Steel Designers Manual - TheBestFriend.org

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Steel Designers' Manual - 6th Edition (2003)
Y P
=
K
- Ê È ˆ ˘
Í
Î
Ë ¯
˙ +
˚
Ê
1
Ï
2
Ô
¸
ˆ Ô
Ì 1 2
Ë ¯
˝
ÓÔ
˛Ô
2
2
w
x
w
w
n w n
magnification factor
10
Fundamentals of dynamic behaviour 359
0
— 0.05
— 0.10
— 0.20
(note at resonance in lightly damped
structures magnification = 1
0
frequency ratio
0 1.0 2.0 3.0 W
(12.7)
P/K is the deflection of the structure under a static force P, and the multiplying term
can be regarded as a magnification factor.The variation of magnification factor with
frequency ratio and damping ratio is shown in Fig. 12.4.
When the dynamic force is applied at a frequency much lower than the natural
frequency of the system (w/wn > 1), the response is governed by the mass (inertia) of the structure;
the amplitude is less than the static deflection.
When the dynamic force is applied at a frequency close to the natural frequency,
the stiffness and inertia forces in the vibrating system are almost equal and opposite
at any instant, and the external force is resisted by the damping force. This is
the condition known as resonance, when very large dynamic magnification factors
are possible. When damping is low the maximum steady-state magnification occurs
when w � wn, and then
Y P 1
=
K 2x
Fig. 12.4 Steady-state response to harmonic loads
1
2
(12.8)
2