why x-y cross-coupling in shaft consistent-M and consistent-K matrices

[electricpete]

Refer to Maurice Adams' "Rotating Machinery Vibration..."
Definition for shaft consistent mass matrix (2.48) and shaft stifness matrix (2.51)

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

With [q''] = [x1'' y1'' [θ]x1'' [θ]y1'' x2'' y2'' [θ]x2'' [θ]y2''] the shaft element consistent mass matrix thus obtained is as follows.....

Mic = Mis/420*[156 0 0 22*L....] [Equation 2.48]

[electricpete's note: coordinate 1 is left end of shaft finite element and coordinate 2 is right end of shaft finite element. x and y are the two radial directions. [θ]x and [θ]y are slopes (d/dz where z is axial) of displacement in these two radial directions.]

I bolded the non-zero entry 22*L in the (row,col) = (1,4) position because using the stated order of coordinates ([x1 y1 [θ]x1 [θ]y1 x2 y2 [θ]x2 [θ]y2]), this would seem to represent a force generated in the x direction as a result of slope in the y direction.

At first glance, this seems non-physical. There is nothing modeled here that should introduce cross-coupling between x and y. Note that gryoscopic effects are treated separately. Also note that similar apparent cross-coupling terms appear in the K matrix shown in 2.51 on the next page.

QUESTION 1 - Am I missing something? Are these cross-coupling terms correct for the stated ordering of variables?

Now an added complication. If we compare to "Rotordynamics Prediction in Engineering", adjusting their notation sligthly, we have a different definition for the state coordinate:
delta = ([x1 y1 [θ]y1 [θ]x1 x2 y2 [θ]y2 [θ]x2]

Note that he has reversed the ordering of [θ]x,[θ]y in the coordinate to a less natural order of [θ]y, [θ]y. I think it might have been done because it keeps the matrices in a form where that is easier to solve.

In Rotordynamics Prediction in Engineering's shaft M (classical) and K matrices, the non-zero terms appear in the same position (for example row 1, column 4) as in Adams, and in the case of the Rotordynamics Prediction in Engineering coordinates, they would not represent cross-coupling between x and y directions (the 1,4 position couples x to thetax).

QUESTION 2 - Perhaps Adams used the same coordinate ordering as Rotordynamics Prediction in Engineering , but simply listed his ordering wrong? (after all we only look at displacement results and don't use the slopes, so it is an easily overlooked error)

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

 

[electricpete]

Mystery solved (I think).

Adams uses a (seemingly) oddball definition for thetaX and thetaY.
ThetaX = -1* slope of shaft in the Y-Z plane
ThetaY = +1* slope of shaft in the X-Z plane


This is shown graphically in another context in Fig 2.5 on page 43 here:

http://books.google.com/books?id=QW...an 8-DOF system, with all generalized&f=false

also it might possibly be inferred (if you squint) in the present context of shaft mass and stiffness matrices from the table on page 60 here:

http://books.google.com/books?id=kJ...abulated sets of boundary conditions"&f=false

Maybe this definition has some logic when considering angular momentum and gryoscopic type effects ? ... the logic is not immediately evident to me.

Rotordynamics Prediction in Engineering at least follows a convention that the post-script to the angle describes the plane in which it lies. They accomplish a similar thing to Adams not by adopting an oddball (swapped) definition of ThetaX and ThetaY, but instead by swapping the order that these angles are listed in the coordinate vector. They also define their angles such that one is a positive slope and one is a negative slope (another oddball choice imo, but common to both books). The end result is that the non-zero elements end up in the same location of the matrix and have the same magnitudes, but there are some sign differences.


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