Page 385 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

372 Classical Methods Chap. 12

Let M and K be the mass and stiffness matriees, respectively, and X the
assumed displacement vector for the amplitude of vibration. Then for harmonic
motion, the maximum kinetic and potential energies can be written as
( 12.1- 1)
and
( 12.1-2)
Equating the two and solving for cd^, we obtain the Rayleigh quotient:
2 X'^KX

CO = ----- ----------- (12.1-3)
X^MX
This quotient approaches the lowest natural frequency (or fundamental
frequency) from the high side, and its value is somewhat insensitive to the choice
of the assumed amplitudes. To show these qualities, we express the assumed
displacement curve in terms of the normal modes X^ as follows
X = x^ r- C2X 2 + C,A^3 t • • • (12.1-4)
Then

x ^ K x = x ; k x ^ + CjX^KX^ CjXlKX^
and
X^MX = X MX, CjX^MX, 2 ^ CiX^ MX, ^ •••
where cross terms of the form X- KX- and X^MX- have been eliminated by the
orthogonality conditions.
Noting that
X KX; cofX/MX^ (12.1-5)
the Rayleigh quotient becomes
XjMX,
( 12.1-6)
iü, XjMX^
If X-MXj is normalized to the same number, this equation reduces to

1 + - 1 + ■ (12.1-7)
w,
It is evident, then, that co^ is greater than co] because col/co] > 1. Because C2
represents the deviation of the assumed amplitudes from the exact amplitudes X^,
the error in the computed frequency is only proportional to the square of the
deviation of the assumed amplitudes from their exact values.
This analysis shows that if the exact fundamental deflection (or mode) X^ is
assumed, the fundamental frequency found by this method will be the correct
frequency, because C2, C,, and so on, will then be zero. For any other curve, the
frequency determined will be higher than the fundamental. This fact can be