Walter Lacarbonara

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.

Resonant non-linear normal modes. Part II: activation=orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.

Nonclassical Responses of Oscillators with Hysteresis

The responses and codimension-one bifurcations in Masing-type andBouc–Wen hysteretic oscillators are investigated. The pertinent statespace is formulated for each system and the periodic orbits are soughtas the fixed points of an appropriate Poincaré map. The implementedpath-following scheme is a pseudo-arclength algorithm based on arclengthparameterization. The eigenvalues of the Jacobian of the map, calculatedvia a finite-difference scheme, are used to ascertain the stability andbifurcations of the periodic steady-state solutions. Frequency-responsecurves for various excitation levels are constructed consideringrepresentative hysteresis loop shapes generated with the two models inthe primary and superharmonic frequency ranges. In addition to knownbehaviors, a rich class of solutions and bifurcations, mostly unexpectedfor hysteretic oscillators – including jump phenomena,symmetry-breaking, complete period-doubling cascades, fold, andsecondary Hopf – is found. Complex (mode-locked) periodic andnonperiodic responses are also investigated thereby allowing to draw amore comprehensive picture of the dynamical behavior exhibited by thesesystems.

Nonlinear Modeling of Cables with Flexural Stiffness

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.

Elastodynamics of Nonshallow Suspended Cables: Linear Modal Properties

A mechanical model describing finite motions of nonshallow cables around the initial catenary configurations is proposed. An exact kinematic formulation accounting for finite displacements is adopted, whereas the material is assumed to be linearly elastic. The nondimensional mechanical parameters governing the motions of nonshallow cables are obtained via a suitable nondimensionalization, and the regions of their physically plausible values are portrayed. The spectral properties of linear unforced undamped vibrations around the initial static configurations are investigated via a Galerkin-Ritz discretization. A classification of the modes is obtained on the basis of their associated energy content, leading to geometric modes, elastostatic modes (with prevalent transverse motions and appreciable stretching), and elastodynamic modes (with prevalent longitudinal motion). Moreover, an extension of Irvine s model to moderately nonshallow cables is proposed to determine the frequencies and mode shapes in closed form.

Flutter of an Arch Bridge via a Fully Nonlinear Continuum Formulation

A fully nonlinear parametric model for wind-excited arch bridges is proposed to carry out the flutter analysis of Ponte della Musica under construction in Rome. Within the context of an exact kinematic formulation, all of the deformation modes are considered (extensional, shear, torsional, in-plane, and out-of-plane bending modes) both in the deck and supporting arches. The nonlinear equations of motion are obtained via a total Lagrangian formulation while linearly elastic constitutive equations are adopted for all structural members. The parametric nonlinear model is employed to investigate the bridge limit states appearing either as a divergence bifurcation (limit point obtained by path following the response under an increasing multiplier of the vertical accidental loads) or as a Hopf bifurcation of a suitable eigenvalue problem (where the bifurcation parameter is the wind speed). The eigenvalue problem ensues from the governing equations of motion linearized about the in-service prestressed bridge con...

Flutter Control of a Lifting Surface via Visco-Hysteretic Vibration Absorbers

In this paper, a visco-hysteretic vibration absorber (VA) is proposed to increase the flutter speed of an airfoil and enhance damping in the pre- and post-flutter regimes. The passive system consists of a parallel arrangement of a dashpot and a rateindependent hysteretic element, represented by the Bouc-Wen differential model. The equations of motion are obtained and various tools of linear and nonlinear dynamics are employed to study the effects of the visco-hysteretic VA in the pre- and postflutter ranges.

Hysteresis of Multiconfiguration Assemblies of Nitinol and Steel Strands: Experiments and Phenomenological Identification

AbstractAn in-house–built rheological device, made of assemblies of custom-made nickel titanium-Naval Ordnance Laboratory (Nitinol) strands, wires, and steel wire ropes, is experimentally tested in different configurations corresponding to three distinct constitutive behaviors: a strong hardening pinched hysteresis, a quasi-linear–softening behavior, and an intermediate behavior in the range of interest. The nonlinear features of the hysteretic rheological device are the result of geometric hardening of the ropes, interwire friction, and dissipation caused by the phase transformations of the Nitinol wires. These different mechanisms determine a pinching at the origin of the hysteresis force-displacement loops. A phenomenological representation of the experimentally acquired constitutive responses is obtained by an extension of the Bouc-Wen model incorporating a pinching function that depends on two parameters. The responses of the Nitinol wires and strands under uniaxial tension are instead well identifie...

Damage detection by modal curvatures: numerical issues

This work is concerned with modal curvature-based damage detection in slender beam-like structures, whereby the modal curvatures are computed by numerical differentiation of noisy mode shapes sampled at a finite number of measurement points. Within this framework, most common techniques greatly amplify the measurement errors and their application leads to unreliable outcomes, especially when a large set of measurement points is considered. Preliminary signal processing, even if beneficial for reducing the noise level, does not solve the problem in that neither the detection of damaged zones nor the reduction of false alarms exhibit significant improvements. In a comparative fashion, we herein demonstrate that a modified Savitzky–Golay filter and the cubic smoothing spline method can provide a more affordable way for detecting damages when using numerically obtained modal curvatures. In doing so, the robustness against the measurement errors and the role of the adopted formulation for the modal curvature-based damage index are considered. A simple statistical procedure that can further improve the detection of damaged regions together with the identification of possible false positives is also presented.

Nonlinear aeroelastic formulation for flexible high-aspect ratio wings via geometrically exact approach

The nonlinear aeroelastic modeling and behavior of HALE wings, undergoing large deformations and exhibiting dynamic stall, are presented. A fully nonlinear three-dimensional structural model, based on an exact kinematic approach, is coupled with the incompressible unsteady aerodynamic model obtained via a reduced-order indicial formulation accounting for viscous effects, in term of dynamic stall and flow separation. To this end, a modified Beddoes-Leishman model is employed. Aeroelastic simulations are performed by reducing the governing equations to a form amenable to numerical integration. Space and time integrations are conducted using a numerical scheme that includes PDE, associated with the equation of motion of the flexible wing, and ODEs, associated with the lag-state formulation pertinent to the unsteady aerodynamic loads, in a hybrid solution form. The numerical investigations show that the proposed approach is suitable for studying the aeroelastic behavior of highly nonlinear wings, for an improved understanding of the nonlinear phenomena occurring particularly in the neighborhood of the flutter boundary and in the post-critical regime.