Page 47 - Dynamic Loading and Design of Structures

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Figure 2.3 SDOF oscillator.

With reference to Figure 2.3, the equation of motion of the SDOF oscillator is

(2.1)

implying that the inertia, damping and restoring forces balance the applied force. Specifically,
M is the mass (kg), k is the stiffness (N/m), and c is the damping coefficient (N-sec/m).
Furthermore, y(t) is the displacement (m), ÿ(t) the velocity (m/ sec), ÿ(t) the acceleration
2
(m/sec ), F(t)=F1f(t) the externally applied force (N) withf(t) its dimensionless time variation.
Finally, dots denote time derivatives d/dt. Obviously, eqn (2.1) is a second order differential
equation that needs to be solved for the displacement y(t).

2.2.1 Motion without damping

2.2.1.1 Free vibrations
The equation of dynamic equilibrium of an SDOF system in the absence of both damping and
external force is given below as

(2.2)

Thus, the oscillator undergoes free vibrations under the influence of an initial displacement
y(0)=y and/or initial velocity ÿ(0)=ÿ . The solution is simply
0
0

(2.3)

and implies a periodic, harmonic motion as shown in Figure 2.4. At this point, we respectively
define the circular frequency, the natural period and the frequency as