-
Modeling of Joints and Interfaces forSimulation and Design of
Structural Systems
K. C. Park, Carlos A. Felippa, Y. Xue, G. ReichCenter for
Aerospace Structures andDepartment of Aerospace Engineering
SciencesUniversity of Colorado, Boulder, CO 80309-0429email:
[email protected]: 303-492-6330/fax: 303-492-4990
Presented at Workshop on Modeling of Structural Systems with
Jointed Interfaces, 25-26 April 2000, Albuquerque, NM
-
Ideal vs. Realistic Joint
(a) Assembled State
uB
uA
uB uAuA
Joint ( d)
uA uB 6D 0; dA dB 6D 0uA dA D 0; uB dB D 0
dA dBdB
Ideal joint : uA uB D 0(b)
Realistic joint:(c)
Two substructures fastened by a bolt. (a) Asembled state; (b)
displacements are always in contact; (c) Realistic joint whose iin
contact, thus creating rocking motions. One approach to modelto
introduce filler or joint elements that undergo nonlinear bthe
uneven rocking motions plus friction.
-
DIFFICULTIES IN JOINT MODELING
Exclusivity in Model Selection
Non-Scalability of Experimental Results
One-Dimensionality
Ambiguous and Uncertain Sources of Nonlinearities
Non-Smoothness
Stiffness Mismatch and Soft Materials
-
Exclusivity in Model Selection:
The structural modeler may find a limited number of models
available in aprogram. As a result, considerable ingenuity may be
required to obtainreasonable results. Sometimes the modeler may
eventually find, afterexhaustive simulation studies, that none of
the available models is adequatefor the application on hand.
Non-Scalability:
A joint model may have been developed in conjunction with a
specificexperimental validation setup. The scalability of a joint
model for other lengthscales, varying loading levels and forcing
frequency ranges is often notdocumented or not understood.
-
One-Dimensionality:
A large class of existing joint models, notably for damping and
friction, havebeen derived for one dimensional scenarios. Their
validity for multidimensionalmotions is open to question.
Ambiguous and Uncertain Sources of Nonlinearities:
Many existing joint models have been developed using
phenomenologicalrepresentations or experiments. As a consequence,
the physical source ofnonlinearities, for example friction and
slip, may be masked or poorlyunderstood. In conjunction with
non-scalability this can lead to erroneouspredictions.
-
Non-Smoothness:
Incorporation of non-smooth joint behavior poses challenges in
that tangentsurfaces are difficult to obtain objectively, or may
not exist. In practice, the non-smooth joint models are available
only in vectorial forms. Hence, a consistentlinearization procedure
must be developed for use in tangent-stiffness methods ofdynamic
response.
Stiffness Mismatch:
Bolted and welded joints may have high intrinsic stiffness (as
3D bodies)compared to attached lightweight structural components,
such as beam profilesor thin-wall plates. This makes the energy and
interface force transmissiondifficult to model in so far as
capturing the dominant physical behavior. Robustregularization
methods must be developed to alleviate these problems.
-
Soft Materials:
In many complex structural systems, particularly aerospace,
nonstructural materials(foams, polymers, etc) are used as
vibration/shock absorbers or as impact attenuators.While these
components, strictly speaking, are not joints, their modeling
presents a newand emerging challenge since their roles are becoming
increasingly important. Forexample, electronic packages are often
tied to load-carrying substructures through acombination of
fasteners and foam-like padding.
-
JOINT MODELING APPROACHES Hierarchically Modulated Joint
Models
Exclusivity in Model Selection Non-Scalability of Experimental
Results One-Dimensionality
Matrix-free Interfacing with Joints Non-SmoothnessVectorial
Representation of Ambiguous Sources of Nonlinearities
Interface Regularization and Localization Stiffness Mismatch and
Soft Materials
-
Model construction
Modelassessment
Overall structuralmodel fidelity evaluation
Stochastic & uncertainty
parametrization
Joint models by hierarchically
modulated characterization
Stochastic characterization ofjoints and interface
conditions
Validation through simulation & experimentalcorrelations
Development of joint constitutive,
phenomenological &analytical models
Interfacingjoint models by
localized Lagrange multipliers method
Feedback from model updates
Roadmap of Joint Modeling
-
Identiflcation of Joint Flexibility -
24 BT FB Rb LbRTb 0 0LTb 0 0
358>>>:
fi
ub
9>>=>>; D8>:
BT Ff0RT f0
0
9>=>;F D
24 F21 F22F23
35 ; F21 D '21121 'T21; etc.LF2LT D F FBLFCb BTL F C PT R.RbFCb
R/1RT P
P D I BLFCb BTL FFb D BTL FBLRb D BTL R; BL D null.LS/
22Finally, obtain the flexibility F of the isolated joint
Continued
Generalized Riccati equation
Center for Aerospace Structures
-
(a)
(b)
(c) S1 S3S2
Center for Aerospace Structures
Partition of example structure into three substructures: (a)
schematics of a stepped bonded joint (b) finite element model in
the vicinity of the joint (c) model for experimental
correlationNote: measurements are typically made along the
substructural boundaries between S1 and S2, and between S2 and S3;
no sensors are collocated directly at the joint boundary
-
(a)
(b)
Substructure S2
S22
S21
S23
Center for Aerospace Structures
Further partition of example structure containing stepped joint
into three subdomains: (a) Analytically and experimentally verified
substructure model S2 (b) Further partition of S2 to model S22 by
the Partitioned Direct Flexibility Method
-
(a)
(b)
Center for Aerospace Structures
Derivation of stochastic boundary flexibilty of glue lap
joint(a) Background FEM model to get stochastic free-free
stiffness(b) Reduction to free-free boundary flexibility
-
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
(b)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
(a)
+
Hierarchical Decomposition of Joint Models
Nonlinear Behavior
Linear part Next hierarchical level
-
Joint Model Scales
P
h
y
s
i
c
a
l
P
h
e
n
o
m
e
n
o
n
S
c
a
l
e
s
equivalent linearized model via structural
identification
micromechanics
nonlocal strain theory
continuum frictionmaterials damping
Hierarchical Model Candidate
-
xA0 dA0 D 0
xB0 dB0D 0
A
B
xA0
xB0
dA0
dB0
A0
A0
B0
B0KB
KA
KJ
AJ
BJ
J
x
y
Modeling Joint via Global or Tightly Coupled Lagrange
Multipliers
-
xA 0 uA 0 D 0
dB0 uB0 D 0
u
u
dA0 uA0A0
D 0
xB0 uB0 D 0
A
B
xA0
xB0
B0
dA0
dB0
A0
B0
KB
KA
KJ
JA
A
JB
B
J
x
y
A0frame
B0frame
Modeling Joint via Localized Lagrange Multipliers
-
Localized Model Equations:
26664PX 0 BX 0 00 QJ 0
BJ 0BTX 0 0 0 LX0 B
TJ 0 0 LJ
0 0 LTX LTJ 0
377758>>>:
xdXJu
9>>=>>;=8>>>:
fXfJ000
9>>=>>;x =
xAxB
; X =
AB
; J =
JAJB
; u =
uAuB
BX = BXDX ; BJ = BJDJ ; BX =
BA 00 BB
; BJ =
BJA 0
0 BJB
Suppose that an existing model is found to be inadequate in
lightof new constitutive laws, new interface mechanisms,
inadequacyfor frequency range under study, new experimental data,
or somecombination of these factors.
-
To incorporate new model features without overwriting the
ex-isting model, the fleld variables are assumed to consist of
theoriginal ones that are designated with subscript 0 plus
additionalcontributions containing subscript 1, in the form:
x = x0 + x1; d = d0 + d1; = 0 + 1
so that PX and QJ can be expressed as
PX (x0 + x1) = P00x0 + P01x1 + P10x0 + P11x1QJ (d0 + d1) = Q00d0
+ Q01d1 + Q10d0 + Q11d1
-
The preceding operators are called hierarchically modulated if
theysatisfy the conditions:
xTk Pijxj = 0; i 6= k
Hierarchically Modulated Level-1 Equation:
266664P11 0 B11 0 00 Q11 0 B11 0
BT11 0 0 0 L110 BT11 0 0 L110 0 LT11 LT11 0
3777758>>>>>>>:
x1d1X1J1u1
9>>>>=>>>>;=8>>>>>>>:
f1fJ1000
9>>>>=>>>>;which indicates that if new
model scales can be made to complywith the modulation criterion,
then signiflcant streamlining inimplementing new joint models can
be achieved. This is becausewe only need to build P11 and Q11,
etc., while preserving theexisting model scales.
Note that not all new models and/or reflnements can be
expectedto satisfy the hierarchical modulation criterion.
-
References[1] G. Adomian, Stochastic Systems , Academic Press,
New York, 1983.[2] K. F. Alvin and K. C. Park, A second- order
structural identification procedure via system theory- based
realization, AIAA J. , 32( 2), 1994, 397 406, 1994.[3] K. F. Alvin,
K. C. Park, and L. D. Peterson, L. D., Extraction of undamped
normal modes and full modal damping matrix from complex modal
parameters,AIAA J. , 35( 7), 1187 1194, 1997.[4] K. F. Alvin,
Finite element model update via Bayesian estimation and
minimizationm of dynamic residuals, Proc. International Modal
Analysis Conference ,Ann Arbor, Mich., 561 567, 1996.[5] K. F.
Alvin and K. C. Park, Extraction of substructural flexibility from
global frequencies and modal shapes, AIAA J. , 37( 11), 1444 1451,
1999.[6] A. C. Aubert, E. F. Crawley, and K. J. ODonnell,
Measurement of the dynamic properties of joints in flexible space
structu res, MIT SSL Report No. 35- 83 ,Sept. 1983.[7] C. F. Bears,
Damping in structural joints, Shock and Vibration Digest , 11( 9),
1979, 35 44, 1979.[8] T. M. Cameron, L. Jordan and M. E. M. El-
sayed, Sensitivity of structural joint stiffness with respect to
beam properties, Computers & Structures , 63( 6),1037 41,
1997.[9] L. B. Crema et al, Damping effects in joints and
experimental tests on rivetted specimens, AGARD Conference
Proceedings , No. 277, April 1979.[10] M. P. Dolbey and R. Bell,
The contact stiffness of joints at low apparent interface
pressures, Ann. CIRP , XVIV 9, 67 79, 1971.[11] C. A. Felippa and
K. C. Park, A direct flexibility method, Comp. Meth. Appl. Mech.
Engrg. , 149, 319 337, 1997.[12] M. Goland and E. Reissner, The
stresses in cemented joints, J. Appl. Mech. , 11( 1), 17 27,
1944.[13] J. W. Ju and K. C. Valanis, Damage mechanics and
localization, ASME Winter Annual Meeting , Anaheim, California,
Nov. 1992.[14] M. Morimoto, H. Harada, M. Okada, and S. Komaki, A
study on power assignment of hierarchical modulation schemes for
digital broadcasting, IEICETrans. , E77- B/ 12, 1495 1500,
1994.[15] K. J. ODonnell and E. F. Crawley, Identification of
nonlinear system parameters in space structure joints using the
force- state mapping technique, MITSSL Report No. 16- 85 , July
1985.[16] K. C. Park and C. A. Felippa, A variational framework for
solution method developments in structural mechanics, J. Appl.
Mech. , 65, 242 249, 1998.[17] K. C. Park and C. A. Felippa,
Aflexibility- based inverse algorithm for identification of
structural joint properties, Proc. ASME Symposium on
ComputationalMethods on Inverse Problems , Anaheim, CA, 15- 20 Nov
1998.[18] K. C. Park and C. A. Felippa, A variational principle for
the formulation of partitioned structural systems, Int. J. Numer.
Meth. Engrg. , 47, 395 418, 2000.[19] K. C. Park, Modeling of
nonlinearities, in Lecture Notes for Aero Course 16.299 , Fall
1999, MIT, Cambridge, MA, 1999.[20] B. Pattan, Robust Modulation
Methods and Smart Antennas in Wireless Communications , Prentice
Hall, 1999.[21] M. D. Rao and S. He, Vibration analysis of
adhesively bonded lap joint, Part II: numerical solution, J. Sound
Vibr. , 152( 3), 417 425, 1992.[22] A. N. Robertson, K. C. Park and
K. F. Alvin, Extraction of impulse response data via wavelet
transform for structural syst em identification, ASME J.
Vibr.Acoust. , 120( 1), 252 260, 1998.[23] A. N. Robertson, K. C.
Park and K. F. Alvin, Identification of structural dynamics models
using wavelet- generated impulse response data, ASME Journalof
Vibrations and Acoustics , ASME J. Vibr. Acoust. , 120( 1), 261
266, 1998.[24] J. W. Sawyer and P. A. Cooper, Analytical and
experimental results for bonded single lap joints with preformed
adherends, AIAA Journal , 19( 11) 1981,pp. 1443 1453.[25] H. S.
Tzou, Non- Linear joint dynamics and controls of jointed flexible
structures with active and viscoelastic joint actu ators, J. Sound
Vib. , 407- 422, 1990.[26] E. E. Ungar, Energy dissipation at
structural joints; mechanisms and magnitudes, Air Force FDL- TDR-
64- 98 , Aug. 1964.[27] E. E. Ungar, The status of engineering
knowledge concerning the damping of built- up structures, J. Sound
Vib. , 26 (1), 141- 154, 1973.[28] W. N. Waggener, Pulse Code
Modulation , Artech House Inc., Boston, 1999.[29] Y. K. Wen, Method
for random vibration of hysteretic systems, J. Eng. Mech. ASCE ,
102- EM2, 249 263, 1976.