# Control and Vibration of mechanical systems

• knight92
In summary, the question asks for the value of the absorption spring constant (Kabs) that will result in the smallest amplitude X2 and lowest acceleration at half the maximum ground frequency of 16.7hz. The equation of motion for each mass is provided, and using Cramer's rule, X2 is found to be a function of K(abs). However, simply setting K(abs) to zero is not a viable solution. The frequency and mass of the absorber are also given, and using a formula, K(abs) can be calculated to be 6928.3. However, this may not necessarily be the lowest amplitude and further verification is needed.
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

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

#### Attachments

• System A.png
2.6 KB · Views: 425
Last edited:
I still can't work it out ...

## What is the purpose of controlling and reducing vibrations in mechanical systems?

The purpose of controlling and reducing vibrations in mechanical systems is to improve the overall performance and efficiency of the system. Vibrations can cause wear and tear on components, reduce accuracy and stability, and lead to premature failure of the system. By controlling vibrations, the system can operate more smoothly and reliably.

## What are the main sources of vibrations in mechanical systems?

The main sources of vibrations in mechanical systems can include unbalanced forces, rotating or reciprocating components, friction, and external disturbances such as wind or uneven surfaces. These vibrations can be amplified by the natural frequencies of the system, leading to potential issues.

## How can vibrations be controlled in mechanical systems?

Vibrations in mechanical systems can be controlled through various methods such as using vibration isolators, dampers, or absorbers. These devices can absorb or dissipate the energy of the vibration, reducing its effect on the system. Additionally, proper design and balancing of components can also help to minimize vibrations.

## What is the role of feedback control in controlling vibrations in mechanical systems?

Feedback control is an important tool in controlling vibrations in mechanical systems. By using sensors to measure the vibrations and feeding that information back to a controller, adjustments can be made to minimize or eliminate the vibrations. This can be done through active control techniques such as vibration cancellation or passive control techniques such as altering the stiffness or damping of the system.

## What are some potential consequences of not controlling vibrations in mechanical systems?

If vibrations are not properly controlled in mechanical systems, it can lead to a range of consequences such as reduced accuracy and stability, increased wear and tear on components, and potential system failure. Vibrations can also create excessive noise and discomfort for operators or users of the system. In extreme cases, uncontrolled vibrations can even pose a safety hazard to people or the surrounding environment.

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