Types of Damping: Damping can be classified into the following types: I. Viscous damping II. Hysteretic or Material Damping III. Dry Friction of Coulomb Damping IV. Damping using Electromagnetic Fields Refer to section 3.7, page 70, for more information.
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Types of Damping:
� Damping can be classified into the following types:
I. Viscous damping
II. Hysteretic or Material Damping
III. Dry Friction of Coulomb Damping
IV. Damping using Electromagnetic Fields
� Refer to section 3.7, page 70, for more information.
� The particular integral ��(�) under the excitation force
� � = �����in Eq.2.1 is of the form
��(�) = �(�� − ∅) Eq.2.17
� It can be shown that the amplitude �of the steady-state
response is
=�
����� ��(��)�Eq.2.20
� Dividing both sides by �
=�
��
�������
��(�
�! )�Eq.2.21
and ∅ = tan����
�����or ∅ = tan��
�� �!
����� �!Eq.2.22
� is the amplitude of the steady-state response and −∅ is
the phase angle of ��(�) relative to the excitation ����.
� The last two equations can be written using �%& =
�
�,
��
�= 2ξ� �%! and ) =
�
�*as
+�
��=
+
+,=
�
��-� ��(&.-)�= / Eq.2.23
and ∅ = tan��&.-
��-�, Eq.2.24
where / is called the magnification factor and ) the frequency
ratio of the excitation frequency � to the natural frequency
�% of the system.
� Eq.2.23 and 2.24 are plotted below with damping factor ξ�as
a parameter.
Fig. Phase Angle Φ Versus
Frequency Ratio r
Fig. Magnification Factor K Versus
Frequency Ratio r
Note:
See comments on these 2 diagrams given in
Textbook of “Machine Vibration Analysis” by
Prof. Dr. Abdul Mannan Fareed
1st Edition, 2007
� The general solution of Eq. 2.1 represents the system
response to a harmonic excitation and the initial conditions.
Substituting Eq.2.16 and 2.19 into Eq.2.2, the general
solution becomes
� � = ��(�)+ ��(�)
� � = 01�.�*2 sin �5� + 7 + �(�� − ∅), Eq.2.25
where =�
����� ��(��)�and ∅ are calculated from
Eq.2.23 and 2.24.
� Find the transient response and the steady-state
response of the system in Example 1, if the excitation