Why am I not getting the correct natural frequency?

In summary: I got 49.4 Hz for angular frequency, 7.86 for the natural frequency. still off from 27.44...In summary, the group got a theoretical natural frequency of 27.44 Hz from lab's computer with the same mass, dimensions and Young's modulus. However, when they tried to solve it manually, they got 1.195 as the natural frequency and 7.51 for the angular frequency. This is way off from the actual frequency of 24.94
  • #1
EastWindBreaks
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Homework Statement


upload_2017-11-23_1-5-6.png

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
upload_2017-11-23_1-5-34.png

ξi=3.516 for Mode 1

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I=bh^3/12

The Attempt at a Solution

 

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  • #2
EastWindBreaks said:

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.
 
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  • #3
haruspex said:
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.
 
  • #4
EastWindBreaks said:
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.
 
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  • #5
haruspex said:
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...
 
  • #6
EastWindBreaks said:
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?)
 
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  • #7
haruspex said:
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..
 
  • #8
EastWindBreaks said:
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.
 
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  • #9
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.
upload_2017-11-24_3-11-8.png
 

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FAQ: Why am I not getting the correct natural frequency?

Why is the natural frequency important?

The natural frequency is important because it is a fundamental property of a system that determines how it will behave when disturbed. It is the frequency at which the system will vibrate most easily and with the largest amplitude.

How is the natural frequency calculated?

The natural frequency of a system can be calculated using the equation f = √(k/m), where f is the natural frequency, k is the spring constant, and m is the mass of the system. This equation applies to simple harmonic motion systems, such as a mass-spring system.

What factors can affect the natural frequency?

The natural frequency of a system can be affected by a variety of factors such as the stiffness of the system, the mass of the system, and any external forces acting on the system. In addition, the shape and material of the system can also impact its natural frequency.

How can I determine if I am not getting the correct natural frequency?

If you are not getting the correct natural frequency, you can compare your calculated natural frequency to the theoretical value or use experimental methods, such as using a frequency analyzer, to measure the actual natural frequency of the system. If there is a significant difference between the two values, then there may be an error in your calculations or an external factor affecting the system.

How can I adjust the natural frequency of a system?

The natural frequency of a system can be adjusted by changing the parameters that affect it, such as the stiffness, mass, or external forces. For example, if you want to increase the natural frequency of a mass-spring system, you can either decrease the mass or increase the spring constant. Alternatively, you can also use a frequency controller to adjust the natural frequency of certain systems.

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