Amplification ratio for forced,damped motion

[John 123]

Hi there
I am new to this forum but I am a regular contributor to the SOS maths forum.
I am working my way through a book on Ordinary Differential Equations and this book defines
the
Amplification Ratio,M, of the system under the above motion as:
<br /> M=\frac{\omega_0^2}{\sqrt{(\omega_0^2-\omega^2)^2+(2r\omega)^2}}<br />
I then solved the following problem in the book.
For what value of
<br /> \omega<br />
will the amplification ratio be a maximum? Find this maximum value?
MY ANSWER
I found the value of
<br /> \omega<br />
For which
<br /> \frac{dM}{d\omega}=0<br />
This value is
<br /> \omega=\sqrt{\omega_0^2-2r^2}<br />
which agrees with the book.
Then when you substitute this value into the formula for M you get
<br /> M_{max}=\frac{\omega_0^2}{2r\sqrt{\omega_0^2-r^2}}<br />
However the book answer gives:
<br /> M_{max}=\frac{1}{2r\sqrt{\omega_0^2-r^2}}<br />
The numerator has become 1 but their defintion gives
<br /> \omega_0^2<br />
in the numerator?
Best regards
John

 

[Thaakisfox]

It must be a typo mistake in the book, it happens quite a few times..
Even after 3-4 editions there might be mistakes in it.

 

[John 123]

Hi Thaakisfox
Do you mean that the correct definition is with the numerator = 1?
Regards
John

 

[John 123]

Hi again
The book quite categorically states:
The amplification ratio,M, as:
<br /> M=\frac{Amplitude of steady state output function}{\frac{Amplitude of input function}{\omega_0^2}}<br />
and so
<br /> M=\frac{\omega_0^2}{\sqrt{(\omega_0^2-\omega^2)+(2r\omega)^2}}<br />
Then an example is calculated where
<br /> F=40:<br /> Amplitude of steady state motion=\sqrt5:<br /> \omega_0^2=12<br />
Then
<br /> M=\frac{\sqrt5}{\frac{40}{12}}=\frac{3\sqrt5}{10}<br />
I am confused.com!
John

 

[Thaakisfox]

The thing is, he just writes it into the definition.

The answer for the M_max the book gave, is a typo, there should be \omega_0^2 in the numerator not 1. (you can also check the units, the units of the M differ from that of M_max)

 

[John 123]

Many Thanks Thakiisfox
I came to the same conclusion myself as I have seen the same formula in a book on Structural Engineering. Thus the typing errors must be in the answers[there are two answers that have 1 in the numerator and another two with the correct omega(0) squared in the numerator].
Regards
John