Fabrizio Vestroni

Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

Planar non-linear free vibrations of an elastic cable

Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.

Monitoring of structural systems by using frequency data

The present work evaluates the possibility of using dynamic data to assess structural integrity. It addresses the problem of understanding when it is sufficient to measure and use only natural frequencies, thus avoiding the need to measure modal shapes. The classic problem of detecting damage in beams, or beam assemblies, due to concentrated cracks, or damage spread over a structural member is dealt with. Damage is represented as a decrease in stiffness and linear behaviour before and after the event assumed to have caused damage is considered. Damage is restricted to a few unknown sections or elements, so that only the modification of few parameters of the system need to be determined. This study thus rejects assumptions unrelated to the physical aspects of the problem, in contrast to many papers on the subject. The amount of data to locate and quantify damage correctly is discussed; general considerations lead to the conclusion that a unique and reliable estimate of the damage can be obtained using only few additional frequency data with respect to the number of damaged zones. Continuous and discrete (finite element) models are examined. Finally the paper considers the applications to both analytical and experimental data of the procedure developed, which takes account of the peculiar characteristics of damage detection problem. Copyright

Parametric analysis of large amplitude free vibrations of a suspended cable

Abstract Partial differential equations of motion suitable to study moderately large free oscillations of an clastic suspended cable arc obtained. An integral procedure is used to eliminate the spatial dependence and to reduce the problem to one ordinary differential equation which shows quadratic and cubic nonlincarities. The frequency-amplitude relationship for symmetric and antisymmetric vibration modes is studied and a numerical investigation is performed to describe the nonlinear phenomenon in a large range of values of the cable sag-to-span ratio. Softening and hardening behaviour is evidenced dependent on both the cable properties and the amplitude of oscillation.

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.

Damage Evaluation in Cracked Vibrating Beams Using Experimental Frequencies and Finite Element Models

The paper focuses on the problem of locating and quantifying damage in vibrating beams due to cracks. The problem is shown to have certain peculiarities that, to some extent, make it easier to solve than classical situations of structural identification. The solution of the problem is based on the minimization of the objective function that compares analytical and experimental data. A fairly general automated procedure is developed using a finite element code as a routine to evaluate modal quantities. The data necessary to locate and quantify damage correctly are discussed. General considerations lead to the conclusion that at least one measurement more than the expected number of cracked sections is necessary to obtain a unique solution. The procedures developed are applied to the study cases: a supported beam and a clamped beam with one or two cracks, using both simulated and experimental data. Satisfactory results are obtained. Although only beams were considered, the methodology developed can be extended to any kind of framed structure.

Periodic response of a class of hysteretic oscillators

Abstract Steady-state oscillations of hysteretic systems are studied by referring to a class of oscillators that are simple but representative of the various models introduced to describe material non-linearity of structures. Frequency-response curves for systems with different restoring force are obtained, and a general stability analysis allows definition of the regions of stable and unstable solutions in the plane of the intensity and frequency of the excitation. The role of the parameters of the hysteretic model in the periodic response is investigated and a range of hitherto unknown behaviour is demonstrated.

On Nonlinear Dynamics of Planar Shear Indeformable Beams

The planar forced oscillations of shear indeformabie beams with either movable or immovable supports are studied through a unified approach. An exact nonlinear beam model is referred to and a consistent procedure up to order three nonlinearities is followed. By eliminating the longitudinal displacement component through a constraint condition and assuming one mode, the problem is reduced to one nonlinear differential equation. A perturbational solution in the neighborhood of the resonant frequency is determined and the stability of the steady-state solutions is studied. The dependence of the phenomenon on the geometrical and mechanical characteristics of the system is put into light and the frequency-response curves for different boundary conditions are furnished.

MONOFREQUENT OSCILLATIONS OF A NON-LINEAR MODEL OF A SUSPENDED CABLE.

Abstract A two-degree-of-freedom non-linear elastic model is considered to analyze the effects of non-linearities on the free motion of a suspended cable. The discretized model is obtained by referring to simplified kinematics of the cable; the equations of motion which show quadratic and cubic non-linearities are solved through the multiple time scale perturbation technique. The monofrequent oscillations of the system are studied in order to analyze the modifications of frequency and motion amplitude of the modal oscillations due to geometric non-linearities in the absence of internal resonance. The possibility that effects arise due to non-linear coupling is examined. A numerical analysis is made for the first symmetric mode for different amplitudes of motion by parametrically varying the geometric and mechanical properties of the cable. The correction of frequency for the in-plane oscillation varies appreciably with the cable properties due to prevalence of either the quadratic or cubic term. In the out-of-plane monofrequent oscillation non-linearities establish a coupling between the two components of motion which strongly influences the frequency correction.

Non-isothermal oscillations of pseudoelastic devices

Abstract The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking into account the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchange with the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency–response curves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response is computed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to trace out multivalued frequency–response curves. The accuracy of the results is checked by comparison with the results of the numerical integration of the basic equations governing the dynamics of the system. The response is investigated for various excitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical response shows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. It turns out that the temperature variations induced by the phase transformations influence the mechanical response and that the results obtained under the simplifying assumption of isothermal behavior can be rather different from those obtained in a fully thermomechanical setting.