Fall 2008 MET 440 FEA-HW-1: 1-DOF

Prof. Dave Johnson, dhj1@psu.edu

Penn State - Erie, The Behrend College

Simple FEA model (lumped system): one mass, one spring, no damping.
This model was constructed in the FEA-HW preparation for the vibration FEA lab classes.  [a spring (COMBIN14) and a lumped mass (MASS21)]

The spring rate, k = 10 lb_f/in, and the mass is 2 lb_m.

a) Use a MODAL Analysis to determine the natural frequency of this single-DOF system.  

• Report the model results compared to a hand calculation of the natural frequency.

b) Add "discrete damping" to the spring element.  

• Determine the damping coefficient, c, such that the damping will be 25% of the critical damping level.  

• Use a Damped Modal Analysis to determine the natural frequency of this single-DOF system. 

  •  "The imaginary part of the eigenvalue, Ω, represents the steady-state circular frequency of the system. The real part of the eigenvalue, σ, represents the stability of the system. If σ is less than zero, then the displacement amplitude will decay exponentially, in accordance with EXP(σ). If σ is greater than zero, then the amplitude will increase exponentially. (Or, in other words, negative σ gives an exponentially decreasing, or stable, response; and positive σ gives an exponentially increasing, or unstable, response.) If there is no damping, the real component of the eigenvalue will be zero."
    

• Report the model results compared to a hand calculation of the damped natural frequency.

c) Use the single-DOF model with 25% of critical damping in a Harmonic Analysis.  

• Shake the base of the model with an amplitude of 0.1" and sweep the frequency from 1 to 10 Hz in steps of 0.1 Hz.  

• Plot the amplitude of the mass deflection vs. frequency.  Annotate the resonance peak on the plot.  

• Plot the reaction force at the base vs. frequency

• Compute the "transfer function" (i.e., the output/input) and graph that result vs. frequency.

d) Use the single-DOF model with 25% of critical damping in a Harmonic Analysis. 

• Apply an harmonic force to the mass in the vertical direction with an amplitude of 0.1 lbf and sweep the frequency from 0 to 10 Hz in steps of 0.1 Hz.  

• Plot the amplitude of the mass deflection vs. frequency.  Annotate the resonance peak on the plot.

• Plot the reaction force at the base vs. frequency

Turn in:

For the initial static analysis (HW-0):

• an element plot showing ALL loads and boundary conditions (and NO reactions). 
  Include element TYPE numbering on the plot.
• hand calculation of static deflection and weight COMPARED to the FEA solution results (PRNSOL) and reactions (PRRSOL) tables.

For the modal and harmonic analysis (HW-1):

• the undamped modal analysis natural frequency and supporting hand calc. 
• the damped modal analysis natural frequency and supporting hand calc. of the damping factor, c, as well as the damped natural frequency
• for the case of "base excitation," turn in the plots of amplitude, reaction force, and the transfer function
• for the case of excitation on the mass, turn in the plot of displacement amplitude for the mass and reaction force at the base.