Mechanical math

Lecture 5

Remember, in Unit 7,for SHM,

i : -crzx Notice that if we can express the matrix equation in the form,

e: Ax : l'x we can then write down the characteristic equation of the dynamic matrix A and find the two distinct real eigenvalues l,r and l,z such that,

0t=Ja and @z=m Note that the eigenvalues of the dynamic matrix of an oscillating mechanical system (without damping and forcing) are always real and Ag@.

We also learnt n Unit 15, that the general solution to this type of equations of motion can be written in the form,

x(/): v 1 (D 1 cos(co 1 r)+D2s in(rrl 1 /))+ v2(D3 cos (rD 2 r)+D4sin(ot 2/))

where, v1 and v2 0,ro the eigenvectors colresponding to the eigenvalues l,r and 1,2 of the dynamic matrix A

and Dr, De, D3 and Da are the arbitrary constants.

The general solution can also be rewritten as,

x(0 : Cr v1 cos(o)r, + 0r) + Czvzcos(co2l + $z)

where, Cu Cz,$r and 0, are the arbitary constants which can be found from known initial conditions.

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