Building Design and Construction Handbook - Merritt - Ventech!

�
W 1 2
STRUCTURAL THEORY 5.145
k11 �
g
� k12 k21 k22 � W ��� k1N 2 2 �
g
��� k2N N
.....................................................
k N1 k N2 ���
W
g
k NN � � 2
� �
0 (5.245)
Solution of this equation for � yields one real root for each normal mode. And the
natural period for each normal mode can be obtained from Eq. (5.239).
If � for a normal mode now is substituted in Eqs. (5.244), the amplitudes A 1,
A 2,...,A N for that mode can be computed in terms of an arbitrary value, usually
unity, assigned to one of them. The resulting set of modal amplitudes defines the
characteristic shape for that mode.
The normal modes are mutually orthogonal; that is,
N
� r rn rm
r�1
WA A � 0 (5.246)
where W r is the rth mass out of a total of N, A represents the characteristic amplitude
of a normal mode, and n and m identify any two normal modes. Also, for a
total of S springs
S
� s sn sm
s�1
ky y � 0 (5.247)
where k s is the constant for the sth spring and y represents the spring distortion.
When there are many degrees of freedom, this procedure for analyzing free
vibration becomes very lengthy. In such cases, it may be preferable to solve Eqs.
(5.244) by numerical, trial-and-error procedures, such as the Stodola-Vianello
method. In that method, the solution converges first on the highest or lowest mode.
Then, the other modes are determined by the same procedure after elimination of
one of the equations by use of Eq. (5.246). The procedure requires assumption of
a characteristic shape, a set of amplitudes A r1. These are substituted in one of Eqs.
(5.244) to obtain a first approximation of � 2 . With this value and with A N1 � 1,
the remaining N � 1 equations are solved to obtain a new set of A r1. Then, the
procedure is repeated until assumed and final characteristic amplitudes agree.
Because even this procedure is very lengthy for many degrees of freedom, the
Rayleigh approximate method may be used to compute the fundamental mode. The
frequency obtained by this method, however, may be a little on the high side.
The Rayleigh method also starts with an assumed set of characteristic amplitudes
A r1 and depends for its success on the small error in natural frequency produced
by a relatively large error in the shape assumption. Next, relative inertia forces
acting at each mass are computed: F r � W rA r1/A N1, where A N1 is the assumed
displacement at one of the masses. These forces are applied to the system as a