Civil Insight: A Technical Magazine Volume 3 | Page 30

Bhatt M.R.
Civil Insight (2019) 29-36
energy to the system. This system may be considered the idealization of the one-story structure, where each
structural member, such as beam, column, and wall, contributes to the inertial (mass), elastic (stiffness or
flexibility) and energy dissipation (damping) properties of the structure. If the system in Fig. 1 is acted by
an external force ‫݌‬ሺ‫ݐ‬ሻ in lateral direction with a relative displacement ‫ݑ‬ between the mass and the base of
the structure or ground, it may be called the SDOF system. From the equilibrium of the forces, the equation
of the motion can be expressed as Eqn (1).
‫݌‬ሺ‫ݐ‬ሻ ൌ ݂ሺ‫ܫ‬ሻ ൅ ݂ሺ‫ܦ‬ሻ ൅ ݂ሺܵሻሺͳሻ
where ‫݌‬ሺ‫ݐ‬ሻ is applied force, ݂ሺ‫ܫ‬ሻis inertial force, ݂ሺ‫ܦ‬ሻis damping force and ݂ሺܵሻis spring force.
Using D’Alembert’s principle and assuming the inertial mass of the system as ݉, linear viscous damping
coefficient as ܿ and system linear lateral stiffness coefficient as ݇, the equation of motion can be expressed
as Eqn (2) (Clough and Penzien, 2003).
‫݌‬ሺ‫ݐ‬ሻ ൌ ݉‫ݑ‬ሷሺ‫ݐ‬ሻ ൅ ܿ‫ݑ‬ሶ ሺ‫ݐ‬ሻ ൅ ݇‫ݑ‬ሺ‫ݐ‬ሻሺʹሻ
where‫ݑ‬ሷሺ‫ݐ‬ሻ, ‫ݑ‬ሶ ሺ‫ݐ‬ሻ and ‫ݑ‬ሺ‫ݐ‬ሻ are acceleration, velocity and displacement of the center of the mass of the
system in the direction of applied force respectively.
Fig. 1. Single-degree-of-freedom system: (a) applied force ‫݌‬ሺ‫ݐ‬ሻand (b) Earthquake-induced ground
motion (‫ݑ‬ ௚ ) (Chopra, 2007)
1.2) Free and forced vibrations
If the system is disturbed from its initial equilibrium and then allowed to vibrate without any external
dynamic excitation, it is known as free vibration. However, in the real field, some sort of the determinant
or probabilistic loads may act upon the structure in time-dependent nature, and the resulting vibration is
known as forced vibration. External force may be periodic and non-periodic in nature, for example a rotating
machine represents simple periodic loading, while earthquake comes under non-periodic loading.
2) Methodology and Assumptions
The external excitation assumed is simple harmonic sinusoidal loading with loading frequency as ߱ in
rad/sec and amplitude of the force as ‫݌‬, as expressed in Eqn (3).
‫݌‬ሺ‫ݐ‬ሻ ൌ ‫݌‬ ‫݊݅ݏ‬ሺ߱‫ݐ‬ሻሺ͵ሻ
2.1) Response of undamped system
If the undamped 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 (4), which is expressed below
(Chopra, 2007; Clough & Penzien, 2003).
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