Which of these spring systems is equivalent?

[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.

 

[Doc Al]

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?)

 

[kerimek]

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.