Building Design and Construction Handbook - Merritt - Ventech!

5.158 SECTION FIVE
2W� 2W� kW
c � � � 2 (5.291)
g g g
d �
Damping sometimes is expressed as a percent of critical (� as a percent of �).
For small amounts of viscous damping, the damped natural frequency is approximately
equal to the undamped natural frequency minus 1 ⁄2� 2 /�. For example,
for 10% critical damping (� � 0.1�), � d � �[1 � 1 ⁄2(0.1) 2 ] � 0.995�. Hence, the
decrease in natural frequency due to small amounts of damping generally can be
ignored.
Damping sometimes is measured by logarithmic decrement, the logarithm of
the ratio of two consecutive peak amplitudes during free vibration.
2��
Logarithmic decrement � (5.292)
�
For example, for 10% critical damping, the logarithmic decrement equals 0.2�.
0.2�
Hence, the ratio of a peak to the following peak amplitude is e � 1.87.
The complete solution of Eq. (5.288) with initial displacement y o and velocity
v o is
� d o d�
vo � �y
��t
o
y � e sin � t � y cos � t
� d
2 �
t
��(t��)
�d 0
� e� � ƒ(�)e sin � (t � �) d� (5.293)
where e� is the deflection that the applied force would produce under static loading.
Equation (5.293) is identical to Eq. (5.274) when � � 0.
Unbalanced rotating parts of machines produce pulsating forces that may be
represented by functions of the form F o sin �t. If such a force is applied to an
undamped one-degree system. Eq. (5.274) indicates that if the system starts at rest
the response will be
� �� �
Fg o
2 1/� �
2 2
y � sin �t � sin �t (5.294)
W 1 � � /� �
And since the static deflection would be F o/k � F og/W� 2 , the dynamic load factor
is
� �
1 �
D � sin �t � sin �t (5.295)
2 2 1 � � /� �
If � is small relative to �, maximum D is nearly unity; thus, the system is practically
statically loaded. If � is very large compared with �, D is very small; thus, the
mass cannot follow the rapid fluctuations in load and remains practically stationary.
Therefore, when � differs appreciably from �, the effects of unbalanced rotating
parts are not too serious. But if � � �, resonance occurs; D increases with time.
Hence, to prevent structural damage, measures must be taken to correct the unbalanced
parts to change �, or to change the natural frequency of the vibrating mass,
or damping must be provided.
The response as given by Eq. (5.294) consists of two parts, the free vibration
and the forced part. When damping is present, the free vibration is of the form of
d