Page 62 - Dynamic Loading and Design of Structures

Page 43

Figure 2.13 Free vibration in the presence of Coulomb damping.

We observe that for every complete cycle of oscillation (t=T), the total dynamic displacement
y(t) reduces by an amount equal to 4Ff/k until all motion ceases.

2.2.2.4 Damped harmonic vibrations
The equation of motion for this case is

(2.41)

and the part of the solution which corresponds to forced vibration with frequency is

(2.42)

i
where γs a phase angle. We mention here that the free vibration part with frequency ω;
dampens out rather quickly, hence it can be ignored. Since the maximum value of the sine is
unity and the static displacement is the maximum value the DLF attains is

(2.43)

We observe that the amplitude of the vibrations is no longer infinite during resonance
when there was no damping. Specifically, we have that

(2.44)

Figure 2.14 plots the maximum value of the DLF as a function of the ratio . We observe
that when the DLF approaches the static value, while as