Building Design and Construction Handbook - Merritt - Ventech!

5.144 SECTION FIVE
�
1 1 kg
ƒ � � (5.240)
T 2� W
If, at time t � 0, the mass has an initial displacement y 0 and velocity v 0, substitution
in Eq. (5.237) yields A � v 0/� and B � y 0. Hence, at any time t, the mass
is completely located by
v 0
y � sin �t � y0 cos �t (5.241)
�
The stress in the spring can be computed from the displacement y.
Vibrations of Lumped Masses. In multiple-degree systems, an independent differential
equation of motion can be written for each degree of freedom. Thus, in
an N-degree system with N masses, weighing W 1, W 2,...,W N lb, and N 2 springs
with constants k rj (r � 1,2,...,N; j � 1,2,...,N), there are N equations of
the form
2 W d y
N
� rj j
r r
� k y � 0 r � 1,2,...,N (5.242)
2 g dt j�1
Simultaneous solution of these equations reveals that the motion of each mass can
be resolved into N harmonic components. They are called the fundamental, second
third, etc., harmonics. Each set of harmonics for all the masses is called a normal
mode of vibration.
There are as many normal modes in a system as degrees of freedom. Under
certain circumstances, the system could vibrate freely in any one of these modes.
During any such vibration, the ratio of displacement of any two of the masses
remains constant. Hence, the solution of Eqs. (5.242) take the form
N
r � rn n n
n�1
y � a sin � (t � � ) (5.243)
where a rn and � n are constants to be determined from the initial conditions of the
system and � n is the natural circular frequency for each normal mode.
To determine � n, set y 1 � A 1 sin �t; y 2 � A 2 sin �t ....Then, substitute these
values of y r and their second derivatives in Eqs. (5.242). After dividing each equation
by sin �t, the following N equations result:
W1 2 � 11 � 1 12 2 1N N
k � � A � k A � ���� k A � 0
g
W2 k21A1��k22 � � A2 � ���� k2NAN � 0
g
.................................................
WN 2 � �
kN1A1 � kN2A2 � ���� kNN � � AN � 0
g
(5.244)
If there are to be nontrivial solutions for the amplitudes A 1, A 2,...,A N, the
determinant of their coefficients must be zero. Thus,