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.

Each frequency of the system has a corresponding eigenmode. The mode shapes are also displayed graphically (see picture # extbookmark and to switch between the eigenmodes (see picture # extbookmark 002253 "> manual|image027595|Select mode shape #). You can also select the corresponding row in the table to display a particular mode shape in the work window.

Natural Frequencies

The table 'natural frequencies' (see image image027927 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 given with the specified # extbookmark manual|eigenvalue solver|Solution method # calculated.

M depends on the type of # extbookmark manual|mass matrix|Mass matrix #.

The eigenvalue λ [1/s²] is connected 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 with T = 1/f.

For a system with several degrees of freedom (MDOF) there are several eigenvalues λ_i and associated eigenmodes u_i .

Effective Modal Masses

The 'Effective Modal Masses' tab contains 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:

The eigenvector u_i of an eigenmode i is shown in equation formula001058 eigenmodes . M depends on the type of # extbookmark manual|mass matrix|Mass matrix #.

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

The effective modal masses m_ij^eff describe the masses that are accelerated in the j-direction, where j = 1, 2, 3 for translation and j = 4, 5, 6 for rotation - for each individual eigenmode i. These masses are independent of the scaling of the mode shapes . They are directly related to the TABLE_PARTICIPATION_FACTORS participation factors Γ_{i, j} (see equation formula001060 participation factor ).

The matrix T exists for every FE node.

The effective modal masses are defined as follows:

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, for example 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 needed to decide whether a specific shape needs to be considered for the response spectrum method. For example, EN 1998-1, Section 4.3.3.3 specifies that “the sum of the effective modal masses of the modal contributions to be considered is at least 90% of the total mass of the structure” and that “all modal contributions are taken into account whose effective modal masses are more than 5% of the total mass amount ".

The effective modal mass factors f_me are defined as follows:

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

Participation Factors

The following results are listed in the 'Participation Factors' tab:

• 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:

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

Masses in mesh points

In the register 'masses in network points' the following results are listed:

• 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 were 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 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 the network points on the model. To do this, use the Mass category in the Navigator.

In the # extbookmark manual|resultFactorsTab|Control panel # allows you to adjust the superelevation factors for the graphical representation of the masses.