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The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

The aim of this paper is to investigate experimentally the effect of large vibration of a cantilever and a fully free rectangular plate made by a Fused Filament Fabrication process. Furthermore, this investigation attempts to compare our measurements and those obtained in the literature experimentally. For this purpose, a test rig was designed and manufactured for all experimental trials. The plate was excited randomly and harmonically at large displacement respectively, to obtain the linear and non-linear frequencies parameter. The non-linear dynamic behaviour of our structure at forced vibration is figured by exciting the plate at large displacement. The dependence of frequency and amplitude vibration are examined for the first, second, and third mode shapes. The non-linear dynamic behaviour of cantilever plates is compared with literature to illustrate the convergence of our results by using our specific mechanical properties, printing parameters, and process. Furthermore, the non-dimensional comparison is shown by 33.38%, 5.83%, and 20.58% for the first, second, and third mode shapes, respectively. Experimental tests will be performed on a 3D-printed metal plate to improve the present work. This work is intended to determine the dynamic proprieties of our parts in order to manufacture a safe and comfort machine. Actually, the dynamic behaviour of our 3D printing plates is compared with the obtained in the case of the isotropic plate for the aim to predict the convergence of both structures.