(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

2. Relevant equations

a_{11}= a_{21}= a_{31}= a_{12}= a_{13}

a_{22}= a_{32}= a_{23}

$$

\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}

$$

$$x_{1} = a_{11}mω^2x_{1}+a_{12}mω^2x_{1}+a_{13}mω^2x_{1}$$

$$x_{2} = a_{21}mω^2x_{2}+a_{22}mω^2x_{2}+a_{23}mω^2x_{2}$$

$$x_{3} = a_{31}mω^2x_{3}+a_{32}mω^2x_{3}+a_{33}mω^2x_{3}$$

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 a_{11}. 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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# Homework Help: Modes of Vibration of 3-DOF Spring Mass System

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