The rod shown in the figure (home.attbi.com/~josephsieh/Picture3.jpg) is
aluminum with properties L=0.5m and A=0.02m^2.Find the first two natural 
frequencies if 
              rho A L/m=0.07656 and k/(AE/L)=0.07656
ps:(1) rho is the density of the rod
   (2) k is the spring constant
   (3) E is the young's modulus

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Clarification of Question by wjs-ga on 30 Apr 2003 16:40 PDT

The rod shown in the figure (http://home.attbi.com/~josephsieh/Picture3.jpg) is 
aluminum with properties L=0.5m and A=0.02m^2.Find the first two natural  
frequencies if  
              rho A L/m=0.07656 and k/(AE/L)=0.07656 
ps:(1) rho is the density of the rod 
   (2) k is the spring constant 
   (3) E is the young's modulus

Hello -

The formula for calculating the natural frequencies is:

           F       1 - cos nL
delta L = --- * ( ------------ )
          Krod    nL * sin nL


delta L = change in length F = frequency Krod = AE/L  n = wavenumber L
= length

Note that this formula fails at periodic values of nL (pi, 2pi, 3pi,
etc) - the frequency associated with these values are defined as the
natural frequencies.

The wavenumber is given by

n = 2*pi*F * sqrt(rho/E)
 
so, nL = 2*pi*F*sqrt(rho/E)*L  and solving for F for the first natural
frequency gives

                 Krod                          0.07656
F = 1/2 * sqrt( -------- )     =  1/2 * sqrt ( -------- ) = 1.58
                rho*A*L                   0.07656*0.5*0.2


The second natural frequency occurs at nL = 2pi so

F = sqrt( Krod/rho*A*L ) = 3.16


A nice reference for these formulas:
http://www.efunda.com/DesignStandards/springs/spring_frequency.cfm


Search strategy
Google : solid rod natural frequency calculations

synarchy

Hello wjs-ga,

A problem similar to yours can be found here:

Louisiana State University
http://me.lsu.edu/~ram/TEACHING/ad_vib/vibration5.pdf
Example 5.1.1

However, this problem doesn’t have the mass.


Another one with a configuration close to yours can be found here:

University of Aberdeen
http://www.eng.abdn.ac.uk/~eng289/3525t4.htm


Googlenut

I believe that the frequency provided in synarchy's answer is for the rod only.

Googlenut

Googlenut's comment is correct.
  The question should state that it is not asking
for natural frequencies of a rod, but for frequqncies of a more
complex system, shown on the figure and consisting of the 
 rod, another mass, and a spring.
It is not clear if the system is fully described: ratio of masses
is given, but absolute mass masses may be needed.. It would help
if wjs would clarify this. The way it is posted now, the question is not
clearly formulated.