NASA / TP--2003-212257

Statistical Evaluation and Improvement

of Methods for Combining Random

and Harmonic Loads

A.M. Brown and D.S. McGhee

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

February 2003
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NASA/TP--2003-212257

Statistical Evaluation and Improvement

of Methods for Combining Random

and Harmonic Loads

A.M. Brown and D.S. McGhee

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

National Aeronautics and

Space Administration

Marshall Space Flight Center • MSFC, Alabama 35812

February 2003
Acknowledgments

The author wishes to acknowledge the technical contribution of Eric Fox, formerly of Pratt & Whitney,

for his preliminary work applying the Monte Carlo method to random-sine loads combination.

TRADEMARKS

Trade names and trademarks are used in this report tbr identification only. This usage does not constitute an official

endorsement, either expressed or implied, by the National Aeronautics and Space Administration.

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 2216 I

(703) 487-4650

ii
TABLE OF CONTENTS

1. INTRODUCTION ....................................................................................................................

2. LOADS COMBINATION EQUATIONS .................................................................................

3. LOADS COMBINATION USING PROBABILITY DENSITY FUNCTIONS ......................

4. LOADS COMBINATION USING MONTE CARLO .............................................................

5. ACCURACY AND SPEED OF MONTE CARLO ..................................................................

6. IMPLEMENTATION IN MICROSOFT ® EXCEL ..................................................................

7. RESULTS ..................................................................................................................................

8. EXAMINATION OF CLOSED-FORM METHODS ...............................................................

9. CONCLUSION AND AREAS OF FUTURE WORK ..............................................................

APPENDIX A--MONTE CARLO TECHNIQUE ..........................................................................

APPENDIX B--INVERSE-INTEGRATION TECHNIQUE ..........................................................

APPENDIX C--APPLYING CURVE FIT OF THE PROBABILITY DENSITY FUNCTION .....

REFERENCES .................................................................................................................................

1

2

4

6

10

11

12

13

17

18

20

23

24

iii
LIST OF FIGURES

I,

2.

3.

4.

5.

6.

7.

8.

Typical results from random response analysis ..................................................................... 1

Probability density function f(y)=sin(x) ................................................................................ 4

Sample time history of random loading, rms=l .................................................................... 6

Sample time history of sine loading ...................................................................................... 7

Sum of sine and random signals ............................................................................................ 8

CDF of combined loadings .................................................................................................... 8

Percent overshoot from baseline using traditional methods .................................................. 13

Curve fit of overshoot of 3xssMS method over CDF of 99.865 percent .............................. 14

LIST OF TABLES

l° Comparison of all methods .................................................................................................... 12

Comparison of derived closed-form equation with 99.865-percent baseline ........................ 15

iv
LIST OF ACRONYMS

CDF

HCF

PDF

PSD

rlTIS

SSME

TP

cumulative distribution function

high-cycle fatigue

probability density function

power spectral density

root mean square

Space Shuttle main engine

Technical Publication

V
A

A sin

F(z)

2

N

P sin

2"

O"

amplitude

amplitude of sine load

cumulative distribution function

tolerance band

normal distribution

number of samples

probability of failure

harmonic load

variable load

very small value

standard deviation

rms of random load

rms of sine load

phase relationship

NOMENCLATURE

vi
TECHNICAL PUBLICATION

STATISTICAL EVALUATION AND IMPROVEMENT OF METHODS

FOR COMBINING RANDOM AND HARMONIC LOADS

1. INTRODUCTION

Many structural components in dynamic environments experience both sinusoidal and random

loadings at the same time. One example of such a structure is a component of a rocket engine. There are

several sources that produce the random component of the excitation, including the gas generator and main

combustion chamber. These excitations are characterized by a power spectral density (PSD) spectrum of

accelerations. These accelerations are used in a base excitation random response analysis, which results in

a PSD response spectrum at any desired location, as shown by the example of the gas generator axial force

response for NASA Marshall Space Flight Center's MC-I engine in figure 1. j Although it can be in any

desired quantity, the mean of this spectrum, defined simply as a load, is zero, and since the process can be

assumed to be Gaussian, 2 the root mean square (rms) of the response is defined to be equal to one standard

deviation, cy. Simultaneously, the turbomachinery in the engine generates large harmonic loads due to the

unavoidable unbalance in their rotors. The frequency of the harmonic excitations are at the shaft speed of

the turbines and their multiples. The excitation from this sinusoidal loading, which can also be determined

by previous acceleration measurements, is applied independently in a frequency response analysis, which

generates the amplitude of a sinusoidai response for any desired location for a specified range of harmonic

excitation frequencies.

3.5x105

3x105

2.5x105

,,, 2x105

-- 1.5x105

lx105

5x 104

o

20 40 60 80 100 120

Frequency(Hz)

140

Figure 1. Typical results from random response analysis.
2. LOADS COMBINATION EQUATIONS

In general, the structural dynamic response analyses for the random and harmonic cases are

performed separately, and the results are then combined to obtain a design load. These loads are used to

determine a level of alternating stress in the component for design to ultimate or yield strength and for

high-cycle fatigue (HCF). The calculation of this load has been a subject of some dispute in industry.

Although the development of methods for combining different types of random loads has been a subject of

extensive research, 3 detailed study of the use of similar equations for the combination of random and

harmonically generated loads has not been performed. Perhaps because of this lack of examination, an

industry standard loads combination equation has not been chosen. There are, therefore, several different

methodologies that have been applied by various users. They are all based on the method used for obtaining design loads for purely random environments. Although there have been studies suggesting that the

extreme-value distribution should be used for limit load design, 4 a widely accepted practice is to use

3 times the rms of the random load, Gran, for the design load where the distribution is assumed to be

Gaussian. This value will exceed the response 99.865 percent of the time. It has also been suggested that a

value less than 99.865 percent is acceptable for HCF design, but this Technical Publication (TP) will

assume that both the Gaussian distribution and the 99.865-percent excedance level are valid.

The most frequently used method for combining the two dynamic load components, referred

to here as the standard method, simply uses the sum of the amplitude of the sine load Asi n and 3×Crran :5

design load = Asi n + 3O'ra n . ( 1)

This technique assumes the sine peak value always occurs simultaneously with the peak random value.

Another frequently applied method is to multiply 3 by the square root of the sums of mean square values of

both the random and the sines, called the 3xssMS method: 6

design load= 3_/(O'sin) 2 *(Crran) 2 , (2)

where _sin, the rms of the sine wave, is equal to the sine amplitude divided by ,f2. This method treats the

sine as if it had a normal probability distribution, which has a much larger range than a sine function. The

3xssMS approach is similar to equations written to combine loads on launch vehicles, which do not have a

substantial harmonic component but do have random loads from several sources. 7 A third approach,

proposed by Steinberg, 8 is to take the root sum square of the peak values (called the peak method):

design load -- _/(Asi n )2 + (3O.ra n)2 (3)
Although it is stated in reference 8 that the standard method is too conservative and that it is convenient to

use the suggested peak method, no mathematically rigorous justification is presented for using it for either

ultimate/yield or HCF design. None of these closed-form methods produce a design load that can be consistently associated with a particular probability level.

Although the methodology discussed in this TP is relatively straightforward, it has not been generally applied in industry. In a paper by DebChaudhury et al., advanced probabilistic techniques were used to

evaluate the stochastic nature of the parameters of the loading and the structure for the Space Shuttle main

engine (SSME), but the sine and random loads were combined using the simple closed-form 3xssMS

method. 2 A similar study by Newlin et al., in which stochastic techniques were used to evaluate HCF

problems in the SSME, generated a simulated random time history and added it to a harmonic time

history. 9 The combined loading was then used for each time step. This methodology is similar to that

presented in this TP but is too complex to use in a routine manner. Schock 1° was apparently able to analytically calculate the joint cumulative density function for ty= 1 and was able to generate charts for different

cummulative distribution function (CDF) values for this level of random or. Ferebee II added to this work;

however, the results do not appear to be consistent with the work presented below, and the methodology for

performing the integration is not discussed.

3
3. LOADS COMBINATION USING PROBABILITY DENSITY FUNCTIONS

Since a sine wave considered by itself is not a random signal, it has always been unclear how to

combine harmonic analysis results with output from a random excitation analysis to obtain a design load.

The harmonic signal can, however, be defined as a stationary random process when combined with an

independent Gaussian random signal since its phase relationship, 0, with the random signal is random.12

If the harmonic signal is defined as

Psin = A sin(ogt + _) ,

(4)

then the argument (ogt + q_)is therefore a random variable with a uniform distribution over the range -7t/2

to re/2. The probability density function of the sine function of this distribution is

shown graphically in figure 2.13

-/I

v

A

Figure 2. Probability density function f(y)=sin(x).

4
TheequivalenteventtechniquecannowbeusedtocalculatetheCDF,F(z), of functions of random

variables. 14 For the function z = x + y, where x and y are random variables, the resultant F(z) is defined in

terms of the probability density functions (PDFs))'I_ andfy as

fy (y)dy . (6)

If x represents the random load, it can be characterized by a normal distribution with mean zero and

standard deviation equal to the rms from the random analysis. Substituting the normal PDF and the PDF

shown in equation (5) into equation (6) results in

F(z) =

A

f

-A

O'ran _ exp

--oo

dy , (7)

where the integral over y is only evaluated from -Asi n to Asi n since the function is undefined outside of that

range. A closed-form solution for this integral cannot be obtained; however, recent improvements in the

software code Mathematica ® 4.1 have enabled it to be evaluated numerically for any value of A, z, and

O'ran. The routine crashes if the integration is carried out over the range exactly from -,4 to A, so the limits

are instead set at -A+e to A-e, where e is a very small value. The value of e was obtained by performing a

series of runs with decreasing e until the solution converged; the final value obtained was 1× 10 -ll

It is also of interest to obtain the solution for the inverse problem from that described above; i.e.,

what would the design load be that would yield a specific CDF value. In particular, if the 99.865-percent

level has traditionally been chosen as an acceptable level of probability of exceedance for designing to

random loads, then it would be consistent to use that level to design for the combination of all dynamic

loads. The Mathematica Runge-Kutta "FindRoot" command can be implemented to calculate this load (z)

given the probability F(z), Asi n, and O'ran.

5
4. LOADS COMBINATION USING MONTE CARLO

Monte Carlo simulation can also be used to obtain the CDF value of interest. The random load is

first simulated by generating a vector of points, {r}, that fall within a Gaussian (or normal) distribution of

mean zero and standard deviation equal to the rms value resulting from the random analysis. This can be

expressed as {r}-N(0.0,O'ran). For illustrative purposes, a simulated history for such a loading is shown in

figure 3. The vectors representing the results from the harmonic analysis are then generated. The method

can account for analysis results from multiple independent harmonic excitations {y }i, i=1 ..... n. Since the

frequencies of each sine wave are different and the relative phasing is unknown, each sine wave should be

considered independently. 15Examples of sine loading time histories are shown in figure 4. This method is

only applicable for independent harmonic sources, such as excitations from different turbopumps. The

loads coming from multiples of the primary shaft rotational speed for a single turbopump would be

correlated and, therefore, not independent.

3

I

Figure 3. Sample time history of random loading, rms = I.
4

-2

-3

-4

q£££££££

VVVVVVVV

2 4 6 8

3

2

I 2 4 6 8

-3 t

Figure 4. Sample time history of sine loading.

For the independent case, a vector for each wave of the same length as {r} is therefore created using the

PDF defined previously, resulting in

{y}i = Aisin(2:rr{x}i) ,

(8)

where {x} is a random vector uniformly distributed between zero and 1. The resulting vector, {y}, will

contain points that have the correct probability distribution of a sine wave, but are related randomly in

phase (unlike the sequential time history shown in fig. 4). This procedure allows the frequencies of different harmonic signals to be rigorously removed from the calculation. The summation of all the harmonic

signals and the random signal is performed for each sample to create a new vector, {z }, which is equal to a

sample set of total responses of the structure to the multiple simultaneous excitations,

{z}={r}+{Y}l +{Y}2 + "'" +{Y},, (9)

A sample of such a loading is shown in figure 5, which is formed by adding the curves in figures 3 and 4.

The CDF of {z} is then generated for the sample case, as shown in figure 6. A search procedure is

performed to obtain the value from this CDF that is greater than 99.865 percent of the responses.

7
I I I

2 4 6 8

Figure 5. Sum of sine and random signals.

0.8

0.6

0.4

0.2

I

-4 -2 0

I I I I I 1 I I I I I I

2 4 6

Figure 6. CDF of combined loadings.
This procedure was implemented in a FORTRAN program running on an SGI workstation. 16 It was

applied successfully in the MC-1 engine program in 1998 to determine design loads used to evaluate

stresses in interface brackets, fasteners, lugs, and other structural members. Since a large number of combinations were needed for each of a large number of load cycles, the speed of the program was increased by

using efficient numerical routines found in standard FORTRAN libraries, such as IMSL, and by automating the process for multiple combinations. Using these methods, 100,000 samples for each of 792 separate

sine-random load combinations were calculated in a total of ! ,573 CPU s (26 min) on a Cray T90. For more

general use, a generic, user-friendly computer program implementing this method was also written and

incorporated into the NASA/Techtrac national software library.

9
5. ACCURACY AND SPEED OF MONTE CARLO

The accuracy of the Monte Carlo simulation is dependent on the number of samples selected. A

convergence study performed here for a single harmonic wave with a random signal indicates that the

values converge for =400,000 samples. There are several analytical methods for determining accuracy.

Applying the method described by Ang and Tang,17 the variation in the probability of failure estimate, Pt"

for a given number of samples, n, with a specified confidence is

pf + k_ n

(10)

Charts for various confidence levels are used to determine k_2. For a 95-percent confidence; i.e., 95/100

simulations would yield a probability estimate within the tolerance band, ka./2=1.96. The accuracy using

400,000 samples is therefore 0.99865 +0.000114 (99.854-99.876 percent). If 200,000 samples are used

instead (doubling the speed), the error would be 0.99865 +0.000161, showing only a small loss of

accuracy.

10
6. IMPLEMENTATION IN MICROSOFT EXCEL

To facilitate ease of use, both the PDF inverse-integration technique, discussed in Section 3, and

the Monte Carlo technique were implemented as Visual Basic macros launched from Microsoft Excel. A

complete listing of the macros and associated batch runstreams are presented in appendices A and B. The

listings are also available in electronic format from the authors. In each case, any number of different

combinations of random 1cr values and sine amplitudes are listed in columns, and these columns, along

with a blank column adjacent, are chosen, the macro is selected, and then launched. The inverse-integration

technique automatically writes the input values to a text file and launches the Mathematica kernel in the

background, which reads the data, performs the root search for the chosen probability level (hard-coded as

99.865 percent), and writes the result into a text file that is then read into the result column in Excel. This

process takes =4,5 s for each combination on an Intel Pentium 4 1.8 GHz workstation with 500 Mb RAM

running Windows ® 2000.

The Monte Carlo macro is based upon the FORTRAN code written previously, but can only

combine a single sine load with the random load. It is run entirely within Visual Basic for Excel to enable

maximum portability, and takes =51 s per combination.

11
7. RESULTS

The integration method was used to evaluate the actual CDF values using each of the three closedform methods for a range of sine and random inputs chosen from actual MC-I analysis results as well as

some hypothetical cases. A representative subset of these CDF results is shown in table 1. The combination

load values are also compared with loads obtained for a 99.865-percent probability level using the inverseintegration method and the Monte Carlo method, with the inverse-integration method being chosen as the

baseline. The Monte Carlo method shows very close agreement resulting from the large number of samples

arrived at from the convergence study. As expected, both the standard and the 3×ssMS methods always

exceed the baseline value by varying amounts. The peak method always yields a value less than the baseline.

Table !. Comparison of all methods.

Amp.Sine

10

5

5

26

97

5O

64

la

Random

5

5

2O

4

14.67

98.7

109.33

Random/

Sine

0.5

1

4

0.154

1.151

1974

1.708

LoadUsing

Integration

Methodfor

99.865%

(baseline)

22.034

17.668

60.915

34.760

129.081

313.047

352.240

Load, Monte

Carlo Method

for 99.86%

400,000

Samples

22.031

17.653

60.888

34.772

129.195

313.422

353.079

Load,

Standard

Method

A+3o

25

20

65

38

141.01

346.1

391.99

g

i

: Overshoot

:

i

From

i

, Baseline i

i (%)

i

i

13.5

i 13.2

|

: 6.7

: 9.3

i 9.2

', 10,6

i

: 11.3

t

i

: CDFValue

i From

i Integration

: (%)

i 99.970

: 99.957

! 99,919

J

: 99.983

: 99.986

! 99.951

! 99.955

i

; ; , ;

: CDFValue Load, : : CDFValue

i ! From Peak : From

la Load, i Overshoot :Integration Method i Overshoot i Integration

| i

Amp. Sine Random 3xssMs i (%) i (%) ss(o', A) i (%) i (%)

i J J i

10

5

5

26

97

50

64

5

5

20

4

14.67

98.7

109.33

25.981 :

i

18,371 :

i

60.930 :

56.4451

210.422 :

314.524 :

354.978 i

17.9

4.0

0.03

62.4

63.0

0.5

0.8

i

: 99.990

e

: 99.912

i

'

l 99.865

: I00.000

J

: 100,000

; 99.871

i

: 99.875

18.028

15.811

60.208

28.636

106.517

300.292

334.176

i

: -18.2

B

' -10.5

l

: -1.2

', -17.6

i

', -17.5

' -4.1

" -5.1

: 93.930

i 94.896

i

: 95.758

; 94.291

i 94.316

i 95.534

: 95.443

12
8. EXAMINATION OF CLOSED-FORM METHODS

If a user chooses to apply one of the simple closed-form techniques, it would be desirable to use the

technique that yields an answer with a CDF above 99.865 percent, but not excessively so. As seen in

table 1, for some combinations, this rationale would point to the standard method, while for others, the

3×ssMS would apply. The exact ratio defining which is better to use can be calculated. First, equate equations (1)

and (2):

Asin + 30.ra n = 3_/(O.si n)2 + (O.ra n )2 (11)

Using the relationship Asi n = _ _sin, equation ( 11 ) is satisfied when

O'sin = 1.21218O-ra n , or Asi n = (1.71429)O-ra n .

(12)

Therefore, if Asi n > 1.71429 O'ran, the standard method will yield a probability value greater than

99.965 percent but less than the value obtained using the 3×ssMS method, and vice-versa. This result is

demonstrated in figure 7, which is a plot of the percentage overshoot versus the ratio, X=Gran/Asin, for each

of the above methods.

8O

70

60

i

4O

30

2O

10

0

--*--3×ssMS OvershootFromBaseline(%)

--I-Asin+3cr OvershootFromBaseline(%)

0 1 2 3 4 5 6 7 _, cJ 10

Ratio (aran/Asi.)

Figure 7. Percent overshoot from baseline using traditional methods.

13
Thefunctionalrelationshipsidentified in figure 7 were examined for various values of the parameters and were seen to be only a function of the ratio. This independent functional relationship identified

the possibility of creating a closed-form equation that would exactly match the values obtained using the

inverse-integration method for this specific probability level. The Mathematica 4.1 curve fit routine was

used to examine different categories of functions for fitting the overshoot, resulting in equation (l 3), with

the comparison of the function with the data shown in figure 8:

4.60691 x 10-1 ] x 7 _ 1.193679 x 10 -8 x 6

overshoot = e- !.53546 /x[+ I. 19197 X 10--6 X 5 _ 6.04117 X 10 -5 X4

+ 1.69965 X 10-3 X 3 _ .0271317X 2 + .241154X

(13)

The desired closed-form equation is therefore

3 - + O'r2an

design load- i + overshoot (14)

1

'i

1 2 3 4 5

Ratio(x=-aran/Asi.)

Figure 8. Curve fit of overshoot of 3xssMS method over CDF of 99.865 percent.

14
A comparison of the baseline values with those obtained using this new closed-form solution is shown in

table 2. The agreement is excellent, with only one of the points showing an overshoot of greater than

1 percent. This equation can be directly used in a spreadsheet. If the ratio O'ran to Asi n is outside of the range

0.025 to 10, or if either component is zero, the curve fit is not valid; therefore, an Excel user-defined

function has been written (see app. C), which will automatically apply the curve fit within the acceptable

range or the appropriate traditional method outside of it (where they are accurate).

Table 2. Comparison of derived closed-form equation with 99.865-percent baseline.

Amp.

Sine

1

2

3

4

6

8

10

20

40

60

80

100

200

300

400

50

la

Random

10 1

10

10

10

10

10

10

10

10

10

10

10

10

10

10

1

Ratio

Crrawtksin

0.000

5.000

3.333

2.500

1.667

1.250

1.000

0.500

0.250

0.167

0.125

0.100

0.050

0.033

0.025

0.020

99.865% CDF

Usinginverse

Integration

(Baseline)

30.0811

30.2986

30,6500

31.1169

32.3175

33.7557

35.3340

44.0679

62.8036

82.0509

101.5086

121.0833

219.7239

318.9018

418.3045

51.7833

Closed-form

Curve Fit

Basedon

3xssMs

Method

30.0749

30.2979

30.6547

31.1152

32.2508

33.5979

35.1116

44.0599

63.1242

82.0168

101.2031

120.8167

219.6545

319.8996

417,9850

51.7960

Overshoot

From

Baseline

(%)

-0.02

0.00

0.02

-0.01

-0.21

-0.47

-0.63

-0.02

0.51

-0.04

-0.30

-0.22

-0.03

0.31

-0.08

0.02

15

9. CONCLUSION AND AREAS OF FUTURE WORK

The probability values associated with several widely-used methods for combining harmonic and

random loads have been determined. In addition, two new Microsoft Excel macros have been written that

calculate the combined load value for a specified probability level. Because of the speed of desktop

computers, these macros can be executed easily and quickly. A closed-form solution approximating the

value for a level of 99.865 percent has also been determined. The comparisons presented in this TP can

assist the analyst in choosing a method for combining these loads; obtaining an accurate value can be

critical for the design of weight-sensitive aerospace structures. A version of the macro using Monte Carlo

simulation was successfully applied to the determination of loads for 44 separate structural components in

the FASTRAC/MC-I engine program for 10 or more load cycles, resulting in large weight savings. One

avenue of future work in this area is incorporating the effect of the correlation of the sine loads for

harmonics of a single rotating shaft. Another will be to implement the methodology discussed here to the

extreme-value distribution.

17
APPENDIX A--MONTE CARLO TECHNIQUE

This appendix details the Microsoft Excel macro for performing combination of random and sine

loads using Monte Carlo technique. Place Asi n in column 1, Crran in column 2, leave column 3 blank, select

all three columns, and run macro "montecarlo". If a probability level other than 99.865 percent is chosen

for use, this value can be inserted into line 24.

Option Base 1

Sub montecarlo ()

maxsam = i000000

Dim combo(1000000) As Double

Dim pdf(10001) As Double

Dim div(10001) As Double

Dim cdf(10001) As Double

Dim cum(10001) As Double

Dim xlo As Double

Dim xhi As Double

Dim delta As Double

Dim rnorm As Double

numsam = 400000

Pi = 4.0*atn(l.0)

current selection = Selection.Address

For I = 1 To Range(current selection).Rows. Count

a = Range(current selection).Cells(I, i)

sigma = Range(current selection) .Cells(I, 2)

For J = 1 To numsam

rnorm =Application. WorksheetFunction. NormInv(Rnd,

combo(J) = a * Sin(2 * Pi * Rnd) + rnorm

Next J

Call Sort(numsam, combo)

ivalue = Int(0.99865 * numsam)

Range (current selection) .Cells(I, 3) = combo(ivalue)

Next I

End Sub

Public Function Sort(N,

Dim RRA As Double

Dim L As Long

Dim I As Long

RA) As Integer 18

0#, sigma)

18
Dim IR As Long

Dim J As Long

Sort = 0

L = N / 2 + 1

IR = N

CONTI 0 :

If (L > i) Then

L = L 1

RRA = RA (L)

Else

RRA = RA(IR)

RA(IR) = RA(1)

IR = IR 1

If (IR = I) Then

RA(1) = RRA

Sort = 1

Exit Function

End I f

End I f

I = L

J = L + L

CONT20:

If (J <= IR) Then

If (J < IR) Then

If (RA(J) < RA(J + i)) Then J = J + 1

End If

If (RRA < RA(J)) Then

RA(I) = RA(J)

I = J

J = J + J

Else

J = IR + 1

End If

GoTo CONT20:

End If

RA(I) : RRA

GoTo CONTI0:

End Function

19
APPENDIX B--INVERSE-INTEGRATION TECHNIQUE

This appendix details the Microsoft Excel macro, DOS batch runstream, and Mathematica Kernal

text input file for performing combination of random and sine loads using inverse-integration technique. Place

Asi n in column I, ¢3ran in column 2, leave column 3 blank, select all three columns, and run macro "eq3sig".

Microsoft Excel Macro eq.slg

Sub eq3sig()

current selection = Selection.Address

For i = 1 To Range(current selection).Rows.Count

a = Range (current selection) .Cells(i, I)

sigma = Range (current selection) .Cells(i, 2)

Open "C:\Program Files\Wolfram Research\Mathematica\4.1

\excel to math.txt" For Output As #i

Open file for output.

Print #i, a

Print #I, sigma

Close #i

shell_sync ("C:\Program Files\Wolfram Research\Mathematica\4.1

\mat_autorun.bat")

Open "C:\Program Files\Wolfram Research\Mathematica\4.1

\math to_excel.txt" For Input As #3

Input #3, y

Range(current selection).Cells(i, 3) = y

Close #3

Kill "C:\Program Files\Wolfram Research\Mathematica\4.1

\math_to_excel.txt"

Kill "C:\Program Files\Wolfram Research\Mathematica\4.1

\excel_to_math.txt"

Next i

End Sub

Public Function shell sync(cmd As String) As Integer

'shell function runs application asynchronously

'for short duration apps where assurance of completion is desired this

'function enters a do loop activating the application until an error

'occurs signifying the app is no longer there and therefore complete

ireturn = Shell(cmd, vbMinimizedFocus)

If (ireturn <> 0) Then

2O
'cmd successfully initiated

'set error trap and activate app in loop

On Error Resume Next

Do While Err.Number = 0

'idle time loop

For i = 1 To i000

Next i

'check for system events that require action

nfrm = DoEvents()

'attempt to activate app

AppActivate ireturn

Loop

'turn off error trap

On Error GoTo 0

'shell-sync successful

shell sync = 0

Else

'cmd initiation unsuccessful

'shell-sync unsuccessful

shell sync = 1

End If

End Function

DOS batch runstream "mat_autorun.bat". Place file in directory with Mathematica Kernel executable.

C:

chdir C:\Program Files\Wolfram Research\Mathematica\4.1

Math <randsin_calc.txt >math_output.txt

Mathematica Kernel text input file "randsin_calc.txt". Place file in directory with Mathematica Kernel

executable". If a probability level other than 99.865 percent is chosen for use, this value can be inserted

into line 6.

AppendTo[$Echo, "stdout"]

Off [General : :spelll]

v=Import ["excel to math.txt","List"] ;

a=v[ [i] ] ;sigma=v[ [2] ] ;

\!\(max = 3*sigma\ + a; eps=l.0*10^-ll; min : .85*max;\n

FindRoot [ .99865 == \ (l\/\ (a\ *

sigma\ \@\ (2 \ [Pi] \_3\)\) \) \ (\ [Integral] \_\ (-\ ( (a

\eps) \) \) \%\ ( (a eps)\)\((\[Integral]\ \ (\ (-6\) *sigma\) \%\ (z -

\y\) \ (\ [ExponentialE] \_\ (\ (-\ ((x\/sigma) \)\_2\) \/2\) \)

\ [DifferentialD]x) \) \

21
\ (i\/\@\ (I - \ ((y\/a) \)\A2\) \) \ [DifferentialD]y\) ,

equiv3sigma=z/.%

Export["math_to_excel.txt",equiv3sigma,,,List,,] ;

Quit[ ];

{z, {rain,max}}]\)

22
APPENDIX C--APPLYING CURVE FIT OF THE PROBABILITY DENSITY FUNCTION

This appendix details the Microsoft Excel user-defined function applying curve fit of PDF Integration. Create macro as shown below, then in spreadsheet, type function "curvefit(a,s)" with Asi n

as the first argument and O'ran as the second argument.

Function curvefit(a, s)

If a = 0 Then

combo = 3 * s

ElseIf s = 0 Then

combo = a

ElseIf (s >= 0.02 * a) Then

x = s / a

combo = (3 * Sqr(a A 2 / 2 + s ^ 2)) /

(I + Exp(-l.53546*x) * ((4.60691E-ll/x^7)- (l.19367935364834E-08/x^6)+

(0.00000119197 /x_5) - (0.0000604117 /x_4) + (0.00169965 /x^3) -

(0.0271317 /x_2) + (0.241154/x)))

Else

combo = a + (3 * s)

End I f

If (s > (I0 * a)) Then

combo = 3 * Sqr(a ^ 2 / 2 + s ^ 2

End I f

curvefit = combo

End Function

23
REFERENCES

° Christensen, E.; Frady, G.; Mims, K.; and Brown, A.: "Structural Dynamic Analysis of the

X-34 Rocket Engine," AIAA Paper No. 98-2012, 39th AIAA Structures, Structural Dynamics, and

Materials Conference, Long Beach, CA, April 20-23, 1998.

. DebChaudhury, A.; Rajagopal, K.; Ho, H.; and Newell, J.: "A Probabilistic Approach to the Dynamic

Analysis of Ducts Subjected to Multibase Harmonic and Random Excitation," AIAA Paper

No. 90-110(OCP, Proceedings of the 31 st Structures, Structural Dynamics, and Materials Conference,

Long Beach, CA, pp. 104-1061, April 2--4. 1990.

3. Gaver, D.R: "On Combinations of Random Loads," U.S. Navy Report No. AD-A085489, 1980.

4. Merchant, D.H.: "Methods for Combining Payload Parameter Variations with Input Environment,"

NASA/CR-149939, Marshall Space Flight Center, AL, June 30, 1976.

5. Rocketdyne Structural Dynamics Manual, Rocketdyne Corporation, Vol. 2, Sec. 2.2-6, 1989.

6. Wen, Y.K.: Structural Load Modeling and Combination for Pelformance and Safe_. Evaluation,

Elsevier, Amsterdam, p. 21 I. 1990.

o Macheske, V.; Womack J.; and Binkley, J.: "A Statistical Technique for Combining Launch Vehicle

Atmospheric Flight Loads," AIAA Paper No. 93--0755, 31 st Aerospace Sciences Meeting, Reno, NV,

January I 1-14, 1993.

8. Steinberg, D.S.: Vibration Analysis for Electronic Equipment, John Wiley & Sons,

New York, p. 249, 1988.

. Newlin, L.; Sutharshana, S.; et al.: "Probabilistic High Cycle Fatigue Failure Analysis with Application

to Liquid Propellant Rocket Engines," AIAA Paper No. 90-1141-CP, Proceedings of the 31st AIAA

Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, pp. 1105-1114,

April 2-4. 1990.

10. Schock, R.; and Tueli, L.: "An Investigation into the Probabilistic Combination of Quasi-Static

and Random Accelerations," AIAA Paper No. 84-0908, 25th Structures, Structural Dynamics, and

Materials Conference, Palm Springs, CA, May 14-16, 1984.

11. Ferebee, R.: "Loads Combination Research at Marshall Space Flight Center," NASA/TM--2000-

210331, Marshall Space Flight Center, AL, 2000.

12. Wirshing, P.; Paez, T.; and Ortiz, K.: Random Vibrations: Theory and Practice, John Wiley

& Sons, New York, p. 81, 1995.

24
13. Rheinfurth,M.; andHowell, L.W.: "Probability andStatisticsin AerospaceEngineering,"NASA/

TP--1998-207194, Marshall Space Flight Center, AL, p. 53, March, 1998.

14. Rheinfurth, M.; and Howell, L.W.: "Probability and Statistics in Aerospace Engineering," NASA/

TP--1998-207194, Marshall Space Flight Center, AL, p. 51, March, 1998.

15. Johnson, P.: and Ayyub, B.: Probabilistic Structural Mechanics Handbook, Sudararajan, C. (ed.),

Chapman & Hall, New York, p. 652, 1995.

16. Brown, A.M.: "Combining Loads From Random and Harmonic Excitation Using the Monte Carlo

Technique," Journal of Spacecraft & Rockets, Vol. 37, No. 4, pp. 541-543, July-August 2001.

17. Ang, A.; and Tang, W.: Probability Concepts in Engineering Planning and Design, Vol. 1,

John Wiley & Sons, New York, p. 254, 1975.

18. Press, W.H.; et al.: Numerical Recipes, The Art of Scientific Computing (FORTRAN), Cambridge

University Press, Cambridge, England, pp. 229-232, 1989.

25
REPORT DOCUMENTATION PAGE Form Approved

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gathering and maintaining the data needed, and completing and rewewmg the collection ol information. Send comments regarding this burden estimate or any other aspecl of this

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1. AGENCY USE ONLY (Leave Blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

February 2003 Technical Publication

4."n'n.eANDSUBTI_.E _. FUNDINGNUMBERS

Statistical Evaluation and Improvement of Methods

for Combining Random and Harmonic Loads

6. AUTHORS

A.M. Brown and D.S. McGhee

7,PERFORMINGORGANIZATIONNAMES(S) ANDADDRESS(ES)

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

9. SPONSORING/MONITORING AGENCYNAME(S)ANDADDRESS(ES)

' National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PERFORMING ORGANIZATION

REPORT NUMBER

M-1062

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TP--2003-212257

11. SUPPLEMENTARYNOTES

Prepared for Structural, Mechanics, and ThermalDepa_ment, Engineering Directorate

: 12a. DISTRIBUTION/AVAILABILI'T_ STATEMENT

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12b. DISTRIBUTION CODE

13, ABSTRACT (Maximum 200 words)

Structures in many environments experience both random and harmonic excitation. A variety of closedform techniques has been used in the aerospace industry to combine the loads resulting from the two

sources. The resulting combined loads are then used to design for both yield/ultimate strength and highcycle fatigue capability. This Technical Publication examines the cumulative distribution percentiles

obtained using each method by integrating the joint probability density function of the sine and random

components, A new Microsoft Excel spreadsheet macro that links with the software program Mathematica

to calculate the combined value corresponding to any desired percentile is then presented along with a

curve fit to this value. Another Excel macro that calculates the combination using Monte Carlo simulation

is shown. Unlike the traditional techniques, these methods quantity the calculated load value with a

consistent percentile. Using either of the presented methods can be extremely valuable in probabilistic

design, which requires a statistical characterization of the loading. Additionally, since the CDF at high

probability levels is veD, flat, the design value is extremely sensitive to the predetermined percentile;

therefore, applying the new techniques can substantially lower the design loading without losing any of the

identified structural reliability.

14, SUBJECT TERMS 15. NUMBER OF PAGES

loads, random vibration, structural dynamics, rocket engines, 32

probabilistic mechanics

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35812