Silvio Sorrentino

A new analytical technique for vibration analysis of non-proportionally damped beams

Abstract Vibrating linear mechanical systems, in particular continuous systems, are often modelled considering proportional damping distributions only, although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy. In this paper an analytical method is given to take into account the effects of a more general viscous damping model, referred to as non-proportional damping, on a class of vibrating continuous systems. A state-form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form, i.e., complex modes corresponding to non-synchronous motions. Numerical examples are included in order to show the efficiency of the proposed method; non-proportional damping distributions of different type, such as internal and external lumped or distributed viscous damping, are tested on non-homogeneous Euler–Bernoulli beams in bending vibration with different boundary conditions. Finally, a discussion on root locus diagrams behaviour and on modal damping ratio significance for non-proportionally damped systems is presented.

Frequency domain analysis of continuous systems with viscous generalized damping

This paper deals with the effects of generalized damping distributions on vibrating linear systems. The attention is focused on continuous linear systems with distributed and possibly non-proportional viscous damping, which are studied in terms of modal analysis, defining and discussing the orthogonality properties of their eigenfunctions.

Analytical Modelling and Experimental Identification of Viscoelastic Mechanical Systems

In the present study non-integer order or fractional derivative rheological models are applied to the dynamical analysis of mechanical systems. Their effectiveness in fitting experimental data on wide intervals of frequency by equivalent damping ratio valid for fractional derivative models is introduced, making it possible to test their ability in reproducing experimentally obtained damping estimates. A numerical procedure for the experimental identification of the parameters of the Fractional Zener rheological model is then presented and vibrations.

Experimental Identification of a Fractional Derivative Linear Model for Viscoelastic Materials

Non integer, fractional order derivative rheological models are known to be very effective in describing the linear viscoelastic dynamic behaviour of mechanical structures made of polymers [1]. The application of fractional calculus to viscoelasticity can be physically consistent [2][3][4] and the resulting non integer order differential stress-strain constitutive relation provides good curve fitting properties, requires only a few parameters and leads to causal behaviour [5]. When using such models the solution of direct problems, i.e. the evaluation of time or frequency response from a known excitation can still be obtained from the equations of motion using standard tools such as modal analysis [6]. But regarding the inverse problem, i.e. the identification from measured input-output vibrations, no general technique has so far been established, since the current methods do not seem to easily work with differential operators of non integer order. In this paper a frequency domain method is proposed for the experimental identification of a linear viscoelastic model, namely the Fractional Zener also known as Fractional Standard Linear Solid [5], to compute the frequency dependent complex stress-strain relationship parameters related to the material. The procedure is first checked with respect to numerically generated frequency response functions for testing its accuracy, and then to experimental inertance data from a free-free homogeneous beam made of High Density Polyethylene (HDPE) in plane flexural and axial vibration.Copyright

Spectral modeling of vibrating plates with general shape and general boundary conditions

The analysis and design of lightweight plate structures require efficient computational tools, because exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. This paper deals with vibration analysis of Kirchhoff plates of general shape with non-standard boundary conditions, adopting a Rayleigh-Ritz approach. Three different coordinate mappings are considered, using different kinds of functions: 1) trigonometric and polynomial interpolation functions for mapping the shape of the plate, 2) trigonometric and polynomial interpolation functions for mapping a constraint domain of general shape, 3) products of linearly independent eigenfunctions evaluated from a standard beam in flexural vibration for describing the transverse displacement field of the plate. Flexural free vibration analysis of different shaped plates is then performed using the same approach: skew, trapezoid and triangular plates, plates with parabolic curved edges, sectors of circular plates, circular and elliptic annular plates. Purely elastic plates are considered, but the method may also be applied to the analysis of viscoelastic plates. The results are compared with those available in the literature and using standard finite element analysis.

A Condensation Technique for Finite Element Dynamic Analysis Using Fractional Derivative Viscoelastic Models

Fractional derivative rheological models are known to be very useful for describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems, the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. The increase in matrix dimensions produced by this expansion, however, is often so fast as to make the calculations too cumbersome for finite element applications. This article presents a condensation technique based on the computation of two reduced-size eigenproblems. The rheological model adopted is the fractional Zener (fractional standard linear solid) model, but the same methodology can be applied to problems using different fractional derivative linear models.

Discrete Spectral Modelling of Continuous Structures With Fractional Derivative Viscoelastic Behaviour

Fractional derivative rheological models were recognised to be very effective in describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. But the growth in matrix dimensions brought about by this expansion is in general so fast as to make the calculations too cumbersome. In this paper a discretization method for continuous structures is presented, based on the Rayleigh-Ritz method, aimed at reducing the computational effort. The solution of the equation of motion is approximated by a linear combination of shape-functions selected among the analytical eigenfunctions of standard known structures. The resulting condensed eigen-problem is then expanded in a low dimension fractional state-space. The Fractional Standard Linear Solid is the adopted rheological model, but the same methodology could be applied to problems involving different fractional derivative linear models. Examples regarding two different continuous structures are proposed and discussed in detail.Copyright

A study on the stability of a motorcycle wheel–swingarm suspension with chain transmission

ABSTRACT The present study describes a possible driving mechanism for a self-excited oscillation observed in motorcycle dynamics, often referred to as chatter. This phenomenon, affecting the performance of road racing motorcycles, has been simulated in straight running braking manoeuvres with multibody motorcycle models. It involves rear suspension bounce and driveline oscillation in the frequency range between 17 and 22 Hz. A simplified model of a motorcycle rear suspension with chain transmission is proposed and its stability in equilibrium configurations is studied via eigenvalue analysis. The sensitivity with respect to all its governing parameters is analysed by means of stability maps and the self-excitation mechanism is explained with the aid of energy balance analysis and phase diagrams. It is found that the key role for the instability onset is played by the gradient of the nonlinear characteristic slip function of the tyre.

Rayleigh-Ritz Analysis of Vibrating Plates Based on a Class of Eigenfunctions

In the Rayleigh-Ritz condensation method the solution of the equation of motion is approximated by a linear combination of shape-functions selected among appropriate sets. Extensive literature dealing with the choice of appropriate basis of shape functions exists, the selection depending on the particular boundary conditions of the structure considered. This paper is aimed at investigating the possibility of adopting a set of eigenfunctions evaluated from a simple stucture as a general basis for the analysis of arbitrary-shaped plates. The results are compared to those available in the literature and using standard finite element analysis.Copyright

Experimental Validation of Non-Conventional Viscoelastic Models via Equivalent Damping Estimates

Non-conventional rheological models based on non-integer order differential operators can be used to describe the viscoelastic behavior of materials, especially of polymers. These models are usually selected and then validated by means of creep and relaxation tests. However, engineers dealing with structural dynamic problems may need to obtain model identification from vibration measurement data. In this case, however, the direct identification of an optimal set of parameters of a viscoelastic model from time or frequency domain measurements is a difficult task, especially if the structural dissipative contributions are slight. In this paper, an indirect approach is adopted, based on the concept of damping ratio. When dealing with standard linear viscous dissipative models, a damping ratio modal parameter ζn can be analytically defined and experimentally estimated. But this theoretical parameter shows a dependency from the modal frequency that may dramatically fail in fitting the experimental data. On the contrary, it is known that a better agreement between theory and experiments can be achieved by means of non-integer order differential models, even though in this case analytical expressions for ζn are difficult to find. To overcome this difficulty, a method of general validity for viscoelastic models is developed, based on the concept of equivalent damping ratio and on the circle-fit technique. The proposed method is applied to experimental damping estimates from plane flexural vibrations of clamped-free beams, obtained from specimens of different size made of materials such as Polyethylene, Polyvinyl-chloride and Delrin.Copyright