Asymmetric vibrations and chaos in spherical caps under uniform time-varying pressure fields

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


Introduction
Thin-walled structures like plates, panels, shells, and caps are important structural elements in engineering; their applications can be found in civil engineering (roofs, vaults, tensile structures), aerospace (airplanes, missiles and rockets), and mechanics (membranebased microsensors and energy harvesters).
These structures are strong and lightweight at the same time, but they are extremely sensitive to perturbations, present a complicated instability behaviors and are very difficult to model. They could buckle under the action of critical loads, following subcritical post-buckling paths; moreover, they can exhibit nonlinear dynamic phenomena, such as chaotic vibrations, when the amplitude of vibration is moderately large.
Nowadays, many theories and simplified models are available for studying shell systems, even in the presence of composite materials, fluid-structure interaction, or thermal fields [1][2][3]. Nevertheless, new challenges come from the new frontiers of the engineering, which asks for even more reliable models where complicating effects are taken into account for exploiting the nonlinearities: for example, phenomena such as multi-stability or the pull-in can be desired features through which designers can achieve structural optimization and develop high performance devices.
A short literature review is reported here for introducing the reader to the most important and recent scientific contributions to the study of thinwalled structures, with a specific focus to spherical caps dynamics.
Concerning the elastic stability of shells, buckling problems are classified into: static buckling when loads are applied extremely slowly and dynamic buckling when the loads are suddenly applied (step loads).
From a literature review, there is a discrepancy between experimental data and theoretical results. The primary sources of inconsistencies that lead to an experimental lower buckling load than the one theoretically predicted are (i) the high sensitivity of shells to geometric imperfections and non-uniform material distribution and (ii) the post-buckling behavior is strongly affected by nonlinearities.
Let us first focus on the static instability of the spherical caps under an external pressure load. Krenze and Kiernan [4] showed the importance of producing high quality specimens for performing experimental tests. In the same period, Huang [5] and Weinitsche [6] used Marguerre's theory with possibility of having non-symmetric buckling. They showed how, for deeper caps, the wavelength of the buckling modes was higher compared to shallow caps, and the numerical results agreed with the experimental ones available at that time. These results were experimentally confirmed by Yamada et al. [7] two decades later.
The role of geometric imperfections on critical static loads of caps was investigated in Refs. [8,9]. The results pointed out how the shape of the geometric imperfection affects the decrement of the critical buckling load; often the snap-through phenomenon disappeared due to imperfections, and continuous and stable buckling paths were shown by the pressuredeflection diagrams.
Since the measurement of imperfections is not always possible for large scale applications, NASA proposed an empirical formula based on the lower envelope of a series of experimental data [10]. Nowadays, the specimens quality is higher and other techniques have been proposed for improving the NASA empirical formula [11,12].
A further reduction of the load-carrying capacity can be observed when the time dependency of the load is considered, i.e., in the case of dynamic buckling.
Lock et al. [13] experimentally analyzed the buckling of shallow domes under a pressure step loading. They discussed the difference between ''direct'' and ''indirect'' snapping phenomena. The direct snapping is a catastrophic phenomenon and involves only symmetric vibrations; conversely, the indirect snapping occurs after a transient and the contribution of the non-symmetric modes is not negligible after the snapping.
Stricklin et al. [14] used nonlinear Novozhilov's theory for investigating the static stresses in shells of revolution and improved their model for studying the dynamic buckling in Ref. [15]. The equilibrium equations were obtained through Castigliano's theorem. The numerical results were compared with experimental ones, and an excellent agreement was proved. The dynamic model was derived employing of the Lagrange equations by considering only axisymmetric modes, and the results confirmed the previous analyses [16,17].
Ball and Burt [18] numerically investigated the dynamic buckling of clamped shallow spherical caps under symmetric and nearly symmetric step pressure loads. Asymmetric modes were considered, and the buckling load of geometrically perfect structures of different shallowness was given.
The asymmetric dynamic buckling of shallow spherical caps was investigated even by Akkas [19], who showed that the asymmetric buckling under step pressure load results in cusps in phase-plane diagrams.
Further results concerning the dynamic buckling of imperfect caps can be found in Refs. [20][21][22], where the possibility of having plastic deformations was considered as well.
In the framework of spherical caps under harmonic loads, the literature is not as vast as for the buckling. Reasons must be sought in the fact that: (i) spherical caps are a particular case of doubly curved shells, they are modeled through equations that are more complex with respect to plates and cylindrical shells; (ii) the high computational cost related to the numerical integration of the equations of motion limited for long time the analysis to low-dimensional models and axisymmetric vibrations.
Using a theory proposed by Yu [23], Grossman et al. [24] investigated the axisymmetric nonlinear vibrations of shallow spherical caps with different boundary conditions. This study compared flat plates to curved caps, and the results pointed out the transition from hardening to softening nonlinearity when the surface curvature is increased.
Evensen and Iwanovsky [25] were the first to perform both analytically and experimental analyses on shallow spherical caps under a combination of static and sinusoidal external pressure loads. The analytical model was based on the Marguerre's nonlinear shell theory. Axisymmetric deflections and uniform load distribution were considered. A detailed scheme of the experimental setup was reported and discussed. The numerical results concerning free vibrations were in excellent agreement with experiments. Unfortunately, differences were shown in several nonlinear forced cases. Such discrepancies were mainly attributed to the interaction between static and dynamic loads and to the asymmetric vibrations observed during the experiments.
Yasuda and Kushida [26] studied the forced vibration of caps under harmonic point loads. The activation of subharmonic motion due to internal resonances was observed. In order to validate the numerical model, experiments were performed on a bent circular plate clamped at its edges. The structure was loaded by a concentrated force induced by two electric magnets, and the experimental results agreed with the numerical ones.
The axisymmetric vibrations of preloaded shallow spherical caps were investigated by Gonçalves [27,28] and Soliman and Gonçalves [29]. For obtaining a reduced-order model (ROM), the Galerkin method was considered. The displacement fields were expanded by using the Bessel functions, and the resulting equations of motion were solved through the Newton-Rapson method. The results showed a strong influence of geometric imperfections and their spatial shape. Softening nonlinearity can be turned to hardening by imposing a suitable initial imperfection of a given shape and amplitude, as shown by the reported backbone curves. Moreover, assuming the excitation frequency as a control parameter, the bifurcation analysis pointed out the existence of period-doubling cascades and chaotic oscillations. The onset of such phenomena is due to energy given by the harmonic pressure to the shell, which leads to multiple backand-forth jumps between potential wells.
Thomas et al. [30,31] studied the response of a free-edge shallow spherical cap under harmonic excitation. Using the multiple-scale perturbation method, the results showed that, having integer or quasi-integer ratio between natural frequencies is not a sufficient condition for having internal resonances activation. This is due to the body symmetry, which leads to the canceling of some nonlinear coefficient in the ODEs. Experiments were carried out by forcing the specimen using an electromagnetic coil. The occurrence of an internal resonance between two conjugate asymmetric modes and one axisymmetric mode (1:1:2) was proven, a good qualitative fitting between theory and experiments was shown for small forcing amplitude. Touzè et al. [32,33] used the nonlinear normal modes approach (NNMs) for predicting the trend of nonlinearity for each mode as a function of the spherical cap geometric aspect ratios. In particular, the transition from hardening to softening nonlinearity was addressed.
Chaotic vibrations in shallow shells with circular planform were investigated by Krysko et al. [34]: The role of size-dependent parameters on vibrations of nanoshells were analyzed. The system of PDE was reduced using a finite difference method (FDM), and the resulting system was solved through a Runge-Kutta scheme. By comparing Fourier's spectra, Poincaré maps, Lyapunov exponents, and Morlet wavelet, the authors showed that considering the size-effect shells exhibits regular vibrations whereas with the same load conditions neglecting the size-effect one obtains chaotic vibrations.
The present work aims to address to some questions arisen recently in Ref. [35] on pressure-loaded spherical caps, where the limits of axisymmetric models were shown using continuation techniques. Here the Novozhilov's geometrically nonlinear theory is considered. For the analysis of the linearized equations, the Rayleigh-Ritz approach is considered to obtain the mode shapes in a semi-analytical way. Lagrange equations are used for reducing the system of nonlinear partial differential equations, PDEs, to a system of ordinary differential equations, ODEs. A bifurcation analysis is carried out by directly integrating the equations of motion. The results are presented and discussed with the help of bifurcation diagrams and other useful tools, such as Poincaré maps and Fourier's spectra. The superimposition of a static and a dynamic pressure yields to non-periodic and chaotic oscillation related to the activation of asymmetric modes.

Problem formulation
A spherical cap having radius R, base radius a, cap height s, and thickness h, is considered, see Fig. 1a-c. A spherical coordinate system ðO; u; #; zÞ is centered at the top of the cap O. The curvilinear coordinates ðu; #Þ identify a point P of the shell middle surface, z is the radial distance of a generic point of the shell from the middle surface. Three displacement fields, meridional u g; #; t ð Þ, circumferential v g; #; t ð Þ, and radial w g; #; t ð Þ, describe the deformed configuration of the middle surface; t is the time variable.
Limiting the analysis to shallow spherical caps, the Lamé parameters of the undeformed middle surface are For describing the relationships between strains and displacements, the Novozhilov's nonlinear shell theory [36] is considered. Such theory is based on the Kirchhoff-Love hypothesis, which states that: (i) the shell is thin h ( R and h ( a, (ii) strains, (iii) transverse normal stresses are small, and (iv) the normal to the undeformed middle surface remains normal after deformation, and no thickness stretching occurs. The hypothesis of small displacements is relaxed in the nonlinear analysis.
Because of the aforementioned hypothesis, the strainsê i ;ĉ ij at an arbitrary point of the cap linearly vary along the thickness; moreover, the plane-stress hypothesis is considered. The strains are given by: where e g , e # , and c g# are the middle surface strains,k g , k # , and k g# are the changes in curvatures and torsion of the middle surface of the shell, which depend on the middle surface displacement fields through the following relationships: c g# ¼ e 12 þ e 21 þ e 11 e 21 þ e 12 e 22 þ e 12 e 23 ; ð2cÞ where the strain components e ij are: For the sake of clarity, the nonlinear terms in (2d-f) do not play a relevant role in thin shells dynamics, thus have been neglected [37].
Considering an elastic linear, homogeneous and isotropic continuum, one can use the Hooke's law, i.e., the following stress-strain relationships: where E and m are the Young's modulus and the Poisson's ratio, respectively. By considering the strains (1a-d) and the stresses (4a-b), the elastic strain energy U S [38] of a thin shallow spherical cap is given by whilethe kinetic energy T S , under the hypothesis of negligible rotary inertia [38], is given by where q S is the material mass density, g 0 is the half opening angle of a hole assumed a the cap pole for avoiding the singularity due to the spherical reference system [39].

Approximate eigenfunctions
In order to develop a ROM for studying the cap nonlinear dynamics, in this section the eigenfunctions of the linearized operator are obtained through the Rayleigh-Ritz approach [40].
In the present study, clamped boundary conditions are considered at the circular edge while no boundary conditions are considered at the cap pole.
The Rayleigh-Ritz approach requires that a trial function set respects the geometric boundary conditions only [41]; on the other hand, the stress-free boundary conditions at the cap pole (where the small hole is present) are neglected.
The generic mode of vibration can be described by considering three displacement fields uðg; #; tÞ; vðg; #; tÞ; and wðg; #; tÞ, which obey to the same time law f ðtÞ, i.e., the variable separation can be considered: where P Ã m ðgÞ ¼ P m ð2g À 1Þ is the mth Legendre polynomial of the first kind shifted in the domain g 2 0; 1 ½ ; m is related to the number of meridional wavelength; n is the number of nodal diameters.
Because of the axial symmetry, spherical caps exhibit conjugate modes, called driven and companion mode shapes or conjugate modes [42,43]. These modes have the same natural frequency and shape, but Fig. 3 Bifurcation diagrams of the Poincaré section for an increasing excitation frequency: a first axisymmetric mode, b second axisymmetric mode, c driven, and d companion asymmetric modes (1,2) the displacement fields are angularly shifted of p=2n. Conjugate modes describe standing waves, but circumferential traveling waves could arise when nonlinear mode coupling occurs [44][45][46]. Therefore, companion modes should be considered when a nonlinear analysis is carried out.
It is worth noting that asymmetric modes are not associated with multiple eigenvalues; therefore, they have not companion modes.
By imposing the set of boundary conditions (7) to the discretized eigenfunctions, a system of algebraic equations is obtained: The solution of this linear system allows to express ðŨ 0;n ;Ṽ 0;n ;W 0;n ;W 1;n Þ in terms of the remaining coefficients ðŨ m;n ;Ṽ m;n ;W m;n Þ; the latter coefficients can be reordered in a vectorq [47] with a number of elements equal to N max ¼ ðM u þ M v þ M w þ 3 À bÞðN þ 1Þ, where b = 4 for a clamped circular cap [35].
Considering only the linear terms in the straindisplacement relations (2a-f), the eigenvalue problem for approximating the natural frequencies and mode shapes of the structure is obtained by imposing the stationarity of the Rayleigh's quotient Rq ð Þ ¼ U Sq ð Þ spaceT Ã Sq ð Þ, where U Sq ð Þ is the maximum potential energy during a ''modal'' oscillation, and Fig. 4 Bifurcation diagrams of the Poincaré section for a decreasing excitation frequency: a first axisymmetric mode, b second axisymmetric mode, c driven, and d companion asymmetric modes (1,2) x is the circular frequency of the harmonic motion; M and K are the mass matrix and the stiffness matrix of the discrete linearized system, respectively.
The ith solution of Eq. (13), x ðiÞ ;q ðiÞ , gives the approximation of the ith natural frequency and mode shape, respectively. To improve the results readability and the numerical accuracy, the approximated mode shapes are normalized using the approach of Ref. [47], and the following condition is sought max abs U ðiÞ Â Â ðg; #Þ; abs V ðiÞ ðg; #Þ Â Ã ; abs W ðiÞ ðg; #Þ Â Ã ¼ 1.

Nonlinear vibrations
Synchronous motion and small amplitude displacement hypotheses are now relaxed, as well as the absence of external excitation. In such conditions we cannot claim anymore that the vibration is harmonic or periodic.
The approach used for analyzing the nonlinear dynamics of the cap is based on the spectral theorem, i.e., taking advantage from the completeness of the eigenfunctions calculated on the previous section, the displacement fields are expanded as follows: where d and c are related to the driven and companion modes, respectively; i and j identify the number of meridional and circumferential wavelengths; f ðÁÞ k;i;j are the time-dependent unknown generalized coordinates.
For thin-walled bodies under external pressure load, two assumptions are common in the literature: (i) the pressure is considered as a radial non-follower load; (ii) the load distribution is applied to the middle surface [36].
The former approximation simplifies the numerical calculations and reduces the numerical effort; however, it could underestimate the safety factor in structures that undergo to large deflections. The latter assumption is a valid approximation for thin shells and should be removed for thicker structures.
Considering a configuration-dependent pressure distribution that always acts orthogonal to the surface (follower force distribution), the expression of the jth generalized force is given by Amabili and Breslavsky, where only the linear strain terms are retained [48] oW p oq j ffi The external pressure consists of a static and a dynamic component pðtÞ ¼ p s þ p d cosðXtÞ is the external pressure. The pressure is positive when inflates the structure.
Taking into account the full expression of the strains (2a-f) and replacing them into the energies and virtual work formulae, the equations of motion are derived by the Lagrange equations d dt ; for j ¼ 1; 2; :::; N dofs ð16Þ N dofs indicates the number of degrees of freedom of the nonlinear ODEs. Such set could be rewritten into statespace form.
The set of nonlinear ODEs could be rewritten into the following first-order form: Note that M À1 C ¼ diagð2f j x j Þ, where f j and x j are the damping ration and the natural frequencies of the jth generalized coordinate; p s and p d are the generalized force vectors due to the static and dynamic pressure, respectively; X is the frequency of the external excitation; y is the generalized velocity vector.
In the following analysis, the equations of motion are reduced to a non-dimensional form: the amplitudes are divided by the shell thickness; the time is divided by the period of the first axisymmetric resonant mode.
The pressure is normalized through to the Zoelly's critical buckling pressure of a complete, isotropic, and homogeneous sphere, see Ref. [49] For the sake of completeness, the expression of the parameter k is here reported due to its important meaning: k includes information on the thinness and the shallowness of the investigated structure.

Numerical results
Consider a clamped shallow spherical cap, having a uniform thickness, made of steel. The nonlinear dynamic response of the structure under a timevarying harmonic pressure is investigated by using the high-dimensional nonlinear model developed in Ref [35] and details about the mode selection criteria and convergence analysis are given below. The asymmetric mode (m = 1,n = 2) is of primary importance for building the nonlinear model because it is the first buckling mode for caps having k ¼ 6 [4,5]. For considering the possibility of having nonlinear interaction, modes with multiple numbers of both meridional and circumferential waves ðk Â 1; p Â 2Þ, where k and p are integers, have been considered into the model. Regarding the axisymmetric modes (n = 0), their role is fundamental for describing the cap deformation because of the structural symmetry and is proven that interactions among axisymmetric and asymmetric modes could significantly affect the shell nonlinear behavior [30,31]. The re-expansion series convergence was based on the static buckling loads. First, the critical loads provided by four nonlinear models with 17, 21, 25, and 28 dofs were compared. In particular, the convergence analysis (Table 5 Ref. [35]) proved that increasing the number of modes used in the expansion does not lead to significant variation in the results. In addition, the results given by the model with 25 dofs were compared to numerical and experimental analyses and a good agreement was proven (see Table 6 in Ref. [35]). Then, for studying the dynamic behavior of the spherical cap, the asymmetric companion modes (n = 0) were considered in the ROM. This last step increased the dofs number from 25 to 38. For the sake of clarity, the linear mode shapes retained into the nonlinear ROM are listed in Table 1.
The geometrical and structural data are listed as follows: The natural frequency of the first axisymmetric mode,x 1;0 , is considered for the time non-dimensionalization, as already stated in (18), and a modal damping factor 1 j ¼ 0:012; j ¼ 1; 2; :::N dofs is assumed.
A static pressure load p s ¼ À0:40 Á p cr (lower than the critical buckling pressure) acts on the shell while a dynamic component, of amplitude p d ¼ 0:020 Á p cr , is superimposed to the static one.
The set of nonlinear ODEs (17) is numerically solved by using the Fortran routine for time integration RADAU5 [50]. This integrator was developed for solving stiff ODEs and is based on the implicit Runge-Kutta method of order 5, three stages, with step-size control. To carry out the bifurcation analysis, the excitation frequency is varied forward and backward in the frequency range X x 1;0 2 ½1:07400; 1:10500, where the occurrence of dynamic instabilities was proven through a path-following analysis in Ref. [35].
The parameters used for setting the time-response analysis are the following: 125 excitation frequency steps with a step-size of DX x 1;0 ¼ AE0:00025; a sampling frequency equal to 40 times the excitation frequency; 600 excitation periods of integration, where only 300 periods are retained for getting rid of the transient response. When the simulation starts, homogeneous initial conditions are considered, and then, for further steps (different frequencies), the initial conditions are assumed to be the final state of the previous step, with a perturbation of amplitude 0.01 (dimensionless) applied to every generalized coordinate. For the frequencies where the system is sensitive to small perturbations and prone to exhibits chaotic motion, the perturbation allows the system to leave an almost unstable orbit and find remote attractors.
In Fig. 2a, b, the frequency-response curves obtained by directly integrating the ODEs are compared to the continuation method results [35]. Starting from X x 1;0 ¼ 1:0740 and considering an increasing forcing frequency (red dashed line), f w;1;0 follows the stable solution path 1 (solid black line) and switch on branch 2 after the period-doubling (PD) at X x 1;0 ¼ 1:08275, see Fig. 2a, where the bifurcation leads to the onset of asymmetric oscillations, see  persists until a second PD bifurcation at X x 1;0 ¼ 1:10125. Considering now a backward frequency variation (solid blue line), the frequency-response curve trend is almost the same obtained by considering an upward frequency variation. However, when the harmonic pressure acts on the structure with a frequency X x 1;0 2 ½1:07650; 1:08400, both the coordinates f w;1;0 and f ðdÞ w;1;2 follow secondary solution branches not shown by the path-following analysis [35]. A further reduction of the forcing frequency leads to a sudden response jump that restores a purely axisymmetric overall motion of the cap.
In order to provide further information for understanding the path-following analysis results, bifurcation diagrams of the Poincaré maps are here presented and discussed.
In Fig. 3a-d, the bifurcation diagrams obtained for an increasing excitation frequency are shown. The response is fully axisymmetric until X display 2-T periodic oscillations. An additional excitation frequency increment gives rise to a perioddoubling bifurcation at X x 1;0 ¼1:10125, in agreement with the finding of [35]. Beyond the perioddoubling, the response becomes periodic with the same frequency of the excitation and the contribution of the asymmetric modes on the overall oscillation becomes null, as already pointed out form the analysis of the frequency-response diagrams in Fig. 2a, b. Bifurcation diagrams of the Poincaré sections are now analyzed by considering a decreasing excitation frequency, Fig. 4a-d.
Starting from X x 1;0 ¼ 1:10500, the structural response undergoes sequentially to a period-doubling  Moon [51], in order to detect nonperiodic or chaotic oscillations it is not sufficient considering only frequency-response or bifurcation diagrams. To this end, other mathematical tools Fig. 8 Chaotic response of the driven modal coordinate (1,2). Decreasing excitation frequency. a Time history, b Fourier spectrum, c Poincaré map, and d phase portrait Fig. 9 Poincaré map of chaotically modulated oscillations deserve to be simultaneously considered, e.g., time histories, Fourier's spectra, Poincaré sections, and phase portraits.
Without loss of generality, only the case of decreasing excitation frequency is here deeply investigated.
In Fig. 5a the spectrogram of f w;1;0 is shown. The energy content is localized at the same frequency of the excitation until the instability onset, where the energy spreads on a broad frequency range. On the other hand, the response of the asymmetric mode f ðdÞ w;1;2 is mainly -subharmonic, see Fig. 5b. When the frequency of the harmonic pressure is decreasing and crosses X x 1;0 ¼ 1:0850, 5 T-subharmonic components of the response are clearly visible from both spectra. It is worthwhile to note that in the spectrum of f ðdÞ w;1;2 the main 1 T-harmonic is almost absent except in the frequency range of strong subharmonic vibrations.
In the following, the development of chaotic oscillations is shown and the behavior of the driven asymmetric mode f ðdÞ w;1;2 is deeply addressed to complete the description of the dynamic scenario.
The driven asymmetric mode f ðdÞ w;1;2 shows asubharmonic: Only odd harmonics appear in the spectrum because of the symmetry of the time waveform, Fig. 6a, b; two points are present in the Poincaré map, Fig. 6c; the regular limit cycle shown by the phase portrait confirms the periodicity of the vibration, Fig. 6d.
The forcing frequency is now reduced to X x 1;0 ¼1:09375, and the system is in the unsteady region, as depicted in Fig. 4. The Neimark-Sacker bifurcation gives rise to quasiperiodic oscillations, and thus, the response can be seen as a sum of many periodic functions, where two or more frequencies are incommensurate [52]: In this case the time response is amplitude-modulated, Fig. 7a; the carrier frequency is x=X ¼ 1=2 and sidebands (modulation frequency Dx=X ¼ 0:11) are present, Fig. 7b; the Poincaré map displays two closed non-connected sets; therefore, the response is two-period quasiperiodic with modulation of the amplitude [53], and the orbit does not close on itself, Fig. 7c, d.
The case at X x 1;0 ¼1:0900 is now analyzed. Chaotic vibrations can be observed: The time history exhibits intermittency of the response bursts, Fig. 8a; the spectrum is characterized by a spreading of energy over a broadband around the carrier frequency (and multiples) x=X ¼ 1=2, Fig. 8b; the Poincare section shows a set of randomly distributed points, where the dimension of the set does not appear integer, Fig. 8c, and the trajectory is completely irregular, Fig. 8d.
Maps of chaotic motion need a larger number of points. Therefore, an additional Poincaré section obtained by considering 10,000 forcing periods is shown in Fig. 9. This map clearly shows chaotically modulated oscillations (weak chaos): The central dense pattern is due to the high-frequency vibration, while the outer sparse region is caused by intermittent bursts governed by a slow dynamic. Such set distribution is justified by the Fourier spectrum where, despite its broad energy distribution, the subharmonic components and sidebands give a significant contribution to the overall dynamic of the asymmetric modal coordinate.
After a further reduction of the excitation frequency, the system exits form the chaotic region even though it is still inside the ''instability region,'' where an alternance of periodic and non-periodic regions is present. More specifically, the case X x 1;0 ¼1:0827 is now analyzed. Here the cap response becomes 5-Tsubharmonic: The time history appears asymptotically stable, Fig. 10a; the fundamental frequency is x=X ¼ 1=5, Fig. 10b; the Poincaré map shows five dots, Fig. 10c; the solution follows a closed regular orbit, Fig. 10d.
The last case to be investigated is X x 1;0 ¼1:07735. The coordinate f ðdÞ w;1;2 exhibits quasiperiodic oscillations, where the superposition of several periodic functions can be noted by simply observing the time history, Fig. 11a. The vibration is strongly characterized by a 1/5-subharmonic contribution Fig. 11b; the phase portrait and the Poincaré section confirms the character of the response, Fig. 11c, d. As already shown by the frequencyresponse curves and the bifurcation diagrams, a further Fig. 11 Quasiperiodic response of the driven modal coordinate (1,2). Decreasing excitation frequency. a Time history, b Fourier spectrum, c Poincaré map, and d phase portrait decrease in the excitation frequency restores a periodic oscillation with a null contribution of the nonsymmetric modes.

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