Undamped vibrations (only spring present)

[kaminho]

Homework Statement

An exhaust fan mass 90 kg operates at 1100 rpm and is supported by an UNDAMPED vibration isolator i.e a spring that has a stiffness of 220 kN. If the harmonic force produced by the fan has a magnitude of 12 kN determine:
a) The transmitted force through the isolator to the foundations(TR)
b) The static deflection of the exhaust fan (δst)

Homework Equations

how do we work out the trasmitted force?

The Attempt at a Solution

I'm able to find trasmission ratio (Tr) as it becomes Tr=1/(1-r^2) with r being the frequency ratio. that is Tr= 0.2258 ( i first found out the frequency of excitation and natural frequency and devided the first by the second to find r and from there i found Tr). however i know Tr=Ft/Fo (Ft transmitted force and Fo applied) but i don't know how to get Ft as the value of Fo is still unknown and i don't think its 12000 mentioned in the question (because that's magnitude of harmonic force). or maybe it is i don't know. can anyone help?

 

[rock.freak667]

The Attempt at a Solution

I'm able to find trasmission ratio (Tr) as it becomes Tr=1/(1-r^2) with r being the frequency ratio. that is Tr= 0.2258 ( i first found out the frequency of excitation and natural frequency and devided the first by the second to find r and from there i found Tr). however i know Tr=Ft/Fo (Ft transmitted force and Fo applied) but i don't know how to get Ft as the value of Fo is still unknown and i don't think its 12000 mentioned in the question (because that's magnitude of harmonic force). or maybe it is i don't know. can anyone help?

Yes, F₀ is the magnitude of the harmonic force, F₀=12kN.

 

[kaminho]

Yes, F₀ is the magnitude of the harmonic force, F₀=12kN.

so after i found r (frequency ratio) and subsequently Tr which is 0.2258 then the rest is pretty straight forward ?
am i right in saying the magnitude of the harmonic force is always the applied force (Fo) and also the amplitude ?
and thanks loads for the help too.

 

[rock.freak667]

so after i found r (frequency ratio) and subsequently Tr which is 0.2258 then the rest is pretty straight forward ?
am i right in saying the magnitude of the harmonic force is always the applied force (Fo) and also the amplitude ?
and thanks loads for the help too.

I think that would be correct.

 

[DeeNos]

Based on the given information, we can determine the transmitted force through the isolator to the foundations by using the following equation:

Ft = F0 * Tr

Where:
Ft = Transmitted force
F0 = Applied force (in this case, the harmonic force produced by the fan with a magnitude of 12 kN)
Tr = Transmission ratio (which you have correctly calculated as 0.2258)

Therefore, the transmitted force can be calculated as:
Ft = 12 kN * 0.2258 = 2.71 kN

To find the static deflection of the exhaust fan, we can use the formula:
δst = Ft / k

Where:
δst = Static deflection
Ft = Transmitted force (calculated above)
k = Stiffness of the spring (given as 220 kN)

Substituting the values, we get:
δst = 2.71 kN / 220 kN = 0.0123 meters

Therefore, the static deflection of the exhaust fan is 0.0123 meters.