Page 5 - Combined_41_OCR
P. 5
it damping were increased (lower pat
tern) , motion would decrease more
rapidly.
The Language of Vibrations
Forced Vibration: If the top support
of the idealized system vibrates with
a constant D over some range of Frequency f is the number of oc For any wave form, there are cer
frequency, how much vibration will currences of an event in a unit of tain fundamental measurements de
occur at weight W? The answer de time. If a motion completes one scribing points on the curve. For
pends upon: 1. the frequency of “in cycle in 0.1 second (its period, T), undistorted sinusoidal motion, rms
put” vibration (the forcing function), its frequency is 10 cycles per second, (root-mean-square) value = 0.707
and 2. the natural frequency and (or hertz, abbreviated Hz). X peak value (or peak = 1.414 X
damping of the system. A mass M suspended on a spring, rms); average value — 0.637 X peak.
when disturbed by a vertical force Which quantities are usually meas
Example: If the system has a passing through the center of gravity, ured? Not instantaneous values x, v,
natural frequency of 1 Hz, with an will oscillate sinusoidally. or a. Usually, the maximum values,
input frequency of 0.1 Hz, (ratio of X, V, or A are required. In practi
input/output frequencies = 1/10), W cal use, the working value for dis
has about the same amplitude as the placement is not amplitude X, but the
input. In other words, transmissibil / / / peak-to-peak value, double amplitude
ity, or the ratio of output vibration Displacement, x D, often called DA.
divided by input vibration, is 1/1 For the rare case where motion is
= 1, as shown on the chart below. — W M— I------- 1 LUJ sinusoidal, V = 'zrfD. A sample
+ — M calculation; let frequency be 100 Hz
and D be 0.1 in. V = w (100) (0.1)
= 31.4 in./sec. If D remains the
tave), V also doubles. If it is neces
sary to keep V constant while f
T Efficiency (%) *1* Amplification* l I— same but frequency doubles (one oc
Transmissibility, Acceleration, a + i 0 360 For nonsinusoidal motion, exact cal
doubles, D must be reduced by half.
culation of V and A is much more
difficult, but general relationships are
similar.
For sinusoidal motion, A = 2fir2f2D.
| 90
270
If the example values given above are
< Isolation '=0 Time (deg) 2tf2(100)2(0.1) = 19,800 in./sec2. If
used, A can be calculated: A =
D remains constant and frequency
Instantaneous position or displace
ment x of the mass, as a function doubles, acceleration quadruples.
of time, is shown in the top graph. Now a question: Is 19,800 in./sec2
Velocity v, as a function of time, is a large or a small acceleration? A
As vibration frequency rises, ampli a plot of the instantaneous slope of “feel” for this unit of acceleration,
tude of W increases. The amount de the displacement curve. Acceleration in./sec2, is easily acquired: if 19,800
pends upon the amount of damping a, in turn, is a plot of the instan in./sec2 is compared with the accelera
in the system. Assume the system is taneous slope of the velocity curve. tion due to gravity, 386 in./sec2, A
very lightly damped. When the in is found to be about 51.3 gravitational
put vibration reaches 1 Hz, weight W units or g units.
has an amplitude about 10 times It is important that the units of
greater than the input amplitude. At displacement values be explicitly
this maximum-response frequency, stated. Common units are inches
resonance exists; the forcing frequency (double amplitude, D), inches per
equals the natural frequency of the second (peak velocity, V), and g
system. As the forcing frequency is units (peak acceleration, A).
further increased, the response ampli
tude decreases to much less than the
input. Transmissibility is reduced to
about 0.1 at a forcing frequency of
3 Hz.
The region above 1.41 fN (where
transmissibility is less than 1) is
called the region of isolation. Weight
W has less vibration than the input;
therefore, it is isolated from the vi Multiple Degrees of Freedom and have many possible motions. For
bration of the support. This illus Continuous Systems: The idealized tunately, only a few are likely to oc
trates the potential for use of vibra system is highly simplified. Real sys cur in any range of frequencies that
tion isolators, which are rubber ele tems can be more accurately repre might reasonably be expected from
ments or springs that reduce the vibra sented by diagrams showing many rotating machinery.
tion input to the isolated unit. Prop masses, many springs, and many Whenever the frequency of a vibra
er selection of isolators can set the dampers, with many natural fre tory force coincides with one of the
natural frequency considerably below quencies. Transmissibility plots of natural frequencies of a structure,
the expected frequency of the forcing real systems will show many peaks resonance exists; high stresses, forces,
vibration, thus reducing the vibration and many notches, rather than the and motions may result. The figure
amplitude of the unit. simple plot shown. Complex systems Continued
May 29, 1969 119
