this is my working:
i have no real idea what i'm doing
coupled pendula Watch
- Thread Starter
- 10-02-2016 22:52
- 11-02-2016 09:02
- 11-02-2016 09:58
A "mode" is a motion where, say, all of the masses are moving in the same direction at the same time, then in terms of your problem, the expression is useful, or when 1/2 the masses are moving in the -ve direction and 1/2 in the +ve direction, then the expression is useful.
These expressions are "useful" because they allow you to write DEs purely in terms of those expressions e.g. you get say:
and you can introduce new variables and get the decoupled DEs
When you do this, the (or their equivs) are called the normal coordinates of the system. They are the combinations of the original coords that allow you to write solveable DEs.
To formalise this, you usually write things in matrix form so you turn your eqns into, say,
and then take a trial solution of:
When you plug this into (*), you get an eigenvalue problem, whose eigenvectors are the A,B column vector, and whose eigenvalues give you the fundamental frequencies of vibration of the system (i.e. you will get a set of possible values for - those are the fundamental frequencies of the normal modes).
It turns out that the eigenvectors specify the combinations of which give you your normal modes, so for this problem, you would expect to find an eigenvector:
You'll note that the process of solving for the eigenvectors only gives the ratio of A to B, rather than their specific values.
At the end, you form a general linear combination of the normal modes to give the general solution, and at that point you can combine the complex s into real sine and cosine solutions.Last edited by atsruser; 11-02-2016 at 11:52.
- 11-02-2016 11:30
- 11-02-2016 11:54