Vibration: find the stiffness of the spring

[bigbang42]

Homework Statement

a machine of mass 100 tonne generates a simple harmonic disturbing force when operating at a speed of 200rpm. to protect the floor and surrounding machinery it is proposed tp mount the machine on helical springs so that the transmissibility ratio is reduced to 0.1

Q1 determine the total stiffness of the springs required assuming the damping to be negligible ??

I'm having with this question because apparently certain sections you have to make "assumptions" which to me is more like guess work

Homework Equations

ωn = sqrt ( k/m)

tr = \sqrt{(1 + ( 2ζ ω/ωn) / ((1 - (ω/ωn)^2)^2 + (2ζ ω/ωn)^2) }

The Attempt at a Solution

right what I've done is I've said that ζ=0 which gets rid of anything in the tr formula with ζ in as obviously anything multiplied by 0 is 0

so that leaves sqrt [ 1 / ((1-(ω/ωn)^2)^2) ]

i found ω is 200/60 x 2pi so you get something like ω=(20/3)pi

i then rearranged the tr formula two lines above to get ωn = 20

then used the formula ωn= sqrt (k/m)

rearranged to get k = ωn^2 x m and got a stupidly high answer so i imagine i done all the above wrong

can anyone give me any steps on what to do or where I'm going wrong, i have been tearing my hair out over it :(

 

[Simon Bridge]

What makes you think the result is "stupidly high"?
It is a 100T machine, after all. So what are you using as your benchmark for "high"?

 

[bigbang42]

i got something silly like 3.9x10^8 for my k value in the end, is that not too high ?

 

[Simon Bridge]

"too high" compared with what?

what sort of value were you expecting and why?
(get into the habit of justifying your hunches.)

i.e.
How far would that 100T machine compress the springs under gravity alone?
What would happen to that compression under the kind of value you were expecting?
i.e. how tall would the springs have to be just to support the machine off the floor?
i.e. how long are industrial springs ... normally?

 

[Simon Bridge]

Letting you off the hook:
With no damping, the transmissibility ratio is $$Tr=\frac{1}{\left | 1-\frac{\omega^2}{\omega_0^2}\right |}$$ ... This is the same as yours, but written more clearly so you can see what is going on better.
Bearing this in mind, you told us:

i found ω is 200/60 x 2pi so you get something like ω=(20/3)pi
i then rearranged the tr formula two lines above to get ωn = 20

Notice that ω = 20π/3 ≈ 20 also ... which suggests a frequency ratio close to 1 - or resonant transmissibility (Tr → ∞): the opposite of what is wanted. The question calls for Tr=1/10 ...