The Influence of the Mesh Element Size …
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hexahedral shapes. For each element, special features are used that interpolate the
unknown function. The FEM use procedure involves several steps. First of all, it
is intended to generate a discrete representation of the studied object. The mesh
generation stage is the most important step in this analysis, affecting the robustness
and accuracy of the simulation results. Numerical simulation results are very sensitive
to choose the type of material behavior, but also to select appropriate boundary
conditions. For each element, a stiffness matrix and a mass matrix are established,
and when internal friction is considered, also the damping matrix. At this stage,
the choice of interpolation functions, number of elements, and number of nodes per
element leads to an increase in the accuracy of the calculation, but also to the time
consumption of the computer resources. The stiffness matrices, mass matrices, and
damping matrices of the elements are then summed to form the global matrices [K],
[M], and [C]. In the case of damping neglect, for a dynamic analysis, the integration
scheme leads to solving the following system of linear differential equations.
[M]{ ¨
u} + [K ]{u} = {0},
(1)
where {u} are nodal displacements. The vibrations of the nodes can be expressed by
harmonic functions such as:
{u} i = {U } i cos( p i t + ϕ i ),
(2)
where {U } i are the eigenvectors and p i are corresponding circular frequencies.
From these values, the natural frequencies, respectively, and the critical speeds are
determined. The eigenvalues problem leads to solve the determinant:
[K ] − p
2
[M]
= 0.
(3)
3 Numerical Results
The shapes of natural modes of vibration and natural frequencies of a rotor considered
as a disk centered on a shaft were determined using the software provided by ANSYS
14.5. For the 3D modeling of the shaft and the rotor, the geometry of a solid element
called SOLID 186, which is defined by 20 nodes, each having three degrees of
freedom representing displacements in three perpendicular directions, was chosen.
This type of finite element exhibits square displacement behavior. To solve the
eigenvalues problem, given by (3), the block Lanczos method is used because it is
efficient to extract a large number of modes in most cases. The shape of the first
bending vibration mode in the OXY plane is shown in Fig. 1, and the shape of the
first bending vibration mode in the OXZ plane is shown in Fig. 2.
525
hexahedral shapes. For each element, special features are used that interpolate the
unknown function. The FEM use procedure involves several steps. First of all, it
is intended to generate a discrete representation of the studied object. The mesh
generation stage is the most important step in this analysis, affecting the robustness
and accuracy of the simulation results. Numerical simulation results are very sensitive
to choose the type of material behavior, but also to select appropriate boundary
conditions. For each element, a stiffness matrix and a mass matrix are established,
and when internal friction is considered, also the damping matrix. At this stage,
the choice of interpolation functions, number of elements, and number of nodes per
element leads to an increase in the accuracy of the calculation, but also to the time
consumption of the computer resources. The stiffness matrices, mass matrices, and
damping matrices of the elements are then summed to form the global matrices [K],
[M], and [C]. In the case of damping neglect, for a dynamic analysis, the integration
scheme leads to solving the following system of linear differential equations.
[M]{ ¨
u} + [K ]{u} = {0},
(1)
where {u} are nodal displacements. The vibrations of the nodes can be expressed by
harmonic functions such as:
{u} i = {U } i cos( p i t + ϕ i ),
(2)
where {U } i are the eigenvectors and p i are corresponding circular frequencies.
From these values, the natural frequencies, respectively, and the critical speeds are
determined. The eigenvalues problem leads to solve the determinant:
[K ] − p
2
[M]
= 0.
(3)
3 Numerical Results
The shapes of natural modes of vibration and natural frequencies of a rotor considered
as a disk centered on a shaft were determined using the software provided by ANSYS
14.5. For the 3D modeling of the shaft and the rotor, the geometry of a solid element
called SOLID 186, which is defined by 20 nodes, each having three degrees of
freedom representing displacements in three perpendicular directions, was chosen.
This type of finite element exhibits square displacement behavior. To solve the
eigenvalues problem, given by (3), the block Lanczos method is used because it is
efficient to extract a large number of modes in most cases. The shape of the first
bending vibration mode in the OXY plane is shown in Fig. 1, and the shape of the
first bending vibration mode in the OXZ plane is shown in Fig. 2.