Modes of Vibration of 3-DOF Spring Mass System

AI Thread Summary
The discussion centers on determining the modes of vibration for a 3-degree-of-freedom (3-DOF) spring-mass system after calculating the natural frequencies as 0, 1, and √3 rad/s. The user attempts to apply the influence coefficient method but encounters difficulties due to the absence of a spring on one side of the left-most mass. They explore setting up a ratio of amplitudes and derive equations based on assumed solutions for the displacements. The user calculates values for the amplitudes A, B, and C but finds inconsistencies in the results, prompting further questioning of their approach. The conversation emphasizes the need for clarification on the method and potential alternative strategies for solving the problem.
Reefy
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Homework Statement


20171119_135837.jpg


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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If you are seriously constrained to this particular method, you had better talk with your teacher some more.
 
Is there another method that you know of? Can I set up a ratio of the amplitudes? My equations after assuming a solution of
X1,2,3 = Asinωt, Bsinωt, and Csinωt are

$$ A(1-ω^2) - B = 0$$
$$
B(2-ω^2) - A - C = 0
$$
$$
C(1-ω^2) - B = 0 $$
 
For a particular value of omega (0,1,root3), choose A=1 and solve for B and C. That will give you the mode vector for the particular omega value.
 
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Ok, if I do that then I get A = C = 1, and then B will equal 1, 0, and -2 for the three different ω
 
Something seems off about that because if I say A = 1, then

$$A(1-ω^2) - B = 0 $$ $$→$$ $$B=(1-ω^2)$$

And

$$C(1-ω^2) - B = 0$$ $$→$$ $$C = B/(1-ω^2) = (1-ω^2)/(1-ω^2) = 1 $$

But I still have the equation for $$B(2-ω^2) - A - C = B(2-ω^2) - 2 = 0$$ $$→$$ $$B=2/(2-ω^2)$$ which will give me slightly different values for B
 
For omega = 0, I get for A=1, then B = C = 1. No inconsistency there, I think.
 
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