Overview

Test Case

This test case models the vibration of a pinned-pinned slender bar with a concentrated mass at its mid-length. The objective is to validate the fundamental natural frequency of the structure. The beam has a length of 80 in and a square cross-section with 2 in sides. The structure supports a mass of 0.5 lbf-s²/in at its midspan. Figure 163 illustrates the domain dimensions and boundary conditions.

Figure 163: Test Case Schematic, including domain geometry, main dimensions, and boundary conditions

The following table lists the main parameters of the test case, which uses the following system of units: length in in, time in s, mass in lbf-s²/in, force in lbf, and pressure in psi.

For a uniform, linear elastic beam with a concentrated mass, the approximate formula to calculate the fundamental natural frequency is given based on the mass location and boundary conditions. For a center mass and pinned-pinned support, the fundamental frequency of the structure can be calculated as:

(16)

Where is the Young's modulus, is the area moment of inertia around the neutral axis, is the beam length, is the concentrated mass, and is the mass of the beam. The area moment of inertia around the neutral axis for a square is a function of its side :

(17)

For the current test case, the area moment of inertia is 1.333 in⁴ and the mass of the beam is 0.2346 lbf-s²/in. Therefore, the fundamental natural frequency is 12.43 Hz.

Analysis Assumptions and Modeling Notes

One part is defined in the model to represent the slender beam, being meshed with 1D beam elements of length 1 in. These beam elements use the Hughes-Liu formulation with cross section integration (ELFORM=1) and an elastic material card (*MAT_ELASTIC) with a density of 7.33⋅10^-4 lbf-s²/in⁴, a Young's modulus of 3.0E⋅10⁷ psi, and a Poisson's ratio of 0.3. A mass element of 0.5 lbf-s²/in is defined for the central node of the structure (NID 41) using *ELEMENT_MASS. The keyword *BOUNDARY_SPC_NODE_ID is used to define the motion constraints of the two end nodes. The keywords *CONTROL_IMPLICIT_GENERAL (IMFLAG=1), *CONTROL_IMPLICIT_DYNAMICS (IMASS=0), and *CONTROL_IMPLICIT_EIGENVALUE (NEIG=2) are used to activate the implicit eigenvalue, static analysis with two eigenvalues to be extracted.

Figure 164: Model setup in LS-DYNA of the 1D modal analysis of a slender beam with a central mass

Results Comparison

The slender beam configuration for the fundamental mode is shown in Figure 165. The visualization of the fundamental mode is performed by reading the d3eigv file, generated for the modal analysis. Note that the first two modes have the same natural frequency and are orthogonal to each other due to the squared cross-section of the beam.

Figure 165: Lateral view of the fundamental mode of the slender beam

To quantify the error between the theoretical and LS-DYNA results, the fundamental frequency of the slender beam with central mass and its relative error are calculated and shown in the table below. This comparison verifies the excellent agreement between the fundamental frequencies.