Complex Representation of Free Vibration

[koab1mjr]

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

 

[tiny-tim]

hi koab1mjr!

(have an omega: ω and try using the X² and X₂ icons just above the Reply box )

i'm not familiar with the use of complex numbers in this way, but i think the original C₁ and C₂ are allowed to be complex, so when you take the real parts you do get a combination of cos and sin (alternatively, you get a phase)

 

[AlephZero]

What you really have is

x = Real part of {(C1+ C2)cos(wnt) + (C1-C2)isin(wnt)}

where C1 and C2 are complex. If you unpick that expression, it amounts to

x = A cos (wnt) + B sin (wnt)

where A and B are real constants. That can be written in a simpler form using complex numbers, namely

x = Real part of {C exp(iwnt)}

where C is a complex constant.

This still represents the complete solution to the differential equation, with two indepedent arbitrary (real) constants, namely the real and imaginary parts of C.

You can then write
x' = Real part of {iwC exp(iwmt)}
etc

However, the "Real part of" is just assumed almost all the time, except in situations where you need to be explicit about exactly what real part you mean. So you would normally just write

x = C exp(iwnt)
x' = iwC exp(iwmt)
etc.

As a general principle, don't try to "discard the imaginary parts" too soon in the math. It is usally simpler to keep all the math in complex variables, and only take the real part at the end to relate the math back to the "real world".