Civil Insight: A Technical Magazine Volume 3 | Page 31

Bhatt M.R.
Civil Insight (2019) 29-36
‫ݑ‬ሺ‫ݐ‬ሻ ൌ ቈ‫ݑ‬ሺͲሻ …‘•ሺ߱ ௡ ‫ݐ‬ሻ ൅ •‹ሺ߱ ௡ ‫ݐ‬ሻ ቊ
‫݊݅ݏ݌‬ሺ߱‫ݐ‬ሻ
‫ߚ݌‬
‫ݑ‬ሶ ሺͲሻ
െ
ሺͶሻ
ቋ቉ ൅
ଶ
݇ሺͳ െ ߚ ଶ ሻ
݇ሺͳ െ ߚ ሻ
߱ ௡
where‫ݑ‬ሺͲሻ is initial displacement of the system, ‫ݑ‬ሶ ሺͲሻ is initial velocity of the system, ߱ ௡ is the natural
frequency of the system, and ߚ is the ratio of forcing frequency to the natural frequency, mathematically
ఠ
ߚൌ
. Eqn (4) is the combination of steady state response and transient response of the system, where
߱
݊
transient state is related to the initial conditions of the system while the steady state response is related to
the nature of external excitation.
2.1.1) Response of undamped system when ߚ = 1
The solution of Eqn (4) yields to Eqn (5) for the case of similar natural and forcing frequency and
considering zero initial condition, meaning the static case (Chopra, 2007; Clough & Penzien, 2003).
‫ݑ‬ሺ‫ݐ‬ሻ ൌ െ
‫݌‬
ሾ߱ ‫ݐ‬ …‘•ሺ߱ ௡ ‫ݐ‬ሻ െ  •‹ሺ߱ ௡ ‫ݐ‬ሻሿሺͷሻ
ʹ݇ ௡
2.2) Response of damped system
If the viscous damped SDOF system shown in Fig. 1 is considered to be acted by an external harmonic
excitation Eqn (3), then the solution of the equation of the motion Eqn (2) yields to Eqn (6) (Chopra, 2007;
Clough & Penzien, 2003).
‫ݑ‬ሺ‫ݐ‬ሻ ൌ ݁
ିకఠ ೙ ௧ ሺ‫߱ݏ݋ܿܣ‬
‫݌‬ሼሺͳ െ ߚ ଶ ሻ •‹ ߱‫ݐ‬ െ ʹߦߚ …‘•ሺ߱‫ݐ‬ሻሽ
ሺ͸ሻ
ௗ ‫ݐ‬ ൅ ‫߱݊݅ݏܤ‬ ௗ ‫ݐ‬ሻ ൅
݇ሼሺͳ െ ߚ ଶ ሻ ൅ ሺʹߦߚሻ ଶ ሽ
where ‫ܣ‬ and ‫ܤ‬ can be found by adopting initial conditions, ߱ ௗ is damped frequency, which
depends on the damping coefficient ratio of critical damping ߦ.
2.2.1) Response of damped system when ߚ = 1
The solution of Eqn (6) yields to Eqn (7) for the case of same natural and forcing frequency and considering
zero initial condition (Chopra, 2007; Clough & Penzien, 2003).
‫ݑ‬ሺ‫ݐ‬ሻ ൌ
‫݌‬
ߦ•‹ሺ߱ ௗ ‫ݐ‬ሻ
ቈ݁ ିకఠ ೙ ௧ ቆܿ‫߱ݏ݋‬ ௗ ‫ݐ‬ ൅
ቇ െ ܿ‫߱ݏ݋‬ ௗ ‫ݐ‬቉ሺ͹ሻ
ʹߦ݇
ඥͳ െ ߦ ଶ
2.3) Dynamic response factors for harmonic excitation
The ratio of maximum ampliture in dynamic state to the maximum static amplitude, known as the
deformation response factor R d using Eqn (6) can be represented in the form of Eqn (8).
ܴ ௗ  ൌ
ͳ
ඥሾͳ െ ߚ ଶ ሿ ଶ ൅ ሾʹߦߚሿ ଶ
ሺͺሻ
The velocity and accelertion dynamic response factors yield to be R v = ߚR d and R a = ߚ ଶ R d respectively.
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