Lanczos mode models represent low-frequency forced response better than do normal mode models and can be developed for both continuous and finite element structural representations. It was recommended that Lanczos mode models for systems with multiple input and/or rigid body modes should be developed; numerical stability of the Lanczos algorithm should be assessed; and control system designs employing the Lanczos mode models should be attempted

Craig, Roy R., Jr.

Turner, Russell M.

English

NASA Technical Reports Server

N87- 22739 !
LANCZOS MODES FOR REDUCED-ORDER
CONTROL OF FLEXIBLE STRUCTURES
by
Roy R. Craig, Jr.
Russell M. Turner
ASE-EM Department
The University of Texas at Austin
999
https://ntrs.nasa.gov/search.jsp?R=19870013306 2020-03-20T10:32:36+00:00Z
LANCZOS ALGORITHM - FE VERSION
Yj = K-1Mqj
rj =Yj - o_jqj - _jqj-1
_j = qjTMfj
l_j = (rj_lTMrj_l) 1/2
qj+ = (1/13j+0ri
13j+1 = (rjTMrj)I/2
The algorithm for recursively forming Lanczos vector qj+l from Lanczos vector qj
for the case of nonsingular K is shown. It involves an orthogonalization step and a nor-
malization step.
STARTING VECTOR
F= be(t)
ro = K-lb
ql = (1/_l)ro
131 = (roTMro)1/2
Lanczos modes are most useful when the spatial distribution of the excitation is
constant. This load distribution then determines the starting vector.
1000
LANCZOS EQUATION FORMAT
MiJ + Ku = F
MK-1Mi_ + Mu = MK-1F
u - Qm z
Qm = [qlq2""qm]
QTMK-1MQm_, + QTMQmZ = QmTMK-1F
Lanczos vectors are used in a mode-superposition manner which is not exactly the
same as the famqiar Rayleigh-Ritz version of mode-superposition. Multiplication by the
equation of MK- "smooths" the loading.
LANCZOS EQUATION FORMAT - CONT.
TroT. + z =gm
gm= Qm TMK-1F
Tm --
tXl _2
_m O_m
The final Lanczos equations have a tridiagonal generalized mass matrix and have a
unit matrix for the generalized stiffness matrix. The form of the generalized force vector
gm is very special.
i001
EXAMPLES
• Lanczos Modes and Equations
• Comparison of Normal Mode Models and
Lanczos Mode Models
+ Poles and Zeros
+ Frequency Response Functions
+ Transient Response
[< L=l __
FINITE ELEblENT blODEL
--_1--_ u[_ )
I _ _1 _c_.._
_ k _
CONTINUOUS HODEL
1002
FE LANCZOS MODE MODEL
T6 =
Tm_ + z = gm
"0.39770 0.04361 0.0
0.04361 0.04234 0.01238
0.0 0.01238 0.01564
0.0 0.0 0.00516
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
0.00516 0.0 0.0
0.00830 0.00221 0.0
0.00221 0.00505 0.00083
0.0 0.00083 0.00320
o.57735.
o:1i:i
•fft_
oo U.U
0.0
0.0
Note that there is tridiagonal inertia coupling of the Lanczos generalized coordi-
nates, but note that the only Lanczos coordinate directly excited by the external force is
the first coordinate.
1003
oo RUN - MQ31111 o RUN - MQ31111
,4 1ST NORMAL VECTOR ,.; 2ND NORMAL VECTOR
8
I--
Z
,<--
ag
o
6"
O0 1. O0 2. O0 3. O0 4. O0 5. O0 S. O0
LOCATION
g
laJ
"So
a_.
T"
0 :3. O0 4. O0 S. O0 IL O0
LOCATION
o RUN - M031111
i IST LANCZOS VECTOR
i,,-t °
a_
-0, O0 t. O0 2. O0 3. OO 4.00 5. O0 II • O0
LOCAT]OH
RUN - M031111
2ND LANCZO$ VECTOR
/
Typical normal modes and Lanczos vectors are shown here for axial deformation
of a clamped-free rod. The starting Lanczos vector is based on a single force applied at
the "free" end. Although finite element normal modes do have some stain near the free
end, "exact" normal modes would all be strain-free at the end where the excitation force
is applied.
1004
TRANSFER FUNCTIONS IN MODAL COORDINATES
s2+2.48152 0 0 0
0 s2+23.36993 0 0
0 0 s2+70.87551 0
0 0 0 s2+156.16108
A
rl(s)
.m
1.42231
-1.48875
• .v_J vv
-1.85632
1005
TRANSFER FUNCTIONS - NORMAL MODE MODELS
t_t(s)/f(s) = 10.34151 [s2+3.28270) 2][s2+(6.82929) 2]
[s2+( 10.94259)2]/A4
An(s ) = [s2+(1.57528)2][s2+(4.83424) 2]
[s2+(8.41876) 2] Is2+(12.49644) 2]
Ol(s) If(s) = 4.23934[s2+(3.52834)2]/A2(s)
A2(s ) = [s2+(1.57528)2][s2+(4.83424) 2]
fit (s)/f(s) = 2.02296/[s2+(1.57528) 2]
TRANSFER FUNCTIONS - LANCZOS MODE MODELS
fil (s)/t'(s) = 18.29893[s2+(3.17763)2][s2+(6.59014)21
[s2+(10.97092)E]/A4(s)
A4(s ) = [$2+( 1.575285)2][s2+(4.83427) 2]
[s2+(8.48668) 2] [$2+( 15.21009) 2]
t] l (s)/l'(s) = 7.14344[s2+(3.06112)2]/A2(s)
A2(s) = [s2+(1.57529) 2] [s2+(5.19414) 2]
ill(s)/f(s) = 2.51446 / [s2+(1.5857) 2]
1006
POLES AND ZEROS OF REDUCED-ORDER MODELS
EXACT 4DOF 4DOF 2DOF
POLES NORMAL LANCZOS NORMAL
(6DOF)
1.57528 1.57528 1.57529 1.57528
4.83424 4.83424 4.83427 4.83424
8.41876 8.41876 8.48668
12.49644 12.49644 15.21009
2DOF
LANCZOS
1.57529
5.19414
TABLE 2. POLES OF REDUCED-ORDER MODELS
EXACT
POLES
(6DOF)
4DOF 4DOF 2DOF 2I)OF
NORMAL LANCZOS NORMAL LANCZOS
3.17759
6.57266
10.39230
14.69693
18.85315
3.28270 3.17763
6.82929 6.59014
10.94259 10.97092
3.52834 3.06112
TABLE 3. ZEROS OF REDUCED-ORDER MODELS
STATIC RESPONSE COMPARISON
Normal Lanczos No. of
Mode Mode Modes
Model Model
0.96961 0.99951 4
0.91005 0.99980 2
0.81521 1.00000 1
1.00000 1.00000 6(EXACT)
TABLE 1. TRANSFER FUNCTIONS EVALUATED AT s--0
These two tables compare poles and zeros of normal mode models and Lanczos
vector models and the static response of normal mode and Lanczos vector models.
1007
--S-'l-l I! ........... I ..... -1 .... l---'l+'r'+l+l--I ............ I ........ I .... I' " !
T
00E+03 _+- .... _ '__I.
E
I
FP
...... _............ _,_ .............. _--- -
i J 'I .L_ /
E + \]k
F tb
1. OOE-O 1L-_J_ J_t J.. J....... J___J____.t_ LJ...L_
5.00E-02 FREOUENCY (HZ) 5.00E+UO
A2: FRF (2 NORMAL) + + + FREORESP-BODE
A3:FRF (B NORMAL) -- IY+ ]Y+ # 3
A4:FRF (4 NORMAL) o o o 092485-001408
092685-152B24
1.00E+03_
i i i i
F
R-
F
I I i I i i i _ ;
................L++ zr +T-:
-m
o0
0
0 7
5. OOE+O0
+
'÷
) +
,+
em_
! ! I• I
FREOUENCY (HZ)
1. OOE-01
5. OD'E'D'2
At, FRF (6 LANCZOS) --- FREORESP-BODE
A3:FRF (4 LANCZOS) o o o IY+ I Y+ # 4
A4=FRF (2 LANCZOS) + + + 092485-002412
021786-215523
Direct frequency response functions (tip displacement/tip force) are shown for nor-
mal mode models and Lanczos vector models. The Lanczos models have improved
low-frequency performance.
1008
clCt)
r
w _ _ _ w
/
t r = T 1
Control maneuvers, such as slewing, are transient response problems. The next
examples compare transient response solutions of normal mode an Lanczos vector
models when the structure is subjected to step and ramp excitation. Two force distribu-
tions are illustrated.
1009
RUN - UDl1112 _ RUN - WSl1112
oo
o
o
O. O0 I. O0 2. O0 3. O0 4. O0 S. O0 6. O0
LOCATION
I-
(/1
g f ...... *
.... t
oo _:oo :_oo _'.oo 4:o, 5'.oo
L OCAT T ON
,_ oo
RUN - LDl1112
(n
g
e,,P
)..
01
RUN - LSl1112
g
t,_................................. p ......
o
o
O. O0 1.O---'--"r--O - 2'. O0 3_ O0 4. O0 S, O0 6. O0 °O I. O0 !'. O0 2'. O0 3'. O0 4" O0 5'. O0 *'. O0
LOCATION LOCATION
These are modal and Lanczos solutions for a step force applied at the tip. Both
displacement and strain solutions are shown. Note that the Lanczos solutions converge a
little better than the normal mode solutions do.
i010
RUN - MDl1323
g
,4
I--
"7
1"4 o
QO
'+,_o I_oo 2"00 "-_Loe-" _ "_._'---_ .... ,:oo
LOCATION
o
RUN - M$11323
g_
o,
o
Z o
.o. ....
o
i
I
i
...... .A
g
_! 0,
.... 1
I
I
I
......
I ,"......
[..... I
I"oo 2'.oo 3.00 4.00 s.oo e_oo
L OCAT Z ON
g RUN - LD11323
g
.o
,4.
g
I-
2:
+oloo,oo ,oo(ocI:00o.
,J
4"oo 5"oo ,_oo
o
g
o
o
r,v.o
F.,-
(/I
o
RUN - LSll323
c_'
oo
_0_00 I'.oo 2.oo 3.oo 4.oo s'.oo ,,'.oo
LOCAT I ON
These are modal and Lanczos solutions for opposing forces applied at nodes 1 and
3 with a ramp time history. Note that the modal solutions are very poor, while the Lanc-
zos solutions show excellent displacement and strain convergence.
1011
CONCLUSIONS AND RECOMMENDATIONS
Lanczos mode models represent low-
frequency forced response better than do nor-
mal mode models.
Lanczos mode models can be developed for
both continuous and finite element structural
representations.
Lanczos mode models for systems with multi-
ple inputs and/or rigid body modes should be
developed.
• Numerical stability of the Lanczos algorithm
should be assessed.
• Control system designs employing Lanczos
mode models should be attempted.
1012


Lanczos modes for reduced-order control of flexible structures

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