The &#8220;torpedo&#8221; pile is a foundation system that has been recently considered to anchor mooring lines and risers of floating production systems for offshore oil exploitation. The pile is installed in a free fall operation from a vessel. However, the soil parameters involved in the penetration model of the torpedo pile contain uncertainties that can affect the precision of analysis methods to evaluate its final penetration depth. Therefore, this paper deals with methodologies for the assessment of the sensitivity of the response to the variation of the uncertain parameters and mainly to incorporate into the analysis method techniques for the formal treatment of the uncertainties. Probabilistic and &#8220;possibilistic&#8221; approaches are considered, involving, respectively, the Monte Carlo method (MC) and concepts of fuzzy arithmetic (FA). The results and performance of both approaches are compared, stressing the ability of the latter approach to efficiently deal with the uncertainties of the model, with outstanding computational efficiency, and therefore, to comprise an effective design tool.

Denise Margareth Kazue Nishimura Kunitaki

Beatriz Souza Leite Pires de  Lima

Alexandre Gon&#231;alves Evsukoff

Breno Pinheiro Jacob

Probabilistic and Fuzzy Arithmetic Approaches for  the Treatment of Uncertainties in the Installation of  Torpedo Piles

A geometrically exact formulation of cables suffering axis stretching and flexural curvature is presented. The dynamical formulation is based on nonlinearly viscoelastic constitutive laws for the tension and bending moment with the additional constitutive nonlinearity accounting for the no-compression condition. A continuation method, combined with a mixed finite-difference spatial discretization, is then employed to path-follow the static responses of cables subject to forces or support displacements. These computations, conducted in the quasistatic regime, are based on cables with linearly elastic material behaviors, whereas the nonlinearity is in the geometric stiffness terms and the no-compression behavior. The finite-difference results have been confirmed employing a weak formulation based on quadratic Lagrangian finite elements. The influence of the flexural stiffness on the nonlinear static responses is assessed comparing the results with those obtained for purely extensible cables. The properties of the frequencies of the linear normal modes of cables with flexural stiffness are also investigated and compared with those of purely extensible cables.

Walter Lacarbonara

Arnaud Pacitti

Nonlinear Modeling of Cables with Flexural Stiffness

The local instability of 2 degrees of freedom (DOF) weakly damped systems is thoroughly discussed using the Li&#233;nard-Chipart stability criterion. The individual and coupling effect of mass and stiffness distribution on the dynamic asymptotic stability due to mainly infinitesimal damping is examined. These systems may be <?bhlt ?>as follows<?ehlt ?>: (a) unloaded (free motion) and (b) subjected to a suddenly applied load of constant magnitude and direction with infinite duration (forced motion). The aforementioned parameters combined with the algebraic structure of the damping matrix (being either positive semidefinite or indefinite) may have under certain conditions a tremendous effect on the Jacobian eigenvalues and then on the local stability of these autonomous systems. It was found that such systems when unloaded may exhibit periodic motions or a divergent motion, while when subjected to the above step load may experience either a degenerate Hopf bifurcation or periodic attractors due to a generic Hopf bifurcation. Conditions for the existence of purely imaginary eigenvalues leading to global asymptotic stability are fully assessed. The validity of the theoretical findings presented herein is verified via a nonlinear dynamic analysis.

Dimitris S. Sophianopoulos

George T. Michaltsos

Anthony N. Kounadis

The Effect of Infinitesimal Damping  on  the Dynamic Instability Mechanism of  Conservative Systems

The effect of geometric imperfections and viscous damping on the type of nonlinearity
(i.e., the hardening or softening behaviour) of circular plates and shallow spherical shells with free
edge is here investigated. The Von K&#225;rm&#225;n large-deflection theory is used to derive the continuous
models. Then, nonlinear normal modes (NNMs) are used for predicting with accuracy the coefficient,
the sign of which determines the hardening or softening behaviour of the structure. The
effect of geometric imperfections, unavoidable in real systems, is studied by adding a static initial
component in the deflection of a circular plate. Axisymmetric as well as asymmetric imperfections
are investigated, and their effect on the type of nonlinearity of the modes of an imperfect plate is
documented. Transitions from hardening to softening behaviour are predicted quantitatively for imperfections
having the shapes of eigenmodes of a perfect plate. The role of 2:1 internal resonance in
this process is underlined. When damping is included in the calculation, it is found that the softening
behaviour is generally favoured, but its effect remains limited.

Cyril Touz&#233;

C&#233;dric Camier

Ga&#235;l Favraud

Olivier Thomas

Effect of Imperfections and Damping on the Type of Nonlinearity of Circular Plates and Shallow Spherical Shells

This paper is concerned with the sliding mode control for uncertain stochastic neutral systems with multiple delays. A switching surface is adopted first. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to ensure the global stochastic stability of the stochastic system in the sliding mode for all admissible uncertainties. The synthesized sliding mode controller guarantees the existence of the sliding mode.

Dilan Chen

Weidong Zhang

Sliding Mode Control of Uncertain Neutral Stochastic Systems with Multiple Delays

The homotopy perturbation method is applied to solve reaction-diffusion equations. In this method, the trial function (initial solution) is chosen with some unknown parameters, which are identified using the method of weighted residuals. Some examples are given. The obtained results are compared with the exact solutions, revealing that this method is very efficient and the obtained solutions are of high accuracy.

Yu-Xi Wang

Hua-You Si

Lu-Feng Mo

Homotopy Perturbation Method for Solving Reaction-Diffusion Equations

The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

Z. H. Wang

Numerical Stability Test of Neutral Delay Differential Equations

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Mathematical Problems in Engineering is a broad-based journal publishing&nbsp;results of rigorous engineering research across all disciplines, carried out using mathematical tools.

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Chief Editor, Professor Guangming Xie, is currently a full professor of dynamics and control with the College of Engineering, Peking University. His research interests include complex system dynamics and control and intelligent and biomimetic robots.

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering

Uncertainties in Nonlinear Structural Dynamics

Uncertainties in Nonlinear Structural Dynamics

Articles

Probabilistic and Fuzzy Arithmetic Approaches for the Treatment of Uncertainties in the Installation of Torpedo Piles

Nonlinear Modeling of Cables with Flexural Stiffness

The Effect of Infinitesimal Damping on the Dynamic Instability Mechanism of Conservative Systems

Effect of Imperfections and Damping on the Type of Nonlinearity of Circular Plates and Shallow Spherical Shells

Sliding Mode Control of Uncertain Neutral Stochastic Systems with Multiple Delays

Homotopy Perturbation Method for Solving Reaction-Diffusion Equations

Numerical Stability Test of Neutral Delay Differential Equations