• 

NASA/TM-20 13-216320 

V AB Sway Investigation 

Robert C. Youngquist 

NASA Kennedy Space Center 

Curtis M. lhlefeld 

NASA Kennedy Space Center 

Stephen M. Simmons 

EASi, Engineering Services Contract, Kennedy Space Center 

John E. Lane 

EASi, Engineering Services Contract, Kennedy Space Center 

Stanley 0. Starr 

NASA Kennedy Space Center 

January 2013 
NASA/TM-20 13-216320 

NASA/TM-20 13-216320 

V AB Sway Investigation 

Robert C. Youngquist 

NASA Kennedy Space Center 

Curtis M. lhlefeld 

NASA Kennedy Space Center 

Stephen M. Simmons 

EASi, Engineering Services Contract, Kennedy Space Center 

John E. Lane 

EASi, Engineering Services Contract, Kennedy Space Center 

Stanley 0. Starr 

NASA Kennedy Space Center 

National Aeronautics and 

Space Administration 

Kennedy Space Center 

January 20 13 
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NASA/TM-20 13- 216320 

Acknowledgments 

The research team gratefully acknowledges the contributions made by the following individuals: 

a. Bobby Williams, VAB Operations Leader, Dept. 93510, United Space Alliance 

b. James Bolton, V AB Operations Manager, NASA KSC, GP-02-B 

c. Jeffrey Brink, AST Tech Engineer Ops Mgmt, NASA KSC, LX-D1 

d. Robert Cox, Engineering Designer, QinetiQ North America 

e. Edward Muktarian, PE, System Engineering & Integration, SGT Inc. 

Available from : 

NASA Center for AeroSpace Information (CASI) 

7121 Standard Drive 

Hanover, MD 21076-1320 

(30 1) 621-0390 

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5285 Port Royal Road 

Springfield, VA 22161-2171 

(703) 605-6000 

This report is also available in electronic form at URL http://www.sti.nasa.gov/ and 

http://ntrs.nasa.gov/search.jsp. 

II 
NASA/TM-20 13- 2I6320 

Contents 

BACKGROUND ............................................................................................... I 

2 TEST PROCEDURE ......................................................................................... 2 

3 DATA ANALYSIS ............................................................................................ 5 

4 DISCUSSION AND SUMMARY ..................................................................... ? 

APPENDIX A. RAW DATA CORRESPONDING TO TABLE 2 ............................................ ! 

APPENDIX B. DEVELOPMENT OF THE SWAY MATH MODEL .................................... 23 

Figures 

Figure 1. 

Figure 2. 

Figure 3. 

Figure 4. 

Figure 5. 

Figure 6. 

Tables 

Table I. 

Table 2. 

Locations for Data Collection ............................................................................ ! 

Telescope Used for Data Collection .................................................................. 2 

Telescope Target ..................................... ........... ................................................ 4 

Data from August 8, 2012 Event ....................................................................... 5 

Example Thunderstorm Event and Associated Wind Direction Change over 

the VAB ....................................................................................................... ...... 6 

Building Displacement Correlated with Wind Speed ........................................ ? 

Equation (I) Evaluated for Various Confidence Levels Cat h =300ft ......... vii 

Correlation of Wind Data with V AB Displacement .................................... ...... 3 

111 
NASA/TM-20 13-216320 

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IV 
KSC 

LED 

UTC 

SO DAR 

VAB 

ABBREVIATIONS AND ACRONYMS 

John F. Kennedy Space Center 

light-emitting diode 

Coordinated Universal Time 

sonic detection and radar 

Vehicle Assembly Building 

v 

NASA/TM-20 13-216320 
NASA/TM-20 13-216320 

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VI 
NASA/TM-20 13-216320 

ABSTRACT 

The Vehicle Assembly Building (VAB) was constructed in the mid- I 960s to house the Saturn V moon 

rocket while it was being assembled. Designed to withstand hurricanes and tropical storms, the V AB has 

a foundation consisting of 30,000 cubic yards of concrete strengthened by 4,225 steel rods driven 160 feet 

into limestone bedrock. The goal of the V AB Sway Investigation, which began collecting data in April 

2010 and ended in November 2012, was to quantify the displacement or sway of the VAB as a function of 

wind loading. Unlike other tall buildings whose height is much greater than their average cross-section 

dimension, the V AB is roughly a box with a height L = 526 ft. In the case of skyscrapers, building sway is 

characterized by a damped oscillation that may be on the order of 0.1 Hz to 0.5 Hz. The V AB, on the 

other hand, is an over-damped harmonic oscillator characterized by a simple displacement in the direction 

of the wind load. This conclusion is based on the fact that no oscillatory behavior was ever observed in 

the entire V AB sway data set. Even though the investigation and data collection continued for over 2 

years, the wind speeds observed during that time did not exceed 50 kts. Therefore, the final data analysis 

consists of an extrapolation formula, which expresses the magnitude of the displacement d as a function 

of wind speed v [kts], height above the ground h [ft], and number of standard deviations n above the 

mean: 

d(n,v,h)=a ebnu (hv / L) 2 [in] (1) 

where a= 0.0001591 [in·kts-2

] , b = ..J5 In1 0 = 5.1487 [in -

1

] , and u= 0.092 [in], are determined from the 

data analysis. Table 1 is an evaluation of Equation (1) for various wind speeds and for n = 0, 1 , and 2. 

The meaning of confidence level is discussed in the Discussion and Summary section. 

Table 1. Equation (1) Evaluated for Various Confidence Levels Cat h =300ft 

v [kts] O·cr l ·cr 2·cr 

(I 0-min peak) (C = 50%) (C = 84%) (C = 98%) 

10 0.005 0.008 0.013 

20 0.021 0.033 0.054 

30 0.047 0.075 0.12 

40 0.083 0.13 0.21 

50 0.13 0.21 0.34 

60 0.19 0.30 0.48 

70 0.25 0.41 0.66 

80 0.33 0.53 0.82 

90 0.42 0.67 l.l 

100 0.52 0.83 1.3 

110 0.63 1.0 1.6 

120 0.74 1.2 1.9 

vii 
NASA/TM- 20 13-216320 

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• 

viii 
NASA/TM- 20 13-216320 

V AB SWAY INVESTIGATION 

1 BACKGROUND 

The Vehicle Assembly Building (originally named the Vertical Assembly Building) was used to assemble 

American manned launch vehicles from 1968 through 20 II. Completed in 1966, the V AB was originally 

built to allow for the vertical assembly of the Saturn V rocket for the Apollo program. It was then used for 

housing Space Shuttle external fuel tanks and flight hardware, and was where Space Shuttle orbiters were 

mated with their solid rocket boosters and external fuel tanks. The VAB is 526 feet ( 160.3 m) tall, 716 

feet (218.2 m) long, and 518 feet (157 .9 m) wide. It covers 8 acres and encloses 129,428,000 cubic feet 

(3,665,000 m

3

) of space. Entry to the bays inside the building is made through the four largest doors in the 

world. Each door is 456 feet (139.0 m) high and takes 45 minutes to completely open or close. Another 

large door on the north entry that leads to the transfer aisle was widened by 40 feet (12.2 m) to 

accommodate the Shuttle orbiter. A slot was added at the center of this north entry to allow for passage of 

the orbiter's vertical stabilizer. 

It is well known that buildings sway in response to wind, as well as to earthquakes, and the taller the 

structure, the more they can sway. Because Kennedy Space Center (KSC) is occasionally assaulted by 

tropical or hurricane force winds, data on the VAS's sway, collected from instrumentation positioned as 

indicated in Figure I, is essential to help characterize the VAB facilities so that they can be used by 

commercial customers in the future. 

Figure 1. Locations for Data Collection 

Data from the nearest wind towers (http://trmm.ksc.nasa.gov/) is used to 

correlate wind velocity and direction to position data from the telescope. 
NASA/TM-20 13-216320 

2 TEST PROCEDURE 

The data acquisition strategy for this study was implemented with a Meade RCX400 telescope, positioned 

on the V AB floor in High Bay 4 as shown in Figure 2, along with a target fixed to the ceiling (see 

Figure 3). The position of the target is recorded once per second with a computer (also shown in Figure 2) 

attached to a camera mounted at the end of the telescope. The telescope data is then compared to data 

from the wind towers in the vicinity of the V AB (Figure 1 ). Instrumentation is located on three elevations 

on the wind towers; data from the highest, at 54 ft, is used in this analysis. The primary goal of this 

project is to correlate the movement of the building as a result of wind only. Table 2 is a list of the 22 

best-correlated data points extracted from telescope-target position data and the wind tower dataset. 

Figure 2. Telescope Used for Data Collection 

Meade RCX400 telescope pointed toward the ceiling in High Bay 4 with a PC 

running Lab View image processing software. 

2 
NASNTM- 20 13-216320 

Table 2. Correlation of Wind Data with V AB Displacement 

(The correlation between wind data and V AB displacement is also shown as 

circles in Figure 6.) 

Date Wind speed Displacement Average Wind 

(10-min peak) d [in] Direction [ deg] 

10/27/2012 39 0.40 0 

09/06/2012 30 0.12 260 

09/09/2012 27 0.12 273 

08/28/2012 32 0.08 226 

08/27/2012 35 0.19 167 

08/08/2012 34 0.21 0 

07/22/2012 41 0.23 167 

07/ 18/2012 32 0.15 250 

06/25/20 12-a 37 0.18 167 

06/25/20 12-b 38 0.23 250 

06/06/2012 42 0.18 250 

05/28/2012 40 0.18 178 

04/06/2012 29 0.18 0 

03/31 /2012 44 0.54 300 

03/04/2012 36 0.38 310 

02/22/2012 26 0.27 330 

09/ 14/2011 23 0.17 47 

09/05/2011 31 0.13 250 

08/28/2011 45 0.13 214 

06/04/2010 31 0.09 226 

05/ 17/2010 28 0.18 0 

05/06/2010 35 0.13 250 

3 
NASA/TM- 20 13- 216320 

Figure 3. Telescope Target 

The telescope's target is mounted on the ceiling above the telescope in High Bay 4. 

As shown in the lower left, the target consists of three bright LEOs, in a rotated 

L-shaped pattern. The dimensions of the LED pattern are I inch x 3 inch. 

4 
NASA/TM-20 13-216320 

3 DATA ANALYSIS 

An example of correlated wind tower and telescope data is shown in the graph of Figure 4. At 

approximately 6 PM local time on August 8, 2012, a strong wind event occurred, lasting for well over an 

hour. The. peak displacements (the differences between telescope and target positions) were only Y.. inch, 

but were very well correlated to the wind tower data. The peak wind, as shown in Figure 4, was only 

35 kts, but the duration and broad impulsive character combined to generate a set of conditions that 

allowed the building position data to track the wind data. It is believed that a doubling of wind speed 

(70 kts in this case) will produce a quadrupling of the building sway, or 1 inch. This has not yet been 

confirmed, however; tropical force (70 kt) winds have not occurred since the hurricanes of 2004, or about 

5 years before the deployment of the sway-tracking telescope. 

In addition to wind-induced displacement, movement resulting from other sources, such as sun-induced 

heating and temperature cycling, as well as crane and heavy equipment operations, has been recorded. It 

is often difficult to distinguish the uncorrelated sources of telescope displacement noise, such as wind 

blowing through the large high bay doors. Since the telescope-target optical measurement reports a 

relative position, this measurement represents the maximum possible building sway, because it is 

statistically unlikely that the telescope and building would move in a way that would cancel out the real 

displacement. This sets a lower limit on displacement (but not an upper limit). In other words, the actual 

building displacement is equal to or Jess than the value measured by the displacement magnitude (d/ + 

d/ )

1

n. in the raw data plots, such as that shown in Figure 4. 

VAB Sway Wind Tower: AUG 2012 

0.4 

250 80 

0.3 0 

""Sij 

"' .E! ~ -250 

6 .§ ... 1l ... 

~ 0.2 ... 0 0 -500 

60 

2J - Target-x 

~ - Target-y 

1 40 IJJ 

"0 ~ "0 ..... .E! 

:s ~ 

- 750 20 

0.1 

0 

- 1 

-1000 0 ,.,. -2 

8.5 8.6 8.7 

Time from AUG 1, [days] 

Figure 4. Data from August 8, 2012 Event 

This wind event produced a well-correlated building displacement as measured by 

the telescope-target system. The wind direction was from the north, i.e., 0 degrees. 

.9 

In most of the good correlations between V AB displacement and wind events, the wind events are short 

and usually associated with thunderstorms or other phenomena related to passing weather fronts. The 

August 28, 2011 data shown in Figure 5 is a prime example of a passing convective thunderstorm. When 

5 

.s 

~ "§ 

c. 

... 

"' bO 

~ 
NASA/TM- 20 13-216320 

this occurs, it is common for the wind speed to increase dramatically, with an abrupt change in direction. 

(Directional change is usually characterized by a rotation of wind direction.) After the front (sometimes 

referred to a gust front or outflow boundary) passes, the wind speed and direction return to their previous 

synoptic state. The entire length of this event is on the order of one hour or less, with the peak wind state 

lasting only 10 to 15 minutes. 

.5 

:s 

.. ... 

til 

0 

til 

....l 

VAB Sway Wind Thv.-er: AUG 2011 

4 

250 - Spd 311 

80 

3 0 

60 

be ., 

::2. -250 ::: 

- Target-x 

- Target-y 

.2 

2 ~ i5 -500 

- LED Err 

---- ----------------- A "0 

~ -750 20 

-1000 0 -2 

Figure 5. 

28.3 

Time from AUG 1, [days] 

Example Thunderstorm Event and Associated Wind Direction Change over the 

VAB 

The thunderstorm event occurred on August 28, 20 II, I7:58 UTC. Note that the 

time scale on the plot corresponds to local time. 

6 
NASAffM-20 13- 216320 

4 DISCUSSION AND SUMMARY 

Figure 6 is a graph of Equation (1) as a function of wind speed for various values of n, and for h = 300 

(green lines) and h = L (red lines). When the data (red circles) is viewed on a log-log plot, such as 

Figure 6, the scatter appears to be uniformly distributed about the mean (O-cr) line, shown as the bold red 

line. cr is calculated by first transforming the wind speed v and displacement building d to new variables 

w = log(v) and u = log(d). The transformed data {w, u(w)} in the coordinate system is then rotated by 

() = tan-1 2 , since the slope of the log-log plot is 2, corresponding to the initial model, which assumes 

that displacement is proportional to v

2

• The displacement data in the rotated coordinate system can be 

modeled as a random variable x, which is approximately uniformly distributed about x = 0: 

_!{ x)2 

P(x) = 1 

e -2\-;;: , 

a-& (2) 

where the standard deviation cr is computed in the standard way in this rotated coordinate system. In order 

to use cr in the original coordinate system of {v, d(v)} , an inverse rotation is performed followed by an 

anti-log transformation, arriving at: 

d(n, v,h) =a ebnuv 2 [in] (3) 

where a and b are fit parameters. By multiplying Equation (3) by the height function, (hiLi, which 

accounts for the quadratic bending mode of the building, Equation (1) is obtained. 

0.1 

Figure 6. 

2010-2012 VAB SWAY 

6 

Wind Speed (knots) 

r 10 Min-Peak WindTelescope Data. h"' 526 ft 

- 0-o, h "'526ft 

- 1-o, h = 526 ft 

- - - 2-o. h .. 526 ft 

- 0-o. h = 300 ft - 1-o. h•300ft 

- - - 2-o. h = 300 ft 

100 

Building Displacement Correlated with Wind Speed 

Figure 6 shows Equation (I) evaluated at the (red) top of the VAB (h = L = 526ft) and 

(green) at h = 300ft. The solid bold lines are the average response (C = 50% confidence 

level), the solid line is one standard deviation above the mean (C = 84% confidence level), 

and dotted lines are two standard deviations above the mean (C = 98% confidence level). 

7 
NASA/TM-20 13- 216320 

The confidence levels C in Table I are obtained by integrating Equation (2) from - CXJ to ncr: 

na 

C(n) = J P(x)dx 

The confidence level C can then be interpreted in this context as the percent fraction of the distribution 

based on the telescope-target and wind tower data, such that a building displacement at a given 

confidence level (based on n) is less than or equal to d of Equation (I), corresponding to the fraction of 

P(x) from - CXJ to na. 

(4) 

For this to have a more physical explanation, we need to discuss the error sources that contribute to the 

distribution and the measured standard deviation a. The raw data can be decomposed into two main 

sources: (1) displacement that is correlated to the wind speed and direction at the tower, and 

(2) everything else. The second category corresponds to the sources mentioned in the previous section. 

The data used in this analysis (shown by circles in Figure 6) are only from the first category, i.e., where 

displacement is proportional to the wind speed and direction at the tower. For the purpose of this analysis, 

everything else was discarded. So the next question might be, What sources control the standard deviation 

a; especially since all uncorrelated data was discarded? Many possibilities must be considered. 

The first source that contributes to the probability distribution width is the likelihood that not all of the 

uncorrelated noise was removed from the data used in Figure 6. This is simply because the correlated 

portion of the displacement is riding on a background signal that is uncorrelated. So even though attempts 

have been made to remove that background noise, some of it will remain, or the attempt at removing (or 

filtering) leads to additional error. 

However, we believe that the largest contribution of data scatter, quantified by a(probability distribution 

width), can be investigated by answering the question, What is the actual wind speed acting on the VAB? 

Even though the wind towers are part of a quality-controlled weather data network for KSC and Cape 

Canaveral Air Force Station, they measure the wind at a specific point in space. Three wind towers were 

used for this analysis, as shown in Figure 1. The distance from the V AB to the towers is in a range from 2 

km to 3 km. The wind tower measurement used in this analysis was taken at a height of 54 ft (the highest 

available sensor on the three towers used). 

Another source of data scatter is the width of the time interval of the wind acting on the V AB. If the time 

interval is smaller than the response time of the building, the displacement will be smaller than the 

average trend. This could explain the circles under the main trend line (shown as the bold solid red line) 

in Figure 6. The circles above the line could be explained by wind forces at 300 ft that were greater than 

those measured by the wind towers at 54 ft. This is exemplified in the August 28, 2011 event shown in 

Figure 5, where the entire event lasted less than one hour and the peak winds were prevalent for less than 

15 minutes. In this case, the building response time was longer than the wind event duration, and 

corresponds to a data point below the main trend line. 

So the primary uncertainty is in determining actual wind speed across the surface of the V AB. 

Conceptually, this could be improved by wind towers at 0.5 km (not too close, else the building-induced 

wind circulation would perturb the measurement), and at a height of 300 ft. A SO DAR wind measurement 

system near the VAB could, in theory, produce an ideal wind measurement dataset. On the other hand, the 

primary data source used to monitor wind speeds relevant to V AB operations should be the ones used to 

make the associated predictions of building displacement. If that primary source is wind towers near the 

VAB (within 2 km to 3 km), then the resulting data scatter shown in Figure 6 must be dealt with in an 

appropriate statistical manner for useful displacement predictions, as has been attempted in this report. 

8 
NASA/TM-20 13-216320 

APPENDIX A. RAW DATA CORRESPONDING TO T ABLE 2 

The following 22 plots correspond to the entries of Table 2. Each plot contains wind speed and direction 

measured at the three towers shown in Figure 1. The horizontal and vertical target positions are also 

plotted, in black and grey. In order to extract a building sway direction, we need to approximate a local 

background position and apply it to the specific event. Flat lines in the telescope data usually indicate the 

crane blocking the telescope. The relative displacement of the building can then be converted into an 

angle for comparison with the wind direction recorded at the tower. Lastly, the LED error is plotted in 

red. 

The target position is the average of the three LED positions, (x0,y0) , (x ~.y , and (x2,y2) over a 5-minute 

period: . 

(A- 1) 

(A-2) 

where c x0 and cy0 are constant offsets for the graph. 

The LED errorE (red line) is computed as the maximum of the I sec error over a 5-minute period, so that 

N = 300: 

(A-3) 

(A-4) 

(A-5) 

(A-6) 

where the LED separations are assumed to be I, 3, and 10112 inch. A high error (under 0.5 inch) might 

indicate telescope movement, possibly from wind blowing past it from the high bay doors. An error 

higher than 0.5 inch usually indicates a missing LED (i.e., the camera does not detect it for a number of 

possible reasons). In some cases the error disappears from the plot for many days or weeks at a time. This 

occurs when the LabView imaging software fails to detect one or two LEOs. Fortunately, the 

measurement of the relative position of the target is not severely affected and useful building sway 

information can still be extracted. 
VAB Sway Wind Tower: OCT 2012 

0.4 

250 80 1 

0.3 0 

60 

bO 

QJ 

.s .32.. 

- 250 ... = 0 0 :;; ... 

I:;EJ 

... ~ 0.2 .... 

~ i5 -500 

"'0 ~ Q 

ti 

'W 0 .5 

~ Q 

"i .sa 

.... 

8. 40 ·;; 

en & 

"'0 .... 

QJ c ~ ti - 1 :a H 

- 750 20 

0.1 

- 1000 0 

- 2 

20 22 24 26 28 

Time from OCT 1, [days] 
VAB Sway Wind Tower: SEP 2012 

0.4 

250 0 1 

Spd 50 

0.3 0 

- Dir 311 - Dir 412 

.8 

... 

0 

t: 

~ 0.2 

~ ~ 

'iO 

Q) 

.32... -250 £:l 

.9 

...., 

as 

... 

0 -500 '1:) 

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> 

6 6.25 6.75 7 r:n 

~ 

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6.5 

Time from SEP 1 [days) 

0 

w 

tG 

0"\ 

w 

N 

0 
VAB Sway Wind Tow

er: SEP 2012 

0.4 

250 80 1 

Sp 

0.3 0 

- Dir 311 - Dir 412 

.a 

.... 

0 

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~ 0.2 

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5 6 7 8 9 10 

Time from SEP 1, [days] 
0.4 

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60 Q) 

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= .... 0 0 

.... 

... Vl w 0.2 

~ ...:l 

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... 

a -500 

"'0 

.!3 

E: 

- 750 20 

0.1 

- 1000 0 

Spd 509 

- Dir 311 - Dir412 - Dir509 - Target-x - Target-y 

28.4 

VAB Sway Wind Tower: AUG 2012 

2 .5 28.6 28.7 28.8 28.9 

Time from AUG 1, [days] 

1 

0 .5 

c 

0 

·~ 

~ ~ 

~ 

- 1 ... 

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- 2 

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1::: 

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51 

~ 

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·= 0 -500 

"0 .5 

~ 

-750 

0.1 

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M ii!Blt::i?f:ir 1 - Dir 412 

60 - Dir 509 

~ - Target-x 

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"i - LED Err 

0. 40 

en 

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20 

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26.5 

VAB Sway Wind Tower: AUG 2012 

1 

0 .:: 

s:l 

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"+3 

"iii 

0 

p.. 

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-2 

27 27.5 28 

Time from AUG 1, [days] 
VAB Sway Wind Tower: AUG 2012 

0.4 

250 80 1 

Spd 509 

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- Dir 311 

0.3 

- Dir 412 

60 

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d 

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8.4 8.6 8.8 9 

Time from AUG 1, [days] 

9.2 C/) 

~ 3: 

1 

0 

\.H 

tG 

0"1 

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N 

0 
VAB Sway Wind Tower: JUL 2012 

0.4 

250 80 1 

0.3 0 

bO 

Ql 

.s .:2. -250 ~ 

... 0 0 

I: ~ 00 ~ 0.2 .!::l 

~ Cl -500 

"' ....:I .!3 

~ 

60 

- Dir 509 

~ 

- Target-x 0 .9 

C!... 

- Target-y ~ 

0 

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- LED Err ·.:; 

~ 40 

·r;; 

0 CJ) l:l. 

"' .. .!3 ~ 

~ 

- 1 ... 

~ 

-750 20 

0.1 

- 1000 0 

22 22.25 22.5 22.75 23 

Time from JUL 1, [days] 
VAB Sway Wind Tower: JUL 2012 

0.4 

250 0 1 

Spd 509 

0.3 0 

- Dir 311 - Dir 412 

.!3 

.... 

~ 0.2 

~ ~ 

bO 

v 

.32. -250 s::1 

0 

·~ 

~ ... 

0 

- 500 ~ 

.!3 

:s: 

60 

- Dir 509 

~ 

- Target-x 

~ 

- Target-y 

"i 

v 

0. 40 

en 

~ .!3 

:s: 

0 ·= 

0 = 

~ 

~ 

ll. 

.. 

v 

bO 

- 1 ~ 

- 750 20 

0.1 

- 1000 0 

- 2 

z 

> 

1 .25 19 en 

~ 1 .5 

Time from JUL 1, [days] 

N 

f 

0 

w 

rG 

0\ 

w 

N 

0 
0.4 

250 80 

0.3 0 

bO 

60 

.s Q.l 

.:Q. -250 d 

"E 

~ 

... 

0 

t: ....... ~ 0.2 0 

,.J 

~ 

0 

:;:; 

~ 

Cl ·= -500 

"tl 

c 

~ 

~ 

~ 40 

en 

"tl 

.a 

~ 

- 750 20 

0.1 

- 1000 0 

VAB Sway Wind Tower: JUN 2012 

- Target-x 

- Thrget-y - LED Err 

25 25.25 25.5 25.75 26 

Time from .TUN 1, [days] 

1 

0 - 1 - 2 

.5 

c 

0 

·.;:; 

-~ 

0.. 

~ 

<II 

bO 

... 

~ 

z 

> (/) 

....... 

0 

~ w 

tG 

....... 

0\ 

w 

N 

0 
VAB Sway Wind Tower: JUN 2012 

0.4 

250 0 1 

0.3 0 

.!3 

.... 

-,...... ~ 0.2 

0 

~ 

'iii 

Q) 

.3!.. -250 .:: 

0 

:;3 

a! 

.... 

0 -500 ~ 

.s 

~ 

60 

ti) 

- 'I'ar-get--x 0 .6 .... 

~ 

- Target-y c 

.9 al - LED Err .... 

·;;; 8. 40 0 rJ) 0.. 

~ .... .!3 ~ 

s: 

- 1 ~ -750 20 

0.1 

- 1000 0 

- 2 

z 

> 

6 6.25 

6.75 7 C/) 

e: 6.5 

'"""l 

I Time from JUN 1, [days] 

0 

w 

~ 

0\ 

w 

N 

0 
VAB Sway Wind Tower: MAY 2012 

0.4 

250 0 1 

Spd 509 

0.3 0 

- Dir 311 - Dir412 

.s 

.... 

0 

.... 

.... 

N 

....... ~ 

0.2 

53 

~ 

bii 

Q) 

.:2. 

- 250 d 

0 

·~ 

~ .... 

iS 

- 500 

"tl 

.!3 

~ 

60 

- ir 509 

00 

- Target-x ~ 

~ 

] 

- LED Err 

Q, 40 {/) 

"tl .!3 

~ 

0 .5 

d 

.9 

~ ·;n 

& 

.... 

~ 

- 1 ~ 

- 750 20 

0.1 

- 1000 

0 

- 2 

2 28.25 28.75 29 

Time from MAY 1, [days] 
VAB Sway Wind Tower: APR 2012. 

0.4 

250 0 1 

0.3 0 

.9 

.. 

0 

-w ~ 0.2 

~ ...:l 

bO 

<U 

3!. -250 s::l 

~ .!:l 

0 -500 

"1:1 .9 

~ 

- 750 

60 

~ 

- Target-x 0 

.!3 

.=!... 

- Target-y s:: 

-g 

- LED Err .9 

... 8. 40 ·;; 

0 rn p... 

"1:1 ... .!3 <U 

1:,0 ~ - 1 ~ 

20 

0.1 

- 1000 0 - 2 

z 

> 

8 r:/) 

e: 5 6 7 

Time from APR 1, [days] f 

0 -w 

t0 

0"1 

w 

N 

0 
VAB Sway Wind Tower: MAR 2012 

0.4 

250 0 1 

0.3 0 

- Dir 311 - Dir 412 

60 - Dir 509 

.!3 

.... 

2 

-~ ~ 0.2 

~ ..:I 

to 

::Q. -250 d 

0 

·~ 

~ .... 

iS -500 

"0 

.!3 

:::: 

'W - Target-x 

~ - Target-y 

"i - LED Err 

8. 40 

ell 

"0 

.!3 

:::: 

0 .5 

d 

0 

·~ 

'iii 

0 

ll. 

.w 

- 1 ~ 

~ 

- 750 20 

0.1 

- 1000 0 

- 2 

31 31.25 31.5 31.75 32 

Time from MAR 1, (days] 
0.4 

250 80 

0.3 0 

bO 

60 Q) 

.s .:£. 

- 250 Q .... 0 0 

-

VI JS 0.2 

~ ~ 

·;; 

~ .... 

i5 -500 

"'0 

Q 

~ 

- 750 

0.1 

- 1000 

4 

Spd 509 

- Dir 311 - Dir 412 - Dir 509 

4.25 

VAB Sway Wind Tower: MAR 2012 4.5 4.75 5 

Time from MAR 1, (days] 

1 

0 .!3 

Q 

0 

:;:; 

·;; 

0 

p.. 

..... 

- 1 ~ 

~ 

- 2 

z 

> TJ) 

~ 

0 

f -w 

tG -0\ 

w 

N 

0 
VAB Sway Wind Tower: FEB 2012 

0.4 

0 1 

Spd 509 

0.3 0 

- Dir 311 - Dir 412 

.s 

B 

- ~ 0.2 0'\ 

~ ~ 

bO 

Q) 

.:2. -250 s:: 

.9 

~ 

·= 0 -500 

"" .!3 

~ 

60 - Dir 509 

~ 

- Target-x 

::s 

- Ta 

~ 

- LED Err 

8. 40 

00 

"" .!3 

~ 

0 .5 

s:: 

0 

·~ 

·;;; 

0 

p.. 

.... 

~ -1 .. 

~ 

- 750 20 

0.1 

- 1000 0 

- 2 

22 22.25 22.5 22.75 23 

Time from FEB 1, !days] 
VAB Sway Wind Tower: SEP 2011 

1 

250 80 

• 

1 

0.75 0 

- Dir 412 

bO 

Q) 

.s .32.. -250 s:: ... 0 0 ·z 

- JS 0.5 ~ -....J ·= 0 0 -500 ~ -o ..:I .5 

~ 

- 750 

60 

- Dir 509 

~ 

- Thrget-x 

~ 

- Target-y 

"i 

- LED Err Cl) 

Q, 40 

en 

-o 

.!3 

~ 

20 

0 .s 

c 

0 

·z 

·;; 

~ .... 

Cl) 

-1 l 

0.25 

- 1000 0 

14 14.25 14.5 14.75 15 

Time from SEP 1, [days] 
1 

250 0 

0.75 0 

bD 

60 Q) 

.8 .:Q.. 

s:: -250 .. .9 0 .... I:: 0.5 ~ 

- ~ .!:l 00 

~ 0 -500 

-c ,....:) s:: 

~ -750 20 

0.25 

-1000 0 

5 

Spd 311 

Spd 509 

- Dir 311 - Dir 412 - Dir 509 - Target-x - Target-y 

VAB Sway Wind Tower: SEP 2011 

LE~~--~------~~~------------~ 

5.25 5.5 

Time from SEP 1, [days] 

5.75 

1 

0 .s 

0 = 

:;3 

·~ 

p.., 

.... 

-1 

fc 

~ 

-2 

6 
VAB Sway Wind Tower: AUG 2011 

4 

250 80 1 

3 0 

bD 

Q) 

.9 .:!2.. -250 ... = .9 0 .., I:: 2 ~ 

- IJ;:I ... 1.0 .... Q Q 

- 500 

w "1j 

~ ..... 

.::::: 

~ 

60 - Dir 509 

~ 

- Target-x 0 .9 - Target-y 0 = 

-s 

- LEDErr ~ 

Q) 40 

·;;; 

0. 0 

en 0.. 

"1j .... .9 ~ 

~ 

- 1 ~ 

- 750 20 

1 

- 1000 0 

- 2 

z 

> 

28 28.25 28.5 28.75 29 C/1 

~ Time from AUG 1, [days] ,..., 

1 

0 -

v.> 

~ -0'1 

v.> 

N 

0 
0.4 

250 

0.3 0 

bO 

Q.l 

.!3 ::2. 

l:l -250 

"" 0 0 

N ~ 0.2 0 

~ ~ 

·~ 

g 

.!::: 

0 -500 

.s "' 

~ 

- 750 

0.1 

- 1000 

311 

0 pdl"d n.~ 

Spd 509 

- Dir 311 - Dir412 

60 - Dir 509 

20 

0 

4 4.25 

VAB Sway Wind Tower: JUN 2010 

1 

r 

- 2 

4.5 4.75 5 

Tim from JUN 1, [days] 
VAB Sway Wind Tower: MAY 2010 

0.4 

250 Spd 311 

0 1 

Spd 509 

0 

- Dir 311 

0.3 

- Dir 412 

60 

- Dir 509 ""60 .9 Q) 

~ 0 .a .:!2.. 

= 

-250 .=!. 

0 = ... 0 "i - LED Err ";1 0 ";1 ·~ N ~ 0.2 ~ 0. 40 - .~ {/) ll.. 

~ 0 -500 " ... " .!3 ~ ~ .9 ~ -1 

~ ~ -750 20 

0.1 

-1000 0 -2 

z 

:> 

17 17.25 17.5 17.75 18 (f.J 

Time from MAY 1, !days] ~ 

f N 

0 

....... 

VJ 

~ 

0'1 -

VJ 

N 

0 
0.4 

250 

0.3 0 

~ 

.s .:2. -250 s:l .. 

0 

N 

N ~ 0.2 

ffi 

...:l 

0 

·~ 

.~ 

Q 

- 500 

"0 .!3 

~ 

- 750 

0.1 

- 1000 

80 

Spd 

- Dir311 - Dir 412 

60 - Dir 509 

20 

0 

4 

VAB Sway Wind Tower: MAY 2010 

5 6 

Time from MAY 1 [days] 

1 

0 .s 

c 

.S! 

.... 

·~ 

p,.. 

.... 

Q) 

- 1 

bO 

~ 

- 2 

7 8 
NASA!fM- 20 13- 216320 

APPENDIX B. DEVELOPMENT OF THE SWAY MATH MODEL 

Development of Equation (1) begins with the following physical principles as starting points: the sway 

displacement of the building d is a quadratic function of the wind speed v and is separable from the 

quadratic dependence of the vertical height h of the sway measurement: 

d(v,h) = f(v)g(h) , (B-1 ) 

f (v ) = av2 ' (B-2) 

g(h) = fJh 2 ' (B-3) 

where a is parameter dependent on the building structure. The parameter fJ is equal to L"2 where L is the 

building height. 

The data from Table 2 is plotted in Figure B-1, along with Equation (B-1 ) with g(L) = 1. The parameter a 

is chosen to fit the data. The key to the strategy leading to the sway model represented by Equation (1) is 

finding the standard deviation of the data, corresponding to Table 2. 

E:5.0 

·-.......... 

5 2.0 

E 

Q) 1.0 ~ 

Q..0.5 

1:1) 

·Q -

00 0.2 

s:: 

:.a -0.1 ·s 

~ 

• 

~ 

20 

~ 

/ 

• ...... / 

.. - / 

• --~ • ~ -~· ~· • 

.. 

30 50 70 100 

Wind Speed [kts] 

Figure B-1. The data from Table 2 with Equation (B-1) with g(L) = 1. 

The first step is to rotate the coordinate system so that the mean of the data equals zero. The following 

variable substitution will simplify this procedure (where " log" is base 1 0): 

u = logd - u0 , (B-4) 

w = logv , (B-5) 

where u0 = loga . 

23 
NASA/TM-20 13-216320 

Figure B-2 shows Table 2 data plotted using the transformed variables (u w). Note that both plots are 

practically identical, where only the axis labeling and plot range has changed, as well as vertical offset, 

uo. 

8.o 0.5 1.0 1.5 2.0 

w 

Figure B-2. Data from Table 2 plotted using variable transformations, Equations (B-4) and (B-5). The 

solid line has a slope of 2 and is just u(w) = 2w. 

The vector formed by (w u) is now rotated by an angle() = tan- 1 2, which has the effect afforcing the 

slope of the fit in Figure B-1 to zero: 

(

w'J (cos(} sinBJ (wJ u' = -sin(} cos(} . u 

Figure B-3 is Table 2 data after application of Equation (B-6). 

24 

(B-6) 
1.0 

0.5 

-::s 0.0 

-0.5 

0.2 

0.1 

-0.1 

-0.2 

3.0 

t 

1 

4~ 

Ia l 

-.-•!. ~ 

4t 

3.2 3.4 

NASA/TM-20 13- 216320 

-~-· ·~ 

2 3 4 

w' 

~ 

4t 

• L-

[ • 

3.6 3.8 4.0 

w' 

Figure B-3 . Data from Figure B-2 plotted after undergoing the rotation transformation of Equation (B-6). 

The bottom plot is a zoomed-in version of the top plot. The line fit from the previous plot, u(w) = 2w, is 

,........, 

0.2 

0.1 

~ 0.0 

-0.1 

-0.20 

T ... • 

tit 

now u'(w') = 0. 

• ••• l~ 

5 10 

k 

4 

• t 

T T 

• ~ 

t 

15 20 

Figure B-4. The u; sequence from Figure B-3 with same ordering shown in Table 2. 

25 
NASA/TM-20 13-216320 

The next step is to compute the mean and standard deviation of the data in Figure B-3. Before doing so, it 

is helpful to replot Figure B-3 as a sequence of data u~ , dropping the w~, since the random variable of 

interest is u~. Figure B-4 is the u~ sequence from Figure B-3, in ascending order, the same ordering 

shown in Table 2. 

Now it is a simple matter to calculate the mean f.1 and standard deviation a of the sequence of N data 

points shown in Figure B-4: 

(B-7) 

( 

1 N )l/2 

0'= -:L(u-u~Y 

N -1 k=l 

(B-8) 

The mean must equal zero, and it is necessary to iteratively adjust a , where uo = loga , so that f.1 ~ 0. 

The result is shown in Table B-1. 

26 
NASA/TM-20 13-216320 

Table B-1. Adjusted data plotted in Figure B-4, with a= 0.0001591. 

k I 

uk 

0.0976349 

2 --0.0342895 

3 0.0066372 

4 --0.13811 

5 --0.0049171 

6 0.0257814 

7 --0.0292712 

8 --0.0160197 

9 --0.037004 

10 0.000245142 

11 --0.08624 

12 --0.0672877 

13 0.0576297 

14 0.109065 

15 0.118765 

16 0.178798 

17 0.13657 

18 --0.0314805 

19 --0.176244 

20 --0.102901 

21 0.0712608 

22 --0.0786225 

f.J -1.905 X 10-10 

u 0.0922 

Table B-1 and Figure B-4 depict a transformed coordinate system where the sway measurement "noise" is 

independent of wind speed and only dependent on the errors resulting from random unknowns such as the 

wind correction factor between wind tower height and V AB height. Figure B-3 hints that as the horizontal 

variable w' increases, the running average mean may not be zero. However, without higher wind speed 

measurements than what was observed during the period of this investigation, it is not constructive to 

infer a wind speed dependence in Figure B-3. But this could be worth reinvestigating for future analysis 

refinement. 

To get back to the original coordinate system where building displacement is a function of wind velocity, 

all of the previous transformations must be reversed. The first step on the way back is to express the 

function in Figure B-4 as a vector composed of a random variable x with zero mean, and a value above 

the mean given by na. This vector is then acted upon by a rotation matrix that is the inverse of the 

original rotation: 

27 
NASA/TM-20 13-216320 

(

w:) = (c~s (} -sin(}). ( x ) 

u sm (} cos(} na-

= (xcos B-na-sin (}) 

na- cos(}+ x sin(} (B-9) 

The two components can be combined, eliminating x: 

u" = na- cos(}+ w" tan(} (B-10) 

The last step is to add Uo and take the anti-log (and dropping the double primes): 

d(n, v) = 10 nal cosO+wtanO+u0 

= 1 o nasecO+IogvtanO+Ioga 

(B-11) 

Equation (B-11) is now Equation (B-2). Note that b=.J51n10, since secB=sec(tan-' 2)=.JS. 

Including the height dependence, Equation (B-3): 

(B-12) 

yields the desired result, Equation (1 ). 

28