This chapter addresses the prediction of vibratory resonances in nonsmooth structural systems via Nonsmooth Modal Analysis. Nonsmoothness in the trajectories is induced by unilateral contact conditions in the governing (in)equations. Semi-analytical and numerical state-of-the-art solution methods are detailed. The significance of nonsmooth modal analysis is illustrated in simplified one-dimensional space semi-discrete and continuous frameworks whose theoretical and numerical discrepancies are explained. This contribution establishes clear evidence of correlation between periodically forced and autonomous unilaterally constrained oscillators. It is also shown that strategies using semi-discretization in space are not suitable for nonsmooth modal analysis. The spectrum of vibration exhibits an intricate network of backbone curves with no parallel in nonlinear smooth systems. Terminology and acronyms The purpose of this chapter is to provide a general picture of the state-of-the-art vibratory analysis of nonsmooth systems. This topic lies at the interface between modal analysis of smooth nonlinear systems and nonsmooth contact dynamics dedicated to the time-evolution of nonsmooth systems, undergoing impact or dry friction, for instance. Some elementary concepts are succinctly recalled for the purpose of completeness. Unless otherwise stated, the epithet discrete (as in "discrete systems" or "discrete setting") designates semidiscretization in space, while continuous refers to everything else.
The influence of prestrain on viscoelastic properties E Ã and tan d of four different HTPB composite propellants was measured using Dynamic Mechanical Analysis (DMA). Nonlinear behaviour in terms of prestrain was observed and then modelled using a modified generalized Maxwell model. Prestrain was introduced as a variable of the stiffness of each Maxwell element using simple two-parameter relationships. An algorithm was proposed and numerically implemented to identify the model parameters from the measurements. The performance of the identification method is discussed in terms of accuracy and robustness. The good match between the predictions and the experimental measurements shows that the dependence on the uniaxial prestrain of the complex modulus of the studied composite propellants can be described with few Maxwell elements but with good accuracy.
International audienceHighly filled elastomers exhibit a complex nonlinear mechanical behaviour that is difficult to characterize experimentally. This paper presents a Dynamic Mechanical Analysis (DMA) method coupled with orthogonal prestrains, applied in two distinct steps. A localization operator between measurements at the arms of a cross shaped specimen and the stress and strain fields at its center was determined using elastic small strain finite element computations. The operator makes estimating the storage and loss moduli at the center of the specimen possible. A mathematical model is then fitted to the moduli values. These results are compared to DMA measurements of highly filled elastomers under uniaxial prestrain. Although the storage and loss moduli increase with the prestrain under both loadings, the nonlinear behaviour is quantitatively modified by adding an orthogonal prestrain. In addition, the modification of the behaviour under a horizontal prestrain is cancelled out by an increase of the vertical prestrain, which may be explained by fillers aligning in the direction of the prestrain
In turbomachinery, it is well known that tighter operating clearances improve the efficiency. However, this leads to unwanted potential unilateral and frictional contact occurrences between the rotating (blades) and stationary components (casings) together with attendant thermal excitations. Unilateral contact induces discontinuities in the velocity at impact times, hence the terminology nonsmooth dynamics. Current modeling strategies of rotor-stator interactions are either based on regularizing penalty methods or on explicit time-marching methods derived from Carpenter's forward Lagrange multiplier method. Regularization introduces an artificial time scale in the formulation corresponding to numerical stiffness which is not desirable. Carpenter's scheme has been successfully applied to turbomachinery industrial models in the sole mechanical framework, but faces serious stability issues when dealing with the additional heat equation.This work overcomes the above issues by using the MoreauJean nonsmooth integration scheme within an implicit Â-method. This numerical scheme is based on a mathematically sound description of the contact dynamics by means of measure differential inclusions and enjoys attractive features. The procedure is unconditionally stable opening doors to quick preliminary simulations with time-steps one hundred times larger than with previous algorithms. It can also deal with strongly coupled thermomechanical problems.
Finite elements in space with time-stepping numerical schemes, even though versatile, face theoretical and numerical difficulties when dealing with unilateral contact conditions. In most cases, an impact law has to be introduced to ensure the uniqueness of the solution: total energy is either not preserved or spurious high-frequency oscillations arise. In this work, the Time Domain Boundary Element Method (TD-BEM) is shown to overcome these issues on a one-dimensional system undergoing a unilateral Signorini contact condition. Unilateral contact is implemented by switching between free boundary conditions (open gap) and fixed boundary conditions (closed gap). The solution method does not numerically dissipate energy unlike the Finite Element Method and properly captures wave fronts, allowing for the search of periodic solutions. Indeed, TD-BEM relies on fundamental solutions which are travelling Heaviside functions in the considered one-dimensional setting. The proposed formulation is capable of capturing main, subharmonic as well as internal resonance backbone curves useful to the vibration analyst. For the system of interest, the nonlinear modeshapes are piecewise-linear unseparated functions of space and time, as opposed to the linear modeshapes that are separated half sine waves in space and full sine waves in time.
In turbomachinery, it is well known that tighter operating clearances improve the efficiency. However, this leads to unwanted potential unilateral and frictional contact occurrences between the rotating (blades) and stationary components (casings) together with attendant thermal excitations. Unilateral contact induces discontinuities in the velocity at impact times, hence the terminology nonsmooth dynamics. Current modeling strategies of rotor-stator interactions are either based on regularizing penalty methods or on explicit time-marching methods derived from Carpenter's forward Lagrange multiplier method. Regularization introduces an artificial time scale in the formulation corresponding to numerical stiffness which is not desirable. Carpenter's scheme has been successfully applied to turbomachinery industrial models in the sole mechanical framework, but faces serious stability issues when dealing with the additional heat equation.This work overcomes the above issues by using the MoreauJean nonsmooth integration scheme within an implicit Â-method. This numerical scheme is based on a mathematically sound description of the contact dynamics by means of measure differential inclusions and enjoys attractive features. The procedure is unconditionally stable opening doors to quick preliminary simulations with time-steps one hundred times larger than with previous algorithms. It can also deal with strongly coupled thermomechanical problems.
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