Fall 2008 MET 440 FEA-HW-2: beam model

Prof. Dave Johnson, dhj1@psu.edu

Penn State - Erie, The Behrend College

Intermediate FEA model (limited* 3D model)
This model was constructed in the FEA-HW0 preparation for the vibration FEA lab classes.  [a 3D solid cantilever beam.  Using SI units.]

The beam is 4 m. long and the cross-section is round, hollow with a radius of 100 mm. and the wall thickness of 10 mm.
The material is steel (modulus = 2.11x10¹¹ Pa, density = 7800 kg/m³, and Poisson's ratio = 0.3)

* the model was split to permit a frictionless surface BC which prevents left-to-right (and twisting) movement while allowing up-and-down (and axial) deflections.

a) Use a MODAL Analysis to determine the first 5 natural frequencies of this multi-DOF system.  

• Report the model results compared to hand calculations of the first five natural frequencies.  Observe an animation of each mode shape to describe the vibration (bending, longitudinal, torsional, etc.)

b) Add "distributed damping" to the beam model.  

• Define the damping ratio (DMPRAT) to be 3% of the critical damping level.

c) Use the beam model with damping in a Harmonic Analysis.  

• Shake the base of the model in the transverse direction with an amplitude of 1 mm and sweep the frequency from 0 to 500 Hz in steps of 2 Hz.  
• Plot the amplitude of the tip deflection vs. frequency.  Observe the resonance peaks on the plot and compare to the hand calcs.
• Explain why ONE of the first five modes is missing from this plot.  

d)  Use the beam model with damping in a Harmonic Analysis.  

• Apply a sustained harmonic forcing function ALONG the beam's axis.  
• Use a force amplitude of 1000 N and sweep the frequency from 0 to 500 Hz in steps of 2 Hz. 
• Plot the amplitude of the tip deflection vs. frequency.  Observe the resonance peak on the plot and compare to the hand calcs.  
• Explain why only ONE of the first five modes appears on this plot.  

Turn in:

For the initial static analysis (HW-0):

• an element plot showing the mesh
• a geometry plot showing ALL loads and boundary conditions.
• plots of deformation and bending stress
• hand calculation of static tip deflection, maximum stress, and weight COMPARED to the FEA maximum deflection, normal stress, reaction solution, and mesh error (SEPC).

For the modal and harmonic analysis (HW-2):

• the undamped modal analysis natural frequencies and supporting hand calculations
• show your instructor a mode shape animation for one of the modes during the lab meeting
• for the case of "base excitation," turn in the plot of tip amplitude UY
• for the case of force excitation on the beam tip, turn in the plot of tip displacement amplitude UX
• Explanation of the Harmonic analysis results, as requested above.