Why am I not getting the correct natural frequency?

[EastWindBreaks]

Homework Statement

To find the natural frequency of a uniform cantilever beam, the area moment of inertia, cross-sectional area and density are not given, everything else is given by the lab, our group got a theoretical natural frequency( it might be angular frequency) of 27.44 Hz from lab's computer with the same mass, dimensions and Young's modulus. however, when I try to solve it manually using the natural frequency equation, I got 1.195 as the natural frequency and 7.51 for the angular frequency. which is way off from 27.44, but I should be getting the same value. can someone confirm if my calculation is correct base on the data? I have tried countless times and its driving me crazy. our experimental natural frequency is 24.94, I am not sure how am I gona justify the huge error for the lab report.
2. Relevant equation

ξi=3.516 for Mode 1

I=bh^3/12

The Attempt at a Solution

 

[haruspex]

Homework Statement

View attachment 215477
To find the natural frequency of a uniform cantilever beam, the area moment of inertia, cross-sectional area and density are not given, everything else is given by the lab, our group got a theoretical natural frequency( it might be angular frequency) of 27.44 Hz from lab's computer with the same mass, dimensions and Young's modulus. however, when I try to solve it manually using the natural frequency equation, I got 1.195 as the natural frequency and 7.51 for the angular frequency. which is way off from 27.44, but I should be getting the same value. can someone confirm if my calculation is correct base on the data? I have tried countless times and its driving me crazy. our experimental natural frequency is 24.94, I am not sure how am I gona justify the huge error for the lab report.
2. Relevant equation
View attachment 215478
ξi=3.516 for Mode 1

View attachment 215479
I=bh^3/12

The Attempt at a Solution

Please post your whole calculation.

 

[EastWindBreaks]

Please post your whole calculation.

angular frequency of mode 1= (3.516/12.4^2)* sqrt（(1*10^7）*(0.000183)/42.86/0.13)
ω= (3.516/12.4^2) * sqrt(328.44)
ω= 0.02287* 18.1229
ω=0.4144

where 0.000183 is the area moment of inertia which came from (bh^3)/12, (1)(0.13^3)/12=0.000183
density=mass/volume=148/(0.13*1*26.56)=42.86

I forgot to square root when I initially post the thread, however, this time, the angular frequency is even smaller.

 

[haruspex]

angular frequency of mode 1= (3.516/12.4^2)* sqrt（(1*10^7）*(0.000183)/42.86/0.13)
ω= (3.516/12.4^2) * sqrt(328.44)
ω= 0.02287* 18.1229
ω=0.4144

where 0.000183 is the area moment of inertia which came from (bh^3)/12, (1)(0.13^3)/12=0.000183
density=mass/volume=148/(0.13*1*26.56)=42.86

I forgot to square root when I initially post the thread, however, this time, the angular frequency is even smaller.

I do not see any units conversion. All the distances are in inches, so that's ok, but you have force in pounds weight and density using grams.

 

[EastWindBreaks]

I do not see any units conversion. All the distances are in inches, so that's ok, but you have force in pounds weight and density using grams.

oh wow, yeah, I didnt do units conversion because they were all in inches but I totally forgot about grams and lbs. ok so this time, the mass is 0.3263 pound, density= 0.3263/(0.13*1*26.56)= 0.0945 lb/in^3, the new angular frequency I got using this new density is 8.828 Hz, which is still way off...

 

[haruspex]

the mass is 0.3263 pound,

In E, it is pounds weight, i.e. a force. That has dimension MLT^-2. To be consistent, that L needs to be in inches, so you need a factor g in in/s².
(Why does anyone use a system other than MKS?)

 

[EastWindBreaks]

In E, it is pounds weight, i.e. a force. That has dimension MLT^-2. To be consistent, that L needs to be in inches, so you need a factor g in in/s².
(Why does anyone use a system other than MKS?)

oh right, you mean convert gram to slug right? so, mass= 0.0101412 slug, density = 0.0101412/(0.13*26.56*1)=0.003016 slug/in^3
and I got 49.4 Hz for angular frequency, 7.86 for the natural frequency. still off from 27.44...
yeah, I wish the world can use a single unit system...we have the U.S unit system, SI, and British unit system..

 

[haruspex]

convert gram to slug right? so, mass= 0.0101412 slug,

I don't think that quite fixes it. The ratio of slug to pound mass, 32 roughly, is based on the value of g when expressed in ft/s². Your change is equivalent to multiplying the E value by 32, which would be right for converting from pounds weight /sq into lb ft s^-2in^-2. But everywhere else in your calculation distance are in inches, so you need E in lb s^-2 in^-1. I.e. rather than express the density in slugs multiply E by 32 x 12.

 

[EastWindBreaks]

Thank you very much! I finally got it, I sometimes completely forgot which unit system is which...the pound force, pound mass, slug are so confusing and unfriendly for students.