Equivalent Spring Constant (3 springs attached to a swinging rod)

[qwerty007]

Homework Statement

Q) Derive the expression for the equivalent spring constant that relates the applied force F to the
resulting displacement x of the system shown in Fig. 1.86. Assume the displacement of the
link to be small.

Homework Equations

I have solved the question but when i checked it on the solution manual, the answer was different. In the solution manual they have approximated the extensions of all the springs as the same but i think they will be different.
Please let me know whether my method is correct or the one in the solution manual.

MY WORKING:
http://www.mediafire.com/view/2bdhs46w9igg83d/IMG_20140510_214819.jpg

solution manual:
http://www.mediafire.com/view/ks0t5btpg5y7d13/IMG_20140510_214607.jpg

The Attempt at a Solution

There are 2 methods and each leading to different values of Keq. Which method should i follow?

 

[paisiello2]

Figure 1.86 wasn't posted but based on the information you provided, I think I like your solution better.

 

[qwerty007]

Thanks for your reply.
Figure 1.86 is the same as what is drawn in my working.

 

[AlephZero]

The only thing that is not absolutely clear from your two attachments is whether the different stiffness of k1, k2, k3 are meant to be the "real" stiffness, or the effective stiffness including effect of the different displacements as the arm rotates.

Assuming they are the real stiffness of the 3 springs, I think your answer is correct.

 

[rezeew]

I cannot provide an opinion on which method is correct without knowing the specific details of your problem and the solution manual's approach. However, I can offer some general insights on how to approach this problem.

Firstly, it is important to understand the concept of equivalent spring constant. This is a value that represents the stiffness of a system that is made up of multiple springs connected in parallel or in series. In this case, the system consists of three springs connected to a swinging rod. To find the equivalent spring constant, we need to consider the combined effect of all the springs on the system.

In your working, you have correctly identified the individual spring constants and their extensions. However, your approach involves solving for the total force and the total displacement of the system, which may not necessarily be the same as the applied force and resulting displacement. This is because the springs are connected to a swinging rod, which may act as a lever and change the effective forces and displacements in the system.

The solution manual's approach may involve approximating the extensions of the springs as the same, which may be a reasonable assumption depending on the specific details of the problem. This approach simplifies the problem and allows for a simpler calculation of the equivalent spring constant. However, it may not be entirely accurate and may lead to a different value compared to your approach.

In conclusion, both methods may be valid depending on the specific details of the problem. It is important to carefully consider the assumptions and approximations made in each method and choose the one that is most appropriate for your problem. If you are unsure, you can always consult with your instructor or a peer for clarification.