30 DYNAMICS OF STRUCTURES

FIGURE 2-13Damping ratio vs. number of cycles required toreduce peak amplitude by 50 percent.Damping ratio

No.

cycl

esto

redu

cepe

akam

plitu

deby

50%

6

5

4

3

2

1

00 0.05 0.10 0.15 0.20

For lightly damped systems, greater accuracy in evaluating the damping ratiocan be obtained by considering response peaks which are several cycles apart, say m

cycles; then

lnvn

vn+m=

2mπξ√1 − ξ2

(2-58)

which can be simplified for low damping to an approximate relation equivalent toEq. (2-57):

ξ.=

vn − vn+m

2mπ vn+m(2-59)

When damped free vibrations are observed experimentally, a convenient methodfor estimating the damping ratio is to count the number of cycles required to give a 50percent reduction in amplitude. The relationship to be used in this case is presentedgraphically in Fig. 2-13. As a quick rule of thumb, it is convenient to remember that forpercentages of critical damping equal to 10, 5, and 2.5, the corresponding amplitudesare reduced by 50 percent in approximately one, two, and four cycles, respectively.

Example E2-1. A one-story building is idealized as a rigid girder sup-ported by weightless columns, as shown in Fig. E2-1. In order to evaluate thedynamic properties of this structure, a free-vibration test is made, in which theroof system (rigid girder) is displaced laterally by a hydraulic jack and thensuddenly released. During the jacking operation, it is observed that a force of20 kips [9, 072 kg] is required to displace the girder 0.20 in [0.508 cm]. Afterthe instantaneous release of this initial displacement, the maximum displace-ment on the first return swing is only 0.16 in [0.406 cm] and the period of thisdisplacement cycle is T = 1.40 sec.

From these data, the following dynamic behavioral properties are deter-mined:

ANALYSIS OF FREE VIBRATIONS 31

⎯k2

⎯k2

c

Weight W = mg v

p = jacking force

FIGURE E2-1Vibration test of a simple building.

(1) Effective weight of the girder:

T =2πω

= 2π

√W

g k= 1.40 sec

Hence

W =(1.40

2π

)2

g k = 0.0496200.2

386 = 1, 920 kips [870.9 × 103 kg]

where the acceleration of gravity is taken to be g = 386 in/sec2

(2) Undamped frequency of vibration:

f =1T

=1

1.40= 0.714 Hz

ω = 2πf = 4.48 rad/sec

(3) Damping properties:

Logarithmic decrement: δ = ln0.200.16

= 0.223

Damping ratio: ξ.=

δ

2π= 3.55%

Damping coefficient: c = ξ cc = ξ 2mω = 0.03552(1, 920)

3864.48

= 1.584 kips · sec/in [282.9 kg · sec/cm]

Damped frequency: ωD= ω√

1 − ξ2 = ω(0.999)1/2 .= ω

(4) Amplitude after six cycles:

v6 =(v1

v0

)6

v0 =(4

5

)6

(0.20) = 0.0524 in [0.1331 cm]