- #1
Reefy
- 63
- 1
Homework Statement
Homework Equations
a11 = a21 = a31 = a12 = a13
a22 = a32 = a23
$$
\begin{bmatrix}
x_{1} \\
x_{2} \\
x_{3}
\end{bmatrix}
=ω^2m
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\begin{bmatrix}
x_{1} \\
x_{2} \\
x_{3}
\end{bmatrix}
$$
$$x_{1} = a_{11}mω^2x_{1}+a_{12}mω^2x_{1}+a_{13}mω^2x_{1}$$
$$x_{2} = a_{21}mω^2x_{2}+a_{22}mω^2x_{2}+a_{23}mω^2x_{2}$$
$$x_{3} = a_{31}mω^2x_{3}+a_{32}mω^2x_{3}+a_{33}mω^2x_{3}$$
The Attempt at a Solution
I have already completed part a. The natural frequencies are 0, 1 and square root of 3 rad/s.
I'm having difficulty finding the modes of vibration. I wanted to use the influence coefficient method where I select the left-most mass to undergo a unit force while keeping the other masses fixed. This would cause a deflection of the left-moss mass and give me my first influence coefficient a11. However, there is no spring to the left of this mass and I'm having trouble figuring out how to apply the influence coefficient method.
$$ a_{22} = a_{32} = a_{23} = \frac {1}{k}$$
$$ a_{33} = \frac {2}{k}$$
$$ a_{11} = a_{21} = a_{31} = a_{12} = a_{13}= ? $$
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