Beam Benchmarks for Dynamic Analysis Process Validation by uws18949

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									Beam Benchmarks for
Dynamic Analysis
Process Validation


Jim Olmstead
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
   Coupled Loads
   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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