Design and Implementation of a TLDfor Experimental Structural Dynamics SS2011

Mohammed Metwally (101102)Peter Olney (100783)

Supervisor: Dr.-Ing. Volkmar Zabel

July 12, 2011

Contents

1 Introduction 2

2 Numerical Modeling 22.1 Simplified Hand Calculations . . . . . . . . . . . . . . . . . . . . . . 32.2 SLANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Experimental Test 1 43.1 MACEC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Preliminary TLD Design 4

5 Experimental Test 2 6

6 TLD re-Design 66.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

7 Experimental Test 3 87.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.2 Modal Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Numerical Model Calibration 158.1 Modeling of TLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

9 Conclusions 17

A Test 3 with Excitation Frequency of 3.23 Hz 20

List of Figures

1 Test Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Simplified Lumped Mass Model . . . . . . . . . . . . . . . . . . . . . 33 Test 1 Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . 54 Description of a TLD ( from [5] ) . . . . . . . . . . . . . . . . . . . . 65 TLD frequency plotted over b = 2a [m] and h [m] . . . . . . . . . . . 76 Test 2 Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . 87 Test 2 - Channel 1 Accelerations [g] . . . . . . . . . . . . . . . . . . . 98 Test 2 - Channel 1 Accelerations Zoom [g] . . . . . . . . . . . . . . . 109 TLD Dimensions ( for use with [6] ) . . . . . . . . . . . . . . . . . . . 1010 TLD Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Dynamic Amplification Function for Equivalent System . . . . . . . . 1112 Test 3 Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Calibrated Dynamic Amplification Function . . . . . . . . . . . . . . 1314 Calibrated Dyn. Amp. Function Zoomed to 20.29 rad/sec (3.23 Hz) . 1315 Structure Excited at 2.9 Hz . . . . . . . . . . . . . . . . . . . . . . . 1516 Structure Excited at 2.9 Hz with TLD C . . . . . . . . . . . . . . . . 1517 Structure Excited at 2.9 Hz with TLD C Zoom . . . . . . . . . . . . 1618 Equivalent TMD Modeling Schematic . . . . . . . . . . . . . . . . . . 1719 SAP Acceleration Response at the Top Floor . . . . . . . . . . . . . . 1820 SAP Response Spectrum (FRF) . . . . . . . . . . . . . . . . . . . . . 18

1

List of Tables

1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Frequencies of the Structure [Hz] . . . . . . . . . . . . . . . . . . . . 53 TLD Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 94 TLD Effectiveness - Predicted vs. Observed . . . . . . . . . . . . . . 125 MAC between MACEC and Simplified Calculations . . . . . . . . . . 146 MAC between MACEC and SLANG . . . . . . . . . . . . . . . . . . 147 MAC between MACEC and SAP . . . . . . . . . . . . . . . . . . . . 148 MAC between SLANG and SAP . . . . . . . . . . . . . . . . . . . . . 149 Modified Parameters for Model Calibration . . . . . . . . . . . . . . . 16

1 Introduction

Our task for Experimental Structural Dynamics consisted of the design and imple-mentation of a Tuned Liquid Damper (TLD) for a laboratory structural model. Thestructural model under consideration was a 5 story steel moment frame building asdepicted in Figure 1.

Figure 1: Test Structure

2 Numerical Modeling

The task began with numerical modeling of the structure shown in Figure 1. Thepurpose of this modeling was to serve as a baseline of comparison for our experi-mental tests. It was also developed to give us a better idea of what to expect fromthe experimental results and which mode shapes and frequencies to look for.

2

2.1 Simplified Hand Calculations

We began our numerical modeling by simplifying the structure into a lumped masssystem with 5 DOFs. Using Matlab, we wrote a set of code that does a simplifieddynamic analysis based upon the FEM to obtain the stiffness matrix of the structure.After assembling the active global stiffness matrix and forming the lumped massmatrix, the eigenvalue problem given by equation (1) was solved by means of thenon-trivial solution given by (2) to obtain the circular frequencies, ωk, and thecorresponding mode shapes, ϕk. Figure 2 shows this simple model along with thefirst two normalized mode shapes.

([K]− ω2k[M ])ϕk = 0 (1)

det([K]− ω2k[M ]) = 0 (2)

m1

m2

m3

m4

m5

simplified model 1st mode normalized 2nd mode normalized

1

-1

1

Figure 2: Simplified Lumped Mass Model

2.2 SLANG

After completing the simplified calculations as described above, we began a si-multaneous rigorous analysis with two commercial software packages, SLANG andSAP2000. The simplified calculations served as the baseline whereas the commercialpackage models were intended to produce results closer to reality. We will beginwith a discussion of our SLANG analysis followed by SAP2000 in the next section.The material properties used for the analysis are summarized in Table 1.

Table 1: Material Properties

Steel

Mass Density 7850 kg/m3

Elastic Modulus 210 GPaPoisson’s Ratio 0.2

3

Each piece of the structure was modeled in SLANG using beam elements. Thecolumns were modeled as 2.5 mm radius rod elements. In order to achieve theproper separation between the double floor plates around the perimeter, two stringerelements were used per edge, with a width of 60 mm, height of 5 mm and ‘e h’eccentricity of 5 mm. The floor braces were regular rectangular elements with awidth of 60 mm and height of 5 mm. The discretization used was five nodes persection. This means that for the columns five nodes equated to one story (0.2 m).For the floors, five nodes were used along each edge (0.3 m). The diagonal bracesalso had five nodes.

2.3 SAP2000

The strategy for the SAP2000 modeling was similar to that of the SLANG analysisas described above. The goal was to have detailed models with similar assumptionsso that the results could be compared. Similar to the SLANG modeling, eachpiece of the structure was modeled in SAP2000 using beam elements. In order toachieve the proper separation between the double floor plates around the perimeter,additional nodes were placed at each floor instead of using a special element type aswas done in SLANG. This way two elements could be drawn around the perimeter.The results of this analysis are summarized with the others in Table 2.

3 Experimental Test 1

The main goal of our first experiment was to obtain the modes and mode shapesof the structure. These were to be used to calibrate our numerical models and todesign a TLD to the proper frequency. The sensor placement for this test is shownin Figure 3, with the reference sensors, channels 1 and 2, shown in red. To beable to extract the modes from the experimental data, we excited the structure twotimes for each setup. The first was a noise base excitation in the x direction andthe second was with a hammer in the x direction at the top floor at the node nextto the reference sensor.

3.1 MACEC Analysis

Following our first experiment, the MACEC Matlab tool was used to anaylize ourdata and complete a modal analysis. Table 2 contains the modes we were able toextract from our data using MACEC along with a comparison of those found bynumerical analysis as described in section 2. Except for the fifth mode, our extractedmodes seemed reasonable and comparable to the ones obtained from the numericalmodels. We also checked the animated mode shapes as produced by MACEC andconcluded that they were also comparable to the numerical ones.

4 Preliminary TLD Design

Our preliminary TLD design was based upon equations (3) and (4) along with Figure4 from [5] for a given liquid, liq, where ν is the viscosity. Our goal was to designthe damper for the first mode of the structure as found from the MACEC analysis,a frequency of 3.23 Hz or circular frequency of 20.29 rad/sec.

4

Figure 3: Test 1 Sensor Placement

Table 2: Frequencies of the Structure [Hz]

Modes

Method 1st 2nd 3rd 4th 5th

MACEC 3.23 9.15 14.69 19.03 31.81SAP2000 3.49 10.19 16.10 20.74 23.69SLANG 3.33 9.78 15.55 20.14 23.12Simplified Calculations 3.53 10.31 16.25 20.88 23.81

ωliq =

√√√√πg

2atanh

(πh

2a

)(3)

ξliq =1√2h

√ν

ωliq

(1 +

h

b

)(4)

Plotting equation (3) for a range of liquid heights, h, and container widths,b = 2a, produces Figure 5. As can be seen in the plot, h has a limited influence onthe obtainable frequency range since the limit of tanh is 1.

We decided to follow the TMD design procedure as given by Zabel [7] and adaptit with equations (3) and (4) for a TLD. Using this procedure, water at 25C asour liquid (ν = 0.9 mPa·s), and an effective damper mass equal to the mass ofthe volume of water, we calculated a required damper circular frequency ωd to be19.97 rad/sec or a frequency of 3.18 Hz. It is clear from Figure 5 that to obtainthe desired frequency, b must be less than 8 cm. The TLD we decided to use was aplastic food container found at a local store with a height of 18 cm, large enough toprevent water from splashing everywhere, and a width of 7.5 cm, small enough tomeet our desired frequency with a given water height. Based on these calculations as

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Figure 4: Description of a TLD ( from [5] )

described above, our container needed a water height of 5 cm to obtain the desiredfrequency.

5 Experimental Test 2

Our main tasks for the second experimental test were to test the TLD and completeour data with the missing DOFs to obtain more complete mode shapes in MACEC.The sensor layout for this is shown in Figure 6. To test the TLD, we loaded thestructure with a sinusoidal base excitation at the natural frequency obtained fromMACEC, 3.23 Hz. The results from using the container as described in section 4 canbe seen in Figures 7 and 8, where the solid blue curve represents the accelerationswith the TLD and the dotted black curve represents the accelerations without theTLD. These plots show that our TLD was ineffective at reducing the amplitudes ofacceleration. The goal of our TLD was to reduce the amplification of the structureat its natural frequency, but these results show that our TLD just increased theamplification. Since the computers for performing the MACEC analysis were inhigh demand and seemingly never available, we decided to focus our energy onsolving the problem of our ineffective TLD rather than completing our MACECmode shapes.

6 TLD re-Design

Since our TLD turned out to be more of a tuned liquid amplifier, or more generallyan additional mass for the structure to vibrate with as was shown in the previ-ous section, we decided to investigate our TLD design and consider redesigning it.Following the procedures of Tait [6], we used equations (5), (6) and (7).

ω2 =πg

Ltanh

(πh

L

)(5)

ζw =(

1

2h

)√ν

2ω

(1 +

2h

b+ SC

)(6)

meq =8ρbL2

π3tanh

(πh

L

)(7)

6

00.05

0.10.15

0.20.25

0.3

b 0.050.1

0.150.2

0.250.3

h

0

2

4

6

8

Figure 5: TLD frequency plotted over b = 2a [m] and h [m]

These equations are similar to the ones we used before (see section 4) with slightlydifferent variable definitions, which are shown in Figure 9. Equation (5) is identicalto (3), which we used before to calculate the frequency of the first sloshing mode.The new equation for damping, (6), is slightly different from (4), and includes anadditional term, SC, which is a surface contamination factor and can be consideredas unity according to Tait [6]. The new equation, (7), describes the effective massof the liquid damper. Before, we took the mass of the damper to be ρV , where ρwas density of water, 981 kg/m3, and V was the volume of water in the damper. Inreality, only a portion of the total liquid mass acts in the system [4]. The importanceof the container width also becomes clear from equation (7). It shows that increasingthe liquid depth, h, has only a limited increase on the effective mass when 0 < h < Land nearly no increase when h > L. It is also clear to see that the more importantfactor, after the tank length, L, which is usually fixed by the desired frequency, isthe width, b, of the container. This was not a factor in any of our equations beforeand it helps to explain why our first TLD showed poor results.

Along with these equations, Tait [6] also includes a TLD design procedure whichdraws on similarities to TMD design. The calculations for our new TLD design aresummarized in the following section.

6.1 Calculations

We began with an effective damping, ζeff = 0.05, required to achieve the targetstructural response level. From equation (8), we could determine the mass ratio,µ = meq

M ′s

, which would satisfy ζeff . Having determined µ to be 0.04, we were able to

calculate the optimal tuning ratio, Ωopt, given in equation (9).

ζopteff =1

4

√√√√µ+ µ2

1 + 3µ4

(8)

7

Figure 6: Test 2 Sensor Placement

Ωopt =

√1 + µ

2

1 + µ(9)

We found the optimal tuning ratio to be 0.9711. Using this in Ω = ωa

ω′s, we obtained

a desired TLD sloshing frequency of 19.71 rad/sec. Using this along with our fixedcontainer length from before, L = 7.5 cm, we determined the required water heightto be, h = 4.2 cm from equation (5). Finally, using equation (7) and our containervertically, we calculated meq to be 0.146 kg. Comparing this with the optimal massratio and required effective damping, it quickly became evident that our design wasinsufficient.

L was fixed by the containers that we had available, since we purchased the oneswith the smallest length with reasonable height. We needed more effective massand lower effective damping. We realized that our container had a tight lid with agood seal so we could rotate it and let the former height become the width, b. Thiswould allow us to achieve a much greater effective mass. We also realized that byincreasing the water depth we could benefit from lower effective damping, while notchanging the frequency of the first sloshing mode by much.

Based upon iterating through our available containers, we were able to come upwith four reasonable configurations. These configurations are shown in Figure 10with the relevant design parameters in Table 3. Since our initial TLD test was sucha huge failure and to confirm our conclusions about proper TLD design, we decidedto test all four configurations in the next test. Figure 11 was obtained by plottingthe dynamic amplification function as described in Zabel [7] for a system with andwithout dampers with these configurations. From this plot, it is clear to see thatTLD C should produce the best results.

7 Experimental Test 3

The main task of test 3 was to check our new TLD calculations to see if we couldobtain damping of the structure when it is excited at its first natural frequency. To

8

Figure 7: Test 2 - Channel 1 Accelerations [g]

Table 3: TLD Design Parameters

TLD L [cm] h [cm] b [cm] ω[radsec

]ζw [%] meq [kg] ρV [kg]

A 7.5 4 11 19.57 16.3 0.146 0.324B 7.5 4 18 19.57 14.7 0.239 0.530C 7.5 6 18 20.14 10.5 0.253 0.795D 8.5 4 27 18.08 14.3 0.445 0.901

do this, we used a sinusoidal base excitation again, with the frequency of 3.23 Hzand a sensor configuration shown in Figure 12. Plots of these results can be foundin appendix A.

During the test, it could be seen from the real-time data that the TLDs werenot decreasing the amplitudes of the acceleration by much. We then experimentedwith the excitation frequency, under the assumption that we had performed ourMACEC analysis incorrectly and that the first natural frequency was actually abit lower than 3.23 Hz. We changed the frequency until we arrived at the highestamplification of accelerations, therefore at the natural frequency of the structure.This frequency was approximately 2.9 Hz. We then tested TLD C once more at thisfrequency, even though it was designed towards damping a frequency of 3.23 Hz, tosee if we would obtain better results. The summary of what we found in both partsof our third test follows.

7.1 Results

If we calibrate the dynamic amplification function for the structure without adamper from Figure 11 using the amplification of acceleration from the base to the

9

Figure 8: Test 2 - Channel 1 Accelerations Zoom [g]

top of the structure when excited at its natural frequency (2.9 Hz), we determinethat structural damping must be 4% instead of 1.85% as obtained from MACECand we obtain the dynamic amplification functions shown in Figure 13. It is clearfrom this figure that our TLDs were not properly tuned for the structural frequencyof 2.9 Hz and that when the system is excited at a higher frequency, 3.23 Hz (20.29rad/sec), then the effect of the dampers would be very limited, if visible at all. Thisis confirmed by a zoom of the dynamic amplification function in Figure 14 and withthe experimental results found in appendix A.

As a final test, we decided to see what reduction in amplitudes could be acheivedby using our best damper from the previous tests, TLD C, when the structure isexcited at 2.9 Hz. Figure 15 shows amplification of the acceleration amplitudeswhen the structure is excited at this frequency, where the blue line is a sensor at the

L

h

direction

of excitation

b

1

Figure 9: TLD Dimensions ( for use with [6] )

10

Figure 10: TLD Configurations

0 5 10 15 20 25 300

5

10

15

20

25

30

omega

am

plif

ication

without TLD

TLD A

TLD B

TLD C

TLD D

Figure 11: Dynamic Amplification Function for Equivalent System

top of the structure and the black line is a sensor at the base. The result of addingTLD C to the system can be seen in Figure 16, where the plot is of acceleration atthe top of the structure and black represents without a TLD while blue representswith TLD C. Figure 17 is a zoom of this plot to show the effect of TLD C moreclearly. Even though the frequency of 2.9 Hz with an amplitude of 0.1 m/s2 atthe base caused erratic sloshing, possibly beyond the assumptions behind the TLDequations, this was the best result that we were able to achieve.

A summary of predicted and achieved results for all four TLDs can be foundin Table 4. The tabulated effectiveness is the amplification with a specific TLDdivided by the amplification without any TLD.

7.2 Modal Validation

Since it became clear that our natural frequency was not extracted properly fromthe test 1 experimental data, we decided to check our extracted mode shapes fromMACEC with the Modal Assurance Criterion (MAC). Allemang [2] gives a briefoverview of performing a MAC test for a pair of modal vectors. Equation (10)is one way of writing the modal assurance criterion as found in [2], where ψrepresents the modal vectors to be compared, distinguished by the subscripts c andd. The subscript r represents the mode under investigation and ψ∗ is the complex

11

Figure 12: Test 3 Sensor Placement

Table 4: TLD Effectiveness - Predicted vs. Observed

Predicted for 3.23 Hz Observed Predicted for 2.9 Hz Observed

TLD Fig. 11 Fig. 13 at 3.23 Hz Fig. 13 at 2.9 Hz

A 0.73 0.97 1.14 0.86B 0.60 0.95 0.81 0.79C 0.50 0.91 0.79 0.79 0.73D 0.54 0.99 0.89 0.62

conjugate of ψ.

MACcdr =

∣∣∣ψcrTψ∗dr∣∣∣2

ψcrTψ∗crψdrTψ∗

dr(10)

We wrote a Matlab function from this equation to perform the check. Allemangstates, “The function of the modal assurance criterion (MAC) is to provide a measureof consistency (degree of linearity) between estimates of a modal vector.” [2] Hegoes on to write that this measure can be used as, “a method of easily comparingestimates of modal vectors originating from different sources.” The range of MACis from 0 to 1, where 1 represents consistent modal vectors and 0 inconsistent modalvectors. In our case, we compared our modal vectors from our numerical FEMmodeling with the modal vectors from the MACEC analysis of the test 1 data.Since relating DOFs between results from MACEC and a computer simulation istime consuming and because our mode shapes from MACEC are missing the 4thfloor DOFs, we did a preliminary comparison of the modal vectors from MACECand our simplified Matlab calculations. These results are summarized in Table 5.

These rough calculations show that our first mode from MACEC was a bit off,and that in general our extracted modes weren’t very good, especially in the ydirection. Since this is a comparison of only 4 DOFs, and the results are a bit

12

0 5 10 15 20 25 300

2

4

6

8

10

12

14

omega

am

plif

ication

without TLD

TLD A

TLD B

TLD C

TLD D

Figure 13: Calibrated Dynamic Amplification Function

19.6 19.8 20 20.2 20.4 20.6

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

omega

am

plif

ication

without TLD

TLD A

TLD B

TLD C

TLD D

Figure 14: Calibrated Dyn. Amp. Function Zoomed to 20.29 rad/sec (3.23 Hz)

better than expected for the x direction, we decided to adapt our code to compareMACEC and SLANG with all 17 experimental DOFs. These results can be foundin Table 6, where the last two columns represent MAC between comparable modesdenoted by (ψMACEC ,ψSLANG). The same procedure was repeated for the modesobtained from SAP with the results give in Table 7. Finally, the MAC was computedcomparing SAP and SLANG for 120 DOFs. These results, shown in Table 8, provethat the MAC test is prone to errors from different signs. The first and secondmode shapes from SAP and SLANG should be identical and produce a result nearto 1. The values obtained initially of 0 show that these first two modes shapes areperpendicular. The structure is symmetric, therefore, there is no real reason whythe first mode is in one direction and the second in the other. On this basis, theMAC test was repeated comparing the first from SAP with the second from SLANGand vice versa. This is shown in the second set of row data in Table 8 and gives aresult very near to 1.

13

Table 5: MAC between MACEC and Simplified Calculations

Mode Shapes

dir. sensors 1st 2nd 3rd 4th 5th

x left 0.9795 0.9914 0.9714 0.8273 0.0406y left 0.9867 0.3840 0.2192 0.1386 0.0336x right 0.9808 0.9849 0.9484 0.8024 0.0572y right 0.7886 0.3413 0.0706 0.0864 0.1058

Table 6: MAC between MACEC and SLANG

Mode Shapes MAC(ψMACEC ,ψSLANG)

dir. 1st 2nd 3rd 4th 5th (1,1) (2,4)

both 0.3209 0.0097 0.0016 0.0219 0.5955 0.3209 0.6925x 0.9754 0.0249 0.0020 0.0282 0.5935 0.9754 0.9877y 0.8581 0.0407 0.0143 0.1157 0.6667 0.8581 0.3635

Table 7: MAC between MACEC and SAP

Mode Shapes MAC(ψMACEC ,ψSAP )

dir. 1st 2nd 3rd 4th 5th (1,1) (2,4)

both 0.5627 0.0097 0.0016 0.0122 0.1407 0.5632 0.4556x 0.9817 0.0187 0.0020 0.0300 0.5980 0.9817 0.9858y 0.8620 0.0361 0.0143 0.1148 0.6675 0.8620 0.3544

Table 8: MAC between SLANG and SAP

Mode Shapes

1st 2nd 3rd 4th 5th

initial 0 0 0.0421 0.0297 0.0298

MAC(ψSLANG,ψSAP )

(2,1) (1,2) (3,3) (5,4) (4,5)

switched 0.9993 0.9993 0.0421 0.9671 0.9671

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.15

−0.1

−0.05

0

0.05

0.1

t [sec]

accele

ration [g]

Measurment 10

Ch. 5

Ch. 1

Figure 15: Structure Excited at 2.9 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.15

−0.1

−0.05

0

0.05

0.1

t [sec]

acce

lera

tio

n [

g]

Measurment 10 vs 12

noDamp

Damp

Figure 16: Structure Excited at 2.9 Hz with TLD C

8 Numerical Model Calibration

For calibration of the numerical models, we decided to only update the SAP model.In order to obtain the same first eigen frequency as the experimentally observedvalue (2.9 Hz), the model was calibrated by:

1. Increasing the mass by 22%

2. Reducing the stiffness by decreasing the modulus of elasticity by 15%

This is a compromise between changing only one parameter. Only increasing themass or only decreasing the stiffness would result in percentage changes that couldnot be justified. By changing both parameters, we were able to keep the changesin reasonable limits and still obtain the observed natural frequency. Increasing themass by 22% can be justified mostly by considering the mass of the sensors. Our

15

0.38 0.4 0.42 0.44 0.46 0.48 0.50.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t [sec]

acce

lera

tio

n [

g]

Measurment 10 vs 12

noDamp

Damp

Figure 17: Structure Excited at 2.9 Hz with TLD C Zoom

estimation is that they would account for most of this 22%. Decreasing the stiffnessby 15% also can be justified because originally the system was modeled as a com-pletely rigid moment frame. It is well know that less rigidity exists in steel momentconnection systems than in idealized models [1]. The moment frame assumption isthat rotations at the joints are zero, i.e. members remain perpendicular. In reality,rotations are nonzero. While it is hard to quantify this lower stiffness, we acceptthis percentage decrease because it achieves the calibration while keeping the massincrease reasonable. The original and updated values of the mass and the elasticmodulus can be found in Table 9.

Table 9: Modified Parameters for Model Calibration

Steel without calibration with calibration

unit weight [N/m3] 76973 93907Elastic Modulus [GPa] 200 170

8.1 Modeling of TLD

The tuned liquid damper (TLD) can be modeled in SAP by considering an equivalenttuned mass damper (TMD). Tait [6] establishes equivalences between TLD andTMD parameters. Based upon this and using the calculated TLD parameters,an equivalent TMD model can be used. CSI Berkeley confirms that, “A TMDcan be modeled in SAP2000 using a combination of friction isolator link and aviscous damper link in series” [3]. Figure 18 shows a schematic of this modeling.Following the example models given on the CSI webpage [3] and considering thedifferent characteristics of the links related to the effective TMD parameters, TLDC could be modeled. Most of the used parameters for modeling TLD C are alreadysummarized in Table 3, with the main ones being meq and ζw. The additionalparameter necessary for modeling the equivalent TMD is the equivalent stiffness,

16

keq, which is also given by Tait as equation (11). The calculated keq for use inmodeling TLD C was 102.6 N/m.

FrictionIsolatorLink

DamperLink

TunedMass

1

Figure 18: Equivalent TMD Modeling Schematic

keq =8ρbLg

π2tanh2

(πh

L

)(11)

Nonlinear time history analysis has been conducted on both of the calibratedmodels, with and without TLD, considering base excitation dynamic action. Thebase excitation used was a sin function of frequency 2.9 Hz and amplitude 0.1 m/s2,the same as the final experimental test. Figure 19 shows the results to this excitationwith and without the TLD. With SAP, we were also able to generate Figure 20,which is a response spectrum using a damping of 4% for both models. Since aresponse spectrum is the plot of a frequency response function, this is comparableto our dynamic amplification functions plotted in earlier sections.

9 Conclusions

The first conclusion that can be drawn from this project is that certain parametersare essential in proper tuned liquid damper design. Our initial calculations provedthat the length of the container in the direction of desired damping, L, sets a majorparameter in the TLD design, ωd. Our failed initial TLD design proved that thiswasn’t the only crucial parameter. Effective mass was discovered to be just asimportant as described by Meskouris [4] and Tait [6]. Since the container lengthin the direction of desired damping, L, is usually set by the required frequency ofthe damper, ωd, the perpendicular dimension, b, becomes the crucial parameter forobtaining enough effective mass. It was also seen that the effective damping of aliquid is largely influenced by it viscosity and its depth. Fortunately in our case,

17

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

time [t]

acce

lera

tio

n [

m/s

2]

without TLD

with TLD

Figure 19: SAP Acceleration Response at the Top Floor

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2000

4000

6000

8000

10000

12000

14000

16000

frequency [Hz]

psuedo s

pectr

al accele

ration

without TLD

with TLD

Figure 20: SAP Response Spectrum (FRF)

damping screens weren’t required to obtain additional damping in the liquid. Onthe contrary, we tried to increase the water depth so that we might decrease thedamping of the TLD and bring it into a more desirable configuration.

The second major conclusion from this project is that proper TLD design ismeaningless if the frequency which the system is being designed to damp is incor-rectly identified. While our TLD in the second test showed bad results because ofan improper design, our TLDs in the third test designed to damp a frequency of3.23 Hz where nearly ineffective since the natural frequency of the structure wasactually 2.9 Hz. This error can be attributed to an improper extraction of modesfrom the experimental data and not realizing that these modes were wrong becauseof an eyeball check of mode shapes and misleading proximity to numerical frequen-cies. For this reason, modal validation via the MAC test was studied. It couldbe seen for our MAC results that the mode shapes obtained from MACEC didn’tentirely match the numerical ones. We still question the MAC numbers obtained

18

because the unidirectional results produced reasonable numbers. It could be seenfrom the results that simple sign differences can have a drastically adverse effecton the MAC number. With the complexity of relating DOFs between models, thedegree of accuracy of the results remain unclear when simple sign errors could exist.We can conclude that a check must be done of extracted modes from experimentaldata, especially when those modes factor into design. However, MAC has its ownlimitations and difficulties. MAC is a purely mathematical check and can only givean idea if two vectors are parallel, perpendicular, or somewhere in between.

The last major conclusion from this project is that numerical calculations canachieve successful results. Comparable results could be seen in the prediction ofamplification from simplified models and in the complex calibrated SAP systems.It was also proven that the effectiveness of TLDs could be predicted.

References

[1] AISC. Steel Construction Manual. American Institute of Steel Contruction, Inc.,13th edition, 2005.

[2] Randall J. Allemang. The modal assurance criterion - twenty years of use andabuse. Sound and Vibration, 2003.

[3] CSI Berkeley. Tuned mass damper. https://wiki.csiberkeley.com/display/tutorials/Tuned+mass+damper, 2011.

[4] Konstantin Meskouris. Structural Dynamics: Models, Methods, Examples. Ernstand Sohn, 2000.

[5] Setra. Footbridges: Assessment of vibrational behaviour of footbridges underpedestrian loading, 2006.

[6] M.J. Tait. Modelling and preliminary design of a structure-tld system. Engi-neering Structures, 30(10):2644–2655, October 2008.

[7] Volkmar Zabel. Lecture notes for: Structural Dynamics. Bauhaus-Universitat Weimar, Winter Semester 2010/11.

19

A Test 3 with Excitation Frequency of 3.23 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t [sec]

acce

lera

tio

n [

g]

Channel 1

Without Damper

Damper 4x18

Damper 6x18

Damper 6x27

Damper 4x11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t [sec]

acce

lera

tio

n [

g]

Channel 3

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t [sec]

accele

ratio

n [

g]

Channel 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.015

−0.01

−0.005

0

0.005

0.01

t [sec]

acce

lera

tio

n [

g]

Base Excitation

Ch. 5

Ch. 6

Ch. 7

0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.440.005

0.01

0.015

0.02

0.025

0.03

0.035

t [sec]

accele

ration [g]

Channel 1

Without Damper

Damper 4x18

Damper 6x18

Damper 6x27

Damper 4x11

21

0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.440.01

0.015

0.02

0.025

0.03

0.035

t [sec]

accele

ration [g]

Channel 3

0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.440.01

0.015

0.02

0.025

0.03

0.035

t [sec]

accele

ration [g]

Channel 7

0.32 0.33 0.34 0.35 0.36 0.37 0.381

2

3

4

5

6

7

8

9x 10

−3

t [sec]

acce

lera

tio

n [

g]

Base Excitation

Ch. 5

Ch. 6

Ch. 7

22