# Beam Benchmarks for Dynamic Analysis Process Validation by uws18949

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```									Beam Benchmarks for
Dynamic Analysis
Process Validation

MDA

June 10-12, 2008

1
Overview
Beam Benchmarks
for Primary Structure Loads Process
Validation
1.   Purpose
2.   Symbolic Craig-Bampton Beam Benchmark
3.   Test Case
4.   Current Model Checks
5.   Test Case Results New Model Checks
6.   Compare Predicted Loads Versus Requirement
7.   Future Work
8.   Conclusions

2
Purpose

To develop a quick and accurate loads calculate process
Load Factor Calculation Method/Program
Analysis Steps        FE     User Program within FE      External
Build Model
Normal Modes
Sine
C-B Reduction
Post Processing
Inaccuracies                             Simple Beam Benchmark
•Method                Solution          •Validate Sine and C-B/OTM Process
•Programming Errors    Process           •Identify/Catch Errors
•User Input                              •Predict Preliminary Loads

3
Beam Benchmark:
Craig Bampton Modes

⎛ y = 1 or ⎞
⎜ θ =1 ⎟
⎜          ⎟
⎝          ⎠

4
Cantilever Beam Benchmark:
Craig Bampton Matrix
EI
Frequency :      ωk = λ2
k
M L3
1       ⎡       x           x               x           x ⎤
Mode Shape :    φk ( x ) =           ⎢ cos(λk ) − cosh(λk ) − σ k (sin(λk ) − sinh(λk ))⎥
M       ⎣       L           L               L           L ⎦
L
Craig Bampton Matrix:

Constrained DOF

Normal Modes DOF

5
Cantilever Beam Benchmark:
Static Acceleration

4σ k
λk

6
Cantilever Beam Benchmark:
Static Shear

Base Shear =
% Effective Mass

4 σ k2       (PF )
k,y
2

=
λ   2
k        M

7
Cantilever Beam Benchmark:
Static Moment

Base Moment =
% Effective Moment

8σ k     PFk , y PFk ,θ
=
λ3
k
ML
2

8
Cantilever Benchmark:
Beam Participation Factor Properties
∞ Number of Modes

[PF ][PF ]T =
⎡       4 σk2
ML      8σ             ⎤
⎢M ∑ 2             ∑ λ 3k            ⎥ ⎡ Mass ⎤
⎢        λk    2        k            ⎥ =⎢       ⎥
⎢             ML       12            ⎥ ⎣ Matrix ⎦
2

⎢ sym.           3
∑ λ4            ⎥
⎣                        k           ⎦
Finite Number Modes
Det [C − B Mass Matrix ]> 0
⎛      4 σk2
⎞
Residual = M ⎜ 1 − ∑ 2
2
y   ⎜                       ⎟
⎟
⎝       λk              ⎠
(
Sine Sweep θ base = θ&base = 0
&            )
⎡          4σ 2 ⎛ ω ⎞ ⎤
&&(ω ) ⎢1 + ∑ 2 k TR⎜ ⎟ ⎥
Fbase (ω ) = M y                         ⎜ω ⎟
⎢ n =1.., k λk `
⎣                  ⎝ k ⎠⎥
⎦
ML          ⎡          8σ    ⎛ ω ⎞⎤
M base (ω ) =      &&(ω ) ⎢1 + ∑ 3 k TR⎜ ⎟ ⎥
y                       ⎜ω ⎟
2          ⎢ n =1.., k λk
⎣                ⎝ k ⎠⎥
⎦

9
Test Case:
1st Lateral Mode Sine Notching
Momentbase

PFi , y PFi ,θ
ML
Q &&base ≈
y
ML
(accQuasi − static )
2                2
⎡ ξ λ1 ⎤
3
&&base ≈ ⎢
y                ⎥ accQuasi − static
⎣ 4 σ1 ⎦

Base Acceleration (g's)
3%
.2

2%
.1
1%

.0
1               2             3
Quasi-Static Acceleration (g's)

10
Current Model Checks
Finite Element Model Checks
•Rigid Body Strain Energy
•Mass
•K6ROT insensitive
•6 Free-Free modes
•Element Quality Checks
•Max. Ratio
•Epsilon
•Unit g Cases
•…
Craig Bampton Matrix Checks
•Free-Free Frequencies after reduction
•Transient Response, Original versus C-B model

11
Test Case:
1st Lateral Mode Notching Results       New
Checks

Base Moment
Limit

New Process Checks
1) Structural Damping
2) Asymptotic Limits
3) Notch Depth

12
Load Prediction versus Requirement:
1st Lateral Mode Sine Test Notching
Acceleration at top due to sine notching ≈ Constant

Sine test : θ base = 0 , tip acceleration higher

13
Method Extension: Fixed-Fixed Beam

Improve prediction by constructing          Participation Factors
Fixed-Fixed beam C-B matrix using                     Reaction k , DOF
PFk , DOF =
constrained and normal modes
(− ω )[Modal Mass]
2
k
,

Limit     ∂ 3 φk (x )
Vk (x = 0 ) =       −EI
x →0         ∂ x3
Limit  ∂ 2 φk (x )
M k (x = 0) =      −EI
x →0        ∂ x2
Craig Bampton Mass Matrix Checks
Determinant 〉 0 , ( positive definite )
Participation Factors
Reaction k , DOF
⇒
(− ω )[Modal Mass]
2
k

14
Future Work

• Determine connection between primary structure load
curves and secondary structure load/MAC curves
•   Compare benchmark to system/subsystem level test data
• Use the model for variational loads prediction

15
Summary/Conclusion
Dynamic Analysis Process Evaluation Beam Benchmark
•    Developed a Beam Benchmark to Validated our C-B Reduction Process
•    Tested Process
•    Suggested New Model Checks
–   FEM Checks
• Structural Damping
• Asymptotic Limits
• Notch Depth
–   Craig Bampton Matrix Checks
• Determinant >0 (Positive Definite)
•                               Reaction k , DOF
Participation Factors =
(− ω )[Modal Mass]
2
k

•    Compared predict loads to requirement for a slender payload
•    Suggested Future Work

16

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