Asymmetric Vibrations and Chaos in Spherical Caps under Uniform Time-varying Pressure Fields

27 This paper presents a study on nonlinear asymmetric vibrations in shallow 28 spherical caps under pressure loading. The Novozhilov’s nonlinear shell theory is 29 used for modelling the structural strains. A reduced-order model is developed 30 through the Rayleigh-Ritz method and Lagrange equations. The equations of 31 motion are numerically integrated using an implicit solver. The bifurcation 32 scenario is addressed by varying the external excitation frequency. The 33 occurrence of asymmetric vibrations related to quasi-periodic and chaotic motion 34 is shown through the analysis of time histories, spectra, Poincaré maps, and 35 phase planes. 36 37


Introduction
Nowadays, many theories and simplified models are available for studying shell 52 systems, even in the presence of fluid-structure interaction or thermal fields. 53 Nevertheless, new challenges come from the new frontiers of the Engineering, 54 which asks for even more reliable models where complicating effects are taken 55 into account for exploiting the nonlinearities: for example, phenomena such as 56 multi-stability or the pull-in, can be desired features through which designers can 57 achieve structural optimization and develop high performance devices. 58 A short literature review is reported here for introducing the reader to the most 59 important and recent scientific contributions to the study of thin walled structures, 60 with a specific focus to spherical caps dynamics. 61 Concerning the elastic stability of shells, buckling problems are classified into: 62 7 exhibit regular vibrations whereas with the same load conditions neglecting the 175 size-effect one obtains chaotic vibrations. 176 The present work aims to address to some questions arisen recently in Ref.
[32] on 177 pressure loaded spherical caps, where the limits of axisymmetric models were 178 shown using continuation techniques. Here the Novozhilov's geometrically 179 nonlinear theory is considered. For the analysis of the linearized equations, the 180 Rayleigh-Ritz approach is considered to obtain the mode shapes in a semi-181 analytical way. Lagrange equations are used for reducing the system of nonlinear 182 partial differential equations, PDEs, to a system of ordinary differential equations, 183 ODEs. A bifurcation analysis of is carried out by directly integrating the equations 184 of motion. Results are presented and discussed with the help of bifurcation 185 diagrams and other useful tools, such as Poincaré maps and Fourier's spectra.
where E and  are the Young's modulus and the Poisson's ratio, respectively.   It is worth noting that asymmetric modes are not associated to multiple 277 eigenvalues, therefore, they have not companion modes. 278 By imposing the set of boundary conditions (7) to the discretized eigenfunctions,  Considering only the linear terms in the strain-displacement relations (2.a-f), the 287 eigenvalue problem for approximating the natural frequencies and mode shapes of 288 the structure is obtained by imposing the stationarity of the Rayleigh's quotient 289 . 300 301

Nonlinear vibrations 302
Synchronous motion and small amplitude displacement hypotheses are now 303 relaxed, as well as the absence of external excitation. 304 In such conditions we cannot claim anymore that the vibration is harmonic or 305 periodic. 306 The approach used for analyzing the nonlinear dynamics of the cap is based on the 307 spectral theorem, i.e., taking advantage from the completeness of the 308 eigenfunctions calculated on the previous section, the displacement fields are 309 expanded as follows: 310 The former approximation simplifies the numerical calculations and reduces the 319 numerical effort; however, it could underestimate the safety factor in structures 320 that undergo to large deflections. The latter assumption is a valid approximation 321 for thin shells and should be removed for thicker structures. 322 Considering a configuration-dependent pressure distribution that always acts 323 orthogonal to the surface (follower force distribution), the expression of the j-th 324 generalized force is given by Amabili   where the system is sensitive to small perturbations and prone to exhibits chaotic 395 motion, the perturbation allows the system to leave an almost unstable orbit and 396 find remote attractors. 397 In Fig. 2(a,b), the frequency-response curves obtained by directly integrating the 398

422
In order to provide further information for understanding the path-following 423 analysis results, bifurcation diagrams of the Poincaré maps are here presented and 424 discussed. 425 In Fig. 3(a-d) Fig. 3(a, b). Nonperiodic vibrations arise for  Fig. 2(a,b). 441 Bifurcation diagrams of the Poincaré sections are now analyzed by considering a 442 decreasing excitation frequency, Fig. 4(a-d).

473
As suggested by Moon [47], in order to detect non-periodic or chaotic oscillations 474 it is not sufficient considering only frequency-response or bifurcation diagrams. 475 To this end, other mathematical tools deserve to be simultaneously considered, 476 e.g. time histories, Fourier's spectra, Poincaré sections, and phase portraits.  Fig. 6(a,b); two points are present 497 in the Poincaré map, Fig. 6(c); the regular limit-cycle shown by the phase portrait 498 confirms the periodicity of the vibration, Fig. 6(d). , and the system is in 504 the un-steady region, as depicted in Fig. 4. The Neimark-Sacker bifurcation gives 505 rise to quasi-periodic oscillations, thus the response can be seen as a sum of many this case the time response is amplitude-modulated, Fig. 7 Fig. 7(b); the Poincaré map displays two closed non-connected sets, 510 therefore the response is 2-period quasiperiodic with modulation of the amplitude 511 [49], and the orbit does not close on itself, Fig. 7(c,d). , is now analyzed. Chaotic vibrations can be observed: 519 the time history exhibits intermittency of the response bursts, Fig. 8(a); the 520 spectrum is characterized by a spreading of energy over a broad-band around the 521 carrier frequency (and multiples) 12  = , Fig. 8(b); the Poincare section 522 shows a set of randomly distributed points, where the dimension of the set does 523 not appear integer, Fig. 8(c), and the trajectory is completely irregular, Fig. 8(d).

529
Maps of chaotic motion need a larger number of points. Therefore, an additional 530 Poincaré section obtained by considering 10000 forcing periods is shown in Fig.  531 9. This map clearly shows chaotically modulated oscillations (weak chaos): the 532 central dense pattern is due to the high-frequency vibration, while the outer sparse 533 region is caused by intermittent bursts governed by a slow dynamic. Such set 534 distribution is justified by the Fourier spectrum where, despite its broad energy 535 distribution, the subharmonic components and sidebands give a significant 536 contribution to the overall dynamic of the asymmetric modal coordinate. 537 is now analyzed. Here the cap response becomes 5-T 544 subharmonic: the time history appears asymptotically stable, Fig. 10(a); the 545 fundamental frequency is 15  = , Fig. 10(b); the Poincaré map shows five 546 dots, Fig. 10(c); the solution follows a closed regular orbit, Fig. 10(d). can be noted by simply observing the time history, Fig. 11(a). The vibration is 556 strongly characterized by a 1/5-subharmonic contribution Fig. 11(b); the phase 557 portrait and the Poincaré section confirms the character of the response, Fig.  558 11(c,d). As already shown by the frequency-response curves and the bifurcation 559 diagrams, a further decrease in the excitation frequency restores a periodic 560 oscillation with a null contribution of the non-symmetric modes.

567
The problem of a shallow spherical cap exhibiting asymmetric oscillations when 568 subjected to a uniform harmonic pressure has been investigated. The 569 Novozhilov's nonlinear shell theory has been considered for defining the strain-570 displacement relations. The partial differential equations are reduced to a finite 571 dimension by using an energy formulation based on Rayleigh-Ritz approach and 572 Lagrange equations. For describing the cap deformation, the set of displacement 573 field trial functions have been expressed by means of Legendre polynomials and 574 trigonometric functions. A static compressive pressure has been superimposed to 575 a harmonic one. Bifurcation diagrams are investigated against the excitation 576 frequency. The dynamic scenario shows that the spherical cap vibrations turned out to be often asymmetric, non-periodic, with multiple jumps among 578 subharmonic, quasi-periodic, and chaotic vibrations. 579