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Identification of a bolted-joint model with fuzzy parameters loaded normal to the contact interface
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 29 (2002) 177–187 www.elsevier.com/locate/mechrescom
Identification of a bolted-joint model with fuzzy parameters loaded normal to the contact interface Michael Hanss, Stefan Oexl, Lothar Gaul
*
Institute A of Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany Received 7 February 2002
Abstract Modeling and identification of a joint with load normal to the contact interface of two connected rods is discussed in this paper. An experimental setup for the analysis of the joint is proposed and measurement results are presented. The perception that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that can hardly be modeled motivates the use of a rather simple model, but with fuzzy-valued model parameters, instead of crisp ones. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers that can be identified on the basis of the measured data. The identification of the fuzzy parameters proves to be a non-trivial problem which can be solved by applying the transformation method as a special implementation of fuzzy arithmetic. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction As it is well known, modeling can always be considered as a problem that can never be solved to complete satisfaction but only to a certain extent. This statement results from the fact that there is usually a number of effects which either are completely unknown or can hardly be taken into account for they would lead to models of high complexity, which prove to be unsuitable for further use. Against this background, models can always be considered as more or less uncertain, either in an unintentional manner due to the lack of knowledge or, intentionally, as a result of simplification. As a problem of structural dynamics, a model of a normally–loaded bolted joint which connects two rods in axial direction is considered in this paper. In fact, there are a number of unknown or hard-to-model effects occurring in this problem which have significant influence on the damping behavior and on the stiffness of the joint. As for the damping, there is friction in the screw thread, gas pumping or impactinduced damping in local micro-gaps between the surfaces of normal contact, material damping in the asperities of the contact surfaces, or plastic deformation. The stiffness of the joint, instead, is very much influenced by the quality of the contact surfaces, i.e. by factors like hardness, roughness and waviness, as
*
Corresponding author. Tel.: +49-711-685-6278; fax: +49-711-685-6282.
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 4 5 - 8
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well as by their shape and their relative position. And finally, regarding the large-scale production of mechanical components, there is always a high degree of uncertainty and variability in the material properties and geometry parameters. With respect to the future use of joint models as parts of more complex and higher-dimensional system models, the connected rods will be expressed by one-dimensional continua in this paper where the joint is described by a Kelvin–Voigt element with one stiffness and one damping parameter (Gaul, 1981). Since various effects are not covered by this model, they might express themselves by a nonlinear behavior of the model or some time-dependency of the model parameters, and the provision for only two model parameters does not seem to be sufficient. To solve this limitation, a novel concept is presented in the paper which consists of extending the simplified linear model in a special way by substituting the crisp model parameters of stiffness and damping by fuzzy-valued parameters. Formerly unmodeled effects and uncertainties in the model parameters can thus be taken into account, leading to an extended model with fuzzy parameters. As a result of the simulation of this fuzzy model, worst-case scenarios can be obtained for any output variable of interest, as e.g. for the frequency response function of the coupled rods. For the nontrivial problem of identifying the fuzzy-valued stiffness and damping parameters from measured data for the eigenfrequencies and the damping ratio, a new methodology is presented in the paper, which uses a special implementation of fuzzy arithmetic based on the so-called transformation method.
2. Experimental setup and measurement results The experimental setup is given by two cylindrical rods of case hardened steel 16MnCr5 (lengths l1=2 ¼ 365 mm, diameters a1=2 ¼ 40 mm) which are centrically connected by a threaded bolt M 12 (Fig. 1). The contact surfaces of the rods have been machined by turning. To protect them from fretting, a polyester washer (thickness b ¼ 50 lm) is embedded between the surfaces. Finally, to minimize the influence of external bearings, the rods are suspended at 3/7 and 4/7 of their overall length. The experiments are performed by using three rods (a), (b) and (c) with presumably identical properties which then are combined in turn, i.e. rods (a) + (b), (a) + (c) and (b) + (c). On one side, the system is excited by means of an impact hammer, whereas on the other side, the velocity is measured using a laser vibrometer. The eigenfrequencies f and the damping ratios D of the natural vibrations can then be determined by analysis of the velocity signal in the time domain (Medusa-Manual, 2001). The results for the eigenfrequency f and the damping ratio D in dependency of the magnitude of the velocity at the ends of the rods are presented in Fig. 2. Here, only the first longitudinal eigenmode of the system has been considered and the bolted joint connection has been tightened by applying a torque of Mt ¼ 25 Nm. It can be seen from the results that the eigenfrequency f and the damping ratio D show some dependency on the magnitude of the velocity which reflects a slightly nonlinear behavior of the joint. This effect, however, can obviously be
Fig. 1. Experimental setup (left-hand side) and joint loaded normal to the contact interface (right-hand side).
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Fig. 2. Eigenfrequency and damping ratio for the first longitudinal eigenmode of the system depending on the magnitude of the velocity at the rod ends.
neglected compared to the differences that can be observed as a consequence of the tolerances in manufacturing (see line plots , þ, in Fig. 2). From additional experiments, the material properties as well as the damping ratio of the single rods can be determined. They show an elasticity of E ¼ 2:1084 1011 N/m2 and a mass density of q ¼ 7812 kg/m3 . The damping ratio D , determined for a rod of length l1 þ l2 at the first longitudinal eigenfrequency, amounts to an average value of D ¼ 2:5 105 which obviously proves to be of negligible extent compared to the damping ratio D of the overall system of coupled rods. For this reason, it is tolerable to consider the measured damping ratio D of the system as being caused by the joint only. In contrast to the conventional way of proceeding where the resulting data for the eigenfrequency f and the damping ratio D are averaged and only the mean values are considered for further calculations, the entire information included in the uncertainty of the measurements of f and D can be used by applying the fuzzy arithmetical concept described below.
3. Structural dynamics modeling The system presented in the experimental setup can be modeled by two rods which are considered as linear continua and a two-parameter joint model consisting of a Kelvin–Voigt element (Fig. 3). Making use of the method of transfer matrices (Pestel and Leckie, 1963; Waller and Krings, 1975), the relationship between the Laplace transforms of the normal forces NL and NR and the Laplace transforms of the displacements UL and UR at the outer ends of the rods can be described in the form
Fig. 3. Two rods connected by a two-parameter joint model.
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UL ðsÞ a ðsÞ ¼ 11 NL ðsÞ a21 ðsÞ
a12 ðsÞ a22 ðsÞ
UR ðsÞ NR ðsÞ
ð1Þ
with
coshðsa1 Þ sinhðsa1 Þ þ sb2 sinhðsa2 Þ; k þ sd sb1 coshðsa1 Þ sinhðsa2 Þ coshðsa1 Þ sinhðsa1 Þ þ a12 ðsÞ ¼ þ coshðsa2 Þ; sb2 k þ sd sb1 sb1 sinhðsa1 Þ a21 ðsÞ ¼ sb1 sinhðsa1 Þ coshðsa2 Þ þ þ coshðsa1 Þ sb2 sinhðsa2 Þ; k þ sd b1 sinhðsa1 Þ sinhðsa2 Þ sb1 sinhðsa1 Þ þ a22 ðsÞ ¼ þ coshðsa1 Þ coshðsa2 Þ; b2 k þ sd a11 ðsÞ ¼ coshðsa1 Þ coshðsa2 Þ þ
the complex eigenvalue s and rffiffiffiffiffi rffiffiffiffiffi q1 q2 a1 ¼ a2 ¼ l1 ; l2 ; E1 E2
b1 ¼ A 1
pffiffiffiffiffiffiffiffiffiffi E1 q1 ;
b 2 ¼ A2
pffiffiffiffiffiffiffiffiffiffi E2 q 2 :
ð2Þ
ð3Þ
Including the boundary conditions NL ¼ NR ¼ 0 for free outer ends of the rods, Eq. (1) is reduced to a21 UR ¼ 0;
ð4Þ
leading to the nontrivial solution a21 ¼ 0, i.e. sb1 sinhðsa1 Þ sb1 sinhðsa1 Þ coshðsa2 Þ þ þ coshðsa1 Þ sb2 sinhðsa2 Þ ¼ 0: k þ sd
ð5Þ
For the resulting free vibrations, the complex eigenvalues s are assumed to be of single-degree-of-freedom form s ¼ d þ ix
D with x ¼ ImðsÞ ¼ 2pf and d ¼ ReðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x; 1 D2
ð6Þ
where f are the eigenfrequencies and D the corresponding damping ratios of the system. To determine the stiffness k and the damping parameter d for a specific eigenfrequency f and its corresponding damping ratio D, Eq. (5) can be solved for d and k, leading to 1 sb1 b2 Im d¼ ; ð7Þ ImðsÞ b1 cothðsa2 Þ þ b2 cothðsa1 Þ k ¼ Re
sb1 b2 b1 cothðsa2 Þ þ b2 cothðsa1 Þ
d ReðsÞ:
ð8Þ
4. Fuzzy arithmetical concept A very practical approach for the numerical implementation of uncertain model parameters is to represent them as fuzzy numbers and to carry out arithmetical operations by using an extended or generalized arithmetic for fuzzy numbers, which is referred to as fuzzy arithmetic.
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4.1. Fuzzy numbers Basically, fuzzy numbers belong to a special class of fuzzy sets (Zadeh, 1965) showing some specific properties (Kaufmann and Gupta, 1991). The fuzzy sets themselves result from a generalization of conventional sets––where the elements of a universal set can either belong entirely to a specific set or are completely excluded from the set––by allowing the elements now to belong to a set to a certain degree (Zadeh, 1965). Thus, fuzzy sets can be expressed by the elements x of a universal set X with a certain degree of membership lðxÞ 2 ½0; 1 assigned. The elements x belonging to conventional sets, alternatively, are characterized by degrees of membership that can only be equal to zero or unity, i.e. by a membership function lðxÞ 2 f0; 1g. On this basis, closed intervals and crisp numbers of the form ½a; b ¼ fx j a 6 x 6 bg c ¼ fx j x ¼ cg;
and
x2R
can be considered as conventional subsets of the universal set R which can also be expressed by 1 for a 6 x 6 b l½a;b ðxÞ ¼ and 0 for all other x lc ðxÞ ¼
1 0
for x ¼ c for all other x
ð9Þ ð10Þ
ð11Þ
ð12Þ
when using the membership function lðxÞ 2 f0; 1g, x 2 R (Fig. 4). Fuzzy numbers, instead, are defined as convex fuzzy sets over the universal set R with membership functions lðxÞ 2 ½0; 1 , and where lðxÞ ¼ 1 is . As an example, symmetric fuzzy numbers of quasi-Gaussian shape can be true only for a single value x ¼ m defined by the membership function ! Þ2 ðx m j 6 3r and lðxÞ ¼ 0 for x > m þ 3r or x < m 3r ð13Þ lðxÞ ¼ exp for jx m 2r2 and r denoting the mean value and the standard deviation of the Gaussian distribution (Fig. 4). with m
, standard deviation Fig. 4. Closed interval ½a; b , crisp number c and symmetric fuzzy number of quasi-Gaussian shape (mean value m r) expressed by their membership functions.
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4.2. Transformation method To carry out arithmetical operations between fuzzy numbers, a ‘‘standard fuzzy arithmetic’’ has been defined (Kaufmann and Gupta, 1991), where each fuzzy number is decomposed into a set of intervals for the different levels of membership so that conventional interval arithmetic (Alefeld and Herzberger, 1983) can be applied separately for each level of membership. Upon closer examination, however, it is clear that the application of standard fuzzy arithmetic to the simulation of real-world systems with uncertain model parameters is undesirable. It has been shown that, as a serious drawback of this method, the results calculated by standard fuzzy arithmetic normally do not reflect the real result of the problem, but a result of much higher uncertainty (Hanss, 2002; Hanss and Willner, 2000). This drawback, however, can effectively be avoided by using the transformation method proposed by Hanss (2000, 2002) as a practical implementation of fuzzy arithmetic. In the following, the transformation method is presented only in its reduced form which is sufficient for the problem discussed. It can be used for both the simulation and the analysis of systems with uncertain parameters. 4.3. Simulation of systems with uncertain parameters Given a problem with n independent parameters, which are assumed to be uncertain, the parameters can ðjÞ be represented by fuzzy numbers p~i , i ¼ 1; 2; . . . ; n, each decomposed into a set Pi of m þ 1 intervals Xi , j ¼ 0; 1; . . . ; m, of the form n o ð0Þ ð1Þ ðmÞ P i ¼ Xi ; Xi ; . . . ; Xi ð14Þ with ðjÞ
Xi
h i ðjÞ ðjÞ ¼ ai ; bi ;
ðjÞ
ðjÞ
ai 6 bi ; i ¼ 1; 2; . . . ; n; j ¼ 0; 1; . . . ; m:
ð15Þ
For the purpose of decomposition, the l-axis is subdivided into m segments, equally spaced by Dl ¼ 1=m (Fig. 5). The m þ 1 levels of membership lj are then given by lj ¼
j ; m
j ¼ 0; 1; . . . ; m:
Fig. 5. Implementation of the ith uncertain parameter as a fuzzy number p~i decomposed into intervals.
ð16Þ
M. Hanss et al. / Mechanics Research Communications 29 (2002) 177–187
183
ðjÞ
Now, instead of applying standard interval arithmetic directly to the intervals Xi , i ¼ 1; 2; . . . ; n, for each biðjÞ of the following level of membership lj , j ¼ 0; 1; . . . ; m, the intervals can be transformed into arrays X form: 2i1 pairs
biðjÞ X
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ¼ ðai ; bðjÞ i ; ai ; bi ; . . . ; ai ; bi Þ
ð17Þ
with ðjÞ
ðjÞ
ðjÞ
ai ¼ ðai ; . . . ; ai Þ; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ðjÞ ai
2ni elements ðjÞ and bi are
ðjÞ
ðjÞ
bðjÞ i ¼ ðbi ; . . . ; bi Þ: |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð18Þ
2ni elements
the lower and upper bounds of the interval at the membership level lj for the ith Note that uncertain model parameter. Assuming that the system which is to be simulated is given by an arithmetical expression F of the functional form q~ ¼ F ð~ p1 ; p~2 ; . . . ; p~n Þ;
ð19Þ
its evaluation is then carried out by evaluating the expression separately at each of the positions of the arrays using the conventional arithmetic for crisp numbers. Thus, if the output q~ of the system can be b ðjÞ , j ¼ 0; 1; . . . ; m, the kth element k ^zðjÞ of expressed in its decomposed and transformed form by the arrays Z ðjÞ b the array Z is then given by ðjÞ ðjÞ k ðjÞ ^z ¼ F k x^1 ; k x^2 ; . . . ; k x^ðjÞ ð20Þ ; k ¼ 1; 2; . . . ; 2n ; n ðjÞ biðjÞ . Finally, the fuzzy-valued result q~ of the problem can be where k x^i denotes the kth element of the array X achieved in its decomposed form Z ðjÞ ¼ aðjÞ ; bðjÞ ; j ¼ 0; 1; . . . ; m; ð21Þ ðjÞ b by retransforming the arrays Z according to
aðjÞ ¼ minðk ^zðjÞ Þ; k
bðjÞ ¼ maxðk ^zðjÞ Þ;
j ¼ 0; 1; . . . ; m:
ð22Þ
k
4.4. Analysis of systems with uncertain parameters Until now, the fuzzy-valued result for the problem only shows the overall combined influence of all the uncertain parameters. However, it is possible to determine the proportions to which the n uncertain parameters of the system separately contribute to the overall uncertainty of the system output. Instead of b ðjÞ to the interval Z ðjÞ , as done in the retransformation step (Eq. (22)) of the transreducing the array Z formation method, the supplementary information given by the values and the arrangement of the elements b ðjÞ can be used. For this purpose, the coefficients gðjÞ in Z i , i ¼ 1; 2; . . . ; n, j ¼ 0; 1; . . . ; m 1, are to be determined according to 2ni X 2i1 X 1 ðjÞ gi ¼ ðk2 ^zðjÞ k1 ^zðjÞ Þ ð23Þ ðjÞ ðjÞ n1 bi ai 2 k¼1 l¼1 with k1 ¼ k þ ðl 1Þ2niþ1 ðjÞ values ai ðjÞ b
ðjÞ bi
and
k2 ¼ k þ ð2l 1Þ2ni :
ð24Þ ðjÞ Xi ,
The and denote the lower and upper bound of the interval and k ^zðjÞ is the kth element of ðjÞ the array Z . The coefficients gi can be interpreted as gain factors that express the effect of the uncertainty
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of the ith parameter on the uncertainty of the output q~ of the problem at the membership level lj . More explicitly, within the range of uncertainty covered at the membership level lj , deviations DzðjÞ from the central value of the output fuzzy number q~ can be considered as being related to the corresponding deðjÞ viations Dxi from the central value of the fuzzy parameters p~i , i ¼ 1; 2; . . . ; n, by the approximation n X ðjÞ ðjÞ gi Dxi : ð25Þ DzðjÞ i¼1
In case of nonlinear dependencies of the output q~ on the parameters p~1 ; p~2 ; . . . ; p~n , the method of system ðjÞ ðjÞ analysis can be extended by the introduction of single-sided gain factors giþ and gi which express the effect of the uncertainty of the ith parameter p~i on the uncertainty of the output q~ when only positive deviations from the central value, and negative deviations respectively, are considered. The single-sided gain factors can be determined on the basis of Eqs. (23) and (24) by using in turn only the right branch, and the left branch respectively, of the membership function of the ith fuzzy parameter p~i while the other fuzzy parameters remain unmodified. Finally, as an overall measure of influence, normalized values qi can be determined for i ¼ 1; 2; . . . ; n according to Pm1 ðjÞ ðjÞ ðjÞ n X j¼1 lj jgi ðai þ bi Þj qi ¼ Pn Pm1 qi ¼ 1: ð26Þ ; satisfying the condition ðjÞ ðjÞ ðjÞ i¼1 q¼1 j¼1 lj jgq ðaq þ bq Þj These values quantify the proportional influence of the ith varying parameter p~i on the overall variation of the problem output q~, assuming every parameter to be varied relatively to the same extent. Thus, they represent a measure for the sensitivity of the problem output with respect to each model parameter (Hanss et al., 2002).
5. Estimation of the stiffness and the damping parameter as an inverse fuzzy arithmetical problem Starting from the measured data available for the eigenfrequency f and the damping ratio D, fuzzy e can be defined to reflect the variability and uncertainty in the measured quantities. Then, numbers f~ and D the problem of identification consists of determining fuzzy values k~ and d~ for the stiffness and the damping parameter in such a way that a numerical re-simulation of the model using the uncertain parameters e . Since in terms of the transformation method preferably yields the original fuzzy-valued quantities f~ and D ~ ~ the parameters k and d can be considered as the n ¼ 2 independent parameters, which initiate the uncertainty in the model, the determination of the unknown (input) variables k~ and d~ on the basis of the given e ðk~; d~Þ represents an inverse problem which proves to be nontrivial. It can, (output) variables f~ðk~; d~Þ and D however, be solved according to the scheme (Hanss and Selvadurai, 2002) which is presented in the ensuing, involving the measured data: e : In order to cover the worst case of uncertainty in the model, Definition of the fuzzy numbers f~ and D e are defined as envelopes of the appropriate membership functions lf and lD for the fuzzy numbers f~ and D ~ e measured data. In the ensuing, f and D are assumed to be of symmetric quasi-Gaussian shape (Fig. 4, Eq. f and m D and the standard deviations rf and rD as follows: (13)) with the mean values m f ¼ 3385 Hz; m 3
D ¼ 1:0 10 ; m
rf ¼ 7 Hz; rD ¼ 0:1 103 :
ð27Þ
This corresponds to the worst-case ranges of 3385 21 Hz for the eigenfrequency f~ and 0:001 0:0003 for e , which is equivalent to worst-case deviations from the mean values of about 0.6% for the damping ratio D e. f~ and 30% for D
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k and m d of the fuzzy-valued stiffness and damping parameter k~ and d~: Determination of the mean values m As a result from the evaluation of Eqs. (7) and (8) for s ¼ d þ ix
D m f and d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x; with x ¼ 2pm 2D 1m
ð28Þ
k and m d are obtained as the mean values m k 7:07 109 N=m; m
ð29Þ
d 6:85 103 N s=m: m ðjÞ
ðjÞ
ðjÞ
Computation of the gain factors: For the determination of the single-sided gain factors gfkþ , gfdþ , gDkþ , ðjÞ ðjÞ ðjÞ ðjÞ and gfk , gfd , gDk , gDd , which quantify the influence of the uncertainty of the model parameters k~ ~ ~ e at the m levels of membership lj , j ¼ 0; 1; . . . ; m 1, and d on eigenfrequency f and the damping ratio D the model has to be simulated for some notionally uncertain parameters k~ and d~ using the transformation k and m d (Eq. (29)), and the method as defined above. The mean values of k~ and d~ have to be set to m assumed uncertainty should be set to a large value, so that the expected real range of uncertainty in k~ and d~ is preferably covered. In the present case, both k~ and d~ are chosen as symmetric fuzzy numbers of quasiGaussian shape with a worst-case deviation of 20% from the mean values. The gain factors can then be determined by evaluating the input/output data of the simulated uncertain model by means of Eqs. (5) and (6). Assembly of the uncertain parameters k~ and d~: Recalling the representation of a fuzzy number in its e , k~ and d~ at the levels decomposed form (Fig. 5), the lower bounds of the intervals of the fuzzy numbers f~, D ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ of membership lj , j ¼ 0; 1; . . . ; m, shall be defined as af , aD , ak and ad , and the upper bounds as bf , bD , ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ bk and bd , respectively. The parameters ak and ad as well as bk and bd of the unknown fuzzy-valued model parameters k~ and d~ can then be determined on the basis of Eq. (25) through 2 ðjÞ 3 2 ðjÞ 3 f 2 31 af m k ak m ðjÞ ðjÞ 6 ðjÞ 7 6 ðjÞ 7 Hfk j Hfd 6 bk m k 7 f 7 bf m 7 6 6 7¼6 6 7 5 6 ðjÞ 6 ðjÞ 7 4 7 d 5 4 ad m 4 aD m 5 ðjÞ ðjÞ D j H H Dk Dd ðjÞ ðjÞ d bd m D bD m ðjÞ gDdþ
with ðjÞ Hfk
" ðjÞ ðjÞ 1 gfk ð1 þ sgnðgfk ÞÞ ¼ ðjÞ 2 gðjÞ fk ð1 sgnðgfk ÞÞ
# ðjÞ ðjÞ gfkþ ð1 sgnðgfkþ ÞÞ ; ðjÞ ðjÞ gfkþ ð1 þ sgnðgfkþ ÞÞ
ð30Þ
.. . ðjÞ HDd
" ðjÞ 1 gðjÞ Dd ð1 þ sgnðgDd ÞÞ ¼ ðjÞ 2 gDd ð1 sgnðgðjÞ Dd ÞÞ
# ðjÞ ðjÞ gDdþ ð1 sgnðgDdþ ÞÞ ; ðjÞ ðjÞ gDdþ ð1 þ sgnðgDdþ ÞÞ
j ¼ 0; 1; . . . ; m 1: ðmÞ ak
ðmÞ bk
ð31Þ ð32Þ
ðmÞ ad
ðmÞ bd
k and m d. ¼ and ¼ are already determined by the mean values m The peak values The fuzzy-valued model parameters k~ and d~ that finally result for the given problem are presented in Fig. 6. Especially the membership function of the damping parameter d~ indicates asymmetric shape which gives evidence of the nonlinear behavior of the model within the covered range of uncertainty. The stiffness
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Fig. 6. Uncertain stiffness parameter k~ and damping parameter d~ of the joint model.
parameter k~ exhibits a worst-case range of 1:8 109 N/m, which corresponds to a worst-case deviation k between 10 % and þ15%. The resulting worst-case range of the damping pafrom the mean value m d between rameter d~ amounts to 661 Nm/s, corresponding to a worst-case deviation from the mean value m 7% and þ3%. Considering the fact that the original uncertainty has been assumed rather unequally e , it is worth to be noted that this undistributed between the eigenfrequency f~ and the damping ratio D certainty has obviously been assigned at a more or less balanced rate to each of the model parameters k~ and d~. Vice versa, it shows that the uncertainties in the stiffness and damping parameters have a rather large effect on the damping ratio while the effect on the eigenfrequency is quite moderate. To validate the results of the inverse fuzzy arithmetical problem, the eigenfrequency and the damping ratio can be simulated by evaluating Eqs. (5) and (6) by means of the transformation method and with the fuzzy-valued stiffness and damping parameters computed above. Indeed, the so-calculated fuzzy-valued e. outputs differ only slightly from the originally assumed fuzzy numbers f~ and D 6. Conclusions In the present paper, modeling and identification of a bolted joint with normal load on the contact interface of two connected rods has been discussed. The major problem of modeling can be seen in the fact that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that are unknown or can hardly be modeled, as well as by factors, like variability in the material properties and the geometry, which are often difficult to control. Against this background, the use of a rather simple model, but a model with fuzzy-valued parameters, instead of crisp ones, has been proposed. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers which can be identified on the basis of measured data. Making use of fuzzy arithmetic instead of the conventional arithmetic for crisp numbers, the model can be evaluated, and it becomes possible to quantify the overall uncertainty of any system variable, involving the fact that the model parameters can vary independently within some predefined fuzzy bounds. This allows the determination of worst-case scenarios which are extremely useful in practice, e.g. for the definition of safety factors. Acknowledgements The authors gratefully acknowledge the support of this work by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB 543 ‘‘Ultraschallbeeinflusstes Umformen metallischer Werkstoffe’’.
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References Alefeld, G., Herzberger, J., 1983. Introduction to Interval Computations. Academic Press, New York. Gaul, L., 1981. Zur D€ ammung und D€ampfung von Biegewellen an F€ ugestellen. Ingenieur-Archiv 51, 101–110. Hanss, M., 2000. A nearly strict fuzzy arithmetic for solving problems with uncertainties. In: Proceedings of the 19th International Conference of the North American Fuzzy Information Processing Society, NAFIPS 2000, Atlanta, GA, USA, pp. 439–443. Hanss, M., 2002. The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems 130 (3), 1–13. Hanss, M., Selvadurai, A.P.S., 2002. Influence of fuzzy variability on the estimation of hydraulic conductivity of transversely isotropic geomaterials. In: Proceedings of the NUMOG VIII International Symposium on Numerical Models in Geomechanics, Rome, Italy, pp. 675–680. Hanss, M., Willner, K., 2000. A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters. Mechanics Research Communications 27, 257–272. Hanss, M., Oexl, S., Gaul, L., 2002. Simulation and analysis of structural joint models with uncertainties. In: Proceedings of the International Conference on Structural Dynamics Modelling, Madeira, Portugal, pp. 165–174. Kaufmann, A., Gupta, M.M., 1991. Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York. Medusa-Manual, 2001. Vielkanal-Messdatenerfassung, Schwingungs- und Schallanalyse. Mahrenholtz + Partner, Ingenieurb€ uro f€ ur Maschinendynamik GmbH, Hannover. Pestel, E.C., Leckie, F.A., 1963. Matrix Methods in Elastomechanics. McGraw-Hill, New York. Waller, H., Krings, W., 1975. Matrizenmethoden in der Maschinen- und Bauwerksdynamik. B.I.-Wissenschaftsverlag, Z€ urich. Zadeh, L.A., 1965. Fuzzy sets. Information and Control 8, 338–353.
Mechanics Research Communications 29 (2002) 177–187 www.elsevier.com/locate/mechrescom
Identification of a bolted-joint model with fuzzy parameters loaded normal to the contact interface Michael Hanss, Stefan Oexl, Lothar Gaul
*
Institute A of Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany Received 7 February 2002
Abstract Modeling and identification of a joint with load normal to the contact interface of two connected rods is discussed in this paper. An experimental setup for the analysis of the joint is proposed and measurement results are presented. The perception that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that can hardly be modeled motivates the use of a rather simple model, but with fuzzy-valued model parameters, instead of crisp ones. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers that can be identified on the basis of the measured data. The identification of the fuzzy parameters proves to be a non-trivial problem which can be solved by applying the transformation method as a special implementation of fuzzy arithmetic. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction As it is well known, modeling can always be considered as a problem that can never be solved to complete satisfaction but only to a certain extent. This statement results from the fact that there is usually a number of effects which either are completely unknown or can hardly be taken into account for they would lead to models of high complexity, which prove to be unsuitable for further use. Against this background, models can always be considered as more or less uncertain, either in an unintentional manner due to the lack of knowledge or, intentionally, as a result of simplification. As a problem of structural dynamics, a model of a normally–loaded bolted joint which connects two rods in axial direction is considered in this paper. In fact, there are a number of unknown or hard-to-model effects occurring in this problem which have significant influence on the damping behavior and on the stiffness of the joint. As for the damping, there is friction in the screw thread, gas pumping or impactinduced damping in local micro-gaps between the surfaces of normal contact, material damping in the asperities of the contact surfaces, or plastic deformation. The stiffness of the joint, instead, is very much influenced by the quality of the contact surfaces, i.e. by factors like hardness, roughness and waviness, as
*
Corresponding author. Tel.: +49-711-685-6278; fax: +49-711-685-6282.
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 4 5 - 8
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well as by their shape and their relative position. And finally, regarding the large-scale production of mechanical components, there is always a high degree of uncertainty and variability in the material properties and geometry parameters. With respect to the future use of joint models as parts of more complex and higher-dimensional system models, the connected rods will be expressed by one-dimensional continua in this paper where the joint is described by a Kelvin–Voigt element with one stiffness and one damping parameter (Gaul, 1981). Since various effects are not covered by this model, they might express themselves by a nonlinear behavior of the model or some time-dependency of the model parameters, and the provision for only two model parameters does not seem to be sufficient. To solve this limitation, a novel concept is presented in the paper which consists of extending the simplified linear model in a special way by substituting the crisp model parameters of stiffness and damping by fuzzy-valued parameters. Formerly unmodeled effects and uncertainties in the model parameters can thus be taken into account, leading to an extended model with fuzzy parameters. As a result of the simulation of this fuzzy model, worst-case scenarios can be obtained for any output variable of interest, as e.g. for the frequency response function of the coupled rods. For the nontrivial problem of identifying the fuzzy-valued stiffness and damping parameters from measured data for the eigenfrequencies and the damping ratio, a new methodology is presented in the paper, which uses a special implementation of fuzzy arithmetic based on the so-called transformation method.
2. Experimental setup and measurement results The experimental setup is given by two cylindrical rods of case hardened steel 16MnCr5 (lengths l1=2 ¼ 365 mm, diameters a1=2 ¼ 40 mm) which are centrically connected by a threaded bolt M 12 (Fig. 1). The contact surfaces of the rods have been machined by turning. To protect them from fretting, a polyester washer (thickness b ¼ 50 lm) is embedded between the surfaces. Finally, to minimize the influence of external bearings, the rods are suspended at 3/7 and 4/7 of their overall length. The experiments are performed by using three rods (a), (b) and (c) with presumably identical properties which then are combined in turn, i.e. rods (a) + (b), (a) + (c) and (b) + (c). On one side, the system is excited by means of an impact hammer, whereas on the other side, the velocity is measured using a laser vibrometer. The eigenfrequencies f and the damping ratios D of the natural vibrations can then be determined by analysis of the velocity signal in the time domain (Medusa-Manual, 2001). The results for the eigenfrequency f and the damping ratio D in dependency of the magnitude of the velocity at the ends of the rods are presented in Fig. 2. Here, only the first longitudinal eigenmode of the system has been considered and the bolted joint connection has been tightened by applying a torque of Mt ¼ 25 Nm. It can be seen from the results that the eigenfrequency f and the damping ratio D show some dependency on the magnitude of the velocity which reflects a slightly nonlinear behavior of the joint. This effect, however, can obviously be
Fig. 1. Experimental setup (left-hand side) and joint loaded normal to the contact interface (right-hand side).
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Fig. 2. Eigenfrequency and damping ratio for the first longitudinal eigenmode of the system depending on the magnitude of the velocity at the rod ends.
neglected compared to the differences that can be observed as a consequence of the tolerances in manufacturing (see line plots , þ, in Fig. 2). From additional experiments, the material properties as well as the damping ratio of the single rods can be determined. They show an elasticity of E ¼ 2:1084 1011 N/m2 and a mass density of q ¼ 7812 kg/m3 . The damping ratio D , determined for a rod of length l1 þ l2 at the first longitudinal eigenfrequency, amounts to an average value of D ¼ 2:5 105 which obviously proves to be of negligible extent compared to the damping ratio D of the overall system of coupled rods. For this reason, it is tolerable to consider the measured damping ratio D of the system as being caused by the joint only. In contrast to the conventional way of proceeding where the resulting data for the eigenfrequency f and the damping ratio D are averaged and only the mean values are considered for further calculations, the entire information included in the uncertainty of the measurements of f and D can be used by applying the fuzzy arithmetical concept described below.
3. Structural dynamics modeling The system presented in the experimental setup can be modeled by two rods which are considered as linear continua and a two-parameter joint model consisting of a Kelvin–Voigt element (Fig. 3). Making use of the method of transfer matrices (Pestel and Leckie, 1963; Waller and Krings, 1975), the relationship between the Laplace transforms of the normal forces NL and NR and the Laplace transforms of the displacements UL and UR at the outer ends of the rods can be described in the form
Fig. 3. Two rods connected by a two-parameter joint model.
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UL ðsÞ a ðsÞ ¼ 11 NL ðsÞ a21 ðsÞ
a12 ðsÞ a22 ðsÞ
UR ðsÞ NR ðsÞ
ð1Þ
with
coshðsa1 Þ sinhðsa1 Þ þ sb2 sinhðsa2 Þ; k þ sd sb1 coshðsa1 Þ sinhðsa2 Þ coshðsa1 Þ sinhðsa1 Þ þ a12 ðsÞ ¼ þ coshðsa2 Þ; sb2 k þ sd sb1 sb1 sinhðsa1 Þ a21 ðsÞ ¼ sb1 sinhðsa1 Þ coshðsa2 Þ þ þ coshðsa1 Þ sb2 sinhðsa2 Þ; k þ sd b1 sinhðsa1 Þ sinhðsa2 Þ sb1 sinhðsa1 Þ þ a22 ðsÞ ¼ þ coshðsa1 Þ coshðsa2 Þ; b2 k þ sd a11 ðsÞ ¼ coshðsa1 Þ coshðsa2 Þ þ
the complex eigenvalue s and rffiffiffiffiffi rffiffiffiffiffi q1 q2 a1 ¼ a2 ¼ l1 ; l2 ; E1 E2
b1 ¼ A 1
pffiffiffiffiffiffiffiffiffiffi E1 q1 ;
b 2 ¼ A2
pffiffiffiffiffiffiffiffiffiffi E2 q 2 :
ð2Þ
ð3Þ
Including the boundary conditions NL ¼ NR ¼ 0 for free outer ends of the rods, Eq. (1) is reduced to a21 UR ¼ 0;
ð4Þ
leading to the nontrivial solution a21 ¼ 0, i.e. sb1 sinhðsa1 Þ sb1 sinhðsa1 Þ coshðsa2 Þ þ þ coshðsa1 Þ sb2 sinhðsa2 Þ ¼ 0: k þ sd
ð5Þ
For the resulting free vibrations, the complex eigenvalues s are assumed to be of single-degree-of-freedom form s ¼ d þ ix
D with x ¼ ImðsÞ ¼ 2pf and d ¼ ReðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x; 1 D2
ð6Þ
where f are the eigenfrequencies and D the corresponding damping ratios of the system. To determine the stiffness k and the damping parameter d for a specific eigenfrequency f and its corresponding damping ratio D, Eq. (5) can be solved for d and k, leading to 1 sb1 b2 Im d¼ ; ð7Þ ImðsÞ b1 cothðsa2 Þ þ b2 cothðsa1 Þ k ¼ Re
sb1 b2 b1 cothðsa2 Þ þ b2 cothðsa1 Þ
d ReðsÞ:
ð8Þ
4. Fuzzy arithmetical concept A very practical approach for the numerical implementation of uncertain model parameters is to represent them as fuzzy numbers and to carry out arithmetical operations by using an extended or generalized arithmetic for fuzzy numbers, which is referred to as fuzzy arithmetic.
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4.1. Fuzzy numbers Basically, fuzzy numbers belong to a special class of fuzzy sets (Zadeh, 1965) showing some specific properties (Kaufmann and Gupta, 1991). The fuzzy sets themselves result from a generalization of conventional sets––where the elements of a universal set can either belong entirely to a specific set or are completely excluded from the set––by allowing the elements now to belong to a set to a certain degree (Zadeh, 1965). Thus, fuzzy sets can be expressed by the elements x of a universal set X with a certain degree of membership lðxÞ 2 ½0; 1 assigned. The elements x belonging to conventional sets, alternatively, are characterized by degrees of membership that can only be equal to zero or unity, i.e. by a membership function lðxÞ 2 f0; 1g. On this basis, closed intervals and crisp numbers of the form ½a; b ¼ fx j a 6 x 6 bg c ¼ fx j x ¼ cg;
and
x2R
can be considered as conventional subsets of the universal set R which can also be expressed by 1 for a 6 x 6 b l½a;b ðxÞ ¼ and 0 for all other x lc ðxÞ ¼
1 0
for x ¼ c for all other x
ð9Þ ð10Þ
ð11Þ
ð12Þ
when using the membership function lðxÞ 2 f0; 1g, x 2 R (Fig. 4). Fuzzy numbers, instead, are defined as convex fuzzy sets over the universal set R with membership functions lðxÞ 2 ½0; 1 , and where lðxÞ ¼ 1 is . As an example, symmetric fuzzy numbers of quasi-Gaussian shape can be true only for a single value x ¼ m defined by the membership function ! Þ2 ðx m j 6 3r and lðxÞ ¼ 0 for x > m þ 3r or x < m 3r ð13Þ lðxÞ ¼ exp for jx m 2r2 and r denoting the mean value and the standard deviation of the Gaussian distribution (Fig. 4). with m
, standard deviation Fig. 4. Closed interval ½a; b , crisp number c and symmetric fuzzy number of quasi-Gaussian shape (mean value m r) expressed by their membership functions.
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4.2. Transformation method To carry out arithmetical operations between fuzzy numbers, a ‘‘standard fuzzy arithmetic’’ has been defined (Kaufmann and Gupta, 1991), where each fuzzy number is decomposed into a set of intervals for the different levels of membership so that conventional interval arithmetic (Alefeld and Herzberger, 1983) can be applied separately for each level of membership. Upon closer examination, however, it is clear that the application of standard fuzzy arithmetic to the simulation of real-world systems with uncertain model parameters is undesirable. It has been shown that, as a serious drawback of this method, the results calculated by standard fuzzy arithmetic normally do not reflect the real result of the problem, but a result of much higher uncertainty (Hanss, 2002; Hanss and Willner, 2000). This drawback, however, can effectively be avoided by using the transformation method proposed by Hanss (2000, 2002) as a practical implementation of fuzzy arithmetic. In the following, the transformation method is presented only in its reduced form which is sufficient for the problem discussed. It can be used for both the simulation and the analysis of systems with uncertain parameters. 4.3. Simulation of systems with uncertain parameters Given a problem with n independent parameters, which are assumed to be uncertain, the parameters can ðjÞ be represented by fuzzy numbers p~i , i ¼ 1; 2; . . . ; n, each decomposed into a set Pi of m þ 1 intervals Xi , j ¼ 0; 1; . . . ; m, of the form n o ð0Þ ð1Þ ðmÞ P i ¼ Xi ; Xi ; . . . ; Xi ð14Þ with ðjÞ
Xi
h i ðjÞ ðjÞ ¼ ai ; bi ;
ðjÞ
ðjÞ
ai 6 bi ; i ¼ 1; 2; . . . ; n; j ¼ 0; 1; . . . ; m:
ð15Þ
For the purpose of decomposition, the l-axis is subdivided into m segments, equally spaced by Dl ¼ 1=m (Fig. 5). The m þ 1 levels of membership lj are then given by lj ¼
j ; m
j ¼ 0; 1; . . . ; m:
Fig. 5. Implementation of the ith uncertain parameter as a fuzzy number p~i decomposed into intervals.
ð16Þ
M. Hanss et al. / Mechanics Research Communications 29 (2002) 177–187
183
ðjÞ
Now, instead of applying standard interval arithmetic directly to the intervals Xi , i ¼ 1; 2; . . . ; n, for each biðjÞ of the following level of membership lj , j ¼ 0; 1; . . . ; m, the intervals can be transformed into arrays X form: 2i1 pairs
biðjÞ X
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ¼ ðai ; bðjÞ i ; ai ; bi ; . . . ; ai ; bi Þ
ð17Þ
with ðjÞ
ðjÞ
ðjÞ
ai ¼ ðai ; . . . ; ai Þ; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ðjÞ ai
2ni elements ðjÞ and bi are
ðjÞ
ðjÞ
bðjÞ i ¼ ðbi ; . . . ; bi Þ: |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð18Þ
2ni elements
the lower and upper bounds of the interval at the membership level lj for the ith Note that uncertain model parameter. Assuming that the system which is to be simulated is given by an arithmetical expression F of the functional form q~ ¼ F ð~ p1 ; p~2 ; . . . ; p~n Þ;
ð19Þ
its evaluation is then carried out by evaluating the expression separately at each of the positions of the arrays using the conventional arithmetic for crisp numbers. Thus, if the output q~ of the system can be b ðjÞ , j ¼ 0; 1; . . . ; m, the kth element k ^zðjÞ of expressed in its decomposed and transformed form by the arrays Z ðjÞ b the array Z is then given by ðjÞ ðjÞ k ðjÞ ^z ¼ F k x^1 ; k x^2 ; . . . ; k x^ðjÞ ð20Þ ; k ¼ 1; 2; . . . ; 2n ; n ðjÞ biðjÞ . Finally, the fuzzy-valued result q~ of the problem can be where k x^i denotes the kth element of the array X achieved in its decomposed form Z ðjÞ ¼ aðjÞ ; bðjÞ ; j ¼ 0; 1; . . . ; m; ð21Þ ðjÞ b by retransforming the arrays Z according to
aðjÞ ¼ minðk ^zðjÞ Þ; k
bðjÞ ¼ maxðk ^zðjÞ Þ;
j ¼ 0; 1; . . . ; m:
ð22Þ
k
4.4. Analysis of systems with uncertain parameters Until now, the fuzzy-valued result for the problem only shows the overall combined influence of all the uncertain parameters. However, it is possible to determine the proportions to which the n uncertain parameters of the system separately contribute to the overall uncertainty of the system output. Instead of b ðjÞ to the interval Z ðjÞ , as done in the retransformation step (Eq. (22)) of the transreducing the array Z formation method, the supplementary information given by the values and the arrangement of the elements b ðjÞ can be used. For this purpose, the coefficients gðjÞ in Z i , i ¼ 1; 2; . . . ; n, j ¼ 0; 1; . . . ; m 1, are to be determined according to 2ni X 2i1 X 1 ðjÞ gi ¼ ðk2 ^zðjÞ k1 ^zðjÞ Þ ð23Þ ðjÞ ðjÞ n1 bi ai 2 k¼1 l¼1 with k1 ¼ k þ ðl 1Þ2niþ1 ðjÞ values ai ðjÞ b
ðjÞ bi
and
k2 ¼ k þ ð2l 1Þ2ni :
ð24Þ ðjÞ Xi ,
The and denote the lower and upper bound of the interval and k ^zðjÞ is the kth element of ðjÞ the array Z . The coefficients gi can be interpreted as gain factors that express the effect of the uncertainty
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of the ith parameter on the uncertainty of the output q~ of the problem at the membership level lj . More explicitly, within the range of uncertainty covered at the membership level lj , deviations DzðjÞ from the central value of the output fuzzy number q~ can be considered as being related to the corresponding deðjÞ viations Dxi from the central value of the fuzzy parameters p~i , i ¼ 1; 2; . . . ; n, by the approximation n X ðjÞ ðjÞ gi Dxi : ð25Þ DzðjÞ i¼1
In case of nonlinear dependencies of the output q~ on the parameters p~1 ; p~2 ; . . . ; p~n , the method of system ðjÞ ðjÞ analysis can be extended by the introduction of single-sided gain factors giþ and gi which express the effect of the uncertainty of the ith parameter p~i on the uncertainty of the output q~ when only positive deviations from the central value, and negative deviations respectively, are considered. The single-sided gain factors can be determined on the basis of Eqs. (23) and (24) by using in turn only the right branch, and the left branch respectively, of the membership function of the ith fuzzy parameter p~i while the other fuzzy parameters remain unmodified. Finally, as an overall measure of influence, normalized values qi can be determined for i ¼ 1; 2; . . . ; n according to Pm1 ðjÞ ðjÞ ðjÞ n X j¼1 lj jgi ðai þ bi Þj qi ¼ Pn Pm1 qi ¼ 1: ð26Þ ; satisfying the condition ðjÞ ðjÞ ðjÞ i¼1 q¼1 j¼1 lj jgq ðaq þ bq Þj These values quantify the proportional influence of the ith varying parameter p~i on the overall variation of the problem output q~, assuming every parameter to be varied relatively to the same extent. Thus, they represent a measure for the sensitivity of the problem output with respect to each model parameter (Hanss et al., 2002).
5. Estimation of the stiffness and the damping parameter as an inverse fuzzy arithmetical problem Starting from the measured data available for the eigenfrequency f and the damping ratio D, fuzzy e can be defined to reflect the variability and uncertainty in the measured quantities. Then, numbers f~ and D the problem of identification consists of determining fuzzy values k~ and d~ for the stiffness and the damping parameter in such a way that a numerical re-simulation of the model using the uncertain parameters e . Since in terms of the transformation method preferably yields the original fuzzy-valued quantities f~ and D ~ ~ the parameters k and d can be considered as the n ¼ 2 independent parameters, which initiate the uncertainty in the model, the determination of the unknown (input) variables k~ and d~ on the basis of the given e ðk~; d~Þ represents an inverse problem which proves to be nontrivial. It can, (output) variables f~ðk~; d~Þ and D however, be solved according to the scheme (Hanss and Selvadurai, 2002) which is presented in the ensuing, involving the measured data: e : In order to cover the worst case of uncertainty in the model, Definition of the fuzzy numbers f~ and D e are defined as envelopes of the appropriate membership functions lf and lD for the fuzzy numbers f~ and D ~ e measured data. In the ensuing, f and D are assumed to be of symmetric quasi-Gaussian shape (Fig. 4, Eq. f and m D and the standard deviations rf and rD as follows: (13)) with the mean values m f ¼ 3385 Hz; m 3
D ¼ 1:0 10 ; m
rf ¼ 7 Hz; rD ¼ 0:1 103 :
ð27Þ
This corresponds to the worst-case ranges of 3385 21 Hz for the eigenfrequency f~ and 0:001 0:0003 for e , which is equivalent to worst-case deviations from the mean values of about 0.6% for the damping ratio D e. f~ and 30% for D
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185
k and m d of the fuzzy-valued stiffness and damping parameter k~ and d~: Determination of the mean values m As a result from the evaluation of Eqs. (7) and (8) for s ¼ d þ ix
D m f and d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x; with x ¼ 2pm 2D 1m
ð28Þ
k and m d are obtained as the mean values m k 7:07 109 N=m; m
ð29Þ
d 6:85 103 N s=m: m ðjÞ
ðjÞ
ðjÞ
Computation of the gain factors: For the determination of the single-sided gain factors gfkþ , gfdþ , gDkþ , ðjÞ ðjÞ ðjÞ ðjÞ and gfk , gfd , gDk , gDd , which quantify the influence of the uncertainty of the model parameters k~ ~ ~ e at the m levels of membership lj , j ¼ 0; 1; . . . ; m 1, and d on eigenfrequency f and the damping ratio D the model has to be simulated for some notionally uncertain parameters k~ and d~ using the transformation k and m d (Eq. (29)), and the method as defined above. The mean values of k~ and d~ have to be set to m assumed uncertainty should be set to a large value, so that the expected real range of uncertainty in k~ and d~ is preferably covered. In the present case, both k~ and d~ are chosen as symmetric fuzzy numbers of quasiGaussian shape with a worst-case deviation of 20% from the mean values. The gain factors can then be determined by evaluating the input/output data of the simulated uncertain model by means of Eqs. (5) and (6). Assembly of the uncertain parameters k~ and d~: Recalling the representation of a fuzzy number in its e , k~ and d~ at the levels decomposed form (Fig. 5), the lower bounds of the intervals of the fuzzy numbers f~, D ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ of membership lj , j ¼ 0; 1; . . . ; m, shall be defined as af , aD , ak and ad , and the upper bounds as bf , bD , ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ bk and bd , respectively. The parameters ak and ad as well as bk and bd of the unknown fuzzy-valued model parameters k~ and d~ can then be determined on the basis of Eq. (25) through 2 ðjÞ 3 2 ðjÞ 3 f 2 31 af m k ak m ðjÞ ðjÞ 6 ðjÞ 7 6 ðjÞ 7 Hfk j Hfd 6 bk m k 7 f 7 bf m 7 6 6 7¼6 6 7 5 6 ðjÞ 6 ðjÞ 7 4 7 d 5 4 ad m 4 aD m 5 ðjÞ ðjÞ D j H H Dk Dd ðjÞ ðjÞ d bd m D bD m ðjÞ gDdþ
with ðjÞ Hfk
" ðjÞ ðjÞ 1 gfk ð1 þ sgnðgfk ÞÞ ¼ ðjÞ 2 gðjÞ fk ð1 sgnðgfk ÞÞ
# ðjÞ ðjÞ gfkþ ð1 sgnðgfkþ ÞÞ ; ðjÞ ðjÞ gfkþ ð1 þ sgnðgfkþ ÞÞ
ð30Þ
.. . ðjÞ HDd
" ðjÞ 1 gðjÞ Dd ð1 þ sgnðgDd ÞÞ ¼ ðjÞ 2 gDd ð1 sgnðgðjÞ Dd ÞÞ
# ðjÞ ðjÞ gDdþ ð1 sgnðgDdþ ÞÞ ; ðjÞ ðjÞ gDdþ ð1 þ sgnðgDdþ ÞÞ
j ¼ 0; 1; . . . ; m 1: ðmÞ ak
ðmÞ bk
ð31Þ ð32Þ
ðmÞ ad
ðmÞ bd
k and m d. ¼ and ¼ are already determined by the mean values m The peak values The fuzzy-valued model parameters k~ and d~ that finally result for the given problem are presented in Fig. 6. Especially the membership function of the damping parameter d~ indicates asymmetric shape which gives evidence of the nonlinear behavior of the model within the covered range of uncertainty. The stiffness
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Fig. 6. Uncertain stiffness parameter k~ and damping parameter d~ of the joint model.
parameter k~ exhibits a worst-case range of 1:8 109 N/m, which corresponds to a worst-case deviation k between 10 % and þ15%. The resulting worst-case range of the damping pafrom the mean value m d between rameter d~ amounts to 661 Nm/s, corresponding to a worst-case deviation from the mean value m 7% and þ3%. Considering the fact that the original uncertainty has been assumed rather unequally e , it is worth to be noted that this undistributed between the eigenfrequency f~ and the damping ratio D certainty has obviously been assigned at a more or less balanced rate to each of the model parameters k~ and d~. Vice versa, it shows that the uncertainties in the stiffness and damping parameters have a rather large effect on the damping ratio while the effect on the eigenfrequency is quite moderate. To validate the results of the inverse fuzzy arithmetical problem, the eigenfrequency and the damping ratio can be simulated by evaluating Eqs. (5) and (6) by means of the transformation method and with the fuzzy-valued stiffness and damping parameters computed above. Indeed, the so-calculated fuzzy-valued e. outputs differ only slightly from the originally assumed fuzzy numbers f~ and D 6. Conclusions In the present paper, modeling and identification of a bolted joint with normal load on the contact interface of two connected rods has been discussed. The major problem of modeling can be seen in the fact that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that are unknown or can hardly be modeled, as well as by factors, like variability in the material properties and the geometry, which are often difficult to control. Against this background, the use of a rather simple model, but a model with fuzzy-valued parameters, instead of crisp ones, has been proposed. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers which can be identified on the basis of measured data. Making use of fuzzy arithmetic instead of the conventional arithmetic for crisp numbers, the model can be evaluated, and it becomes possible to quantify the overall uncertainty of any system variable, involving the fact that the model parameters can vary independently within some predefined fuzzy bounds. This allows the determination of worst-case scenarios which are extremely useful in practice, e.g. for the definition of safety factors. Acknowledgements The authors gratefully acknowledge the support of this work by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB 543 ‘‘Ultraschallbeeinflusstes Umformen metallischer Werkstoffe’’.
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