Envelope frequency Response Function Analysis of Mechanical Structures with Uncertain Modal Damping Characteristics

Transcription

1 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 Envelope frequency Response Function Analysis of Mechanical Structures with Uncertain Modal Daping Characteristics D. Moens 1, M. De Munc and D. Vandepitte Abstract: Recently, an interval finite eleent ethodology has been developed to calculate envelope frequency response functions of uncertain structures with interval paraeters. The ethodology is based on a hybrid interval ipleentation of the odal superposition principle. This hybrid procedure consists of a preliinary optiization step, followed by an interval arithetic procedure. The final envelope frequency response functions have been proved to give a very good approxiation of the actual response range of the interval proble. Initially, this ethod was developed for undaped structures. Based on the theoretical principles of this approach, this paper introduces a new ethod for the analysis of structures with uncertain odal daping factors. In the proposed procedure, next to the classical interval paraeters that affect stiffness and ass properties of the odel, additional independent odal daping paraeter intervals describe the uncertainty on the daping factor of each ode that is taen into account in the response function. The algorith adopts the general odal superposition strategy of the undaped procedure. The effect of the introduction of the odal daping intervals in the interval arithetic part of the procedure is studied analytically, and a new interval arithetic procedure is developed according to the observations fro this study. In order to validate the procedure, it is applied to a realistic industrial odel with geoetric as well as odal daping uncertainty. Keyword: envelope frequency response functions, interval finite eleent analysis, uncertainty 1 Postdoctoral research fellow of the Research Foundation - Flanders FWO - Vlaanderen) 1 Introduction Non-deterinistic approaches are gaining oentu in the field of finite eleent analysis. The ability to include non-deterinistic properties is of great value for a design engineer. It enables realistic reliability assessent that incorporates the uncertain aspects of the design. Furtherore, the design can be optiized for robust behavior under varying external influences. Recently, criticis arises on the general application of the probabilistic concept in this context. Especially when objective inforation on the uncertainties is liited, the subjective probabilistic analysis result proves to be of little value, and does not justify its high coputational cost see e.g. [Elishaoff 2000); Moens and Vandepitte 2005)]). Consequently, alternative non-probabilistic concepts are used for non-deterinistic finite eleent analysis, as interval of fuzzy finite eleent analysis. Fuzzy finite eleent analysis fors the core of possibilistic analysis, which has been applied for nonprobabilistic reliability studies [Stroud, Krishnaurthy, and Sith 2002)]. The Interval FE IFE) analysis is based on the interval concept for the description of nondeterinistic odel properties, and so far has been studied only in liited publications see e.g. [Köylüoǧlu, Çaa, and Nielsen 1995); Mullen and Muhanna 1999); Dessobz, Thouverez, Laîné, and Jézéquel 2001); Qiu, Wang, and Chen 2006)]). Recently, Muhanna introduced a penalty-based solution procedure for the IFE analysis [Muhanna, Mullen, and Zhang 2005)]. Also, new developents in the area of linear systes with interval paraeters have been proposed by Neuaier [Neuaier 2006)]. The Fuzzy FE FFE) analysis is basically an extension of the IFE analysis. It was first introduced by

2 130 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 Rao [Rao and Sawyer 1995)]. It has been studied in a nuber of specific research doains, as e.g. static structural analysis see [Hanss 2003); Massa, Tison, and Lalleand 2006)]) and dynaic analysis see [Chen and Rao 1997); Wasfy and Noor 1998); Donders, Vandepitte, Van de Peer, and Deset 2005)]). Recently, an interval finite eleent ethodology to calculate envelope frequency response functions FRF) of uncertain structures has been developed by the authors [Moens and Vandepitte 2004)]. This procedure fors the basis for the ipleentation of the fuzzy finite eleent ethod. The goal of the interval analysis is to calculate the envelope of the FRF taing into account that the input uncertainties can vary within the bounded space defined by their cobined intervals. For this purpose, a hybrid procedure involving both a global optiization step and an interval arithetic step has been developed. The resulting envelope response function gives a clear view on the possible variation of the response in the frequency doain. While the efficiency and accuracy of the envelope FRF procedure has been proved, the ethod in its current for has an iportant shortcoing, as it cannot handle uncertainty on the daping properties of the analyzed odel. Perforing a realistic quantification of daping properties of a physical echanical structure is generally not straightforward. This is ainly due to unnown internal characteristics of the used aterials. But also the connecting parts in asseblies of coponents tend to have a high yet fairly unpredictable effect on the daping characteristics of a echanical structure. Generally, the lac of inforation on daping properties fors a ajor source of nondeterinis, especially in the dynaic response analysis of structures. Therefore, the possible extension of the proposed envelope FRF procedure toward the analysis of structures with iprecise daping properties constitutes a fundaental extension of the applicability and practical value of the dynaic interval finite eleent analysis. Section 2 of this wor first suarizes the undaped envelope FRF procedure as it was initially developed in [Moens and Vandepitte 2004)]. The effect of the introduction of independent odal paraeter intervals on the procedure is then analyzed in section 3. Finally, section 4 illustrates the newly developed procedure by applying it on a realistic case study. 2 Envelope FRF analysis of undaped structures The goal of the envelope FRF analysis is to calculate the bounds on the dynaic response of a structure in a specific frequency region given that a set of odel paraeters x is uncertain but bounded. The intervals on these paraeters are specified in an interval vector x I. The ethodology for the envelope dynaic response analysis as developed by the authors is based on a hybrid interval solution strategy, consisting of a preliinary optiization step, followed by an interval arithetic step. In the first part of this procedure, the optiization is used to translate the interval properties defined on the finite eleent odel to the exact interval odal stiffness and ass paraeters of the structure. The calculation of the envelope FRFs in the second part is done by applying the interval arithetic equivalent of the odal superposition procedure on these interval odal paraeters. The final envelope FRFs have been proved to contain only a very liited aount of conservatis. A brief overview of the basic principles of the ethod is given in this section. The coplete atheatical description of the ethod can be found in [Moens and Vandepitte 2004)]. 2.1 The deterinistic odal superposition principle For undaped structures, the deterinistic odal superposition principle states that, considering the first n odes odes, the frequency response function between degrees of freedo j and equals: n odes FRF j = i=1 FRF j i n odes = i=1 φ i j φ i φ i T Kφ i 2 φ i T Mφ i 1) with φ i the i th eigenvector of the syste and φ i j the j th coponent of the i th eigenvector. Siplifi-

3 Envelope frequency Response Function Analysis 131 cation of Eq. 1 yields: n odes 1 FRF j = 2) i=1 ˆ i 2 with ˆ i and the odal paraeters defined as: ˆ i = φ i T Kφ i φ i j φ i = 1 φi K j φi K 3) = φ i T Mφ i φ i j φ i = 1 φi M j φi M 4) with φi K and φi M the stiffness and ass noralized eigenvectors of the syste. 2.2 Interval finite eleent FRF analysis The odal superposition principle has been translated into an interval finite eleent ethod for FRF analysis. Figure 1 gives a graphical overview of the translation of the deterinistic algorith into an interval procedure. On the left-hand side is the deterinistic algorith as described in the previous section. On the right-hand side is the sae procedure translated to an equivalent interval algorith. The interval ethod consists of the calculation of the result ranges of the sub functions appearing in the consecutive steps of the deterinistic algorith. Therefore, the deterinistic algorith is split into three sub functions. In the first step, step 1.1, the odal stiffness ˆ i and ass are calculated for each considered ode. Step 1.2 then consists of the calculation of the odal FRF contributions FRFj i. Step 1.1 and 1.2 have to be perfored for each ode that is taen into consideration in the odal superposition. Therefore, it is referred to as the odal part. Instep2,thesuperposition is perfored by a suation of the odal FRF contributions. The interval procedure follows the sae outline as the deterinistic algorith. Each step now concentrates on the derivation of the range of the sub functions in the deterinistic algorith: step 1.1 For all n odes taen into account, the ranges of possible values that the odal stiffness and ass can adopt have to be deterined, taing into account that the uncertain paraeters in x can vary within their respective intervals. These correct ranges of the odal paraeters denoted by ˆ i S and ˆ S i are deterined using a iniization and axiization over the uncertain interval space x I. For nuerical convenience, the global optiization is perfored on the inverted odal paraeters, after which the obtained intervals are inverted in order to obtain the actual odal paraeter ranges: [ ˆ i S = in x x I [ ˆ S i = in x x I φ K i j φ K φ M i j φ M i ), ax x x I i ), ax x x I φ K i j φ K i ) ] 1 φ M i j φ M i ) ] 1 5) 6) If the inverted odal paraeter range contains zero, the inversion results in the union of two interval ranging fro respectively plus and inus infinity to a finite value. These odes are referred to as switch odes. The odes for which the inverted odal paraeter interval has a constant sign, are referred to as strict odes, and classified in either positive or negative odes, based on the sign of the interval. step 1.2 The odal envelope FRF is calculated by substituting the ranges of the odal paraeters in the deterinistic expression of the odal FRF contribution: ) FRF i I 1 j = ˆ i S 2 ˆ S 7) i This is an analytical procedure perfored using the interval arithetic approach. step 2 Finally, the total interval FRF is obtained by the suation of the contributions of all considered odes: FRF I j = n i=1 FRF i j ) I 8) Also this final step is perfored using interval arithetics.

4 132 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 deterinistic algorith interval algorith step 1.1 step 1.1 odal part ˆ i = φ i T Kφ i φ i j φ i = φ i T Mφ i φ i j φ i = 1 φ K i j φ K i 1 = φ M i j φ M i ˆ S i = in x x I φ K i j φ K i, ax x x I ˆ S i = in x x I φ M i j φ M i, ax x x I φ K i j φ K i φ M i j φ M i 1 1 } optialisation suation step 1.2 FRFj i 1 = ˆ i 2 step 2 FRF j = n i=1 FRF i j step 1.2 step 2 FRF i j FRF I j = I 1 = ˆ i S 2 ˆ S i n i=1 FRF i j I } interval arithetic Figure 1: Translation of the deterinistic odal superposition algorith to an equivalent interval procedure 2.3 Eigenvalue interval correction The ethod as described above is enhanced based on a graphical interpretation of the odal part of the interval algorith. For each ode, consider the doain of odal ass and stiffness pairs that are achieved by considering the coplete range of odels defined by the interval uncertainty space x I : { ) )} ˆ i, = ˆ i, x x I 9) This doain defines a bounded area in a ˆ i, - worspace. The exact bounds of this doain however, are generally unnown. The odal part of the interval algorith now is interpreted in this worspace. Fro the optiization as described in step 1.1, it is clear that, for a strict ode, the calculated ranges on the odal paraeters ˆ i S and ˆ S i represent a rectangular approxiation of the actual ˆ i, -doain. Therefore, this ethod is referred to as the Modal Rectangle MR) ethod. Figure 2 shows a general ˆ i, -doain and its approxiation using the MR ethod. The interval arithetic procedure for the calculation of the odal envelope FRF contributions in step 1.2 is interpreted in the sae graphical do- ˆ i ˆ S i ˆ i, Figure 2: Graphical illustration of a ode s ˆ i, -doain and its approxiation using the odal rectangle ethod ain. The goal in this step is to derive the bounds on the deterinistic odal FRF, taing into account that ˆ i and are located anywhere inside their intervals. By considering the odal FRF contribution as defined in Eq. 7 as an analytical function of ˆ i and, the bounds on this function over the odal rectangle have to be deterined. It has been shown that this can be done analytically for the coplete frequency doain, by considering only the function evaluations at the upper left and the lower right corner points of the rectangle. For switch odes, the odal doain spans ˆ S i

5 Envelope frequency Response Function Analysis 133 out over the first and third quadrant of the odal paraeter space. Still, a siilar interpretation is possible see [Moens and Vandepitte 2004)]). Based on these observations, it becoes clear that the calculation based on the odal rectangle introduces conservatis in the procedure if the actual ˆ i, -doain differs strongly fro the approxiate rectangle. This is often the case, as the odal paraeters are generally strongly coupled through the global syste and therefore show a high degree of correlation. Therefore, an enhanced procedure has been introduced. The enhanceent is based on an iproved approxiation of the ˆ i, -doain. This is achieved by using inforation on the eigenvalue ranges, which are obtained using an additional eigenvalue optiization step in the odal part of the algorith. An eigenvalue interval λi I introduces an extra restriction on the quotient of possible cobinations of the odal paraeters. This restriction is atheatically expressed as: λ i ˆ i λ i 10) Graphically, the eigenvalue bounds represent lines through the origin of the ˆ i, -space tangent to the actual ˆ i, -doain. These lines are extra deliiters for the ˆ i, -doain approxiation, and therefore give rise to an iproved ˆ i, - doain approxiation as illustrated in figure 3. This doain is referred to as the Modal Rectangle with Eigenvalue interval correction MRE). ˆ i ˆ i = λ i c 2 c 1 ˆ i, c 4 c 3 ˆ i = λ i Figure 3: Effect of the introduction of the exact eigenvalue interval in the ˆ i, -doain approxiation of a positive ode It has been shown that the conservatis in the odal envelope FRF contributions derived in step 1.2 is substantially reduced by considering the MRE doain instead of the MR doain as area of possible odal paraeter pairs. The corresponding odal envelope FRF contributions are deterined analytically by calculating the deterinistic odal FRFs at the vertex points of the MREdoain indicated with c i, i = in figure 3). This yields: ) FRF i I j = λ i λ i ), ˆ i λ i 2 ˆ i λi 2) for 2 λ i λ i 1 ), ˆ i λ i 2 λi 2) for λ i < 2 < λ i 1 1 ), λ i 2 λi 2) for λ i 2 for positive odes, and: 11) ) FRF i I j = λ i λ ˆ i λi 2), i ) for 2 λ i ˆ i λ i 2 1 λ i λi 2), ) for λ i < 2 < λ i ˆ i λ i λi 2), ) for λ i 2 λ i 2 12) for negative odes. A siilar procedure was derived for the bounds on the odal FRF contributions of switch odes. It has been shown that after applying the final suation step, this enhanceent on the odal level of the algorith leads to a close and guaranteed outer approxiation of the actual odal envelope FRF contribution [ Moens and Vandepitte 2004)].

6 134 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , Introduction of independent odal daping intervals in the envelope FRF procedure 3.1 Overview of the algorith For daped structures with odal daping paraeters ĉ i defined for each ode taen into consideration, the odal superposition procedure states that the total FRF equals: FRF j = = n i=1 n i=1 1 ˆ i + jĉ i 2 13) FRF i j 14) extended in a third diension in which the odal daping paraeter range is represented. As this odal daping interval is ipleented as an independent odel uncertainty, it has no relation with the other odal paraeters. Hence, the extension in this third diension coes down to an extrusion of the existing MRE ˆ i, -doain over the interval defined for the corresponding odal daping paraeter. Figure 4 illustrates the resulting polyhedron in the new three diensional odal ˆ i,,ĉ i ) -space. ˆ i with FRF i j the ith odal FRF and ˆ i and the noralized odal paraeters as defined in equations Eq. 3 and Eq. 4, and ĉ i the corresponding odal daping paraeter, expressed as: ĉ i ˆ i,, ĉ i ĉ i = 2η i ˆ i 15) using η i in its classical definition as the odal daping ratio. Equation 13 shows that the daped FRF calculation through the odal superposition principle is very siilar to the undaped case. Copared to the undaped FRF, the ain difference caused by the addition of daping is that the odal FRF contributions FRFj i are coplex functions. Also, the odal daping paraeter ĉ i appears explicitly in the expression of the odal contributions. Again, the strategy for the calculation of the range of the odal contributions is based on the definition of a doain of possible odal paraeters. The evolution of the real and iaginary part of a odal contribution over this doain then deterines the envelope bounds of both parts. For the odal ass and stiffness paraeters, the feasible doain is inherited fro the undaped MRE-concept as described in section 2.3. The MRE ˆ i, -doain has to be extended in order to tae the daping uncertainty into account. By placing an interval on the odal daping ratio for each ode, an interval on the odal daping paraeters is obtained using Eq. 15. This eans that the two-diensional MRE ˆ i, -doain is Figure 4: Graphical representation of the approxiation of the ˆ i,,ĉ i -doain after addition of the independent odal daping interval to the MRE ˆ i, -doain approxiation Now that we have an outer approxiation of the odal ˆ i,,ĉ i -doain, the range of the odal contributions FRF j as defined in Eq. 13 have to be calculated. The algorith processes the real and coplex parts of the response separately. For both parts, the response range is calculated for each ode. The superpositionthen constructsthe range of the real and iaginary parts of the total response by adding together all real respectively iaginary odal FRF contributions. Finally, based on these results, the aplitude and phase are directly derived fro the total real and iaginary envelope FRFs. In suary, the outline of the MRE algorith extended to structures with odal daping uncertainties is the following: 1. for all considered odes: a) calculate the MRE ˆ i, -doain approxiation by calculating the range of

7 Envelope frequency Response Function Analysis 135 the odal ass and stiffness paraeters and the eigenfrequency b) derive the intervals on the odal daping paraeters based on odal daping ratio intervals c) calculate the range of the odal real and iaginary FRF based on the MRE ˆ i,,ĉ i -doain approxiation 2. su the odal real and iaginary envelope FRFs to obtain a conservative approxiation of the total real and iaginary envelope FRFs 3. post-process the real and iaginary parts to obtain the total aplitude and phase envelope FRF Step 1a is copletely siilar to the undaped case see section 2.3 or [Moens and Vandepitte 2004)] for details). Also, step 1b is straightforward, and results directly fro the definition of the treated proble. The core of the theoretical developent is in step 1c which consists of the analytical derivation of the odal real and iaginary envelope FRFs given the MRE ˆ i,,ĉ i - doain approxiation as illustrated in figure 4. Sections and describe these in detail for a positive ode. Sections and describe how to obtain the equivalent procedure for negative respectively switch odes. The final processing of the real and iaginary parts to the aplitude and phase envelopes is briefly suarized in section Modal real envelope FRF calculation for positive odes The real part of the odal FRF in Eq. 13 equals: R FRFj i ) ˆ i 2 = ) ˆ i ) +ĉ 2 i 2 This is further referred to as the odal real FRF. At a specific frequency, it can be regarded as an analytical function of the three odal paraeters ˆ i, and ĉ i. The range of this function taing into account that the odal paraeters are inside the used polyhedral ˆ i,,ĉ i -doain approxiation is now derived analytically. A first iportant observation is ade regarding the ) evolution of the odal real FRF for ˆ i,,ĉ i -locations on a line parallel to the ĉi - axis in the odal paraeter space. By taing the derivative of the real part function in the ĉ i - direction at constant values for and ˆ i,itis easily shown that the real part function always evolves onotonically over these lines. Therefore, in order to find the bounds on the range of this function over these lines, only the lower and upper bound of the odal daping paraeter interval have to be taen into consideration. This eans that the global bounds of the real part response function over the polyhedral ˆ i,,ĉ i -doain can be calculated by analyzing the ˆ ) i,,ĉ i -locations in the two-diensional MRE doains at ĉ i = ĉ i and ĉ i = ĉ i. The behavior of the real part response function over horizontal, vertical and sloped lines in the two-diensional MRE doain at a specific odal daping value c is now first analyzed. This will lead to the the procedure for the calculation of the odal real envelope FRF described at the end of this section. Lines parallel to the ˆ i -axis For the analysis of the behavior of the odal real FRF for odal paraeter cobinations on vertical lines parallel to the ˆ i -axis) in the odal paraeter space, the constant values and c are introduced for the odal ass and daping paraeters ) in Eq. 16. This results in a function R FRFj i which,c has only ˆ i as variable. The analysis now focuses on the variation of this function when the odal stiffness [ ] paraeter varies over a positive interval,. It can be shown that the odal real FRF as a function of ˆ i has exactly one iniu and one axiu as illustrated in figure 5. The extrea are reached for ˆ i equal to respectively: + ) = 2 +c 17) ) = 2 c 18) The explicit function of indicates that the locations of the extrea depend on the frequency.

8 136 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 R FRF i j,c ) +) Figure 5: Evolution of the odal real FRF over the ˆ i doain for a constant odal ass and daping value ˆ i + ) ) Figure 6: Evolution of ) and + ) in the frequency doain Figure 6 illustrates the global for of both the location of the axiu + ) and the iniu ) as a function of the frequency. Both functions start in the origin of the ˆ i -axis. This eans that for low frequencies, both extrea in figure 5 are located to the left-hand side of any positive [, ] interval on the ˆ i -axis. For increasing frequency, both extrea approach and possibly cross the interval. This can be interpreted as if the response function of figure 5 evolves globally to the right relative to the [, ] interval. The iplications of this evolution for the lower and upper bound of the response is discussed by analyzing the frequency doain in increasing direction. ˆ i The lower response bound: For sall frequencies, both extrea are situated to the left of the [, ] interval. Therefore, the lower bound on the real response corresponds to the upper bound of the interval as indicated in figure 7a). When the frequency increases, the response function shifts to the right and the response axiu enters the [, ] interval. Further, a frequency is reached for which the response values are equal for and figure 7b)). Fro this frequency on, the response iniu [ ] is located on the lower bound of the, interval figure 7c)). When the frequency increases further, the response iniu enters the [, ] interval for soe frequency. Starting fro this frequency, the response iniu is located in the extreu inside the [, ] interval figure 7d)) and the inial value of the response is obtained for the corresponding ) values. Finally, the response iniu reaches the upper bound of the interval for soe frequency. Fro this frequency on, the function over the interval becoes onotonically decreasing and the response iniu is located in the upper bound of the interval figure 7e)). The upper response bound: For sall frequencies, both extrea are situated to the left of the [, ] interval. Therefore, the upper bound on the response corresponds to the lower bound of the interval as indicated in figure 7a). Since + ) is onotonically increasing, the location of the response axiu always enters the [, ] interval for soe frequency. Starting fro this frequency, the response axiu is located in the extreu inside the [, ] interval figure 7b)). For these frequencies, the axial value of the response is obtained for the corresponding + ) values. Once the response axiu has reached the upper bound of the [, ] interval, the response axiu is located in the upper bound of the interval figure 7c)). When the frequency increases further, the response iniu enters the [, ] interval. A frequency is reached for which the response values are equal for and figure 7d)). Fro this frequency on, the response axiu is located on the lower bound of the [, ] interval figure 7e)). This description shows that the exact upper and lower bounds on the odal real FRF above an

9 Envelope frequency Response Function Analysis 137 R FRF i j,c R FRF i j,c ˆ i ˆ i a) b) R FRF i j,c R FRF i j,c ˆ i ˆ i c) d) R FRF i j,c ˆ i e) Figure 7: a)-e) Evolution of the odal real FRF above an interval on a vertical line in the odal paraeter space for increasing frequencies interval on a vertical line in the odal paraeter space follow directly fro an analytical procedure. The procedure consists of selecting the correct ˆ i value which, cobined with the value at the vertical line, yields the bounds of the odal real FRF at c. This correct ˆ i value always lies either on one of the bounds of the [, ] interval, either in an extreu of the odal real FRF inside the interval. The evolution of this ˆ i value as a function of is illustrated in figure 8a) for the lower bound and in figure 8b) for the upper bound. The frequencies 1 l, l 2, u 1 and u 2 represent the points where respectively ) and + ) cross with ˆ i = and ˆ i =. Therefore, they satisfy: = l 12 c l 1 19) = 22 l c 2 l 20) = 1 u2 +c 1 u 21) = u 2 2 +c u 2 22) Furtherore, l = and u = follow directly fro satisfying the equation: R FRF i j =) ),,c = R FRF i j =) ),,c 23) This eans that the curves of ˆ i values that yield the bounds on the response at any frequency as

10 138 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 ˆ i + ) ) given in figure 8 are copletely described analytically. Hence, the bounds on the real FRF over the vertical line can be calculated by substituting the values on this curves into Eq. 16. ˆ i l = l 1 l 2 + ) u 1 u 2 a) lower bound u = b) upper bound ) Figure 8: Evolution of the ˆ i value corresponding to the extrea of the odal real FRF at a vertical line in the odal paraeter space - a) lower bound b) upper bound R FRF i j,c ) +) Figure 9: Evolution of the odal real FRF over the doain for a constant odal ass and daping value + ) ) Figure 10: Evolution of ) and + ) in the frequency doain Lines parallel to the -axis Theprocedureto calculate the bounds on the odal real FRF above a horizontal line in the odal paraeter space is very siilar to the one for the vertical lines described above. Constant values and c are introduced for the odal stiffness and daping paraeters in ) Eq. 16. This results in a function R FRFj i which has only as variable.,c When the odal ass paraeter varies over a positive interval [,], the odal real FRF as a function of has exactly one iniu and one axiu as illustrated in figure 9. The extrea are reached for equal to respectively: + )= 2 + c )= 2 c 24) 25) Also here, the locations of the extrea depend on the frequency as illustrated in figure 10. Both functions tend to infinity at = 0, and tend to zero when tends to infinity. The iplications of the evolution of ) and + ) for the lower and upper bound of the response is siilar to the description for the vertical boundary in figure 7. The difference is that in this case the locations of the extrea decrease for increasing and that the iniu and axiu have switched positions. Still, the exact upper and lower bounds on the odal real FRF above an interval on a horizontal line in the odal paraeter space follow directly fro a siilar analytical procedure. The procedure consists of selecting the correct value which, cobined with the and c value at the horizontal line, yields the lower and upper bound of the odal real FRF. The evolution of this correct value as a function of is illustrated in figure 11a) for the lower bound and in figure 11b) for the upper bound. Again, the curves in figure 11 are fully described analytically, as the frequencies 1 l, l 2, u 1 and

11 Envelope frequency Response Function Analysis 139 l = u 1 u 2 + ) l 1 l 2 a) lower bound + ) u = b) upper bound ) ) Figure 11: Evolution of the value corresponding to the extrea of the odal real FRF at a horizontal line in the odal paraeter space - a) lower bound. b) upper bound u 2 represent the points where respectively +) and ) cross with = and =,and l = and u = follow directly fro satisfying: R FRF i j = ) ),,c = R FRF i j = ) ),,c 26) Lines with a constant slope in the odal paraeter space The bounds on the odal real FRF above a line with a constant slope are derived analytically. In Eq. 16, a constant value c is introduced for the odal daping paraeter. The sloped line is represented by introducing ˆ i = λ into the sae equation. This results in the function: R FRFj i ) λ 2 ) = ˆ 2 i λ 2 ) 2 +ĉ 2 27) i 2 which has only as variable. The analysis now focuses on the variation of this function when the odal ass paraeter varies over a positive interval [,]. Siilar as for the horizontal and vertical lines in the odal paraeter space, it can be shown that the odal real FRF as a function of has exactly one iniu and one axiu. The extrea are reached for equal to respectively: c + λ )= λ 2 28) )= λ c 2 λ 29) Alsohere, the explicitfunctionof indicates that the locations of the extrea vary with the frequency. In this case however, the exact evolution of these extrea strongly depends on the gradient of the slope λ. Siilar to the horizontal and vertical lines, the extrea of the odal real FRF will be reached either on the bounds of the interval [,], or on the extrea for the frequencies for which the functions + ) and ) are located within the interval [, ]. Therefore, the following procedure is applied: FRF i j 1. a) calculate the odal real FRF given in Eq. 27 for all frequencies at the lower and upper bounds of, resulting respectively in R ) and R FRFj i b) calculate the odal real FRF given in Eq. 27 at λ + + ) and λ ) with: + = { < + λ ) < } 30) = { < ) λ < } 31) ) resulting respectively in R ) FRFj i + λ and R FRFj i 2. calculate the lower bound on the odal real FRF using: in R FRFj i ),R FRF i ) ) j 32) for /,and in R FRFj i ),R FRFj i ),R FRFj i ) ) λ 33) for λ )

12 140 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , calculate the upper bound on the odal real FRF using: ax R FRFj i ),R FRF i ) ) j 34) for / +,and ax R FRFj i ),R FRFj i ),R FRFj i ) ) + λ 35) concluded that if a local axiu is reached inside the doain, it will also be present on the boundary lines of the doain. This observation can be generalized toward local inia. The line of constant local odal real FRF iniu is illustrated in figure 12, and yields: ˆ i = 2 c 38) for + Fro these equations, the bounds on the odal real FRF are now deterined for the coplete frequency doain. ˆ i ˆ i = 2 + c + Miniu and axiu response locations in MRE ˆ i, -doain The analytical analysis of the odal real FRF over vertical and horizontal lines in the MRE ˆ i, -subdoain of the ˆ i,,ĉ i -doain indicates that local optia are possibly present inside the doain. Figure 12 shows the locations + and of local optia of the odal real FRF on a vertical line at =. The response value at the local optia locations on the line, however, proves to be independent of the location of the line. Indeed, as the local axiu of the odal real FRF at the frequency on a vertical line at,c ) is found at + ), substituting + ) as defined in Eq. 17 together with and c in Eq. 16 yields the response value at the local axiu: R FRFj i = ) 1,c, + 2c 36) Since this value is independent fro,theresponse value at the local axiu found on a vertical line in the doain does not depend on the location of the line. Consequently, the sae axiu response value is reached at ˆ ) i, - cobinations located on the line represented by + ): ˆ i = 2 +c 37) This equation represents a straight line in the ˆ i, -doain, as indicated in figure 12. As this line crosses with the MRE ˆ i, -bounds, it is ˆ i = 2 c Figure 12: Locations of the local axiu of the odal real FRF at a vertical line at = and the corresponding lines of constant optia In conclusion, we can state the if local optia are present inside the two-diensional MRE subdoain, they are also present on the boundary lines of this subdoain. Therefore, the final procedure to derive the odal real FRF bounds focuses exclusively on the behavior of the odal real FRF above the MRE ˆ i, -doain boundary lines. Modal real envelope FRF procedure In order to find the range of values that the odal real FRF can reach inside the polyhedral ˆ i,,ĉ i -doain illustrated in figure 4, the following procedure applies: 1. for both the MRE ˆ i, -subdoains at ĉ i and ĉ i : a) analyze the optia of the odal real FRF for the vertical boundary lines b) analyze the optia of the odal real FRF for the horizontal boundary lines

13 Envelope frequency Response Function Analysis 141 c) analyze the optia of the odal real FRF for the sloped boundary lines representing the eigenvalue interval bounds 2. envelope the ranges of the odal real FRF obtained in the previous step The ipleentation is done based on the definition of the MRE ˆ i, -doain as illustrated in figure 3. Step 1a is ipleented by applying the vertical line procedure on the line segents at = and = with ˆ i ranging respectively fro ˆ i to λ i and fro λ i to ˆ i. Step 1b is ipleented by applying the horizontal line procedure on the line segents at ˆ i = ˆ i and ˆ i = ˆ i with ranging respectively fro to ˆ i /λ i and fro ˆ i /λ i to. Step 1c is ipleented by applying the sloped line procedure with λ respectively equal to λ i and λ i,and ranging respectively fro ˆ i /λ i to, and fro to ˆ i /λ i. Finally in step 2, the odal real FRF range for the coplete ˆ i,,ĉ i -doain is derived by taing the union of the odal real FRF envelopes found in the MRE ˆ i, -subdoains at both ĉ i and ĉ i Modal iaginary envelope FRF calculation for positive odes The iaginary part of the odal FRF in Eq. 13 equals: I FRFj i ) ĉ i = ) ˆ i ) +ĉ 2 i 2 This is further referred to as the odal iaginary FRF. Also here, the range of this function taing into account that the odal paraeters are inside the used polyhedral ˆ i,,ĉ i -doain approxiation is derived analytically by considering the behavior of the function over specific lines in the odal paraeter space. In this case, the doain of possible locations of extree values of the odal iaginary FRF is reduced significantly by first considering radial lines in the ˆ ) i, -space at a specific ĉi. It can be shown that on these lines, the odal iaginary FRF has a onotonic behavior with respect to the radial distance to the origin of the ˆ ) i, - space. Referring to the global polyhedral for of the ˆ i,,ĉ i -doain approxiation as illustrated in figure 4, this eans that no local optia are present inside the doain, neither on the planes resulting fro extruding the MRE eigenvalue deliiters in the odal daping paraeter direction. The analytical search for optia therefore is liited to the horizontal and vertical boundary planes of the ˆ i,,ĉ i -doain. These planes are now analyzed by considering the lines on these planes parallel to the odal paraeter axes. Lines parallel to the ĉ i -axis For the analysis of the behavior ) of the odal iaginary FRF for ˆ i,,ĉ i -locations on a line parallel to the ĉi - axis, constant values and are introduced for the odal ass and stiffness paraeters ) in Eq. 39. This results in a function I FRFj i which, has only ĉ i as variable. The analysis focuses on the variation of this function when ĉ i varies over a positive interval [c,c]. It can be shown that the odal iaginary FRF as a function of ĉ i has exactly one axiu and one iniu as illustrated in figure 13. The axiu and iniu are reached for ĉ i equal to respectively: c + )= 40) c )= 41) It is clear that c ) is negative for all frequencies. Therefore, it is not of iportance in the derivation of the response bounds over a positive [c,c] interval. Figure 14 illustrates the global for of the evolution of c + ) over the frequency doain. The exact upper and lower bounds on the odal iaginary FRF above an interval on the analyzed line in the odal paraeter space follow directly fro an analytical procedure. The ĉ i -values for which the upper and lower bounds on the response are reached are always located either on one of the bounds of the [c,c] interval, or in the extreu locations c + or c inside the interval. The evolution

14 142 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 I FRF i j, ĉ i c + ) c c + ) c ) ĉ i c Figure 13: Evolution of the odal iaginary FRF over the ĉ i doain for a constant odal stiffness and ass value ĉ i c l 1 l 2 l 3 l 4 c + ) a) lower bound ĉ i c + ) c Figure 14: Evolution of c + ) in the frequency doain u =1 u =2 b) upper bound Figure 15: Evolution of the ĉ i value corresponding to the extrea of the odal iaginary FRF at the analyzed line in the odal paraeter space - a) lower bound b) upper bound. of this optial ĉ i value as a function of is illustrated in figure 15a) for the lower bound and in figure 15b) for the upper bound. The frequencies 1 l and l 2 represent the lower points where c + ) crosses respectively with = and =. Siilarly, the frequencies 3 l and 4 l represent the upper points where c +) crosses respectively with = and =. Therefore, they satisfy: c = 1 l 1 l 42) c = 2 l 2 l 43) c = 3 l 3 l 44) c = l 4 l 4 45) Furtherore, =1 u and u =2 follow directly as the two roots resulting fro the equation: I FRF i j u = )) c,, = I FRF i j u = )) c,, 46) This eans that the curves of optial ĉ i values given in figure 15 are copletely described analytically, and the bounds on the iaginary FRF over the analyzed line can be calculated by substituting the values on these curves into Eq. 39. Lines parallel to the ˆ i -axis The introduction of constant values and c for the odal ass and daping paraeters ) in equation 39) results in a function I FRFj i which has only ˆ i as,c variable. If ˆ i varies over a positive interval [, ], the odal iaginary FRF as a function of ˆ i has exactly one iniu as illustrated in figure 16. This iniu is reached for ˆ i equal to: 1 )= 2 47)

15 Envelope frequency Response Function Analysis 143 I FRF i j,c 1) ˆ i ˆ i 1 ) Figure 16: Evolution of the odal iaginary FRF over the ˆ i doain for a constant odal ass and daping value ˆ i l 1 l 2 a) lower bound 1 ) The location of the iniu 1 ) is a onotonically increasing function of the frequency. Again, the exact upper and lower bounds on the odal iaginary FRF above an interval on the analyzed ˆ i -line in the odal paraeter space follow directly fro an analytical procedure. The evolution of the ˆ i values that yield the bounding functions of the response range is illustrated in figure 17a) for the lower bound and in figure 17b) for the upper bound. The frequencies 1 l and l 2 represent the points where 1 ) crosses respectively with ˆ i = and ˆ i =. Therefore, they equal: 1 l = 48) 2 l = 49) Furtherore, u = follows directly fro satisfying the equation: I FRF i j u =) ),,c = I FRF i j u =) ),,c 50) which yields: u = = ) This eans that the curves of optial ˆ i values given in figure 17 are copletely described analytically, and the bounds on the iaginary FRF u = b) upper bound Figure 17: Evolution of the ˆ i value corresponding to the extrea of the odal iaginary FRF at the analyzed line in the odal paraeter space - a) lower bound. b) upper bound over the analyzed line can be calculated by substituting the values on these curves into equation 39). Lines parallel to the -axis For the analysis of the behavior ) of the odal iaginary FRF for ˆ i,,ĉ i -locations on lines parallel to the ˆi - axis, constant values and c are introduced for the odal stiffness and daping paraeters in equation ) 39). This results in a function I FRFj i which has only as variable.,c Again, it can be shown that when varies over a positive interval [,], the odal iaginary FRF as a function of has exactly one iniu as illustrated in figure 18. This iniu is reached for equal to: 1 )= 2 52) The location of the iniu 1 ) starts fro infinity at = 0 after which it tends to zero when

16 144 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 I FRF i j,c 1) 1 ) Figure 18: Evolution of the odal iaginary FRF over the doain for a constant odal stiffness and daping value l 1 l 2 1 ) a) lower bound the frequency tends to infinity. The exact upper and lower bounds on the odal iaginary FRF above the analyzed line in the odal paraeter space follow directly fro an analytical procedure. The evolution of the values that yield the bounding functions of the response range is illustrated in figure 19a) for the lower bound and in figure 19b) for the upper bound. The frequencies 1 l and l 2 represent the points where 1 ) crosses respectively with = and =. Therefore, they equal: 1 l = 53) 2 l = 54) Furtherore, u = follows directly fro satisfying the equation: I FRF i j u =) ),,c = I FRF i j u =) ),,c 55) which yields: u = = ) This eans that the curves of optial values given in figure 19 are copletely described analytically, and the bounds on the iaginary FRF over the analyzed line can be calculated by substituting the values on this curves into equation 39). u = b) upper bound Figure 19: Evolution of the value corresponding to the extrea of the odal iaginary FRF at the analyzed line in the odal paraeter space - a) lower bound b) upper bound. Miniu and axiu response locations on horizontal and vertical planes The analytical analysis of the odal iaginary FRF over lines parallel to the ĉ i -axis in the ˆ i,,ĉ i -doain indicates that a local iniu is possibly present inside the considered [ ] ĉ i,ĉ i interval. The response value at the local iniu, however, proves to be independent of the location of the line. Indeed, if the location of the local iniu is not on the boundary of the doain, it is found at c + ). Substituting c + ) as defined in Eq. 40, and in Eq. 39 yields: I FRFj i = ) 1,,c + 2c 57) Since this value is independent fro both and, the response value at the local iniu found on a line parallel to the ĉ i -axis does not depend on the location of the line. Hence, in order to find the range of the odal iaginary FRF over a horizontal or vertical plane in the ˆ i,,ĉ i ) -space,

17 Envelope frequency Response Function Analysis 145 only the boundary lines of these planes have to be analyzed. Modal iaginary envelope FRF procedure The procedure for the analytical response range analysis of the odal iaginary FRF is the following: 1. analyze the optia of the odal iaginary FRF for the ˆ i,,ĉ i -doain boundary lines parallel to the ĉ i -axis 2. analyze the optia of the odal iaginary FRF for the vertical boundary lines of the polyhedral ˆ i,,ĉ i -doain approxiation 3. analyze the optia of the odal iaginary FRF for the horizontal boundary lines of the polyhedral ˆ i,,ĉ i -doain approxiation 4. envelope the ranges of the odal iaginary FRF obtained in the previous steps The ipleentation is done based on the definition of the MRE ˆ i, -doain as illustrated in figure 3 and its extrusion in the ĉ i -direction as illustrated in figure 4. Step 1 is ipleented based on the analysis on lines parallel to the ĉ i -axis described above at the lines with the following, )-locations: ˆ i, ), λ i, ), ˆ i, ˆ i /λ i ), ˆ i, ), λ i, ) and ˆ i, ˆ i /λ i ). Step2isipleented by applying the vertical line procedure on the line segents at = and = with ˆ i ranging respectively fro ˆ i to λ i and fro λ i to ˆ i, both at ĉ i and ĉ i. Step 3 is ipleented by applying the horizontal line procedure on the line segents at ˆ i = ˆ i and ˆ i = ˆ i with ranging respectively fro to ˆ i /λ i and fro ˆ i /λ i to, both at ĉ i and ĉ i. Finally in step 4, the odal iaginary FRF range for the coplete ˆ i,,ĉ i -doain is derived by taing the union of the odal iaginary FRF envelopes found in the previous steps The odal envelope FRF for negative odes The analytical procedures in sections and consider only positive ranges for the odal paraeters. They require that the ˆ i,,ĉ i -doain approxiations of the analyzed odes are located in the first quadrant of the odal paraeter space, and, therefore, are liited to positive odes. The goal is to apply these procedures also for negative odes. This can be achieved by considering the MRE ˆ i,,ĉ i - doain approxiation of a negative ode irrored into the first quadrant of the odal paraeter space. This is equivalent with switching the sign of the odal paraeters. The corresponding horizontal and vertical boundaries on the irrored odal paraeters are then: ˆ i = ˆ i ) 58) ˆ i = ˆ i ) 59) = ) 60) = ) 61) ĉ i = ĉ i ) 62) ĉ i = ĉ i ) 63) It is easily shown that for negative odes, the correct odal real and iaginary envelope FRFs are obtained by copensating for the odal paraeter sign inversion in the following way: R FRFj i ) = ˆ i,,ĉ i ) R FRFj i ˆ i,, ĉ i ) 64) ) R FRFj i = R ) FRF i ) j ˆ i,,ĉ i ˆ i,, ĉ i 65) for the real part, and: I FRFj i ) ) ) = I FRF ˆ i i,,ĉ i j ˆ i,, ĉ i ) I FRFj i = I ) FRF i ) j ˆ i,,ĉ i ˆ i,, ĉ i 66) 67) for the iaginary part. The conclusion is that with an appropriate pre- and post-processing of

18 146 Copyright c 2007 Tech Science Press CMES, vol.22, no.2, pp , 2007 the data used in and obtained fro the procedures for the positive odes, the equivalent procedure for the negative odes is easily ipleented The odal envelope FRF for switch odes For a switch ode, the MRE ˆ i, -doain approxiation ranges over infinity in the odal paraeter space. It is easily shown that both the odal real and iaginary FRF tend to zero when the odal paraeters tend to infinity inside the MRE ˆ i, -doain approxiation. Therefore, the ranges of the odal real and iaginary FRF should contain zero for all frequencies. Therefore, the procedure consists of the following steps: 1. the calculation of the upper and lower bounds on the odal real and iaginary FRF for the MRE ˆ i,,ĉ i -doain approxiation both in the first and the third quadrant 2. taing the axiu of all upper bounds and iniu of all lower bounds of the envelope functions resulting fro the previous step 3. the calculation of the odal real and iaginary envelope FRFs by correcting positive lower bounds and negative upper bounds in the result of the previous step to zero The derivation of the range of the odal FRFs in step 1 results fro applying the procedure for positive and negative odes described above to the finite boundaries in respectively the first and third quadrant of the MRE ˆ i,,ĉ i -doain approxiation of the switch ode. 3.2 Total aplitude and phase envelope FRF calculation After the calculation of the odal real and iaginary envelope FRF of all odes, the total aplitude and phase envelope FRF are calculated using the total real and iaginary envelope FRFs. These result fro the suation of the odal contributions derived in the previous sections: R FRF j ) I = n i=1r FRF i j) I 68) I FRF j ) I = n i=1i FRF i j) I 69) The result of the suation is an interval range for the real and iaginary part of the coplex response for every frequency. This eans that it defines a rectangle in the coplex space in which the response vector is contained. Based on this rectangle, an approxiation of the aplitude range of the coplex response is easily obtained by taing the points on the rectangle which are respectively the nearest and ost distant fro the origin. 4 Nuerical exaple The presented ethodology is now illustrated on a realistic case study. The odel under investigation represents a car windshield in free-free conditions. The windshield is coposed of five layers: two outer glass layers, and three inner layers of various polyers. During anufacturing, the thicnesses of these layers are subject to tolerances. The tolerances are defined as a range of thicness for the total of the five layers, and a range of total thicness for the inner polyer layers. Furtherore, daping variation has been detected fro experients. The ai of this case study is to analyze the effect of the anufacturing design tolerances in cobination with the daping uncertainty on the dynaical behavior of the windshield. Figure 20 shows the noinal finite eleent odel of the windshield courtesy of Renault, France). The odel contains a total of 798 eleents. In the dynaic response analysis, a direct drive point FRF along the Z-direction is considered, with the input and response location as indicated in figure 20. At this location, the coponents of the force and displaceent noral to the surface are considered. The response is analyzed up to a frequency of 180 Hz, which corresponds to the range covered by the 15 first odes.

19 Envelope frequency Response Function Analysis 147 input and output direction x 10 5 drive point location Rx 1,3 /F 1,3 ) x 10 5 Ix 1,3 /F 1,3 ) 1 2 Figure 20: Noinal finite eleent odel of the windshield, with indication of the response location of the direct FRF The windshield is odeled using layered coposite eleents, the individual layer thicnesses of which are paraetrized in order to perfor the global optiization on the odal paraeters, necessary for the MRE ˆ i, -doain approxiation. The daping uncertainty is added to the analysis as an individual uncertain odal daping paraeter for each ode under consideration in the odal superposition. The odal daping factors are derived fro a total of three experients perfored on three different realizations of the windshield. In the interval analysis, the odal daping uncertainty is quantified as the interval enveloping these experiental odal daping factors. This brings the total of independent uncertain paraeters to 17 2 layer thicnesses, 15 odal daping factors). Figure 21 illustrates the odal real and iaginary envelope FRF for the third ode resulting fro the procedure as described in section 3 of this paper. As a verification, a total of 1000 saples has been generated, resulting in the cloud of response curves grey lines in the figure). The saples result fro a Monte Carlo-siulation using a unifor distribution over the tolerance and odal daping intervals. This figure clearly indicates that for this ode, the procedure introduced above finds conservative but very tight bounds on the dynaic response range under the given interval uncertainty Hz Figure 21: Predicted odal real and iaginary envelope FRF of the third ode blac), copared to 1000 saples randoly generated over the interval uncertainty space grey) Figure 22 finally shows the total aplitude envelope response function obtained with the presented ethodology. Again, the total envelopes are copared to the saples generated with the Monte Carlo-procedure. Also here, the predicted bounds are conservative, but give a good indication of the upper bound on the feasible response range. Especially in the critical area where the response of the structure is high, the envelope describes very tight bounds around the sapled response functions. Fro these results, conclusions can be drawn regarding the possible dynaic behavior of the structure, given the uncertain interval bounds on specific odel properties. This can be very valuable inforation for a designer who is interested in the cobined effect of defined tolerances and daping variability on the dynaic behavior of his design. 5 Conclusions This paper explains how the hybrid optiization interval arithetic procedure developed for dynaic response analysis of undaped structures can be extended to the analysis of structures with uncertain odal daping factors. The developed procedure adopts the hybrid interval translation of the odal superposition principle, including