error in shaft consistent mass and stiffness matrices?

[electricpete]

I am trying to derive the shaft element mass and stiffness matrices listed in:
1 – Adams = Rotating Equipment Vibration..
2 – RPIE = Rotordynamics Prediction in Engineering.

I believe Lagrange’s method suggests that we can find these from:
Mr*W = dT/dw_r
and
Kr*W = dV/dw_r
where Mr is the r’th row of M matrix and Kr is the r’th row of the K matrix and w_r is the r’th element of the W vector.

So the recipe is:
Determine displacement y(z) and x(z) in terms of coordinate vector W
Express V in terms of W and T in terms of W’ = d/dt(W)
Compute dV/dw_r for each r to determine K matrix
Compute dT/dw_r’ for each r to determine M matrix

Slide 1 attached has embedded pdf’s that accomplish the above recipe using Maple
Slide 2 shows the Maple results agree with the RPIE K matrix
Slide 3 shows the Maple results agree with the RPIE M matrix
Slide 4 shows the Maple results agree with the Adams K matrix
Slide 5 shows the Maple results do not agree with the Adams M matrix....

Specifically, the Adams’ M matrix has 0 in the following positions: (3,6), (3,7), (6,3), (7,3)

The Maple solution indicates these elements should not be zero.

I would suggest that common sense suggests these elements should not be zero. Specifically the x-z plane is given in rows/cols 1,4,5,8 and the y/z plane is given in columns 2,3,6,7. Since x-z and y-z are uncoupled and represent the same behavior (with possibble sign differences in slope components of coordinate vector), we expect similar elements in 1,4,5,8 positions as in 2,3,6,7 positions. All cells which have row and column both belonging to the x-z set {1,4,5,8} have non-zero entries. Why does the same not hold for the set of cells with rows and colums both belonging to the y-z set {2,3,6,7} ? (that set of cells has 4 zero-value cells.)

I think Adams made an error. But like the last thread, I am open to the possibility that I am missing something. Am I missing something?


=====================================
(2B)+(2B)' ?

 

[electricpete]

I would suggest that common sense suggests these elements should not be zero. Specifically the x-z plane is given in rows/cols 1,4,5,8 and the y/z plane is given in columns 2,3,6,7. Since x-z and y-z are uncoupled and represent the same behavior (with possibble sign differences in slope components of coordinate vector), we expect similar elements in 1,4,5,8 positions as in 2,3,6,7 positions. All cells which have row and column both belonging to the x-z set {1,4,5,8} have non-zero entries. Why does the same not hold for the set of cells with rows and colums both belonging to the y-z set {2,3,6,7} ? (that set of cells has 4 zero-value cells.)

Another way to say it:

For the Adams mass matrix shown on slide 5 RHS and shown in Eqn 2.48 here:

http://books.google.com/books?id=QW...&resnum=1&ved=0CBsQ6AEwAA#v=onepage&q&f=false

If we look only at the elements whose row and column indices both fall within {1,4,5,8} (these are the cells corresponding to the x,z plane), we see a single-plane beam stiffness matrix.

If we look only at the elements whose row and column indices both fall within {2,3,6,7} (these are the cells corresponding to the y,z plane), we see a single plane beam stiffness matrix WITH 4 ELEMENTS MISSING. Where did those 4 missing elements go?

=====================================
(2B)+(2B)' ?