SOLDADURA RIVESTIMENTO
7. I TERPRETACIÓ DE LOS RESULTADOS 1 Plan de Acción con el método RULA
transmissibility is found to be [50, 143]
"
T =
Urn "=
1 ( WgmO QgmW )? . W 1
U g
-
--wgl110 - + J-- --wgmo Qgl11 (3. 19) Equation (3.19) can be analyzed at the three frequency bands of interest; the extremes of very low and very high frequency and the resonance frequency. For low frequencies W « wgmo the transmissibility is 1 which means that there is no vibration isolation and the mass follows the ground movements. At resonance and for high quality factors Qgm » 1 the transmissibility is(3.20) The system amplifies the ground movements by a factor of Qgm . The transmis sibility in equation (3.20) is complex and the factor -j means a phase shift of -90° .
In the band of frequencies
WgmO « W Q gm WgmO
the transmissibility is found to be
2
T
ex _ , (3.21)while at very high frequencies W » Qgm wgmo the transmissibility only decreases
like
T
ex -J -- -- . . WgmO 1W Qgm (3.22)
Schmid and Varga [168] mention that elastomers have internal damping, which makes them not as stiff as the pure viscou damper assumed in equation (3. 12), so that equation (3.21) also holds for these materials at very high frequencies.
The vibration isolation is most effective at high frequencies. A compromise has to be made when choOSing a quality factor, since a high quality factor increases the transmissibility at resonance but decreases it at high frequencies. A small transmissibility at high frequencies is highly desirable, but with a high Q factor an accidentally excited vibration of the system will not die away rapidly. To reduce the transmissibility further, more than one damping isolation stage was used to isolate the probe head from the ground. The individual transmissibilities of the stages are the multiplied to give the total transmissibility of all stages [88] . At very high frequencies the total transmissibility will vary with the number of stages as
(3.23) which has been also found by Schmid and Varga [168] . Schmid and Varga con clude that multiple stages improve the vibration isolation at high frequencies, but that they may also increase the transmitted vibrations at the resonance fre quencies of the individual stages, depending on the location of these frequencies and th quality factors.
Tip-Sample Transmissibility
Although there are many possible vibrational modes for the probe head and sample holder the system can be approximated by a simple spring-mass system with a resonance at the lowest vertical vibrational frequency of the tip-sample system. The transfer function can again be found from equation (3. 13) . For the probe head, however, it is the difference between tip movement (Ut) and sample movement (us) that is important. This difference is used to define the tip-sample transfer function (5) as
5
=
Ut - Us . UsThis type of transfer function is known as relative transmissibility as opposed to absolute transmissibility, which was the definition of T [50] . The sample movement (us
)
is equal to the mass movement (urn) of the last isolation stage(Tn ) ·
If in equation (3. 13) the ground and mass extensions are replaced by those for the tip and sample then a simple relation between 5 and T is found, which
is
5 = Ut -1
=
T -l .Us
Note that both transfer functions are complex.
From equation (3. 19) the tip-sample transmissibility is found to be 5 =
( � )2
1 -( "' )2 + . '" "'toO 1
J "" .0 Q,.
(3.24)
For low frequencies W « WtsO the transmissibility is proportional to the square of the driving frequency w, i.e.
(3.25)
At high frequencies W » WtsO the transmissibility tends to unity, while it is approximately equal to the quality factor Qts at resonance. In summary, the tip-sample system behaves like a mechanical two-pole high-pass filter.
Overall Transmissibility The overall transmissibility (V) is
V =
5T = Ut - Us Us=
Ut - Us .Us ug ug
This can be interpreted as the product of the transfer functions of a low-pass filter and a high-pass filter. A plot of the overall transmissibility is shown in figure 3.13. For an effective vibrational damping system the natural frequency of the high-pass filter, WtsO of the tip-sample system, should be much higher than the natural frequency of the low-pass filter, which depends on the vibration isolation stages.
At low frequencies the overall transmissibility rises with the square of the frequency in accordance with equation (3.25) until the low-frequency cut-off at
wgrno is reached. If this frequency and the resonance frequencies of the damp ing stack are different, which is the case in the system described here, then
3.3. VIBRATION ISOLATION
59there is a flat area of overall transmissibility between these two frequencies. Be tween the stack-resonance frequency and the high frequency cut-off at WtsO the transmissibility falls with
F ex: 1 w2( n- l )
The number of stages is n = 4 in the current design with three plates comprising the damping stack plus the air table. After WtsO the overall transmissibility F
falls even more steeply with
1
V ex: s . W
The most critical frequency region is the low-frequency cut-off point, where the overall transmissibility reaches its highest values at the resonance frequencies of the air table and the damping stack.
Schmid and Varga [168] analyzed vibration isolation systems and calculated the overall transmissibility. According to equation (3.23) the transmissibility of a damping stack with n plates falls with w-2n . Schmid and Varga point out that due to the maximum amplitude being at the resonance of the stack, already two plates should provide enough damping. Two plates means the number of isolation stages is n = 3, which is only one less than in this project. The
third plate will increase the overall transmissibility at around the resonance frequency of the damping stack, which is not desirable. However, at the same time it provides an extra amount of damping at higher frequencies. The number of plates can easily be changed to suit the encountered noise spectrum of the laboratory; several experiments to that extent would need to be made in the future.
3.3.4 Air Table and Damping Stack
A pneumatically damped table, or air table, is the first stage of the vibrational isolation system. A stack of metal plates as a second isolation stage sits on top of the air table. The probe head including the sample holder sits on top of this damping stack. A photo of the air table and damping stack can be seen in figure 2. 13.
The air table is a commercial vibration isolation table, model Micro-g 63-531
from TMC [180] . It fulfills the soft spring and large mass requirements. The elasticity is provided by pneumatically damped membranes in the table legs with a spring constant of approximately 4 kN/m• The heavy table top weighs about 100 kg. The manufacturer states a vertical natural frequency of 1 Hz and also specifies a horizontal frequency of 1 .2 Hz. The pneumatic isolators installed by the manufacturer have two air chambers interconnected with a small orifice to provide damping. A quality factor of Q � 3 is cited for such a system.
The manufacturer specifies a vertical isolation efficiency at 5 Hz of 60% to 95% and at 10 Hz of 80% to 97% .
The damping stack consists of three concentric brass plates spaced apart by viton o-rings. The stack is part of the vibration isolation and should in principle have large mass and soft springs. The mass of each brass plate is 800 g. The o-rings are of the type 100 x 3 V80 from Engineering Plastics [66] , with an outer
diameter of 100 mm and a thickness of 3.0 mm. Judging from equation (3. 1 1 ) the 3.0 m m thick viton rings cannot provide a very low resonance frequency.
However, the criteria are not as stringent as they are for the first stage, since the stack is already isolated by the first stage. The main purpose of the rings is to absorb a passing shock or vibrational movement by the high internal damping.
Viton is a fluorocarbon elastomer that has been designed to be incompress ible to fulfill its purpose as a seal [158] . Under pressure the o-ring as a whole deforms and consequently the viton o-rings do not have a large compliance. To be able to seal fluid-containing containers under high pressure the rubber even has to have a minimum hardness [66] . Viton behaves like a viscous fluid with high surface tension, the viscosity providing high internal damping.
In their analysis, Schmid and Varga [168] point out that the spring constant of viton changes with the load placed upon the o-ring. They approximate the change as being a linear increase with the load and come to the conclusion that it is advantageous to have a damping stack with plates decreasing in mass from bottom to top. However, in this project three plates of equal mass have been used so far. Viton o-rings provide a spring constant of the order of 100 kN/m [168, 147] . The static extension of the 0-rings was measured and found to be less than
0.05 mm, which means a spring constant of less than 160 kN/m . With a mass of 0.8 kg the resonance frequencies of the damping stack will therefore be at around 70 Hz, although each plate will vibrate at a slightly different frequency due to the different loading. Using a softer type of rubber will shift the resonance of the damping stack to lower frequencies, but possibly afford less internal damping. The longer the total length of rubber is, the stiffer the equivalent spring constant. To decrease the stiffness of the damping elements shorter pieces of rubber can be used as well as pieces of a tube instead of a full-bodied ring.
3.3.5 Vibrational Modes of P robe Head and Central Ac tuator
Since the probe head is a mechanical structure it can be excited into vibration. It is important to know the longitudinal vibration frequencies of the probe head assembly, since the lowest order vertical vibration determines the maximum possible acquisition speed of the microscope. The resonant frequency should be as high as feasible, which was assured by designing the probe head and the sample holder so as to be small and rigid.
Figure 3.16 shows a section through probe head and sample holder. The probe head consists of five pieces, the disk, three outer actuators and one center actuator, where the actuators are clamped into the disk. The disk is supported by the outer actuators, while the center actuator is free.
The probe head can be seen as a simple spring-mass system with the disk as the mass and the three outer actuators jointly as the spring. Against the mass of the disk the masses of the actuators are negligible.
Consider a piece of material to be used for the outer actuator with length
l, section A and Young's modulus Y . To achieve a high spring constant it is favorable to have a short thick piece of high elasticity Y . This can be seen from equation (A. 19)
k = AY
1 '
derived in the appendix. The mass of the probe head should be as small as possible, while still providing sufficient static friction for the ball bearings.