Building Design and Construction Handbook - Merritt - Ventech!

5.146 SECTION FIVE
static load and displacements B r1 due to them calculated. Then, the natural frequency
can be obtained from
N
r r1
r1
N
2 WB r r1
r1
g F B
2 (5.248)
where g is the acceleration due to gravity, 386 in/s 2 . For greater accuracy, the
computation can be repeated with B r1 as the assumed characteristic amplitudes.
When the Rayleigh method is applied to beams, the characteristic shape assumed
initially may be chosen conveniently as the deflection curve for static loading.
The Rayleigh method may be extended to determination of higher modes by the
Schmidt orthogonalization procedure, which adjusts assumed deflection curves to
satisfy Eq. (5.246). The procedure is to assume a shape, remove components associated
with lower modes, then use the Rayleigh method for the residual deflection
curve. The computation will converge on the next higher mode. The method is
shorter than the Stodola-Vianello procedure when only a few modes are needed.
For example, suppose the characteristic amplitudes A r1 for the fundamental mode
have been obtained and the natural frequency for the second mode is to be computed.
Assume a value for the relative deflection of the rth mass A r2. Then, the
shape with the fundamental mode removed will be defined by the displacements
where c 1 is the participation factor for the first mode.
a A cA (5.249)
r2 r2 1 r1
N
WA r r2Ar1 r1
1 N
2 WA r r1
r1
c (5.250)
Substitute a r2 for B r1 in Eq. (5.248) to find the second-mode frequency and, from
deflections produced by F r W ra r2, an improved shape. (For more rapid covergence,
A r2 should be selected to make c 1 small.) The procedure should be repeated,
starting with the new shape.
For the third mode, assume deflections A r3 and remove the first two modes:
The participation factors are determined from
A A cA cA (5.251)
r3 r3 1 r1 2 r2
N N
WA r r3Ar1 WA r r3Ar2 r1 r1
1 N 2 N
2 2
WA r r1 WA r r2
r1 r1
c c (5.252)
Use a r3 to find an improved shape and the third-mode frequency.
Vibrations of Distributed Masses. For some structures with mass distributed
throughout, it sometimes is easier to solve the dynamic equations based on distributed
mass than the equations based on equivalent lumped masses. A distributed