G,]: it is the difference in the full flexibility of the structure and the flexibility based on a set of retained or measured 

modes. That is, it provides an approximation of mode shapes not retained in a reduced model or not 

measured in a modal test. If desired, this procedure can be carried a step further to include second-order 

residual effects.12 but these will not be shown until later in this TM. 

Both MacNeal l1 and Rubinl* developed a stiffness matrix formulation for component mode 

synthesis using residual flexibility. Martinez, Carne, and Miller13 discussed the MacNeal and Rubin 

representation expressed in a form similar to the Craig-Bampton component synthesis approach.28 

This was a significant step in facilitating practical use of the residual flexibility method, due to wide use 

of the Craig-Bampton method for model reduction in the aerospace industry. Following the approach 

of reference 13, but utilizing quantities for boundary DOFs with mass loading, structure displacements 

can be written in the partitioned form 

where [ @] is the (N x n) matrix of retained mass-additive modes, and [G rb 1 is the boundary partition 

of the (N x N) residual flexibility matrix defined in equation (4). Note that N is the size of the original 

or unreduced model, and n is the number of retained modes. If the lower partition of equation (5) is 

solved for the boundary forces {Fb}, and the resulting expression is substituted back in to equation (3, 

it can be shown that 

This form of displacement for the mass-loaded structure results in modified free-free elastic modes 

(coefficient of (9) in eq. (6)) and modified residual attachment modes (coefficient of {ub}). Assembly 

of equation (6) with the identity {ub} = {ub} results in the final expression for displacements of the 

mass-loaded structure, 

8 
and the overbar is used in [F] to indicate mass-added conditions. 

2.1.2 Free-Free Unloaded Model in Terms of Mass-Added Modes and Residuals 

In the process of deriving constrained-boundary mode shapes and frequencies from a free-free 

mass-added model, it is first necessary to express the equations of motion for the free-free structure with 

unloaded boundaries in terms of mass-additive modes and residual flexibility. This step is essential since 

constraining the boundary DOFs without first analytically removing the masses would yield incorrect 

results in most cases. For example, for a structural model of a Shuttle payload that has only certain 

translational boundary DOFs constrained in flight, the unconstrained mass-loaded DOF would experience undesired inertia loading influencing the mode shapes and frequencies. 

The unloading process is accomplished by subtracting the added mass terms, [MI, from equation (2), yielding an alternate expression for the model without boundary masses, 

which can also be written in partitioned form: 

In equation (9), the mass addition affects all six DOFs at an attachment point, with the possibility arising 

that a boundary mass can load interior DOFs, and not just the DOFs partitioned to the matrix boundary. 

The implications of this are discussed further later in this section. The partitioned form of the residual 

flexibility matrix corresponding to equation (9) is given by 

Substitution of equation (7) into equation (8) allows the equations of motion to be written 

in terms of generalized coordinates and physical boundary coordinates: 

9 
I, 

(1 1) 

-T Premultiplication of equation (1 1) by [ T ] yields the desired reduced model with unloaded boundary 

! DOFs in terms of mass-additive modes and residual flexibility, 

which can be written in the form 

0 

Fb 

where [z] = [TIT[M] [TI and [z] = [TIT[K] [TI. For mass-additive modes normalized to unit modal 

mass, it can be shown using the approach of reference 13 that 

[z]= 

-@$G-' 

'bb 

G-' 

%'bb 

where SZ,, is the diagonal matrix of retained frequencies con, and Qnh is the boundary partition ~ of the .. . -1 Tretained -- modes. Also in equation (14), [Jbb]=[Grbb] [Hbb][Grbb]-' and [Hbb]=[Grb] [M][Grb], 

where Grb contains the two partitions of the residual flexibility matrix shown in equation (5). 

The second-order residual effects for the mass-loaded configuration, or residual inertia effects, 

are contained in [Hbb]. The mass and stiffness matrices in equation (14) represent the full-residual or 

Rubin method, with only residual damping effects neglected. If the residual inertia effects are neglected 

in equation (14), the MacNeal formulation is obtained and the generalized mass matrix for the structure 

I1 

10 
with mass-loaded boundaries, [z], becomes the identity matrix. Neglecting residual inertia is a commonly used approximation to simplify the technique.l3?l4 In subsequent references to mode shapes in 

this TM, it is understood that retained modes are under discussion, and the rz subscript is dropped. 

2.1.2.1 Development of the Generalized Added-Mass Matrix. It remains to evaluate the 

generalized added mass term in equation (13). Use of the transformation defined in equation (7) with 

[AM] from equation (9) yields the following expression: 

['l'[dM][T]=[dM]= 

which can also be written in the form 

where 

Mii Mib 

[" [AM] [ .]= [ ..I= 

[ibi ibbl ' 

These equations for the generalized added-mass matrix are for the general case where only a portion 

of the DOFs at a given mass attachment point are to be constrained or coupled to another structure, 

which is the case for Shuttle payloads. The implication of this is that some of the interior DOFs have 

mass loading, as can be seen in equations (17)-(20). For the case where all six DOFs at each mass 

attachment point are constrained or used in coupling, the problem is greatly simplified, since all massloaded DOFs are partitioned to the matrix boundaries, and [AMji] = [AMib] = [AMbi] = 0. In the case 

where no interior DOFs are mass loaded, it can be seen from equations (17)-(20) that the generalized 

added-mass matrix reduces to 

11 
I. 

Also for equation (21), [dMbb) has dimensions (nbxnb), where nb= 6 n,, and n, is the number of boundary masses. It should be noted that in the context of this discussion, interior DOFs can include part of the 

mass attach point DOF, such as rotational coordinates. The term interior does not imply that the DOFs 

are actually on the physical interior of the structure, far removed from the physical boundaries, but rather 

that the DOFs have been partitioned in the mathematical model to the interior of the matrices. 

Although equations (17)-(20) are very useful for seeing the effects of constraining or coupling 

all six DOFs at each mass attach point versus using only a portion of the mass attach point DOFs, they 

do not show which added-mass terms correspond to residual inertia. This is important because generally 

the residual inertia terms are not utilized due to difficulty in accurately measuring them, and the 

MacNeal formulation is used. In order to observe which terms in equation (16) (or eqs. (17)-(20)) are 

residual inertia effects, it is necessary to perform the transformation shown in equation (15) in two steps. 

As shown in reference 13, the transformation defined in equation (7) can be written as the product of 

two transformation matrices, 

, 

I 

where [TI] is given in equation (3, and [GI is the transformation that allows the elimination of boundary forces in going from equation (5) to equation (7). 

If the transformation of the equations of motion for the structure with unloaded boundaries, 

equation (S), is done in two steps using [TI] and [G] separately, the following expression is obtained: 

[.I’ [ TIT {[MI-[ I} [ TI] [ ..I 

The resulting transformed mass and stiffness, [z] and [F], respectivley, are identical to equation (14), 

but the transformed added-mass matrix [dM] shown in equations (24)-(28) looks different than 

equations (16)-(20) because the original added-mass matrix [AMI appears in full form without 

partitioning: 

12 I 
where 

[dMii]=~TAM@-%TG-lGTAM@-~TAMGr G-’%+% T G -1 G T AMG, G-’% 

‘bb ‘b ‘bb ‘bb ‘b ‘bb 

‘bb 

[Ejb]=pTAMGrbG-’-@b T G -1 G T AMG,G- 

‘bb ‘bb ‘b 

[Ebi]=[G-’GTAM@-G- 1T G AMG, G-I 

‘bb ‘b ‘bb ‘b ‘bb 

Analogous to the discussion following equation (14) and similar to an approach discussed 

in references 12-14, the residual added inertia effects can be found in equations (25)-(28). Defining 

[dlbb] = [Grbh]-’[AHbb][Grbb ]-’, where [AHbb] = [Grb]T[AM] [Grb], it can be seen that the secondorder residual effects for the added-mass matrix, [ AM], are contained in [ AHbb] just as the residual 

inertia effects for the mass-loaded configuration are contained in [ Hbb I. Using the expressions for 

[AHbb] and ldlbb], equations (25)-(28) can be written in more compact form: 

2.1.2.2 Equations of Motion in Terms of Mass-Added Modes and Residuals. Equations (29)- 

(32), along with equations (12)-( 14), define the hybrid method for the general case where not all mass 

attach points are to be constrained or coupled to another structure, and mass addition can occur for 

13 
interior terms of the mass matrix. Both residual flexibility and residual inertia effects are also included at 

this stage. Repeating equation (13) as equation (33) and utilizing equations (29)-(32) for [dM] results in 

the following general formulation for the free-free unloaded model equations of motion in terms of 

mass-additive modes and residuals: 

where 

[E] - [rn] = 

(34) 

and the generalized stiffness is unchanged from equation (14), 

sym. ‘bb 

(35) 

The general hybrid method formulation given in equations (33)-(35), which corresponds to the 

Rubin method, can be simplified considerably if the residual inertia effects corresponding to [ 1 and 

[MI, which are [ Jbb 1 and [AJbb 1, respectively, are neglected. In that case, 

r - 

I-~~AM~+%~G-I G~ M~+~~AMG G-’ q, -Q~AMG G-’ 

‘bb ‘b ‘b ‘bb ‘b ‘bb 

-G-’G AM@ 0 

‘bb ‘b 

which corresponds to the MacNeal approach. The generalized stiffness is unchanged and is given 

by equation (35). 

I 14 
Finally, for the case where all six DOFs at each mass attachment points are to be constrained 

or coupled to another structure, and all mass-loaded DOFs are partitioned to the matrix boundary, 

the formulation is simplified further. In that case, the transformed added-mass matrix[dM] is given 

by equation (21), and the transformed mass for the unloaded model in terms of mass-additive modes 

and full residuals is given by 

which differs only in the boundary partition from the basic Rubin formulation with no mass loading. 

If residual inertia is neglected in equation (37), the transformed mass matrix for the unloaded model 

becomes 

[E]-[dM]=l I . 

Neglecting [htfbb] is justified on the basis of equations (20), (28), and (32), which show that 

[AJbb]=[mbb] for the case of mass loading limited to boundary DOFs. Further, [dMbb] is small 

in comparison to the diagonal terms of the identity matrix in equation (38). An alternative formulation 

neglecting [Jbb] in equation (37) but retaining I&kfbb] could also be investigated. Numerical results for 

this case were not investigated in the context of the current study. In either case described in equations 

(37) and (38) for mass loading limited to boundary DOFs, the generalized stiffness is unchanged and is 

defined in equation (35). 

2.1.3 Constrained Unloaded Model in Terms of Mass-Added Modes and Residuals 

The reduced hybrid model defined by equation (33) is the form used for coupling to other structures or components. For cases where it is desired to derive fixed-boundary modes and frequencies, 

another step remains in the analysis. Starting with the most general formulation given by equations (33)- 

(35), the hybrid reduced model with constrained boundaries is obtained by setting boundary displacements to zero and striking all matrix rows and columns with boundary terms. This operation results in 

the following expression: 

where for use of full residuals and mass loading possible for interior and boundary DOFs, the generalized mass matrix is, from equation (34), 

15 
and the generalized stiffness is given by 

If residual inertia is neglected, the constrained-boundary generalized mass reduces to 

[(M-M)nn]=[I-QTdM~+~TG-l 'bb GT 'b dMQ+QTdMGrbG-' 'bb cQ, 1 . 

For the case of mass loading limited to boundary DOFs, but including full residuals, the constrainedboundary generalized mass becomes 

Further, if residual inertia is neglected, the generalized mass becomes the identity matrix. In all of these 

cases, the constrained-boundary generalized stiffness is unchanged and given by equation (4 1). 

Interestingly, equation (43) has the same form as the constrained-boundary generalized mass 

matrix for the basic Rubin method with no mass loading. However, it must be kept in mind that the 

mode shapes shown in equation (43) are mass-additive modes, so the effect of mass loading does appear 

upon the dynamics of the structure. The observation that no AM terms appear in equation (43) does point 

out an interesting fact, however. If mass loading is limited to boundary DOFs, that is, DOFs partitioned 

to the matrix boundary, it is not necessary to analytically remove the added mass before constraining the 

model. However, as shown in equations (40) and (42), if mass loading of interior DOFs occurs, the 

added mass must be removed before constraining the model. If the model is to be used for substructuring 

and coupling to other components, the added mass must always be analytically removed before the 

coupling is done. 

The frequencies obtained from eigensolution of equation (39) are the constrained frequencies 

of interest. However, to obtain physical constrained-boundary mode shapes, a back-transformation 

of the modes obtained in the eigensolution of equation (39), [@,,,I, must be done. This is accomplished 

by using the transformation matrix defined in equation (7) and forming the following matrix equation 

for constrained physical modes, 

Zeros must be added in the last nb rows of the free-free mode shape matrix to make it compatible 

with [TI, since nb DOFs were eliminated by constraining the boundaries in equation (39). 

16 
2.2 Frequency Response Formulation of Hybrid Method 

The hybrid matrix equations of motion developed in the preceding sections are extremely 

useful for pretest analysis to determine which residual terms are needed and how many mode shapes 

are required in dynamic testing of the mass-loaded test article. If it is desired to develop a test-verified 

constrained model, these equations allow convergence studies to determine if a selected number of mode 

shapes and residual terms is sufficient for obtaining accurate constrained modes and frequencies. However, when it comes to experimental implementation of the hybrid method, an FRF formulation is more 

practical because test residual functions and mode shapes are obtained from measured FRFs. Further, it 

can be seen from the matrix formulation in equation (4) that computation of residual flexibility requires 

the full flexibility (inverted stiffness) matrix. Obviously, the full flexibility matrix is not available 

experimentally. 

By developing a frequency response formulation, a practical implementation for developing a 

test-verified model involves the following: (1) Comparison of free-free modes for model and test, with 

appropriate model updates; (2) comparison of measured and predicted boundary FRFs and residual 

functions (obtained from FRFs), with model updates as required; and (3) constraining of the model 

boundary DOFs or coupling to another structural component. The response function equations are 

discussed in the following paragraphs. 

As described by Rubin in reference 12, displacement can be written as a function of frequency, 

where [Y] is the displacement/force FRF matrix and (F} is the applied force vector, both as functions 

of frequency. However, it must be noted that in contrast to reference 12, all terms in equation (45) and 

following equations in this section refer to the mass-loaded structure. The residual FRF matrix, or 

residual function matrix as it will be designated here, is obtained by subtracting from the full FRFs in 

equation (45) the modal FRFs containing rigid-body and elastic free-free modes that are to be measured 

or retained. The undamped modal FRF matrix is given by 

where [M,] is the generalized mass associated with the measured or retained modes [ @,I,and [A,l is the 

diagonal matrix [ o, - m2]. The residual function matrix becomes 

which can be approximated over the frequency range of interest by the undamped form, 

17 
corresponding to the residual flexibility matrix given by equation (4). The residual inertia matrix in 

equation (48) is given by [H,] = [G,] [MI [G,]. For comparison of residual flexibility values, the 

undamped forms of equations (46)-(48) should be sufficient. However, if analytical and test FRF results 

are being compared, inclusion of damping may be desirable in some instances. In that case, 

[An] = [o,' + i2~,wn - w in equation (46) and the residual function takes the form 

T- 

'I 

where [B, I is the residual damping matrix. 

For practical computations, residual functions are obtained individually as functions of frequency I rather than in matrix form. Residual flexibility for a particular residual function is the value of the 

function at zero frequency, as can be easily seen from equations (48) and (49). Each value of [ G,] 

obtained in this manner is equal to the corresponding value from equations (4) and (10). 

It is noted that the second-order term [H,] in equations (48) and (49), the residual inertia, was 

only used in this TM for obtaining smooth curve fits of residual functions. Residual inertia was not 

included in reduced models or in the model correlation work described in subsequent sections. In section 2.3, the curve-fitting procedure for estimating residual flexibility values from experimental residual 

functions is described. 

2.3 Statistical Least-Squares Curve-Fitting Procedure for Identification 

of Experimental Residual Flexibility Values 

As will be shown later in this TM, noisy frequency response and residual test data are often 

observed at low frequencies, particularly in antiresonance regions of the FRF. In addition, undesired 

peaks or spikes generally occur in experimental residual functions at system resonances. These spikes 

are observed due to inaccuracies in approximating damping in the synthesized FRF, which are generated 

using system identification software in modal testing. Because of these effects, a second-order polynomial curve fit of experimental residual functions is required to determine the residual flexibility (constant coefficient) and the residual inertia (second-order coefficient) shown in equations (48) and (49). 

The procedure for determining experimental residual terms is presented for the damped form of the 

residual function, equation (49), for generality. The approach presented in this section is the curve-fitting 

method developed by Bookout,22 and also used by Tinker and Bo~kout.~~ First, it can be seen that 

the residual function appears in the general complex form a+ib. This general form can be equated 

to equation (49), 

a + ib = [ G, ] + m2 [ H, ] - iw[B,] , (50) 

and like terms can be separated to yield 

[B,] = -b/o 
directly from the imaginary term. Real terms are given by 

a = [G,]+02 [ H,-] 

and must be determined by curve fitting. The least-squares conditions are obtained by expressing 

equation (52) in matrix coefficient form, 

where 

[R]=[l 02] 

and 

(54) 

The residual flexibility and residual inertia terms are then obtained as 

[ X] = [ R]- ' a . (56) 

These residual values are compared to analytical residuals in the test/model correlation process. 

A theoretical residual function in displacement/force format is relatively flat at low frequencies 

and has slight upward curvature at higher frequencies. Variations of consecutive values of the residual 

function should therefore be small. When examining the residual functions produced from test data, 

these characteristics can be seen in an overall sense. However, in regions of poor or noisy data, consecutive residual values can have large variations in magnitude. It is apparent that a weighting function is 

needed that applies low weighting to data points having large variation with respect to neighboring 

points and high weighting to data in regions of small variations. 

The required weighting function can be expressed statistically in terms of sample variance, where 

each sample consists of two or more neighboring data points. Since the variance represents the amount 

of variation between data points in the sample, regions of the residual function with high variation 

(containing spikes and noise) can be given low weight by defining the weighting function as the inverse 

of the sample variance, 

19 
i=l 

where s2 is the sample variance, k is the total number of data points, p is the number of data points 

in the sample, and x is the mean of the sample.22 

Premultiplying both sides of equation (53) by the weighting function W gives 

Next, premultiplying both sides of equation (58) by [RITand solving for [Xl results in the final form 

of the residual flexibility and residual inertia terms, 

In this form, there is no need to normalize the weighting function W. 

By stepping through the test data, the variance of each data point with respect to the neighboring 

points can be calculated. The weighting value for each data point is set equal to the inverse of the variance assigned to that data point. This gives the desired effect that, when the variances of neighboring 

data points are high, the weighting function value is low, and vise versa. Incorporating the weighting 

function of equation (57) into the least-squares curve fit allows determination of smooth experimental 

residual functions having the characteristics previously described for theoretical displacement/force 

functions. 

In previous studies, different weighting matrices generated by examining samples with two, 

three, and four data points were used in the curve fit process. The error range for the residual flexibility 

term produced by examining different sample sizes was found to vary considerably, with best results 

achieved using three data points. Based on these earlier findings, sample sizes of three data points were 

used in the curve fitting described in this TM. In addition, the frequency range of curve-fitted data was 

varied to determine the effect on accuracy of the residual flexibility value. This was done because 

experimental residual functions typically are very noisy at low frequencies. Further description 

of the statistically weighted curve-fitting procedure is given in reference 22. 

20 

1 
3. ANALYTICAL APPLICATION OF HYBRID METHOD 

TO SHUTTLE PAYLOAD SIMULATOR 

A simple Shuttle payload simulator was designed specifically for studies of free-suspension 

modal test techniques. This payload simulator has frequency content somewhat similar to real payloads, 

with a number of well-spaced global mode shapes, and it also has prominent flexible interfaces (simulating Shuttle payload trunnions which interface with the orbiter). This structure, shown in figures 5-7, was 

also designed to be simple enough that high confidence could be obtained for the model through correlation to free-free test data. Dimensions of the beam-like interfaces (trunnions) were chosen to provide 

prominent stiffness lines, or well-defined linear regions in the frequency responses at higher frequencies, 

for enhancing the measurement of residual flexibility values. 

Initially, the basic residual flexibility method,1° based on equation (14) but with no mass loading 

of the trunnion interfaces, was applied to the payload simulator. However, the procedures are the same 

when mass loading of the boundaries is utilized; such mass loading is discussed later in this section. The 

procedure followed in this approach was to (1) measure a set of free-free modes and frequencies using 

shaker excitation (fig. 6); (2) measure the acceleration/force frequency response functions in X, Y, Z 

directions at the ends of all four trunnions (fig. 7), this time using impact hammer excitation; (3) modify 

the model in a global sense to obtain the best possible agreement with test frequencies and mode shapes, 

(4) modify the model in the trunnion regions to match the experimental response functions in both the 

minima (antiresonances) and maxima (peaks or resonances); and (5) compare the measured and predicted residual flexibility functions and values. It is noted that normally a 5-percent frequency error or 

less is the goal for such model correlation activities, but in this case a 1-percent goal was established so 

that insight could also be gained into accuracy of residual measurements. In addition, hammer impact 

excitation was used for the small, lightweight trunnions because it was believed that connecting a shaker 

to them would unacceptably modify their dynamic behavior. 

3.1 Correlation of Free-Boundary Mode Shapes and Frequencies 

The procedures for mode shape correlation have little difference for residual flexibility and 

hybrid method testing compared to any free-suspension test. It should be noted, however, that a higher 

number of modes may need to be measured and used in model correlation than for a standard free-free 

test. Also, it is possible that the frequency errors should be lower than the standard 5-percent limit 

currently used in model correlation. These potential differences, which are still in the process of being 

fully verified and quantified, are due to the synthesis process described in the previous section of this 

TM, where the free-suspension modes and boundary residual flexibility values are used to derive 

constrained-boundary modes. 

21 
I 

Cross-Sections. Materials 

3 in 

Trunnions: @ 0.75-in Diameter 

(2) AI 6061-T6 

All Other Members: 

0.1 88 in 

3 in 

5 in 

Weld Lines: - 1 

3 in 

f 

t -- 

I 

I 

I 

37.75 in 

36.75 in 

I 

I 

36.75 in 

7 37.75 in 

End Beam 

59.25 in 

2.1 25 in 

170 in 

164 in 

1 

Figure 5. Space Shuttle payload simulator structure. 

I 22 
Figure 6. Shuttle payload simulator in free-suspension modal test configuration. 

Figure 7. Trunnion interface region of payload simulator. 

23 I 
To correlate the model to the measured modal data, the analyst compares the predicted and 

measured frequencies and mode shapes to identify errors in geometry, material properties, and modeling 

assumptions. These comparisons are done based on engineering judgment, either alone or in combination with the use of analytical tools, to find errors and shortcomings in the model. In the cases described 

in this TM, the mode shape correlations were done based on engineering judgment and visual inspection 

of the modes. For the payload simulator, the welded intersections of the hollow box beams comprising 

the model (fig. 5) were found to be the critical areas for global model updating. This was due to the 

nature of the welds-they were outer surface welds, having widely varying stiffness from one weld to 

another and having lower stiffness than the box beam cross sections. Beam elements were used to model 

the welds, and a trial-and-error approach was used to find proper material and geometric properties. 

Table 1 contains the first 10 test/analysis frequency comparisons and mode shape orthogonalities after 

model correlation. For this simple structure, it was possible to obtain frequency errors of 1 percent 

or less. 

Table 1. Comparison of test and analytical free-free modes for payload simulator, after model correlation (zero mass loading). 

Mode 

Number 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

Test 

Frequency 

16.56 

21.21 

46.49 

51.40 

75.73 

83.80 

96.78 

98.68 

105.72 

11 2.87 

135.70 

136.70 

138.70 

167.36 

Analytical 

Frequency 

16.52 

21.29 

46.10 

51.57 

75.33 

84.32 

96.1 1 

98.37 

105.98 

113.03 

136.02 

137.79 

139.39 

167.41 

Percent 

Error 

0.28 

-0.37 

0.86 

-0.33 

0.52 

-0.62 

0.70 

0.32 

-0.24 

-0.14 

-0.25 

-0.80 

-0.50 

-0.03 

XOtihog. 

Diagonal 

0.99498 

0.99469 

0.99384 

0.99003 

0.99905 

0.99015 

0.98078 

0.99246 

0.98155 

0.98598 

0.94888 

0.93058 

0.99716 

0.98405 

3.2 Model Updating for Boundary Frequency Response 

Using initial test data for drive point FRF at one of the trunnion simulators, a large discrepancy 

was observed between the predicted and measured antiresonance frequency (at the function minimum). 

Parameter studies were performed to identify modifications to the model that would improve the agreement. It was found that drastic, unreasonable changes in geometry and material properties of the 

trunnion still did not produce acceptable agreement with the measured data. Finally, additional measurements were taken to determine if errors existed in the test data. It was found that by using various impact 
hammer tips of different hardnesses, the measured antiresonance frequency shifted by several hertz. Use 

of the softest tip available provided the data considered most accurate. This is explained by the fact that 

soft tips provide more energy at lower frequencies, while hard tips excite higher frequencies of the 

structure. It is also quite difficult in general to measure clean antiresonances in FRFs due to the very low 

response amplitudes in the vicinity of a function minimum. Antiresonance response amplitudes with 

hammer excitation can be near the “noise floor” or response limit of the accelerometers. 

After more accurate frequency response measurements had been obtained, model correlation 

for the trunnion regions of the payload simulator was done. By varying the trunnion stiffness (modulus 

or area moments of inertia), the antiresonance frequencies were shifted to obtain the best possible 

agreement with test. The problem encountered in this process is that when good antiresonance agreement was obtained, the FRF peaks did not agree well. In the test configuration, a considerable amount 

of lumped mass was present on each trunnion due to accelerometers at the midpoints and ends, tape, 

adhesive, and wires connected to the accelerometers. By varying the amount of mass lumped at the 

center and end of each trunnion in the model, it was found that the FRF peaks and stiffness lines could 

be shifted to provide better agreement with measured data. A justification for the final net increase in 

lumped mass on each trunnion is the fact that the wires, adhesive, and tape were initially ignored in the 

model. It can be seen in figure 8 that excellent agreement was finally obtained between the test and 

model trunnion interface FRFs. 

l.OOE+OO 1 i i 

I ANFRF iter16 

1.00E-01 1 - I 

1 .OOE-03 

1.00E-05 1 

0 50 100 150 200 250 300 

Frequency (Hz) 

Figure 8. Test/analysis trunnion response functions, after model updates (zero mass loading). 

3.3 Comparison of Residual Flexibility Values for Boundary Degrees of Freedom 

The next step in the process was to compare the boundary residual flexibility values from test 

and the model that had been correlated to both the global free-free modes and trunnion response data. 

As discussed previously, residual functions, which are obtained by subtracting interface FRFs based 

25 
on the measured modes from full measured interface FRFs, are typically very noisy. To obtain accurate 

estimates of measured residual flexibility, the statistical least-squares curve-fitting approach described in 

a previous section was used.22 This curve-fitting technique provides an estimate of both residual flexibility and residual inertia. However, only the residual flexibility values were used in model correlation. 

Initially, the percent error was not as low as desired (within a few percent of test), so the trunnion stiffness and mass properties were further modified as described in the previous section to obtain acceptable 

correlation. As a note of interest, in error analysis for the Materials Science Laboratory, the Shuttle 

payload structure in figure 2 showed that up to 10-15 percent error in the residual values can be present 

and still yield constrained frequencies within 5 percent error. However, for this simple payload simulator. higher accuracy was desired. 

Test 

Residuals, 

(innb) 

1.1597E-05 

1.2277E-03 

1.2853E-03 

1.7254E-05 * 

1.2335E-03 

1.2784E-03 

_-------- 

1.2368E-03 

1.3057E-03 

1.0012E-05 * 

1.2359E-03 

Table 2 compares the analytical residual values (both before and after final model correlation to 

FRF data) and test results for each trunnion in the X, Y, and Z directions. It is noted that the X direction 

was along the main axis of each trunnion, while Y and Z were both bending directions. Excellent agreement with test (near 2 percent error and below) was obtained for all eight residual flexibility values in 

bending directions, but poor agreement was obtained for the axial direction. This is a problem that will 

not likely be overcome in hybrid and residual flexibility testing of Shuttle payload-type structures in the 

near future due to several reasons, including the following: (1) Dominant stiffness lines are not observed 

in trunnion axial-direction data, making the curve-fitting process for test data generally inaccurate; 

(2) the residual flexibility values are even smaller for the axial direction than for bending; and (3) the 

trunnion axial FRF test data are typically very noisy, which, when combined with the approximations 

in modal parameter estimation, yields very noisy residual functions that are difficult to interpret. Fortunately for Shuttle payloads, inaccurate axial-direction residuals for trunnions is not an issue since such 

payloads are only constrained in bending directions of the trunnions and keel. At this point, the analytical model with zero mass loading had been correlated with test as well as could be done. 

Analytical Residuals 

Before 

Updates (innb) 

4.8099E-06 

1.3852E-03 

1.4049E-03 

4.9603E-06 

1.3856E-03 

1.4051 E-03 

4.831 8E-06 

1.4083E-03 

1.4269E-03 

4.7943E-06 

1.3632E-03 

Table 2. Residual flexibility values for Shuttle payload simulator, before and after 

model updates to trunnions (zero mass loading, 14 free-free modes). 

Initial 

Percent 

Error 

---_----- 

12.83 

9.31 

-----_--_ 

12.33 

9.91 

--------- 

13.87 

9.28 

--------- 

10.30 

~ 

Model 

Location 

51 X 

Y 

Z 

53x 

Y 

Z 

55x 

Y 

Z 

57x 

Y 

Z 

Analytical Residuals Final 

Updates (innb) Error 

After Percent 

4.7566E-06 -__------ 

1.2420E-03 1.16 

1.261 1 E-03 -1.88 

4.9068E46 --------- 

1.2424E-03 0.72 

1.2614E-03 -1.33 

4.7784E-06 ---_---_- 

1.2635E-03 2.16 

1.2815E-03 -1.85 

4.741 OE-06 ______-__ 

1.221 6E-03 -1.16 

1.2250E-03 I 1.3830E-03 ' 12.90 1 1.2408E-03 1.29 

+ Confidence in experimental residual flexibility values for the X-direction is low because stiffness lines are not 

present in the response functions and cutve-fitting breaks down. These directions are unimportant for Shuttle 

payloads. 

I 26 
3.4 Use of Payload Simulator Model for Hybrid Method Parameter Studies 

Investigation of the equations presented in this TM has been done through parameter studies 

using the payload simulator model. For these parameter studies, the model had been updated to agree 

with measured free-free modes and frequencies but had not been correlated with trunnion interface 

response functions and residual flexibility values. However, as shown in table 2, percent error between 

model and test residual values was under 14 percent for the trunnion bending directions, which for some 

cases is within the error bounds identified in the Materials Science Laboratory error analysis mentioned 

in the previous section. Further, the parameter studies were mainly comparative in nature for (1) different boundary mass configurations, (2) different residual terms, and (3) different numbers of mode 

shapes, so that the partially correlated model was sufficiently accurate for the objectives of the studies. 

It is noted that for the parameter studies described in this section, rectangular masses were added 

symmetrically to all four trunnions of the payload simulator model. 

3.4.1 Hybrid Method With Full Residuals (Modified Rubin Method) 

Initially the payload simulator model was used for investigations of the hybrid Rubin method, 

which includes both residual flexibility and residual inertia, along with mass loading of the structure 

boundaries. This is the most general technique described in this TM, as stated previously, though not 

necessarily the most practical for test implementation, due to the requirement of measuring residual 

inertia effects. These second-order effects are very small, and differences between test and model can be 

greater than one or two orders of magnitude. 

For all the hybrid/Rubin method cases discussed, 20 mass-additive mode shapes, including 

6 rigid-body modes, were retained in the modal synthesis. In the first cases studied, all 6 DOFs at the 

end of each payload simulator trunnion (fig. 5), a total of 24 DOFs, were placed in the boundary partitions of the matrices in the equations of motion, equations (33)-(35). As discussed in section 2. I .2.1, 

near equation (21), mass loading is limited to the boundary DOFs in this case, since all mass loaded 

DOFs are to be constrained. Three different mass loading conditions were evaluated: (1) Zero mass 

loading reference case, (2) 0.5 Ib attached to each trunnion, and (3) 5-lb masses on each trunnion. 

In table 3, derived constrained frequencies and mode shape cross-orthogonality values are compared for the unloaded, 0.5-lb mass, and 5-lb mass loading conditions. As stated in the previous paragraph, 24 boundary DOFs were constrained in each case, and 20 free-free mass-additive modes were 

used. With the exception of mode 15 for the unloaded case, all derived frequencies are nearly exact, with 

errors below 0.1 percent, and the diagonal orthogonality values are near 1. It can be seen from table 3 

that considerably more constrained-boundary modes could be derived for the zero mass loading case 

compared to either of the loaded-boundary cases. This is due to the fact that the added boundary masses 

lowered the bending frequencies of the trunnions, with the result that in the selected sets of 20 modes 

there were fewer global modes and more localized trunnion modes than for the unloaded case. A thorough pretest analysis must always be done when using boundary mass loading to determine the best 

mass properties for a given structure and test configuration. This will assure that the desired constrained 

modes are derived from the measured or retained mass-additive modes. 

27 I 
Table 3. Effect of different boundary mass loadings on derived constrained frequencies 

and mode orthogonality for hybrid Rubin method (24 boundary DOFs, 20 modes). 

Exact 

requency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

07.9900 

16.7489 

26.1755 

38.8683 

41.6920 

42.5079 

Derived 

Frequency 

(0 Ib) 

9.2768 

18.7627 

20.6366 

23.9800 

32.8236 

40.71 46 

40.8734 

56.3896 

60.0862 

79.881 0 

91.6815 

96.6936 

107.9949 

11 6.7606 

129.1 603 

138.881 6 

141.7253 

142.5607 

XOrthog. 

Diagonal 

(0 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

.oooo 

.oooo 

.oooo 

.oooo 

.oooo 

.oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9964 

0.9999 

0.9991 

0.9990 

Derived 

Frequency 

(0.5 Ib) 

9.2768 

18.7627 

20.6366 

23.9800 

32.8247 

40.7146 

40.8734 

56.3900 

60.086 1 

79.8814 

91.6869 

96.7022 

xorthog. 

Diagonal 

(0.5 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9999 

Derived 

Frequency 

(5 la) 

9.2768 

18.7627 

20.6366 

23.9800 

32.8235 

40.7503 

40.8733 

56.3893 

60.0861 

XOrthog. 

Diagonal 

(5 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

It was also desirable to observe the effects of having mass loading of interior DOFs; i.e., cases 

where not all DOFs affected by the added masses are to be constrained. This more general case was 

investigated for the hybrid Rubin method by constraining only translations at the end of each trunnion 

(12 DOFs total) for the payload simulator. Results are shown in table 4 for 0.5-lb mass loading at each 

trunnion and 20 modes retained. Of course, the constrained frequencies are generally lower than for the 

six-DOF-constrained cases, but no loss in accuracy of derived modes is seen when mass loading of 

interior DOFs occurs. Again, the derived modes and frequencies are almost exact when both residual 

flexibility and residual inertia are included. 

These results point out the great accuracy achievable for reduced models and derived constrained 

modes using the hybrid full residual or Rubin approach. However, the disadvantage of the method is 

the difficulty of measuring residual inertia as stated earlier in this section. For this reason, the hybrid 

residual flexibility method, or hybrid MacNeal formulation, will be emphasized for the remainder of this 

TM. However, estimates of residual inertia are required when curve-fitting experimental residual functions to obtain residual flexibility values. 

28 
Table 4. Derived constrained modes for hybrid Rubin method 

(0.5-lb mass, 12 boundary DOFs, 20 modes). 

Exact 

Frequency 

6.9501 

9.6823 

12.4921 

13.7879 

22.8302 

28.6450 

34.1 967 

50.5303 

53.6326 

77.4622 

87.1 822 

96.2350 

(Hz) 

Derived 

Frequency 

(Hz) 

6.9501 

9.6823 

12.4921 

13.7879 

22.8302 

28.6450 

34.1968 

50.5305 

53.6330 

77.4636 

87.1 881 

96.2400 

XOrthog. 

Diagonal 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9999 

3.4.2 Hybrid Residual Flexibility Technique (Modified MacNeal Approach) 

Due to the great practical importance of the hybrid residual flexibility formulation for experimental implementation, the effects of the following variables were investigated in a highly detailed 

parameter study: (1) Size and weight of added masses, (2) number of retained mass-additive modes, 

(3) residual flexibility terms retained, and (4) number of constrained DOFs. In addition, results for basic 

residual flexibility method with no mass loading and for basic mass-additive approach with no residual 

terms were obtained as reference cases for (1) and (3). 

The studies are broken down into two broad categories: 

(1) Six DOFs constrained at each mass attach point (24 DOFs total), described by equations (33), 

(33, and (38) 

(2) Three DOFs constrained at each mass attach point (translations only, total 12 DOFs), 

governed by equations (33) and (35)-(36). 

Convergence characteristics of the derived constrained modes for these two configurations 

are described in the following sections. 

3.4.2.1 Results for Six Degrees of Freedom Constrained at Each Mass Attach Point. The 

first parameter to be discussed, weight of added masses, was investigated for three cases: (1) No mass 

loading, (2) 0.5-lb masses, and (3) 5-lb masses. Of course, the case for no mass loading represents the 

basic residual flexibility approach. Table 5 shows a comparison of derived constrained frequencies and 

mode cross-orthogonalities (with exact modes) for the three mass loading cases, using 20 retained massadditive modes and full residual flexibility matrices for all cases in the reduced mass and stiffness 

matrices. The residual flexibility matrices referred to here can be seen in equations (7) and (35). In all 

three boundary mass configurations, accurate constrained modes were obtained, though the 5-lb case 

29 
Table 5. Comparison of derived constrained modes for hybrid MacNeal method with different mass 

loadings (20 mass-additive modes, full residual flexibility matrix, 24 boundary DOFs). 

Exact 

Frequency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

107.9900 

1 16.7489 

126.1755 

138.8683 

141.6920 

142.5079 

Derived 

Frequency 

(0 Ib) 

9.2774 

18.7836 

20.6493 

24.01 17 

32.9330 

40.8183 

41.3649 

56.5504 

60.4672 

80.1 406 

92.7889 

96.7142 

108.3370 

11 7.2540 

148.491 1 

139.4997 

143.7859 

144.9400 

XOrthog. 

Diagonal 

(0 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9987 

1 .oooo 

0.9997 

0.9997 

0.9907 

0.9919 

0.9718 

0.9709 

Derived 

Frequency 

(0.5 Ib) 

9.2773 

18.7754 

20.6415 

23.9865 

32.8794 

40.7687 

41.31 34 

56.4205 

60.2583 

80.3566 

95.5041 

96.7356 

XOrthog. 

Diagonal 

(0.5 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9999 

1 .oooo 

0.991 4 

0.9926 

Derived 

Frequency 

9.2774 

18.7648 

20.641 0 

24.0000 

32.881 2 

42.7748 

41.4633 

56.4502 

60.1910 

(5 Ib) 

XOrthog. 

Diagonal 

(5 Ib) 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9998 

1 .oooo 

0.9999 

was less accurate and fewer modes could be derived. This occurred because the retained set of 20 massadditive modes had more localized trunnion modes and fewer global structure modes than the cases for 

zero and 0.5-lb mass loading. These results point out the importance of pretest analysis and selecting a 

proper mass size and weight that will allow the desired number of constrained modes to be derived with 

accuracy. Of the mass loading configurations presented here for the Shuttle payload simulator, the 0.5-lb 

case is more desirable. Though more modes could be derived with no mass loading, the rotations at the 

trunnions, particularly torsion, cannot be accurately measured without boundary masses. 

Investigation of the second parameter, number of retained mass-additive modes, is described 

next. Again, the full residual flexibility matrices that appear in equations (7) and (35) were utilized. In 

tables 6-8, derived constrained frequencies are shown with mode shape cross-orthogonality for the 0.5- 

Ib, 5-lb, and zero mass loading cases, respectively. Each table lists the derived constrained modes for 10, 

15, and 20 retained mass-additive modes. In each mass loading case, the progress of convergence of the 

constrained modes can clearly be seen as a function of the number of retained mass-additive modes. All 

three mass loading cases start out with the same number (7) of derived modes and have comparable 

accuracy, though the 5-lb case lags behind the other cases slightly. The zero added-mass case and 0.5-lb 

case both increased to 12 derived constrained modes for 15 retained mass-additive modes, while the 5-lb 

case shows no increase in the number of derived modes. However, the accuracy of the constrained 

frequencies did increase significantly for the 5-lb case in going from 10 to I5 retained mass-additive 

modes. A similar situation is seen for the 0.5-lb mass loading case when going from 15 to 20 retained 

mass-additive modes. 

30 
Table 6. Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (0.5-lb loading, full residual flexibility, 24 boundary DOFs). 

Exact 

:requency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

10 Mass-Added Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2779 1 .OOOO 

18.8367 1 .OOOO 

20.6532 1 .OOOO 

24.0514 1 .OOOO 

33.0968 1 .OOOO 

42.8722 1 .OOOO 

42.1046 0.9993 

15 Mass-Added Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2774 1 .OOOO 

18.7823 1 .OOOO 

20.6531 1 .OOOO 

24.0369 1 .OOOO 

32.9077 1 .OOOO 

40.7955 1 .OOOO 

41.3294 1 .OOOO 

56.7825 0.9997 

60.91 73 0.9999 

80.7216 1 .OOOO 

96.91 88 0.9946 

97.4750 0.9943 

15 Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2774 1 .OOOO 

18.8314 1 .OOOO 

20.7148 1 .OOOO 

24.0459 0.9999 

32.9186 0.9998 

43.0194 0,9999 

41.6791 0.9982 

20 Mass-Added Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2773 1 .OOOO 

18.7754 1 .OOOO 

20.6415 1 .OOOO 

23.9865 1 .OOOO 

32.8794 1 .OOOO 

40.7687 1 .OOOO 

41.31 34 1 .OOOO 

56.4205 1 .OOOO 

60.2583 0.9999 

80.3566 1 .OOOO 

95.5041 0.9914 

96.7356 0.9926 

20 Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2774 1 .OOOO 

18.7648 1 .OOOO 

20.641 0 1 .OOOO 

24.0000 1 .OOOO 

32.8812 1 .OOOO 

42.7748 1 .OOOO 

41.4633 0.9998 

56.4502 1 .OOOO 

60.1 91 0 0.9999 

Table 7. Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (5-lb loading, full residual flexibility, 24 boundary DOFs). 

Exact 

Frequency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 1 

10 Modes 

Derived 

Frequency 

9.2776 

18.8727 

20.9275 

24.7170 

34.01 92 

43.2223 

44.6197 

XOrthog. 

Diagonal 

1 .oooo 

1 .oooo 

0.9997 

0.9992 

0.9999 

1 .oooo 

0.9968 

31 
Table 8. Effect of retained mass-added modes on derived constrained modes for hybrid 

MacNeal method (zero mass loading, full residual flexibility, 24 boundary DOFs). 

Exact 

requency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

07.9900 

16.7489 

26.1 755 

41.6920 

42.5079 

38.8683 

10 Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2780 1 .OOOO 

18.8362 1 .OOOO 

20.6564 1 .OOOO 

24.0546 1 .OOOO 

33.0833 1 .OOOO 

42.8673 1 .OOOO 

42.0271 0.9994 

15 Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2777 

18.7836 

20.6564 

24.0546 

33.0045 

40.81 84 

41.3650 

56.7381 

60.6858 

80.1407 

92.7893 

96.7575 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9998 

0.9999 

1 .oooo 

0.9987 

0.9997 

20 Modes 

Derived XOrthog. 

Frequency Diagonal 

9.2774 1 .OOOO 

18.7836 1 .OOOO 

20.6493 1 .OOOO 

24.01 17 1 .OOOO 

32.9330 1 .OOOO 

40.81 83 1 .OOOO 

41.3649 1 .OOOO 

56.5504 1 .OOOO 

60.4672 1 .OOOO 

80.1406 1.0000 

92.7889 0.9987 

96.7142 1.0000 

108.3370 0.9997 

117.2540 0.9997 

148.491 1 0.9907 

139.4997 0.991 9 

143.7859 0.971 8 

144.9400 0.9709 

The conclusion to be drawn from tables 6-8 is that the 5-lb mass loading case performs worse 

than the other cases because the masses are too heavy for the payload simulator structure. The 0.5-lb 

case performs very well in comparison. However, if there is no need to measure rotations, and the 

boundary FRFs and residual functions are satisfactory in the desired bandwidth, mass loading should not 

be utilized. The basic residual flexibility method should be used in such cases because more constrained 

modes can be derived using a given number of free-free modes in the synthesis process. 

The third parameter investigated was the number of residual flexibility terms retained in the 

reduced stiffness matrix, as shown in equation (35). Note that the reduced mass matrix in equation (38) 

has no residual flexibility terms. It is desirable to reduce the number of residual flexibility terms in the 

formulation if possible, in order to reduce the number of measurements required for experimental implementation. Previous investigations have shown that the diagonal terms of Grbb are much easier to 

measure with accuracy than the off-diagonal terms. It is noted that these parameter studies apply only to 

the reduced stiffness matrix. The transformation matrix in equation (44) was used with full Grbb 

[G~J for all cases. 

[I 

[ I and 

Initially, for the 24 boundary DOF cases, only the diagonal values of [Grbb] were used, but it was 

found that a fairly large number of mass-additive modes (25-30) were required for acceptable accuracy 

of derived constrained modes. It can be difficult to develop a test-correlated model that accurately 

represents that many mode shapes. For comparison to these results, the full [Grbb] matrix was used 

in equation (35). This was found to work very well, even for small numbers of mass-additive modes, 

I 32 
and representative results are shown in table 9 for the 0.5-lb mass loading case, where 20 mass-additive 

modes (14 elastic) were retained. Derived constrained modes in table 9 are compared for diagonal [ Grbb] 

and full [Gbb 1 in equation (35). The disadvantage of using the full [Gbb 1 matrix is that it may be impossible to verify the off-diagonal terms experimentally; measurement of the diagonal terms is much 

easier, as stated in the previous paragraph. Engineering judgment is required in determining whether to 

use only the diagonal terms of Grbb with a larger set of mass-additive modes. [I 

Table 9. Effect of retained residual terms on derived constrained modes in hybrid 

MacNeal method (0.5-lb loading, 20 mass-added modes, 24 boundary DOFs). 

Exact 

Frequency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

Diagonal [Grbd 

Derived X Orlhog. 

Frequency Diagonal 

7.9465 0.9777 

16.0974 0.9922 

15.7103 0.9679 

16.341 5 0.9300 

26.1 688 0.9503 

37.6595 0.9909 

34.2264 0.9908 

54.01 15 0.9776 

53.1553 0.8987 

82.7598 0.9762 

103.3862 0.8857 

96.41 99 0.9967 

Full [Grbd 

Derived 

Frequency 

9.2773 

18.7754 

20.641 5 

23.9865 

32.8794 

40.7687 

41.31 34 

56.4205 

60.2583 

80.3566 

95.5041 

96.7356 

XOrthog. 

Diagonal 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9999 

1 .oooo 

0.9914 

0.9926 

3.4.2.2 Results for Three Degrees of Freedom Constrained at Each Mass Attach Point. 

Similar studies to those described in the previous section were done for the Shuttle payload simulator 

with only translations to be constrained at each boundary mass attachment location (12 DOFs 

constrained total). Mass loading of 0.5 Ib at each trunnion was utilized, and a zero mass loading case 

was analyzed for comparison. Similar to the case for six DOFs constrained at each mass attach point, 

these parameter studies apply to the reduced mass and stiffness matrices only. For the transformation 

matrix in equation (44), the full [Grbb] and [Grib] partitions were used in all cases. 

For the case of full residual flexibility terms in the reduced mass and stiffness matrices, a pattern 

similar to that shown in table 6 for convergence of derived constrained modes was observed. Constrained frequency errors were very low in comparison to exact results. Further, there was virtually no 

loss in accuracy when neglecting [Gqb] and using only the full [Grbb] residuals in the reduced mass 

and stiffness matrices, equations (35)-(36). Again, the frequency errors were very low. However, when 

the residual flexibility terms were reduced to only the diagonals of [Grbb], a clear loss in accuracy of 

derived constrained modes was observed. This is seen in table 10, where results are shown for derived 

constrained modes when retaining 10, 15, and 25 mass-additive modes. As stated in the previous 

33 
section, it can be difficult to correlate a model to test data for 25 modes. However, accuracy of constrained modes for 20 retained mass-additive modes (not shown in table 10) was actually worse than for 

15 retained modes, showing that up to 25 modes could be required. 

Exact 

:requency 

I 

Table 10. Derived constrained modes for different numbers of mass-added modes in hybrid MacNeal 

method, with only diagonal [Grbb] residual terms (0.5-Ib loading, 12 boundary DOFs). I 

Derived XOrthog. 

Frequency Diagonal 

10 Modes 

6.9501 

9.6823 

12.4921 

13.7879 

22.8302 

28.6450 

34.1967 

50.5303 

53.6326 

77.4622 

87.1822 

96.2350 

107.1969 

1 14.3559 

126.1380 

137.9789 

138.7379 

7.01 81 1 .OOOO 

9.7548 1 .OOOO 

12.2298 1 .OOOO 

13.7778 1.0000 

23.1873 1.0000 

29.0594 1 .OOOO 

52.4894 1 .OOOO 

50.5625 1 .OOOO 

15 Modes 

Derived XOrthog. 

Frequency Diagonal 

6.5489 0.9969 

9.3592 0.9998 

13.9098 0.8735 

14.0869 0.8714 

21.7631 0.9900 

29.1923 0.9952 

32.5939 0.9991 

53.5009 0.9931 

54.1 238 0.9994 

77.7251 0.9996 

93.2500 0.9788 

97.01 81 0.9859 

___ ~~ 

25 Modes 

Derived XOrthog. 

Frequency Diagonal 

6.9476 1 .OOOO 

9.6728 1 .OOOO 

12.5757 1 .OOOO 

13.7871 1 .OOOO 

22.8220 1 .OOOO 

28.6230 1 .OOOO 

33.7316 0.9991 

50.61 10 0.9998 

53.6327 1 .OOOO 

77.51 86 1 .OOOO 

87.1 747 1 .OOOO 

96.2351 1 .OOOO 

107.1979 1.0000 

114.3576 1 .OOOO 

166.0426 0.9287 

238.5921 0.2929 

138.7391 0.9990 

Table IO also shows that use of 15 retained mass-additive modes gave worse results for derived 

constrained modes than the case for 10 retained modes. These results show that there were modes in the 

larger set that did not contribute to the constrained-boundary configuration, and actually hurt convergence. This also points out the potential value of selecting mass-additive modes that contribute most to 

the constrained-boundary modes through use of modal participation calculations. However, such an 

analysis is not included in the context of this TM. 

For comparison to the 0.5-lb mass-loaded configuration, analysis was done for the 12-DOFconstrained model using the basic MacNeal or residual flexibility approach with zero mass loading of 

the boundaries. Table 11 shows the convergence characteristics of derived constrained modes for use of 

10, 15, and 20 retained mass-additive modes, where only diagonal [Grbb] residuals were utilized. The 

results of table 11 compared to table IO point out clearly that use of mass loading in the residual flexibility approach can result in slower convergence of constrained modes, or alternatively, require additional 

residual flexibility terms. As stated previously, if rotations are not to be constrained and the boundary 

FRFs and residual functions are satisfactory for obtaining residual flexibility values, mass loading 

should not be used, and either the basic residual flexibility method or basic Rubin method should be 

used. 

, 

34 

I 
Table 1 1. Derived constrained modes for different numbers of mass-added modes in hybrid MacNeal 

method, with only diagonal [Grbb] residual terms (zero mass loading, 12 boundary DOFs). 

Exact 

Frequency 

6.9501 

9.6823 

12.4921 

13.7879 

22.8302 

28.6450 

34.1967 

50.5303 

53.6326 

77.4622 

87.1822 

96.2350 

107.1969 

1 14.3559 

126.1380 

137.9789 

138.7379 

140.3294 

10 Modes I 15 Modes 

Derived 

:requency 

6.9780 

9.7726 

12.3795 

13.8117 

23.0062 

28.9372 

53.2095 

50.6756 

XOrthog. 

Diagonal 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

Derived 

Frequency 

6.9673 

9.6470 

12.5119 

13.9772 

22.9567 

28.5938 

33.4080 

50.7973 

53.8477 

77.6671 

87.6263 

96.2649 

X Orthog. 

Diagonal 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9998 

1 .oooo 

1 .oooo 

1 .oooo 

0.9998 

0.9999 

20 Modes 

Derived XOrthog. 

Frequency Diagonal 

6.9614 

9.6709 

12.5165 

13.6573 

22.9258 

28.9203 

33.3839 

50.7661 

53.6969 

77.7103 

87.8136 

96.2576 

107.5268 

114.7188 

169.4028 

138.2839 

140.5087 

141.9454 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9998 

1 .oooo 

1 .oooo 

1 .oooo 

0.9997 

1 .oooo 

0.9998 

0.9999 

0.9944 

0.9973 

0.9973 

0.9977 

3.4.3 Variation of Residual Flexibility Values for Different Numbers of Retained Modes 

and Boundary Mass Sizes 

It is interesting to observe the effect upon the magnitude of residual flexibility values of varying 

(1) boundary mass size (weight) and (2) number of mass-additive modes. Of course, the residual flexibility decreases as a function of number of retained modes, since it represents an approximation of 

modes not retained in the reduced model. In addition, for a given frequency bandwidth, one would 

expect residual flexibility associated with boundary DOFs to be smaller in the case of mass-loaded 

boundaries, since localized boundary modes are lowered into the bandwidth. These trends are indeed 

observed and are of importance because the smaller the residual flexibility values are, the more difficult 

it becomes to measure them with accuracy. 

Table 12 shows the variation of the residual flexibility values for all six DOFs at the mass-loaded 

end of the -X, -Y trunnion of the Shuttle payload simulator. It is noted, however, that it is unimportant 

which trunnion is under discussion due to symmetry of the structure. Results are shown as a function 

of added mass and number of retained mode shapes. As expected, little variation is seen in the residual 

flexibility values with the number of retained modes for zero mass loading because the localized trunnion modes occur at high frequency. However, residual flexibility values generally decrease with 

increased mass and number of retained mass-additive modes. 

35 
Table 12. Variation of residual flexibility values for different mass loadings 

and number of retained mass-additive modes. 

Exact 

Frequency 

Added 1 Modes I XResidual 

Mass (Ib) Retained (innb) 

30 Modes 40 Modes 

Derived XOrthog. Derived XOrthog. 

Frequency Diagonal Frequency Diagonal 

0.0 

0.5 

5.0 

0.0 

0.5 

5.0 

0.0 

0.5 

5.0 

10 

10 

10 

15 

15 

15 

20 

20 

20 

1.0322E-05 

1.0296E-05 

1.0533E-05 

4.9568E-06 

5.0768E-06 

1.0201 E-05 

' 4.7650E-06 

4.9033E-06 

9.8922E-06 

Y Residual 

(innb) 

1.3980E-03 

1.3843E-03 

1.2924E-03 

1.3856E-03 

1.3483E-03 

5.8527E-04 

1.3785E-03 

6.2687E-04 

2.1 21 3E-06 

ZResidual 1 RXResidual 1 RYResidual 1 RZResidual 

(innb) (radfin-l b) (radlin-lb) (radlin-l b) 

1.4309E-03 

1.3346E-03 

9.1884E-04 

1.3932E-03 

1.191 1 E-03 

1.2984E-04 

1.3157E-03 

6.6237E-04 

2.5148E-06 

7.4436E-05 

7.4433E-05 

7.441 2E-05 

7.441 1 E-05 

7.4400E-05 

7.4332E-05 

7.4391 E-05 

7.4399E-05 

7.2830E-05 

5.5845 E-05 

5.5347E-05 

5.21 36E-05 

5.2996E-05 

2.6923E-05 

5.4923E-05 

3.9352E-05 

5.5649E-05 

1.71 82E-05 

5.5702E-05 

5.5685E-05 

5.5572E-05 

5.5680E-05 

3.5435E-05 

5.56 1 8E-05 

3.4554E-05 

5.5347E-05 

1.5776E-05 

3.4.3.1 Results for Basic Mass-Additive Method Without Residual Flexibility. Analysis 

was also done to provide a reference for the improved accuracy of the hybrid method in comparison 

to the basic mass-additive appr~ach.~.~ Results are first presented for the case of 0.5-Ib mass loading 

at each trunnion of the payload simulator, with three translational DOFs to be constrained at each point 

(12 boundary DOFs total). It was found that 30 or more mass-additive modes are required in the absence 

of residual terms to derive accurate constrained modes, as shown in table 13. The precise number of 

mass-additive modes needed depends upon the number of constrained modes required and upon the 

mass size. Clearly the use of residual flexibility reduces the number of mass-additive modes that must be 

retained, measured, and used for model correlation. 

Table 13. Derived constrained modes for the basic mass-additive method 

with no residual terms (0.5-lb loading, 12 boundary DOFs). 

6.9501 

9.6823 

12.4921 

13.7879 

22.8302 

28.6450 

34.1 967 

50.5303 

53.6326 

77.4622 

87.1 822 

96.2350 

107.1969 

114.3559 

126.1 380 

137.9789 

138.7379 

6.9592 

9.8883 

12.4951 

13.7987 

22.8595 

28.6501 

45.1418 

50.5326 

53.6362 

184.9571 

87.1 841 

96.2370 

107.2051 

11 4.3574 

139.4956 

139.4956 

138.7394 

1 .oooo 

0.9996 

1 .oooo 

1 .oooo 

0.9997 

1 .oooo 

0.9788 

1 .oooo 

1 .oooo 

0.4288 

1 .oooo 

1 .oooo 

0.9999 

1 .oooo 

0.871 1 

0.4613 

0.9994 

6.9508 

9.6990 

12.4947 

13.7906 

22.8327 

28.6477 

35.461 2 

50.5322 

53.6334 

79.6931 

87.1830 

96.2351 

107.1 971 

1 14.3561 

139.5815 

138.0498 

138.7382 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9995 

1 .oooo 

1 .oooo 

0.9985 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.91 73 

0.9325 

1 .oooo 

36 
For the case of 24 boundary DOFs (6 DOFs at each mass attach point), the findings are more 

dramatic. It was found for the 0.5-lb masses that over 100 mass-additive modes are required. If 5-lb 

boundary masses are used, the number of required modes is reduced, but is still on the order of 50 modes 

(table 14). The basic mass-additive approach without residual flexibility thus becomes impractical as the 

number of boundary DOFs to be constrained increases. Further, large masses may be required depending on the structure geometry and weight and, again, on the number of boundary DOFs. 

Table 14. Derived constrained modes for basic mass-additive method (24 boundary DOFs). 

Exact 

Frequency 

9.2768 

18.7627 

20.6366 

23.9799 

32.8235 

40.7146 

40.8733 

56.3890 

60.0858 

79.8798 

91.6779 

96.6930 

107.9900 

11 6.7489 

126.1 755 

138.8683 

141.6920 

142.5079 

0.5-lb Mass 

80 Modes 

Derived XOrthog. 

Freauencv Diaaonal 

9.5214 

50.2338 

21 A980 

26.1960 

35.6061 

103.1966 

43.5301 

5 7.4 7 0 5 

61.3925 

50.2338 

93.2684 

96.8202 

108.2658 

11 7.6004 

140.1 308 

103.1 966 

142.1686 

143.5388 

0.9996 

0.8195 

0.9998 

0.9986 

0.9990 

0.7874 

0.9991 

0.9987 

0.9997 

0.5709 

0.9991 

0.9998 

0.9996 

0.9996 

0.9828 

0.5971 

0.9306 

0.9303 

120 Modes 

Derived 

Frequency 

9.3666 

19.8629 

21.1 155 

24.7410 

33.7720 

42.3876 

41.9371 

56.7345 

61.1922 

80.7078 

92.2803 

96.7348 

108.0823 

117.0174 

135.0474 

139.2480 

141.9843 

142.8434 

XOrthog. 

Diagonal 

1 .oooo 

1 .oooo 

0.9999 

0.9998 

0.9999 

0.9996 

0.9997 

0.9999 

0.9998 

0.9999 

0.9999 

1 .oooo 

1 .oooo 

1 .oooo 

0.9900 

0.9965 

0.9958 

0.9958 

5-lb Mass 

40 Modes 

Derived 

Frequency 

9.7064 

37.8934 

24.3270 

27.0747 

45.1629 

37.8934 

48.481 7 

66.2439 

81.6551 

160.7461 

81.6551 

97.3461 

109.2487 

122.2673 

160.7461 

160.7461 

128.4337 

152.4853 

XOrthog. 

Diagonal 

0.9956 

0.8474 

0.9585 

0.8692 

0.7567 

0.4814 

0.8387 

0.8393 

0.5665 

0.4393 

0.7764 

0.9863 

0.9539 

0.8914 

0.0322 

0.2484 

0.6333 

0.6525 

60 Modes 

Derived 

Frequency 

9.2779 

18.7899 

20.6458 

23.9906 

32.8350 

41.4989 

40.9150 

56.3938 

60.1071 

80.2266 

91.701 3 

96.6935 

107.991 2 

116.7526 

137.0175 

138.9920 

141.6965 

142.51 27 

XOrthog. 

Diagonal 

1 .oooo 

0.9999 

1 .oooo 

1 .oooo 

1 .oooo 

0.9993 

1 .oooo 

1 .oooo 

1 .oooo 

0.9995 

1 .oooo 

1 .oooo 

1 .oooo 

1 .oooo 

0.9852 

0.9968 

1 .oooo 

1 .oooo 

Two other observations can be made from tables 13 and 14. First, it appears that the larger 5-lb 

mass, if used in the 12 boundary DOF case for the basic mass-additive method, could reduce the number 

of required modes to a practical level. However, this case was not investigated, and it has merit for 

further study. Second, it is possible that the type of boundary DOFs (rotational or translational) may be 

more important than the number of boundary DOFs. It seems logical that more modes would be needed 

for deriving constrained modes for rotational boundary DOFs than for translational boundary DOFs. 
4. EXPERIMENTAL IMPLEMENTATION OF HYBRID METHOD FOR PAYLOAD 

SIMULATOR WITH MASS-LOADED BOUNDARIES 

- 

Demonstration of the experimental implementation of the hybrid method was accomplished 

by using the Shuttle payload simulator modified at one trunnion. A rectangular 0.5-lb aluminum mass 

having dimensions 3.5 by 3 by 0.5 in was attached to the end of the (-X, -Y) trunnion, as shown in 

figures 9 and 10. Three triaxial accelerometers were placed at the rectangular mass center, upper right 

comer, and lower left corner when viewed along the positive X axis as seen in figure 11. Hammer impact 

excitation was applied at several points to allow measurement of both translational and rotational FRFs 

and residual flexibility values. In all cases, the objective was to obtain the response of the mass center 

point, or response point 3, as labeled in figure 11. 

5in 

II I 

II I 

II I 

25 in 

CI II 

I II 

I I1 

+ 

I 

I 

37.75 in 

I 

I 

I 

AI 

I 

I 

- 

3 in 

t 

i Rectangular Mass 

7- 

Figure 9. Payload simulator with rectangular mass attached to one trunnion. 

38 

I 
---- 

Weld 

Accelerometers 

Trunnion ---- 

Payload Simulator Simulator 

Test Article 

L X 

Figure 10. Detailed view of added mass and trunnion region of payload simulator. 

T I legend 

@ Triaxial response points 

dl = 1.75 

x X-direction excitation points -- 1 

-+ Y- or Z-direction excitation points 

Y +Az 

1 6 

+ d2=1.5in 

r 

05 

I 

I 

I 

I 

X 

Figure 11. Rectangular mass with excitation and response points. 

7 

39 

1 
4.1 Translational Frequency Response Function and Residual Flexibility Measurements 

Compared to Analysis 

For translational response of the mass center (point 3) in the X direction, excitation was applied 

at 6X, and for translational Yand Z response, excitation was applied at 5Y and 5Z, respectively. The 

excitation and response points are numbered in figure 11. The terminology for the three translational 

FRFs is given by the ordered pairs (6X, 3X), (5 Y, 3 Y), and (5Z, 32) respectively. In these ordered pairs, 

the first entry represents the excitation point number and direction, while the second entry refers to the 

response point number and direction. 

Figure 12 shows initial comparison of the test and analytical trunnion Y-bending response functions (5Y, 3Y). A fairly large discrepancy (=8 percent for resonant frequency) existed between the model 

and test FRF, which required further updates to the payload simulator model. As described previously, 

the model without mass loading had already been updated to agree very well with measured free-free 

modes, frequencies, and trunnion FRFs (residuals). However, addition of the boundary mass introduced 

new errors and uncertainties in the model that had to be corrected. It was discovered that the discrepancy 

shown in figure12 was primarily due to inaccurate modeling of the trunnion-to-mass intersection. While 

in reality the surface of the rectangular mass was attached to the end of the trunnion, the model initially 

had the attach point at the center or mid-plane of the mass plate elements. For this reason, the mass 

attach point was in error by 0.25 in, or half the thickness of the mass. Once this was corrected by adding 

a 0.25 in offset in the -X direction, much better agreement was obtained between the measured and 

predicted FRFs (fig. 13). Other refinements reflected in figure 13 include the addition of X-direction 

offsets for the lumped mass representations of the accelerometers, which were mounted to the outer 

(-X) surface of the rectangular mass. 

I 

Model updates for the mass-loaded trunnion FRFs were done at this stage for the two bending 

directions only. Previously, while investigating the basic residual flexibility method, it was found that 

X-direction trunnion responses were dominated by the global mode shapes rather than the stiffness 

of the trunnion. Very little improvement is possible by realistically modifying the trunnion axial stiffness. The Z-bending FRFs are compared for test and analysis in figure 14, and the axial (X direction) 

responses are shown in figure 15. It is noted that the test data at lower frequencies, below 50 Hz for the 

X direction and below 35 Hz for 2-bending, are not reliable near antiresonances or FRF minima. This 

is due to general difficulty in measuring the low response near FRF minima, which can be near the noise 

floor of the instrumentation. Also, the hammer tip was not soft enough to give the best response near 

anti resonances. 

After completion of model updates for the mass-loaded trunnion to obtain acceptable agreement 

with test FRFs, the boundary (trunnion) residual flexibility values were compared for test and analysis. 

As described earlier in this TM, a residual function is defined as the difference between a full FRF based 

on all the structural modes and a synthesized FRF based on a subset of the modes. For example, consider 

a case where the full frequency response is computed or measured up to 200 Hz, but only the global 

modes to =85 Hz are of interest. The residual function is obtained by subtracting the synthesized FRF 

(based on the modes to 85 Hz) from the full FRF, as can be seen in figure 16 for the analytical (5Y, 3Y) 

response. Note that the FRF in displacementlforce format must be used. Further, as explained previously, 

residual flexibility is defined as the value of the residual function at zero frequency. Analytically, a 

I residual function has zero slope at zero frequency. 
e 

.- 

. 

c - 

1.00Et00 

1.00E-01 

1 .OOE-02 

1.00E-03 

1 .OOE-04 

1.00E-05 4 ' I 

0 50 100 150 200 

Frequency (Hz) 

Figure 12. Initial tesdanalysis comparison of Y-bending responses for attached mass, 

excitation at mass corner point 5, response at mass center point 3 (5 Y, 3 Y). 

1.00Et01 

0 50 100 150 200 

Frquency (Hz) 

Figure 13. Comparison of tesdanalysis Y-bending responses (5 Y, 3 Y) for attached 

mass after model updates. 

41 
1.00E-01 

1.00E-02 

1.00E-03 

1.00E-04 

-- . ~- ~- ~ ~ ~- 

1.00E-06 

0 50 100 150 200 

Frequency (Hz) 

Figure 14. Comparison of test and updated analytical Z-bending responses (52, 32) 

for attached mass. 

1.00Et01 

100Et00 

1.00E-01 

1 .OOE-02 

100E-03 

- ANALlTERl 

E .- $ 1.00E-04 

100E-05 

1.00E-06 

1.00E-07 

1.00E-08 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

Figure 15. Measured and updated analytical axial responses (6X, 3X) for attached mass. 

42 
1 .OOE-01 

1.00E-02 

P 

1.00E-03 

s 

1 .OOE-04 

1.00E-05 

~___ __ ~ ~ ~ . 

1.00E-06 

0 50 100 150 200 

Frequency (Hz) 

Figure 16. Analytical residual function as difference between full and modal response 

functions (5 Y, 3 Y). 

Test data for the (5Y, 3Y) response functions and residual function are shown in figure 17. At low 

frequencies, the region most critical for determining residual flexibility values, the full FRF and the 

residual function are very noisy. This is overcome by performing a weighted second-order least-squares 

curve fit as discussed earlier in this TM. The curve-fitting procedure utilizes the fact that a theoretical 

residual function, or a perfectly clean measured residual function, must approach zero slope as it 

approaches zero frequency. In figure 17, both the residual function and a smooth curve-fitted function 

are shown. Statistical weighting of the data allows the noisy regions to be excluded from the smoothed 

curve, which therefore follows the “backbone” of the residual function. Figure 18 presents a comparison 

of experimental and analytical residual functions for the trunnion Y-bending (5Y, 3 Y) response, including 

a smooth curve-fit of the test function. The noisy test residual data that has been weighted out by the 

curve fit can be clearly seen near 10 Hz. In figures 19 and 20, measured and predicted residual functions 

are compared for the trunnion Z-bending (5Z, 32) and axial (6X, 3X) directions, respectively. The 

experimental and analytical residual flexibility values for all three trunnion translational directions 

are given in table 15. 

It must be noted at this point that a limitation exists in the current state of the art for this type of 

testing. In figures 20 and 21, for the trunnion X direction residual functions, the test data are very noisy 

for frequencies below =70 Hz. A curve approaching zero slope at zero frequency cannot be discerned, 

which is a requirement for obtaining accurate residual flexibility values. Figure 21 shows that even 

though the experimental residual flexibility value is reasonable and close to the analytical value, the 

least-squares curve-fitting procedure broke down. This occurred because the noisy test data actually have 

an upward trend below 60 Hz, not the gradual downward trend which correctly describes analytical and 

clean experimental residual functions. In such cases, only the FRFs can be compared with confidence in 

the current state of the art. Possibly, higher order curve fitting could be used to alleviate this problem. 

43 
... .......... ... -. ... .... .... . .- .... .- .. . - -. ... ... .. 

.. .- .. 

1.00Et01 

.. 

1.00E-01 

1 .OOE-02 

1.00E-03 

1.00E-04 

- 

1 

'. / 1 OOE-051 TEST MODAL / -i -/- 

Figure 17. 

0 20 40 60 

Frequency (Hz) 

80 100 

Test residual function and smooth curve-fitted function, shown as difference 

between full and synthesized response functions (5Y, 3 Y) for attached mass. 

l.OOE+Ol 

1.00E-01 

1.00E-02 

1.00E-03 

0 20 40 60 80 100 

Frequency (Hz) 

Figure 18. Comparison of test and updated model residual functions, 

Y-bending (5 Y, 3 Y) for attached mass. 

44 
1.00Et01 

l.OOEtOO{ -\ / TEST RES --. _. __ _. -. 

_- 

~ 

~ 

. 00 E-0 1 

~ 

_~ __- - 

. 00 E-0 2 

.00E-03 

. 00 E-04 

Figure 19. 

0 20 40 60 80 100 

Frequency (Hz) 

Measured and updated analytical residual functions for trunnion 

2-bending at mass attachment location (5Z, 32). 

1.00Et01 

.OOE-O3 

.00E-04 

.00E-05 

.00E-06 

1.00E-07 

1.00E-08 

0 50 100 150 

Frequency (Hz) 

200 

Figure 20. Residual functions from test and updated model for mass-loaded trunnion 

in axial direction (6X, 3X). 

45 
Table 15. Residual flexibility values for test and correlated model 

at trunnion mass attachment location. 

Direction 

RX 

RY 

RZ 

Location of 

Drive, Response 

Test 

Residuals 

(infib, Win-lb) 

Analysis 

Residuals 

(in/lb, l/in-lb) 

6X, 3X 

5Y, 3Y 

5z. 32 

8.7579E-06 

1.3233E-03 

9.5622E-04 

6.1 642E-05 

3.41 06E-05 

4.1580E-05 

1.0294E-05 

1.4295E-03 

6.0524E-05 

5.0201 E-05 

5.6278E-05 

1.1052E-03 

1.00Et01 _- .. . .- I 

l.OOE+OO 

1 .OOE-02 

1.00E-03 

$ 1.00E-04 

1.00E-05 

1 .OOE-06 

1.00E-07 

0 20 40 60 80 100 

Frequency (Hz) 

Figure 21. Residual functions from test and analysis for X direction (6X, 3X), showing 

an example of breakdown of second-order curve fit for noisy test data. 

Inspection of figures 1618 for trunnion Y bending shows that the residual function closely tracks 

the FRF except at the lowest frequencies, where the FRF is dominated by rigid-body behavior. This 

phenomenon points out the physical significance of residual flexibility for the payload simulator trunnions. Residual flexibility in a given coordinate is an approximation of the physical flexibility (inverse 

of stiffness) of the trunnion in that direction. Closed-form solutions for the stiffness and flexibility of a 

simple cantilever beam verified this result. This observation holds true for any structural boundary DOE 

However, it is particularly obvious for a prominent flexible interface whose FRF has a dominant stiffness 

line, or upward-sloping trend, toward the fundamental mode of the interface structure. Such stiffness 

lines are especially apparent for the trunnion Y-bending FRF in figures 13 and 16-18. 

46 
4.2 Approximate Rotational Frequency Response Functions Derived From Translational Test 

Data and Comparison With Analysis 

In this and the following section, the main advantages of combining mass loading of structure 

boundaries with the residual flexibility technique will be demonstrated. It has been stated previously in 

this TM that rotational measurements typically are not attempted in modal testing, mainly due to difficulties in performing the measurements in the absence of rotational sensors and due to the expense of 

such sensors if they are utilized. In addition, even if rotational accelerometers were available, rotational 

responses are typically small and difficult to measure accurately. This is particularly true for torsion if 

the structure has beam-like boundary regions, which is the case for the payload simulator structure. 

The problem of rotational FRF and residual flexibility measurements can be at least partly 

alleviated using the hybrid test/analysis approach presented in this TM. The rectangular mass excitation 

and response points shown in figure 11 were utilized to obtain good approximations of the trunnion 

rotational FRF and residual functions for the payload simulator. Redundant excitation and response 

points were used to provide at least two approximations for each rotational FRF. In all cases, the objective was to measure response about the mass center, labeled point 3 in figure 11. 

The rotational data were extracted from translational measurements using the simple arc length 

formula, s = r 8. In this expression, r is the length of a line extending from a reference point to a point on 

a curve, 8 is the angle about the reference point swept out by the line, and s is the arc length. In the case 

of the rectangular mass shown in figures 10 and 11, the reference point is the mass center response point, 

or point 3. For rotational response about an axis through the mass center and parallel to the X axis, the 

excitation at 5Y was used initially, along with the Y translational responses at points 1 and 3 (mass center 

and upper right corner, respectively): 

It is noted that small rotations are assumed in this formulation, which is a valid assumption for the low 

input forces used in most modal tests. Equation (60) gives the rotational (torsional) response about an 

axis parallel to the X axis through the mass center point, but the standard normalized form of the FRF is 

needed, which is expressed as 

In equation (61), the moment about the mass center is FSYdl, the force in the Y direction applied at 

excitation point 5 of figure 11, multiplied by the distance’to the mass center. It was desired to verify 

experimental results obtained by use of equation (61) by developing a second expression for the rotational FRF. In this case, excitation applied at point 5 in the Z direction was utilized along with translational responses in Z at points 2 and 3: 

22 -23 

(62) -- MX Rx - ?[&]=a* 

47 
Rotational FRF about the rectangular mass center for the other two axes were obtained in a 

similar manner. In the case of rotations about an axis parallel to the Y axis, excitation in the X direction 

at point 6 (fig. 11) was used, along with X responses at all three accelerometers, to obtain redundant 

expressions given in equation (63): 

Finally, rotational responses about an axis parallel to the Z axis and through the rectangular mass 

center were expressed in terms of X-direction excitations at points 4 and 7 along with X responses at all 

three accelerometers, as shown in equation (64): 

To validate the accuracy of the approximate expressions in equations (6 1)-(64), analytical rotational FRF extracted from translational responses were compared to exact 8/M analytical responses. 

Figure 22 presents such a comparison for 

to equation (6 I). 

rotations about the mass center (RXIMX), corresponding 

1.00E-06 4 I 

0 50 100 150 200 250 300 350 400 

Frequency (Ht) 

Figure 22. Rotational responses of mass (3RX, 3RX), direct and calculated 

from translational responses. 

48 
I 

, 

Very little difference can be seen between the derived (dashed line) and exact (solid line) analytical 

response functions in figure 22. Comparisons of exact and derived OY and 8,analytical rotational 

responses, corresponding to equations (63) and (a), are presented in figures 23 and 24. 

100E-01 

1.00E-02.) ... I 

1.00E-03 

1.00E-04 

1.00E-05 

1.00E-06 

1.00E-07 

1.00E-08 

0 50 100 150 200 

Frequency (Hz) 

Figure 23. Comparison of direct and calculated rotational responses (3RY, 

for attached mass. 

1 .00E-01 

0 50 100 

Frequency (Hz) 

150 200 

Figure 24. Direct and calculated rotational response functions (3RZ, 3RZ) 

for attached mass. 

49 
I 

Differences in the derived and exact FRFs are observed in the antiresonance regions 

of figures 23-24 for higher frequencies. This can be explained in terms of the physical significance 

of a drive-point (excitation, response in same location, and DOF) response function. Antiresonances 

of drive-point FRF for a particular DOF correspond to the frequencies and modes that would be obtained 

if the structure were constrained in that DOE Since the approximate rotational FRFs are derived from 

translational functions for which the excitation and response points are not coincident (non-drive-point 

FRFs), it should be expected that differences would exist in the antiresonances when compared to exact 

8/M response functions, which are drive-point FRFs. However, since in all cases the derived rotational 

FRFs are virtually identical to the exact results below 140 Hz, it was concluded that the approximate 

rotations are sufficiently accurate. In particular, the derived functions appear accurate enough to allow 

estimation of residual flexibility values, which is the primary goal. 

Having established acceptable accuracy of approximate rotational FRFs based on the arc length 

formula, experimental functions for the payload simulator were compared to results from the model. In 

figure 25 (a) and (b), test RXIMX functions derived using equations (61) and (62) are compared to the 

direct or “exact” analytical OJM,rotational FRFs. The computational results in figure 25 are for the 

final updated or correlated model that had been corrected using all available test data, including 

rotations. Initial model-to-test comparison of translational responses used to determine torsion (fig. 26, 

for example) had revealed that the finite element model in-plane torsional stiffness of the rectangular 

mass plate elements was drastically too low. Once this was corrected, along with increasing the trunnion 

torsional rigidity by about 26 percent, good agreement was achieved between model and test frequency 

response (fig. 27). The correlation approach was to verify all the analytical translational FRFs used in 

approximating the rotational responses, perform model updates as necessary, and then compare the 

rotational responses as the last step. Figures 28 and 29 show excellent test/analysis agreement for the 

RYIMY and RZIMZ functions. Experimental rotations were obtained using both redundant calculations 

I in equations (63) and (64). 

4.3 Experimental Rotational Residual Flexibility Results and Comparison With Analysis 

The final step in the implementation of the hybrid method was to estimate the rotational residual I 

I functions and residual flexibility values for the attachment point of the rectangular mass. It was stated 

earlier in the TM that free-suspension modal testing was done for the payload simulator structure without mass loading. Since no rotational DOFs were included in the test modal database, the normal approach for obtaining residual flexibility, subtracting a synthesized modal FRF from a full measured FRF 

(fig. 17, for example), could not be done for rotations as it was for translations. However, the presence of 

prominent stiffness lines (ramping up toward dominant local trunnion modes) in the experimental rotational FRF, as seen in figures 25,28, and 29, made it possible to estimate residuals in an alternative 

fashion. A by-product of this property of the FRF is that the response function and residual function 

are virtually identical except at the lowest frequencies. This is also true for the translational functions, 

as shown in figure 17. 

50 
. -. . 

... ...... ... . .. ... .. ._ .- . .- .. __ 1.00Et01 1 . 

1.00Et00 

1.00E-01 

A 

1.00E-02 

s* 

1.00E-03 

1.00E-04 

1.00E-05 

1.00E-06 

. TEST 5 Y DR-- _.__~__________.~ ____ 

~~~~ .- .. 

........ . 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(a) Test rotational response computed from test translational data using equation (6 1). 

1.00Et01 

1.00Et00 

1.00E-01 

P 

c: 

1.00E-02 

1.00E-03 

1.00E-04 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(b) Test rotational response computed from test translations using equation (62). 

Figure 25. Comparison of tesdanalysis rotational responses about X axis at mass 

center (3RXIMX). 

51 
.. 

1.00Et01 I 

1.00E-01 

1.00E-02 

1.00E-03 

1.00E-04 

I.""L "V . 

0 50 100 

Frequency (Hz) 

150 200 

Figure 26. Testlanalysis Z-bending responses (52,22) at attached mass before model updates. 

l.OOE+Ol 

1.00Et00 

1.00E-01 

1.00E-02 

1.00E-03 

1.00E-04 

0 50 100 150 200 

Frequency (Hz) 

Figure 27. Test and analytical trunnion 2-bending responses (52,227 after model correlation. 

52 
1 .OOE-03 

1 .OOE-04 

1 .OOE-O5 

1.00E-06 

1.00E-07 - ~_______~ 

I.""_ "" . 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(a) Test rotational response computed from test translational data using 

second part of equation (63). 

1.00Et00 

1.00E-01 - 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(b) Test rotational response computed from test translations of attached 

mass, using first part of equation (63). 

Figure 28. Comparison of test and updated analytical rotations about Y axis (3RYIMY) 

at mass center. 

53 
. . . -. . .- . -. 

. .. .. .- .. .. .. . .. .. 1.00E-01 -1 -- . 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(a) Test rotational response computed from test translational data 

using second part of equation (64). 

1.00Et00 

1.00E-02 

1.00E-03 

1 .OOE-04 

1.00E-05 

1.00E-06 

1.00E-07 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(b) Test rotational response computed from test translations, 

using first part of equation (64). 

Figure 29. Test and updated analytical trunnion rotations about Z axis (3RZIMZ) 

at mass center. 

54 
Because of the prominent stiffness lines in the rotational FRF, and the fact that the FRF are 

essentially the same as the residual functions when this occurs, the second-order least squares curvefitting procedure that was used for translational test residual functions was applied directly to rotational 

FRF. Results comparing the estimated test residual functions to model predictions for RX/MX (8,/M,) 

at mass point 3 are given in figure 30 (a) and (b). Figure 30 (a) corresponds to the 5Y excitation on the 

rectangular mass and equation (61), while figure 30 (b) corresponds to the 5Z excitation as shown in 

equation (62). In both parts of figure 30, the test FRFs are shown, along with a smooth curve fit of the 

FRF based on the prominent stiffness line region around the center of the frequency bandwidth. The 

analytical residual function is overlaid on the test data for comparison. For both the 5Y and 5Z excitation 

cases, the estimated experimental residual functions, and thus residual flexibility values, compare 

extremely well. Recall that residual flexibility is the value of the residual function at zero frequency. 

Figure 3 1 compares the estimated test residual functions with analysis for RY/MY(8,,/M,)) at the 

mass center (point 3 in figure 11). Figure 3 1 (a) corresponds to the first part of equation (63), where 

responses at mass points 1 and 3 are used, and figure 3 1 (b) corresponds to the second part of the equation. The agreement between test and analysis residuals is fairly good for RYIMY and is considered 

acceptable, especially in view of the good test/analysis agreement for the FRFs shown in figure 28 for a 

large bandwidth. Finally, the results for RZIMZ at the mass center (point 3) are shown in figure 32 (a) 

and (b). Analysis compares extremely well with test for the 4X excitation case (first part of eq. (64) and 

fig. 32 (a)), and the comparison is acceptable for the 7X excitation case. 

Finally, the analytical and estimated experimental residual flexibility values for rotational DOFs 

are shown in the lower portion of table 15. The test values are averages based on the redundant excitation and response data shown in figures 30-32 and described by equations (61)-(64). Also, the translational residual flexibilities are given in the top portion of table 15. It is noted again that in the table for 

the X direction, low confidence existed for the experimental residual value due to the lack of a prominent 

stiffness line and the resulting breakdown of the second-order curve fitting. 

4.4 Using Models Correlated to Free-Suspension Modes and Residual Flexibility 

Once the model has been updated to agree with measured free-free modes and residual terms in 

the mass-loaded configuration, it is a simple matter to obtain the constrained configuration. The boundary masses are removed from the finite element model, and the boundary DOFs to be constrained in 

service are analytically constrained. 

This approach applies to the use of full models which have not been subjected to dynamic reduction. Alternatively, reduced residual flexibility models corresponding to equations (33)-(44) can be used 

that contain all or predominantly test-verified quantities or actual test data. Finally, as discussed previously, test-verified free-free models can be coupled to other components rather than constrained completely. This situation arises when coupled system loads analyses are to be performed. 

55 
- -. .. I ... 

. .. .- . . - -. 

.. . .. . -. . .. 

. 

-. l.OOE+OO~~ .. ... - -- . .. 

. __ . . .- 

I.""_ 1" . 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(a) Test rotational response and residual functions based on equation (61). 

l.OOE+Ol 

1.00E-01 

1.00E-02 

1.00E-03 

1 .OOE-04 

1.00E-05 

0 50 100 150 200 250 300 350 400 

Frequency (Hz) 

(b) Test rotational response and residual functions based on equation (62). 

Figure 30. Test and updated analytical residual functions about X axis (3RXIMX) 

for attached mass. 

56 
l.OOE+OO 

1.00E-01 

1.00E-02 

n 

P 

c 

S 1.00E-03 w 

s 

2 

1.00E-04 

1.00E-05 

1.00E-06 

1.00Et00 

1.00E-01 

1.00E-02 

n 

= 1.00E-03 E 

\ r 

.- 

w 

$ 1.00E-04 

k 

P 

1 .OOE-05 

1.00E-06 

1.00E-07 

4 I 

0 20 40 60 80 100 

Frequency (Hz) 

Test rotational response and residual functions obtained 

from translations in first part of equation (63). 

I 

0 20 40 60 80 100 

Frequency (Hz) 

(b) Test rotational response and residual functions from test 

translations using second part of equation (63). 

Figure 31. Test and updated analytical rotational residuals about Y axis (3RYIMY) 

at mass center. 

57 
1.00Et00 

1.00E-01 

1.00E-02 

1 .OOE-03 

1.00E-04 

I.""& V" . 

0 20 40 60 

Frequency (Hz) 

80 100 

(a) Test rotational response and residual functions obtained 

from translations in first part of equation (64). 

20 40 60 

Frequency (Hz) 

80 

(b) Test rotational response and residual functions from test 

translations using second part of equation (64). 

100 

Figure 32. Test and updated analytical rotational residuals about Z axis (3RZIMZ) 

at mass center. 

58 
5. SUMMARY 

This TM has presented a thorough description and investigation of a new hybrid approach for 

free-suspension modal testing that allows verification of constrained-boundary structural models. The 

approach combines some of the best characteristics of both the residual flexibility and mass-additive 

techniques. The residual flexibility method alone involves the measurement of global free-free modes 

of the structure, along with boundary residual flexibility values, while the basic mass-additive approach 

involves mass loading of the structure boundaries and measurement of the free-free modes in that configuration. The hybrid approach presented in this TM combines the excellent convergence characteristics 

of the basic residual flexibility method with the enhancement provided by mass loading that allows 

measurement of rotational properties of the boundaries. 

Results have been presented for a payload simulator structure showing that the hybrid method 

converges in a manner comparable to the basic residual flexibility method and also showing the effect 

of mass loading on response functions and residuals. Good agreement was demonstrated between test 

and analysis translational response functions and residual flexibility for the payload simulator with 

mass-loaded boundaries. A technique for obtaining rotational FRFs and residual functions based on 

the arc length formula was developed and presented in detail. Excellent agreement was obtained overall 

between analytical and experimental rotational FRFs and residuals. 

In practical implementation of the method, the following approach should be taken: (1) Perform 

a free-suspension modal test in the mass-loaded configuration to obtain global modes and natural frequencies; (2) measure translational and rotational drive-point FRF and residual functions; (3) update the 

finite element model to agree with test modal, FRF, and residual data, typically striving for frequency 

errors under 5 percent for modes and FRF and diagonal cross-orthogonality values 0.90 or greater; and 

(4) constrain the test-correlated model or couple it to other structural components at the boundary DOE 

Alternatively, and perhaps ideally, a reduced model (residual flexibility model) consisting mostly or 

entirely of test-verified modes and residual terms could be developed after step (3) and before step (4). 

This approach could be needed in cases where large finite element models are being coupled to other 

components and then used for transient loads analysis. 

The work presented in this TM should advance free-suspension residual flexibility testing by 

making it more applicable to general boundary conditions and situations where rotational DOFs must 

be included. Though the development was based on the use of translational measurements to estimate 

rotational data, the technique could be used somewhat more easily in cases where rotational accelerometers are available. However, even as rotational sensors become more reliable and affordable, it is likely 

that the use of mass loading will always be needed to amplify the rotational responses and allow application of moments at the structure boundaries. 

59 
~ 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

REFERENCES 

Tinker, M.L.: “Modal Vibration Test Facilities and Methods for Space Station Modules,” AIAA95-1 295, Proceedings of 36th Structures, Structural Dynamics and Materials Conference, New 

Orleans, LA, pp. 1190-1 197, April 10-13, 1995. 

Coleman, A.D.: “A Mass Additive Technique for Modal Testing as Applied to the Space Shuttle 

ASTRO-1 Payload,” Proceedings of the Sixth International Modal Analysis Conference, 

Kissimmee, FL, pp. 154-159, February 1988. 

Kinney, T.L.: “Material Science Laboratory (MSL) Carrier Modal Survey Test Summary and 

Model Correlation,” PMIC-RPT4925, Teledyne Brown Engineering, June 1990. 

Admire, J.R.; Tinker, M.L.; and Ivey, E. W.: “Mass-Additive Modal Test Method for Verification 

of Constrained Structural Models,” AIM Journal, Vol. 3 1, No. 11, pp. 2148-2153, November 

1 993. 

Goldenberg, S.; and Shapiro, M.: “A Study of Modal Coupling Procedures for the Space Shuttle,” 

NASA CR-112252, April 1973. 

Gwinn, K.W.; Lauffer, J.P.; and Miller, A.K.: “Component Mode Synthesis Using Experimental 

Modes Enhanced by Mass Loading,” Proceedings of the Sixth International Modal Analysis 

Conference, Kissimmee, FL, pp. 1088-1093, February 1988. 

Yasuda, C.; Riehle, P.J.; Brown, D.L.; and Allemang, R.J.: “An Estimation Method for Rotational 

Degrees of Freedom Using a Mass-Additive Technique,” Proceedings of the Second International 

Modal Analysis Conference, Orlando, FL, pp. 877-886, 1984. 

Barney, P.S.: “Investigation of Mass Additive Testing for Experimental Modal Analysis,” Ph.D. 

Dissertation, University of Cincinnati, College of Engineering; Final Report for NASA Grant 

NAGS- 170M, 1992. 

Benfield, W.A.; and Hruda, R.E: “Vibration Analysis of Structures by Component Mode 

Substitution,” AIAA Journal, Vol. 9, No. 7, pp. 1255-1261, July 1971. 

Admire, J.R.; Tinker, M.L.; and Ivey, E. W.: “Residual Flexibility Test Method for Verification 

of Constrained Structural Models,” AIAA Journal, Vol. 32, No. 1, pp. 170-175, January 1994. 

MacNeal, R.H.: “A Hybrid Method of Component Mode Synthesis,” Computers and Structures, 

Vol. 1, pp. 581-601, 1971. 

61 
12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

62 

Rubin, S.: “Improved Component Mode Representation for Structural Dynamic Analysis,” AIM 

Journal, Vol. 13, Palm Springs, CA, pp. 995-1006, 1975. 

Martinez, D.R.; Carne, T.G.; and Miller, A.K.: “Combined ExperimentallAnalytical Modeling 

Using Component Mode Synthesis,’’ Proceedings of the 25th Structures, Structural Dynamics 

and Materials Conference, Palm Springs, CA, pp. 140-152, May 14-18, 1984. 

Craig, R.R., Jr.; and Chang, C.-J.: “On the Use of Attachment Modes in Substructure Coupling 

for Dynamic Analysis,” Proceedings of the 18th Structures, Structural Dynamics and Materials 

Conference, San Diego, CA, pp. 89-99, March 21-23, 1977. 

Craig, R.R., Jr.: “A Review of Time-Domain and Frequency-Domain Component Mode Synthesis 

Methods,” Combined ExperimentallAnalytical Modeling of Dynamic Structural Systems, 

AMD-Vol. 67, American Society of Mechanical Engineers, New York, pp. 1-30, 1985. 

Kammer, D.C.; and Baker, M.: “A Comparison of the Craig-Bampton and Residual Flexibility 

Methods for Component Substructure Representation,” Proceedings of the 26th Structures, 

Structural Dynamics and Materials Conference, Orlando, FL, pp. 699-706, April 15-17, 1985. 

Blair, M.A.: “Space Station Module Prototype Test Report: Free-Free Testing Methods,” 

EM ATTIC 001, Lockheed Missiles and Space Company, Sunnyvale, CA, June 1991. 

Chandler, K.O.; Driskill, T.C.; and Lindner, J.L.: “Space Station Module Alternative Testing 

Techniques for Interface Characterization: Free-Free Modal Survey Test Report,” SS-DEV-ED9& 

061, NASA Marshall Space Flight Center, Huntsville, AL, October 1990. 

Flanigan, C.C.: “Test-Analysis Correlation of the Transfer Orbit Stage Modal Survey,” 

Report 40864-8, SDRC, Inc., San Diego, CA, October 1988. 

Smith, K.; and Peng, C.Y.: “SIR-C Antenna Mechanical System Modal Test and Model 

Correlation Report,” JPL 0-10694, Jet Propulsion Laboratory, Vol. 1, April 1993. 

Klosterman, A.L.; and Lemon, J.R.: “Dynamic Design Analysis via the Building Block 

Approach,” Shock and Vibration Bulletin, No. 42, Part 1, pp. 97-104, January 1972. 

Bookout, P.S.: “Statistically Generated Weighted Curve Fit of Residual Functions for Modal 

Analysis of Structures,” NASA TM 108481, February 1995. 

Tinker, M.L.; and Bookout, P.S.: “Measurement of Residual Flexibility for Substructures Having 

Prominent Flexible Interfaces,” Proceedings of 35th Structures, Structural Dynamics and 

Materials Conference, Hilton Head, SC, pp. 1944-1958, April 18-20, 1994. 

Tinker, M.L.: “Free-Suspension Residual Flexibility Testing of Space Station Pathfinder: 

Comparison to Fixed-Base Results,” AIM-98-1 791, Proceedings of 39th Structures, Structural 

Dynamics and Materials Conference, Long Beach, CA, April 20-23, 1998. 
25. 

26. 

27. 

28. 

Tinker, M.L.; and Cutchins, M.A.: “Model Correlation Issues in Residual Flexibility Testing,” 

Proceedings of I997 ASME Design Engineering Technical Conferences, DETC97MB-4262, 

Sacramento, CA, Sept. 14-17, 1997. 

Blair, M.A.; and Vadlamudi, N.: “Constrained Structural Dynamic Model Verification Using Free 

Vehicle Suspension Testing Methods,” Proceedings of the 29th Structures, Structural Dynamics 

and Materials Conference, Williamsburg, VA, pp. 1187-1 193, April 18-20, 1988. 

Chandler, K.O.; and Tinker, M.L.: “A General Mass-Additive Method for Component Mode 

Synthesis,” AIAA-97-1381, Proceedings of 38th Structures, Structural Dynamics and Materials 

Conference, Kissimmee, FL, April 7-10, 1997. 

Craig, R.R.; and Bampton, M.C.C.: “Coupling of Substructures for Dynamic Analyses,” AZAA 

Journal, Vol. 6, No. 7, pp. 1313-1319, 1968. 

63 

I 
REPORT DOCUMENTATION PAGE 

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OMB NO. 0704-0188 

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 

March 2003 Technical Memorandum 

I 

PublE reporting burdm lor mis wllectwn 01 inlormabon is eshmated to average 1 hour per response. including the time lor reviewing instwctions. searching existing data sources. 

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4. TITLE AND SUBTITLE 

Hybrid Residual Flexibility/Mass-Additive Method for Structural 

Dynamic Testing 

6. AUTHORS 

M.L. Tinker 

5. FUNDING NUMBERS 

7. PERFORMING ORGANIZATION NAMES(S) AND ADDRESWES) I 

14. SUBJECT TERMS 

structural dynamics, modal testing, free-suspension methods, modal 

coupling, component mode synthesis, dynamic model verification 

George C. Marshall Space Flight Center 

Marshall Space Flight Center, AL 35812 

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NASA/TM - 2003-2 1 2343 

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13. ABSTRACT (Maximum 200 words) 

A large fixture was designed and constructed for modal vibration testing of Oirernariotial Space Station elements. This fixed-base 

test fixture. which weighs thousands of pounds and is anchored to a massive concrete floor, initially utilized spherical bearings and 

pendulum mechanisms to simulate Shuttle orbiter boundary constraints for launch of the hardware. Many difficulties were 

encountered during a checkout test of the common module prototype structure. mainly due to undesirable friction and excessive 

clearances in the test-article-to-fixture interface bearings. Measured mode shapes and frequencies were not representative of orbiterconstrained modes due to the friction and clearance effects in the bearings. As a result, a major redesign effort for the interface 

mechanisms was undertaken. The total cost of the fixture design. construction and checkout, and redesign was over $2 million. 

Because of the problems experienced with fixed-base testing. alternative free-suspension methods were studied, including the 

residual flexibility and mass-additive approaches. Free-suspension structural dynamics test methods utilize soft elastic “bungee” 

cords and overhead frame suspension systems that are less complex and much less expensive than fixed-base systems. The cost of 

free-suspension fixturing is on the order of tens of thousands of dollars. as opposed to millions, for large fixed-base fixturing. In 

addition, free-suspension test configurations are portable. allowing modal tests to be done at sites without modal test facilities. For 

example, a mass-additive modal test of the ASTRO-1 Shuttle payload was done at the Kennedy Space Center launch site. In this 

Technical Memorandum, the mass-additive and residual flexibility test methods are described in detail. A discussion of a hybrid 

approach that combines the best characteristics of each method follows and is the focus of the study. 

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