# Vibration of a mass connected via preloaded spring

• sfensphan
In summary, if the preloaded spring force is less than the inertial force of the small mass, then the two will separate. However, if the system vibrates at resonance frequency, it becomes impossible to determine where Δx is such that F_spring = F_inertia. Additionally, the larger mass may be displaced depending on how much it vibrates.
sfensphan
The setup: I have a mass (m1)connected to a much, much larger mass (m2) via a preloaded spring. They start out in contact because the preloaded spring holds them together. Now suppose the large mass is subject to vibrations, possibly at the resonance of the structure. Will the two masses separate? If they do, what is the maximum distance between the two?

Will the two masses separate?
My original chain of thought was that if the spring force is less than the inertial force of the small mass, then the two will separate.
F_spring = -k(Δx)
k = spring constant
Δx = displacement from spring resting state. Since there is a preload, this is non-zero

F_inertia = m1 * a
m1 = mass of the small mass
a = acceleration. This should be sinusoidal, since it's a vibration

I would then compare the two and see which one is larger. I would use the max value of F_inertia during this comparison.

If they do, what is the maximum distance between the two?
This is where I get a little lost. A static calculation would be to find where Δx is such that F_spring = F_inertia.
But what if the system vibrated at resonance frequency (which I believe would be sqrt(k/m1) for this case)?
How do I deal with the preload?
Is the Force body diagram such that the sinusoidal force acts directly on the m1?
Do I need to know how much the larger mass is displaced?

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sfensphan said:
I have a mass (m1)connected to a much, much larger mass (m2) via a preloaded spring. They start out in contact because the preloaded spring holds them together. Now suppose the large mass is subject to vibrations, possibly at the resonance of the structure. Will the two masses separate? If they do, what is the maximum distance between the two?

just as a help to your approach one can see the given reference-

actually ...In engineering practice, we are almost invariably interested in predicting the response of a structure or mechanical system to external forcing.

For example, we may need to predict the response of a bridge or tall building to wind loading, earthquakes, or ground vibrations due to traffic. Another typical problem you are likely to encounter is to isolate a sensitive system from vibrations. For example, the suspension of your car is designed to isolate a sensitive system (you) from bumps in the road.

Electron microscopes are another example of sensitive instruments that must be isolated from vibrations. Electron microscopes are designed to resolve features a few nanometers in size. If the specimen vibrates with amplitude of only a few nanometers, it will be impossible to see! Great care is taken to isolate this kind of instrument from vibrations. That is one reason they are almost always in the basement of a building: the basement vibrates much less than the floors above.

reference-http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations_forced.htm

sfensphan
Thanks very much. From your link, I see that it is a base excitation case. Greatly appreciate you leading me in the right direction.

drvrm

## What is the definition of "Vibration of a mass connected via preloaded spring"?

The "Vibration of a mass connected via preloaded spring" refers to the movement or oscillation of a mass that is connected to a spring that has already been compressed or stretched before the mass is attached to it.

## What is the importance of studying the "Vibration of a mass connected via preloaded spring"?

Studying the "Vibration of a mass connected via preloaded spring" is important because it can help us understand and predict the behavior of various mechanical systems, such as car suspensions, building structures, and even musical instruments. It is also essential in designing and optimizing these systems for better performance and stability.

## What factors affect the "Vibration of a mass connected via preloaded spring"?

The "Vibration of a mass connected via preloaded spring" is affected by several factors, including the mass of the object, the stiffness of the spring, the amount of preload, and the frequency of the applied force. Other factors such as damping, friction, and external forces can also have an impact on the vibration.

## What are the equations used to calculate the "Vibration of a mass connected via preloaded spring"?

The two main equations used to calculate the "Vibration of a mass connected via preloaded spring" are Hooke's Law and Newton's Second Law of Motion. Hooke's Law relates the force exerted by a spring to its displacement, while Newton's Second Law relates the acceleration of an object to the net force acting on it.

## How can the "Vibration of a mass connected via preloaded spring" be controlled or minimized?

The "Vibration of a mass connected via preloaded spring" can be controlled or minimized by adjusting the stiffness of the spring, the amount of preload, and the damping of the system. Additionally, adding additional supports or dampers can also help reduce unwanted vibrations.

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