- #1
Valeron21
- 8
- 0
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 that m1 = m3 and m2 = βm1.
Summing the forces on each mass and using Newton's second law I have obtained the following matrix system:
$$
\begin{bmatrix}
βm & 0 & 0\\0 & m & 0\\0& 0& βm\\
\end{bmatrix}
\begin{bmatrix}
\ddot x_{1}\\\ddot x_{2}\\\ddot x_{3}\\
\end{bmatrix}
+
\begin{bmatrix}
k & -k & 0\\-k & 2k & -k\\0& -k& k\\
\end{bmatrix}
\begin{bmatrix}
x_{1}\\x_{2}\\x_{3}\\
\end{bmatrix}
=
\begin{bmatrix}
0\\0\\0\\
\end{bmatrix}
$$
which is $$\underline{M}\underline{\ddot X} + \underline{k}\underline{X}=0$$
After assuming a solution of the form:
$$ \underline{X}={U}[A_{}cos(ω_{}t)+B_{}sin(ω_{}t)]$$
it can be shown that
$$\underline{\ddot X}=-ω^{2}\underline{X}$$
and then that:
$$( \underline{k}-ω^{2} \underline{M}) \underline{X}=0$$
and for there to be a non-trivial solution,
$$det( \underline{k}-ω^{2} \underline{M})=0$$
here I have a problem, which might not even be a problem, but the characteristic equation I get only has two positive roots, where I am asked for three. Is there a) something obviously wrong with my method;b) a potential error in my calculations;c) is it possible for there to be only two frequencies?
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 that m1 = m3 and m2 = βm1.
Summing the forces on each mass and using Newton's second law I have obtained the following matrix system:
$$
\begin{bmatrix}
βm & 0 & 0\\0 & m & 0\\0& 0& βm\\
\end{bmatrix}
\begin{bmatrix}
\ddot x_{1}\\\ddot x_{2}\\\ddot x_{3}\\
\end{bmatrix}
+
\begin{bmatrix}
k & -k & 0\\-k & 2k & -k\\0& -k& k\\
\end{bmatrix}
\begin{bmatrix}
x_{1}\\x_{2}\\x_{3}\\
\end{bmatrix}
=
\begin{bmatrix}
0\\0\\0\\
\end{bmatrix}
$$
which is $$\underline{M}\underline{\ddot X} + \underline{k}\underline{X}=0$$
After assuming a solution of the form:
$$ \underline{X}={U}[A_{}cos(ω_{}t)+B_{}sin(ω_{}t)]$$
it can be shown that
$$\underline{\ddot X}=-ω^{2}\underline{X}$$
and then that:
$$( \underline{k}-ω^{2} \underline{M}) \underline{X}=0$$
and for there to be a non-trivial solution,
$$det( \underline{k}-ω^{2} \underline{M})=0$$
here I have a problem, which might not even be a problem, but the characteristic equation I get only has two positive roots, where I am asked for three. Is there a) something obviously wrong with my method;b) a potential error in my calculations;c) is it possible for there to be only two frequencies?