Can someone show me how they got from 1 to 2 (I have marked in red).
4.12 Determine the equation of motion in matrix form, then calculate the natural frequencies and mode shapes of the torsional system of Figure P4.12. Assume that the torsional stiffness values provided by the shaft are equal (k, = k, ) and that disk 1 has three times
the inertia as that of disk 2(J, = 3J,)
Flgure 4.12 Torsional system with two disks and hence two degrees of freedom
Solution: Let k = k, = k, and J, = 3.J,. The equations of motion are J0+2ko-ko,=0 J0-kO+ko=0 So, .[ ;] Calculate the natural frequencies: 32J, +2k det-J+K= -6
k , = 0.482, J K
Calculate the mode shapes: mode shape 1:
30.2324k+2k
=0.7676
0.7676
mode shape 2:
=0.434i
o.u