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Steel Designers' Manual - 6th Edition (2003)
360 Applicable dynamics
Since in many structural systems x is of the order of 0.01, magnification factors
of the order of 50 may result. The force in the structure is proportional to the displacement
so the same magnification factor applies to structural forces.
Human perception of motion is usually related to acceleration levels rather than
displacement. The peak acceleration amplitude at steady state is given by
2
â = w Y
2
(12.9)
= ( 2p
f) Y
In practice, there is usually advantage in avoiding the possibility of resonance
whenever possible by ensuring that structural frequencies are well away from the
frequencies of any known sources of substantial dynamic force.
12.2.5 Response to an impact
Another dynamic loading case of interest is the response of a structure to an impact,
say from an object falling on to the structure. A full discussion of impact loading is
given in Reference 1, but a simple approximate method is useful for many practical
situations when the mass of the impacting object is small compared with the
mass of the structure, and the impact duration is short compared with the natural
period of the structure. In these cases the effect of the impact can be assessed as an
impulse I acting on the structure. The magnitude of I may be calculated as m Dv,
where m is the mass of the falling object and Dv its change in velocity at impact.
If there is no rebound Dv can be taken as the approach velocity. For the simple
system discussed in previous sections conservation of momentum at impact requires
the initial velocity of the structural mass to be I/M. A lightly damped system then
displays damped free vibration corresponding to an initial displacement amplitude
of approximately
I I
Y = =
w M 2pfM
n n
12.2.6 Response to base motion
(12.10)
The previous sections have illustrated the behaviour of a dynamic system with a
fixed base subject to applied forces. When the dynamic excitation takes the form of
base motion, as for example in an earthquake, the formulation of the equation
of motion and the solutions are slightly modified. Detailed treatment of this type of
excitation is beyond the scope of this chapter and References 1 and 2 are recommended
for discussion of the solutions.
Although not correct in detail, the general form of response indicated by Fig. 12.4
for harmonic applied forces is still relevant. Resonance occurs for those components
of the base motion close in frequency to the natural frequency of the structure.