Online Manuals RFEM 6 / RSTAB 9 | Dynamic Analysis

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2023-10-30

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Natural Frequencies

The Natural Frequencies result table category contains the natural frequencies of the undamped system. In the table title bar, you can switch between the results of the modal load cases.

Result Category "Natural Frequencies" for Modal Analysis

Each frequency of the system has a corresponding eigenmode. The mode shapes are also displayed graphically (see the image Results of Modal Analysis ). Select the mode shape in the navigator or use the buttons and to switch between the mode shapes (see the image Selecting Mode Shape ). You can also select the corresponding line in the table to display a particular mode shape in the work window.

Natural Frequencies

The "Natural Frequencies" table (see the image Result Category "Natural Frequencies") provides an overview of the following results of the undamped system:

Eigenvalue
Angular frequency
Natural frequency
Natural period

The equation of motion of a multiple degree of freedom system without damping is calculated with the specified solving method .

M_{i} = u_{i}^{T} \cdot M \cdot u_{i}

u_i	Eigenvector of the mode shape i
M	Mass matrix

Modal Mass

M depends on the type of a mass matrix .

u = (^{u_{X}, u_{Y}, u_{Z}, φ_{X}, φ_{Y}, φ_{Z}) T}

u_j	Translational component (j ... direction X/Y/Z)
𝜑_j	Rotational component (j ... direction X/Y/Z)

Mode Shapes with Translational and Rotational Components

The eigenvalue λ [1/s²] is related to the angular frequency ω [rad/s] with λ_i = ω_i². The natural frequency f [Hz] is then derived with f = ω / (2π). The natural period T [s] is the reciprocal of the frequency, which is determined as T = 1 / f.

For a structural system with more degrees of freedom (MDOF – multiple degrees of freedom), there are several eigenvalues λ_i and the corresponding mode shapes u_i.

Effective Modal Masses

Table "Effective Modal Masses"

The "Effective Modal Masses" tab provides an overview of the following results:

Modal mass M_i
Effective modal mass for translational directions m_eX, m_eY, m_eZ
Effective modal mass for rotational directions m_eφX, m_eφY, m_eφZ
Effective modal mass factor for translational directions f_meX, f_meY, f_meZ
Effective modal mass factor for rotational directions f_mφX, f_mφY, f_mφZ
Sums of results

The effective modal masses describe how much mass is activated in each direction by each eigenmode of the system.

The modal mass is defined as follows:

M_{i} = u_{i}^{T} \cdot M \cdot u_{i}

u_i	Eigenvector of the mode shape i
M	Mass matrix

Modal Mass

The eigenvector u_i of an eigenmode i is given in the equation Mode Shapes. M depends on the type of mass matrix .

The modal mass M_i is independent of the direction. However, it changes depending on the mode shape scaling.

The effective modal masses m_ij^eff describe the masses that are accelerated in the j-direction, where j = 1, 2, 3 applies to the translation and j = 4, 5, 6 to the rotation – for each individual mode shape i. These masses are independent of the mode shape scaling. They are directly related to the participation factors Γ_{i, j} (see the equation Participation Factor).

T = [\begin{matrix} 1 & 0 & 0 & 0 & (Z - Z_{0}) & - (Y - Y_{0}) \\ 0 & 1 & 0 & - (Z - Z_{0}) & 0 & (X - X_{0}) \\ 0 & 0 & 1 & (Y - Y_{0}) & - (X - X_{0}) & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) \begin{matrix}  \end{matrix}]

X, Y, Z	Global coordinates of the considered FE node
X₀, Y₀, Z₀	Coordinates of the center of gravity

Axis Transformation

The matrix T exists for every FE node.

The effective modal masses are defined as follows:

m_{i j}^{e f f} = M_{i} \cdot Γ_{i j}^{2}

M_i	Modal Mass
Γ	Participation factor

Effective Modal Mass

The sum of the effective modal masses ∑m_e is given at the end of the table. In translational directions, these sums are equal to the total mass of the structure ∑_M. Exceptions are masses that are not activated, such as masses in fixed supports. The full mass is only achieved if all eigenvalues of the model are calculated.

The effective modal mass factor f_me is required to decide whether to consider a specific shape for the response spectrum method. For example, Section 4.3.3.3 of EN 1998‑1 specifies that "the sum of the effective modal masses for the modes taken into account amounts to at least 90% of the total mass of the structure” and that “all modes with effective modal masses greater than 5% of the total mass are taken into account."

The effective modal mass factors f_me are defined as follows:

f_{m e} = \frac{m_{e}}{\sum m_{e}}

m_e	Effective modal mass

Effective Modal Mass Factor

For more information about the modal analysis and the effective modal mass factors, see Meskouris [2] and Tedesco [3].

Participation Factors

Table "Participation Factors"

The "Participation Factors" tab lists the following results:

Modal mass M_i
Participation factor for translational directions Γ_X, Γ_Y, Γ_Z
Participation factor for rotational directions Γ_φX, Γ_φY, Γ_φZ
Sums of results

The participation factor is defined as follows:

Γ_{i, j} = \frac{1}{M_{i}} \cdot u_{i}^{T} \cdot M \cdot T_{j}

M_i	Modal mass
u_i	Eigenvector of the mode shape i
M	Mass matrix
T_j	j^th column of the matrix T (axis transformation)

Participation Factor

The participation factors, which also define the degrees of freedom of rotation, are described in more detail in [1]. The participation factor Γ_i,j is dimensionless for translations; for rotations, it has the unit [m].

Masses in Mesh Points

Table "Masses in Mesh Points"

The "Masses in Mesh Points" tab lists the following results:

Dimensions for translational directions m_X, m_Y, m_Z
Mass for rotational directions m_φX, m_φY, m_φZ
Sums of masses

These values represent the masses that have been assigned in the modal analysis load case and applied to the nodes of the FE mesh during the calculation. They also depend on the modal analysis settings. Further information can be found in the chapter Masses.

The sum of the masses for each direction is given at the end of the table.

You can graphically display the masses in mesh points on the model. To do this, use the Mass category in the Navigator.

Graphical Display of Masses m-x

Use the control panel to adjust the scaling factors for the graphical display of masses.

Parent Chapter

Results