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I. Cr⸠stiu et al.
1 Introduction
At the bottom of the finite element analysis (FEA), the whole system is “fractured”
in small parts that are incomparable with the dimensions of the analyzed system.
From a mathematical point of view, the behavior of the whole system, i.e., the model
response, is considered to be approximated by that of the discrete model obtained by
connecting or assembling all elements. These pieces are called finite elements, and
according to the FEA theory, the precision of the numerical calculations is all the
greater as the dimensions of these elements are smaller, respectively, and the number
of the finite elements is larger, but the time allocated to the numerical calculations is
increasing.
In [1], static and modal analyses are performed on a rectangular steel plate to
discuss the effects of the element size in the analysis of the finite elements. Bending
deformation and von Mises stress for each model were calculated and compared,
regarding the influence of the element size on the static analysis results. The finite
element was used in the paper [2] for a study on the failure by compressing similar
concrete geometry columns using different mesh sizes. Deflections and stress analysis of beam on different elements using ANSYS were the purpose of the paper
[3]. The need for experimental verification of porous materials for the validation of
the biomechanical behavior analysis by numerical calculations for different types,
shapes, and dimensions of the finite elements is presented in paper [4]. Other studies
show the influence of mesh density on the results of finite element analysis of the
mechanical biocompatibility of dental implants [5], respectively, on the sensitivity of
structural responses in fracture of coastal impact bones [6]. Some papers are oriented
toward the dynamic behavior of mechanical structures, on modal analysis or impact
analysis on a rectangular plate [1] as well as the influence of mesh size on the accuracy of the results [7]. The effect of the finite element size on the modal analysis of
a heavy machine is analyzed in [8]. In [9], a study is made on finite element analysis
of a cracked structure, and in [10], the natural frequency vibrations of a crankshaft of
an internal combustion engine are simulated. In this paper, the influence of the mesh
size on the natural frequencies, respectively, on the corresponding critical speeds of
a rotor-shaft subjected to edge conditions, the simply supported ends, is analyzed.
2 Modal Analysis
Mechanical systems whose behavior is governed by differential equations with partial
derivatives, perfect analytical solutions can be determined only if they are defined on
a simple geometry. For complex geometries and nonlinear behaviors, approximate
analytical methods and numerical methods have been developed. The last category is
the FEM method, which consists of the approximation of the complex geometry of the
mechanical system in a lot of other geometric entities with simple geometric shapes,
called finite elements. The most used finite elements are triangular, tetrahedral, or
I. Cr⸠stiu et al.
1 Introduction
At the bottom of the finite element analysis (FEA), the whole system is “fractured”
in small parts that are incomparable with the dimensions of the analyzed system.
From a mathematical point of view, the behavior of the whole system, i.e., the model
response, is considered to be approximated by that of the discrete model obtained by
connecting or assembling all elements. These pieces are called finite elements, and
according to the FEA theory, the precision of the numerical calculations is all the
greater as the dimensions of these elements are smaller, respectively, and the number
of the finite elements is larger, but the time allocated to the numerical calculations is
increasing.
In [1], static and modal analyses are performed on a rectangular steel plate to
discuss the effects of the element size in the analysis of the finite elements. Bending
deformation and von Mises stress for each model were calculated and compared,
regarding the influence of the element size on the static analysis results. The finite
element was used in the paper [2] for a study on the failure by compressing similar
concrete geometry columns using different mesh sizes. Deflections and stress analysis of beam on different elements using ANSYS were the purpose of the paper
[3]. The need for experimental verification of porous materials for the validation of
the biomechanical behavior analysis by numerical calculations for different types,
shapes, and dimensions of the finite elements is presented in paper [4]. Other studies
show the influence of mesh density on the results of finite element analysis of the
mechanical biocompatibility of dental implants [5], respectively, on the sensitivity of
structural responses in fracture of coastal impact bones [6]. Some papers are oriented
toward the dynamic behavior of mechanical structures, on modal analysis or impact
analysis on a rectangular plate [1] as well as the influence of mesh size on the accuracy of the results [7]. The effect of the finite element size on the modal analysis of
a heavy machine is analyzed in [8]. In [9], a study is made on finite element analysis
of a cracked structure, and in [10], the natural frequency vibrations of a crankshaft of
an internal combustion engine are simulated. In this paper, the influence of the mesh
size on the natural frequencies, respectively, on the corresponding critical speeds of
a rotor-shaft subjected to edge conditions, the simply supported ends, is analyzed.
2 Modal Analysis
Mechanical systems whose behavior is governed by differential equations with partial
derivatives, perfect analytical solutions can be determined only if they are defined on
a simple geometry. For complex geometries and nonlinear behaviors, approximate
analytical methods and numerical methods have been developed. The last category is
the FEM method, which consists of the approximation of the complex geometry of the
mechanical system in a lot of other geometric entities with simple geometric shapes,
called finite elements. The most used finite elements are triangular, tetrahedral, or