Static bending of granular beam: exact discrete and nonlocal solutions

Meccanica - This study is an attempt towards a better understanding of the length scale effects on the bending response of the granular beams. To this aim, a unidimensional discrete granular chain...     

Dynamics of mechanisms: Some intrinsic properties

We propose in this paper to give some intrinsic results about, dynamics of mechanisms by using Lie group approach. This language allows to obtain two main results. The first one concerns a new definition of Lagrange's multipliers which is more conform to intuition. The second one gives the explicit linear combinations of the dynamic equations allowing their elimination.     

Kinematics of Mechanisms to the Second Order – Application to the Closed Mechanisms

The degree of freedom of a closed mechanism is the dimension of a subset M of R ⁿ , M being the inverse image of the unity by the closure function f : (q ₁, ..., q _n ) f(q ₁, ..., q _n ), where q ₁, ..., q _n are the articular coordinates. We first study the regular points for the mapping f from R ⁿ into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.     

Some insights into structure instability and the second-order work criterion

This paper is an attempt to extend the approach of the second-order work criterion to the analysis of structural system instability. Elastic structural systems with a finite number of freedoms and subjected to a given loading are considered. It is shown that a general equation, relating the second-order time derivative of the kinetic energy to the second-order work, can be derived for kinetic perturbations. The case of constant, nonconservative loadings are then investigated, putting forward the role of the spectral properties of the symmetric part of the tangent stiffness matrix in the occurrence of instability. As an illustration, the case of the generalized Ziegler column is considered and the case of aircraft wings subjected to aeroelastic effects is investigated. In the both cases, the consequences of additional kinematic constraints are discussed.     

General equations of curvilinear systems: A coordinate-free approach by using Lie group theory

Using Lie group theory, it is proposed to give the intrinsic most general equations of any curvilinear system (Σ), the only hypothesis being that each section is rigid. After giving these equations, we shall illustrate the power of the method by proposing some elements of the automatic generation of the corresponding scalar equations.     

Some explicit relations in kinematics of mechanisms

Using Lie group theory, we propose to give some explicit relations in kinematics of mechanisms without introducing coordinates. After a presentation of the mathematical tools, the kinematics of closed mechanisms is presented using Lie group language. We give first a family of relations between dependent and independent variables and their derivatives which may be useful for linearisation or for numerical integration. Two examples are then proposed to illustrate these relations.     

Singular divergence instability thresholds of kinematically constrained circulatory systems

Static instability or divergence threshold of both potential and circulatory systems with kinematic constraints depends singularly on the constraints? coefficients. Particularly, the critical buckling load of the kinematically constrained Ziegler?s pendulum as a function of two coefficients of the constraint is given by the Plücker conoid of degree n=2$n=2$. This simple mechanical model exhibits a structural instability similar to that responsible for the Velikhov–Chandrasekhar paradox in the theory of magnetorotational instability.     

About the automatic generation of equations of curvilinear systems

Following other papers devoted to intrinsic formulations of curvilinear systems, we develop here the Maple procedures and some additional calculations in Lie group of displacements which yield explicit scalar equations.