Comparison of machinability of various types of curves using
parametrization and chord error reduction
A PROJECT REPORT
Submitted by
Rathod Abhijeet
140950119568
Shah Mitul
140950119578
Shah Parth D.
140950119581
Shah Parth K.
140950119582
In fulfillment for the award of the degree of
BACHELOR OF ENGINEERING
in
Mechanical Department
Institute of technology and management universe , Vadodara
Gujarat Technological University, Ahmedabad
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Institute of technology and management universe
Vadodara
Mechanical Department
CERTIFICATE
Date:
This is to certify that the dissertation entitled “Comparison of machinability of various types of curves using parametrization and chord error reduction “ has been carried out by “Rathod Abhijeet,
Shah Mitul,Shah Parth D.,Shah Parth K” under my guidance as fulfillment of the degree of Bachelor of Engineering in Mechanical
Engineering (8th Semester) of Gujarat Technological University, Ahmedabad during the academic year 2018-19.
Guide: Ujjwell Trivedi
(Assoc. Prof.)
Head of the Department .
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ACKNOWLEDGEMENT
We take this opportunity to express our gratitude to them who have rendered cooperation and guidance that supported us during our project work.
We would like to express our heartiest gratitude to our Prof. Ujjwell sir, who is our mentor, project guide and dawn of hope. He has expanded his horizon of technical expertise and has attained immerse pleasure in sharing his experience with us. He has been very supportive, inspirational and also guided us in our work throughout the project.
This is the right moment to express our sincere gratitude towards Him for his valuable time, guidance and sharing his knowledge with us, at every stage during our project work.
We thanks from bottom of our heart to all the faculties and staff members for their cooperation and supportive nature.
Last but not the least we express our inner gratitude to our family members and friends for their everlasting support and welcome social distraction.
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Abstract
To machine curved surface with higher accuracy,large number of linear segments has to be produced using linear part programming on CNC machine.
Due to Repetitive commands the length of program increases which is extremely difficult to modify. The problem of increase in storage and runtime requirements makes it even worse.
So to nullify this problem we are using macro to get optimal solution. As it does not follow repetitive method
The parametric curves will be generated by segment method so that curve will be divided into small parts and by this smoothness,accuracy of surface finish will be maintained.
Number of points will be reduced by optimisation using chord error tolerance.
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LIST OF TABLES
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Page No
Table No
4.1
Macro variable
21
5.1
Curve segment(10)
25
5.2
Curve segment (20)
27
5.3.1
Conventional part programming
28
5.3.2
Conventional part programming continue
29
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LIST OF IMAGES
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Page No
1.1
Parametric representation of circle
10
1.2
Hermit curve
11
1.3
Bezier curve
12
1.4
B-Spline curve
13
1.5
Bezier surface
13
1.6
Hermit surface
14
1.7
B-Spline surface
14
6.1
Images of machined part
31
7.1
Screenshot of simulator
32
7.2
Screenshot of simulator
32
7.3
Error tolerance
33
8.1 CMM Result
38
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LIST OF GRAPH
PAGE NO
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5.1
Comparison of Hermit and Bezier curve with 10 segments
26
5.2
Comparison of Hermit and Bezier curve with 20 segments
27
8.1
Curve with different chord errors
36
8.2
Curve with different chord errors
36
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TABLE OF CONTENTS
PAGE NO
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Acknowledgement
i
Abstract
ii
List of Figures
iv
List of Tables
vii
List of Graphs
x
Table of Contents
xi
Chapter : 1 Introduction to the curves
10
1.1 Representation of Curves
1.2 Types of curves
1.3 Application of curves
Chapter : 2 Part programming
15
2.1 Integration of CAD/CAM
2.2 Conventional part programming
2.3 Conversational part programming
2.4 Part programming using CAM
2.5 Macro programming
Chapter : 3 Literature Review
19
3.1 Real time interpolator for parametric curves
3.2 Feed optimisation for five axis CNC machine tools with drive constraints
3.3 A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining
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3.4
Machining of Bezier curve by macro programming
Chapter : 4 Macro Programming
21
4.1 Variables and Expressions
4.2
Macro Functions
4.3 Branches and Loop
Chapter : 5 Design Specification
25
5.1
Comparison Of Bezier and Hermit curves
5.2
Increment in segments with four control points
5.3
Convention programming for chosen curve
Chapter : 6 Experimental details
30
6.1
Details of Work piece And Machining
6.2
Images of Machined Work piece.
Chapter : 7 Parameter study
32
7.1
Compatibility of macro coding
7.2
Error tolerance
Chapter : 8 Result and conclusion
36
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Chapter : 1 Introduction to the curves
1.1 Representation of curves
• Curves can be mathematically represented by two methods 1. Non-parametric representation
In non-parametric representation the curve is represented as a relationship between x,y,z
i.e f(X,Y,Z) = 0
(x )2 + (y )2 = r2, (circle)
2.Parametric representation
In Parametric representation curves are not represented by variable X,Y,Z. but these variables are depended on one variable i.e u or say t.
i.e X=f(u or t)
Y=f(u or t)
Z=f(u or t)
i.e
Here, X and Y are the function of t.
Image-1.1 parametric representation of circle
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1.2 Types of curve.
Analytic curves:
• These curves are defined by analytical equations and one can not modify the shape of any analytical curve.
• This curves’s shapes are identified i.e circle,ellipses,parabolas,hyperbolas
Synthetic curves:
• These curves are defined by the set of data points .
• They are represented by polynomials.
• They are used to represent the car bodies,air plane wings,propeller blades,shoe insoles etc.
• Major CAD/CAM softwares use three types of synthetic curves.
1. Hermite Cubic splines
1.2 hermit curve
This is interpolated using two end points and the tangents.
Here p(0) and p(1) are start and end points and p(0)’ and p(a)’ indicates the tanget.
p(u) =
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One can change the starting and end points as well as the knots to change the shape.
2. Bezier curves.
1.3-Bezier curve
This is also a synthetic curve whose shape can be controlled or modified.
Here p0,p1,p2,p3 are the control points. by changing them one can control the shape.
It’s equation is ,
p(t) =
3. B-spline curves
• It allows local control over the shape of curve as against the global control in case of bezier curve.
• One can add on weight on the point so that the curvature of the curve can also be controlled.
• The degree of polynomial representing the curve can be set independently of the number of control points.
• Overall,It gives better control
• It permits to add or delete any number of control points without changing the degree of polynomial.
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1.4-Bspline curve
1.3 Application of curves.
• Basically these curves form various surfaces.
• Few surfaces having intricate shapes are made up with this curves.
1.5-Bezier surface
This image is showing the bezier surface. bezier curve is utilised to generate such surfaces.
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1.6 : Hermit surface
1.7-B-spline surface
The image above shows the usage of b spine surface. some more intricate shapes can be machine using B-spline curve.
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chapter : 2 Part Programming
2.1 Integration of CAD/CAM
The field of computer, development has been rapidly increasing in past few decades. which can be undoubtedly considered as the single most significant factor responsible for the boost in the technological development.
The computer has entered in each and every field. It has became the most important tool in all the technological areas including design and manufacturing.
CAD: Computer Aided Design can be defined as the use of computer system to assist in the creation, modification, synthesis or optimization of a design.
CAM: Computer Aided Manufacturing is defined as use of computer systems to plan, manage and control the manufacturing operations through the direct/indirect computer interface with the manufacturing machine.
1. Advent of CAD/CAM:
The first major innovation in machine control was the hardwired Numerical Control (NC System) which used paper tape as an input medium and tape reader as an output , gradually upgrading it with mainframe computers to control a group of NC machines called as Direct Numerical Control (DNC) system. Today the innovation and progress in technology have given us soft wired Computer Numerical Control (CNC) with the facilities of mass program storage, offline editing and software logic control and processing.
With the integration of hardware and software in CAD/CAM systems, manufacturing industries have witnessed a paradigm shift in the methodology of NC programming. Like, CNC program can be directly generated from a CAD model, availing design and modelling of fixture setup, design of blank, process plan, and selection of optimum process parameters and so on. Capability of producing acceptable components can now be ensured by
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simulating machining operations on the CAD/CAM workstations. Also, demand for complex shapes are achieved with shape generating capabilities of CNC systems. CNC technology was adapted in the development of co-ordinate measuring machine’s (CMMs) which automated inspection. Robots were introduced to automate several tasks like machine loading, materials handling, welding, painting and assembly. All these developments led to the evolution of flexible manufacturing cells and flexible manufacturing systems.
2. CAD Application:
For the suitability number of design can be checked. Here, calculations are done by the computer. So errors are minimised. This improves the accuracy
Every time consuming and iterative phases of the design process are carried out on the computer, hence time required to design can be reduced.
CAD reduces engineering personnel requirements. The model of object is available in database, so the modifications become easier to make.
CAD helps a lot in preparation of documentation. It eliminates prototype testing which is costly and time consuming in place of it simulation is done.
The manufacturing drawings are prepared automatically by CAD.
3. CAM Application:
It includes Automation in manufacturing such that:
1. Automatic path control.
2. It controls feed, speed, depth of cut.
3. It controls coolant and spindle.
4. Automation in tool changing.
• The CNC machine is as good as the part programming. As it follows what ever the instruction is given to it. It just does the process as directed. So any improvement
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can be done by the programmer. CNC controller would only be following the instructions of program. it works just like an “obedient slave” in a very precise manner.
2.2 Conventional part programming.
simple G-code,M-code programming is a conventional part programming. it has several limitations and has confined scope. if one wants to machine of certain limited shapes repeatedly this programming type can be employed. this program is suffering for efficiency and productivity.there is no variables, no mathematical calculations, functions,logical statements,looping. In other words it does not have very basic features such as PASCAL. it is just one rigid program for a particular requirement. it only serves the limited purpose for which it was designed.For small job,conventional part programming works well. Increase in complexity leads to longer programs which are difficult to modify
2.3 conversational part programming.
Even conventional part programming is fairly complex. So, for the purpose of simplifying programming for certain common applications, conversational or lead-through programming (referred to as blue print programming or direct dimension programming) was introduced, which enables users to even without having adequate programming experience, easily develop suitable part programs. The programmer need not know the part programming language in detail; he only has to know what is to be done in what sequence, and with what cutting parameters. The control prompts program for all the required information, in an interactive manner. This method of programming, how- ever, suffers from the inherent limitation of being applicable only to certain specific geometries. So, even though it is possible to quickly write efficient programs for some common applications, this method is useless for special requirements.In reality, conversational program- ming is only a small subset of what can be done with conventional part programming technique. In fact, conversational programming is not meant for experienced engineers.
2.4 Programming using CAM software
The main aim for the development of CAM software based programming is it calculates which can not be calculated manually. In conventional part programming certain calculations can not be done. in this design is prepared in CAD software attaching that design with CAM software it prepares the path. And by this the path of tool can be controlled. Some tedious calculations make the conventional programming worse . So for those tedious iterative this software is very useful.
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2.5 Macro programming
This is the higher level of programming. In this type certain logics are used which eliminates some repeated tasks. it also eliminates the tedious calculation. In other programming type, the programme length is too much. To machine one surface with precision it requires one lengthy program and repeated tasks. Using certain
logics ,loops programming is done in this type of programming, program length is reduced. this programming type reduces the repeated tasks and it dose calculations by it self. In Macro program length remains constant if we still increase the accuracy,size of the object etc.Program has to calculate all the points in advance. For the curves with large curvature the faced surface appears which requires smaller increment in parameter and thus more points. The time wasted is perfectly can be better justified by calculating the required accuracy in runtime. Thus a macro is developed to provide more flexibility.
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Chapter : 3 Literature review
3.1 Real time interpolator for parametric curves (Yukui Cai, Wenlong chang
(2017)
As the industry demands high speed machining accuracy of various surfaces , with the help of the interpolator or with use of interpolator the feed rate and acceleration discontinuity can be solved this paper basically focuses on the interpolator in parametric curves used for optimisation and also provide real time solutions and at the end it also concluded that interpolator for parametric curves is much much better than linear interpolator and will increase the productivity of the firm to 10 times
3.2 feed optimisation for five axis CNC machine tools with drive constraints(B Sencer)
(2008)
It requires smooth feed and acceleration machine tools for five axis machine (real time control) this paper tells us that how the machining time can be reduced for five axis machine . Velocity and acceleration in five axes will give us the optimal feed required for the machining so that we can reduce the machining time and increase the productivity . Long tool paths can be handled easily .this paper showed that the machining time can be reduced to greater extent by controlling the feed rate and optimising all the five drives
3.3 a parametric interpolator with confined chord errors, acceleration and deceleration for NC machining (2003)
As we all know that the parametric interpolation is far more better than the linear interpolation as it has solved problems related to acceleration and deccerelation , feed rates and also about the jerks that were affecting machining but in that about chord error is not mentioned so this paper focuses on the chord error constraints . Various velocity planning is done so that the chord error is satisfied within the feed limits Information is stored in 2by2 matrices . In these the value of chord error tolerance is stored and other values are maintained accordingly .
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3.4 Machining of Bezier curve by macro programming (Vratraj K. Joshi Prof K.P. Desai,Prof. H.K. Raval) 2016
In today's great manufacturing world the industries are demanding the various curvilinear shapes and it is difficult to get the curvilinear shapes with the given preparatory codes in the machine controller. If the shape size is more then there will be more and more number of points for that and that will increase the program size and will required more memory.
To overcome this difficulty MACRO PROGRAMMING is introduced to get the curvilinear shapes with greater accuracy and precision and there's no need to write the program again and again as macro can be introduced during run time of the machine . So these papers focuses on the macro programming and result we get from these will help to improve the flexibility and surface roughness that were problem in conventional programming
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Chapter : 4 Macro programming
There are two popular vendors for macro controller in the market. And they are FANUC and SIEMENS.
Even controller also have their own macro system,syntax.
Here details about FANUC controller is explained
4.1 Variables and Expressions
• A macro variable is any mathematical number with confined range.
• Variables are designated with the “#” symbol, followed by a number , in the appropriate range.
i.e #1,#10
• These numbers represent particular memory locations that may have any negative or positive value.
Table-4.1 Macro variables
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There are two types of macro expressions.
Arithmetic expressions
Arithmetic expression is a mathematical formula having variable and/or constants (i.e 0.12, 1.2, 12, etc.), with or without functions (such as SIN, ACOS, SQRT, etc). this expression and the argument of a function must be enclosed by square brackets
i.e
1 + #2
#3 + #4 * SIN[30 * [#5 I 10]]
Conditional expressions.
A conditional expression includes conditional operators (i.e EQ, NE, and LT) between two expressions. It must be enclosed in square brackets too. It evaluates to either TRUE or FALSE . A conditional expression is also called as a Boolean expression.
Examples:
[#1 EQ 0]
4.2 Macro Functions
There are some functions available in macro which promotes the easiness for the programming via macro.
Trigonometric Functions
The available functions are
SIN cos TAN ASIN ACOS ATAN
.#1 SIN[30];
.#2 ACOS [#1];
.#3 TAN[#2*3I4];
.#4 ATAN[SQRT[3]] I [#1 * 2];
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Rounding Functions
Examples:
ROUND [ l O . O ] (Returns 10.0)
ROUND[l0.2] (Returns 10.0)
Miscellaneous Functions
Other functions for evaluating square root (SQRT), absolute value (ABS), natural logarithm (LN), and exponential value (EXP) are available.
SQRT
This calculates the square root of a given positive number. Examples:
SQRT[-2) (Illegal argument)
SQRT [2 ) (Returns 1.4142136)
ABS
Examples: ABS[-2) (Returns 2.000)
ABS [ O] ((Returns 0.000)
ABS[2 ) (Returns 2.00
EXP
This function calculates the antilog of the natural logarithm (i.e., e ). 10x=e"ln 10, that is, EXP[x *LN[lO]]
Logical Functions
The available logical functions are AND, OR, and XOR.
4.3 Branches and Loop
Unconditional Branching
The format is GOTO n;
where n is the wanted sequence number (l to 99999). This make the jump on the n Sequence number
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Conditional Branching
In conditional branching, GOTO is used with a conditional expression.
i.e
IF (<a conditional expression>] GOTO n;
WHILE [<a conditional expression>] DO n ;
Macro Call
A subprogram is called by M98 and M198. A macro can be called using any of the following methods:
• Simple call (G65)
• Modal call (G66)
• Call with user-defined G-eode
• Call with user-defined M-code
Siemens followed the trend. It has some more advanced features and it is closed to any structure programming language. Some useful features are as follow:
1) Variable Names
added concept of data types . Variable can now be widely defined by the programmer to save memory e.g an INT takes 2 bytes while a real variable type takes 4 bytes. thus, a large saving in runtime memory requirement
Variable are not just numbers can also be alphabets as well e.g VAR1 FEED1 TOOL1 etc. are all variable names
2) Jump statement
The GOTO are classified as GOTO, GOTO A and GOTO B. The Jump statement now as a label to be able to locate and identify the position FANUC used block noto jump which may be ambiguous
GOTO B and GOTO A identifies the direction of label in the program
3) FANUC provides a large number of fixed register variables local global and permanent which unnecessary required more memory Siemens has provided small pool which can be expanded by the programmer 'R' prefix is used from 1-99 in Siemens and more can be added
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Chapter 5: Design Specification
5.1 Comparison of Bezier and Hermit curves.
• Taken control points are : P0(1,1) P1(7,4) P2(8,7)
• Equations for bezier and Hermit are :
(Hermit) P(u)=
(Bezier)P(t)=
Taking 10 segments for each curve with given control points the curve coordinates are:
table -5.1 Curve segments (10)
u(segments)
x (hermite)
y(hermite)
x (bezier)
y (bezier)
0
2
3
2
3
0.1
2.7
3.23
3.16
3.52
0.2
3.53
3.33
4.24
3.88
0.3
4.484
3.32
5.24
4.08
0.4
5.48
3.20
6.16
4.12
0.5
6.5
3
7
4
0.6
7.47
2.712
7.76
3.72
0.7
8.35
2.35
8.4
3.28
0.8
9.10
1.94
9.04
2.68
0.9
9.66
1.48
9.56
1.92
1
10
1
10
1
5.1 Comparison of bezier and hermit curve with 10 segments
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x (hermite) x (bezier) control points
7
8, 7
6
5
4
7, 4
3
2
1
1, 1
0
012345678910
•This is the graph of the bezier and hermit curves interpolation of given control points with 10 segments.
•This graph indicates that with same control points we can get smother curve with bezier interpolation with compare to hermit.
5.2 Increment in segments with four control points.
Now taking control points : P0(35,30) P1(25,0) P2(15,25) P3(5,10)
For getting more smother interpolation we need to increase the segments.
Here the table indicates the coordinates of bezier curves with 10 segments and 20 segments .
35
30
35
30
33.5
25.9
32
22.55
32
22.5
29
17.84
30.5
19.89
26
15.28
29
17.84
23
14.32
27.5
16.32
20
14.37
26
15.28
17
14.88
24.5
14.63
14
15.26
23
14.32
11
14.96
21.5
14.25
8
13.39
20
14.375
5
10
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18.5
14.60
17
14.88
15.5
15.12
14
15.26
12.5
15.23
11
14.96
9.5
14.37
8
13.40
6.5
11.96
5
10
20
10
Table- 5.2 Curve segments (20)
Here as shown in graph if we use 10 segments there were some marks of straight cuts on machined surface comparatively with 20 segments.
• And we can get more smother curve with increased segments.
5.2 Comparison of Bezier and Hermit with 20 segments
10 segments
20 segments
control points
30
35, 30
22.5
15, 25
15
7.5
5, 10
0
25, 0
0
10
20
30
40
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5.3 Convention programming for chosen curve.
N
G
X
Y
Z
M
T
N0
G17
N1
G71
N2
G40
N3
G28
U0
V0
W0
N4
G91
Z0
N5
N6
G92
X0
Y0
N7
N8
G00
M06
T01
N9
G90
N10
G43
X0
Y0
N11
G00
Z0
M08
H01
N12
G01
X35
Y30
F20
N13
X33.5
Y25.9
N14
X32
Y22.5
N15
X30.5
Y19.89
N16
X31.52
Y17.84
N17
X27.5
Y16.32
N18
X26
Y15.28
N19
X24.5
Y14.63
N20
X23
Y14.32
N21
X21.5
Y14.25
N22
X20
Y14.37
Table-5.3.1 Conventional part programming
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N23
X18.5
Y14.60
N24
X17
Y14.88
N25
X15.5
Y15.12
N26
X14
Y15.26
N27
X12.5
Y15.28
N28
X11
Y14.96
N29
X9.5
Y14.37
N30
X8
Y13.40
N31
X6.5
Y11.96
N32
X5
Y10
N33
G00
Z0
N34
G28
U0
V0
W0
N35
G91
Z0
N36
G28
U0
V0
W0
N37
G91
X0
Y0
N38
G40
N39
G80
N40
G49
N41
M09
N42
M02
N43
M30
Table-5.3.2 Conventional part programming
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Chapter : 6 Experiment details
6.1 Details of Work piece And Machining
Material of workpiece : Aluminium
Dimensions (mm): 100 (Length),100(Width),15(Thickness)
Depth of machining: 4 mm
Speed of spindle: 3000 rpm
Diameter of cutter: 10 mm
Machine: Vertical milling machine (CNC)
Milling Cutter: End mill
Controller: Siemens controller
Machined Curve: Bezier
Control points: 4
Machining place: ITC Tarsali, Vadodara
6.2 Images of Machined Work piece.
Images shown on the next page are the actual images of the machined work piece . We have machined 5 samples and five curves are machined over the aluminium plates of described dimensions. these all are bezier shapes with 0.1 error intolerance . Home position is also indicated in the images.
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6.1-Images of machined part
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Chapter : 7 Parameter study
In this project we had an urge to integrate design and machining one step further. We looked after certain possibilities and studied for the further scope. Than after we went on this region. So those certain parameters on which we have worked are as follow.
7.1 Compatibility of macro coding
The first task of us was to check the compatibility of our macro coding with controller being used and it was also one important task to check whether the available controller can understand the generated syntax of macro or not. We checked it on the simulator and made some changes accordingly.
1)
7.1
2)
7.2
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7.2 Error tolerance
Normally CNC tool traces the path guided by the controller. But the accuracy is depended on the number of segments. More number of segments increase the accuracy of machining, But accuracy may differ from one curve to another curve as per the requirement of the customer. So we have add one feature to our coding that we can optimise the error. And we can very the tolerance of error. That means we are also dealing with the segment optimisation as well.
Image-7.3 Error tolerance
Here in the figure segment approximation is shown and ideal arc path is also shown. The difference between this two is call chord error. In coding that tolerance can also be controlled. This is another parameter we have studied.
Um
X₁, Y₁: X, Y coordinates at point 1
X2, Y2: X, Y coordinates at point 2
U₁, U2: Parameter valueU1U2
Um: Average parameter value
Xm, Ym: X, Y coordinates at parameter U
Xv,Yu : X, Y average of X1, X2 and Y1, Y2 respectively
Therefore, distance (chord height d ) =
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The value of U2 is to be reduced and set, such that d ≤ chord height tolerance.
To reduce U2, use the following formula:
For this new value of U (Ur) repeat the calculation f chord height to meet the desired constraint value.
It has been found that for the chosen curves a value of
Macro algorithm
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Chapter 8: Result and conclusion
8.1 Concluded points.
• With increased accuracy, number of block would remain same using Macro.
• Control for accuracy is given in program. So if machine generates rough surface, one need not to go back to the CAM system software. Means it can be generated in runtime.
• Feed rate need not to be changed even for a stiff change in curvature.
• The measurement data complies the analytical data.
• Bezier is more flexible and parameterization can be controlled using chord error tolerance.
graphs on the next page indicates that if we decrease the error tolerance(0.1,0.05) no doubt we would get higher accuracy. But these will result in increase in number of segments and if the segments will increase it will increase the length of the program and also more memory storage will be required. So we made the macro programming with 0.1 error tolerance which would have lesser segments and will give us the accuracy that is needed for machining, and also incase we need to change the error tolerance than we need not to write the whole program again , this will not increase the length of the program and memory required will also be less and accuracy will also be maintained .
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Graph 8.1 Curve with different chord errors
Graph 8.2 curve with different chord errors
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Image 8.1 CMM result
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This images are obtained by combining the machining designed data and the CMM data (digital image of machined curve) in the CAD software. It indicates that the machined curve and the data we got from the CMM while comparing both the datas , both the curves overlap on each other , so we can conclude that our objective is fulfilled.
That means we designed bezier curves. And for that we made a Macro program. We run it on CNC milling machine. After that we did Coordinate measurement and got the digital image. And the designed curve and machined curve’s digital data overlaps. That indicates that we have fulfilled our objective of project.
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