Hi Aslam
I can not explain the subject with differential equation form. I was never present at site when spring was fixed.
Still I can try to explain in layman style & the way I understand the fact.
1) How spring is fixed with pipe. We hold the pipe in pipe centerline elevation ( zero movement) then we attach the spring with it, gradually set spring for specified load. Then we remove the holding, release the spring, allow it to work.

2) Now say we have straight line node 10,node 20, node30. Let’s say node 10 & node 30 we put resting support, but middle one node 20 spring. If say when we put all resting support each one node take 5000N load.

3) Spring at piping act as intentional upward force which try to balance downward load (Pipe+fluid weight) acting at that point.

4) If we do not put support at node 20 at all, then there will be sagging & piping will form a bow shape. If say we set the spring with load 4800N at zero level then release. Then spring will not take all load still there will be small sagging. When we release spring to work it will slightly push down ( to settle down) Spring load at sustain will change , same time it will slightly change weight taken care at node 10 & node 30 , resting support.

5) If say we set spring with load 5200 N. Again similar things happen. Piping will try to settle down this time piping will form bow shape in upward direction. This time also spring load at sustain & sustain load at support node 10 & node 30 will disturb & re –distributed. That’s what we call as spring settle down.

6) If you check my test file since I overdesign spring it move up 0.575 mm in sustain to settle down. ( In that particular case I don’t like to call spring overdesign , it’s my conscious decision to set load at spring like this to achieve better nozzle load in sustain & operating case).

7) More I differ to set spring load (then pipe+fluid) there will be more settle down problem.

8) It is basic requirement to put pipe weight, fluid, valve weight accurately while designing a spring. I mention before over design of spring & under design of spring is equally bad. Some times wrong selection of spring may create more harm than good in sensitive piping system.

9) There is also a hidden accuracy load calculation problem. Piping allow 12.5% milling tolerance. That additional or deducted pipe weight could also effect our proper spring setting.

10) Once I use some spring in large bore line where valve was there close to spring but valve weight was not confirmed. We give tentative installed load with clear note that site people to set spring to suit at site for those particular springs.


Regards
Habib