Mechanical math

I.€c.ture 5

As can be seen, the general solution x(0 is a linear combination of linearly independent normal modes of the system. (Theorem 7.7, pg 12)

Go through Procedure 7.1, pg 14 ano'Analysing oscillating mechanical systems".

Coqrditiou$Jor qor{nal moSes to occur i.e. both particles must oscillate sinusoidally with same (0.

. For systems starting from rest i.e. ;L(0) = 0 (pS 19) the two particles must be given displacements equal to the normal mode eigenvector.

. For systems sturting with particles at origin (pS 23) i.e. x1(0) : xz(0) : 0. the two particles must be given initial velocities equal to the normal mode eigenvector.

If the above conditions are not satisfied, then the two particles will exhibit motions that are not in a sinusoidal manner and with different angular frequencies as shown in the two displacement responses, xa and xs below.

An efample.of nor-slnfso#a/ responpe from the 2 padicles

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