# Complex Representation of Free Vibration

by koab1mjr
Tags: complex, free, representation, vibration
 P: 106 Hi all I am struggling with going between various representations of vibrations in paticular the complex form. I am using Rao as my text btw so for a free vibration and making it simple no damping the euqation of motion is $$mx^{..}$$ + kx = 0 with the general solution being x = C1$$e^{iwnt}$$ + C2$$e^{-iwnt}$$ Here is where the confusion starts, I am only suposed to consider the real portion of the solution above and disregard the imaginary. So using the euler identity becomes x = (C1+ C2)cos(wnt) + (C1-C2)isin(wnt) which is Now based on the statement above i would disregard the second piece since its imaginary. but the problem is the book follws up with x = C1'cos(wnt) + C2'sin(wnt) is including the second piece and now considering real. FRom here on I am fine but I am lost on this jump Any help would be much apreciated. Thanks