Multiple Spring system?
A light rod is supported by two springs fixed on ceiling. A mass hangs from centre of the rod with a third spring as shown in the diagram. Find the natural frequency of the system:
(i) When spring 1 and 2 share the weight equally
(ii) When the bar is assumed to stay horizontal all the time
I am not getting the force equations of the springs. Any ideas?
First, two previously known lemmas about multiple spring systems that you can assume to be true or derive for yourself:
(1) In parallel, so
(2) In series, so
Start with , ,
because these are in series, so . Since the load is split evenly between springs #1 and #2, we can assume .
(i) Assume so WLOG . So or . Since , . Solving for gives . By lemma 2,
(ii) By lemma 1, and by lemma 2,
In each instance of course,