If, for example, a spring-mounting system deflects statically 7 mm under the load it supports, the theoretical value for natural frequency of the system is 15.8/7 Hz-6 Hz. If the static deflection were four times the previous value, the natural frequency would be halved.
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Simple formulae are available for the determination of fundamental na- tural frequencies of beams for various end-fixing conditions and types of loading. For a simply supported beam centrally loaded, the expression given in the previous paragraph is applicable; thus, if the deflection at the centre is 7 mm, fn=6 Hz. If the same deflection were given for uniform loading, how- ever, the natural frequency would be higher about 6.7 Hz in this case. Forced vibration is produced in an elastic system, when it is affected by some external source of vibrational energy. Then the system will vibrate at the same frequency as that of the source of vibration (the 'forcing fre- quency') and not at its natural fre- quency. For example, a beam may have a natural frequency of 6 Hz. If it is affected by an electric motor pro- ducing vibration of frequency 50 Hz then it will be caused to vibrate at a frequency of 50 Hz.
Table 1 Calculation of Dieckmann K-values
Vertical vibrations
Below 5 Hz:
5-40 Hz:
Above 40 Hz:
K = 0.001 A.ƒ2 K = 0.005 A.ƒ K = 0,2 A
Horizontal vibrations
Below 2 Hz: K = 0.002 A.ƒ2 2-25 Hz: K = 0.004 A.ƒ Above 25 Hz: K = 0.1 A
(A being amplitude in microns and ƒ the frequency in Hz)
is still accepted for steady-state vibra- tions, but for transient vibrations, e.g. floor vibrations produced by people walking, there is recent evidence that amplitudes much greater than those given by the scale are necessary to produce a given sensation at a given frequency.
The movement (or response) of a system affected by vibration will de- pend on the relationship between the forcing frequency and the natural fre- quency and also on the degree of damping present. In the special case where the frequencies coincide, reson- ance occurs and a significant increase in amplitude and force can result.
In the Reiher-Meister investigation it was noted that vertical vibration was most readily detected when people were standing, whereas horizontal vibration was more noticeable when they were lying down. The sensation produced depends on frequency and amplitude. An amplitude of 100μ constitutes an annoying vibration if the frequency is 5 Hz and a painful vibration if the frequency exceeds 20 Hz. An amplitude of 10u is just per- ceptible at 5 Hz, but would be annoy- ing at 50 Hz.
At this critical stage even a small change in the frequency of the excita tion will usually result in a noticeable diminution of response. Although structural materials do not generally possess a great degree of intrinsic damping, there is usually sufficient to keep amplification at resonance within reasonable bounds. The small amount of damping does not greatly affect re- sponse other than at, or near, reasonance condition however.
Human sensitivity to vibration
the
the
ing the effect on people of building sway in high winds. In this scale, zones of equal intensity are defined and are given K-values ranging from 0.1 to
Expressed in terms of peak velo- city, the threshold of perception cor- responds to a velocity of 0.3 mm/sec and a vibration is annoying if the velo- city exceeds 2.5 mm/sec. The data for vertical vibration (persons standing) are given in Fig. 1.
100, the effects of various intensities being given as follows:
K=0.1; lower limit of perception K=1; allowable in industry for any period of time
More recent data are provided by the Dieckmann scale, which estends into lower ranges of frequency and, hence, may be useful also in determin-
100
50
frequency (Hz)
30
20
05
6
\K=0·1
K 0.5
6 8
K=1
K=10; allowable only for a short time
K=100; upper limit of strain for the average man.
The charts for vertical vibrations and horizontal vibrations are given in Figs. 2 and 3. The Dieckmann K-values for vibrations of given amplitudes and frequencies can be simply calculated,
as shown in Table 1.
Two examples illustrate the calcu- lation of vibration intensity and the effect on people. In the case of a very severe building vibration, due to the operation of reciprocating machines at high level, the amplitude of horizontal vibration was 750μ and the frequency 4 Hz. The Dieckmann K-value is 12 and the Reiher-Meister classification is annoying to unpleasant. A typical re- sult for vertical vibration for street traffic on the other hand is an ampli- tude of 10μ at frequency 20 Hz. This
K=5
Under certain conditions human body can detect amplitudes as small as one micron, whilst amplitudes of the order 0.05μ can be detected with the finger-tips. The basic data concerning 'wholebody' sensitivity to vibration are provided by the. Reiher- Meister scale. Although this was de- veloped over 30 years ago, its validity Fig. 2. Dieckmann values: vertical vibrations
Far East BUILDER, November 1970
0.1
10
6
K=10
K-50
K=100
103
10*4
amplitude (u)
2
B
31
Page 35Page 36
gives a strength of K=1 on the Dieck mann scale and is graded as clearly per- ceptible on the Reiher-Meister scale.
Vibration and damage to buildings
a
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