Paper 104 Misawa

7/30/2019 Paper 104 Misawa

1/8

IOMAC'135th International Operational Modal Analysis Conference2013 May 13-15 Guimares - Portugal

COMPONENT MODAL TEST OF LARGE

STRUCTURES

Masayoshi Misawa1, Jyunichi Nakajima

2, and Hidenori Kawasoe

2

ABSTRACT

This paper describes a new method to predict the natural frequencies and modes of large structures.Component modal test is effective at identifying the dynamic characteristics of the structures

consisting of several components. Select a tested component subjected to modal tests. The rest of thecomponents are untested components. In this method, the effect of the untested components is

considered as several additional masses and stiffnesses to obtain the dynamic characteristics of thestructures through component modal tests. After the completion of the modal test, a different testedcomponent is selected. In this way, modal tests are performed for different tested components. Weconsider identical frequencies obtained by component modal tests as frequencies of the structures.The feature of the proposed method is to obtain the dynamic characteristics of large flexible structures

with component modal tests and to need no component mode synthesis. A numerical example is givento demonstrate the effectiveness of the method in predicting frequencies and mode of large structures.

Keywords: Component Modal Test, Additional Mass and stiffness, Large Structures, Modal, Analysis

1. INTRODUCTIONModal tests are usually performed to confirm the dynamic characteristics of a structure. However, forlarge flexible structures such as deployed antennas and solar paddles for satellite use, gravitational

considerations may prevent a fully assembled ground modal test because these structures are not

strong enough to withstand the force of gravity. Component modal tests can provide a means ofidentifying the dynamic characteristics of the structure without testing the whole structure.

An approach to predict the dynamic characteristics of structures is the mass additive method. Gwinnet al. [1], Karpel et al. [2-4], and Chandler et al. [5] consider the use of mass loading for componentmode synthesis. The masses are added to the boundaries of a tested structure to force localdeformation near boundaries which improves subsequent modal coupling. By subtracting the massfrom the equation of motion of the tested structure, constrained modes are estimated. Therefore, these

methods do not deal with the value of the mass. On the other hand, Coleman et al. [6] and Admire etal. [7] consider the mass additive method for verification of a constrained model. The masses aresized in such a way as to cause the lower frequencies to be within the frequency range of interest.

1Professor, Shizuoka University, [email protected]

2Graduate student, Shizuoka University


2/8

M. Misawa, J. Nakajima, H. Kawasoe

2

A new method has been presented to predict the dynamic characteristics of large structures withcomponent modal tests [8, 9]. In this method, a tested component with an additional mass andstiffness is subject to component modal test. It was shown how to find additional mass and stiffnessand to select a location attached them to the tested component. The feature of this method needs nocomponent mode synthesis to predict the natural frequencies and modes of structures. When usingmultiple additional masses and stiffnesses in component modal tests, additional mass and stiffnessattached to each location cannot be determined as a unique value. This paper introduces the weightingcoefficient considering the effect of the untested component as the i-th additional mass and stiffness.In this paper, we investigate the effects of selected tested component on the prediction of dynamiccharacteristics of structures.

2. METHODOLOGY2.1. Mass additive coordinatesDenote the translational coordinates with the additional mass and stiffness aspi (i=1, 2, , n), and therest as qi. n is the number of mass and stiffness additive coordinates. To find a reduced mass

ipsM ,

and a reduced stiffnessips

K,

, the dynamic equation of the structure is reduced to coordinatepi as

0,,,, iiii pspspsps2

s xKxM (1)

where

11

,11

i

iiii

iiiii

i

i

iiii

iiiii

i

ps,

pps,qps,

pqs,qqs,ps,ps,

ps,

pps,qps,

pqs,qqs,ps,ps, K

KM

MR

K

KKRR

M

MMRTT

(2)

Subscripts shows quantity of structures. Transformation vectorips,

R satisfiesiii ps,psqs,

x Rx , and is afunction of an arbitrary frequency s. Equation (2) shows that the reduced mass

ipsM , is calculated

with sub mass matrices which vary due to a mass additive coordinate. Therefore, the reduced massips

M , is determined by only a mass additive coordinate. In a similar way, one obtains the followingequation by reducing the dynamic equation of the tested component to coordinatepi as

0,,,, iiii ptptptptt xKxM2

(3)

where

11

ii

i

ii

i

pt,it

pt,pt,

pt,it

pt,pt,

1K

1M

RK

RRM

R 11,

TT

(4)

Subscript tshows quantity of tested component. Matrices1i

tM and1i

tK are expressed as

1

1

21

1

1

1

21

1

j

i

jjt

it

j

i

jjt

it

K

M

CKK

CMM

),,2,1( ni (5)

The second term at the right-hand side shows additional masses and additional stiffnesses. Adescription of them is provided in the following. Matrix Ci is 1 at thepi-th diagonal element and 0 forthe other elements. When finding the i-th reduced mass

iptM , , mass matrix Mt includes (i-1)

additional masses. Matrix Mt used in finding the reduced massipt

M , (i=2, 3,, n) varies each time.

The transformation vector ipt,R

also varies because of the same reason. Therefore, it is required to usean appropriate mass matrix, stiffness matrix, and transformation vector when finding the reduced mass

ipt,M and stiffness

ipt,K .


3/8

5th International Operational Modal Analysis Conference, Guimares 13-15 May 2013

3

Reduced massips

M , and stiffness ipsK , include the effect of untested components, while the reducedmass

1pt,M and stiffness

1pt,K do not include it. Clearly, the differences, (

ii ptpsMM ,, ) and

(ii ptps

KK ,, ), show the effect of untested components. Therefore, it is reasonable to consider thesedifferences as the i-th additional mass Mi and stiffnessKi. It is impossible to uniquely determineeach additional mass Mi and stiffnessKi when considering multiple additional masses andstiffnesses. We introduce the weighting coefficient i that the effect of the untested component is

considered as the i-th additional mass and stiffness. Therefore, the i-th additional mass and additionalstiffness are expressed by

)(

)(

ii

ii

pt,ps,ii

pt,ps,ii

KKK

MMM

),,2,1( ni (6)

If1 is equal to one, the whole effect of the untested component is considered in determining the firstadditional mass and stiffness. Since this paper deals with multiple additional masses and stiffnesses,1 should be less than one. On the other hand, n is always one in determining the last additional massand stiffness attached to coordinatepn. Therefore, i takes a value of

ni

ni

i

i

if11,,2,1if10

(7)

A procedure to select mass additive coordinates is basically the same as that described in the reference[9]. The peak frequencies of

ipsM , are determined by a mass additive coordinate alone. Therefore,

we select a proposed coordinate with a large displacement of the target mode for a mass additivecoordinate. On the contrary, peak frequencies of

1pt,M , solutions of Eq. (8), vary with the mass

additive coordinate and coefficientsi.

0xMK iiiii qt,

1iqqt,t

1iqqt,

2)2( (8)

Mass additive coordinates are selected so that the additional mass is positive for the proposedcoordinate. Therefore, it is very important to confirm that the selected mass additive coordinatesnever change for different values of1 because this is closely connected to the effectiveness of theproposed method.

2.2. Selection of tested componentsSince a tested component is selected from interconnected components, the boundary locations of thetested component are free. Displacements at the boundaries are seldom small in component modaltests. Therefore, it will be impossible to obtain the target mode if the boundaries of the testedcomponent are close to the node of the target mode. The only way to obtain small boundarydisplacements is to attach a large mass at the boundaries. However, this means that the peakfrequencies of reduced mass

ipsM , (and additional mass Mi ) approach the target frequency of the

structure. As a result, we must use large additional masses, and accurate frequency of the structurecannot be obtained by component modal tests. Therefore, one needs to select a tested component sothat the boundaries are separated from nodes of the target mode.

2.3. Simulated test resultsBy adding additional masses and stiffnesses to the translational coordinates pi (i=1, 2, , n), oneobtains the dynamic equation of the tested component as

0xKxM ntnt

~~2 (9)

where


4/8


4

(10)

ii ppt,M ( ii ppt,K ) is the pi-th diagonal element ofMt (Kt ). Equation (9) can be considered to be thedynamic equation of the structure because the effect of untested components is taken into account asmultiple additional masses and stiffnesses. Frequencies and modes found by Eq. (9) are treated as themeasured data of component modal test in this paper. This paper assumes that there are no modelingerrors because the purpose of this paper is to confirm whether the proposed method has the potential topredict natural frequencies and modes of structures. Therefore, reference frequencies and modes,hereafter referred to as exact values, are computed directly from a finite element model of structures.These frequencies are used to calculate additional mass and stiffness.

3. RESULTSConsider a cantilever beam as shown in Figure 1. Arabic numbers show the location number. The

boundary location is location 9. Each element length is 100 mm. Figure 1 also shows the testconfiguration of tested components with the additional mass M and additional stiffness K. For

example, Mi,j shows thej-th additional mass attached to a tested component i. Only bending modesin the X-Y plane are considered, so the additional mass and stiffness are attached to the Y coordinate

of a location on a tested component. The simulated test frequencies and modes are obtained by Eq. (9).In the following, the simulated test results are shown mainly for a case when component 1 is the tested

component with two additional masses and stiffnesses.

(1) Cantilever beam (2) Component test configuration

Figure 1 Test configuration

3.1. Mass and stiffness additive coordinatesThe proposed coordinates and the next proposed coordinates with additional mass and stiffness areselected, as listed in Table 1. In this table, 9Y shows the coordinate Y at location 9. The boundary

800mm

Component 1 Component 2

Heig ht = 3mm Width = 10mmCross section

Density = 1.6 X10 - 6

Material properties

Young' s modulus = 7700kg/mm 3

kg/mm2

1000mm

1 2 3 4 5 6 7 8 9 11 13 15 17 19

X

Z

Y

Component 2

M1,2

K1,2

Component 1

K1,1

M1,1M2,2

K2,2K2,1M2,1

nppt,

2ppt,pqt,

1ppt,

qpt,qqt,

nt

nppt,

2ppt,pqt,

1ppt,

qpt,qqt,

nt

KK

KK

KK

MM

MM

MM

nn

22

11

nn

22

11

0

K

0

KK

0

M

0

MM

K

M

~

~


5/8


5

location is selected so that the boundary location is separated from the nodes of the target mode toaccurately measure target modes of the beam.

Table 1 Proposed and selected coordinates for tested component 1

Proposed

coordinate

Selected

coordinate

Proposed

coordinate

Selected

coordinate1 9Y 8Y

2

3

4

5

( ) : Next proposed coordinate

Modenumber

1st coordinate 2nd coordinate

9Y 8Y

9Y,(8Y)

9Y,(8Y)

8Y,(7Y)

8Y,(7Y)

6Y 7Y6Y,(7Y) 7Y,(5Y)

10Y 11Y5Y,(10Y) 4Y,(11Y)

4Y 12Y4Y,(12Y) 12Y,(8Y)

9

9

12

11

12

Boundarylocation

3.2. Additional massWe need to select a tested component so that the boundary is separated from nodes of a target mode to

obtain the target mode with small additional masses. Figure 2 shows variations in the first and secondadditional mass with 1 to obtain the third beam mode. In Figure 2, 7/8 shows that the first massadditive coordinate is 7Y and the second one is 8Y. BL9 shows boundary location of the testedcomponent is location 9. The third mode of the beam has a node near the boundary (location 9) of thetested component, so we select the tested component of new boundary location 12 with a largedisplacement of the third beam mode. For boundary location 9, the proposed method gives massadditive locations 7/8. Although additional masses are not large, mode accuracy is low (see Figure 6).

This is because it is difficult to keep boundary displacement significantly small by attaching masses tolocations 7 and 8. For mode accuracy improvement, we need to attach large masses near the boundary.

Then, mass additive locations 8/9 are selected. In this case, additional mass, especially the second

additional mass M1,2 , is very large, as shown in Figure 2. For boundary location 12, the first andsecond additional masses are small. The same results are obtained for the fifth mode with a node near

boundary location 9. The above result shows that it is effective to select a tested component so that itsboundary can have large displacement for obtaining an accurate target mode with light additionalmasses.

1.21.00.80.60.40.20.00

20

40

60

80

7/8, BL98/9, BL96/7, BL12

1

AdditionalmasM1,1,x10-4g

1.21.00.80.60.40.20.0-200

-100

0

100

200

1

AdditionalmasM1,2,x10-4g

7/8, BL98/9, BL96/7, BL12

(1) First additional mass (2) Second additional mass

Figure 2 Variation in additional mass for the third frequency measurement (tested component 1)

3.3. Frequency accuracyTable 2 shows the simulated test frequencies of tested components when 1=0.5 and 2=1.0. For thefirst frequency measurement, the test frequencies of 1.0, 14.7, 43.8, 112.3 and 219.6 Hz are obtainedwhen component 1 is the tested component. Because we have no information on which the testfrequency is the target frequency, component modal tests are performed for different tested

components. When component 2 is the tested component, the test frequencies of 0.1, 1.0, 18.3, 52.0and 102.5 Hz are obtained. The identical frequency (1.0 Hz) obtained by component modal tests is

considered to be the first beam frequency. Although test frequencies vary with 1, the identical


6/8


6

frequency of 1.0 Hz does not vary with 1. Similarly, the second and higher frequencies of the beamare predicted. The bold numbers in Table 2 show the predicted natural frequencies of the beam. Thepredicted frequencies agree with the exact frequencies of the beam.

Table 2 Simulated test frequencies (1=0.5 and 2=1.0)

Tested

component No. 1st freq. 2nd freq. 3rd freq.

1.0 6.4 17.9

1

14.7 25.1 29.8

43.8 70.0 80.9

9.1

2

16.6

18.7 17.9

Mode No.

1

2

3

1

2

3

Target frequency

(Hz)

0.1 0.5

1.0 6.4

18.3

112.3 135.5 161.5

219.6 236.1 251.1

4

5

52.0 53.5 48.8

102.5 101.1 107.0

4

5

4th freq. 5th freq.

11.0 51.5

35.0 57.9

81.0 90.4

10.5 13.4

28.8 18.4

35.0 38.0

137.5

252.4 263.3

55.2 57.9

105.8 103.2

153.2

1.0 6.4 17.9 35.0 57.9Exact

3.4. Mode accuracyFigure 3 shows the first and second modes. The test modes are obtained from simulated componentmodal tests when boundary location of the tested component is 9. The first mode is identical with the

exact one. Therefore, component modal tests provide the first mode regardless of1. For the second

mode, the effect of coefficient 1 is also negligibly small. Since masses are attached to the tip ofcomponent 1 (locations 8 and 9) the same as we do for the first mode, the curvature of the simulatedtest mode of component 1 is similar to the first mode. For this reason, the simulated test mode is

slightly different from that of the exact mode. However, MAC (Modal Assurance Criterion) of the

second beam mode is greater than 0.9 (see Figure 7).

19161310741-1.2

-0.8

-0.4

0.0

0.4

0.8

Location number

Modalamplitude

ExactTest (1 =0.3)Test (1 =0.5)Test (1 =0.7)

19161310741-0.8

-0.4

0.0

0.4

0.8

1.2

Location number

Modalamplitude

ExactTest (1 =0.3)Test (1 =0.5)Test (1 =0.7)

(1) First mode (2) Second mode

Figure 3 Exact and simulated test modes

Figure 4 shows a comparison of the fourth modes for 1 = 0.5. Since boundary location affects thesimulated test mode, the simulated test mode is calculated when changing boundary location fromlocation 9 to location 11. The simulated test mode of component 2 is improved for boundary location

11. As boundary location of the tested component is free, it is difficult to obtain a curved line similarto the exact fourth mode near the boundary.


7/8


7

-0.8

-0.4

0.0

0.4

0.8

1.2

191715131197531

ExactTest (BL11)Test (BL9)

Location number

M

odalamplitude

Figure 4 Effect of boundary location on the simulated fourth mode (1=0.5)

Figure 5 shows the third mode of component 1 for different boundary locations and mass additivecoordinates. Variation in MAC of the third mode with 1 is shown in Figure 6. The third mode of thebeam has a node near location 9. When the boundary is location 9, it is difficult to keep boundarydisplacement small by attaching masses at locations 7 and 8 because the boundary is free. As shownin Figure 5, the simulated test and exact modes are distinct at the tip of the component 1. MACdecreases with increasing 1. The only way to obtain small boundary displacement is to add largemasses near the boundary (locations 8 and 9). The simulated test mode approaches the exact mode, so

MAC is improved. However, this is not effective for obtaining accurate dynamic characteristics of thebeam through component modal tests. Peak frequency of sM is the natural frequency of the beam withmass additive coordinate fixed. Therefore, peak frequency equals the third frequency of the beam.Variation in additional mass becomes large near the target frequency for small variation in frequencys when one of the peak frequencies of

1psM , approaches the target frequency. For this reason, it is

hard to obtain an accurate target frequency and mode using tested component 1 with boundarylocation 9. It is required to select a new boundary location with large displacement to obtain thesimulated test frequency and mode similar to the exact ones. For boundary location 12, it can be seenfrom Figure 5 that the simulated test mode approaches the exact mode. As a result, MAC is greaterthan 0.9, and MAC variation with 1 is small as shown in Figure 6. Therefore, it is expected that wecan obtain the target mode by component modal tests.

121110987654321

ExactTest (7/8, BL9)Test (8/9, BL9)Test (7/8, BL12)

Location number

Modalamplitude

0.4

0.6

0.8

1.0

7/8, BL98/9, BL97/8, BL12

0.90.70.50.30.1

1

MAC

Figure 5 Mode comparison of the third mode Figure 6 MAC of the third mode

(component 1) (component 1)Figure 7 shows variations in the MAC of the beam modes with 1. The first mode is identical with the

exact mode, so the MAC is very close to one and is constant with 1. For the second mode, althoughthe simulated test mode is slightly different from the exact mode, the MAC is greater than 0.9. For the

third mode, the MAC gradually decreases with the increasing of 1 because there is a differencebetween the simulated and exact modes of component 2. As component 2 is supported by springs(additional stiffnesses) at locations 13 and 19, the mode is similar to the first mode with both edges ofcomponent 2 simply supported. It is difficult to obtain a test mode similar to the exact mode. For thefourth mode, the MAC variation is almost the same as that for the third mode. For the fifth mode, the

effect of1 on the MAC appears for1 >0.5 and the MAC decreases gradually to about 0.8. Thus, we

can conclude that it is possible to obtain the target mode by selecting an appropriate value of1.


8/8


8

0.90.70.50.30.10.6

0.7

0.8

0.9

1.0

1.1

1st2nd3rd4th5th

1

MAC

Figure 7 MAC variation with coefficient 1

4. CONCLUSIONSA component modal testing method with multiple additional masses and stiffnesses is described to

obtain the dynamic characteristics of large structures. The additional mass and stiffness attached toeach location cannot be determined uniquely in this case. This paper introduces the weighting

coefficient considering the effect of the untested component as the i-th additional mass and stiffness.In this study, we investigate the effects of a tested component on predicting dynamic characteristics ofstructures. It is shown that the accurate frequency and mode of structures are obtained through

component modal tests by selecting an appropriate tested component.

REFERENCES

[1] Gwinn, K. W., Lauffer, J. B., and Miller, A. K. (1988) Component Mode Synthesis UsingExperimental Mode Enhanced by Mass Loading. In: 6th International Modal Analysis

Conference, 1988-1993

[2] Karpel, M., and Raveh, D. (1996) Fictitious Mass Element in Structural Dynamics.AIAA Journal34(3): 607-613

[3] Karpel, M., and Ricci, S. (1997) Experimental Modal Analysis of Large Structures bySubstructuring.Mechanical System and Signal Processing11(2): 245-256

[4] Karpel, M., Raveh, D., and Ricci, S. (1996) Ground Modal Tests of Space-Structure ComponentUsing Boundary Masses.Journal of Spacecraft and Rockets 33(2): 272-277

[5] Chandler, K. O., and Tinker, M. L. (1997) A General Mass-Additive Method for ComponentMode Synthesis. In: 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, andMaterials Conference, AIAA-1997-1381

[6] Coleman, A. D., Anderson, J. P., Driskill, T. C., and Brown, D. L. (1988) A Mass AdditiveTechnique for Modal Testing as Applied to the Space Shuttle ASTRO-1 Payload. In: 6th

International Modal Analysis Conference, 154-159

[7] Admire, J. R., Tinker, M. L., and Ivey, E. W. (1993) Mass-Additive Modal Test Method forVerification of Constrained Structural Models.AIAA Journal31(11): 2148-2153

[8] Misawa, M., and Funamoto, K. (2005) Dynamic Characteristic Prediction of Large SatelliteAntennas by Component Tests.Journal of Spacecraft and Rockets 42(5): 845-849

[9] Misawa, M. (2010) Component Modal Tests with Additional Mass and Stiffness. AIAA Journal48(8): 1840-1847

Documents

Paper 104 Misawa