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Entry  Thu Jan 31 14:00:00 2013, Kristen Holtz, Notes, PMC, PMC Longitudinal Testing and Modal Analysis PMCphoto_circclamps.JPGPMC_ringdown_Jan28.JPGPMC_unconstrainedringdown_Jan28.JPGPMC_Excitation_Diagram_1-28-13.JPG
    Reply  Wed Mar 27 01:52:34 2013, tara, Notes, PMC, PMC Longitudinal Testing and Modal Analysis pmc_eigen.png
       Reply  Sun Mar 31 03:03:28 2013, tara, Notes, PMC, PMC Longitudinal Testing and Modal Analysis psl_log.pngpsl_log2.png
          Reply  Sun Mar 31 14:42:05 2013, Evan, Notes, PMC, PMC Longitudinal Testing and Modal Analysis 
             Reply  Sun Mar 31 20:06:16 2013, tara, Notes, PMC, PMC Longitudinal Testing and Modal Analysis 
       Reply  Mon Apr 1 20:59:19 2013, tara, Notes, PMC, PMC Longitudinal Testing and Modal Analysis 
          Reply  Thu Apr 4 23:55:12 2013, Evan, Notes, PMC, PMC Longitudinal Testing and Modal Analysis mirrorsag_coarse.pdfmirrorsag_fine.pdf
    Reply  Thu Apr 4 11:44:14 2013, Evan, Notes, PMC, 270 Hz clamped PMC twisting mode pmc_clamp_twist_mode.jpg
Message ID: 1135     Entry time: Sun Mar 31 14:42:05 2013     In reply to: 1134     Reply to this: 1136
Author: Evan 
Type: Notes 
Category: PMC 
Subject: PMC Longitudinal Testing and Modal Analysis 

I think the analytical formula in terms of rho is going to be (1.57/2*pi) * sqrt(E / rho * L^2), since the Roark formula is (1.57/2*pi) * sqrt(A* E * g / w * L^2) and the weight per unit length is w = m * g / L = rho * A * g. With your values for L, A, E, and rho, this gives f1 = 16 kHz. Since A does not appear in the analytical formula, this also explains why changing the area in the Comsol model doesn't change the frequency.

Quote:

I compared results between COMSOL and analytical solution. The first longitudinal mode from both results differ by an order of magnitude!!

Peter sent me a note from Dennis about PMC longitudinal mode calculation. Dennis mentioned about a book by Young&Roark (here), so I looked it up and see how to estimate body mode frequencies of a simple block/beam.  I tried a simple geometry, a 0.1x0.1x0.175 (m) block. According to the book, cf situation 7b, table16.1 page 771, the first longitudinal mode is

f1 = (1.57/2*pi) * sqrt ( AE/ rho*L^2), where A is the cross section area (0.1x0.1), rho is the mass density of the material (2202 kg/m^3, for SiO2), E is the Young's modulus (72 GPa), L is the length of the block ( I use L = 0.175/2 because 7b situation is a uniform bar vibrates along its longitudinal axis, with upper end fixed, lower end free. This is similar to a whole beam resonate freely on both end because its center will be fix. Thus, to use the formula for our case, we have to use half length of the beam).

The analytical solution gives f1 = 1.6 kHz ,while COMSOL result is ~ 16 kHz.

psl_log2.png

 It is very strange that, according to COMSOL simulation, when the cross sectional area of the block is changed to 0.01x0.01 m^2 instead of 0.1x0.1 m^2, the frequency of the longitudinal mode does not change that much (still close to 16kHz. However, from the analytical solution, the frequency should drop by a factor of 10 ( around 165 Hz).

I'm going to think about this a bit more, but at this point, I think my COMSOL model is not correct. Might be some kind of bdy conditions that I'm missing.

psl_log.png

 

 

 

 

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