Civil Insight: A Technical Magazine Volume 3 | Page 36

Bhatt M.R.
Civil Insight (2019) 29-36
It is clear from Fig. 7, Fig. 8, and Fig. 9 that for frequency ratio ߚ equal to zero, R d is equal to one, while
R v and R a are both zero; this shows that displacement is sensitive. For frequency ratio near to one or slightly
less than one, the R d is maximum. When frequency ratio ߚ = 1, R v is sensitive; this shows that the matching
in frequency is not sensitive for all response factors. For frequency ratio more than one, all the dynamic
response factors have some particular values, but in decreasing order. For frequency ratio near to infinity,
meaning forcing frequency is very much greater than the natural frequency of the system, R a is unity, which
implies that acceleration is sensitive.
4) Conclusions and Findings
Idealized single-degree-of-freedom system may represent the real behavior of the practical structures in the
structural dynamics. If the structural system is subjected to the external loading, then it will attain its initial
equilibrium due to its inherent property of damping. From this study, the following conclusions can be
drawn:
x
x
x
x
x
If the system is considered undamped, the amplitude of the deformation response will go on
increasing when the natural frequency is equal to the forcing frequency. However, in practice, it
will not happen.
Practically, if the system is undamped and amplitude goes on increasing for longer time, then there
will be modification of the natural frequency of the system either by brittle failure (separation of
some components of the system that cause modification of mass) or by ductile failure (yielding of
the components of the system that cause modification of the system stiffness).
For the damped harmonic system, if the frequency of the force is equal to the natural frequency of
the system, the amplitude goes on increasing up to some particular maximum amplitude, and it
remains constant for higher time periods. The response is strongly governed by the damping
properties considered for the system.
When the forcing frequency resembles the natural frequency, it is only then the velocity response
is sensitive. For frequency ratioߚ near to zero, the displacement response factor is sensitive; when
the frequency ratio is more than zero, the velocity response factor is sensitive, and for frequency
ratio near to infinity (theoretically) or frequency ratio much more than one, the acceleration
response factor is sensitive.
Deformation response of the system is controlled by stiffness, mass, and damping when the
frequency ratio ߚ is near to zero, near to one, and much more than one respectively.
References
Chopra, A.K. (2007). Dynamics of Structures: Theory and Applications to Earthquake Engineering. Noida,
India: Pearson Education Inc. ISBN 978-81-317-1329-7.
Clough, R.W., & Penzien, J. (2003). Dynamics of Structures. Berkeley, USA: McGraw-Hill Education.
Paz, M. & Leigh, W. (2004). Structural Dynamics Theory and Computation. Massachusetts, USA: Kluwer
Academic Publisher. ISBN 1-4020-7667-3.
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