Comparison of the Dynamic Characteristics of a Hydrostatic Carriage and a Mechanical Carriage of High-Speed Milling Machines

Comparison of the Dynamic Characteristics of a Hydrostatic Carriage and a Mechanical Carriage of High-Speed Milling Machines

Emmanuel Kabu Oman

Department of Mechanical Engineering, Accra Polytechnic

P.O box GP 561, Accra, Ghana

Stephen Kudadze

Department of Mechanical Engineering, Accra Polytechnic

P.O box GP 561, Accra, Ghana

Abstract

Comprehensive vibration data were generated for a hydrostatic carriage system that carries workpiece of a High-Speed Milling Machine at the Machine Tool Research Center of the University of Florida. Machine tool accuracy is critical if final quality of machined parts is of importance. In high speed machining, easy movement of parts is also a desired feature and the use of a hydrostatic bearing and guides provides a way to achieve this. Data was generated using experimental modal analysis; a technique that has found wide use with vibration analysts. A comparison of this data is made with a carriage built on purely mechanical system on another High-Speed Milling Machine. The data shows good vibration characteristics of the hydrostatic system. It was observed that both the hydrostatic and the mechanical bed have high stiffness with comparable stiffness values for the two high-speed milling machines. The hydrostatic bed exhibited more damping than the mechanical bed in low frequency ranges with damping values of hydrostatic system reaching three times that of the mechanical bed. The dynamic response expected for a hydrostatic system, which is high stiffness and damping, was confirmed

1. Introduction

   Recent requirements of manufacturing have been to increase productivity and the quality of produced parts simultaneously. The demand for high quality means that the overall accuracy of the machine tool must be high. Productivity and quality are contradictory goals

   because easy movement of workpiece or tool and achieving accuracy are conflicting objectives. The challenge facing the machine tool industry is to be able to develop new technologies to meet increasing specifications and requirements from consumers.

   High speed machining (HSM) is revolutionizing mechanical production especially in the area of aircraft manufacturing where high material-removal rate (MRR) is of great importance. The biggest challenge to HSM is to achieve good surface quality of machined parts with high material removal rate. The life span of the High Speed Milling Machine must also be comparable with a traditional or conventional type. These demands require continued improvement of existing equipment used for HSM.

   Surface quality and surface integrity of machine parts are the direct results of a stable, rigid axis design and construction for which vibration effects due to the dynamics of cutting can be eliminated. When machining, it is highly desirable to cut with high forces, (i.e., large depths of cut and at high speeds with minimal errors to the workpiece). For a machine tool to handle these large forces it must be very stiff and should have good damping characteristics. The stiffness of a machine tool allows for small vibrations under heavier forces, whereas the damping helps to dissipate excess energy generated by these forces. In short, high stiffness and damping help to reduce the errors associated with machining and allows for high metal-removal rates and longer tool life

   Undesirable attributes of a machine tool are errors in motion and variable stiffness in different configurations. Many of the errors come from the manufacturing of the machine tool itself. Errors in form

   and surface finish of any of the guideways can lead to undesirable errors of motion in the machine tool and to points in the machinable space where the stiffness is less than desired.

   1. Accuracy of Machine Tools

      Machine tools are made up of linear and rotary axes combined in series with the work and cutting tool. Machine axes inherently have unintended motions, thereby introducing errors. The error motions of each of the axes combine to cause a resultant volumetric error between the work piece and the tool. These can be decomposed into six fundamental degrees of freedom corresponding with the chosen coordinate system. These errors are referred to as displacement, straightness, roll, pitch, and yaw for a linear axis. Figure 1-1 shows the different types of errors associated with a single axis of a machine tool. Displacement error is the error along the intended- motion direction of the axis. Straightness error is in the two directions perpendicular to the motion axis of interest. Rolling, pitching, and yawing are the rotational motions about the coordinate axes.

      Rotational errors are due to inadequate stiffness in the entirety of a machine tool bed. In traditional design of a machine tool, bearings with mechanical contact between elements are used as intermediaries between the moving parts of the machine and the stationary base. The bearings, supporting a carriage, ride along an accurately machined way and act as springs between the moveable carriage and the stationary ways. Roller- pack, re-circulating ball, and sliding element are examples of bearings that use mechanical contact as a method of supporting as well as stiffening the machine tool. While mechanical-contact bearings can be made to allow high stiffness between the carriage and the base, they are susceptible to the before-mentioned errors in the form and surface finish of the ways they ride on.

   2. Hydrostatic Bearings

      An alternative to mechanical contact bearings is the use of hydrostatic bearings. Hydrostatic bearings use a thin film of high-pressure fluid to support a load (in this case the carriage). Hydrostatic systems on machine tools are evolving as next-generation designs to achieve this difficult task of high surface finish. They are known to have such properties as high geometric precision, high speed of response, suppression of

      stick-slip effect at low speed, minimum friction factor, low heat intake, durability, high stiffness, high vibration damping and running accuracy. This research determines the stiffness of a hydrostatic carriage, bearing and guideway on a High Speed Machine tool and comparison is also made to that of a mechanical contact bearing.

2. Theory

   Experimental modal analysis, basically, is a procedure of experimental modeling. The primary purpose is to develop a dynamic model for a mechanical system using experimental data. The data generated is a set of response properties such as measured frequency response functions. Generally a modal model consisting of natural frequencies, modal damping ratios and mode shape vectors is obtained from the frequency response functions. Complex vibrating systems usually consist of components that possess distributed energy-storage and energy dissipative characteristics. In these systems, the inertial, stiffness, and damping properties vary (piecewise) continuously with respect to the spatial location. Consequently, partial differential equations with spatial coordinates, e.g. Cartesian coordinates (x, y, z) and time t as independent variables, are necessary to represent their vibration response.

   A distributed (continuous) vibrating system can be approximated (modeled) by an appropriate set of lumped masses properly interconnected using discrete spring and damper elements. Such a model is called a lumped-parameter model or discrete model. An immediate advantage resulting from this lumped- parameter representation is that the system equations become ordinarily differential equations. Often, linear springs and linear viscos damping elements are used in these models. The method is based on the fact that these idealized systems (models) have preferred frequencies and geometric configurations (or natural modes), in which they tend to execute free vibration. An arbitrarily response of the system can be interpreted as a linear combination of these modal vibrations; and as a result, its analysis can be conveniently done using modal techniques.

   Modal analysis is an important tool in vibration analysis, diagnosis, design, and control. In some systems, mechanical malfunction or failure can be attributed to the excitation of their preferred motion such as modal vibrations and resonances. By modal analysis, it is possible to establish the extent and location of severe vibrations in a system. For this reason, it is an important diagnostic tool. For the same reason, modal analysis is also a useful method for predicting impending malfunctions or other mechanical problems. Structural modification and substructuring are techniques of vibration analysis and design, which are based on modal analysis. By sensitivity analysis methods using a modal model, it is possible to determine what degrees of freedom of a mechanical system are most sensitive to addition or removal of mass and stiffness elements. In this manner, a convenient and systematic method can be established

   for making structural modifications to eliminate an existing vibration problem or to verify the effects of a particular modification. A large and complex system can be divided into several subsystems that can be independently analyzed. By modal analysis techniques, the dynamic characteristics of the overall system can be determined from the subsystem information. This approach has several advantages, including: (1) subsystems can be developed by different methods such as experimentation, or other modelling techniques and assembled to obtain the overall model; (2) the analysis of higher order system can be reduced to several lower-order analyses; and (3) the design of a complex system can be done by designing and developing its subsystems separately. Modal control, a technique that employs modal analysis, is quite effective in the vibration control of complex mechanical systems.

3. Experimental Set-up

   The experiment setup included a computer, an ICP accelerometer, two amplifiers, and an instrumented impact hammer together referred to as dynamic analyzer. The computer ran on the TXF software. The TXF 98 version developed by Manufacturing Laboratory Incorporated was used. It is specifically designed for modal testing of machine tools using impact test methods. It utilizes data acquisition and computing routines to produce a transfer function in 10, 5, 2.5 KHz ranges. TXF monitors two signals to produce transfer functions when the impact is made on the table with the hammer; a high impedance charge signal is generated by the piezoelectric sensing element. The hammer provides a simple means of exciting the table into vibration. It has a set of different tips and heads, which serve to extend the frequency and force level ranges for testing a variety of structures. Integral with the impacting end of the hammer is a force transducer, which detects the magnitude of the force, felt by the impact end, which is assumed to be equal and opposite to the force experienced by the table. The magnitude of the force applied is determined by the mass of the hammer and the velocity with which it is moving when it hits the table. The TXF 98 has an inbuilt program to reject impacts that resulted in an overload.

   An accelerometer is a linear seismic transducer, which produces an electric charge proportional to the applied acceleration. A simple model of an accelerometer is shown in

   A mass is supported on a piece of piezoelectric ceramic crystal which is fastened to the frame of the transducer

   body. Piezoelectric materials have the property that if they are compressed or sheared, they produce an electric potential between their extremities, and this electric potential is proportional to the amount of compression or shear. As the frame experiences an upward acceleration it also experiences a displacement. Because the mass is attached to the frame through the spring-like piezoelectric element, the resulting displacement it experiences is of different phase and amplitude than the displacement of the frame. This relative displacement between the frame and mass causes the piezoelectric crystal to be compressed, giving off a voltage proportional to the acceleration of the frame.

   Selection of an accelerometer for testing was tricky. The Machine Tool Research Center has a wide range of accelerometers with varying sensitivities. Sensitivity ranges are from 5.30mV/g(1851m/s2/V) to 1086mV/g(9.03m/s2/V). High sensitivity is desired to get accurate readings of frequency response functions. The higher the sensitivity the heavier and larger the transducer and the more space required by the accelerometer. After some test using different accelerometer, an accelerometer of 101.3 mV/g (98.7m/s2/V) was selected because of its good response in the frequency range of vibration of the table.

   In order to determine the dynamic characteristic of the hydrostatic bed, transfer functions were taken along and across the entirety of the bed. A transfer function is a function that represents the cause and effect relationship of a force and the vibration it creates. A transfer function can either be a direct transfer function or a cross transfer function. A direct transfer function is obtained by applying a force to a point and measuring the vibration at the same point. A cross transfer function is obtained by applying a force at one point and measuring the response at a different point. When the impact is made on the table with the hammer, a high impedance charge signal is generated by the piezoelectric sensing element. This is then converted by the built-in electronics in the ICP accelerometers into usable low impedance voltage signals that can be readily transmitted over ordinary two wire or co-axial cables. The electronics within the ICP accelerometers require excitation power from a constant current regulated, DC voltage source. So one DC voltage supply is provided to each accelerometer. The cables then run into the FFT analyzer where the computer performs a Fast Fourier Transform (FFT) on both signals. Then a double integration is performed on the accelerometer signal in order to convert from

   acceleration (m/s) to displacement (m). The displacement signal is divided by the hammer signal to give the transfer function across a specified frequency range. The real and imaginary part of the transfer function is plotted against frequency. The natural frequency of the system can be read directly from the transfer function plot. By reading peak points of the imaginary value the natural frequency for various modes of vibration is obtained. The difference between the frequencies f of the two peaks of the real part of the transfer function along with the natural frequency

   can be used to calculate the damping ratio () of the

   mode.

   =

   2

   where fn is the natural frequency of vibration of the mode.

   The dynamic stiffness, k is found from the imaginary peak value, H, referred to as a dynamic flexibility.

   = 1

   2

   The first set of experiments captured modes of vibration along the bed of the hydrostatic table. The High Speed Milling Machine hydraulic system was turned on and allowed for flow of oil into the four bearings supporting the table. The circulating pressure was recorded as 55 bar and some time was allowed for steady conditions to be achieved. The oil temperature was also recorded as 78oF. The accelerometer was placed at one end of the table and the impact hammer was applied at five different points along the table. It includes one direct transfer function and four cross transfer functions. The location of the accelerometer was changed to the other end of the table and an impact was made at an arbtrary point to determine whether a mode has not been recorded. The explanation to this is that if the accelerometer is positioned at or very close to a node of one or more of the tables mode, it will be difficult to make an effective measurement of that particular mode.

   The second set of experiments was performed across the bed of the hydrostatic table. Three impacts were

   made. The position of the accelerometer was placed in the line of impact to capture as much modes of vibration as possible..

   After a careful study of results it was decided that transfer functions should be obtained for the entire table because it was possible that coupling modes were present in the vibration of the table. Each result was stored in the computer for analysis.

   The study of the dynamics of the hydrostatic bed involves comparing to an already existing mechanical bed supported by roller bearings and guideways installed on.

   In order to compare effectively, a set of transfer functions was obtained from the purely mechanical bed. The mechanical bed has its bed held vertically parallel to X and Y axis. Transfer functions were taken along the Y-axis column and the center row of the X- axis. Similarly five transfer functions were taking across the mid-section of the bed.

4. Results and Discussion

   Results are in the form of a frequency response functions and show plots of imaginary and real values against frequency in the range of vibration of the table. Only one frequency response functions have been reported for both hydrostatic bed and mechanical bed. These are transfer functions recorded along the X-axis in the center of the beds. Refer to end of publication for transfer functions. Figure1 and Figure 2.

   In this session, modal parameters that is stiffness and modal damping ratios are calculated for each result. These are summarized in a table below each figure. Also modal damping ratios and stiffness were calculated for these set of experiments and summarized in tables preceding figures. They were obtained when the impact hammer was applied across the bed. The piezo- electric accelerometer was placed at the end of the bed at mid position along the bed to capture as much frequency of excitations as possible.. The chapter also includes results obtained from experiments performed on the purely mechanical bed. These experiments were performed to have a full data of vibration characteristics of the entire hydrostatic bed. Results show that the vibration characteristic of the hydrostatic bed depends on the position of impact. Large operations on hydrostatic bed will require that the whole bed is used. Subsequently it was thought of

   that knowledge of the characteristic of vibration at all points on the bed could be of importance. The position of the accelerometer was kept at the end along the bed as in the case of experiment series A. The transfer functions here shows results as obtained from the purely mechanical. The impact hammer was applied at various points on the table with the accelerometer kept at one point. Results have been shown for transfer function obtained along the bed in the mid section of the table in the direction of the X-axis only. It was decided not to publish all of mechanical bed results because the modes of vibration were consistent over the entire table. The results here are similar to experiment series A of the hydrostatic carriage. Consequently analysis will be done in only one direction of the table in this case the X-axis. The intent of the study is to identify the modal parameters of the hydrostatic table. These include all important natural frequencies, modal damping ratios and mode shapes. The mode shapes of hydrostatic are drawn for each natural frequency. Also the mode shape vector is determined. A summary of all the modes identified in the experiment has been summarized in Table 4-1.

   A dominant mode at a frequency of 185Hz was observed along the bed in the direction of the X-axis. It is noted from the modal shape vector that a bending type of vibration is present. Modal stiffness ranges from 1.24E+09 N/m and 5.99E+09 N/m. Damping ratios are in the order of 0.15. The mode

   shape vector with normalization mid point on the table is

   0.91

   0.36

   1

   0.63

   0.41

   3.66

   2.38

   1

   2.31

   1.95

   Three frequencies were distinct in experiment (b) where the hammer was applied across the bed in the direction of Y-axis. A dominant mode at a frequency of 140Hz was observed along the bed in this direction.. It can be inferred from the modal shape vector that a translational type of vibration is present. Stiffness ranges from 1.87E+08 N/m and 2.33E+08N/m. Damping ratios are in the magnitude of 0.1. The mode shape vector with normalization at point 2 on the table is

   1.12

   1

   1

   1.27

   The second dominant mode in the direction of the Y- axis across the bed is at 540Hz.. A vibration form of the rolling type is identified in this mode of vibration about the x-axis. Calculated stiffness is in the range of 1.72E+09 N/m and 2.24E+10 N/m. Modal damping ratios are in the magnitude of 0.03. The modal shape vector with normalization at a point on the table is

   6.91

   A second distinct and dominant mode was found at a 685 Hz. along the bed in the direction of the X-axis. A vibration form of the pitching type is identified in this mode of vibration. Calculated stiffness is in the range of 8.17E+09 N/m and 3.94E+10 N/m. Modal damping ratios are in the

   magnitude of 0.03.The modal shape vector with normalization at mid point on the table is given as

   1

   1

   9.91

   A third distinct and dominant mode was found at 838 Hz. along the bed in the direction of the Y-axis.. A vibration form of the translational type is identified in this mode of vibration. Calculated stiffness

   are in the range of 5.60E+09 N/m and 9.62E+09 N/m. Modal damping ratios are in the magnitude of 0.03.The modal shape vector with normalization at a point on the table is given by

   1.46

   1

   1

   1.92

   Mode shapes were drawn for mechanical table as well. As indicated, the mechanical table table showed consistency in natural frequency over the entire table.

   A combination of a bending and translational motion is present. Calculated stiffness are in the range of 3.11E+09 N/m and 1.13E+10 N/m. Modal damping ratios are in the magnitude of 0.02. The mode shape vector with normalization at mid point on the table is

   4.52

   2.59

   1

   Subsequently mode shapes have been drawn for transfer functions only along the bed in the direction of the x-axis. This can be compared to experiment series

   (a) of the hydrostatic bed.

   A dominant mode of the mechanical table table is seen

   given by

   1.27

   3.34

   at a frequency of 268 Hz. This is the lowest mode of vibration of the table.. It is representative of a translational vibration motion. . Calculated stiffness is in the 4.07E+08 N/m and 9.00E+09 N/m. Modal damping ratios are in the magnitude of 0.04. The mode perpendicular to the plane of the table. Calculated stiffness is in the range of shape vector with normalization at point on the table is given by

   A bending mode is observed for the last mode of vibration. This occurs at a frequency of 820 Hz.. Calculated stiffness is in the range of 5.35E+09 N/m and 1.05E+10N/m. Modal damping ratios are in the magnitude of 0.04. The mode shape vector with normalization at mid point on the table is given by

   2.53

   0.83

   0.99

   0.98

   1

   1

   1.23

   0.28

   /p>

   0.86

   0.61

   Another dominant frequency is observed at 375 Hz.. Clearly this indicates a pitching about the X-axis or a rolling about the Y-axis. Calculated modal stiffness is in the range of 2.45E+09 N/m

   and 9.91E+09 N/m. Modal damping ratios are in the magnitude of 0.04. The mode shape vector with normalization at a point is given as

   25.54

   18.41

   1

   14.35

   26.76

   Table 4-1 shows a summary of the vibration characteristic of the hydrostatic bed as obtained from experiments. A total of twelve distinct modes are present. Five of these are dominant and the table shows their corresponding stiffness and modal damping ratios.

   High damping is recorded here. At a frequency of 185 Hz, a damping ratio of 0.15 was recorded. It shows a good damping characteristic of the hydrostatic bed. The high stiffness exhibited is also a good attribute of the system.

   A summary of the vibration characteristic of the mechanical bed of mechanical table is shown in Table 4-2. A total of six distinct modes are present. Four of these are dominant and the table shows their corresponding stiffness and modal damping ratios. Though the mechanical system did not record a high damping value, it has high stiffness.

   Each of the two tables exhibited high stiffness as can be seen from the results above. It is interesting to note that the hydrostatic system have high damping values especially at low frequencies. In some modes a modal damping ratio of 0.15 is obtained which is about four times that obtained on a mechanical bed. Overall the hydrostatic system has a better damping characteristic than the mechanical bed.

   To determine the stiffness of the bearings, the rigid body motion of the table is studied. The rigid body motions of the hydrostatic table are at frequencies 540 Hz and 685 Hz. All four hydrostatic bearings have the same size with equal volumetric flow of damping oil, implying that the total bearing stiffness of the table is equal to the sum of individual stiffness of each bearing.

   The rotational plus translational motion of the table is considered. Here two rigid modes are of interest, the rigid mode due to pitching of the table and the mode due to rolling.

   k k mw2 (k L k L )

   1 2 1 1 2 2 0

   (k L k L ) k L2 k L2 Jw2

   1 1 2 2 1 1 2 2

   Since the table is rigid and symmetrical in this equation

   with equal stiffness, that is

   Distances from one bearing position to the other along the table and across the table respectively. The stiffness k is obtained from the rotational part of the equation

   k k k and L L L / 2

   (2k w2m)(kl2 / 2 w2 J ) 0

   1 2 1 2

   Where k = measured stiffness of bearings

   And L = distance between two bearing support points of the table.

   A new determinant is obtained as

   (kl2 / 2 w2 J ) 0 , or

   2w2 J

   , that is

   2k w2m

   0 0

   k l 2

   0 kl2 / 2 w2 J

   And that; (2k w2m)(kl2 / 2 w2 J ) 0

   or 2k2l2 / 2 2kw2 J kl2w2m / 2 w4 Jm 0

   (Jm)w4 (2kJ kl2m / 2)w2 k 2l2 0

   J m*(a2 b2 ) /12

   where a= thickness of table and b = length of table.

   The J calculated is the moment of inertia of table along the table and determines the natural frequency in the pitching mode.J is also determined for the case of rolling motion.

   Measured pitching frequency = 680 Hz. Measured rolling frequency = 540 Hz.

   From the relation

   Mode #	Natural Frequen cy (Hz)	Dynamic Stiffness (N/m)	Modal Dampi ng Ratio	Type of error with respect to X-axis
   1	142	2.05E+08	0.1	Straightness (Z)
   2	185	2.00E+09	0.15	Straightness (Z)
   3	239
   4	267
   5	395
   6	449
   7	474
   8	540	1.72E+09	0.03	Rolling(Y)
   9	602
   10	685	1.48E+10	0.03	Pitching
   11	734
   12	838	5.60E+09	0.03	Straightness (Z)

   Mode #	Natural Frequen cy (Hz)	Dynamic Stiffness (N/m)	Modal Dampi ng Ratio	Type of error with respect to X-axis
   1	142	2.05E+08	0.1	Straightness (Z)
   2	185	2.00E+09	0.15	Straightness (Z)
   3	239
   4	267
   5	395
   6	449
   7	474
   8	540	1.72E+09	0.03	Rolling(Y)
   9	602
   10	685	1.48E+10	0.03	Pitching
   11	734
   12	838	5.60E+09	0.03	Straightness (Z)

   (2k w2m)(kl2 / 2 w2 J ) 0

   The k value obtained for pitching is 1.3386+009N/m. The stiffness of each bearing is k. Estimated bearing stiffness is 0.6693+009N/m. For the case of rolling, obtained is 1.7859+009N/m. Estimated bearing stiffness is 0.8925+009N/m. The variability in stiffness may be due to unequal tolerances in the bearing gaps of individual bearings.

   Table 4-1 Summary of modal analysis of hydrostatic bed.

   ,

   from which the stiffness of the each bearing is determined. A schematic representation of the table is shown in Figure 6-10 and measured table values and calculated table properties are given below.

   a = 0.0762, b = 1.016, m = 280, c = 0.7112.

   1

   1

   J m *(a2 b2 ) / 12,

   l1 0.8128, l2 0.5588,

   J2 m *(a2 b2 ) / 12.

   Table 4-2 Summary of modal analysis of HSM1s mechanical bed.

   Mod e #	Natura l Freque ncy (Hz)	Dynamic Stiffness (N/m	Moda l Damp ing Ratio	Type of error with respect to X- axis
   1	268	7.25E+08	0.03	Straight ness (Z)
   2	375	6.75E+09	0.04	Rolling (Y)
   3	530	2.50E+09	0.04
   4	682	7.45E+09	0.02	Straight ness (Z)
   5	820	2.50E+09	0.04	Straight ness (Z)

5. Conclusion

Results obtained with experimental modal analysis proved to be qualitatively and quantitatively excellent. The dynamic parameters of the hydrostatic table are good, the information content of excitation and response signal is abundantly sufficient, the reliability of modl parameter extraction methods is accepted, the tables constraints respond to good modal analysis practice requirements. It was observed that both hydrostatic and mechanical bed have high stiffness with comparable dynamic stiffness values for the two high-speed milling machines. The hydrostatic bed exhibited more damping than the mechanical bed at low frequencies with damping values of hydrostatic system reaching four times that of the mechanical bed. The dynamic response expected for a hydrostatic system, which is high stiffness and damping has been confirmed. The results obtained from the calculation of stiffness in the hydrostatic bearing are acceptable.

10. References

1. A.B. Smith, C.D. Jones, and E.F. Roberts, Article Title, Journal, Publisher, Location, Date, pp. 1-10.

2. Jones, C.D., A.B. Smith, and E.F. Roberts, Book Title, Publisher, Location, Date.

1. Beards, C., Vibrations and Control Systems, Ellis Horwood, New York, 1988.

2. De Silva C., VIBRATION Fundamentals and Practice. CRC Press, New York, 2000.

3. Ewins, D. 1994. Modal Testing: Theory and Practice, John Wiley, New York, 1994.

4. Fuller, D. Theory and Practice of Lubrication for Engineers. Wiley, New York, 1984.

5. Ginsberg, J., Mechanical and Structural Vibrations, John Wiley, New York, 2001.

6. Kakiel, A,. Process Improvements in Milling Titanium Alloy Ti-6Al-4V, Masters Thesis, University of Florida, Gainesville, 1999.

7. Peterson, D. Design of a Hydrostatic Bearing System for the McMaster 3-Axis Milling Machine, Masters Thesis, University of Florida, Gainesville, 2000.

8. Slocum, A. Precision Machine Design. Prentice Hall, New York, 1992.

9. Tlusty, G., Manufacturing Processes and Equipment. Prentice Hall, New Jersey, 2000.

Figure 4-1 Transfer function of the hydrostatic bed at Mid Point

Figure 4-2 Transfer function of the Mechanical bed at Mid Point