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: kcpark@titan.colorado.eduphone: 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.