# Which of these spring systems is equivalent?

• MarkH748
In summary, Mark is looking at a vibrations problem with 2 springs (System A) and needs to find the overall stiffness K to calculate the natural frequency. There are two suggested solutions, one where the equivalent stiffness is K/2 (System C) and one where it is 2K (System B). Mark is unsure which system is correct and is seeking clarification.
MarkH748
Hi guys. I looking at a vibrations problem with 2 springs in it (System A). I need to find an overall value for the stiffness K to find the natural frequency of the system. I know that normally when 2 springs are in series I should get the sum of (I/Ki)^-1 which would give me an equivilant stiffness of K/2 in this case (System C). But when I take moments about the pivot point og the rod O it makes more sense to me for it to be 2K (System B). Here are some sketches I made.

Can anyone shed some light on this as I'm unsure of which system is correct.

Any help would be greatly appreciated.

Mark.

MarkH748 said:
I know that normally when 2 springs are in series I should get the sum of (I/Ki)^-1 which would give me an equivilant stiffness of K/2 in this case (System C).
Ah... but you don't really have two springs in series, you have one above and one below. Why not derive the effective spring constant for yourself? (Imagine the rod displaced by some Δx. What's the net force on it?)

Both System B and System C are correct representations of the equivalent stiffness for System A. The key difference between the two is the method used to calculate the equivalent stiffness.

System B uses the principle of moments to determine the equivalent stiffness. This method takes into account the individual stiffness values of the two springs and their respective distances from the pivot point. This results in an equivalent stiffness of 2K.

System C, on the other hand, uses the principle of series springs to determine the equivalent stiffness. This method assumes that the two springs are connected in series and therefore the equivalent stiffness would be half of the individual stiffness values, resulting in a value of K/2.

Both methods are valid and will give you the same natural frequency for the system. The choice between System B and System C depends on the specific problem and the assumptions made. In some cases, using the principle of moments may be more appropriate while in others, using the principle of series springs may be more accurate.

Ultimately, it is important to understand the underlying principles and assumptions behind each method in order to determine which one is most suitable for your specific problem. I hope this helps clarify the difference between the two systems and provides some insight into choosing the correct method for your calculations.

## 1. What is a spring system?

A spring system is a mechanical system that is made up of one or more springs. Springs are elastic objects that can be stretched or compressed and have the ability to store potential energy when they are deformed.

## 2. How do you determine if two spring systems are equivalent?

Two spring systems are equivalent if they have the same spring constant, which represents the stiffness of the spring, and the same equilibrium position, which is the point at which the spring is neither stretched nor compressed.

## 3. What are the different types of spring systems?

The three most common types of spring systems are Hooke's Law springs, torsion springs, and leaf springs. Hooke's Law springs are the most common and obey Hooke's Law, which states that the force needed to stretch or compress a spring is directly proportional to the displacement. Torsion springs are used to store rotational energy and are often used in door hinges and clock mechanisms. Leaf springs are commonly used in vehicles to absorb and distribute weight and force.

## 4. Can different types of springs be equivalent?

Yes, different types of springs can be equivalent as long as they have the same spring constant and equilibrium position. The type of spring only affects the way in which it deforms, but the overall behavior and properties of the spring system remain the same.

## 5. Why is it important to determine if two spring systems are equivalent?

Determining if two spring systems are equivalent is important because it allows for the prediction of the behavior and properties of the system. This information is crucial for designing and engineering various mechanical systems that utilize springs, such as in cars, machinery, and medical devices.

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