T1-20
yC.G.
xC.G.
y1
x1
LC1
y
LC2
y2
x
y3
LC3
C.G.
x3
x2
Figure 29 Setup for Experimental
Finding of the C.G. Location
The coupling concept can be illustrated on the example of a simpler "pla-
nar" system shown in Figure 28, which shows a mass supported by springs and
constrained so that it can move only in the plane of the drawing [5]. Such a
system has three coordinates which fully describe its configuration: translational
coordinates x and y, and angular coordinate 3. If the system is symmetrical
about axis y, then when excited by a sinusoidal force Fy, in the vertical direction
along the axis of symmetry, the object will behave as previously shown (Figure
1), namely by vibrating in the vertical (y) direction. However, if the force vector
does not coincide with the axis of symmetry, then the vertical force would excite
vibratory motions not only in the y-direction, but also in x and 3 directions. When
the mass is excited by a horizontal force Fx, both horizontal (y) or longitudinal
mode and pitching (3) vibratory motions are excited. These modes are said to
be coupled when vibrations of one mode can be stimulated by a vibratory force
or displacement in another. Coupling modes are in most cases undesirable. For
example, many vibration-sensitive objects have the highest vibration sensitivity
in a horizontal direction, while the floor vibrations are often more intense in the
vertical direction. Coupling between the vertical and horizontal directions can
be avoided by using vibration isolating mounts at each mounting point whose
stiffness is proportional to the weight load acting on this mount (CNF mount) [1].
6.0 STATIC LOAD DISTRIBUTION CALCULATION
In order to calculate the weight distribution between the mounting points, the position of the CENTER OF GRAVITY
(C.G.) has to be determined first. It is a simple task only for an axisymmetrical object. Position of the C.G. can be obtained by
computation or experiment. The computational approach is feasible in most cases to the manufacturer who has all relevant
drawings containing the data on mass distribution inside the object. The experiment is suggested by the definition of the C.G.
as the point of support at which the body will be in equilibrium. For example, a small object can be supported on a peg; when
in equilibrium, a vertical line drawn through the peg will pass through the C.G. Unfortunately, this method is applicable only
to small objects. For large objects, such as machine tools, the object is mounted, for the C.G. location purposes, onto three
load cells LC1, LC2, LC3, as shown is the plane view in Figure 29. If the weight loads as sensed by these load cells are W1,
W2, W3, respectively, then coordinates of the C.G. are as follows:
xC.G. = ;
(23)
yC.G. = .
After the C.G. position is known, weight distribution between the
mounting points should be calculated. Such a calculation can be
rigorously performed only for the case of an object with three mount-
ing points (a statically-determinate problem). Unfortunately, only a
relatively small percentage of objects requiring vibration isolation
are designed with the "three point" mounting arrangement. If the
number of the mounting points is greater than three, the accuracy
of weight distribution calculations is suffering, unless the mounting
surface of the floor is flat and horizontal and the mounting surface
of the object is also flat. The tolerance on the "flatness" requirement
should be a small fraction of the projected static deformations xst of
the selected vibration isolators.
For example, if the vertical natural frequency of the isolated
object is fn = 20 Hz, then, from Equation (4), xst = 0.0625 cm or
0.625 mm.
Similarly, for fn = 10 Hz, xst = 2.5 mm, and
for fn = 5 Hz, xst = 10 mm.
¨
X1W1 + X2W2 + X3W3
_________________
W1 +
W2 +
W3
y1W1 + y2W2 + y3W3
__________________
W1 +
W2 +
W3
Fx
Fy
T
3
m, I
kxky
kxky
3
x
y
Figure 28 Planar (Three-Degrees-
of-Freedom) Vibration
Isolation System
3
