 Open Access
 Authors : M. A. Nasser , M. A. Asy , K. M. Khedr , R. M. Hashem
 Paper ID : IJERTV9IS040366
 Volume & Issue : Volume 09, Issue 04 (April 2020)
 Published (First Online): 20042020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mechanical Redesign of Egyptian Made Milling Machines for Modification and Updating by using CAD
M. A. Nasser
Professor of Machine Dynamics, Production Engineering and Mechanical Design
Department, Faculty of Engineering, Menoufia University, Shebin El Kom, Egypt.
K. M. Khedr
Associate Professor,
Production Engineering and Mechanical Design Department, Faculty of Engineering, Menoufia University, Shebin El Kom, Egypt.
M. A. Asy
Associate Professor,
Production Engineering and Mechanical Design Department, Faculty of Engineering, Menoufia University, Shebin El Kom, Egypt.
R. M. Hashem
Instructor,
Production Engineering and Mechanical Design Department, Faculty of Engineering, Menoufia University, Shebin El Kom, Egypt.
Abstract: For decades the Egyptian made machine tools have not updated since they were first introduced in 1960. The goal of this study is to introduce a working model for the modification of the present produced milling machine. Modification of the milling machine column is carried out by redistribution of the material were used in the casting of its column. The same mass of the material will be reshaped to be honeycomblike rather than the present hollow structure. Several scenarios for the design of gearboxes are carried out. For the design and updating of milling machines, an integrated set of Ansys and Solidworks beside tailored written Visual Basic codes are used. Gearlab and VB codes are used for calculating the cutting parameters, gearboxes design, kinematic and calculating forced vibration frequencies. The redistribution of the material of the machine column gives light, stiff and welldamped structure. The redesign proposal gives versatile design and more reliable modifications.
Keywords: Milling Machine; Modal Analysis; Redesign; Vibration; CAD; Machine Tools; Structure; Gearbox.

INTRODUCTION
The goal of this study is to introduce a working model for the modification of the present Egyptian made machine tools. For decades the Egyptian made machine tools produced by Hulwan Company for Machine Tools have not been updated since they were first introduced. Hulwan Company for Machine Tools is a Factory 999 at Ein Hulwan region, Cairo. This military factory is affiliated with the Ministry of Military Production. The factory produces several civilian products. Civilian products include lathes, drilling machines, grinders, wood sawing machines, shapers, milling machines. Around the world, the great progress in manufacturing traditional machine tools and a competitive market put the stagnant or old fashion Egyptian made machine tools products under pressure. So, the modification and updating of the Egyptian made machine tools are essential. The universal milling machine 6P80 is selected for this study.
Machine tools fundamentals, theory and application are studied in detail {17]. Computeraided design is widely used
in machine tool design and dynamic analyses by using Ansys software [8 – 9]. References [10 – 44] concerned with the study and application of vibration, modal analysis, static and kinematic analyses of machine tools structures and gearboxes. Several references, [10 – 27] studied milling machines structure's static and dynamic analyses, as well as kinematic analysis, has received a lot of attention. Other applications of computeraided design on other machine tools such as shaper [28] & [29], Gaint [30] are done. References from [32] to [44] studied in detail the static and dynamic behavior of machine tools gearboxes. Bearings as one of the most important parts in machine tools are considered, as the bearing frequencies arising from some faults represents excitation frequencies, [45].
A comparison of the experimental obtained from experimental modal analysis and theoretical data obtained from Ansys revealed that the natural frequencies and corresponding mode shapes are in good agreement for the gearbox structure model. The well correlate results between experimental and theoretical prove the validation of Ansys to construct the good design, [32]. The validated FE model by Ansys could be used in the design process with greater confidence that the calculated mechanical properties of the properties' machine tools structures and/or parts. It is reliable. The accuracy of the results highly depends on user experience and knowledge. The designer should know the physics of his problem and use his solid engineering judgment and intuition to understand what is going on in his design.
This study started with the calculation of cutting speed and power required for the milling process. The conventional methods of calculation with the associated equations of metal cutting in the milling process as well as the nomogram, [3]. A step forward was given by building electronic nomograms and specialpurpose calculators in the VB6 language. These computerized tools make the design process easy, faster, accurate and more reliable. Soon after this, the structure of the machine was modeled in Solidworks and dynamically
analyzed by using Ansys software. In this paper, the milling machine structure is modified by redistribution of the mass of the machine column by using part of this material as ribs in the form of honeycomb, as the reference [14] & [31] done before. Moreover, the study and modification of the milling machine main and feed gearboxes were done. Three scenarios of optimum design of the main gearboxes were proposed and analyzed for the design parameters of gears and shafts. These results of other models of the main gearbox were obtained by using a special code written by a student in (VB6). Moreover, these gearboxes were modeled in Solidworks software and analyzed by using Ansys software. Generally, the results obtained from an integrated set of ANSYS and Solidworks for design modification, besides Gearlab [39] and tailored written Visual Basic codes for calculating cutting process and gearboxes design, kinematic and forced vibration frequencies give versatile design tool and reliable design modifications.

CUTTING FORCES, TORQUE & POWER

Analysis
Software for Calculating Power and Force in slab milling Process using VB6 is written, tested and applied. The cutting parameters are cutter diameter, cutter number of teeth, spindle revolution, feed rate, cutting depth and width. The cutting speed is calculated from the following:
V=.d.n
(1)
where n is spindle revolution number, d is the cutter diameter and V is cutting speed.
The cutter's angle of rotation relies on the cutter diameter and the depth of cut and it can be calculated as follows [5]:
= 360Â°/
(2)
(3)
where is pitch angle of the cutter is the angle of rotation of the is the number of teeth of the cutter, d is cutter diameter and a is the depth of cut.
When the cutter tooth has entered the workpiece material by an angle the chip thickness as follows can be calculated:
ct= (mm)
(4)
where is the feed per tooth.
s = st. nt.n (mm/min)
(5)
s = st. nt.n (mm/min)
(5)
ct = . (mm)
(6)
ct = . (mm)
(6)
The feed pe minute is equal to: And the chip thickness:
Then maximum chip thickness can be calculated as follows:
= . (mm)
(7)
where s is the feed rate and is the maximum chip thickness.
The middle chip thickness can be calculated as follows:
= . sin (mm)
(8)
The mean power required at cutting edge can be calculated as follows:
= .a. b. s (kw)
(9)
where is the specific cutting resistance.
The mean cutting force acting at the cutter can then be calculated:
As well as
the mean torque:
(N.mm)
(11)
the mean torque:
(N.mm)
(11)

Software Code Construction
Nomogram Software is a code constructed by using Visual Basic language. The cutting parameters; workpiece material, cutter speed, a width of cut, depth of cut and feed rates. The relevant parameters either to be read from the given hard copy nomogram, [3] or to be calculated from the given computerized nomogram. The computerized nomogram is written in VB6. Visual Basic code is constructed to be used in slab milling process metal cutting calculations. The code uses the horizontal and vertical scroll bars. Scroll Bar Events are used in this code. The designer uses this event to retrieve the cutting parameter after any changes in the scroll bar. As the user changes the scroll bar value, both the cutting fore and power, you have to calculate this, in integer format and animated lines on the electronic nomogram displayed on the computer screen as shown in Figs. 1 &2.
The formula for cutting force and power is given in the analysis. Scroll eventtriggered continuously whenever the scroll box is being moved. Clicking an end arrow increments the scroll box a small amount, clicking the bar area increments the scroll box a large amount, and dragging the scroll box (thumb) provides continuous motion. Using the properties of scroll bars, the designer can completely specify how one works. The scroll box position is the only output information from a scroll bar.
Another code is written to calculate the vertical and end milling processes as shown in Fig 3. The user will input in scroll boxes the cutting parameters and workpiece material. The outcomes of the code will appear scroll boxes. Case study of cutting speed V= 30480 (mm/min), tool diameter d=
101.6 (mm), number of teeth of the cutter nt = 8, width of cut b= 67.3 (mm), depth of cut a= 4.064 mm and feed rate s=
152.4 (mm/min) is used in this article.

Results and Discussion
The two Softwares were built and tested for calculating the cutting forces and required power for the milling machine. The results obtained from the software were compared with those obtained from manual calculations by using the calculator and nomograms given in textbooks [3]. Many cases of cutting conditions were investigated. Table (1) presents the power & cutting fore estimated by manual nomogram (hard copy), nomogram software (VB( and equations software (VB) for steel alloy, carbon steel, light alloy. The percentage error was 4.5 % in manual nomogram (hard copy). These results proved that the proposed given computerized nomogram written in VB(6) is an accurate, faster, reliable and expandable tool for more new parameters to be added.
Power (kw)
Manual Nomogram (hard copy)
steel alloy
carbon steel
light alloy
3.8
2.2
1
Nomogram Software (VB)
3.8
2.3
1
Equation Software (VB)
3.8
2.3
1
Cutting force (N)
Manual Nomogram (hard copy)
steel alloy
carbon steel
light alloy
7480.3
4331
1968.5
Nomogram Software
7480.3
4527.6
1968.5
Equation Software
7480.3
4527.6
1968.5
Power (kw)
Manual Nomogram (hard copy)
steel alloy
carbon steel
light alloy
3.8
2.2
1
Nomogram Software (VB)
3.8
2.3
1
Equation Software (VB)
3.8
2.3
1
Cutting force (N)
Manual Nomogram (hard copy)
steel alloy
carbon steel
light alloy
7480.3
4331
1968.5
Nomogram Software
7480.3
4527.6
1968.5
Equation Software
7480.3
4527.6
1968.5
Table (1) The Power & Cutting Fore Estimated By Different Methods.
(N)
(10)
Fig. 1. Welcome Screen of the Milling Nomogram.
Fig. 2. Slab Milling Nomogram.

Inputs for Carbon Steel

Results of Carbon Steel.

Fig. 3. Milling Process Calculator.


STRUCTURE MODIFICATION
This study presents four different suggested structure modifications for the milling machine to enhance its performance. These modifications can increase the mitigation of harmful vibration. Harmful vibration adversely affects the milling process. The mass of the milling machine column is 1175 kg. 165 kg is subtracted from the base of the column and added again in the form of honeycomb form to be as reinforcement ribs added to the core of the machine column. The redistribution and the reshaping of the column are shown in Fig 5. The current height of the machine base is 130 mm. This height reduced to 80 mm through the suggested four modifications. Figure (5) summarizes the four suggested modification scenarios. The first one is adding a 66 matrix of ribs with the same thickness equals to (10 mm). Also, the second one is adding a 55 matrix of ribs with the same thickness is (12 mm). While the third one is adding a 33 matrix of ribs with a thickness equals to (12 mm) in addition to one diagonal cross ribs with the same thickness equals to (12 mm). Finally, the fourth one is adding a 33 matrix of ribs with a thickness equals (7 mm) in addition to 16 internal diagonal cross ribs with the same thickness (7 mm). The surface area of the base and column before adding ribs is 5.43 m2. The surface area are 9.33, 8.65, 8.29 & 11.11 m2 for modifications #1 to #4. This means that the increase in surface area are1.72, 1.59, 1.53 & 2.05 times higher than the original one. This increase in the surface area could play an important role in structural heat transfer.

Static analysis
Static, modal and frequency analyses were done by using Ansys software. The evaluation of the static loads' effect on structure deformation is {x}, this deformation can be evaluated as follows:
[Q]{x}={F} (12)
where; {F} denotes to the static force applied to the spindle nose (which equals 4000 N as assumed value) and [Q] is a constant. The displacements {x} can be evaluated under assumptions as; small deflction theory is used, linear elastic material behavior is assumed, there are no varying forces and inertial effects (mass and damping) via considering linear elastic material behavior in addition to some nonlinear boundary conditions may be included.
The total deformation (Utotal in Âµm) can be calculated as follows;
(13)
where; (u) are the deformation components in the orthogonal directions (x, y, and z).
Fig. 4. Physical Universal Milling Machine. Solidworks Model & FEM Ansys Model.
Modification (#1). Modification (#2).
Modification (#3). Modification (#4).

Results of Static Analysis
Original Machine Column & Base.
Fig. 5. The Original and Proposed Modifications in the Universal Milling Machine.
elasticity is recommended. Higher stiffness mitigates the milling machine's deformation. The results show that the
Higher rigidity, higher natural frequencies, and lightweight machine tool structures are the best from static, dynamic and feasibility points of view. These goals have to be considered while designing the column of the milling machine. Stiffness is the degree to which the structure resists deformation as a response to an applied force. The stiffness is defined as the ratio between the applied force and deflection. The stiffness of a structure is of major significance in machine tool structures, so the young's modulus is the most important property considered when selecting a material. Since the deformation in machine tool structures is undesirable, a high modulus of
maximum deflection occurs at the spindle head. Hence, the spindle head can be marked as the weakest location from the characteristic of static behavior.
The static deformation obtained from Ansys is given in Table (2). the percentage change is calculated from the following form:
Rate of Improvement (%) = [Modified value – Current value] / Current Value x 100.
From table (2) it is clear that the maximum static deformation is reduced from 145.1 Âµm for the present machine tool structure to 102.9, 103.7, 104.2 & 99.7Âµm for the
Modification (#1).
Modification (#2).
Modification (#3).
Modification (#4).
Original Column base.
Fig. 6. The Total Deformation Shape of the Orignal & Modifed Structures.
Modification (#1).
Modification (#2).
Modification (#3).
Modification (#4).
Original Column base.
Fig. 6. The Total Deformation Shape of the Orignal & Modifed Structures.
modifications #1 to # 4. The less deformation the higher stiffness. The maximum deformations at spindle head for modifications #1 to #4 decreases by 29%, 28.5%, 28% & 31.2% lower than the original one. This means that modification # 4 reduced the milling machine structure by 31.2%.

Modal analysis
[M]{Ã¼}+[C] {u } + [K] {u} = {F (t)} (14)
[M]{Ã¼}+[C] {u } + [K] {u} = {F (t)} (14)
Modal is carried out on the milling machine column. When exposed to loads or displacements, all true physical buildings act dynamically. The equations of motion can be written depending on solving the EigenValues problem's equation of multidegree of freedom systems as follows;
[M]{Ã¼}+ [K] {u} = {0} (15)
[M]{Ã¼}+ [K] {u} = {0} (15)
where; {u} denotes to the Eigenvectors, [M] is the mass matrix, [C] denotes to the damping matrix, [K] is the stiffness matrix and F(t) is the applied force. The previous equation can be rewritten in the following form by assuming free vibrations and ignoring damping;
([K] 2[M]) {u} = {0}
(16)
([K] 2[M]) {u} = {0}
(16)
Also, the following equation can be formulated considering the harmonic motion, i.e., u=U sin (t), as follows;
f= /(2)
(17)
f= /(2)
(17)
where; the angular frequency () is equal to the square roots of the eigenvalues. Also, the structure's circular natural frequencies f (in cycles/sec) can be calculated as follows;
f= /(2)
(18)
f= /(2)
(18)
The eigenvectors {u}i represent the mode shape when the system is vibrating at fi, where i=1,2,..,n using Ansys software.
Table (2) The Maximum Deformations at Spindle Head.
Current
Deformation (Âµm)
Reduction in Deformation (%)
Current Machine
145.1
–
Modification # 1
102.9
29
Modification # 2
103.7
28.5
Modification # 3
104.2
28
Modification # 4
99.7
31.2

Modal Analysis Results
Modal analysis is the study of the dynamic properties, "mode shapes, natural frequency and damping", of systems in the frequency domain. Structural modal analysis of milling machine structure uses the overall mass and stiffness of the structure to find the modal parameters. The animated display of the mode shapes of the milling machine column is very beneficial to study and understand the vibration and dynamic behavior. Machine tool designer tends to learn from modal analysis, how potential influence could happen in product quality produced by using the machine tool. Modal analysis is also important in the machine tool structure where the engineer should attempt to keep the exciting natural frequencies away from the frequencies of the machine tool structure and its components. From Tables (2) and (3), the dynamic deflection values of the milling machine are smaller than the static deflections. From Table (3) and Fig 7, it is clear that all modifications have lower dynamic deformations. Table (3) shows the maximum deflection (Âµm): frequency, location, and direction. The detailed study of such parameters is essential to understand the milling machine column dynamic behavior. As can be seen in Table (4), the natural frequencies of all ten first modes of all modifications are higher than those of the original structure. The first mode natural frequencies of the original structure and modifications
#1 to #4 are 73.8, 111.2, 110.7, 109.3 & 112.6 Hz
respectively. The improvement is by increasing the
fundamental natural frequency of the proposed scenarios by 50.6, 50, 48 & 52.5% higher than those of the original structure. Modification #4 is the best stiff structure. So, the fourth modification is the versatile design against harmful vibration. The mode shapes of the milling machine and the milling machine with the modifications added to the machine column are thoroughly investigated to see haw significance in structural behavior is obtained. Fig 8, shows the first mode shape of the original milling machine and the milling machine with modifications. The milling machine can safely use wide cutting conditions as the cutting speed without dangerous resonance occurrene. Generally, the fourth modification has the highest natural frequency among all modifications, the widest range of natural frequencies, and the minimum value of deformation. These good dynamic results are a result of reinforcement achieved by redistribution and reshaping the structure through stiffener elements added, which play as multiple ribs.

Harmonic Response Analysis
By selecting the FRF Analysis option, the integrated harmonic analysis tool opens. The frequency response function at X, Y, and Zdirection of the original and the four modifications of the milling machine column are obtained. Many risky effects on the machine tools can be avoided reliant on the harmonic analysis which depends on vibration analysis. Moreover, the Frequency Response Function (FRF) can be considered as an effective design tool to help the designer. Therefore, the designer can depend on the (FRF) to provide him with the necessary details about the frequency of the high response and the time of its occurrence.
(2[M] + j [C] + [K]){x1 + j x2}={F1 + j F2}
(19)
F=
(20)
(2[M] + j [C] + [K]){x1 + j x2}={F1 + j F2}
(19)
F=
(20)
The harmonic response equation can be expressed as follows;
where; () denotes to the excitation frequency, Fo is the amplitude of the force, which is assumed to be 4 N and [M], [C], in addition to [K] are constants under some assumptions as;
Location of Max.
Deformation
Maximum Deflection (Âµm)
FRF
Direction
Present Machine
Spindle head
22.87 (@ 73.8 Hz)
Xdirection
Table
0.262 (@ 247 Hz)
Ydirection
Spindle head
6.16 (@ 121Hz)
Zdirection
Modif. # 1
Spindle head
1.91 (@170 Hz)
Xdirection
Column
0.447 (@ 442 Hz)
Ydirection
Column
0.287 (@ 442Hz)
Zdirection
Modifi. # 2
Spindle head
1.77 (@ 111 Hz)
Xdirection
Spindle head
0.103 (@ 145 Hz)
Ydirection
Spindle head
0.191 (@145 Hz)
Zdirection
Modifi. # 3
Spindle head
3.61 (@169Hz)
Xdirection
Table
0.338 (@340Hz)
Ydirection
Spindle head
0.288 (@215Hz)
Zdirection
Modifi. # 4
Spindle head
1.65 (@ 174Hz)
Xdirection
Spindle head
0.469(@ 146Hz)
Ydirection
Spindle head
0.867(@ 146 Hz)
Zdirection
Location of Max.
Deformation
Maximum Deflection (Âµm)
FRF
Direction
Present Machine
Spindle head
22.87 (@ 73.8 Hz)
Xdirection
Table
0.262 (@ 247 Hz)
Ydirection
Spindle head
6.16 (@ 121Hz)
Zdirection
Modif. # 1
Spindle head
1.91 (@170 Hz)
Xdirection
Column
0.447 (@ 442 Hz)
Ydirection
Column
0.287 (@ 442Hz)
Zdirection
Modifi. # 2
Spindle head
1.77 (@ 111 Hz)
Xdirection
Spindle head
0.103 (@ 145 Hz)
Ydirection
Spindle head
0.191 (@145 Hz)
Zdirection
Modifi. # 3
Spindle head
3.61 (@169Hz)
Xdirection
Table
0.338 (@340Hz)
Ydirection
Spindle head
0.288 (@215Hz)
Zdirection
Modifi. # 4
Spindle head
1.65 (@ 174Hz)
Xdirection
Spindle head
0.469(@ 146Hz)
Ydirection
Spindle head
0.867(@ 146 Hz)
Zdirection
Fig. 7. Natural Frequencies of Current Design and Modified Columns. Table (3) The Maximum Deflection (Âµm): Frequency, Location & direction.
The small deflection theory is applied, the system undergoes a linear elastic material behavior associated with the damping effect via ignoring the nonlinearities. The applied load {F} and its response {x} have a sinusoidal function at a given frequency ().
All frequency response spectra are obtained and analyzed. The sample is shown in Fig 9 which shows the frequency response functions of the original & modified structures in the ydirection. It is helpful in dynamic analyses of the milling machine Both excitation and structure natural frequencies must have different values for avoiding the risky response effects. Sources of forced vibration in machine tools generated mainly from the rotating elements of the machine itself or transmitted from near vibrating machines. Electric motor, belts, pulley, shafts, bearings, gears, spindles are probably sources of forced vibrations.
Table (4) The Natural Frequencies of the Original & the Modified Machine Columns.
#
Frequencies and Rate of Improvement of the Current Machine Column and Modifications # 1 : 4.
Current
#1
%
#2
%
#3
%
#4
%
1
73.8
111.2
50.6
110.7
50
109.3
48
112.6
52.5
2
97.2
145.2
49.3
144.9
49
144.2
48.3
146
50.2
3
120.8
170
40.7
169.4
40.2
168.9
39.8
173.8
43.8
4
134.4
217.7
61.9
217.5
61.8
214.9
59.8
218.4
62.5
5
156.4
243.6
55.7
242.2
54.8
242.3
54.9
250.6
60.2
6
245.1
340.8
39
340.7
39
339.9
38.6
341.3
39.2
7
259.5
369.2
42.2
368.8
42
367.1
41.4
372.5
43.5
8
281.5
426.9
51.6
426.5
51.5
425.8
51.2
429.1
52.4
9
317.4
442
39.2
441.1
38.9
446.7
40.7
468
47.4
10
336.33
478.6
42.3
480.5
42.8
477.6
42
487.6
44.9
Mode 1 at f=111.2 Hz, Modification (#1).
Mode 1 at f=110.7 Hz, Modification (#2).
Mode 1 at f=109.3 Hz, Modification (#3).
Mode 1 at f=112.6 Hz, Modification (#4).
Original column base (Free webbed) Mode 1 at f=73.8 Hz.
Fig. 8. The Mode Shapes of the Orignal & Modified Structures
Modification (#1).
Modification (#2).
Modification (#3).
Modification (#4).
Original column base (Free webbed).
Fig. 9. The Frequency Response Functions of the Orignal & Modified Structures in YDirection.


GEARBOX DESIGN

Gearboxes Design Calculations:
The multiStage main cutting gearbox is designed in several scenarios. Gearboxes are used to obtain a wide variety of speeds. The multistage gearbox has a large size and a high number of elements. The following steps will illustrate the calculations necessary for the gearbox design process. In machine tools, speed regulation is achieved by more than one method using different progression such as geometric, harmonic, logarithmic and differential progression. The best
method to reach the proximity of speed is the differential progression. However, as an international standard, geometric progression is widely used in machine tools. Speeds are valued as the following in geometric progression:
(21)
(21)
(22)
(22)
First speed: Second speed:
Third speed:
The general formula for calculation speed is:
(24)
where z is the speed number, n is the motor speed and r is the geometric progression base.
Theoretical rotational speeds are determined according to standard geometric progression base (r). the values of progression base are 1.06, 1.12, 1.26, 1.41, 1.58, 1.78, and 2.

Speed Chart
The speed chart is described as a flow chart describes the distribution of gearbox transmission ratios which determines the probability equation to design the gearbox. It has the following form:
z=
(25)
(26)
(27)
(28)
(29)
where p is the number of speed steps and x is the characteristic of the transmission group.

Kinematic Diagram
It is a diagram shows the layout of gears and bearings in the gearbox according to the chosen arrangement equation.

Gears Specifications and Reduction Ratios
The reduction ratios are calculated using the following equations:
(30)
where; n=2x 1 and Z is the number of the gear teeth.
The meshed gear group module is calculated as follows:
(31)
where; is the torsion moment, is the gear width factor, is the bending stress, y is the gear teeth factor is a constant.
(32)
(33)
(34)
(35)
where is the addendum is the dedendum diameter, is the pitch diameter and b is the gear width.

Actual, Theoretical Speeds and Speed Errors
Actual speeds are produced in the spindle of machines due to the motor speed and the mechanical movement of the meshed gears. Actual speeds are calculated by the motor speed, belt pulley reduction ratio and the designed gears using the speed chart. Theoretical speed can be obtained from standard tables
some approximation in design processes that must be less than permissible errors, all can be calculated as follows:
(23) and it is an approximate value of the actual speed. The error between speeds (theoretical and actual) produce is a result of
(23) and it is an approximate value of the actual speed. The error between speeds (theoretical and actual) produce is a result of
: is evaluated from the speed chart.
(36)
Allowed error = Â±10 [r1] %
(37)
: is evaluated from the speed chart.
(36)
Allowed error = Â±10 [r1] %
(37)
nth: is achieved from standard tables.

Numerical Case Study of Main Gearbox Design
A multistage gearbox with 12 speed and probability 6×2 has the following data: nmax = 2240 rpm, nmin = 50 rpm, Nm = 3 kw, Dp = 189 mm, dp = 90 mm, nm = 1720 rpm, m = 2 mm, g,bp,b = 95%, where DP is the diameter of driven pulley and dp is the diameter of the driver pulley.
The progression value (r) is estimated from the following equation:
(38)
(
39)
Fig. 10. The Gearbox Kinematic Diagram.
Fig. 11. Speed Chart of Main Gearbox.

The Software
The software is constructed to design a multistage gearbox of probability (6×2) to give (12) speed. The design includes
speed chart, gears design, actual, theoretical speeds, speed errors and gear meshing frequencies. The Input data is for motor and gearbox speeds, motor power 3 kw and 1667 rpm, maximum speed = 2240 rpm, minimum speed = 50 rpm, gear module = 2 mm, driver pulley diameter = 90 mm, belt reduction ratio = 0.47, bearing efficiency = 0.95, gear efficiency=0.95, pulley efficiency = 0.95.
The software is written in Visual Basic 6 language, it works as a graphic user interface (GUI) to be used easily. The gearbox kinematic diagram, (Fig 10), speed chart of the main gearbox, (Fig 11), the welcome screen of the gearbox design program, (Fig 12), the input data, (Fig13), the gearbox speed chart, (Fig 14), the gears data obtained from the software, (Fig 15), theoretical, actual and speeds errors of the gearbox, (Fig 16), the gear meshing frequencies, (Fig –
17) are given.
Fig. 12. Welcome Screen of Gearbox Design Program.
Fig. 13. The Input Data.
Fig. 14. The Gearbox Speed Chart.
Fig. 15. The Gears Data obtained from the Software.
Fig. 16. Theoretical, Actual and Speeds Errors of the gearbox.
Fig. 17. The Gear Meshing Frequencies.
The 1st probability (4×3) of 12 speeds main gearbox is presented as following; The solution of the gearbox consists of three steps. In the first step, choosing the number of speeds and motor speed, maximum and minimum speed, motor power and progression ratio. In the second step, entering the minimum number of teeth, pressure angle, efficiency, width factor and strength of gears, and shaft strength. The third step is sed to enter width and efficiency of bearing, inclined angles and reduction ratio of the belt, pulley diameter and width and distance between pulleys axis. The input data for gearbox design, [39], (Fig 18), theoretical, actual speeds and percentage errors between them (Fig 19), gears specifications (Fig 20), speed and flow chart. (Fig 21), gearbox shafts design (Fig 22), kinematic diagrams of probability 6×2 (Fig 23) are given to show how a friendly user software is. Modal Analysis of the main gearbox
Fig 18 The Input Data for Gearbox Design, [39].
The Kinematic Diagrams of the Minimum and Maximum Gear Meshing of the Milling Machine Gearbox. of Probability, 6×2 is given in Fig 23, while The First Mode Shape of them are shown (Fig 24). The gearbox assembly geometry modeled in Solidworks was imported to Ansys software for studying the dynamic performance. The bearing supports were assumed as a boundary condition for the gear components, which allows rotational motion along the shaft axis but restricts axial motion and radial motion.

Harmonic Response Analysis of the Gearbox
The gearbox exposed to defects in the gears or shafts due to varying magnitude loading. According to the previous calculations, a moment was applied as a boundary condition at the input shaft of each model of the main gearbox. Table
(5) presents the natural frequencies (Hz) of the gearbox with the probabilities (6×2), (4×3) & (3x2x2). The purpose of the harmonic response analysis is to measure the maximum deflection and its corresponding frequency also determining the resonance frequencies. The maximum deflection of 2.33
Âµm of the 6×2 probability occurs at xdirection at a frequency of 276 Hz at the maximum speed. The maximum deflection of 2.3 Âµm of the minimum speed is found in ydirection at frequency 592 Hz. The maximum deflection and operating frequencies of the probability 4×3 are given in Table (). The results show that the maximum deflection 0.702 Âµm at y direction at 188 Hz and 3.16 Âµm at zdirection at 230 Hz of the maximum and minimum speed respectively. The maximum deflection 13.51 Âµm at ydirection at 475 Hz and
5.59 Âµm at zdirection at 210 Hz at the maximum and minimum speed respectively for the probability 3x2x2.
Fig 19 Theoretical, Actual Speeds and Percentage Errors between Them. 
Fig 20 Gears Specifications. 
Fig 21 Speed and Flow Chart. 
Fig 22 Gearbox Shafts Design. 
Table (5) The Natural Frequencies (Hz) of the Gearbox with the Probabilities (6×2), (4×3) &(3x2x2).
Probability 
Speed 
FRF direction 
Maximum Deflection (Âµm) 
Natural Frequency (Hz) 
Position 
6×2 
Maximum 
Xdirection 
2.33 
276 
Gear 5 
Ydirection 
0.532 
400 
Gear 7 
Zdirection 
0.042 
636 
Gear 7 

Minimum 
Xdirection 
1.756 
688 
Gear 5 

Ydirection 
2.3 
592 
Gear 1 

Zdirection 
0.088 
592 
Gear 1 

4×3 
Maximum 
Xdirection 
8.4861e004 
188 
Gear 11 

Ydirection 
0.702 
188 
Gear 11 

Zdirection 
0.229 
140 
Gear 11 

Minimum 
Xdirection 
2.3024e002 
230 
Gear 1 

Ydirection 
1.8912 
230 
Gear 1 

Zdirection 
3.1666 
230 
Gear 1 

3x2x2 
Maximum 
Xdirection 
0.136 
475 
Gear 10 

Ydirection 
13.518 
475 
Gear 10 

Zdirection 
5.833 
402 
Gear 7 

Minimum 
Xdirection 
4.3479e003 
282 
Gear 11 

Ydirection 
0.199 
240 
Gear 7 

Zdirection 
5.593 
210 
Gear 4 

Gear Meshing Frequencies
The frequencies of the gear meshing are defined in Hertz (Hz) or [cycles / sec.]. Gear meshing frequency is a harmonic
phenomenon and it can be calculated by the following equations:
f = z.n/60
(40)
where; n is rotational speed (rpm).
@ Maximum Speed
@ Minimum Speed
Main Gearbox Model
Fig 23 The Kinematic Diagrams of the Minimum and Maximum Gear Meshing of the Milling Machine Gearbox of Probability 6×2.
Mode# 1 @ f=177.7 Hz Mode# 1 @ f=171.6 Hz
Fig 24 The First Mode Shape of the Minimum and Maximum Gear Meshing of the Milling Machine Gearbox.

SUMMARY
The virtual design carried out for Mechanical Redesign of Egyptian Made Milling Machines for Modification and Updating by Using CAD. This is done by using a set of commercial packages like Ansys and Solidworks beside Gearlab and tailored codes written in VB 6 for calculating the cutting forces and required power, as well as codes to design main and feed gearboxes. This technique has several advantages as indicated below. Virtual design helps designers to prepare and conduct their design, modification, and updating without being restricted to factory locations or official work times. It helps the designers to repeat the same and/or modified one more than once, helping the designer in covering all aspects of the design. The virtual design helps managers to solve the problem of lack of instrumentation and experiments fund. It protects designers from the dangers they
face during conducting some laboratory experiments on the first or modified prototype. The designer is allowed to control the inputs of the design, change the different design parameters and observe the changes in the results without being exposed to any hazards. Providing cooperation and interaction between designers and each other, and between designers and production engineers. providing a lot of money for manufacturing institutions. Moreover, it saves time and effort of the designers, as they avoid moving between the places in their factory

CONCLUSIONS

With the increasing demand for higherproductivity machines and devices, shorter design times, and lower machine costs, machine designers need to reconsider their design approaches. An integrated set of software is successfully used
and applied in machine tool design. This machine design guide examines design techniques that successful machine builders are using to improve the productivity and lower the cost and risk of their machines. The following conclusion can be achieved:

The virtual tools enable the design, test, optimize and control machine parts in a computer simulation environment. A virtual machine is proven to be a good tool when it comes to predict and optimize a complete machine tools properties. An efficient optimization strategy is key when facing problems with a large number of design variables.

The need to design faster and more efficient machine tools has changed the way design tools are perceived. Mechanical engineers use finite element technology based tools to study strain and deformation while the machine is operating. With this new approach, engineers can integrate different design tools to achieve a faster view of how design changes affect overall performance. They can then build prototypes not to check the performance but to modify, update and validate the overall design.

The reshaping of the milling machine column increases surface area for modifications #1 to #4. The increase is in surface area 1.72, 1.59, 1.53 & 2.05 times higher than the original one. This increase in the surface area could play an important role in structural heat transfer.

The less deformation the higher stiffness. The maximum static deformations at spindle head for modifications #1 to #4 decrease the deformation 29%, 28.5%, 28% & 31.2% lower than the original one, (145.1 Âµm). This means that modification # 4 improved the milling machine structure by 31.2%.

The reshaping of the milling machine column increases allnatural frequencies for all modified structures. The natural frequencies of all ten first modes of all modifications are higher than those of the original structure. The natural frequencies of the original structure and modifications #1 to #4 are 73.8, 111.2, 110.7, 109.3 & 112.6 Hz respectively. The improvement is by raising the natural by 50.6, 50, 48 & 52.5% up those of the original structure.

Generally, the fourth modification has the highest natural frequency among all modifications, the widest range of natural frequencies, and the minimum value of deformation.

Modification #4 is the best stiff structure. So, the fourth modification is the versatile design against harmful vibration. Thus, the milling machine can safely use wide cutting conditions as the cutting speed without dangerous resonance occurrence.

These results proved that the given computerized nomograms and calculators written in VB(6) are accurate, faster, reliable and expandable design tools.
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Bearing Frequencies, [Online], Available: http://www.ntnamericas.com/en/website/documents/brochuresand literature/techsheetsandsupplements/frequencies.pdf