Frequency of a Slender Beam with Concentrated Mass

Description

An 80" long beam has a square cross-section of 2" X 2". A block of 8" X 2" x 6" is attached to the beam as shown below. Determine the natural frequency of the system for simply supported and fixed end restraints. In separate studies, the block is modeled as a remote mass and as a solid.

File Name

Browse to drive letter:\Users\Public\Public Documents\SOLIDWORKS\SOLIDWORKS version\samples\Simulation Examples\Verification\Frequency_16.SLDASM and open the file.

Study Type

Frequency.

Mesh Type

Solid mesh.

Material Properties

Modulus of elasticity = 10,007,604 psi, Poisson's ratio = 0.33, Density = 0.0975437 lb/in3.

Results

Four studies are created:
  • Study Fixed-Fixed models the block as a remote mass
  • Study Pinned-Pinned models the block as a remote mass
  • Study Solid-Fixed-Fixed models the block as solid
  • Study Solid-Pinned-Pinned models the block as solid

The fundamental frequency for a slender beam with concentrated center mass is:

f1 = (4/π)*(3 E I /L3 (M+0.37 Mb) )1/2 (fixed-fixed end restraints)

f1 = (2/π)*(3 E I /L3 (M+0.49 Mb) )1/2 (pinned-pinned end restraints).

where:
  • E: Modulus of elasticity
  • I: Area moment of inertia of beam (neutral axis)
  • M: Mass
  • Mb: Mass of beam
  • L: Span of beam
  Fundamental Frequency, Hz (Fixed-Fixed) Fundamental Frequency, Hz (Pinned-Pinned)
Theory 48.37 22.27
SOLIDWORKS Simulation (block as solid) 48.80 22.58
SOLIDWORKS Simulation (block as remote mass) 49.13 22.75

Reference

Robert D. Blevins, "Formulas for Natural Frequency and Mode Shape", Krieger Publishing Company, Florida, Reprint Edition 2001, pp. 158-159.