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 DOI: 10.1177/107754639800400302

1998 4: 237Journal of Vibration and ControlGianni Biscontin, Antonino Morassi and Pia Wendel

Asymptotic Separation of the Spectrum in Notched Rods  

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Asymptotic Separation ofthe Spectrum in Notched Rods

GIANNI BISCONTINANTONINO MORASSI

PIA WENDEL

Department of Civil Engineering, University of Udine, Viale delle Scienze 208, 33100 Udine, Italy

(Received 23 May 1996, accepted 16 December 1996)

Abstract: Theoretical analysis of the axial vibration of slender notched rods via localized compliance modelsindicates that when there are N notches, the higher part of the spectrum splits in N + 1 branches. Althoughthe separation property is supposed to be valid only in the asymptotic sense, in this paper the authors show thatthere are cases in which it can be observed and quite accurately measured experimentally, even for relativelylow frequencies.

Key Words: Axially vibrating beams, notches, eigenvalues, modal analysis

1. INTRODUCTION

Vibrational characteristics of slender elastic beams are altered by geometrical disconti-nuities resulting from notches. Several mechanical models have been proposed in thetechnical literature to include the effect of a notch in one-dimensional systems, and anaccount of them can be found in Petroski (1984). A recurring assumption in modeling anotch is that the geometrical discontinuity of the cross-section reduces the effective stiff-ness of the beam in a region close to the discontinuity, but at a certain distance, the system isaccurately described by means of the classical beam theory. Usually, physical experimentsare carried out to establish which is the equivalent step reduction in stiffness for variouswidths and depths of real notches.

A different approach to notch modelization in axially vibrating rods has been presentedby Freddi and Morassi (1996). The basic idea comes from the papers by Acerbi, Buttazzo,and Percivale (1988) and Anzellotti, Baldo, and Percivale (1994), where it is shown thatclassical one-dimensional models of slender beams with a smooth longitudinal profilecan be obtained as the variational limit of the corresponding three-dimensional elasticityproblems when the ratio between a characteristic diameter of the cross-section and thebeam length goes to zero. By adopting a flow of ideas that is typical of the asymptoticdevelopments in the r-convergence of functionals defined on measures, Freddi and Morassi(1996) extend some of the above results to include the occurrence of abrupt geometricalchanges on the longitudinal profile of the beam, such as those induced by a notch.

It turns out that the three-dimensional elasticity problem for a notched rod under axialloads converges to a class of one-dimensional models with localized compliance when the

Journal of Vibration and Control, 4: 237-251, 1998©1998 Sage Publications, Inc.

237-2

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beam becomes very slender and when the ratio between the depth and the width of eachnotch vanishes in a suitable way. In the limit family (indicated by the local compliancemodel [LCM]), the classical one-dimensional model for axial deformations holds betweentwo consecutive discontinuities, and each notch is represented by a translational elasticspring located at the damaged cross-section. This macroscopic description of localizeddamages coincides with the classical one (for stationary cracks) and has been extensivelyused in structural diagnostics (cf. Adams et al., 1978; Dimarogonas and Paipetis, 1983;Gudmunson, 1983), but the novelty is that the notch is now properly considered as ageometrical discontinuity of the beam profile rather than the consequence of some fracturephenomena.

A direct analytical study of vibrating notched rods via LCM shows that the spectrumhas an interesting branching property-namely, the presence of N notches splits highereigenvalues in N + 1 branches that correspond to the asymptotic spectrum of the piecesof bar adjacent to the damaged cross-sections. This shows that the influence of a notchon the higher axial modes is greater than for the lower ones, confirming an observation ofThomson (1949), who adopted a different modelization of the damage. The asymptoticseparation of the spectrum for LCM was pointed out by Morassi (1997), who showed itsusefulness in dealing with the inverse problem of locating notches in a rod from spectraldata. An analogous asymptotic spectral behavior was observed by Chen et al. (1989) instudying the bending vibration of coupled beams with dissipative joints.

Until now, the branching property has been obtained from analytical considerations,first to deduce the LCM as a limit of the three-dimensional notched rod and then to inferfrom it the asymptotic separation of eigenvalues. The situation becomes more complicatedin dealing with real vibrating rods. In fact, analytical one-dimensional models usually offergood precision in estimating the first frequencies, but they lose accuracy for those of ahigher order where, on the contrary, the branching property could be detectable. Accuracyof the analytical model and the asymptotic separation of the spectrum are conflicting facts.Then one can ask for the existence of a frequency range where the branching property canbe experimentally detectable, and the mathematical model turns out to be at the same timesufficiently accurate.

To give general answers to these questions, one should investigate the accuracy bothof the one-dimensional model and of the macroscopic description of the notch. Results

of this kind would be desirable, but no analytical estimates of the approximation resultingfrom the dimension reduction process seem to be available. Taking these aspects intoaccount, dynamic experimentation has been adopted to verify if there are situations inwhich the analytical model is good in a frequency range where the asymptotic separa-tion is detectable. A preliminary series of dynamic tests performed on a slender steelbeam with a solid square cross-section and with a single notch supported the existenceof the asymptotic separation and its practical detection for rather severe levels of damage(cf. Morassi, 1997). In this paper, we present the results of a systematic experimentalstudy of a steel rod with a double T cross-section and with multiple notches of differentdepth. Experiments confirm that the LCM describes accurately the dynamics of the realrod for a relatively large range of frequency and that modelization errors are comparable tothose of the classical model for a vibrating undamaged rod. In particular, the appearance of

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asymptotic branches on the spectrum of the damaged rod has been detected and quiteaccurately measured as well in the case of multiple notches.

The remainder of the paper runs as follows. First, we recall the analytical branch-ing property of the spectrum. Then we present the dynamic tests and comment on theexperimental results.

2. ASYMPTOTIC SEPARATION OF THE SPECTRUM

FOR LOCALIZED COMPLIANCE MODELS OF NOTCHED RODS

To fix the ideas, let us consider the infinitesimal axial vibrations of a free-free slenderbeam of length L and cross-section of constant area A, with a notch at the cross-sectionof abscissa x = dl. To study the axial vibration of the notched rod, we adopt the LCMwhere, according to Freddi and Morassi (1996), the notch is represented by the insertion ofa massless elastic spring of infinitesimal length at the damaged cross-section (cf. Figure 1).The stiffness of the spring is related in a precise way to the geometry of the notch, but sincethe derivation of the LCM holds only in a limit sense, in real situations physical experimentsare carried out to quantify its value. Denoting by X2 the ratio ycon/E, where y is the volumemass density, E the Young’s modulus, and con the nth natural cyclic frequency, the ntheigenpair (un, Àn), n = 0, 1, 2,..., of the notched rod satisfies the following boundaryvalue problem:

where the jump conditions

hold at the notched cross-section. In equation (16), kl is the spring’s stiffness and>1 ~ ~/EA. Moreover, [({J]lx=dl - (~P(di ) - %’(di )), with q;(dt) = lim({J(x) and

x~.dl i

~p(dl ) = 1 m ~p(x), denotes the jump of the function w (x ) evaluated at the point of abscissaxtdix = di .

The eigenvalues (Àn)~o of the notched rod are the countable set of zeros of theWronskian

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Figure 1. The localized compliance model (LCM) of a notched rod.

and it can be shown that they satisfy the asymptotic estimates

where d2 n L - dl, Mi E N and lim (o(1/m;)/(1/m~)) = 0, i = 1, 2. These equationsM, --> 00

.

show that the notch splits the higher eigenvalues into two classes that correspond to theleading term of the asymptotic behavior of the spectrum of two pieces of rod adjacentto the notched cross-section (under suitable boundary conditions). More generally, fora uniform rod with N notches of compliances 1 / kl , 1 / k2, ... , 1 / kN located at abscissas(0 <)dl < di + d2 < ... < dN_1 + dN < dN + dN+l n L, respectively, the asymptoticbranches of the spectrum are

where tt* = k~ /EA(l = 1, N). It can be shown that the branching property holds alsofor rods in which the longitudinal profile is regular enough-that is, when the secondderivative of the cross-sectional area A(x) is a continuous function (cf. Morassi, 1997).There is a quite simple physical interpretation of the asymptotic separation of the spectrum.In fact, as frequency increases, the dynamic stiffness of the rod increases, while the localstiffness of the notch, which is a static quantity, remains constant. Hence, the notchstiffness decreases relatively to the rod dynamic stiffness as the mode number increases.In the limit, the notch stiffness vanishes, and the rod behaves dynamically like a system ofuncoupled rod segments, each leading to a group of eigenvalues.

Estimates (4) allow one to reorder higher eigenvalues into the asymptotic classes(}&dquo;m)~iI according to the closeness of quotients (,k,,/(7r/d,)) to the integers mi, i -1, N + 1. Clearly, the above selection criterion is unambiguous in asymptotic senseonly-namely, for large values of the order of the eigenvalue-but equation (4) showsthat the separation of the damaged rod spectrum could be detected even among vibrating

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modes of relatively low order in the case of severe levels of damage (e.g., for small valuesof A!).

As a remark, we observe that the asymptotic separation of equation (4) is useful indealing with the inverse vibration problem of locating N notches in a rod from spectraldata. In fact, adapting the arguments of Morassi (1996), it can be shown that knowledge ofthe higher part of a spectrum of a damaged rod suffices to determine uniquely the lengthsof the (N + 1 ) segments of bar adjacent to the damaged cross-sections (cf. Colonnello andMorassi, 1998).

3. DYNAMIC TESTS ON A STEEL ROD WITH MULTIPLE NOTCHES

In this section, we give an interpretation of a series of dynamic tests performed on a steelrod with multiple notches via localized compliance models.

3.1. Experimental Results

The model under study is a steel beam of the series HEl00B suspended by two steel wireropes to simulate free-free boundary conditions (cf. Figure 2). The object of the experimentwas to study the effect of multiple notches on the higher axial frequencies of the beam,and to do so, the impulsive technique was adopted. The excitation was introduced at oneend by an impulse force hammer PCB 086B03, while the structural axial response wasacquired by a piezoelectric accelerometer 303A03 (with a mass of 4 grams) fixed in thecenter of the right-end cross-section of the beam, as shown in Figure 2. Both the inputand output signals were acquired by a dynamic analyzer HP3565S, then sampled at theconstant rate of 102.4 KHz and finally decimated in time to obtain aliasing free data in theconsidered frequency range. The input signal was weighted by a &dquo;force&dquo; window with aduration equal to 1/10 of the registration time, while an exponential window (with decayfactor c = 4) was adopted to force to zero the amplitude of the structural response withinthe acquisition time. The signals were then worked out in the frequency domain to measurethe relevant frequency response term (inertance).

To localize the axial frequencies, we performed preliminary band analysis tests, with aresoluti on of 8 Hz and with a number of 10 averages for each measurement, from 0 to 12800Hz both for the undamaged and damaged configurations. The choice to restrict the analysiswithin this frequency range was motivated by the observation that, starting from about13 KHz, the behavior of the inertance term became very irregular (see Figure 3b). Sincethe accelerometer guarantees a quite good response until 20 KHz and since the spectrum ofthe excitation was sufficiently flat in this range, the above behavior was probably due to theinadequacy of the mathematical model to describe the dynamics of the higher frequencies.After these preliminary tests, throughout the experiments we performed zoom analysis inthe interval ( fn - 25Hz, fn + 25Hz) around each proper frequency fn(= wn/2n), with aresolution of 1/8 Hz. In all cases, because of the repetition of single measurements, eachinertance term was evaluated as an average of three impulsive tests.

With this experimental setup, we studied the undamaged bar and a series of damagedconfigurations obtained by introducing two notches of different severeness at two cross-sections of the rod (see Figure 2). The notches were obtained by saw-cutting the cross-

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242

Figure 2. Experimental model and damaged configurations D1-D4.

section at progressive depth, taking care to avoid the introduction of permanent distortionsthroughout the beam.

The measured inertance function for the undamaged configuration was quite regularand close to the expected one for the one-dimensional model of axially vibrating rods (seeFigure 3a). The occurrence of well-separated vibration modes and the very small dampingallowed us to identify the frequencies by the single-mode technique. The third column ofTable 1 shows the measured values for the first 20 modes (the frequency of the Oth &dquo;rigid&dquo;translational mode is omitted).

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Figure 3a-3c. Inertance term (magnitude) for undamaged (a, b) and damaged configurations D1-D4 (c-f).

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Figure 3d-3f. inertance term (magnitude) tor undamaged (a, b) and damaged contigurations u~-D4 (c-f).

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2a)m0M 00 0

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S g-O n£0) Is±

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246

Because of some coupling detected between the axial and the flexural and torsionalvibration modes, the interpretation of the measurements on damaged configurations turnedout to be less simple then the previous one. In fact, in spite of the special care adoptedto control the direction of the excitation, throughout the experiments several &dquo;spurious&dquo;modes with frequencies different from those predicted by the mathematical model of theaxially vibrating beam were present in the selected frequency range. The above character,detected near the frequencies marked with an asterisk in Table 1, was particularly evidentfor the damaged configurations D and D3, probably because of the nonsymmetric positionof the notch in the cross-section (see Figures 3c, 3e). For each of these cases, we performadditional dynamic tests with transverse impulsive excitation to select the axial frequenciesand discard the &dquo;spurious&dquo; ones. Moreover, for configurations D2, D3, and D4, it was foundconvenient to excite the rod at the same cross-section of the accelerometer to measurethe inertance term more accurately. However, the origin of the slight increase of somefrequencies measured from one configuration to another with more severe damage has notbeen clarified (see, e.g., the 11th eigenfrequency from the undamaged configuration to D1 Ior the 2nd frequency from D2 to D3).

Taking the experimental difficulties into account, we decided to restrict the interpre-tation of the measurements within the first 17 modes. Proper frequencies were extractedfrom inertance terms as before, and the results of the modal analysis are summarized inTable 1.

3.2. Interpretation of Measurements via Localized Compliance Model

For the interpretation of dynamic tests on the undamaged configuration, we adopted theclassical one-dimensional model for axial vibrations of slender beams, with mass densityyA = 20.775 kg/m and axial stiffness EA = 5.55078E8 N. The mass density is evaluatedfrom the total mass, M = 83.100 kg, of the specimen under the hypothesis that the rod beuniform, while axial stiffness is calculated by matching the theoretical and experimentalvalues of the fundamental frequency. The latter assumption is motivated by the observationthat the mathematical model is more adequate to describe the dynamics of low frequencies.A comparison between experimental and theoretical frequencies is shown in the first threeleft columns of Table 1. The agreement is very good, with percentage errors less then 1 %up to the 14th mode; starting from the 15th mode, modelization errors quite rapidly increasewith the order of modes, and we measured a deviation of about 5% for the 20th mode.It is worth noticing that, apart from the seventh and eighth frequencies, the mathematicalmodel overestimates the frequencies of the real beam according to general monotonicitytheorems.

Measurements on damaged configurations D1-D4 are interpreted adopting the mathe-matical model of equation (1) with a localized compliance in each notched cross-section.In the case of a single notch, the constant ,u 1 (,u i ) related to the first level of damage Dl 1(D2) is obtained by assuming that the position of the damage is known (d, = 0.70 m) andby taking, as before, the measured value for the analytical fundamental frequency of thenotched rod. The same procedure is used to evaluate ~c2 (,c.~2 ) for the damage configurationD3 (D4). Table 1 compares the measured values and the analytical estimates of the first17 frequencies for all the damage configurations. With few exceptions, as, for instance,

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Table 2. Experimental quotients (X n l(n /d¡ »/..1,2 for damaged configurations 01 and D2.

in correspondence of the 2nd and 9th frequency for D or the 7th frequency for D2-D4,the localized compliance model fits very well with the real notched rod and modelizationerrors turning out to be comparable with those of the classical model of the undamag-ed rod.

We conclude the section, showing that the asymptotic separation of the spectrum of thedamaged rod can be detected within the range of measured frequencies. We consider firstthe case of the single notch with the two levels of increasing damage severeness, D1 andD2. The first 17 quotients (~/(7r/~))<=i,2 evaluated from the experimental frequenciesare given in Table 2. The rearrangement of the eigenvalues of the notched rod suggestedby formula (4) (italicized quotients in Table 2) shows that for both the configurations D land D2, the asymptotic separation is detectable and, moreover, the numerical evaluationsare in agreement with the analytical expectation. Some indeterminacy remains for theconfiguration D1, where it is difficult to establish the proper asymptotic class and the re-lated order for frequencies À6, h7 and X12, ~.13 (marked boxes in Table 2). For example,let us consider the 12th and 13th frequencies of the rod with a single notch. Their relatedquotients, (2.08, 2.17) and (9.82, 10.24) in Table 2, both are quite close to the integernumbers 2 and 10, respectively, and it is not clear how assign to ~.12, h 13 the proper order(2 or 10) in each asymptotic class. Then, we considered the two possible extreme situa-tions that may occur-namely, we assumed an order 2 for both ~.12 and X13 (asymptoticclass (X,/(7r/dl))) and the same for the order 10 (asymptotic class (Àn/(n/d2»). Tak-ing this indeterminacy into account, we obtained a &dquo;band&dquo; of possible variation for theclass (~,n~(n/dl)). On the contrary, for the class (~n/(~7~2))< the indeterminacy can beremoved because the asymptotic behavior is in any case determined from the remainingquotients (see Figure 4). It is worth noting that being the quotient di /d2 &dquo;--’ 0.21, we found

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Figure 4. Rearrangement of the frequencies xn following formulas (4) for damaged configurationsD1 and D2.

five frequencies of the class (,X, / (7r /d2)) between two consecutive frequencies of the otherasymptotic class. As a further remark, we observed that the most part of the quotients fora fixed asymptotic class overestimates the integer numbers corresponding to the new classorder, and the differences are bigger for small damages in accordance with the asymptoticformulas (4).

We consider now the case of two notches. The second notch defines three piecesof rod of lengths 0.70 m, 1.1 m, and 2.2 m, from the left to the right end, respectively(Figure 2). Although all the eigenvalues of the notched rod are simple, from formulas (4)~.m3 ’&dquo; Àm2 for m3 = 2m2 and for &dquo;large&dquo; values of the integers m2, m3, and consequentlywe expect an additional indeterminacy in the choice of the proper asymptotic class ofquotients (~n/(~7~))!-2,3. The measurements and the evaluation of the three classes of

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249

Table 3. Experimental quotients (~.n /(~ /<~)),= 1.2.3 ~or damaged configurations D3 and D4.

quotients confirm this behavior (see marked boxes in Table 3 for two asymptotic classesinvolved). Then, to distinguish the various frequencies, we proceeded as we have donebefore, identifying a band of possible variation for the doubtful cases (see Figure 5). In

spite of the indeterminacy also in the present case, the asymptotic behavior is detectableand quite accurately measurable.

In conclusion, we found that the separation of the damaged spectrum of the notchedrod in multiple asymptotic branches is detectable even for low vibrating modes. This isessentially due to the fact that here the damage is rather severe from the beginning, andthen estimates (4) are accurate enough even for the eigenfrequencies of relatively loworder. There are of course some cases when the above asymptotic behavior is not easilydetectable as it was shown in Morassi (1997) (second experiment).

4. CONCLUSIONS

In this paper, we have presented a series of dynamic tests performed on an axially vibratingbeam with multiple notches, and we have given an interpretation of the measurementswithin a class of one-dimensional models where a notch is represented by a localizedcompliance located at the damaged cross-section. Theoretical analysis indicates that asingle notch splits the higher eigenvalues in two asymptotic classes, and when there areN notches, the spectrum forms N + 1 groups. Although the separation is supposed to bevalid only in the asymptotic sense, we have observed and quite accurately measured the

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250

Figure 5. Rearrangement of the frequencies xn following formulas (4) for damaged configurationsD3 and D4.

branching property experimentally even for relatively low frequencies.

Acknowledgments. We would like to thank Professor Cesare Davini for his useful comments on the text.

REFERENCES

Acerbi, E., Buttazzo, G., and Percivale, D., 1988, "Thin inclusions in linear elasticity: A variational approach," Journal f&uuml;rdie Reine und Angewandte Mathematik 300, 1-17.

Adams, R. D., Cawley, P., Pye, C. J., and Stone, B. J., 1978, "A vibration technique for non-destructively assessing theintegrity of structures," Journal Mechanical Engineering Science 20(2), 93-100.

Anzellotti, G., Baldo, S., and Percivale, D., 1994, "Dimension reduction in variational problems, asymptotic developmentin &Gamma;-convergence and thin structures in elasticity," Asymptotic Analysis 9, 61-100.

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251

Chen, G., Krantz, S. G., Russell, D. L., Wayne, C. E, West, H. H., and Coleman, M. P., 1989, "Analysis, designs, andbehavior of dissipative joints for coupled beams," SIAM Journal of Applied Mathematics 49(6), 1665-1693.

Colonnello, A. and Morassi, A. 1998, "Hearing cracks in a vibrating rod," Proc. 23rd International Conference on Noiseand Vibration Engineering, Belgium, K. U. Leuven.

Dimarogonas, A. D. and Paipetis, S. A., 1983, Analytical Methods in Rotor Dynamics, Applied Science, London.Freddi, L. and Morassi, A., 1996, "Functionals defined on measures and dimension reduction problems in elasticity,"

Preprint.Gudmunson, P., 1983, "The dynamic behaviour of slender structures with cross-sectionals cracks," Journal of Mechanics

and Physics of Solids 31(4), 329-345.Morassi, A., 1997, "A uniqueness result on crack location in vibrating rods," Inverse Problems in Engineering 4, 231-254.Petroski, H. J., 1984, "Comments on ’Free vibration of beams with abrupt changes in cross-section,’

"

Journal of Soundand Vibration 94(2), 157-159.

Thomson, W. T., 1949, "Vibration of slender bars with discontinuities in stiffness," Journal of Applied Mechanics, ASME

16, 203-207.

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