Tube/Beam Vibration

Calculate lateral fundamental vibrations of composite beams.

To begin the vibration analysis, navigate to the Vibration tab in the Tube/beam Analysis window as shown below. The algorithms used in this analysis are taken from Vibration Analysis (Ref. 11.2) and the Shock and Vibration Handbook (Ref. 11.3), and the limitations and assumptions listed therein apply here.

Once the Laminate and Tube Geometry have been defined, complete the following steps in the Vibration tab:

1. Boundary Conditions - The analysis boundary conditions are selected from a drop-down menu. There are eight distinct boundary conditions available (Fixed-Free, Free-Simple, Fixed-Fixed, Free-Free, Simple-Free, Simple-Simple, Bending Fixed-Free with End Mass, and Torsion Fixed-Free with End Mass). The first six options use the formulation given in the Shock and Vibration Handbook (Ref. 11.3) shown in the equation below.

   

   The frequency of vibration shown here is based upon the stiffness (EI), the beam length (L), the mass of the beam per unit length (μ), and a constant (A) which determines the mode of vibration. The last two options, Bending w/ End Mass and Torsion w/ End Mass, are based upon algorithms from Vibration Analysis (Ref. 11.2 pp. 27-37).

   When you select an option, the graphic for that option will be displayed in the window above the drop-down. The dimensions and loading required to define the analysis are depicted in the drawing.

2. Stiffness Calculation Method - Of particular interest on this form are the "Stiffness Calculation Method" options. These options apply to closed cross-sections only (rectangle, circle, and ellipse).
   • The "Ply-By-Ply Tube" option forces all bending analyses to be calculated on a ply-by-ply basis through the thickness of the laminate. That is, the stiffness contribution for each ply is modeled separately.
   • The "Laminate Tube" option uses the structural (or smeared) properties of the entire laminate in bending calculations. Whenever you analyze a closed section beam using the "Laminate Tube" option, turn the "Coupling" option under the Laminate tab to zero. This is critical to generating accurate results.
3. L - Length - Specify the overall length of the beam.
4. End Weight - Specify the weight of the mass on the end of the beam.
5. Weight Moment of Inertia - When using the Torsion with End Mass option, you will have to input a calculated "I" (weight moment of inertia) based upon the geometry of the end mass.
6. Calculate - Once the input has been completed, click Calculate to compute the vibration analysis. The output box on the right side of the window shows the results of the defined analysis. The first three modes of lateral vibration are calculated and displayed. Each mode is conveniently presented in three units of measure: Hertz, radians/second and seconds/cycle. When the Bending w/ End Mass and Torsion w/ End Mass options are selected, the output is for the first fundamental mode of vibration only.