In this paper an experimental study on circular cylindrical shells subjected to axial compres- sive and periodic loads is presented. The setting of the experiment is explained and deeply described along with a complete analysis of the results. The linear and the nonlinear dynamic behaviour associated with a combined effect of compressive static and a periodic axial load has been considered and a chaotic response of the structure has been observed close to the resonance. The linear shell behaviour is also investigated by means of a theoretical and finite element model, in order to enhance the comprehension of experimental results, i.e. the natural frequencies of the system and their ratios

BARBIERI, MARCO

Errede, Vito

PELLICANO, Francesco

STROZZI, MATTEO

ZIPPO, Antonio

In this paper an experimental study on circular cylindrical shells subjected to axial compres-
sive and periodic loads is presented. The setting of the experiment is explained and deeply
described along with a complete analysis of the results. The linear and the nonlinear dynamic
behaviour associated with a combined effect of compressive static and a periodic axial load
has been considered and a chaotic response of the structure has been observed close to the
resonance. The linear shell behaviour is also investigated by means of a theoretical and finite
element model, in order to enhance the comprehension of experimental results, i.e. the natural
frequencies of the system and their ratios

Antonio Zippo

Marco Barbieri

Matteo Strozzi

Vito Errede

Francesco Pellicano

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Linear and nonlinear dynamics of a circular cylindrical shell under static and periodic axial load

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

LINEAR AND NONLINEAR DYNAMICS OF A CIRCULARCYLINDRICAL SHELL UNDER STATIC AND PERIODICAXIAL LOADAntonio Zippo, Marco Barbieri, Matteo Strozzi, Vito Errede and FrancescoPellicanoDipartimento di Ingegneria Meccanica e Civile, Universita´ di Modena e Reggio EmiliaModena, Italy, 41125e-mail: francesco.pellicano@unimore.itIn this paper an experimental study on circular cylindrical shells subjected to axial compres-sive and periodic loads is presented. The setting of the experiment is explained and deeplydescribed along with a complete analysis of the results. The linear and the nonlinear dynamicbehaviour associated with a combined effect of compressive static and a periodic axial loadhas been considered and a chaotic response of the structure has been observed close to theresonance. The linear shell behaviour is also investigated by means of a theoretical and finiteelement model, in order to enhance the comprehension of experimental results, i.e. the naturalfrequencies of the system and their ratios.1. IntroductionCircular cylindrical shells are important elements in many Engineering fields e.g. Aerospace,Nuclear, Civil and the need of more and more efficient structures in terms of strength and weightled to a strong reduction of safe factors; one of the direct consequences of weight reduction is theincreasing of vibration problems. Nowadays, several commercial softwares allow to carry out static,stability and vibration analyses; however, regarding the shell dynamics, such kind of analyses are gen-erally reliable in the linear field only. Difficulties in developing accurate models for shell structureswere the motivations of a large scientific production, withal there are few experimental studies aboutdynamic instabilities and the comparisons between theory and experiments are not yet satisfactory.The fundamental investigation on the stability of circular cylindrical shells is due to Von Karman andTsien [1] that analyzed the static stability (buckling) and the postcritical behavior of axially loadedshells. In this study, it was clarified that the discrepancies between forecasts of linear models andexperimental results were due to the intrinsic simplifications of linear models; indeed, linear analysesare not able to predict the actual buckling phenomenon observed in experiments. Vijayaraghavan andEvan-Iwanowski [2] analyzed, both analytically and experimentally, the parametric instabilities of acircular shell under seismic excitation. The cylinder position was vertical and the base was axiallyexcited using a shaker. Instability regions are found analytically and compared to experimental re-sults. The Donnells nonlinear shallow-shell theory [3] was used by Gonalves and Del Prado [4] toanalyze the dynamic buckling of a perfect circular cylindrical shell under axial static and dynamicloads. Pellicano and Avramov [5] published a paper concerning the nonlinear dynamics of a shellwith base excitation and a top disk. The work was mainly theoretical and only some experimentalICSV19, Vilnius, Lithuania, July 8–12, 2012 119th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012Table 1: Natural frequencies of mode (1,4) at increasing preload.Preload (N ) Natural Frequency (FEM) (Hz) Natural Frequency (Experimental) (Hz) relative error (%)0 1623 1621 0.150 1619 1614 0.3100 1616 1610 0.4150 1612 1604 0.5200 1608 1598 0.6250 1604 1592 0.7results concerning the linear dynamics were presented. The shell was modeled using the nonlinearSanders Koiter theory [3] and a reduced order model was used. Kubenko and Kovalchuk [6] pub-lished an experimental work focused on shells made of composite materials; they pointed out theinadequateness of the linear viscous damping models. Axial loads (base excitation, and free top endof the shell) as well as combined loads were considered. Dynamic instability regions were determinedexperimentally: a disagreement between previous theoretical models (narrower region) and experi-ments (wider) was found, the conjecture made by such scientists was that the disagreement was dueto shells geometric imperfections. Such paper summarizes some of experimental results publishedon a previous book (Kubenko et al.) [7] (see e.g. page 123, figure 2.23 gives an interpretation of theenlargement of instability boundaries)In the present paper, circular cylindrical shells subjected to axial loads having static (compres-sive) and harmonic components are investigated.2. Finite element analysisFinite Element modal analysis has been carried out for each step of the applied preload. Modeshape description for a generic mode (m,n) [8] is characterized with an own mode shape where:m number of longitudinal half-waves (m = 1,2,3,..)n number of circumferential waves (n = 1,2,3,..)In Figure 1 the modal shape of mode (1,4) is shown. In Table 1 a comparison between numerical andexperimental frequency for mode (1,4) is shown.Figure 1: F.E.M. analysis, mode (1,4)3. ExperimentsThis section provides informations about the experimental setup developed to test thin shellstructures under constant axial load and periodical excitation. In Figure 2 a schematic representationof the setup is shown. A shaker is rigidly inserted in a frame designed for the application of thepreload. Connections to the system for data acquisition are shown.219th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 20123.1 DescriptionThe system under investigation consists of a circular cylindrical shell, made of aluminum alloy,clamped both base and top sides by means of two worm gear clamps at two rigid supports. The bottomsupport is an aluminum alloy circular disk rigidly bolted to the shaker. The top disk is connected tothe frame by means of a dynamic load cell, a stinger, and a static load cell. The stinger is introducedin order to reduce the effects of misalignments. A laser vibrometer is used to measure the velocity ofthe side of the shell and its output is routed both to the spectrum analyzer and to the LDS controller.The control system is open-loop in order to avoid control instabilities induced by nonlinear behaviorof the tested structures.The characteristics of the shell are the following: length L = 0:137m, thickness h = 0:1 10 3m, mean radiusR = 32:910 3m, density  = 2796 kgm3, Young’s modulusE = 71:020109 Nm2, Poisson’s ratio = 0:31 and static preload P0 = 0  250N .Figure 2: System setup: (1) structure under test (2) shaker (3) shaker amplifier (4) air cool system (5)static load cell (6) digital load gauge (7) force transducer pcb m231b (8) force transducer amplifier (9)press system to apply static preload (10) laser vibrometer (11)laser controller (12) lds dactron laserusb shaker control system (13) pc (14) spectrum analyzer ono-sokki cf-5220.The press system has been suitably designed for this research. A screw can move up or down aplate which applies the desired preload to the shell, both compressive and tractive. A static load cellAEP transducers TC8 10KN” is interfaced with a digital load gauge that displays the exact preloadapplied to the shell. The Shaker LDS V530 is integrated rigidly in the press system structure, ispowered independently and equipped with a system of air cooling.The laser vibrometer is targeted on the shell surface and plays a significant role as it measuresvelocity data acquisition on the shell side without interfering with it. This point is very significant forthin structures, because the use of a standard accelerometer would introduce an undesired extra mass,thus altering the symmetry of the structure. The signal coming from the laser is splitted and directedboth to the spectrum analyzer, used to detect the regions of instability, and to the LDS controller.3.2 Data PresentationIn this section all tests carried out on the shell are described. Formerly a few tests were doneto calibrate the devices properly, then several combination were investigated for different preloads.The experimental analysis was carried out by sweeping frequencies from 700Hz to 2500Hz, witha step rate of 7.5 Hz per second, starting from the highest frequency (2500Hz) towards the lowerfrequency (700Hz) and backwards, each time varying the magnitude of the oscillations generatedfrom the shaker, from 0.1 V with regular steps of 0.1 V up to the maximum value of 1.0 V. Thesesweeps were repeated, not only by varying the power supplied to the shaker, but also by imposing an319th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012increasing axial static compressive load P0. The static axial load is increased from 0N up to 250N(safety limit to avoid shaker damage). Results are presented for preload values of 0N , 100N and250N and for voltage levels of 0:5V and 0:8V . In Figures 3 and 4, d and u represent the decreasingfrequency sweep (d, down) and the increasing frequency sweep (u, up).3.2.1 External Load Amplitude of 0.5 voltIn Figure 3 the dynamic axial load vs frequency is plotted whereas in figure 4 the amplitude ofvibration of the shell (in terms of velocity) vs.-frequency are shown.(a) (b)Figure 3: Dynamic load-frequency diagram for different preloads at 0.5 volt.Dynamic load-frequency diagrams show a reduction in terms of resonance frequency for in-creasing preloads. The amplitude of the resonance peak has a peculiar behavior: it has a maximumvalue of 350N for the mid value of preload (100N), see Figure 3b. This point needs further investiga-tions. The same behavior can be observed in terms of velocity amplitude at the target point, see Figure4. The maximum velocity, i.e. the maximum amplitude of vibration, is seen for P0 = 100N . Thevelocity frequency diagram shows another resonance between 1300 and 1350 Hz. For this resonance,the amplitude of vibration is minimum for P0 = 100N .1150 1200 1250 1300 1350 140000.511.522.53Frequency [Hz]Velocity [m/s]  P0=0N; Ae=0.5V upP0=0N; Ae=0.5V downP0=100; Ae=0.5V upP0=100; Ae=0.5V downP0=250N; Ae=0.5V upP0=250N; Ae=0.5V downFigure 4: Amplitude-frequency diagram for different preloads at 0.5 volt.3.2.2 External Load Amplitude of 0.8 voltFigure 5 and 6 present the corresponding experimental results for 0.8V excitation amplitude.From Figure 5a and 5b it is noted that increasing the static load P0, the maximum peak in dynamicload is reduced. The system presents the opposite behavior with respect to the case of 0.5V excitation.Moreover, Figure 6 shows again the frequency reduction due to the preload.4. Measured non linear phenomenaThe behavior of the structure close to the mode (1,4) is investigated. Numerical and experimen-tal analyses show that this mode is close to 1623 Hz (see Table 1) and it is quite far from other modes.The amplitude-frequency diagram for such resonance is shown in Figure 7. In order to describe thenonlinear behavior close to the resonance, it is useful to introduce the following dimensionless values:419th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012(a) (b)Figure 5: Dynamic load-frequency diagram for different preloads at 0.8 volt.800 1000 1200 1400 1600 1800 2000 2200 240000.511.522.533.5Frequency [Hz]Velocity [m/s]  P0=0N; Ae=0.8V upP0=0N; Ae=0.8V downP0=100; Ae=0.8V upP0=100; Ae=0.8V downP0=250N; Ae=0.8V upP0=250N; Ae=0.8V down(a)1150 1200 1250 1300 1350 140000.511.522.533.5Frequency [Hz]Velocity [m/s]  P0=0N; Ae=0.8V upP0=0N; Ae=0.8V downP0=250N; Ae=0.8V upP0=250N; Ae=0.8V down(b)1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 165000.10.20.30.40.50.60.7Frequency [Hz]Velocity [m/s]  P0=0N; Ae=0.8V upP0=0N; Ae=0.8V downP0=100; Ae=0.8V upP0=100; Ae=0.8V downP0=250N; Ae=0.8V upP0=250N; Ae=0.8V down(c)Figure 6: Amplitude-frequency diagram for different preloads at 0.8 volt.1. !x = !n=!0;1 , is the ratio between the frequency corresponding to the maximum amplitude forthe n-th excitation amplitude and the frequency at which the maximum occurs for the referencecase (0.1 V);2. vx = vn=v0;1 , is the ratio between the corresponding maximum velocities.0.94 0.96 0.98 1 1.02 1.0402468ωxv x  Figure 7: Amplitude-frequency diagram. mode (1,4); P0 = 100N ; voltage levels 0.1-1VFigure 8: Amplitude-frequency diagram comparison of mode (1,4) with different preloads (0N, 100Nand 250N) at 0.8 voltIn terms of dimensionless variables, the backbone can be drawn by interpolating the maxima,Figure 7 it results that the shell presents a softening behavior. The backbone curve moves to the leftwith respect to the vertical line through the resonance frequency.Figure 8 shows dependence of the non linear behavior on the different values of the preload.The preload of 100N produces a less softening resonance, with respect to the non preloaded case andthe case of 250N preload.519th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 20125. Nonstationary responseTests performed on the shell structure pointed out the existence of certain frequency intervalscharacterized by high amplitude of vibration and complex dynamics; in the following we will callthem ”instability regions”, this gives the clear idea that the periodic orbit has lost stability. Table 2summarizes the instability ranges found during experiments and Figures 9 - 12 shows the time histo-ries and spectra of the lateral shell vibration at different forcing loads and with a static compressionof 100N .Table 2: Frequency ranges of instability regions with different preload.P0 (N ) Voltage level (V ) Instability range (Hz)1000.8 1210.01 - 1217.920.9 1205.61 - 1218.801.0 1201.22 - 1225.841500.7 1208.25 - 1210.010.8 1202.97 - 1212.530.9 1199.46 - 1215.291.0 1196.82 - 1221.442000.7 1209.13 - 1211.770.8 1207.37 - 1213.530.9 1195.06 - 1215.291.0 1191.54 - 1218.802500.5 1208.25 - 1215.290.6 1204.73 - 1221.440.7 1202.92 - 1234.630.8 1198.53 - 1236.390.9 1194.18 - 1238.421.0 1189.78 - 1242.420 1 2 3 4 5 6 7 8 9 100.40.30.20.100.10.20.3time (seconds)m/s450 900 1350 1800 2250 2700 3150 3600 4050 4500 4950 540000.020.040.060.080.10.120.14Frequency (Hz)m/s1797 1798 1799 1800 1801 1802 1803 180400.020.040.060.080.10.12Frequency (Hz)m/sFigure 9: Time history and Spectrum at 900 Hz forcing loadIn Figure 9 the dynamic scenario is represented, the excitation frequency is 900Hz. The re-sponse is quasi-periodic, i.e. there is an amplitude modulation that presents two time scales: i) a slowtime scale with period of 2s (see 0:5Hz sidebands); ii) intermediate time scale with period of 10-2s(see 100Hz sidebands). The response is dominated by the second main spectral harmonics (1800Hz),suggesting an important role of quadratic nonlinearities; this role is also confirmed by the loosing ofsymmetry in the time response (max = 0:2m=s,min =  0:25m=s).619th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 20120 550 1100 1650 2200 2750 3300 3850 4400 4950 5500 6050 6600130001234567x 10−3Frequency (Hz)m/s0 1 2 3 4 5 6 7 8 9 10−0.02−0.015−0.01−0.00500.0050.010.0150.02time (seconds)m/s0.0535 0.0555 0.0575 0.0595 0.0615 0.0635 0.0655 0.0675 0.0695 0.0715−0.015−0.01−0.00500.0050.010.015time (seconds)m/s0.0271 0.0471 0.0671 0.0871 0.1071 0.1271 0.1471−0.015−0.01−0.00500.0050.010.015time (seconds)m/sFigure 10: Time history and Spectrum at 1100 Hz forcing load591 1182 1773 2364 2955 3546 4137 4728 5319 5910 6501 709200.050.10.150.20.250.30.350.40.450.5Frequency (Hz)m/s 0 1 2 3 4 5 6 7 8 9 10−1.5−1−0.500.511.5Time historytime (seconds)m/s0.7693 0.8093 0.8493 0.8893 0.9293 0.9693 1.0093 1.0493 1.0893 1.1293−1−0.500.51time (seconds)m/s0.803045 0.789341 0.80626 0.807868 0.809476 0.811083 0.812691−1−0.500.511.5time (seconds)m/sFigure 11: Time history and Spectrum at 1182 Hz forcing loadAt 1100Hz (Figure 10) the scenario changes; here the energy in frequency domain is mainlyconcentrated at 1100 and 3300Hz. This means that here the linear and cubic behaviors are dominant;note that this is confirmed by the symmetry in the time history.615 1230 1845 2460 3075 3690 4305 4920 5535 6150 6765 738000.20.40.60.811.21.4Frequency (Hz)m/s0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5−2−1.5−1−0.500.511.522.5time (seconds)m/sFigure 12: Time history and Spectrum at 1230 Hz forcing loadAt 1182Hz (Figure 11) an interesting phenomenon takes place, the response is quasi-periodicbut something interesting happens in the periodicity. The main spikes in the spectrum are spaced ofabout 590Hz, i.e. one half the excitation frequency. This means that the periodic part of the responsepresents a fundamental frequency of 590Hz. The straightforward consideration is that the responseis one half sub-harmonic; this is partially contradicted by the spectrum that does not present spikesat 590Hz. However, if one zooms around 590Hz, then will disclose an extremely small peak. Ourconjecture is that the response is sub-harmonic and the amplitude of the fundamental harmonic is719th International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012negligible.6. ConclusionsThe dynamic behavior of a thin cylindrical shell under static axial preload and harmonic externalaxial load has been characterized. An ad hoc setup has been developed in order to measure boththe dynamic load and the amplitude of vibration on the shell surface. Tests performed at differentpreload values and different amplitudes of the external excitation pointed out the existence of nonlinear behavior in certain instability ranges. Chaos is found close to 1230 Hz and a very well definedsoftening behavior can be observed close to the mode (1,4). Non linear dependence of the response onthe preload parameter is observed. Further experiments and correlation with numerical and theoreticalmodels need to be performed in order to fully understand the observed phenomena.REFERENCES1 Von Ka´rma´n, T., and Tsien, H., 1941. “The buckling of thin cylindrical shells under axial compres-sion”. Journal of the Aeronautical Sciences, 8(8), pp. 303–312.2 Vijayaraghavan, A., and Evan-Iwanowski, R. M., 1967. “Parametric instability of circular cylindri-cal shells”. Journal of Applied Mechanics, 34(4), pp. 985–990.3 Amabili, M., 2008. Nonlinear Vibrations and Stability of Shells and Plates. Cambridge UniversityPress.4 Gonc¸alves, P. B., and Del Prado, Z. J., 2002. “Nonlinear oscillations and stability of parametricallyexcited cylindrical shells”. Meccanica, 37, pp. 569–597.5 Pellicano, F., and Avramov, K., 2007. “Linear and nonlinear dynamics of a circular cylindricalshell connected to a rigid disk”. Communications in Nonlinear Science and Numerical Simulation,12(4), pp. 496 – 518.6 Kubenko, V., and Koval’chuk, P., 2009. “Experimental studies of the vibrations and dynamicstability of laminated composite shells”. International Applied Mechanics, 45, pp. 514–533.10.1007/s10778-009-0209-4.7 Kubenko, V. D., Koval’chuck, P. S., and Krasnopol’skaya, T. S., 1984. Nonlinear interaction of theflexural modes of vibration of cylindrical shells. Nauk. Dumka.8 Leissa, A. W., 1973. Vibration of shells. NASA SP-288, Scientific and Technical InformationOffice, National Aeronautics and Space Administration; Washington, DC.8

Linear and nonlinear dynamics of a circular cylindrical shell under static and periodic axial load

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