This paper examines Probabilistic Sensitivity Analysis (PSA) methods and tools in an effort to understand their utility in vehicle loads and dynamic analysis. Specifically, this study addresses how these methods may be used to establish limits on payload mass and cg location and requirements on adaptor stiffnesses while maintaining vehicle loads and frequencies within established bounds. To this end, PSA methods and tools are applied to a realistic, but manageable, integrated launch vehicle analysis where payload and payload adaptor parameters are modeled as random variables. This analysis is used to study both Regional Response PSA (RRPSA) and Global Response PSA (GRPSA) methods, with a primary focus on sampling based techniques. For contrast, some MPP based approaches are also examined

McDonald, Emmett J.

McGhee, David S.

Peck, Jeff A.

NASA Technical Reports Server

 
American Institute of Aeronautics and Astronautics 
 
 
1
Probabilistic Sensitivity Analysis for Launch Vehicles with 
Varying Payloads and Adapters 
for Structural Dynamics and Loads 
David S. McGhee1 and Jeff A. Peck2 
NASA Marshall Space Flight Center, Huntsville, Alabama, 35812, USA 
and 
Emmett J. McDonald3 
Jacobs ESTS Group, P.O. Box 9030, Huntsville, Alabama, 35812, USA 
This paper examines Probabilistic Sensitivity Analysis (PSA) methods and tools in an 
effort to understand their utility in vehicle loads and dynamic analysis.  Specifically, this 
study addresses how these methods may be used to establish limits on payload mass and cg 
location and requirements on adaptor stiffnesses while maintaining vehicle loads and 
frequencies within established bounds.  To this end, PSA methods and tools are applied to a 
realistic, but manageable, integrated launch vehicle analysis where payload and payload 
adaptor parameters are modeled as random variables.  This analysis is used to study both 
Regional Response PSA (RRPSA) and Global Response PSA (GRPSA) methods, with a 
primary focus on sampling based techniques.  For contrast, some MPP based approaches are 
also examined.   
Nomenclature 
Si = Probability Sensitivity of the response with respect to the random variable i distribution mean 
                       parameter  normalized to the standard deviation over the probability of failure 
Si, = Probability Sensitivity of the response with respect to the random variable i distribution standard  
                       deviation parameter normalized to the standard deviation over the probability of failure 
I. Introduction 
ASA Marshall Space Flight Center desires to develop Spacecraft and launch vehicles systems that maximize 
payload to orbit and minimize weight and costs, while achieving schedule milestones and maintaining high 
standards for safety.  To achieve this goal, probabilistic sensitivities may be used to help identify the significant 
input parameters and significant uncertainty drivers on the structural dynamic and loads response of a launch vehicle 
system. Probabilistic sensitivities may be used in design, analysis, manufacturing and testing to help identify the 
important parameters that should be focused on, the tolerances and requirements that are important, and testing that 
should be performed to reduce the significant uncertainties driving loads and dynamic responses.  
 
The purpose of this paper is to demonstrate Probabilistic Sensitivity Analysis (PSA) methods and tools on a launch 
vehicle with varying payloads and adapters. A realistic, but manageable, integrated launch vehicle analysis with 
varying payload and adapter subjected to liftoff type loads is used to demonstrate PSA.  While the current analysis is 
made manageable by reducing its complexity, it is set up with the intent of expanding to more realistic (i.e. 
complex) models, random variable sets and loads analysis simulations in the future so that this method can be used 
                                                          
1 Engineer, Dynamics, Loads and Strength Branch (EV31), NASA Marshall Space Flight Center, Huntsville, 
Alabama, 35812/MS 4600:2217.  
2 Engineer, Dynamics, Loads and Strength Branch (EV31), NASA Marshall Space Flight Center, Huntsville, 
Alabama, 35812/ MS 4600:2127. 
3 Staff Engineer, Spacecraft and Vehicle Systems Loads and Dynamics Analysis, P.O. Box 9030, AIAA Member. 
N
https://ntrs.nasa.gov/search.jsp?R=20120014367 2019-08-30T22:13:36+00:00Z
 
American Institute of Aeronautics and Astronautics 
 
 
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to understand the general sensitivity of the vehicle response to structural parameters.  Several PSA methods are 
investigated, including methods currently available in NESSUS, a Probability Analysis software tool with a GUI 
interface.  Both regional response PSA (RRPSA) and global response PSA (GRPSA) methods will be discussed. 
 
The primary PSA methods discussed herein are sampling based analysis, such as Monte Carlo, since current loads 
processes and heritage data are based on Monte Carlo analyses. Since numerous load responses are generated for 
each run in loads analysis, advanced probabilistic techniques are less attractive since these advanced methods 
generally process only one response variable at a time. There is much interest in defining Probabilistic Sensitivities 
near the enclosure limit (i.e. 99.865/50 limit or 3-sigma limit) for loads analysis.  
 
NASTRAN is used to predict normal modes and load responses.  A Matlab mode tracking tool is used to track 
normal modes.   
 
This paper is organized as follows. The PSA Model is presented in Section II. PSA Methods are discussed in Section 
III.  Results are given in Section IV. Conclusions are given in Section V, including a recommendation of the PSA 
method to use for Launch Vehicle Loads and Dynamic Analysis.  
  
II. The PSA Model 
A realistic, but manageable, integrated launch vehicle system with varying payload and adapter subjected to 
liftoff type loads is used to demonstrate PSA with the intent of expanding to more complex models, random variable 
sets and loads analysis simulations in the future.  A sketch of the model is shown in Fig. 1.  Figure 1a depicts a 3D 
rendering of a 2D generic Apollo launch vehicle. Figure 1b shows a simplified schematic of the model. Payload and 
Adapter terms are varied. Payload and Adapter random variables are shown in Table 1.  The integrated vehicle 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                 
                             Origin (0., 0., 0.) 100” forward of LAS tip       (X – Aft, Y-Starboard, Z-up) 
 
                                               1a – 3D Rendering of 2D NASTRAN Finite Element Model 
                                 1b – Schematic of Varying Payload and Adapter on Integrated Launch Vehicle 
 
Figure 1. PSA Integrated Launch Vehicle System with Varying Payload and Payload Adaptor 
  MRcenter 
   Tcenter
  MRadapter 
     Tadapter 
  fnmode 1   
SE 
 k,m 
 SE 
 k,m 
 SE 
k,m
SE
k,m
 SE 
k,m
 SE 
k,m
 SE 
k,m
 SE 
k,m 
 SE 
 k,m 

 Thrust (T)
In the future, these vehicle superelement (SE) Stiffness (k) and Mass(m) 
terms may be varied in more complex models.
        Adaptor Stiffness 
        kx,ky,kz,kx,ky,kz 
where ky = kz and ky = kz 
 Payload Mass  
     & Inertia 
 m, Ixx, Iyy, Izz   
 where Iyy = Izz  
 Center of Mass (cm) of Payload 
 with respect to Payload/Adapter   
 Interface  
              xcm, ycm, zcm 
    where R and  are used to  
         define ycm and zcm 
 T=8,500,000 lbs
 T
hr
us
t (
T)
, l
bs
 
 Time, s 
Angles between thrust vector and 
components directions:
= 1 
= 90 - 1/   
= 90 - 1/    
Thrust Vector: 
T (0.999848i + 0.012341j + 0.012341k)
 
American Institute of Aeronautics and Astronautics 
 
 
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model normal modes will be evaluated to a 100 Hz cutoff.  Load responses will be evaluated for a liftoff-like thrust 
step load input applied to the vehicle base at an incidence angle. The step load excites all the modes in the model.  
The responses investigated for PSA are the first structural dynamic mode, the maximum resultant My-Mz moment 
and maximum torque at the vehicle center and the maximum resultant My-Mz moment and maximum torque at the 
adapter as shown in Table 2. 
 
 
Table 1. Input Random Variables for Varying Payload and Adapter 
 
Random 
Variable 
 
Units 
 
Mean 
 
Std Dev 
 
COV 
 
PDF 
 
Description 
Mass lb-s2/in 571.012 57.012 0.10 Normal Payload Mass, m 
XCG in -432 64.8 0.15 Normal Payload CM xcm 
I11 lb-in-s2 4.5E6 1.125E6 0.25 Normal Payload Inertia Ixx 
I22 lb-in-s2 3.75E7 0.9375E7 0.25 Normal Payload Inertia Iyy;  Iyy = Izz   
KX lb/in 1.0E7 0.3E7 0.30 Normal Adapter Stiffness kx 
KY lb/in 1.0E7 0.3E7 0.30 Normal Adapter Stiffness ky;  ky = kz  
KXX in-lb/rad 0.35E+12 0.105E+12 0.30 Normal Adapter Stiffness kx 
KYY in-lb/rad 0.35E+12 0.105E+12 0.30 Normal Adapter Stiffness ky;  ky = kz 
RCG in 24.00000 7.2 0.30 Normal Payload CM, Radius R 
THECG degrees 0.0 109.5 -  Normal Payload CM, Angle  
 
 
Table 2. Responses for Launch Vehicle 
 
 
Response 
 
Units 
 
Description 
FREQ1 Hz First Structural Mode Frequency of vehicle – Unconstrained 
MRCTR lb-in/rad Resultant Moment =  at center of vehicle 
TCTR lb-in/rad Torque = Mx at center of vehicle 
MRADPT lb-in/rad Resultant Moment =  at Adapter 
TADPT lb-in/rad Torque = Mx at Adapter 
 
III. PSA Methods 
 
PSA methods may be categorized as regional response PSA (RRPSA) and global response PSA (GRPSA) methods.   
RRPSA is defined as a PSA for the case that the interest is among a partial range of a response distribution, either at 
the tail of the distribution or within a localized range of the distribution.  GRPSA is defined as a PSA for the case 
that the interest is among the entire distribution of a response. 
 
Variance decomposition is a decomposition of the variance of a response to its variation sources.  It highlights the 
difference between the main effect with response to only one random variable vs. the total effect that includes the 
individual effect of the random variable as well as its interaction with other random variables. Variable 
decomposition is a GRPSA.   
 
The following Probability Sensitivity Analyses are performed herein:  
 
1) Using NESSUS, perform RRPSA on Monte Carlo sample-based analysis, averaging the Probabilistic Sensitivities 
over all the failure cases. 
 
2) Using NESSUS, perform RRPSA on an advanced reliability Most-Probable-Point (MPP)-based method, such as 
AMV+, utilizing importance levels derived from the direction cosines inherent in the MPP-based process.  
 
 
American Institute of Aeronautics and Astronautics 
 
 
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3) Using NESSUS, perform GRPSA using Variance Decomposition. Evaluate first-order factors and total factors, 
where total factors incorporate both the first-order factors and the partial variance interaction terms. Perform 
Variance Decomposition using Structured Monte Carlo and for Fourier Amplitude Sensitivity Test (FAST). 
 
4) Perform RRPSA on Monte Carlo sampled-based analysis using several “locally accurate” surrogate model 
formulations, such as moving least squares and Kriging.   
 
IV. Results 
The results of PSA are given herein.  Results, similar to example cases shown in Figures 2-5,  will be given.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                  
 
 
Figure 2. First Structural Mode PSA with respect to Mean, Normalized to (/Pf)  
for Several Pf Conditions for 10,000 Monte Carlo Runs – An Example. 
 
 
Figure 3. First Structural Mode PSA with respect to Standard Deviation,  
Normalized to (/Pf) for Several Pf Conditions  
for 10,000 Monte Carlo Runs – An Example. 
 
American Institute of Aeronautics and Astronautics 
 
 
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Figure 4. First Structural Mode PSA with respect to Mean and  
with respect to Standard Deviation,  Normalized to (s/Pf)  
for Level=19  Pf Condition for 10,000 Monte Carlo Runs – An Example 
 
 
 
Figure 5. First Structural Mode Variance Decomposition  
with First Order Sensitivities and Total Sensitivity Factors  
for Level=1  Pf Condition for 10,000 Monte Carlo Runs – An Example 
 
 First Order Sensitivities 
 Total Sensitivity Factors
 
American Institute of Aeronautics and Astronautics 
 
 
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V. Conclusions 
 
To be completed after results have been analyzed. 
 
Acknowledgments 
We would like to thank Mr. Jonathan L. Band for helping implement the modal tracking tool.  We would like to 
thank Mr. Robert L. Towner for the initial generic launch vehicle NASTRAN FEM.  We would like to thank Dr. 
John McFarland and Dr. Ben Thacker from Southwest Research Institute for their support on the NESSUS software. 
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Probabilistic Sensitivity Analysis for Launch Vehicles with Varying Payloads and Adapters for Structural Dynamics and Loads

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