Free Vibration of Spring system with two DOF

[nerak99]

In some questions I am doing, you set of a pair of simultaneous equations and in the notes we have that . For a non trivial solution of X1 and X2,the determinant of
coefficients of X1 and X2 must be zero.

An equation might typically look like this (m \omega^2 +k_1)x_1 +k_2 x_2=0

Why must the determinant be zero? When solving SEs in general using matrices, the determinant must be non-zero?

 

[AlephZero]

For free vibtration, there are no external forces appplied to the system, so the right hand sides of the equations are 0.

If the determinant is non-zero, the only solution is $x_1 = x_2 = 0$ which is not very interesting!

The determinant is only zero for "special" values of $\omega$, and these are the frequencies at which the system can vibrate.

 

[nerak99]

Thanks for that. I still don't understand "why" but that is v helpfule.