When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.

The paper aims to study a plane system with bars, with certain symmetries. Such problems can be encountered frequently in industry and civil engineering. Considerations related to the economy of the design process, constructive simplicity, cost and logistics make the use of identical parts a frequent procedure. The paper aims to determine the properties of the eigenvalues and eigenmodes for transverse and torsional vibrations of a mechanical system where two of the three component bars are identical. The determination of these properties allows the calculus effort and the computation time and thus increases the accuracy of the results in such matters.

Our study is concerned with a generalization of the principle of Saint-Venant in the context of the theory of elastic bodies with a dipolar structure. It is an extension of the form of this principle proposed by Toupin in the case of classical linear elasticity. We consider a straight cylinder which is occupied by an dipolar elastic body that is only loaded at one end and our main finding proves that the internal energy in a portion of this cylinder, that is located at a given distance regarding the end that is loaded, is exponentially decreasing, regarding the respective distance.

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem.

In this study we want to extend the results of B. Straughan, which quite recently addressed the issue of double porosity structure for classical elastic bodies. In this case, the double porosity structure of the body is not influenced by the displacement field, which is not consistent with reality. For the mixed initial boundary value problem in the context of micropolar bodies with double porosity, we prove the existence of the solution, its uniqueness as well as some considerations on stability of solution.