Nonlinear vibration localisation in a symmetric system of two coupled beams
Filipe Fontanela, Alessandra Vizzaccaro, Jeanne Auvray, Björn Niedergesäß, Aurélien Grolet, Loïc Salles, Norbert Hoffmann
NNonlinear vibration localisation in a symmetric system of two coupledbeams
Filipe Fontanela · Alessandra Vizzaccaro* · Jeanne Auvray · Bj¨orn Niederges¨aß · Aur´elien Grolet · Lo¨ıc Salles · Norbert HoffmannAbstract
We report nonlinear vibration localisation in a system of two symmetric weakly coupled nonlinearoscillators. A two degree-of-freedom model with piecewise linear stiffness shows bifurcations to localised solu-tions. An experimental investigation employing two weakly coupled beams touching against stoppers for largevibration amplitudes confirms the nonlinear localisation.
Keywords
Vibration localisation · Symmetry breaking bifurcation · Clearance nonlinearity
The emergence of localised vibration in symmetric structures is a challenging problem in the aerospace industrydue to high cycle fatigue [6, 16, 25]. Usually, aerospace structures such as bladed-disks, antennas, and reflectorsare composed of ideally identical substructures assembled in a symmetric configuration. In the linear regime,localisation may arise due to structural inhomogeneities originating in the manufacturing process or due to wear[1, 2]. In the aerospace industry, especially in the bladed-disk community, the phenomenon is thus widely referredto as a mistuning problem. The topic has attracted considerable attention in the literature, and research hasmainly focused on effective numerical tools for prediction, experimental investigation, and the use of intentionalmistuning during design stages [3, 4, 13, 14, 24].However, in some cases, due to inherent nonlinear phenomena, the assumption of linear vibration might bemisleading. In the case of structural dynamics, nonlinearity may arise e.g. due to friction induced by internaljoints, or vibro-impacts [18]. It is also known that the emergence of localised vibration might be provokedby nonlinear effects, as an alternative to the linear localisation mechanisms in mistuning [23]. For example, *Corresponding author: A. VizzaccaroE-mail: [email protected]. Fontanela · A. Vizzaccaro · L. Salles · N. HoffmannImperial College London, Exhibition Road, SW7 2AZ London, UKJ. AuvrayEcole Centrale Marseille, 13451 Marseille Cedex 20, FranceA. GroletEcole Nationale Superieure d’Arts et M´etiers ParisTech, 59000 Lille, FranceB. Niederges¨aß · N. HoffmanHamburg University of Technology, 21073 Hamburg, Germany a r X i v : . [ n li n . PS ] S e p Filipe Fontanela et al. Fig. 1: Symmetric system under investigation.even perfectly symmetric structures may experience localised vibrations due to the dependence of mode shapeson amplitude, or due to bifurcations. However, most of the available knowledge on this kind of nonlinearvibration localisation relies on results from minimal models, and only few experimental studies have attemptedto demonstrate the existence of localised vibrations in symmetric structures due to nonlinear interactions [9, 22].This paper thus reports the existence of nonlinear localised vibrations in a symmetric mechanical structure dueto the presence of clearance nonlinearity. First we introduce a conceptual model with two degrees of freedomunder the effect of a harmonically moving base. The bilateral contact phenomenon is assumed perfectly elastic,leading to a nonlinear mathematical model which is piecewise linear [21]. In the free case, a nonlinear modalanalysis is carried out, and we demonstrate that localised states bifurcate from the homogeneous out-of-phasemode. The results are similar to bifurcated states calculated in smooth systems, such as in chains of Duffingoscillators [12, 15, 19]. In the case of externally driven vibration, if the excitation is perfectly in phase, threekinds of stable response states may result. First, a purely linear configuration, where both masses vibrate in lowamplitude and in phase. Second, a nonlinear configuration where both masses vibrate in large amplitude and inphase. And third, just one oscillator, either the first or the second, vibrates in large amplitudes, while the otherone vibrates in small amplitudes and out of phase. An experimental validation of the numerical findings, basedon a test-rig composed of two weakly coupled cantilever beams touching stoppers for large amplitude vibration,is reported.The paper is organised as follows. In Sec. 2 the numerical model is described and the outcomes of a non-linearmodal analysis and a response analysis are presented. Section 3 introduces the experimental test used to validatethe results from the model. Sec. 4 discusses the main findings and suggests directions for future investigation. m coupled to the ground byidentical springs k l and viscous dampers c . The masses are coupled to each other by a coupling spring k c . Inthe subsequent experimental setup we will employ coupled mechanical beams with bilateral contacts as theoscillators and describe there how the parameters can be determined.To excite the system we assume that the ground is connected to a moving base with given periodic displacement y ( t ). We also assume that there is no dissipation of energy when the oscillators touch the stoppers. Denoting onlinear vibration localisation in a symmetric system of two coupled beams 3(a) non li n ea r f o r ce (b) r e s t o r i ng f o r ce Fig. 2: Panel (a) nonlinear forces due to the stoppers; Panel (b) total force-displacement relationship for a singlemass, ignoring the coupling.the horizontal displacement of each oscillator with x and x , the dynamical system can be expressed as m ¨ x + c ˙ x + ( k l + k c ) x − k c x + f nl ( x ) = c ˙ y + k l y, (1) m ¨ x + c ˙ x + ( k l + k c ) x − k c x + f nl ( x ) = c ˙ y + k l y, (2)where f nl represents the non-linear force, depicted in Fig.2. The non-linear force can be expressed as a piece-wiselinear function, f nl ( x i ) = | x i | < sk nl ∗ ( x i − s ) if x i > sk nl ∗ ( x i + s ) if x i < − s (3)where s might be called the gap dimension, characterising the amplitude that the oscillators need to reachbefore touching the stoppers.2.2 Numerical tools To compute periodic solutions of the numerical model, the harmonic balance method is applied [7, 17]. Thesolution is determined in the form of a truncated Fourier series: x ( t ) = A + H (cid:88) k =1 A k cos kωt + B k sin kωt . (4)Substituting the expression into the equation of motion results in a nonlinear algebraic system of equationsfor the coefficients of the series. For a given frequency, the algebraic system can be solved using a Newton likeroot-finding algorithm. Continuation tools [5, 8, 20] allow to follow the results over varying parameters, like forexample the frequency of the base excitation. Filipe Fontanela et al. (rad/s) m a x ( | x n | ) ( m ) P1 P2P3 P4 (a) (rad/s) m a x ( | x n | ) ( m ) -202 x n ( m ) P1 (b) -505 x n ( m ) P2 -505 x n ( m ) P3 t (s) -10010 x n ( m ) P4 Fig. 3: Backbone curves for the conservative nonlinear system, starting from the two linear normal modes.Panel (a) shows the bifurcation diagram, with full lines indicating stable solutions, while dashed lines representsunstable ones. Panel (b) shows the solutions identified in Panel (a) in time domain over two periods. The redlines denote x , the black lines denote x .2.3 Linear and nonlinear modal analysis of the free conservative systemWe first study the linear and non-linear modes of the conservative form of the nonlinear two-degree-of-freedomsystem. The nonlinear modes are the periodic solutions of the undamped and unforced equations of motion, m ¨ x + ( k l + k c ) x − k c x + f nl ( x ) = 0 , (5) m ¨ x + ( k l + k c ) x − k c x + f nl ( x ) = 0 . (6)In the linear case, i.e. for low vibration amplitudes, there are two modes. One mode where the masses move inphase with equal amplitude, at frequency ω = k l m , and one mode where the masses move out of phase withequal amplitude, at a slightly larger frequency ω = k l +2 k c m due to the coupling spring being activated. In the nonlinear case the analysis is carried out numerically.In the following we assume m = 1, k l = 1, k nl = 1, s = 1, k c = 0 .
05, and c = 0 .
005 where applicable.Typical results are presented in Fig.3. The homogeneous in-phase and the out-of-phase modes continue to existin the nonlinear regime, but their resonance frequencies depend on the amplitude of vibration. At amplitudeslower than the gap size, the modes are the modes of the linear system. At higher amplitudes, the oscillatorsstart touching the stoppers and the frequencies increase due to the hardening-type nonlinearity. At very highamplitudes, the gap size becomes negligible compared to the vibration amplitude and the eigenfrequenciesasymptotically approach those of the linear system with zero gap.The stability analysis of the nonlinear normal modes, or periodic solutions, shows that the branch of the in-phase oscillations turns out to be linearly stable for all amplitudes. The out-of-phase oscillation, however, is onlinear vibration localisation in a symmetric system of two coupled beams 5 (rad/s) m a x ( | x n | ) ( m ) P1P1 P2P2 P3P3 P4P4 (a) (rad/s) m a x ( | x n | ) ( m ) -101 x n ( m ) P1 (b) -101 x n ( m ) P2 -202 x n ( m ) P3 t (s) -20020 x n ( m ) P4 Fig. 4: Backbone curves and bifurcating solution branch. Panel (a) depicts the solutions starting from the linearmodes in grey and the bifurcating branch in red. The two red lines correspond to one new bifurcating solutionbranch and denote x and x respectively. Full lines denote stability, dashed ones instability. Panel (b) depictsthe time dependency of solutions on the bifurcating branch of localised vibration.linearly stable only in the linear regime, and again for larger vibration amplitudes, i.e. there is an amplituderange for which it is linearly unstable. It also turns out that in fact qualitatively new solutions bifurcate fromthe out-of-phase oscillations at the points of change of stability. Fig.4 shows the bifurcating solutions. The mostremarkable property of the new solution class is that an extremely strong localisation of vibration amplitudesonto just one of the two oscillators is realised.2.4 Forced responseWe now apply excitation in the form of a harmonically moving base, y ( t ) = Y cos( ωt ), where Y represents theamplitude of y ( t ). First the system is studied assuming Y = 0 . oscillators to reach the stoppers when the excitation frequency is near resonance.Figure 5 depicts the results. First one should note that due to the symmetry of the base excitation, from alinear systems perspective, out-of-phase solutions should not be excited, but merely in-phase solutions. Corre-spondingly, the response function of the in-phase mode shows the typical non-smooth stiffening behaviour forlarge response amplitudes, including its stability characteristics. However, out-of-phase solutions do show upin the form of an isola. These solutions can be obtained in the numerical approach by properly setting initialconditions for the root-finding process. The observed out-of-phase solutions are, however, not symmetric, butstrongly asymmetric, or in other words localised on a single oscillator. As in the case of the free system, theout-of-phase solutions are related to localisation of the vibration amplitude onto one of the two oscillators.One might note that the excitation of the localised solutions on the isola can be understood in terms of Filipe Fontanela et al. (rad/s) m a x ( | x | ) + m a x ( | x | ) ( m ) P1 P2P3P4 (a) -101 x n ( m ) P1 (b) -101 x n ( m ) P2 -101 x n ( m ) P3 t (s) -101 x n ( m ) P4 Fig. 5: Forced response of the system in assuming a harmonic base excitation of amplitude Y = 0 . Y = 0 .
15. Now the isola observed earlier has grown in size and the resulting branches of localised solutionsmerge back to the homogeneous in-phase solution. onlinear vibration localisation in a symmetric system of two coupled beams 7 (rad/s) m a x ( | x | ) + m a x ( | x | ) ( m ) P1P2P3P4 (a) (rad/s) m a x ( | x | ) + m a x ( | x | ) ( m ) -505 x n ( m ) P1 (b) -505 x n ( m ) P2 -202 x n ( m ) P3 t (s) -101 x n ( m ) P4 Fig. 6: Forced response of the system assuming a harmonic base excitation of amplitude Y = 0 .
15. The twopanels show the same quantities as in the previous figure.
Fig. 7: Structure designed to represent a system with two degrees of freedom. Panel (a) depicts a front view ofthe two beams with added masses at the tips. Panel (b) depicts an isometric illustration.The system is designed to show weak coupling between the two oscillators. In the experimental realisationthe coupling is fully controlled by the slender structure connecting the two beams. If the connection is widerthe structure becomes stiffer and the coupling stronger. A similar effect is obtained if the position of theslender connection is changed. If the connecting structure is positioned towards the tip of the beam, wheredisplacements are relatively high, the coupling is also increased. In practice, the connection has been designedto obtain coupling values of a few percent of the beam stiffness.
Filipe Fontanela et al.
Fig. 8: Test rig. Panel (a) depicts the stoppers near one of the two beams, while Panel (b) shows the platformconnected to a shaker.The nonlinear effect is obtained by means of the beams contacting stoppers for large deflections. When reachinga certain vibration level, the beams start to touch the stoppers, see Panel (a) in Fig. 8, the effective beam lengthdecreases due to the change in boundary conditions and the effective stiffness of the system increases.The level of nonlinearity is controlled by the position of the stoppers. If the stoppers are moved towards the clamping position, the equivalent bending stiffness before and after touching the stoppers are not so different.The opposite effect is obtained by moving the stoppers towards the tips of the beams. The base excitation, asassumed in the minimal model, is implemented by means of a moving platform. The two beams are clamped toa relatively rigid frame, and the final assembly is connected to the walls of the platform. A shaker is attachedto the platform, and the two beams are excited indirectly through the moving platform.Panel (b) in Figure 8 shows the test structure attached to the platform where an accelerometer is attachedto the tip of the beam for measurements. The two stoppers near the blades are also illustrated. An impulseresponse test applying hammer excitation in the linear regime yields the in-phase and the out-of-phase naturalfrequencies for the test structure as 11 .
18 Hz and 11 .
62 Hz. These two values can be used to estimate an actualequivalent coupling ratio k c /k l in the test rig of about 4%. onlinear vibration localisation in a symmetric system of two coupled beams 9 f (Hz) | H n | (a) |H ||H | f (Hz) a n ( m / s ) a b ( m / s ) (b) a a a b Fig. 9: Response measured in the linear regime. Panel (a) displays the resulting amplitude response ratio | H n | = | a n | / | a b | , while Panel (b) displays the measured accelerations directly.3.2 Test results and measurementsThe structure is first tested in the linear regime where the imposed force produced by the shaker is not largeenough to drive the two beams into contact with the stoppers. The motivation of this test is to investigatethe level of remaining inhomogeneities. If the system is perfectly symmetric, the response measurements shoulddepict a single resonance frequency only, since the out-of-phase mode should not be excited. The accelerationis measured at the tip of each beam and also at the platform.Panel (a) of Fig. 9 depicts the response of the two beams with H n = a n /a b , where a n is the measured accelera-tion for the respective beam and a b is the corresponding acceleration measured at the platform. The results areplotted in logarithmic scale. The transmissibility displays a single peak centred at f = 11 .
18 Hz, with respectivemodal damping of ξ = 0 . f = 11 .
62 Hz. For completeness, Fig 9 also displays the directly measuredresults in a linear scale.For stronger base excitation, the beams start touching the stoppers and the system becomes nonlinear. As to beexpected from the modelling results, hysteresis effects are to be observed and the excitation is varied upwardsand downwards in frequency. Figure 10 displays the corresponding measured results.
The graph in Panel (a) depicts the acceleration measured at one of the beams, while Panel (b) depicts the samequantities for the neighbouring oscillator. In both panels the blue lines indicate the measurements for the base.The measurements for the first beam, in Panel (a), show in thick black lines the results when the excitationfrequency is increased. The system follows the upper branch of homogeneous in-phase solutions and jumpsback to the low-amplitude configuration at f ∼ . f ∼ .
75 Hz. The solutions linking thetwo turning points are not measured due to their linear instability. The results in Panel (b) follow the samemeasurement strategy, but are measured on the other beam. The results are in excellent agreement, confirmingthe symmetry of the setup. The results are also in excellent agreement with the response characteristics of the
10 11 12 13 f (Hz) a ( m / s ) (a) a b ( m / s )
10 11 12 13 f (Hz) a ( m / s ) (b) a b ( m / s ) Fig. 10: Response measured in the nonlinear regime. Panel (a) displays the acceleration of a single beam,where the thick black lines represent the measurements when the frequency is varied upward, while the thinlines represent the same results obtained when frequency is varied downward. The blue lines indicate the baseacceleration. Panel (b) shows the same quantities as in Panel (a), but measured at the neighbouring beam.nonlinear model treated earlier.Modelling and simulation also predict that a branch of localised solutions may exist. The branches leadingto localised vibrations arise either through bifurcations from the homogeneous branch or in the form of isolas,depending on the excitation level. In order to test the existence of such kind of a nonlinear vibration localisationin the experiment, appropriate initial conditions have been employed, consisting in triggering high amplitudevibration in one of the oscillator. The platform was excited at f = 12 . f = 12 . linear case where both beams do not touch the stoppers and the measured accelerations are nearly sinusoidal.The first regime of localised vibrations observed experimentally is indicated in Panel (c). Within this config-uration only the first beam touches the stopper and keeps vibrating in a nonlinear large amplitude manner,while the neighbouring oscillator remains in low amplitude, not touching the stoppers. The localised solutionmeasured in the time-domain is in excellent agreement with the results predicted numerically.Due to the symmetry inherent in the system, also the analogous vibration localisation on the other oscillatoris expected. To test this hypothesis of the existence of the symmetric state, the neighbouring beam is nowpulled towards the stopper. Panel (d) of Fig. 11 shows the corresponding measured response. Obviously thenonlinear vibration localisation can thus also arise on the other oscillator. It depends merely on the choice ofinitial condition where the localisation will happen. In the supplementary online-material of this paper a video onlinear vibration localisation in a symmetric system of two coupled beams 11 t (s) -100-50050100 a ( m / s ) (a) t (s) -10-50510 a ( m / s ) (b) t (s) -100-50050100 a ( m / s ) (c) t (s) -100-50050100 a ( m / s ) (d) Fig. 11: Accelerations measured in time-domain. Panel (a) depicts the homogeneous large amplitude state whereboth beams touch the stoppers. Panel (b) displays the small amplitude response, where both beams do not touchthe stoppers. Panel (c) depicts the localised state when one of the beams touches the stoppers, the other onedoesn’t. Panel (d) displays the same behaviour, just for the localisation on the other beam. All panels show inblack measurements for the first beam, while the red lines are for the second one. The results in blue depict thebase acceleration.illustrates all states discussed.One should note that in the experimental setup, for the localised state the acceleration values measured forthe beam vibrating in large amplitudes exceed four times the same quantities for the neighbouring oscillator insmall amplitude. Moreover, the observed state seems to be very robust, and the system remains in the localisedbranch even if the beams are slightly perturbed externally.Once one localised state is reached, the whole stable branch of localised states can be traced experimentally.
The approach is similar to the experimental strategy implemented to measure the branch of homogeneous, i.e.in-phase solutions. First, the system is perturbed externally until the desired localised configuration is achieved.Then the excitation frequency is varied in frequency until the physical system jumps back from the localisedbranch to the underlying homogeneous one. This transition delineates the range of existence of the stable lo-calised solutions at hand. The same approach can be carried out varying the excitation frequency downwards.The measured localised responses trace the underlying branch of stable non-homogeneous solutions. Figure 12depicts the experimental results.The results depicted in blue and red lines denote the two localised solutions. One may note that when e.g.the first beam vibrates with touching the stoppers, as depicted by the large amplitude of the red line in Panel(a), the corresponding neighbour oscillator is in low amplitude, as illustrated by the red line in Panel (b). And
10 11 12 13 f (Hz) a ( m / s ) (a)
10 11 12 13 f (Hz) a ( m / s ) (b) Fig. 12: The two localised solutions, in red and blue lines respectively, identified experimentally. Panel (a)displays the response measured for the first oscillator, while Panel (b) depicts the response measured for thesecond one. The grey lines illustrate the homogeneous in-phase solutions.vice versa for localisation on the second beam. Ideally, the measured response in Panels (a) and (b) should beidentical, which is the case to a surprisingly good extent, considering the usual technical difficulties in measuringnonlinear mechanical vibrations.
This work focused on the investigation of a symmetric system with piecewise nonlinearity. A minimal modelwith two degrees of freedom was set up to study the properties of localised vibration due to nonlinearity. Anexperimental setup confirmed and validated the model-based findings.First a conservative analysis was carried out in order to compute the nonlinear normal modes of the modelsystem. The underlying linear regime shows the usual normal modes, where the two oscillators vibrate homo-geneously in phase or out of phase. The nonlinear analysis showed that novel states emerge bifurcating fromthe out-of-phase mode, characterised by vibration localisation on single oscillators. In frequency ranges near thelinear resonance, the out-of-phase homogeneous states may result unstable and the localised response is stableinstead.
The nonlinear localised states also exist in the driven case. The analysis was performed assuming a harmonicbase excitation. The underlying linear analysis suggests that only the in-phase mode should be excited sincethe external forcing and the out-of-phase mode are perfectly orthogonal. However, due to the existence of thebifurcated, asymmetric localised modes, the nonlinear localised modes can be triggered through the base exci-tation. Consequently, localised vibrations, arising from the nonlinear system dynamics, may easily result whenthe nonlinear regime is reached. Depending on the level of external forces they arise in the form of isolas orthrough bifurcations. Moreover, large parts of the branches of localised states turn out as linearly stable andare so to be expected to be observed directly.In order to test and validate the findings from the model, a test setup was designed and built. The experimentalsystem consisted of two weakly coupled beams with the effect of frequency changing contacts for large vibration onlinear vibration localisation in a symmetric system of two coupled beams 13 amplitudes. The configuration was deliberately set up to reproduce the bilinear stiffness behaviour of the model.Each beam vibrates as a simple oscillator with effectively softer or stiffer springs, depending on the vibrationamplitude. Harmonic base excitation of the platform was applied.The system was tested and analysed in the linear and nonlinear regimes. The response in the linear case showsthe expected single-mode excitation, i.e. response in the form of a symmetric in-phase mode. For large ampli-tudes, when the beams start touching the stoppers and thus the system becomes nonlinear, localised responsecan be observed. Within the given configuration the system localises vibrations in either the first or the secondbeam, depending on the initial conditions. Depending on the frequency and the level of excitation, four stableconfigurations may arise and have been observed experimentally. In the first one, both beams touch the stop-pers, resulting in a nonlinear homogeneous state. In the second configuration, none of the beams touches thestoppers, leading to a homogeneous and purely linear response. In the two other configurations the localisedstates are induced by nonlinearity, and vibration localises in either the first of the second beam.The present work has attempted to contribute to the understanding of vibration localisation in symmetric sys-tems caused by nonlinearity. Both modelling and testing show conclusive evidence that the studied symmetric,weakly coupled two degree-of-freedom oscillator may tend to respond in the form of localised vibrations whenthe driving is near the resonance of the individual oscillators, and when the forcing amplitude is strong enough.The findings have close analogies with results in nonlinear localisation from various fields of physics [10, 11]. Thepresent work might thus be considered an attempt to better understand the mechanics of nonlinear vibrationlocalisation in nonlinear structures of engineering. Future work will need to clarify further aspects. Amongstothers, the role of the vibration amplitudes necessary to reach the effects in actual applications, the role ofnonlinear vibration localisation in larger systems with more than two degrees of freedom, the interplay betweenmistuning related and nonlinear vibration localisation.
Acknowledgements
The first author is funded by the Brazilian National Council for the Development of Science and Tech-nology (CNPq) under the grant 01339/2015-3.
Conflict of interest
The authors declare that they have no conflict of interest.
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