1
$\begingroup$

Hi I've been stuck on a problem for one of my exam revisions, we were asked to identify the natural frequency of the following structures

enter image description here

I know that the equation for the stiffness is as follows
Left Column : $k= 12EI/L^3 $
Right Column : $k2=3EI/L^3 $
Equivalent Stiffness : $15EI/L^3$
Hence the natural frequency of the column will be $$w_n=\sqrt{15EI/mL^3} $$ Given $EI:8*10^{11} Nmm^2$, $L: 2.5m$ , $m:250kg$ The problem is when i tried to do the problem without converting the EI into SI unit of $Nm^2$ I will have a different result in terms of radians. i.e $$wn(withoutconverting) :w_n=\sqrt{15(8*10^{11})/250(2500)^3}=1.75 rad/sec $$ $$wn(with conversion) :w_n=\sqrt{15(8*10^{11})*10^{-6}/250(2.5)^3}=55.42 rad/sec $$ The second answer seems to make more sense, as we have a very stiff column with a small mass it should have a really high natural frequency, however the answer key provided insisted that the first one is the correct way to approach it. How do i approach this and is my approach of converting it to SI unit correct? Thankyou so much for the help

$\endgroup$
2
  • $\begingroup$ Sorry can you elaborate? $\endgroup$ May 17 at 13:50
  • $\begingroup$ I agree that 1.75 rad/sec is incorrect because the units under the square root work out to be m/mm/sec^2. There needs to be a conversion of 1000 to cancel m and mm. The result then works out to be 55.42 rad/sec. $\endgroup$
    – JohnHoltz
    May 17 at 18:11

1 Answer 1

2
$\begingroup$

The basic unit of the natural frequency is $Hz$, $cycle/s$, or just $1/s$.

Let's check the results by using "$m$" and "$mm$" as the base unit -

1. Use $m$ as the base unit:

  • $\omega^2 = \dfrac{N-mm^2}{kg*m^3} = \dfrac{N*10^{-6} m^2}{kg*m^3} = \dfrac{N}{kg*10^6m} = \dfrac{kg*m}{kg*10^6 m*s^2} = \dfrac{1}{10^6s^2}$

2. Use $mm$ as the base unit:

  • $\omega^2 = \dfrac{N-mm^2}{kg*m^3} = \dfrac{N-mm^2}{kg*10^9mm^3} = \dfrac{N}{kg*10^9mm} = \dfrac{kg*m}{kg*10^9mms^2} = \dfrac{kg*10^3mm}{kg*10^9mms^2} = \dfrac{1}{10^6s^2}$

Note: The base unit of $N is kgm/s^2, which needs to be converted too.

Your mistake/discrepancy was caused by terminating the conversion too early before getting to the bottom of it.

Check:

$EI = 8*10^{11} N-mm^2 = \dfrac{8*10^{11}kgm-mm^2}{s^2}*\dfrac{10^3mm}{m} = \dfrac{8*10^{14}kg-mm^3}{s^2}$

$\omega = \sqrt{\dfrac{15*8*10^{14}kg-mm^3}{250kg*(2500mm)^3*s^2}} = 55.42Hz$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.