A scheme is presented which allows the modeling of a moving boundary with NASTRAN NOLIN cards. Various aspects and limitations of the approach are explained. Recommendations are given as to the procedure to be used in implementing the method

Frye, J. W.

English

NASA Technical Reports Server

TRANSIENT ANALYSIS OF BODIES WITH
MOVING BOUNDARIES USING NASTRAN
John W. Frye
Naval Underwater Systems Center
SUMMARY
A scheme is presented which allows the modeling of a moving boundary with
STRAN NOLIN cards. Various aspects and limitations of the approach are
plained. Recommendations are given as to the procedure to be used in imple-
_ting the method.
!
1 INTRODUCTION
I
Situations occasionally arise when the boundary conditions of a theoretical
_el change as a result of the response of the model. The case under consider-
on is that where the boundaries of the model move as a function of rigid body
placement. The problem which motivated the development of the technique which
_are presenting is that of a shaft under both axial and transverse loading moving
two stationary bearings. Figure 1 is a diagram of such a shaft.
The reactions at the bearings are dependent on the applied loadings and the
_rtial forces generated by transverse motions of the shaft. The applied loadings
known beforehand. The inertial loadings of the shaft are unknown since the
_nsverse motion of the shaft is an unknown. Therefore, the reactions of the
_rings cannot be predetermined before the start of the analysis.
Constraints cannot be directly applied to the model to account for the
_rings since such constraints would be useful for only a short time of the
_lysis covering a small axial displacement of the shaft. It is possible to solve
problem by dividing the total time of the analysis up into many small time
:rements each with a different constraint condition. The idea here would be to
_rt the analysis with the bearings in one position; run the problem for a short
_e; stop the analysis; change the constraints; and restart the problem with
_tial conditions equal to the state of the model at the end of the first time
gment. This process is repeated until the shaft has moved as far as desired.
is technique, while being possible, is awkward from the standpoint of data
dling, the number of runs required, and the total number of days needed to
ain an answer.
, F2
SYMBOLS
distance between grid points
nonlinear loads at grid point accounting for moving boundary
377
https://ntrs.nasa.gov/search.jsp?R=19750023433 2020-03-19T21:21:56+00:00Z
Fx(t), Fy(t)
i
K
KB
S
Ux, Uy
x,y
applied time dependent axial and transverse loads
index number of grid point
maximum slope of nonlinear loading curve
stiffness of bearing
scale factor
grid point axial and transverse displacements
axial and transverse coordinates of model
integration variable in axial direction
DISCUSSION
The approach which accounts for the moving bearings treated the bearings
as stiff springs which moved along the model of the shaft as the shaft moved wi
respect to the bearings. These springs were fabricated artificially by the use
of the nonlinear load option in NASTRAN. Each grid point along the shaft model
over which the bearings could pass had two nonlinear loadings applied to it fc
each bearing. Figure 2 shows a diagram of these loadings for a typical grid pc
on the shaft model. For simplicity we consider the case of only one bearing.
The force F2 is shown as a positive ramp function with a slope equal to
bearing stiffness and is a function of the displacement ux of the shaft. The
second force Fl is shown as a negative ramp function with a slope equal to the
bearing stiffness and is a function of the shaft displacement ux added to the
transverse deflection of the grid point Uy. Note here that since Uy is severa
orders of magnitude smaller than ux, the magnitude of ux + Uy is mainly determ
by ux alone. Both Fl and F2 are zero until ux is greater than xi_ l, the distal
from the bearing at the start of motion to grid point i-l. Fl and F2 are equa
and opposite and, hence, negate each other when ux is greater than xi+ l, the
distance from the bearing at the start of the motion grid point i+l. Therefor
the loadings Fl and F2 add to zero at all times except those times when the
bearing is between grid points i-l and i+l.
Now consider a case when the bearing is between grid points i-l and i+l
Figure 3 illustrates such a condition.
NET LOAD = Fl + F2 = -K(ux + Uy - Xi_l) + K(u x - xi_ l)
NET LOAD = -KUy
378
s can be seen, the net load on the grid point i is equal to -KUy which is a
estoring force proportional to the bearing spring constant and the lateral
ieflection of the shaft at i. Thus, the combination of two nonlinear loadings at
results in simulating a bearing spring which turns itself on when the bearing
is in the vicinity of grid point i and turns itself off when the bearing is out-
;ide the vicinity of grid point i.
It is possible to allow the spring constant of the bearing to vary as a
Junction of its distance from a grid point. There is someadvantage to this
;ince when a bearing is between two grid points, its stiffness ought to be shared
in someway by the two grid points. See figure 4.
Wehave used the proportional sharing schemewhere the stiffness of the
fearing is distributed in proportion to that of a load shared between two ends of
i pinned ended beam. To establish the loading curve for this kind of stiffness
;haring scheme, notice that
/\
(1) dF: Kd_ :_KB)-_-d_ 0 < _ < a
KB(2a - _)
(2) dF = KdC = a d_ a < _ < 2a
;rom (1) noting F = 0 at _ = 0
(3) F =_oX IKB)_-_ = KB_2
2a
@F _ KBX BF - 0
_x a _x
x 2
0 = KBx
2a
O<x<a
atx=O
@___F_F: KB
@x
F = KBa
2
atx--a
atx=a
:rom (2) noting F = KBa at x = a
2
=_a x KB(2a - _) KBaF a d_+T
L_KB(2a _ _)2 KBaF = 2a + 2 - KBa -
a
KB(2a - x) 2
2a
379
(4) F = KB[a (2a- x)212a a < x < 2a
aF _ KB(2a - x) aF = KB at x : a
ax a @x
aF
-- = 0 at x = 2a
ax
F- KBa
2 atx=a
Equations (3) and (4) establish the shape of the loading functions for
the proportional stiffness loading scheme. Other stiffness sharing schemes
could be devised and their loading curves easily arrived at in a similar manner
to that shown here. Figure 4 shows a plot of the nonlinear load function for
the proportional stiffness sharing scheme we used.
The scheme depends completely on accurately determining the difference
between the quantities K(u x + Uy) and Kux. Theoretically, this presents no
difficulty. If the value of ux is six or seven orders of magnitude larger thar
Uy, a numerical difficulty arises since a computer may only work to eight signi
ficant figures using single precision arithmetic. Even if double precision
arithmetic is used, there will come a point where large values of ux start to
degrade the numerical accuracy of the differencing operation K(u x + Uy) - Kux.
There is a way around this numerical difficulty. The objective of the
differencing operation is to obtain the quantity KUy. If the ux quantity is
multiplied by a scalar constant the differencing operation remains unchanged.
K(su x + Uy) - Ksu x = KUy
Here u was multiplied by the scalar s. To remove the numerical difficulty,
the scalar constant is assigned such that ux is no more than three or four
orders of magnitude greater than values of Uy which can possibly be of interes
su x cannot be too small or else the larger values of Uy will start to affect
the first few significant figures of the quantity su x + Uy. This would result
in inaccuracies in the determination of the spring constant since the spring
constant should be a function of ux only and not su x + Uy. Thus, some judgmen
must be used in the determination of s; that is to say it cannot be made some
very small number without considering the values which Uy are expected to be i
the neighborhood of a bearing.
38o
I Whenusing the scalar multiplier the nonlinear loading diagram changesch that the nonlin ar lo ds start to act at sxi_ l rather than at si_ l, since
e lock up function or independent variable is sux rather than u . Figure 5x
_ows the forcing function set up using the scalar multiplier.
The equation for the nonlinear load as a function of sux or sux + Uy nowecomes
K_s_dF -K O<_<a
ds_ = _
dF = sK+ dE
2
SKBX
F= 2a O<x<a
I F : sKB a (2a2_ x)2
lifferentiating with respect to sx yields
a<x<2a
dF d
dsx dsx KB s2x2))_ 2KBSX KBX-- O<x<a
dF dKB [ (2as 2_ )2 ]sx - sx sa - a sx
: KB [2 (2as - sx) ] = KB'(2a -x)2sa a
a<x<2a
The slope of the nonlinear force curve with respect to the scaled
coordinate remains the same as the slope of the force curve using an unscaled
Coordinate.
Figure 6 shows a typical result for the transverse displacement of a grid
_oint in the analysis of a shaft moving between two bearings. The results show
the expected suppression of transverse displacement while the bearings are in the
,vicinity of the grid point.
The application of nonlinear loads raises questions as to the numerical
Istability of the solution. The numerical stability will depend on the time step
used and the maximum slope of the nonlinear force curves. If the moving boundary
is a rigid one, some compromise must be made on the stiffness of the moving
boundary. An infinite stiffness will always result in an unstable condition.
In our case, we ran several short problems with our model using what we considered
as a reasonable time step. We reduced the slope of the nonlinear load curves
381
until we obtained a stable solution and then reduced the time step until the
solution appeared reasonably converged. This trial and error approach may not
always work out, but it did in our case. We were able to arrive at an acceptabli
time step and loading slope within a few days. It is best when setting up the
tables for the loading curves to set the maximum slope in the table to unity and
to use the NOLIN cards to scale the slope. Thus, the maximum slope values will
appear directly on the NOLIN cards. The tables are tedious to construct as ther(
is a table for each grid point on the model over which the moving boundary must
pass, whereas it is relatively easy to change the slope values on the NOLIN cards!
If the time steps are large, the slope of the nonlinear load curve must b_
very small in order to give good convergence and numerical stability. In such a
case, the moving boundary cannot be accurately modeled using this method. This
is unfortunate but as far as we can see, there is no alternative for solving
the problems of convergence and stability other than to reduce the time step
or the loading slope.
382
Zo_
0
0
I--
I.l_
-r-
!
I..1_
z
l,l
0
l,l
l---
l,l
--J
0
l--
i,
Q_O
0
--J
383
i-l i+l
xi-I
xi+ 1
t Fy(t)
Fl(Ux+Uy)
(Ux+Uy) xi_ l xi+ l
I
F2(u x)
U
X
!
xi -l Xi+l
FIGURE2. - NONLINEARLOADINGDIAGRAMS
384
i+l
K(Ux+Uy-Xi_l )
F1(Ux+Uy)
Ux+Uy xi-I xi_1
' I
F2(u x)
K(Ux-Xi_ l)
U
X
i K
Xi-I xi+l
Net Load = FI+F 2 = -KS(Ux+Uy-Xi_ l) +KS(Ux-Xi_ l) = -KSUy
FIGURE3, - NOi_LINEARLOADSWITHBEARINGNEAR
GRID POINT
38_
i_lXWA
• @ •
g
i+l
= K(a-x)
Ki-l a ; Ki = Kx 0 < x < a
Ki = K(2a-X)a ; Ki+l = K(x-a)a a < x < 2a
1.0
K=I.O
F_(__ .5
a
0
0 .5a a 1.5a 2a
FIGURE4, - PROPORTIONALSTIFFNESSSHARINGSCHEME
386
Fl Fy(t)
I m _ Fx(t)
xi-IE
I- Xi+l •
i
1(SUx+Uy)
(sux+Uy )
sxi_ 1 sxi+ l
k I
F2(su x)
SU
X
| !
sxi_ l sxi+ l
FIGURE5, - ADDITIONOF SCALEFACTORs TO ELIMINATE
ROUNDOFF ERROR
387
Xo
I
o
i.o
| !
o _
o
i i
w_ 'fn 'V _U_0d _e _ua_a0eLds_o _eqs Leaa_e3
O0
_'0
W
oO ILl
ILl
O0 ,=_
.-_
ILl 0
I.--
__1
(-,) 0
Q-
I-- 0
I
388


Transient analysis of bodies with moving boundaries using NASTRAN

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server