A Method for Calculating Accurate PSD Reaction Loads

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A Method for Calculating Accurate PSD Reaction
Loads
Alex Grishin Senior Consulting Engineer Phoenix
Analysis Design Technologies
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About PADT

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Problem Statement

• PADT received a technical support call
• When I do an FSUM on my PSD results my results
  are ridiculously high. This makes no sense.
  ANSYS gives bad results. Fix it
• We looked into it
• ANSYS was summing the forces correctly
• Problem is that all forces are positive
• Sum all positive values, you get a high value
• Very typical issue with post-processing PSD
  results
• Needed to find an accurate way to get reaction
  loads by keeping sign while calculating

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Simple Example Shows Issue

• Make a beam model
• Solve one mode
• Do PSD with one mode
• Look at displacement and reaction loads

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Simple Example Shows Issue
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Simple Example Shows Issue
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Simple Example Shows Issue

• For first mode (Set, 1,1), you see expected
  up/down reaction force to remove moment

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Simple Example Shows Issue

• For PSD results, reactions are positive
• FSUM Results

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Solution 1 Scale Mode by Mode Coefficient

• There is a suggestion in the tech support
  (Knowledge Base) database
• Multiply Dominant Mode From PSD Analysis By Mode
  Coefficient
• Very easy to do
• get,mc1,mode,1,mcoef
• mc1 sqrt(mc1)
• set,1,1,mc1
• cmsel,s,nbase
• fsum,,
• For beam example Fsum(Y) 0.00757
• Great if one mode dominates
• Use shape functions to see if this is true

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Solution 2 Miles Equation

• There is a hand calc for single mode excitation
• Called Miles Equation

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Solution 2 Miles Equation
http//femci.gsfc.nasa.gov/random/MilesEqn.html
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Solution 2 Miles Equation

• Base Reaction Force Estimate Based On Miles
  Equation
• For our beam example
• damping ratio .05
• Grms sqrt(pi/2f1QA)
• Grms sqrt(3.14159/236.6910.0623)
• Grms 5.992g
• Fbase MeffyGrms386.4
• Fbase 3.29e-65.992386.4
• Fbase 0.0076 lbf

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Solution 3 Reaction Force Covariance

• Perform Reaction force calculation with full
  Nodal reaction force covariance matrix to retain
  sign
• Valid for ANY number of nodes (DOFs)
• Works for multiple dominant modes
• The tech support call was just such a case
• Implemented in psdfsum.mac

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Solution 3 Reaction Force
Covariance

• Use PSDFSUM Only On base nodes
• If you are looking at the base nodes of
  canilevered, or long slender members, only the
  diagonals of the covariance matrix are needed
• Diagonal terms are the mode coefficients
• You can only get the diagonal terms
• If you want non-base nodes for any arbitrary
  system you need
• Full modal coordinate covariance matrix
• PSDFSUM macro may be modified to do this

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Solution 3 Reaction Force
Covariance

• The Math

The mean square reaction force at any DOF, u may
be expressed as
(6)
where is the variance of the reaction force
at any DOF u, and are the eigenforces
corresponding to this DOF. C is the modal
covariance covariance matrix, give by
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Solution 3 Reaction Force
Covariance
If, however, one wants a nodal component
reaction, using FSUM for example, equation (6)
becomes
(7)
Which IS WRONG!!
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Solution 3 Reaction Force
Covariance
Equation (7) will always yield a gross
overestimation of the mean square force reaction
because it sums mean square values without regard
to their sign! To remedy this, first rewrite
equation (7) as
(8)
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Solution 3 Reaction Force
Covariance
And then note that an effective fsum
with restored relative signs may be retrieved by
selecting an arbitary row of equation (8) and
dividing by the square root of its diagonal
member!
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Solution 3 Reaction Force
Covariance
And for a given a given spatial direction, fsum
may be calculated according to
Resulting in a reliable value for the RMS
reaction force of the nodal component. Note that
the terms in equation (9) retain their sign
RELATIVE TO THE DIAGONAL MEMBER OF THAT ROW.
This reminds us that we really dont know the
absolute sign of an RMS quantity
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Solution 3 The PSDFSUM Macro

• /post1
• !Select the component that has the nodes to get
  the reaction loads for
• cmsel,s,arg2
• ! Get the number of nodes
• get,nn,node,,count
• !Create the array to hold the force data for a
  mode forces(i,j,k)
• !j 1 will hold node number
• !j 2-7 hold the 6 component forces/moments
• !
• dim,forces,,nn,7,arg1
• !Create the array to hold the forces as you sum
  across modes
• dim,ff,,nn,7
• !Create the pivot array
• dim,pivot,,7
• !Create the array to hold the model coeficients

• !Make sure you are in CSYS,0 and all elements are
  selected
• csys,0
• esel,all
• !Place the node number in the num,1,1 position of
  forces()
• ind0
• do,i,1,nn
• forces(i,1,1)ndnext(ind)
• indforces(i,1,1)
• enddo
• ! Loop on Modes, getting forces for each
  specified node at each mode
• do,j,1,arg1
• set,1,j
• do,i,1,nn
• nsel,s,node,,forces(i,1,1)
• fsum,rsys,all
• get,forces(i,2,j),fsum,0,item,fx
• get,forces(i,3,j),fsum,0,item,fy

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Solution 3 The PSDFSUM Macro

• !Create a pivot vector telling us where there is
  a non-zero force
• ! (avoids a devide by zero)
• do,j,2,7
• pp 1
• do,i,1,nn
• if,forces(i,j,1),ne,0,then
• pivot(j) i
• exit
• else
• pivot(j) i
• endif
• enddo
• enddo
• !Loop on modes, loop on nodes
• ! Sum forces as you go
• do,j,1,arg1
• do,i,1,nn
• ff(i,2) ff(i,2) forces(pivot(2),2,j)for
  ces(i,2,j)mc(j)

• ! zero summed values (just in case)
• ffx 0
• ffy 0
• ffz 0
• mmx 0
• mmy 0
• mmz 0
• ! To get toal forces, loop on each node
• ! Summing the calculated force over the modes
  devided by the sqrt of the pivot value
• do,i,1,nn
• if,ff(pivot(2),2),ne,0,then
• ffx ffx ff(i,2)/sqrt(ff(pivot(2),2))
• endif
• if,ff(pivot(3),3),ne,0,then
• ffy ffy ff(i,3)/sqrt(ff(pivot(3),3))
• endif
• if,ff(pivot(4),4),ne,0,then
• ffz ffz ff(i,4)/sqrt(ff(pivot(4),4))

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Solution 3 The PSDFSUM Macro

• !Calculat absolute values of sum
• ffx abs(ffx)
• ffy abs(ffy)
• ffz abs(ffz)
• mmx abs(mmx)
• mmy abs(mmy)
• mmz abs(mmz)
• !Display
• /uis,msgpop,1
• msg,note,ffx,ffy,ffz,mmx,mmy,mmz
• fx g /
• fy g /
• fz g /
• mx g /
• my g /
• mz g
• allsel,

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Solution 3 Sum Over Modes

• Results for Beam Model
• .00757 lbf

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Summary for Simple Test Case

• Same answer for all three methods
• Verified for dominant mode Case

PSDFSUM