Calculation of Natural Frequencies and Modes of Vibration

FLAC3D is primarily intended to simulate structural failure. However, for full practical application, FLAC3D should also be able to simulate accurately the dynamic response of structures in the elastic range. The performance of FLAC3D models subjected to dynamic excitation in the elastic range can be demonstrated by calculating the natural frequencies and modes of vibration for small amplitude vibrations.

FLAC3D contains the command zone dynamic eigen modes, which can be executed to calculate natural frequencies for different modes of vibration in a system composed of deformable zones. The calculation of natural frequencies and modes of vibration assumes that the structural system is elastic. The kinematic variables of the system are the nodal displacements, or the vector of translations. For solid continuum zones, rotations are implicit through displacement gradients. In a FLAC3D model, the deformability is given by the two independent elastic constants, e.g., bulk and shear moduli. A global stiffness matrix for the system is assembled, which relates the forces applied to each node with the displacements. The mass matrix is assumed to be diagonal and equal to the nodal inertial mass for each degree of freedom.

The global stiffness matrix is formed by assembling the elementary stiffness matrix for each tetrahedron within a zone. Assuming small strains in the elastic regime, the elementary stiffness matrix (a 12x12 matrix in FLAC3D and a 6x6 matrix in FLAC2D) is calculated for each tetrahedron. The global stiffness matrix is obtained by assembling all the elementary stiffness matrices into a single large sparse matrix.

An iterative Krylov-Schur algorithm is used to solve the eigenvalue problem. The algorithm gives the first \(N\) eigenvalues requested. In the case that no boundary conditions are applied, i.e., a free structure, the eigenvalues will all be near zero and they correspond to rigid body modes. In three dimensions, there are six rigid body modes: three translations and three rotations. The first six eigenvalues will be near zero, and the corresponding eigenvectors will represent the rigid body modes. In addition, the dynamic inertial masses must have been calculated. Therefore, the command model dynamic active on must have been given, followed by a model cycle 1 command to force the calculation of dynamic inertial masses. The stiffness matrix requires non-zero elastic moduli, so the zones must have been assigned elastic properties.

Deformable Pillar Example

The response of a FLAC3D model subjected to low level vibration can be verified for the analysis of bending of pillars and walls. Lemos (2007) presents two verification examples: elastic vibration modes of a square pillar and of a wall with variable thickness. A version of the square pillar example is presented here. The verification involves a pillar, 10 m high, with a square section of 1 x 1 m.

The pillar is composed of 640 zones, each having an edge length of 0.25 m. The bottom boundary is fixed in all directions. The mass density of the pillar material is 2500 kg/m3. The Young’s modulus and Poisson’s ratio are 1.0 GPa and 0.25, respectively. The data file for the FLAC3D model is shown below. The command zone dynamic eigen modes 6 is executed to calculate the first six modes of vibration.

The natural frequencies for the first three bending modes of the pillar are compared to values calculated from Timoshenko beam theory (Chopra 1995, and Ferreira and Fasshauer 2006). The natural frequencies for the first three bending modes are listed below. The values compare reasonably closely to the values from Timoshenko beam theory.

Table 1: Natural Frequencies (Hz) of Bending Modes for the Square Pillar

Bending mode

FLAC3D model

Timoshenko beam theory

1st

1.06

1.02

2nd

6.36

6.09

3rd

15.67

16.1

Data File - squarepillar.dat

model new
model configure dynamic
model large-strain off

; --- geometry ---
zone create brick point 0 (0,0,0) point 1 (1,0,0) ...
    point 2 (0,1,0) point 3 (0,0,10) size 4 4 40

; --- properties ---
zone cmodel assign elastic
zone property density 2500 young 1e9 poiss 0.25

; --- boundary condition ---
zone gridpoint fix velocity range pos-z 0

; --- force inertial mass calculation ---
model cycle 1

; --- calculate eigenvalues ---
zone dynamic eigen modes 6

; --- plot ---
zone dynamic eigen setmode