Nonlinear Equations Of Motion Research Articles

Orbit Predictions Using KS Elements With Earth’s Flattening

Predicting the orbit of a satellite is a fundamental requirement in many areas of aerospace, such as mission planning, satellite geodesy, re-entry prediction, maneuver planning, collision avoidance, formation flying, etc. For near-Earth orbits, the forces due to the non spherical nature of the Earth and atmospheric drag plays an important role and are mainly responsible to bring the satellite back to the Earth. Thus inclusion of the effect of these perturbing forces becomes important for precise orbit computation of near-Earth orbits. Many transformations have emerged in the literature to stabilize the equations of motion either to reduce the accumulation of local numerical errors or allowing the use of large integration step sizes, or both in the transformed space. One such transformation is known as KS transformation by Kustaanheimo and Stiefel [1], who regularized the nonlinear Kepler equation of motion and reduced it into linear differential equations of a harmonic oscillator of constant frequency. In this paper a detailed analysis is carried out for orbit prediction using KS differential equations by including the non spherical gravitational potential of the Earth as the perturbing force. Earths zonal and tesseral harmonics, which produce bands of constant deviation from the spherical field along lines of latitude and longitude, are included to precisely model the Earth’s gravitational potential. Higher order Earth’s flattening terms are included by utilizing the recurrence relations of associated Legendre polynomial and its derivatives. To know the effectiveness of the theory, the results are compared with the real IRS-1A satellite data and the results of other existing solutions for a long duration of nearly a month time. The comparison shows that the KS method provides one of the most and accurate techniques for orbit prediction available at present.

A novel approach to the motion equations of null Cartan curves via the compatible Hasimoto map

In this manuscript, we try to understand the links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of null Cartan curves and some physical results. First, we obtain the equations of Hasimoto transformation occurred by applying Hasimoto map for null Cartan curves. Then, based on the Hasimoto map of null Cartan curves, we examine both Landau–Lifshitz system and visco Landau–Lifshitz system. Lastly, we portray the links of findings obtained by geometric physical issues. After all, we propose a new method, known as Khater method, to obtain solutions of fractional order differential equations. Also, these solutions are simulated by graphically using physical constructions.

Applicability of Clohessy-wiltshire Model For Optimal Low-thrust Rendezvous

This paper investigates the applicability of the continuous thrust Clohessy-Wiltshire model for the optimal low-thrust rendezvous in the orbit. While simplifying the non-linear equations of relative orbital motion, certain assumptions were taken in account by Clohessy and Wiltshire to linearize the system of equations. When the same model is used for optimization of the trajectory of a relative transfer in the orbit using low-thrust, because of the assumptions in the dynamic model there are errors in the result which in turn depends upon the initial conditions for the optimal control problem. A study has been done to investigate the possible trend of error in the control input by varying the initial conditions for the relative transfer and comparing the results with those obtained using non-linear equations. Further, the trend of error in control input has also been studied by increasing the altitude of the target spacecraft orbit. Optimization of the relative motion trajectory has been performed to account for minimum propellant expenditure.

Rendezvous and Proximity Operations in Cislunar Space Using Linearized Dynamics for Estimation

As interest in Moon exploration grows, and efforts to establish an orbiting outpost intensify, accurate modeling of spacecraft dynamics in cislunar space is becoming increasingly important. Contrary to satellites in Low Earth Orbit (LEO), where it takes around 5 ms to communicate back and forth with a ground station, it can take up to 2.4 s to communicate with satellites near the Moon. This delay in communication can make the difference between a successful docking and a catastrophic collision for a remotely controlled satellite. Moreover, due to the unstable nature of trajectories in cislunar space, it is necessary to design spacecraft that can autonomously make frequent maneuvers to stay on track with a reference orbit. The communication delay and unstable trajectories are exactly why autonomous navigation is critical for proximity operations and rendezvous and docking missions in cislunar space. Because spacecraft computational hardware is limited, reducing the computational complexity of navigational algorithms is both desirable and often necessary. By the introduction of a linear system approach to the deputy spacecraft motion, this research avoids the computational burden of integrating the deputy relative equations of motion. In this research, the relative CR3BP equations of motion are derived and linearized using a matrix exponential approximation. This research continues the development of the matrix exponential linearized relative circular restricted three-body problem (CR3BP) equations by applying the dynamics model to estimation and control applications. A simulation is performed to compare state estimation results obtained from using the linearized equations of motion utilizing a Kalman filter and for state estimation utilizing an unscented Kalman filter with the full nonlinear equations of motion. The linearized exponential model is shown to be sufficient for state estimation in the presence of noisy measurements for an example scenario. Additionally, a linear quadratic regulator (LQR) controller was added to optimally control a deputy spacecraft to rendezvous with a chief spacecraft in cislunar space. The contribution of this work is twofold: to provide a proof of concept that the matrix exponential solution for the linearized relative CR3BP equations can be used as the dynamics model for state estimation, as well as to simulate an optimal rendezvous maneuver in the presence of measurement noise.

Nonlinear dynamic and vibration characteristics of metamaterial shallow arches

Metamaterial shallow arches have shown promise in enhancing the seismic resistance of structures due to their unique properties. In this study, we present a theoretical model for the nonlinear vibration and dynamic response of metamaterial shallow arches. The nonlinear motion equations consider lateral shear deformation and large deflection based on the third-order shear theory and von Kármán nonlinear theory. The study focuses on three aspects: establishing an effective model to capture the global characteristics of laminated structures with damage, designing members with auxetic and non-auxetic properties using the effective model, and obtaining an asymptotic solution of the motion equations using a two-perturbation approach. The presented model is applied to study the effect of initial thermal stress, negative Poisson's ratio, and internal damage on the forced and larger amplitude vibration of arches with damage. The results show that the auxetic effect can reduce the displacement of arches with and without damage, thus mitigating dynamic damage. Furthermore, the numerical results also reveal that deflection, linear, and nonlinear frequency are significantly dependent on the internal damage and auxetic property. Our findings represent an efficient approach to evaluating the dynamic and vibration response of metamaterial arches and optimally design the seismic resistance of arches.

Energy dissipation mechanisms in three‐dimensional rocking motion of cylindrical structures

AbstractIn recent years, there has been a growing recognition that the rocking motion of structures, especially cylindrical ones, should be evaluated with three‐dimensional (3D) models. However, dissipation mechanisms in 3D models are complex and have yet to be well established. This paper presents an analytical study on an inherent dissipation mechanism based on the rolling friction model and an external dissipation mechanism based on energy dissipators (EDs) for 3D rocking cylindrical structures. Through a variational formulation, the nonlinear equations of motion are derived considering the rolling friction model and the Bouc–Wen model. Compared with the existing inherent dissipation model in literature, the rolling friction model has a noticeable advantage that it can capture the energy dissipation behavior during free vibration under various initial conditions. Additionally, the rolling friction model is more versatile since the contact coefficient is adjustable. On the other hand, external EDs further enhance the energy dissipation of the structures. Roles of the inherent and external mechanisms in the 3D rocking motion of cylindrical structures against earthquakes are assessed using a series of near‐fault pulse‐like ground motions. Results indicate the inherent dissipation mechanism, that is, the rolling friction model, cannot avoid overturning under near‐fault pulse‐like ground motions, leading to unstable structures. This is because an uplifted cylinder will continuously absorb energy from earthquake excitations, while the amount of energy dissipated by the rolling friction model is small. Comparatively, adding external EDs is effective in mitigating seismic responses and thereby avoiding overturning.

Optimal Design and Dynamic Analysis of a New Quasi-Zero-Stiffness Isolation Device

Compared with the linear isolation system, the quasi-zero-stiffness (QZS) nonlinear isolation system has the characteristics of high static stiffness and low dynamic stiffness, which has better low-frequency vibration isolation performance. However, most of the existing QZS isolators only consider the quasi-zero-stiffness characteristic at the static equilibrium position achieved by the parallel connection of positive and negative stiffness structures. To optimize the isolation performance of the QZS system, a new isolation device based on the parallel connection of oblique springs and vertical springs was proposed. The device can not only achieve quasi-zero-stiffness at the static equilibrium position but also expand the interval of quasi-zero-stiffness through parameter optimization design to optimize the stiffness characteristics of the QZS system, thus effectively improving the vibration isolation performance. The QZS nonlinear systems with the optimal parameters were analyzed dynamically, and the nonlinear motion equations were approximately solved based on the fifth-order polynomials fitted by the restoring force curves. A prototype was further designed and fabricated to compare and analyze the vibration isolation performance of the QZS system and the equivalent linear system through a shaking table test.

Cross-Coupled Dynamics and MPA-Optimized Robust MIMO Control for a Compact Unmanned Underwater Vehicle

A compact, 3-degrees-of-freedom (DoF), low-cost, remotely operated unmanned underwater vehicle (UUV), or MicroROV, is custom-designed, developed, instrumented, and interfaced with a PC for real-time data acquisition and control. The nonlinear equations of motion (EoM) are developed for the under-actuated, open-frame, cross-coupled MicroROV utilizing the Newton-Euler approach. The cross-coupling between heave and yaw motion, an important dynamic of a class of compact ROVs that is barely reported, is investigated here. This work is thus motivated towards developing an understanding of the physics of the highly coupled compact ROV and towards developing model-based stabilizing controllers. The linearized EoM aids in developing high-fidelity experimental data-driven transfer function models. The coupled heave-yaw transfer function model is improved to an auto-regressive moving average with exogenous input (ARMAX) model structure. The acquired models facilitate the use of the multi-parameter root-locus (MPRL) technique to design baseline controllers for a cross-coupled multi-input, multi-output (MIMO) MicroROV. The controller gains are further optimized by employing an innovative Marine Predator Algorithm (MPA). The robustness of the designed controllers is gauged using gain and phase margins. In addition, the real-time controllers were deployed on an onboard embedded system utilizing Simulink′s automatic C++ code generation capabilities. Finally, pool tests of the MicroROV demonstrate the efficacy of the proposed control strategy.

NIXO-Based identification of the dominant terms in a nonlinear equation of motion of structures with geometric nonlinearity

The objective of this publication is to propose a novel technique for black-box identification of structures with geometric nonlinearity. The technique is based on a recently introduced frequency-domain system identification method called Nonlinear Identification through eXtended Outputs (NIXO). The proposed algorithm first expresses the nonlinear part of modal equation of motion (EOM) as a general polynomial of high order, and then removes the terms which are determined to be irrelevant to the mechanical system’s response. This division into dominant and irrelevant nonlinear terms relies on the values of two novel indicators that are particular to NIXO. The heuristic presented here was observed to work well only when the tested structure is excited with various swept-sine input signals of different magnitudes (so that the system oscillates at sufficiently distinctive amplitudes in different experimental tests). The technique is first demonstrated on a noise-free numerical case study employing a reduced model of a curved beam. The results obtained are verified via comparing the true Nonlinear Normal Mode (NNM) to the one computed using the modal EOM pointed out by NIXO. Then the method is demonstrated on experimental measurements collected on a 3D-printed flat beam exhibiting significant natural frequency shifts, and the outcomes are verified by overlaying the identified NNMs on the swept-sine responses.

Linear and Nonlinear Models for Drop Simulation of an Aircraft Landing Gear System with MR Dampers

In this study, our focus is on the drop test simulation of an MR (Magnetorheological) damper-based main landing gear (MRMLG), aiming to explore multi-degree-of-freedom (DOF) dynamic models during aircraft landing. Three different 6-DOF dynamic models are proposed in this work, and their drop performances are compared with results achieved by commercial software. The proposed models include a nonlinear aircraft model (NLAM); a linearized approximated aircraft model (LAAM) linearizing from the nonlinear equations of motion in NLAM; and a fully approximated aircraft model (FAAM) which linearizes the MRMLG’s strut force model. In order to evaluate the drop performance of the aircraft landing gear system with MR dampers, a 7-DOF aircraft model incorporating the nonlinear MRMLG was formulated using RecurDyn. The principal comparative parameters are the coefficient of determination (R2) for the system response of each model with the RecurDyn model and root mean square error (RMSE), which is the ensemble of CG displacement data for each model. In addition, the ensemble of time series data is created for diverse drop scenarios, providing valuable insights into the performance of the proposed drop test models of an aircraft landing gear system featuring MR dampers.

Nonlinear Identification through eXtended Outputs (NIXO) with numerical and experimental validation using geometrically nonlinear structures

This work presents a novel technique for nonlinear system identification that operates in the frequency domain and fits a model to measured spectra to estimate the parameters in a modal domain nonlinear equation of motion (EOM). Nonlinear terms are added to the linear EOM in the form of polynomials, and the proposed algorithm estimates the polynomial coefficients as well as the underlying linear Frequency Response Function (FRF). This method is an extension to a popular nonlinear system identification algorithm called NIFO, from Nonlinear Identification through Feedback of the Outputs. However, NIFO identifies the nonlinear parameters as complex numbers that may be different at each frequency line, even though the mechanical system is expected to be governed by an EOM in which the nonlinear parameters are real and constant with frequency. This might be problematic, because any variation in the identified nonlinear parameters will distort the linear FRFs estimated by NIFO, and those linear FRFs are important to tell the user whether all of the significant nonlinearity has been extracted from the system. The proposed algorithm, here dubbed Nonlinear Identification through eXtended Outputs (NIXO), estimates the nonlinear parameters as frequency-independent and real. Additionally, it is demonstrated that for the systems studied here that the algorithm works when random and swept-sine inputs are used to excite the tested structure, while NIFO only worked well when random inputs were used. The method is first evaluated numerically using benchmark case studies, starting with the SDOF equation and then reduced models of a clamped-clamped flat beam, and the results are compared to those obtained with NIFO. Then the algorithm is applied to swept-sine measurements from a 3D-printed flat beam and the results are validated by computing the primary nonlinear normal mode of the identified model and comparing it with measurements.

A Maneuvering Model for an Underwater Vehicle Near a Free Surface—Part II: Incorporation of the Free-Surface Memory

Using energy-based modeling techniques, we propose a nonlinear, time-dependent, parametric motion model for an underwater vehicle maneuvering near an otherwise undisturbed free surface. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving through an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free-surface Lagrangian</i> , which accounts for the instantaneous energy stored within the free surface due to an impulsive vehicle motion as well as fluid memory effects. The system Lagrangian then enables us to derive the six-degree-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, although additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. For the proposed model, the vessel motion is unrestricted. This is in contrast to traditional seakeeping models, which use convolution integrals to incorporate memory effects for a vessel, which experiences only small perturbations from steady, forward motion. The proposed motion model is amenable to real-time simulation, design performance analysis, and nonlinear control design. Other important hydrodynamic effects due to viscous flow, for example, may then be incorporated into a robust, nonlinear, closed-loop control system as lumped parameter effects or model uncertainties.

Research on second level of vulnerability criteria of parametric rolling with stochastic stability theory

The second level of vulnerability criteria of parametric rolling proposed by the International Maritime Organization (IMO) is aimed to evaluate the probability of roll instability and large roll motion of ships in the designated sea areas. Compared with the numerical simulation in random waves, the stability boundary derived from the criteria has large errors. Some additional checks are required in the key region. A new method is proposed to assess ship parametric roll in the designed sea area based on stochastic stability theory. A nonlinear system has the same stability properties as its equivalent linear system. Therefore, the nonlinear parametric rolling motion equation is linearized in order to study its stochastic stability in random waves. Based on the top Lyapunov exponent and the stochastic averaging theory, the expressions for parametric rolling stability boundary are given. The accuracy of these stability boundaries is verified by comparison with numerical simulation. The new method with the stochastic stability boundary can quickly and accurately evaluate the probability of large roll motion of ships in the designated sea areas which illustrates the possibility of improvement to a certain extent.

Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using the nonlocal Euler-Bernoulli beam theory and multiple time scales method are investigated. Lorentz force equation is used to obtain the vertical force due to the applied voltage to the system. The surface effects as well as a combinational non-linear viscoelastic foundation are considered, and finally, the dynamic equilibrium equations are used, and non-linear equations of motion are extracted. In the following, the Galerkin and multiple time scales methods are used, and finally, analytical solutions are extracted as the non-linear free and forced vibrational responses of the system. The relevant coefficients of the extracted analytical solutions are discovered for two both simple support (S-S) and clamped (C-C) boundary conditions. In the following, , and the effects of the different parameters such as non-local parameter as well as electric-magnetic fields, effect of hardness-linear damping parameters of nonlinear considered viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied. As one the results of this study, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. Also, it is predictable that as the load affect position moves away from the center of the single-walled piezoelectric carbon nanotube toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak, and the degree of nonlinearity of the vibrational response of the system.

Primary and subharmonic resonance features of a graphene nanoplatelet strengthened beam in a thermal ambient directly attacking nonlinear integro-partial differential governing equations

The nonlinear aspects of advanced structures may be utilized for design purposes and making use of advantages and avoiding disadvantages of such structures. On the other hand, developing information from secondary resonance examinations of advanced structures is worthwhile for engineering objectives such as mass detection, energy harvesting and damage detection of systems likewise structures. Moreover, a precise solution methodology discards any possibility of qualitative errors and incorrect prejudice. Accordingly, for the first time, the nonlinear primary resonance alongside the subharmonic resonance of an advanced beam structure strengthened by means of graphene nanoplatelets in a thermal ambient is addressed in details employing the direct method of multiple scales. With the aim of defeating to the disadvantageous influences of the discretization procedure on the nonlinear dynamics of the graphene nanoplatelet strengthened beam, the nonlinear integro-partial differential equation of motion in accompanied by the in-homogeneous boundary conditions are attacked directly resorting to the method of multiple scales. It is shown that how an FGX distribution pattern of graphene nanoplatelets for a beam subjected to a subharmonic resonance excitation relative to an FGO distribution pattern effectively extends the range of excitation frequency that leads to a bistable solution. This deduction is very proficient for harvesting energy scopes. A similar finding is observed for a CC beam with respect to a CS beam. It is seen that when an FGX beam is exposed to a subharmonic resonance in a wider range of force amplitude experiences a non-trivial equilibrium state relative to an FGO beam. However, the strengthening of the beam with graphene nanoplatelets decreases the lower bound of the excitation frequency required for the activation of subharmonic resonance more significant for an FGX distribution pattern. Moreover, the primary resonance frequency response shows different trend prior to and after the situation of the external force frequency to the first natural frequency ratio equal to one for an FGX and an FGO graphene nanoplatelet strengthened beams. The presented results open a new view of the future of engineering designs for various engineering scopes such as energy harvesting, mass detection and damage detection purposes employing advanced structures.

Analyzing softening and hardening behavior in vibration of a thin incompressible hyperelastic cylindrical shell

This study focuses on nonlinear vibration of a thin incompressible hyperelastic cylindrical shell subjected to radial harmonic excitation. Mooney–Rivlin model is used to find the Strain Energy Density and model the material’s nonlinear behavior. The independent bending terms are considered for shape deformation of the middle surface. The nonlinear relevant equations of motion are obtained using Donnell’s nonlinear shallow-shell theory and Lagrange’s equation based on the small strain hypothesis. In this study, the linear and nonlinear models are verified by comparing the present theory’s natural frequency results with corresponding results obtained using a finite element method. Subsequently, the derived equations are solved under undamped and free vibration conditions, leading to an equation of motion in the form of a Duffing equation. This equation is then analyzed using the Lindstedt-Poincaré technique. Multiple Scale Method is used to address the system of coupled nonlinear equations in the presence of harmonic force and damping term. Backbone curves and amplitude response for both asymmetric and axisymmetric modes are presented by considering coupled and, non-coupled governed equations. The effects of the geometrical and material characteristics are studied on the free and forced vibration of shell structure as well as the impact of shape function, which describe the deformation of the middle surface, on both backbone and amplitude-response.

Application of Parametric Forced Tuned Solid Ball Dampers for Vibration Control of Engineering Structures

In this paper, parametric forced tuned solid ball dampers (TSBD) are considered for vibration control of engineering structures in an untypical way. The special feature of the presented investigation is to evaluate the potential application of parametric forcing of the rolling cylindrical or spherical body in the runway for reducing the vertical vibrations of a vibration-prone main system. Typically, tuned solid ball dampers are applied to structures that are prone to horizontal vibrations only. The coupled nonlinear differential equations of motion are derived and the phenomenon of parametric resonance of the rolling body in the runway is analyzed. A criterion for avoiding parametric resonance is given to achieve the optimal damping effect of the TSBD. In the second part of the article, a method for the targeted use of parametric resonance to reduce the vertical vibrations of engineering structures is presented and verified, considering a biaxially harmonic excited pedestrian bridge. It is shown that, with a suitable choice of damper parameters, a stable vibration of the rolling body in the runway is formed over the course of the vibration despite the occurrence of parametric resonance and that the maximum vertical vibration amplitudes of the main system can be reduced up to 93%. Hence, the here presented untypical application of parametric forced TSBD for reducing the vertical forced vibrations of vibration-prone main systems could be successfully demonstrated.

Nonlinear dynamic study of non-uniform microscale CNTR composite beams based on a modified couple stress theory

This study aims at investigating the nonlinear dynamic behavior of microscale carbon nanotube reinforced (CNTR) composite Euler–Bernoulli beams with a non-uniform cross-section, based on a modified couple stress theory (MCST). The nonlinear partial differential equations (PDEs) of motion are established based on the Von-Karman nonlinear strain–displacement relationship and Hamiltonian principle. The coupled PDEs are reduced to a single PDE, by neglecting the effects of the axial inertia and considering two different types of boundary conditions (i.e. clamped–clamped and clamped–free). At the same time, the single PDE is reverted to a nonlinear ordinary differential equation (ODE) by means of the Galerkin approach, and it is solved by using a semi-inverse method and the method of multiple time scales (MTS) for a free and forced vibration analysis, respectively. A large systematic numerical analysis is here performed to check for the sensitivity of the nonlinear response of CNTR composite beams to different boundary conditions and reinforcement parameters, with useful scientific insights for further computational investigations on the topic.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

Nonlinear two-dimensional water waves with arbitrary vorticity

We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables.