Recent content by Valeron21

Whoops. erm, so my roots are: $$0, \sqrt\frac{k}m, \sqrt\frac{k(2+β)}{βm}$$ making 0(?) the fundamental frequency, then the square root of k/m the first overtone and the square root of k(2+β)/m the second overtone, each with a corresponding mode shape vector?

Three masses, m1 , m2, m3, on a frictionless, horizontal plane, connected by two springs, both with a spring constant k. The system is set in motion by displacing the middle mass, m2, a distance a to the right, whilst holding the end masses, m1 and m2, in equilibrium. Also should be noted...

Hmm, maybe there's something wrong in how I'm approaching this? I've always previously, when working with 2.D.O.F. systems, assumed a solution of the form: $$ \underline{X}={U}[A_{}cos(ω_{}t)+B_{}sin(ω_{}t)]$$ then through a process of differentiation and substitution obtained this...

Yes, that is correct. Both springs in the system have the same stiffness constant, k.

Sorry. The characteristic equation isn't really the problem - I should have worded that better - it's how I use what will be pretty horrible roots to find the corresponding eigenvectors that I'm having trouble with. EDIT: $$(k-ω_{i}^{2}m)u_{i}=0$$ is what I would usually sub the values of ω...

I've found the characteristic equation of the system I'm trying to solve: $$ω^{4}m_{1}m_{2}-k(m_{1}+2m_{2})ω^{2}+k^{2}=0$$ I now need to find the eigenfrequencies, i.e. the two positive roots of this equation, and then find the corresponding eigenvectors. I've been OK with other examples...

Hmm, I'm not really sure that makes it significantly easier, though. To give a bit more context - I need to find the eigenfrequencies and corresponding eigenvalues, then give a solution of the form: $$ \underline{X}=\sum_{i=1}^{}\underline{U_{i}}[A_{i}cos(ω_{i}t)+B_{i}sin(ω_{i}t)]$$...

The det. of the following matrix: $$ \begin{matrix} 2k-ω^{2}m_{1} & -k\\ -k & k-ω^{2}m_{2}\\ \end{matrix} $$ must be equal to 0 for there to be a non-trivial solution to the equation: $$(k - ω^{2}m)x =0$$ Where m is the mass matrix: $$ \begin{matrix} m_{1} & 0\\ 0& m_{2}\\...