1
Chapter 1 Introduction This thesis describes the design, construction and use of a precision level. There are important applications for such an instrument, and a brief background in precision engineering will demonstrate the relevant design consideration. The objective of this thesis project and how it relates to the field of precision engineering is also explained.
1.1
Background
The level of precision attained by modern machines is constantly increasing.
Today’s
advanced diamond turning machines can produce, with good repeatability, surfaces with peak to valley form errors of just a few microinches. In order to maintain this degree of accuracy, the base table of a precision machine must be extremely flat and level. The base for a precision machine is often made of granite or cast iron because of the low cost and the ability of these materials to be made quite flat by lapping or scraping. A regular master surface plate that would be used as a reference to a scraped machine table used to have flatness of 25
microinch inch
[1]. Today’s precision surfaces can
attain even greater levels of flatness on the order of optically flat surfaces, which are accurate to less than 6
microinch inch
or one quarter the wavelength of visible light. In many cases,
nominally horizontal machine surfaces must also be extremely level. The levelness of a surface can be thought of as its perpendicularity to the Earth’s gravitational field. An affordable machinist level may be able to measure how level is a surface to a resolution of around 10 arc-seconds.
This equates to a levelness of approximately 50
2
microinch inch
.
Commercially available levels can measure to an angular resolution of
approximately one arc-second, but in this work we are attempting to reach finer resolution.
1.2.
Objective
Users of precision machines need a way to determine the level of their machine surfaces. The purpose of this project is to create a device that can measure the levelness of a surface to a tolerance of less than one arc-second. Mr. David Arneson of Professional Instruments suggested a few improvements to a proven technique for measuring how level a surface is. Mr. Arneson’s idea introduces some improvements to a pendular level measured with a capacitance gauge. Using this principal, it seemed possible to create a level that could measure surface angles to less than one arcsecond. This design concept would be the basis for the ultra-precision level. It is explored and analyzed further in the chapters that follow.
3
Chapter 2 Design and Analysis of Ultra-Precision Level A design based on the suggestions of Mr. Arneson is introduced in this chapter. Furthermore, the main properties of this design are shown along with the theoretical calculations of the sensitivity with finite element verification.
2.1.
Original Design Concept
The design concept consists of a vertical beam hanging from a horizontal support. The sensitivity of this design depends on three major parameters: the vertical beam’s mass, length, and the stiffness of the pivot on the end of the vertical beam. A flexural pivot is used to produce the necessary compliance in the pendulum beam.
A high-resolution
capacitance gauge measures the displacement of the vertical beam. In order for the gauge to register a displacement when the level is on a surface level to one arc-second, the vertical beam must displace by at least the minimum resolution of the measuring device. The main initial design objective is to determine parameters for the vertical beam geometry that allow it to displace by a measurable amount while placed on surfaces that are level to less than one arc-second.
2.2
Determination of Design Capabilities
A further explanation is needed for measuring how level a surface is to such a great tolerance using the design mentioned previously. When dealing with such a small target tolerance, one might wonder measurement errors caused by factors such as bending under own weight, humidity, and temperature are accounted for. In this section a simple procedure will be
4
shown that minimizes the inherent errors of the level. A mathematical model for calculation of surface level angle is also outlined. This mathematical proof shows that the angle of levelness can be determined by measuring the displacement of the pendulum beam.
2.2.1
Method of Surface Angle Calculation
The final desired output from the instrument is the angle of levelness that is characteristic of the measured surface. This section will prove that the only measurements needed by the ultra-precision level are two displacements between a reference on the rigid support of the pendulum beam and the beam itself. These measurements are taken before and after the level is reversed 180∞ about its center. Keeping in mind that the only important parameters in this simple design used for mathematical purposes, are length of vertical beam, mass of vertical beam, and torsional stiffness of beam pivot, the following mathematical proof is justified. Figure 2.1 shows a simplified drawing of the level placed on a surface with angle q. Let
a
represent the angle
between the vertical beam and the y-axis. Due to the fact that the flexural pivot has a torsional stiffness (kt) greater than zero,
a
will always be smaller than q. The capacitance
probe, which can be seen on the bottom of the drawing, is used to measure the distance between the probe and the vertical beam. For the first measurement this distance will be denoted by d1.
5
a
kt q
= 9.81 m/s2 d1
L mg
Surface Capacitance Probe
y
x Figure 2.1: Schematic of Ultra-Precision Level on surface with exaggerated angle.
When the level is turned 180∞ about its vertical axis, a new distance between the probe and the beam is measured and designated
d2.
The difference between
d1
and
d2
is the unknown
measurement value needed to determine the angle q, which is the levelness of the surface. The method of self-checking by turning the device 180∞ and taking a new measurement is known as reversal [1]. This technique accounts for any errors that are inherent in the device. By reversing the instrument and taking the difference between the two measurements, the same errors present in the first measurement are reversed and subtracted out.
Only the inherent angle of the surface, which is responsible for the
difference between
d1
and
d2,
remains. Using this measured difference,
Dd= d1
-
d2,
along
with the known design parameters (the flexural pivot stiffness, the mass of the vertical beam, and the length of the vertical beam), the angle of the surface can be determined.
6
2.2.2. Mathematical Proof of Surface Angle Calculation
As mentioned previously only two displacement inputs are needed to determine the final angle output with this design. Consider a free-body diagram of the vertical beam of the precision level shown in Figure 2.1. a
-mg
= 9.81 m/s2
kt[q-(µ-µo)]
y
L mg
x
Figure 2.2: Free-body diagram of vertical beam when level is on surface of angle q.
Recall that the level is measuring a surface of angle q. The inherent error, which may cause the vertical beam to be offset by a certain angle, is denoted by
ao.
As seen in the free-body
diagram of Figure 2.2, the moment acting on the flexural pivot during the first measurement is equal to kt[q-(µ-µo)], where kt is the torsional stiffness of the pivot and
µ
is the angle
between the vertical beam and the y-axis. Assuming that the beam is homogeneous and of uniform shape, the beam can be assumed to have all of its mass concentrated at a point exactly halfway between both ends of the beam.
7
The mathematical analysis for the level at position 1 is as follows.
The sum of moments about the pivot:
∑M p
= 0 = mg
L Sinα + k t (α − α o − θ ) . 2
2.1
Assuming small angles for a: Sinα ≅ α and equation 2.1 becomes, − mg
L α = k t (α − α o − θ ) . 2
2.2
From analysis of Figure 2.2: δ1 =
L sin(α − θ ) .
2.3
Now, assuming small angles for q,
δ =
L(α − θ ) and
α =θ +
δ2
L
.
Substituting for a in equation 2.2, − mg
δ
Simplifying,
and
θ =
L δ δ ( + θ ) = kt ( −αo ) . 2 L L
L
+θ =
− 2k t δ
( −αo ) mgL L
2k t − 2k t ( ). − 1) + α o ( L mgL mgL
δ
Solving for δ 1 with further algebraic simplification, 2
δ1
k t α o L − mg L 2 = L k t + mg 2
θ
.
2.4
8
Upon reversal of the level, the mathematical model changes. Figure 2.3 shows the level after it has been turned 180 on the same surface as depicted in Figure 2.1. ˚
a
Kt
q
= 9.81 m/s2
L Cap Probe
mg
Surface
d2
Figure 2.3: Model of precision level after reversal on same surface of Figure 2.1.
The mathematical analysis for position 2 is shown.
It can be seen that for position 2, δ2 =
L (θ − α ) and
α =θ −
δ2
.
L
Again, taking the sum of moments about the flexural pivot, k t (θ − α − α o ) − mg
L α = 0. 2
2.5
9
Substituting for a, kt (
δ2
− α o ) − mg
L
δ L (θ − 2 ) = 0 . 2 L
Expanding, δ2
L (k
Solving for
t
+
δ2,
L
−
α
mg 2 ) k
t
o
−
L mg 2
θ =
0.
and simplifying further, L2 θ 2 . L k t + mg 2
kα o L + mg δ2 =
2.6
Using equations 2.4 and 2.6 to find the difference between δ 2 − δ 1 = ∆δ =
2mgL2θ 2k + mgL
δ 2 − δ1 ,
we find
.
t
Solving for θ =
θ,
∆δ ( 2k t + mgL )
2mgL2
2.7
.
Equation 2.7 is the underlying equation used to determine the angle of levelness of a surface with the displacement measurements acquired by the reversal method.
2.2.3. Analysis of Mathematical Results
The parameters of Equation 2.7 include design characteristics with two unknown variables. It is now proven that the only necessary design parameters are the torsional stiffness of the flexural pivot (kt), the length of the vertical beam (L), and the mass of the vertical beam ( m). Knowing these three parameters along with the angle, q, can be determined.
d1
and
d2,
obtained from the reversal method,
10
It is important to remember that the desired measured output, angle of levelness, is to be measured at very small resolution of one arc-second. In order to reach this goal, the three design parameters of the level must be chosen carefully.
A limitation of these
parameters is the resolution of the capacitance probe that will be used to measure the values d
d 1
and 2. It will be necessary to measure these displacements within the set of limits created by the precision of the available capacitance gauges.
2.2.4. Minimum Horizontal Displacement for Vertical Beam
In this project non-contact measurement of displacement is the method implemented to find the displacement of the vertical beam. The non-contact measurement device is a C1C Lion Precision capacitance probe. The specifications for the probe are shown in Table 2.1 [2].
Sensor Size
Calibration
Range
Resolution
C
Fine
Ø 0.067" Ø 1.7mm
0.003"-0.005" 75µm-125µm
0.14µinch 3.5nm
Ultrafine
0.0008"-0.0012" 20µm-30µm
0.07µinch 2.0nm
Table 2.1: Specifications for Lion Precision non-contact C1C precision probe [2].
Both the Fine and the Ultrafine capacitance gauges are available for this project. In order for the measurement,
Dd,
to be applicable to equation 2.7, it must be greater than the
resolution of each probe. It is attempted to design the level to be used with both the Fine and Ultrafine probes. This will allow for the range of the level to be maximized. It is the Ultrafine probe, however, that determines the greatest theoretical resolution of the level because it is capable of smaller measurements. It is important to understand that although
11
the level is designed to measure surfaces level to less than one arc-second, it should also be able to measure surfaces with lower accuracy to expand its capabilities. Considering the applications that this level will be used for, it is desirable to keep the size to a minimum. This allows the operator to use it with relative ease. The design parameter that this affects is the length of the vertical beam.
If the beam length is
minimized, it decreases the resolution, but makes the instrument easier to use. It will be necessary to create a design that reaches the target resolution while keeping the size to a minimum.
2.3
Determination of Level Parameters
The torsional stiffness governing the motion of the vertical beam is to be created by using a flexural pivot.
The stiffness of this flexure depends on its material properties and
dimensions. These same material properties and dimensions affect the length and mass of the beam. Careful consideration is taken in choosing the design parameters.
2.3.1
Dimensions and Effects for Pivotal Notch-Hinge Flexure
The design of the flexure incorporates an estimated length and weight for the vertical beam in order to calculate the stiffness. After considering the previously determined mathematical equations and the preferred size for the level, it was found that the desired length of the pendulum beam should be about 2.5 to 3 inches. This is an efficient size range for use with the level. The necessary stiffness of the flexure also depends on the mass of the beam. Using the desired length of between 2.5 and 3 inches, careful calculations are made to find a suitable mass in conjunction with a desired stiffness.
For maximization of the level
12
resolution, the stiffness needs to be fairly low. Depending on how low the stiffness is, the necessary mass of the beam varies. Recall Equation 2.7: θ =
∆δ ( 2k t + mgL) 2
2mgL
From examination of this equation, and knowing that the possible stiffness values that are obtained for a notch hinge flexure of this size are relatively large compared to the length and mass of the beam, the 2k t term in the numerator strongly outweighs the mgL term. For ∆δ ( 2 k t )
design purposes this changes the equation to
θ ≈
small angle, q, the displacement difference,
can be allowed to be above the resolution of
Dd,
2
2mgL
, where it can be seen that for a
the capacitance probe if the denominator term, 2mgL2, is maximized or if the stiffness, kt, is minimized. Since the value of L has already been estimated, the stiffness ( kt) and mass (m) can be arranged to create an acceptable displacement difference ( Dd) for a surface angle (q) of less than one arc-second. Figure 2.4 is a drawing of a notch hinge flexure with all of the parameters that determine its torsional stiffness.
t
R b L h
y z
x
Figure 2.4: Drawing of notch hinge pivotal flexure design.
13
The total width in the x-direction, h, is equal to 2 R + t for this design.
The
dimension, t, is the smallest thickness in the x-direction and R is the radius of the semicircular notches on both sides. The thickness of the flexure in the z-axis direction is denoted b k E by . The only material property necessary for calculations of t is the elastic modulus, . For small deflections, the equation for angular deflection is as follows [3]: 1
θz =
9πR 2 M 2 Ebt
5
.
Solving for kt, which is equal to
kt =
2 Ebt 9πR
5 1
2.8
2
M θz
, we find
2
.
2.9
2
Equation 2.9 will be used in considering the design of the flexure for the vertical beam of the ultra-precision level. This equation affects not only the stiffness of the flexure, but also incorporates the material by its elastic modulus, E, thus affecting the mass of the beam as well.
2.3.2
Flexure Stiffness, Material, and Mass
Aluminum was chosen to be the flexure material because it is easy to machine and its elastic modulus is relatively low compared to other abundant metals. The low elastic modulus allows for the stiffness to be minimized, thereby increasing the tolerance of the level. After determining the basic dimensions of the vertical beam (not all of the flexure dimensions) in conjunction with the available brackets used to construct the level, the estimated weight for the chosen material, aluminum, is chosen to be about 0.23 lbs. In order to mount the
14
flexure to the available structural supports, the length of the vertical beam is determined to be 2.80 inches. As seen in Table 2.1 the ultimate resolution of the Ultrafine capacitance probe is 0.07x10-6 inches. For a mass of 0.23 lbs. and a length of 2.80 inches, the greatest allowable stiffness for minimum displacement, measurable by the Ultrafine capacitance lb − inch
probe, is ktª123
radian
(determined from Equation 2.7). It would be preferred, however,
that level be usable with the Fine probe also. Because the resolution of the Fine probe is exactly twice that of the Ultrafine probe, the maximum stiffness becomes 61.5
lb − inch radian
, half
of its previous value. Realistically, in order for the performance of the level to be optimized, the stiffness should be designed to be significantly lower than these values in order to account for any other possible affects, such as slight error caused during fabrication. If the stiffness can be reduced even further than these calculated values, then the level will execute its measurements with greater accuracy. The limiting property of the flexure design in trying to decrease the stiffness is the minimum thickness, t. Mechanically, the thickness of the aluminum cannot be too low, or plastic deformations and even fractures can occur with careless handling of the level. The minimum allowable thickness for this design was set at about 0.020 inches. The following parameters were determined after many attempts were made to find a suitable stiffness by using the equation 2.9 and checking it with finite element analysis. For R=0.5 inches, t=0.020 inches, and b=0.75 inches, and an elastic modulus of 10.4x106
lb inch
2
for aluminum [4], the corresponding stiffness calculated by Equation 2.9 is
found to be 14.7
lb − inch radian
. This stiffness value is acceptable considering that the length of
15
the beam is 2.8 inches and the mass is about 0.23 lbs. A corresponding solid model for this design is shown in Figure 2.5.
y z
x
Figure 2.5: Solid Model of Flexure and Vertical Beam for Precision Level.
The design of the flexure is created with the solid modeling program, Solidworks‘. The dimensions are the same as those used to calculate the previously mentioned stiffness, except, the maximum thickness in the z-axis direction is 0.80 inches. This thickness does not affect the stiffness of the flexural pivot, but is implemented to maintain the necessary weight of the vertical beam. The length from the middle of the notch, which is essentially the pivot-point, to the bottom of the beam is 3.281 inches. This is the effective length, L, which is used in Equation 2.7. The increased thickness on the upper portion of the flexure provides support when it is bolted to the support bracket used to hold it in place. The
16
weight of the bottom portion is exactly 0.23 lb. This fulfills the sought after weight for the vertical beam. Considering the necessary degree of precision, it is very important to be conscious of the structural characteristics of the instrument. The level should not be allowed to rock or tilt in any way when measurements are being taken. One of the best ways to prevent this from happening is to support it on three legs. Whenever a structure stands upon three supports that do not form a straight line, it cannot rock back and forth and will always stay in place. For this design, however, it is decided that a continuous, flat surface be used to support the level. This incorporates a different idea than that used in the under-constrained three-support mount, because in essence it over-constrains the device. The inherent error that may be caused from a continuous surface, supporting the structure, would be bending under its own weight due to gaps left underneath the support. For a structure of this size, the weight due to bending will be miniscule considering the precision flatness of the surfaces that it will test. In conclusion, for this design application a continuous horizontal support surface will be implemented.
2.4
Analysis of Level Design
The design of the level is analyzed prior to construction using finite element analysis or FEA. The software program I-DEAS‘ if used for the analysis. A plane-stress analysis is implemented on the surface of the flexure perpendicular to the z-axis. A known force is applied to the bottom of the flexure and the displacement recorded.
17
Model with 0.009 inch plane stress elements Input: F=5x10-5 lbs. Output: Displacement.=3.722x10-5 in. Angular disp.=1.135x10-5 rad. Stiffness: kt=14.46
lb − inch radian
Figure 2.6: FEA output from model with input force of 5x10-5 lbs.
The torsional stiffness of the flexure is determined after acquiring the displacement for a known input force. Four different FEA meshes are used to analyze this final design to ensure that the data is not faulty. In addition, the force applied to the bottom of the beam is varied in order to check the consistency of the stiffness. Figure 2.6 is the output from an IDEAS FEA model where the force at the bottom right corner of the flexure was 5x10 -5 lbs.
18
The displacement output at the bottom of the beam is 3.722x10-5 inches. This corresponds to an angular displacement of 1.135x10 -5 radians. The moment acting on the beam is equal to F * L, where L is the distance from the pivot to the bottom of the beam.
Model with 0.0085 inch plane stress elements Input: F=1x10-5 lbs. Output: Displacement.=2.58x10-6 in. Angular disp.=7.863x10-7 rad. Stiffness: kt=14.49
lb − inch radian
Figure 2.7: FEA output for model with input force of 1x10-5 lbs.
This moment, M, is calculated to be 1.641x10 -4 lb-inches. The torsional stiffness of the flexure, kt, is equal to
M θ
, which is 14.46
lb − inch radian
. This value is very close to that predicted
19
by use of Equation 2.9, which is 14.7
lb − inch radian
. Another FEA model is analyzed, this time
with a different input force and slightly different element size. Figure 2.7 is a model with a lower input force than that of Figure 2.6. The element size is also decreased so that the conditions would not be equal to the previous model. This was done to show a convergence in the FEA output data to a specific value. The input force of 1x10-5 lb. creates an output displacement of 2.58x10-6 inches. The resulting stiffness for this data corresponds to 14.49
lb − inch radian
, which is almost exactly the same as that found from
the first model. This shows a convergence in the data between the two models. It also confirms that the stiffness of the flexure is relatively constant for small angles.
2.5
Conclusions about Level Design
From the previously mentioned analysis, the design of the level appears to be suitable for obtaining an accuracy of one arc-second of resolution.
The finite element analysis
theoretically proves the stiffness value which is calculated by equation 2.9. If the analysis is correct, then the stiffness of the flexure is equal to around 14
lb − inch radian
, an acceptable value.
For the minimum measurement attainable by the capacitance gauge being 0.07x10 -6 inches, along with length and mass of the flexure as 3.281 inches and 0.23 lbs, respectively, this stiffness allows for a minimum angle measurement of about 0.1 arc-seconds (equation 2.7). The structural characteristics of the level are designed to support the flexure enough to allow for repeatable data during acquisition. This is done with steel indicator mount brackets. The next step is to fabricate the aluminum flexure and assemble the prototype level for calibration and testing.
20
Chapter 3 Test Method and Results for Constructed Precision Level The flexure used to construct the level is the only part that required a machining operation. This is done at Ultran Laboratories, Inc. in Boalsburg, PA. It is performed with the aid of an experienced machinist, John Ream, who gave invaluable advice.
The dimensional
tolerances are met, and the most critical, minimum thickness, t, between the two semicircular notches is machined to an accuracy of less than one thousandth of an inch. After fabrication, t is measured to be 20.4x10 -3 in. This very accurate measurement is obtained ultrasonically by using a high frequency transducer to send an impulse through the aluminum at the point of minimum thickness, t. The ultrasonic procedure is carried out by Mahesh Bhardwaj of Ultran Laboratories, Inc. Figure 3.1 is a picture of the level assembly mounted to the test surface.
Flexure
Support Structure
Capacitance Probe
Spindle Surface
Figure 3.1: Picture of flexure and level after fabrication and assembly.
21
Several other pictures of the level and spindle assembly can be seen in Appendix A. It may be useful to examine them in order to gain perspective of the size of the components and how they are integrated. After the flexure is fabricated, the complete level is constructed using indicator mount brackets. To calibrate and test the level for its accuracy, a precision level surface is necessary. A precision air-bearing spindle placed on top of a granite block table has been chosen for this purpose. It allows for complete reversal of the level, by simply turning the spindle by 180°. The spindle was made by Professional Instruments and had very high rotational accuracy. However, one of the problems is creating a precision level surface for supporting the spindle. It is found that the currently used granite table surface did not have the required accuracy. In the next section a closer look is taken at the spindle and its base.
3.1
Air Bearing Spindle and Data Capture Instrumentation
The air bearing spindle being used to test the level is a Professional Instruments model 4R BLOCK-HEAD“. This spindle operates with an air pressure of about 110 psi, which gives it characteristics almost equal to that at 150 psi. Its specifications for operation at 150 psi are shown in Table 3.1 [5]:
Load Capacity Radial Axial
Ultimate 100 lb 400 lb
Working 50 lb 200 lb
Tilt 400 lb-In 200 lb-in Radial 0.67 lb/microinch Axial 2.0 lb/microinch Tilt 4.0 lb-in/microradian Radial 1 microinch Error Motion Axial 1 microinch Tilt 0.1 microradians Table 3.1: Specifications for PI air bearing spindle at 150 psi. Stiffness
22
The important specification for the spindle use is the error motion. Specifically, the tilt error is of significance. Table 3.1 shows that the maximum tilt for the spindle at an air pressure of 150 psi is 0.1 microradians.
This is equal to 0.02 arc-seconds of error.
Because the
resolution of this level is attempting to be at 1 arc-second of angle, an error of 0.02 arcseconds would be well below the measurable resolution. Therefore, this spindle can provide the necessary accuracy for the measurements. A Lion Precision non-contact capacitance gauge is used to determine the values of d1 and
d2.
This probe is connected to an amplifier that enhances the recorded voltage drop
between the probe and the pendulum beam of the level. This output signal is received by a signal analyzer called SigLab, which behaves as a virtual oscilloscope when its data is processed through the program MATLAB. SigLab is a data acquisition system that allows for selection of the sampling frequency, record length, and range of the data capture. All three of these options are used to their full advantage during testing.
Figure 3.2: Capacitance gauge amplifier used to create a signal for input into the virtual oscilloscope.
23
Figure 3.2 is a picture of the voltage amplifier that is used in conjunction with the capacitance probe. The figure actually depicts two amplifiers. On the right is the one used for all of the measurements recorded in this project. The measured distance between the flexure and the probe relative to the full range of the probe is indicated by the green light near the dial on the amplifier. To ensure the highest linearity of the readout, it is important to operate as close to the middle of the range as possible.
3.2
Determination of Torsional Stiffness of Flexure Pivot
Recall equation 2.7:
θ =
∆δ (2k t + mgL)
2mgL2
After fabrication of the flexure, the mass and length are easily calculated. They are found to be almost exactly equal to their designed values due to the accuracy of the machining operation for the flexure. The stiffness has been estimated based upon the dimensional and material properties of the flexure by equation 2.9 and by finite element analysis. Although these values are probably close to the actual stiffness of the flexure, they are still theoretical. An experimental method to find the stiffness of the flexure must be devised. It is noticed that the flexure has distinct vibratory characteristics from the output of the response obtained from the data acquisition system. It is noticed to oscillate at a constant frequency when excited by a slight force. This free vibration response is shown in Figure 3.3.
24
Figure 3.3: Free vibration response of flexure pendulum beam after being initially excited.
The natural frequency concerning the motion of the flexure about its flexural pivot is calculated from the response of the flexure shown in Figure 3.3. It is the high frequency dominant mode that is most noticeable. A lower natural frequency of about 2 Hertz can also be seen from the free vibration response. This low frequency mode is most likely caused by the isolation damping supports of the granite table. The air, which provides the pressure for the isolation damping of the granite table, is not used for most of the measurements; it was left on at certain times in order to support the table so that it would not endure shock. These support mounts are observable in Appendix A. The air pressure in the tables had been turned on at the time that the response of Figure 3.3 was captured. This explains the observable low frequency rigid body mode. Although there are other modes associated with it, the one most dominant for the flexure is the pendulum beam’s rotation about the flexural pivot. For the purposes of this project, the system can be modeled with a single degree of freedom in this rotational direction. The frequency of the flexure is dependent on the mass moment of inertia for the beam and the torsional stiffness. The natural frequency is calculated from the time response
25
to be 12.5 Hertz or 78.54
radians second
. The vertical beam of the flexure had a weight equal to
0.23 lb. which corresponds to a mass of 5.876x10 -4
lb − s
2
inch
. By the Central Axis Theorem
the mass moment of inertia, J, for a particle mass about a point is mr2 [6], where r is equal to the distance from the center of mass to the point of revolution. For this purpose r is equal to 1.9 inches. Using this correlation, the mass moment of inertia for the flexure beam is found to be 0.0021 lb-s2-in. The natural frequency in radians is equal to the square root of the torsional stiffness divided by the mass moment of inertia [6]. The mathematical expression for this is
ωn =
kt J
3.1
where wn is the natural frequency in radians and J is the mass moment of inertia. Solving for kt in equation 3.1, the torsional stiffness is calculated to be 12.9
lb − inch
. This
radian
value is slightly less than that calculated by equation 2.9 and from the FEA, which yields a stiffness of around 14.5
lb − inch radian
. A possible reason for the difference may be that the
aluminum used to fabricate the flexure is slightly anisotropic and has a different elastic modulus around the flexural pivot. Another explanation could be that slight dimensional differences between the design model and the actual flexure are causing the change in stiffness. Whatever it may be, a torsional stiffness of 12.9
lb − inch
is perfectly acceptable
radian
since the maximum stiffness that would accommodate the resolution of the Ultrafine capacitance probe is over 100
lb − inch radian
.
26
The vibration associated with the flexure presented a measurement problem for the level. The next section will explore the methods implemented to compensate for the beam vibration.
3.3
Solutions for Vibratory Characteristics of Flexure Beam
A consistent factor regarding the measurement between the capacitance probe and the pendulum beam is the vibration of the beam about the flexural pivot. A flexural bearing will only provide as much damping as the material it is constructed from [7]. In this case it is negligible considering that the material is aluminum, which has very low internal damping. Figure 3.3 clearly shows the vibration that the pendulum beam exhibits while measurements are being taken in an unstable environment. A small bump to the table or a stomp on the ground caused the flexure to vibrate. Steps taken to prevent this vibration are explored further.
3.3.1
Use of Oil as a Damping Solution
A small pool of oil was placed underneath the bottom surface of the flexure beam in order to induce a drag coefficient strong enough to damp the vibration. Mobile Velocite“ oil no. 6, a lubricant for bearings, was used in this application. A comparison was made between the free response of the pendulum with and without oil viscous damping after being initially excited. Figure 3.4 is the time response of the flexure pendulum beam without oil damping after an initial excitation. It can be seen from the figure that the system reduces to about 10% of its srcinal amplitude in approximately nine seconds.
27
Figure 3.4: Free response of initially excited underdamped flexure beam.
The area of the flexure beam where the oil is applied in order to increase damping of the system is shown in Figure 3.5. This figure shows it being applied at the space between the bottom of the pendulum beam and the spindle surface. The oil was placed on top of aluminum foil that covered the spindle surface. Another condition, where an aluminum foil container was placed around and underneath the flexure in order to prevent the oil from spreading around the surface of the spindle, is tested. This condition causes the oil to creep between the probe and the beam and is therefore deemed less effective. The oil is viscous enough that it contains itself relatively well without walls around it. The setup for the container-less condition is shown in Figure 3.5.
28
Area where oil was applied
Figure 3.5: Area where oil is applied for additional viscous damping.
The time response for the oil viscous damped flexure system is shown in Figure 3.6. This system consists of the oil being placed only underneath the bottom of the flexure as shown in Figure 3.5. The plot shows that the system is reduced to about 10% of its initial amplitude after approximately 5 seconds. Obviously, the amount of time it takes for the oildamped system to settle to an acceptable measurement amplitude is shorter than the time it takes for the flexure system without oil damping. Although it is experimentally proven that the system with the oil used as a viscous damper is more effective for reducing vibration, other problems with the oil system have arisen. One of the major problems with the oil damping is the diffusion or creep of the oil from the bottom of the flexure onto the capacitance probe. After this has happened the probe does not function properly and can no longer take displacement measurements when the oil is in between the capacitance probe and the beam.
29
Figure 3.6: Free response of oil viscously damped flexure beam after initial excitation.
It is difficult to keep the oil from spreading up the flexure beam into the capacitance probe when reversing the spindle by 180 ∞, as required when calculating surface angle measurements. This problem is even more apparent when the oil is contained on all sides. A possible solution would be to raise the probe and prevent oil from creeping near to it. However, it would then be measuring the displacement of the flexural beam at a different vertical point. This alters the effective length, L, on which the mathematical measurement calculations depend. This changes the design parameters of the level, and therefore, is not an option at this point. Another method of dealing with the vibration of the flexure needs to be explored in order to continue acquiring test measurements.
30
3.3.2
Signal Averaging and its Effects on Displacement Measurements
Since it is not possible to implement the viscous oil damping on the level assembly without redesign of the flexure, another method will need to be implemented to deal with the beam vibration. If the operator of the level is patient enough to wait several seconds, and careful enough not to disturb the spindle table, the vibration of the flexure reduces itself enough to produce a signal with a fairly low amplitude. Specifically, the amplitude of the signal is be reduced to less than 0.05 Volts. This corresponds to a displacement of 1x10-6 inches. The amplitude of the signal can be further reduced by using signal averaging with the virtual oscilloscope. By taking 30 to 60 averages of the signal, at a sampling rate high enough to avoid aliasing, for about 0.1 to 1.0 seconds per sample, the vibration amplitude of the signal can be reduced to below 5x10 -4 V. This amplitude of vibration is acceptable for measuring the displacement, d, between the probe and flexure. However, a hand-calculated mean displacement of several peak-to-peak values from the averaged signal, are used to obtain the most accurate values for d. Although the viscous damping provided by the oil helps reduce the vibration of the pendulum beam, signal averaging is chosen as the means to retrieve the test measurements for the level. The signal averaging method requires more patience but is easier to use for the purposes of qualifying the level. Oil viscous damping could possibly be something of further consideration for the level design in the future. The next step is to check the degree of accuracy by which the correlation in Equation 2.7 along with the ultra-precision level can measure the angle to which a surface is level. It will initially be tested at angles between 10 and 100 arc-seconds because this range can be checked against the available machining level.
31
3.4
Test for Capability of Surface Angle Measurements
It is already possible to roughly measure the angle of surfaces to increments of 10 arcseconds using the available standard machinist level. This level uses the buoyancy of an air bubble submerged in liquid water to determine the angle of a surface. Such levels are more useful when determining whether a surface is level within the range that they can measure. They are less effective when trying to measure surfaces that are close to their highest resolution. But, because this machinist level is the only reference available to this project, it will be attempted to determine the angle of a surface to the maximum precision of the standard level and then compare it to the value obtained upon the same surface with the ultra-precision level. The only way to use the standard level to its fullest resolution is to line the edge of the air bubble up with the mark on the glass that contains the water with the air bubble. For this reason the accuracy of the measurements acquired by the standard level is not high. However, a useful is still made. The data is acquired by varying the angle of the spindle surface and taking measurements at each angle with the ultra-precision and machinist levels. A plot representing this data is shown in Figure 3.7. When observing this data, it is important to remember that the resolution of the standard level is poor compared to the angles being measured.
32
Figure 3.7: Plot comparing measurements acquired from ultra-precision level and standard level.
The measurements taken by the ultra-precision level are consistently higher than those of the machinist level. This plot shows a fairly linear trend for both the ultra-precision and the machinist levels. The angle increments have been kept fairly constant between data points for this comparison experiment. All of the data used for the plot in Figure 3.7 is shown in Table 3.2. The table includes the percent difference between the ultra-precision and machinist level measurements. This percent difference for both of the levels stays between 22% and 27% for all of the measurements. Since both levels are not compared to a reference, it is not known which one has measured the surface angle more accurately. It is possible that the accuracy of the machinist level is sub-par compared to the ultra-precision level.
An explanation for the consistently higher angles measured by the ultra-precision
33
level could be that the torsional stiffness calculated for the flexure is greater than its actual value. This conclusion is probably unlikely due to the accuracy with which the stiffness of an underdamped system can be measured by the method used in Section 3.2.
Data Point
Ultra-Precision Angle Measurement Standard Level Angle Measurement (arc-seconds) (arc-seconds) Percent Difference
1
12.9
10
22.5
2
28.6
21
26.6
3
40.2
31
22.9
4
59.9
46
23.2
5
76.1
57
25.1
6
87.6
67
23.5
Table 3.2: Measurement data used to compare the ultra-precision level to the standard level.
Even though there is a noticeable difference between the two level measurements, the accuracy with which the ultra-precision level can measure the angle of a surface may be adequate for measuring surface angles at less than one arc-second. The linear trend of the data shows that there is consistency for the measurements. The next section will explore the ultra-precision measurements acquired by the newly constructed level.
3.5
Experimental Results While Attaining Ultra-Precision Levelness of Spindle
In order to prove that the level is capable of measuring the angles of ultra-precision level surfaces, the spindle surface used for the tests had to be level to the same degree of precision. Before this spindle was used to take measurements for the precision level, it sat on a base that was flat upon the supporting granite table. This table is by no means close to being level on the order of arc-seconds. Because of this, a method is devised to alter and control the angle of the spindle base as it is on the granite surface. The idea that a surface
34
cannot rock and can be supported fully by three contacts is mentioned briefly in chapter 2. If three supports are situated near 120 ∞ from each other around the base of the spindle, then the spindle can be supported and held at any angle. By testing the surface with the ultraprecision level, it is possible to achieve arc-second levelness of the spindle surface. At first it was difficult to adhere to the delicacy of this process. A standard machining level with a inch maximum resolution of 50x10-6 had been used to obtain an initial angle of levelness inch for the spindle. After the peak of its measurement precision has been reached it is replaced by the ultra-precision level. This is the most efficient way to create a surface capable of testing the ultra-precision level. Successive measurements are taken while altering the levelness of the base of the air bearing spindle. The trial and error process used to obtain the final and most critical levelness is changed slightly, although the basic method of achieving this precision mentioned above is followed. To achieve the initial degree of levelness measured with the machinist level, measurements are taken along the x and y axes of the spindle. It should be noted that these axes are not in line with all of the three supports upon which the spindle base is sustained. The initial degree of levelness is attained by using the machinist level to measure the spindle surface angle along the x and y axes, while changing the heights at each of the supports. The height at the supports is varied by using standard ¼-20 bolt screws at two of the contact points and a precision gauge block at the third. The screws can be turned to vary the height at their contact points. Figure 3.8 is a drawing of the spindle assembly from a top view. Here it can be seen that the spindle base has contact with the table at three points approximately 120∞ apart. An actual picture of a top view for the assembly can be found in Appendix A.
35
Spindle Surface Direction Line for Spindle Base
Angle Measurement (In line with x-axis)
Level y
Contact Point (3 Places)
x
Figure 3.8: View of spindle assembly from above (z-axis is perpendicular to the surface).
Obtaining the initial degree of levelness, equal to the resolution of the machinist level, is not as difficult as attempting to reach a levelness of less than one arc-second. After the previous method had been found to be less effective, a slightly different technique is implemented to attain angles less than one arc-second for the spindle surface. This new means of adjusting the spindle surface to extremely small angles had to involve the ultra-precision level as the angle-measuring device because no other level with enough precision was available. Instead of measuring the angle in the x and y axes, it is found that a greater accuracy is attained by measuring the angle in the line of direction associated with each support point. This allows for direct adjustment in the angle of the surface plane being measured. Consider the drawing in Figure 3.9.
36
Spindle Base
Spindle Surface
Level
Direction Line of Angle Measurement (In line with contact point)
y
Contact Point (3 Places)
x
Figure 3.9: Drawing that depicts the method for acquiring sub-arc-second levelness of spindle surface.
The drawing above shows the line of angle measurement, which is in line with the contact point, used in this method. Recall that in order for the ultra-precision level to measure the angle of a surface, it must be reversed 180°. This means that two measurements must be taken for a single contact point’s line of direction. This process is iterative and must be repeated for each of the three contact points while changes in the height of the contact points are made continuously between measurements. By making very slight alterations in the height of each contact support, it is possible to reduce the spindle surface angle to less than one arc-second in all three contact point lines of direction (measured by the ultraprecision level). This course of action required many measurements and adjustments. At the very same time that the spindle surface angle is being reduced, the ultra-precision level is being tested for its precision.
37
3.6
Ultra-Precision Angle Measurements
According to the measurements made by the ultra-precision level, the surface angle of the spindle has been reduced to less than one arc-second. This means that the level is making measurements of less than one arc-second, which is the goal of the project. Successive measurements are taken during the process of reducing the spindle surface angle. One series of incremental measurements is recorded and presented in Appendix B.
Select
measurements that are recorded for angles of less than one arc-second are shown in the figures that follow. Figure 3.10 is a comparison plot showing the probe to pendulum distances
δ1
and δ2
taken for the spindle surface at a certain angle. The voltage difference between the average values of
δ1
and
δ2
is 0.027 V. For the ultra-fine probe, this corresponds to a distance, ∆δ,
-7
of 5.4x10 inches. From equation 2.7, the angle of this surface is calculated to be 0.72 arcseconds. This has reached the desired tolerance sought at the beginning of the project. Its accuracy however, is not measurable due to the fact that there are no references for comparison.
As seen in Figure 3.10, there is oscillation noticeable even with the signal
averaged response. Hand calculated average values are made of this response, and used to determine the surface angle of the spindle.
38
Figure 3.10: Superimposed plots of δ1 and δ2 for a measured angle of 0.72 arc-seconds.
Several other measurements are taken for sub arc-second angles, while two more time responses have been provided. Figure 3.11 is the plot of δ1 and δ2 for a measured angle of 0.69 arc-seconds. Again, this particular set of measurements yields an angle calculation of less than one arc-second.
39
Figure 3.11: Comparison plot of δ1 and δ2 for a measured angle of 0.69 arc-seconds.
The last plot included in this section shows the time response of measurements made for probe to flexure distances that correspond to an angle of 0.40 arc-seconds. Figure 3.12 is the plot for the acquired data at this point. This measured angle is less than half of an arc-second. From the plot, it is noticed that
δ1
and
δ2
are very close to each other. A
measurement of this type is approaching the tolerance limits of the ultra-precision level under this set of conditions. It is necessary to reduce the damping further in order to prevent overlap between the time responses of δ1 and δ2, at such low surface angles. Several other sub-arc-second measurements are taken for the spindle surface whose time response plots are not included. Some of these measurement values are provided in Appendix B. The gradual increments made during the process of decreasing the spindle surface angle are noticeable when analyzing the measurement data in the appendix.
40
Figure 3.12: Comparison plot of δ1 and δ2 for a measured angle of 0.40 arc-seconds.
The angle measurements gathered from the ultra-precision level show that the level is capable of prescribing measurements for surface angles below one arc-second. Considering some of the parameters surrounding the level, the maximum tolerance may not have been reached. These results are representative of the potential of this instrument where certain issues such as damping were not completely accounted for. They still show, however, that the desired level tolerance was achieved.
41
Chapter 4 Conclusions and Further Work It has been shown that the constructed ultra-precision level can provide measurement values for surface angles that are below one arc-second. This has been proven since a standard machinist level cannot resolve any difference, whereas the ultra-precision level can, for such surfaces. The accuracy of these sub-arc-second measurements is a topic of debate. From the calibration curve in Figure 3.7, it could be inferred that the linearity of the ultra-precision level would continue into the realm of angles less than a single arc-second. This may likely be the case since there is no mathematical or physical reason to the contrary. The extensive use and experimentation with the level show that it appears to be useful for measuring the surface angles equal to or greater than one arc-second. It is thought that the resolution of the level is acceptable at these angles because the distance between and
δ2
δ1
strongly outweighs the amplitude of vibration for the flexural pivot. At or around
angles of one half arc-second, the amplitude of vibration for an averaged signal begins to approach
∆δ.
To enhance the precision of the level, the vibration must be damped to a
greater degree. The method of using oil in contact with the flexural beam is explored earlier. It proves to be successful in reducing the amplitude of vibration much faster, but is not practical for other reasons. Recall that the oil diffuses to between the capacitance probe and the beam while rotating the spindle. This is a problem that could be solved by redesigning the level. For this design the capacitance probe measures the displacement at a location very close to the bottom of the flexure. This allows for the oil to creep up a small distance in order to come between the probe and the beam. If the length of the beam is increased, and
42
the displacement measured at a higher location, it will give the oil a chance to damp the system and not allow it to creep up the flexure. Of course this changes many of the design parameters that the calculations of which the angle is dependent on are affected. This can also possibly change the effect the oil has on the flexure vibration. Changing the length and mass of the flexure beam alter the moment of inertia that it carries about its pivot point. This alters the angular velocity of the flexure during vibration and therefore the drag coefficient induced by the oil. The effect could be minimal, but requires experimentation before conclusions can be drawn. There are several ways to damp the system and reduce the vibration of the flexure. If the vibration amplitude of the pendulum beam is lessened, then the precision of the level is increased. This may allow the level to measure surface angles lower than half an arcsecond. During the design of the level, the flexure is intended to overcome the resolution of the capacitance probe. With a capacitance probe resolution of 0.07x10-6 inches, a torsional stiffness of 12.9
lb − inch radian
, and the values for the mass and length of the flexure, the
minimum surface angle measurement would be below 0.1 arc-seconds. The largest factor preventing the level from measuring surface angles this low is the vibration amplitude associated with the flexure. Another possible factor concerning measurements on the order of 0.1 arc-seconds could be the thermal effects on the aluminum flexure. When dealing with distances that approach the capacitance gauge resolution of 0.07x10 -6 inches, it only takes a slight change in temperature to affect displacement measurements. Thermal expansion is not a factor in the measurements taken during this project, but it is something worth considering if superior level resolutions are pursued.
43
The results of this experiment have shown that it is possible to measure the levelness of surfaces precise to less than one arc-second. Measurements of such low-angle surfaces are made using the ultra-precision surface level designed and constructed in this project. The accuracy and validity of these measurements may be acceptable, but the precision of the level cannot be confirmed since measurement devices with which to compare do not exist. With further work it may be possible to prove the legitimacy of the precision level measurements. There is exciting research needed to explore and expand the capabilities of this ultra-precision surface level.
44
Appendix A Pictures of Ultra-Precision Level Assembly This appendix includes pictures of the constructed ultra-precision level and other components used during the experiment. This includes the spindle, the spindle base, the granite table, and its isolation mounts.
Spindle Surface
Spindle Base
¼-20 Bolt Contact Points
Gauge Block Contact Point
Figure A1: Top view of spindle and base showing all three contact points.
45
Vertical Support Bracket
Horizontal Support Bracket
Connection Bracket
Figure A2: Structural support brackets used to maintain rigidity of level.
Isolation Support Mounts
Granite Block
Figure A3: Granite table and isolation damping mounts.
46
Spindle Axis of Rotation
Air Bearing Spindle
Air Intake Tube
Figure A4: Air bearing test spindle and spindle base.
47
Lion Precision Capacitance Gauge Amplifier
Level and Spindle Assembly
Surface Table
Isolation Mounts
Figure A5: View of entire test area used for ultra-precision level project.
48
Appendix B Data for Acquiring Precision Level Spindle Surface Several series of measurements are taken while decreasing the angle of the spindle surface. A selected series of these measurements is provided in this appendix.
Measurement Number
d1
(V)
d2
(V)
Dd
(V)
q
(radians)
q
(arc-seconds)
1 2 3 4 5 6 7 8
0.0127 0.475 0.0033 0.178 0.111 -0.036 0.185 0.0325
-2.229 -0.286 0.691 -0.236 0.464 0.28 -0.022 -0.078
2.2417 0.761 0.6877 0.414 0.353 0.316 0.207 0.1105
2.87969E-05 9.77581E-06 8.83419E-06 5.31824E-06 4.53464E-06 4.05934E-06 2.65912E-06 1.41948E-06
59.41 20.17 18.22 10.97 9.35 8.37 5.49 2.93
9 10 11 12 13 14 15 16 17 18
0.725 -0.0732 0.24 0.151 0.034 0.311 0.182 0.119 0.0835 0.121
0.641 0.009 0.282 0.117 0.061 0.337 0.167 0.107 0.0922 0.128
0.084 0.0822 0.042 0.034 0.027 0.026 0.015 0.012 0.0087 0.007
1.07906E-06 1.05594E-06 5.39532E-07 4.36764E-07 3.46842E-07 3.33996E-07 1.9269E-07 1.54152E-07 1.1176E-07 8.9922E-08
2.23 2.18 1.11 0.90 0.72 0.69 0.40 0.32 0.23 0.19
Table B: A series of measurements taken while decreasing the level angle of the spindle surface.
Table B shows the decreasing trend in the level angel for the spindle surface. It can be seen that the measurements gradually decrease and reach values below one arc-second of angle.
49
Appendix C Academic Vita of Anuj Bhardwaj
Anuj Bhardwaj
[email protected] Bachelor of Science Major: Mechanical Engineering Thesis Title: Design and Construction of Ultra-Precision Surface Level Thesis Supervisor: Dr. Eric Marsh Work Experience:
May – August 2002 Engineering Cooperative Education Intern • Research and development of fiber optic pigtails for telecom applications • Mechanical improvement of laser soldering techniques specific to fiber optic components • Design and analysis of mechanical equipment used for fiber metallization process 3M Optical Components Project Austin, Texas January - May 2001 Engineering Cooperative Education Intern • Creation of microcomponents used for production of 3M optical products • Design of components for upgrade of precision diamond turning machine • Extensive experience with optical instrumentation 3M Micro-Replication Technology Center Petaluma, California May – August 2000 Engineering Cooperative Education Internship • Research and development for automobile graphics films • Improvement of electro-powder coating process on high strength commercial banner film • Application of polymer extrusion to test high-grade film 3M Commercial Graphics Division St. Paul, Minnesota
50
May 1996 – July 1998 Laboratory Research Assistant • Characterization of ultrasonic transducers • Material enhancement of transducers to maximize resolution • Application of non-contact transducers to scan for material defects Ultran Laboratories Incorporated Boalsburg, Pennsylvania Awards and Memberships:
Phi Kappa Phi Honors Society, College of Engineering Pi Tau Sigma, National Mechanical Engineering Honors Fraternity Phi Eta Sigma Honor Society National Society of Collegiate Scholars Recognition of Academic Excellence Pennsylvania State University Dean’s list. Awarded seven of seven semesters enrolled Community Service Involvement:
Volunteer work at Foxdale Retirement Home Language Proficiency:
Intermediate level in Spanish Beginner’s level in Hindi Computer Skills:
• Computer Aided Design and Finite Element Analysis Programs: Unigraphics, I-DEAS, and Solidworks • Modal Analysis Programs: STAR Modal and SigLab • Computer language programs: Turbo C++ and MATLAB
51
References
[1] Moore, Wayne R. Foundations of Mechanical Accuracy. Bridgeport, CT: The Moore Special Tool Company, 1970.
[2] Lion Precision Company. Non-Contact Measurement. St. Paul, MN: Precision Company, 1999. http://www.lionprecision.com/modular/3-8-cyl.html
Lion
[3] Smith, S.T. and Chetwynd, D.G. Foundations of Ultraprecision Mechanism Design. Warwick, UK, 1992.
[4] Norton, Robert L. Machine Design An Integrated Approach, Second Edition. Prentice Hall: Upper Saddle River, NJ, 2000.
[5] Professional Instruments Company. Spindle Specifications. Hopkins, MN, 2002.
Model 4R BLOCK-HEAD“ Air Bearing
[6] Rao, Singiresu S. Mechanical Vibrations, Third Edition. Addison-Wesley Publishing Company, Inc: U.S., 1995.
[7] Slocum, Alexander H. Precision Machine Design. Society of Manufacturing Engineers: Dearborn, MI, 1992.
