Control and Vibration of mechanical systems

[knight92]

Homework Statement

So I have this question given to me, basically there is a aero simulator which makes the ground shake so the data logger gets false readings, now to solve this problem a vibration absorption system has been added. Please see the attached System A.png file for the complete diagram. So the question it asks me is what I should adjust the absorption spring constant(Kabs) to, to get the smallest amplitude X2 hence lowest acceleration at half the maximum(16.7hz) ground frequency which is 8.35hz. Now this all takes into consideration mode shapes etc.

2. Homework Equations and The attempt at a solution:

Equation of Motion for each mass:
a = acceleration

m(abs)a1 = -K(abs) (X1-X2) ---------(1)
ma2 = K(abs)*(X1 - X2) - K(X2 - Y) ---------(2)

Now the terms in equations (1) and (2) can be multiplied, simplified and converted into the matrices below:

Stiffness Matrix =[ K(abs) ____ -K(abs),
_______________-K(abs) ____ K(abs) + K ]

Motion Matrix = [ K(abs)-w^2*m(abs) _____________ -K(abs) __________ ][X1] = [0]
______________[-K(abs) _________________________ K(abs) + K -w^2*m][X2] = [KY]

Note: Apologies for the '________' which indicates spaces between the matrix terms as the INDENT function messes things up even more.

Please note that 'm' is the mass of the data logger as shown in "System A.png"

m = 14 kg
m(abs) = 2.52 kg
K = 22900 N/m
Y = 0.0022 m
w = Frequency in Rad/S

Now I use cramer's rule to find X2 =
[ ( K(abs) - m(abs)*w^2 )*22900Y ] / [ 22900*K(abs) - 16.52*K(abs)*w^2 - 57708w^2 - 35.28w^4 ]

As the question says find the K(abs) which gives the lowest amplitude X2, if I do that then I can just plug in zero for K(abs) and get the lowest amplitude, surely that can't be right ?
It seems relatively simple otherwise, I have a formula frequency(w) = SQRT( K(abs) / m(abs) ) and I have been given the frequency which is half the max and the mass of absorber [m(abs)] is 2.52kg, so I can just convert frequency to radians put the numbers back in the formula and get K(abs) = 6928.3 but this seems too simple as its assuming that the ground vibration is the same is the vibration of the absorber and also it asks find the lowest amplitude and if I put it back in the X2 equation then I get the amplitude but how do I verify that is the lowest because like I said I can just keep decreasing the value of K(abs) to zero and reach the lowest amplitude but there's no point of a spring if the constant is going to be zero. I am so confused by this please help. Cheers

 

[knight92]

I still can't work it out ...