ODE 4. Consider the two-mass, three-spring system shown in the figure n2 Figure

Question

ODE

4. Consider the two-mass, three-spring system shown in the figure n2 Figure 7.1.3a © John Wiley & Sons, Inc. All rights reserved. Let m1 = m2 = m and k1 = k3. The system describing the displacements and the corresponding velocities is given by, ki k2 (i) (+5) Determine the eigenvalues and eigenvectors of the system. (ii) (+10) Write down the general solution of the system (iii) (+10) Determine a particular solution of the system 0 0 01 + cos(wt) You can assumew> 0 but you may need to consider the resonant and non resonant cases separately

Explanation / Answer

It can be shown that the eigenvalue eqn for the system is

|x1"| [ -2 1 ] |x1|

|x2"| = [ 1 -2 ] | x2|

the eigenvalues are sqrt(3) and 1, corresponding to masses moving opposite each other and together.

In the present notation: we have the two motions stated as

[V1, V2, V1', V2'] and ou tof phase [ V1 -V2, V1', -V2'] which are corresponding solutions or modes

now id X, Y are from the same mode, the sum is a multiple of the mode and is there fore a solution

The difference [(a-b), (b-a), (a-b)',(b-a)'] also correspnds to a solution when X and Y are from the second mode since the signed differences become positive.

The mode shapes are [1,1], [1,-1] in the 2 mass system 2X2 matrix

in terms of displacement and velocity, [ 1,1,1,1], [ 1-,1,1,-1]

iii) the eigenvalues are W^2 =-1,-3

eigenvectors are [1,1] and[1,-1] in the old notation

in the new [1 1 1 1 ], [1-1 1 -1]

GS x1 = e (iWt), X2 = e(iWt) where W are found above , constant to be found from IC and BC

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