# Modal Analysis in Ansys to get natural frequencies and compare them with theory results

Hi, I'm Pietro, a short term Italian incoming student at VT (virginia tech).

My advisor, in order to start my thesis there about skin friction coefficients in a scramjet combustion chamber, gave me a Modal analysis to do on a gage described as follow:

a cantilever beam  (1mm diameter and 3cm long cylindrical shape) with 1mm depth and 1cm diameter cylindrical flat "hat" attached upon.

Base constraints: zero displacements, zero rotation

On the top is applied a costant, mono direction wall shear stress of 200 Pa, so a 0.0157 N of wall shear force (but i don't think it needs for modal analysis).

Material:  Nickel bellows: density=8880 kg/m^3, Poisson ratio:0.25, E=195 GPa (I ask more:is it correct?)

My problem is:THERE'S NO AGREEMENT BETWEEN THEORY AND SOFT. ANSYS ANALYSIS ABOUT NATURAL FREQUENCIES (e.g. ANSYS gave me a first mode frequency of 300Hz roughly,on the other hand, theory by f=(Kn/2*PI)*sqrt(EI/w*L^4); where w=width=1mm,L=3cm,Kn=3.52 for first mode roughly, gave me E+4 m order of magnitude)

WHAT'S WRONG?...WHAT HAVE I TO DO TO HAVE AGREEMENT?IS THERE SOMETHIN' WRONG IN THEORY FORMULA OR IN SOFT. ANALYSIS?

Thanks a lot, Best Regards.Pietro

### my comment

Hi,

The Problem, as far as I figured out, can be expressed as the flextural free vibration of a cantilever beam with a concentrated mass at the tip... I recommand you to check whether you have considered the tip mass. Also, make sure that you have considered a consistent set of units for mass, stiffness and length

Amir Shakouri

PhD Research Student

Nanyang Technological University

Singapore

### Tip Mass

I agree with Amir, i think not taking into account the tip mass is the issue, although im not sure what you mean exactly by the wall shear component.

By my calculations the first bending mode natural frequency of the cylinder, without the tip mass, should be approximately 1000 Hz, from wn = 1/2/pi*(1.875^2)*sqrt((E*I)/(rho*A*L^4))  with the values you provided.

The effect of the tip mass will reduce the natural frequency by about 74%, giving a natural frequency of the combined system of approx. 270Hz. The reduction in the first bending mode natural frequency can be given approximately by sqrt(1/(1+M/meq)) where, M is the tip mass and meq is the equivalent mass of the system without the tip mass being: meq = 3*rho*A*L/(1.875^4).

You can find an example of how to come up with this reduction in natural frequency due to the mass loading in Appendix D of the following link http://unsworks.unsw.edu.au/vital/access/services/Download/unsworks:8233... .

From memory there is also something on this in the textbook by Rao "Mechanical Vibrations".