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  Application of Eigenvectors-Vibrational Modes

An Application of Eigenvectors:

Vibrational Modes:

One application of ew’s as well as ev’s is in the analysis of vibration problems. A easy nontrivial vibration problem is the motion of two objects with equal masses m attached to each other and fixed outer walls by equal springs with spring constants k.

Let x1 signify the displacement of the first mass and x2 the displacement of the second and note the displacement of the walls is zero. Every mass experiences forces from the adjacent springs proportional to the stretch or compression of the spring. Disregard any friction Newton’s law of motion ma = F, leads to:

m¨x1 = −k(x1− 0) +k(x2− x1) = −2kx1 + kx2
m¨x2 = −k(x2− x1) +k(0 − x2) = kx1− 2kx2

Dividing both sides by m we are able to write these equations in matrix form:

¨x = −Ax,

Where,

815_matrix equation.jpg

For this kind of equation the general solution is:

2012_general solution of matrix.jpg

Where λ1 as well as λ2 are ew’s of B with corresponding ev’s v1 and v2. One can ensure that this is a solution by substituting it into the equation.

We are able to interpret the ew’s as the squares of the frequencies of oscillation. We can locate the ew’s and ev’s of B using Mat lab

> B = [2 -1 ; -1 2]
> [v e] = eig(B)

This must produce a matrix v whose columns are the ev’s of B and a diagonal matrix e whose

299_diagonal matrix.jpg

Two equivalent masses attached to each other and fixed walls by equal springs.

324_equal springs.jpg

Two vibrational modes of a easy oscillating system. In the left mode the weights move together as well as in the right mode they move opposite. Note that the two modes in fact move at different speeds.

Entries are the ew’s of B In the primary eigenvector v1 the two entries are equal. This stands for the mode of oscillation where the two masses move in sync with each other. The second ev, v2 has the similar entries but opposite signs. This signifies the mode where the two masses oscillate in anti-synchronization.

Notice that the occurrence for anti-sync motion is √3 times that of synchronous motion.

Which of the two modes is the largely dangerous for a structure or machine? It is the one with the lowest frequency for the reason that that mode can have the largest displacement. Occasionally this mode is called the fundamental mode.

To obtain the frequencies for the matrix A = k/mB notice that if viis one of the ev’s for B then:

1112_frequencies of matrix.jpg

Therefore we can conclude that A has the same ev’s as B but the ew’s are multiplied by the factor k/m. Therefore the two frequencies are:

905_frequecy.jpg

We are able to do the same for three equal masses the corresponding matrix B would be:

2159_frequency.jpg

Find the ev’s as well as ew’s as above. There are three dissimilar modes Interpret them from the ev’s.

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