Force–displacement relationship in micro-metric pantographs: Experiments and numerical simulations

In this paper, we reveal that the mathematical discrete model of Hencky type, introduced in [1], is appropriate for describing the mechanical behavior of micro-metric pantographic elementary modules. This behavior does not differ remarkably from what has been observed for milli-metric modules, as we prove with suitably designed experiments. Therefore, we conclude that the concept of pantographic microstructure seems feasible for micro-metrically architected microstructured (meta)materials as well. These results are particularly indicative of the possibility of fabricating materials that can have an underlying pantographic microstructure at micrometric scale, so that its unique behavior can be exploited in a larger range of technological applications.     

Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

This paper concerns the quadratically-damped Mathieu equation:

     

Elasto-plasticity of paper

We develop a constitutive model of paper's in-plane biaxial tensile response accounting for the elastic–plastic hardening behavior, and its orthotropic character. The latter aspect is motivated by machine-made papers, which, in contrast to isotropic laboratory handsheets, are strongly oriented. We focus on modeling paper's response under monotonic loading, this restriction allowing us to treat the elastic-plastic response as a physically nonlinear elastic one. A strain energy function of a hyperbolic tangent form is developed so as to fit the entire range of biaxial and uniaxial experiments on a commercial grade paper. This function may then be introduced as the free energy function into a model based on thermomechanics with internal variables.     

载流导线的横向非线性振动

研究二长直电流间载流导线的横向非线性振动，采用广义加权残值法偏偏微运动方程简化。导线载直流时得到Ｄｕｆｆｉｎｇ方程，求出解和频率的精确形式和近似形式。导线载交流时得到非线性Ｍａｔｈｉｅｕ方程，共振解摄动法得到。交直流载的频幅流形都反应出二长直电流间载流导线横向振动典型的非线性。     

A nonlinear oscillatory system subjected to driving forces of elliptic type

The aim of this paper is to show how Jacobi elliptic functions in combination with the averaging and the harmonic balance methods can be applied to obtain the approximate solution of two coupled, ordinary differential equations having a spring with cubic nonlinearity and subjected to driving forces of elliptic type. By an appropriate choice of the system parameter values, it is possible to show that our derived solution represents the exact steady-state solution of the undamped Duffing equation with driving force of elliptic type. At the end of this work, we also demonstrate the validity of our derived solution by comparing the amplitude–time response curves with those of the numerical integration solutions.     

On the finite displacement analysis of quadrangular plates

In this paper, a numerical method for the linear and geometrically non-linear static analysis of thin plates is presented. The method begins with the elasticity equations pertaining to strain components, stresses, displacement components, strain energy and work due to externally applied loads. The plate geometry is defined by a quadrangular boundary with four straight edges and the natural coordinates in conjunction with the Cartesian coordinates are used to map the geometry. The matrix equation of equilibrium is derived using the work-energy principle with the displacement fields expressed by algebraic polynomials, the coefficients of which are then manipulated to satisfy the kinematic boundary conditions. To validate the results from the present method, square plates having all sides fully fixed and all sides simply supported without in-plane movement are analysed. Comparison is made for the uniformly loaded square plate with the results obtained by Levy who solved the non-linear plate bending problem using the Th.von Ka’rma’n’s equations. Rhombic plates are examined and numerical results corresponding to these cases are presented in this paper. Very good comparison of the results regarding deflection and bending stresses with other sources available in the literature is found.     

Active Longitudinal Control of Wind-Induced Oscillations of a Suspended Cable

Abstract. Active control of oscillations of a suspended cable under transversal wind, described by a constant term and a fluctuating turbulent component, is dealt with. Control of transversal in-plane and out-of-plane oscillations is performed by imposing a longitudinal motion to one support; the control law depends, linearly or quadratically, on measures of displacement and velocity of a selected point. First, the steady-state oscillations around the planar static configuration, determined by the wind constant component, under sinusoidal in-plane and out-of-plane loading are studied, showing the nonlinear behaviour in the regions of primary and parametric resonance and the effects of the control action. Afterwards, attention is focused on the response of the cable to transversal wind turbulence, whose fluctuating component is described by a random process with an assigned spectral density. The effectiveness of longitudinal control is analysed by means of numerical investigations.Sommario. Si analizza il controllo attivo delle oscillazioni di un cavo sospeso, sottoposto all'azione di un vento agente in direzione ortogonale al piano del cavo e descritto mediante un termine di carico costante ed una componente variabile di turbolenza. Il controllo delle oscillazioni piane e spaziali viene realizzato mediante spostamento longitudinale di uno degli appoggi; la legge di controllo dipende linearmente o quadraticamente da misure di spostamento e velocità di un punto prescelto. In una prima fase, vengono studiate le oscillazioni stazionarie sotto carichi sinusoidali agenti rispettivamente nel piano e fuori del piano del cavo, intorno alla configurazione di equilibrio statico determinata dalla presenza della componente costante del vento, mostrando il comportamento nonlineare nelle regioni di risonanza primaria e parametrica e gli effetti dell'azione di controllo. Successivamente, si focalizza l'attenzione sulla risposta del cavo alla componente trasversale di turbolenza del vento, la quale è descritta da un processo stocastico a media nulla con densità spettrale assegnata. L'efficacia del controllo viene analizzata attraverso un'indagine numerica.     

Experimental evidence of energy pumping in acousticsMise en évidence expérimentale de pompage énergétique en acoustique

This Note presents an experimental vibro-acoustic set-up that aims to reproduce the energy pumping phenomenon between an acoustic medium and an essentially nonlinear oscillator. It shows a one-way irreversible transfer of energy between the first acoustic mode in a tube and a thin visco-elastic membrane. To cite this article: B. Cochelin et al., C. R. Mecanique 334 (2006).