UPR-RUM

Design of a Rapid Toolbit
Switching Adapter for Rotary
Tools
Submitted as a partial requirement
for the course INME4058: Section 126
Design Course for B.S. in Mechanical Engineering Department
University of Puerto Rico at Mayagüez, Mayagüez, PR
Students:
Borrero Valentín, Susana
Undergraduate Student at Mechanical Engineering Department
University of Puerto Rico at Mayagüez, Mayagüez, PR
Rivera Maldonado, Rodolfo
Undergraduate Student at Mechanical Engineering Department
University of Puerto Rico at Mayagüez, Mayagüez, PR
Faculty Advisor:
Vijay K. Goyal, PhD
Professor at Mechanical Engineering Department
University of Puerto Rico at Mayagüez, Mayagüez, PR
May 2005
Project Report
To:
Dr. Vijay K. Goyal
From:
Borrero Valentín, Susana E.; Rivera Maldonado, Rodolfo
Date:
May 12, 2005
Re:
Engineering Design Project (INME4058: Section 126)
Dear Dr. Goyal:
Greetings. The intention of this document is to propose the design and optimization of the Rapid Toolbit Switching
Adapter (RTSA) for rotary tools. This device is a disk-like adapter for hand-operated rotary devices that will hold,
close and handy, a set of chosen toolbits at a time, allowing for quick and easy switching between them, thus reducing
the time required for a set of tasks. The enclosed document is the most current report with the progress and analysis
completed as of yet.
With over 5 million automobiles produced each year in the U.S. alone, plus many millions more already on the
streets requiring periodic repair and services, a system that could reduce the time, effort, and consequently money
spent in the hand tasks involved in the assembly and repair of such, will have a significant impact on the
Manufacturing Industry, and more specifically in the Automotive sector. These tasks involve a considerable number
of different tools, and toolbits, making the proper access to them a time challenge. The RTSA, a light disk-like
adapter, will eliminate that challenge and the problem of lost tool-bits, by keeping the operator’s chosen set of bits
together at his fingertips in a circular array around the rotary device front end, making his work a lot easier in less
steps.
Our main goal is to deliver the optimized design of the RTSA at a cost that should not exceed $60, to make it an
affordable and attractive solution to our costumer. This will involve a thorough design evaluation which will include a
complete static, dynamic and finite element analysis to achieve the maximum design performance as well as safety
standards associated. Constraints such as weight, attachment and removal of device, visibility, ergonomics and others
will also be taken into consideration.
We appreciate your time and consideration in this project. If we can be of any further help, please do not hesitate to
contact us.
Regards,
Susana Borrero Valentín
Rodolfo Rivera Maldonado
Design of a Rapid Toolbit
Switching Adapter for Rotary Tools
Susana Borrero Valentín and Rodolfo Rivera Maldonado
In any high scale line production, every extra second per task becomes a bottle neck for the
next, causing delays down the line, and subsequently lost money to the business. In the Automobile
Industry, where constant repetitive tasks takes place along the line, success depends on the efficiency
of performing each task in the least amount of steps possible. The problem is that rotary devices, such
as drills, which are one of the most employed tools on assembly lines, don’t provide for the complete
task to be performed in a none-stop single step process. Instead, the time is lost in moving to the side
to look for the next toolbit required, detaching the one on the tool, attaching the next one, moving back
to the operating position to finally performing the drilling activity. Based on this need to increase line
efficiency, the engineering team has come up with the idea of the Rapid Toolbit Switching Adapter
(RTSA), which enables for a reduction in steps, time and effort, by providing quick accessibility and
placement of toolbits by holding them in a disk-like adapter that is placed on the front end of the
rotary device, for a quick switch between bits.
The RTSA is disk-like toolbit holder for regular rotary devices. The disk has a star-shape form
that can fit up to 8 toolbits at its corners, and functions by one hand rotation of the disk about its axis
to click the desired bit in place. It has an adjustable ring clamp for attaching it to the different rotary
devices available. To function, it will not consume any energy from the main device, because its
revolving mechanism will relay on a gear-type mechanism for proper toolbit alignment with the
rotating tip of the main device. The cost of this design should be within the $60 to $80 range, and a
Manufacturers Suggested Retail Price (MSRP) of around $100. The user should be able to perform a
set of related tasks (requiring different toolbits), in an easier and time reduced manner, with the same
tool, without the need to stop activity or move from operating position to find the needed tool.
An evaluation of technology available (toolbelts, tool cabinets, ceiling tool racks, toolbit kits)
show that even though some of these available solutions solve part of the time related problems, they
still present inconveniences to the operator, because of extra man power required for effective usage
of far tool racks for example; space constraints limits the use of racks which are too big; accessibility
from operating position, and weight limitations of tool carriers. But most important, none of them
provide for an extremely quick switch between toolbits, mounted on the tool. The RTSA will satisfy
the need of user friendly, step reducing tools that can increase productivity in any manufacturing line.
In order to achieve the high level design and optimized mechanism promised, a complete
static, dynamic and finite element analysis will be performed, and important constraints such as safety,
ease of use, weight, ergonomics, visibility and others will be considered as well.
Acknowledgements
First and foremost we want to thank GOD for giving us the strength to live through the long
nights and finishing this project.
We would also like to thank Prof. Goyal for his guidance and time throughout the development
of this project. Finally, we would also like to express our appreciations to all those people that in
one way or another helped make this project possible.
List of Contents
List of Figures ...................................................................................................... viii
List of Tables ......................................................................................................... ix
Chapter 1. Preliminary Remarks............................................................................10
1.1 Background ..........................................................................................10
1.2 Literature Survey .................................................................................11
1.3 Problem Description ............................................................................11
1.4 Expected Outcomes .............................................................................13
Chapter 2. Project Description...............................................................................14
2.1 Description of the Project ....................................................................14
2.2 Design Selection ..................................................................................15
2.2.2 Second Idea: Centered Rotating Star Adapter ......................16
2.2.3 Selected Design.....................................................................16
2.3 Methodology ........................................................................................18
Chapter 3. Optimization.........................................................................................20
3.1 Design Problem....................................................................................20
3.3 Objective Function...............................................................................24
3.4 Constraints ...........................................................................................25
3.5 Method of Lagrange Multipliers..........................................................28
3.6 Discussion ............................................................................................34
vi
Chapter 4. Finite Element Analysis .......................................................................35
4.1 Introduction..........................................................................................35
4.2 Geometry..............................................................................................35
4.3 Mesh Creation......................................................................................36
4.3 Loads and Boundary Conditions..........................................................37
4.5 Analysis................................................................................................39
4.6 Additional Analysis (COSMOSWorks)...............................................41
Chapter 5. Results and Discussion.........................................................................43
5.1 Present Design .....................................................................................43
5.2 Discussion ............................................................................................46
5.3 Cost Analysis .......................................................................................46
Chapter 6. Final Remarks ......................................................................................50
6.1 Conclusion ...........................................................................................50
6.2 Recommendations................................................................................51
References..............................................................................................................52
Appendix A. Submitted Proposal ...............................................................53
Appendix B. Method of Lagrange Multipliers ...........................................63
Appendix C. ANSYS Tutorial .....................................................................82
Appendix D. Complementary Information...............................................150
Student Vitae........................................................................................................155
Evaluation Sheet ..................................................................................................156
vii
List of Figures
Figure 2.1 - Preliminary Concept for Idea 1 ..................................................................... 15
Figure 2.2 - Preliminary Concept for Idea 2 ..................................................................... 16
Figure 2.3 - Welding Machine .......................................................................................... 19
Figure 2.4 - Electric Cut-off Saw...................................................................................... 19
Figure 2.5 - Industrial Milling Machine ........................................................................... 19
Figure 3.1 - Star-Shaped Adapter ..................................................................................... 21
Figure 3.2 - Tool bit holder............................................................................................... 21
Figure 3.3 - Bending Moment at the star extension.......................................................... 21
Figure 3.4 - Design Variables Diagram ............................................................................ 23
Figure 3.5 - Variation of Design Variables to fit constraints............................................ 24
Figure 4.1 - 2D Area to be extruded ................................................................................. 35
Figure 4.2 - ANSYS Geometry and Mesh Plot................................................................. 37
Figure 4.3 - Applying Load and Restrictions.................................................................... 38
Figure 4.4 - ANSYS von Mises Stress Plot ...................................................................... 39
Figure 4.5 - ANSYS Deflection Plots............................................................................... 40
Figure 4.6 - COSMOSWorks Meshing and Safety Factor Plot ........................................ 41
Figure 4.7 - COSMOSWorks von Mises Stress Plot ........................................................ 42
Figure 4.8 - COSMOSWorks Deflection Plot .................................................................. 42
Figure 5.1 - RTSA - Isometric View ................................................................................ 43
Figure 5.2 - RTSA - Front View....................................................................................... 43
Figure 5.3 - RTSA - Side View ........................................................................................ 44
Figure 5.4 - RTSA - Top View ......................................................................................... 44
Figure 5.5 - RTSA - Engineering Drawing....................................................................... 44
viii
List of Tables
Table 2.1 - Pugh's Matrix Analysis................................................................................... 17
Table 5.1 - Star-Shaped Adapter Material Comparison.................................................... 47
Table 5.2 - Adapter Support Material Comparison .......................................................... 48
Table 5.3 - Expected Production Costs............................................................................. 48
ix
Chapter 1.
Preliminary Remarks
1.1 Background
There are many existing rotary tools available in the automotive industry aimed for the
assembly/disassembly and repair of parts, focused on the idea of increasing the efficiency of the
overall in-line production processes. In figure 1.1 you can observe examples of a variety of tools used
for different tasks performed in an automobile machine shop. The problem with these tools is that the
continuous changes between them make the work in one task too long and a very uncomfortable one.
The problem of dispersion of toolbits, at-hand type of accessibility, quick and easy switching between
toolbits on a single tool is a common problem to all.
For the hundreds of tasks in the automotive industry that require the use of rotary tools,
whether it is for an assembly line of the millions of vehicles produced annually; for the industrial
machine shops that repair the many other that are on the streets, or simply for personal use, the reality
is that even though there are many rotary tools available for the different tasks, the problem of
dispersion of toolbits, at-hand type of accessibility, quick and easy switching between toolbits on a
single tool is a common problem to all.
The focus of this design team will be developing an adapter for currently existing rotary
devices such as drills and electric or pneumatic wrenches. This adapter will employ a rotating starshaped disk to hold up a set number of toolbits in its corners, allowing for quick switching between
the toolbits. The RTSA disk will function in such a way that it will require only a hand rotation of the
disk, and its revolving mechanism about its own axis will allow the desired toolbit to auto-click in
place. The operator can choose how many toolbits and which types to accommodate in the disk at any
given time. The disk will have a number of holes that accept a variety of standard toolbit types and
sizes, for an even higher user convenience.
10
This venture will be achieved at a cost effective manufacturing cost, but without trading off
the top of the line reliability, robustness, or high quality guaranteed by this design.
1.2 Literature Survey
There is a variety of existing aids for performing part of the tasks we are trying to achieve with
a single mechanism. From the existing devices available, let’s take for example the tool holder waist
strap. This particular tool carrier permits the operator to hold the tools and toolbits on him, but still
requires him to 1) take the current toolbit from the rotary device and put it back in the strap, 2) take the
next toolbit, which requires him to take his eyes away from the job to look at the strap, 3) place it on
the rotary device tip, and finally 4) perform task, which is still a long process. Other available tool
carriers work properly in keeping tools in place, avoiding misplacement, as for example, toolbit kits
(see figure 2 on Appendix C). Still, they don’t solve the specific need of having them ready and
mounted in the rotary tool itself for quick switching. The RTSA, will provide for both of these needs
while minimizing the steps required to perform a given task, by holding the desired set of toolbits
mounted and ready for use in the tool itself.
1.3 Problem Description
After studying the Manufacturing Industry, specifically the Automotive, the team evaluated
different machine shops, and found there was a common need to increase the mechanic’s efficiency
when performing a repetitive type of task. When further observing these tasks, required in processes
such as assembly of parts, we found that rotary devices are the most frequently used, quantifying for
most of the repetitive activities that take place in automotive machine shops and industrial assembly
lines. One way the efficiency of such processes can be increased is by reducing the amount of steps
required and thus, the time a mechanic spends changing toolbits when using rotary tools, which is a
very common and inevitable time-consuming process as already mentioned.
11
A brief analysis was performed to better understand what could be the needs and most
common complaints of mechanics and technicians that use this type of tools. Users of this type of
tools tend to need the same group of toolbits for a specific task over and over again, providing for an
opportunity to optimize the steps taken to perform such a task. The main problem pointed was the
need to change tools or constantly reach out for the correct toolbit to change it and then perform the
action. Second was the fact that in a machine shop, assembly line or personal workshop area, it is
often hard to keep all tools handy and ready for the specific task. Often, tools get misplaced and lost
because they are on the floor or laying around. A toolbit holder that provides the user the ability to see
the set of toolbits in a disk while using them, while having them mounted and ready for quick
placement to the rotary tool, would reduce the problems mentioned above, as well as keep the user
focused on the task without distractions, which is the mayor cause of injuries in the work place.
Given the existence of a wide variety of hand-operated rotary tools in the market (see figure 4
of Appendix C), but no actual device available to provide for rapid toolbit switching, by developing an
adapter which will hold the chosen set of bits handy and ready for quick switching, mounted on the
tool itself, we can achieve this. The solution involves the ability to insert a preset number of toolbits
on the corners a star-shaped disk, which has specially designed holes to accept different sizes and
types of toolbits. These specially designed holes will hold the bits in place, but will permit the toolbit
to rotate inside the disk without abrasion of the toolbit or disk.
In order for this device to be effective and practical it should comply with the following
specifications: (a) light weight; (b) easy to install/remove; (c) fits a wide variety of rotary tools
(universal adapter); (d) enables quick tool bit switching; (e) accepts standard tool bits including
sockets, drill bits, screwdriver bits, etc.; (f) able to fit between 4-8 different tool bits at a time; (g) does
not interfere with the typical operation or capabilities of the rotary tool and (h) provides a user-friendly
type of tool manageability.
12
1.4 Expected Outcomes
The goal of this project is to design an adapter that will provide the user of a rotary tool the
ability to quickly and effortlessly swap between a number of toolbits. The adapter needs to be
universal, fitting a wide variety of rotary tools without any additional parts. Its cost should be within
the $60-$80 range. It will be lightweight and shall not interfere with the typical operation or
capabilities of the tool and it must work with standard toolbits already available in the market. This
adapter will decrease the amount of time needed for performing certain tasks and should help improve
the overall efficiency of the processes in a reliable robust way.
13
Chapter 2.
Project Description
2.1 Description of the Project
In order to complete analysis and optimization of the design for the RTSA, this project will include
several different parts, each of which will be discussed in greater detail in the following chapters.
Now that the problem description has been stated and the needed specifications and characteristic of
the design set, the actual design and analysis process can begin. First, all the different ideas for the design
must be narrowed down and the final design to be analyzed must be selected. This process and the
evaluation parameters used are discussed in section 2.2.3.
The selected idea will be refined and one of its parts will be optimized using the Lagrange
Multipliers Method based on a number of constraints and design requirements. The Star-shaped Adapter
was selected as the part to be optimized mainly because it the most prominent and main feature of the
design. The optimization involves minimizing the adapter’s size/weight, while at the same time, making
sure that its strength and resistance to possible common failure causes like impact or constant use/abuse is
not compromised. For this, certain parameters, like yield strengths and safety factors will be set and the
design will revolve around them, providing for a final design that complies with the requirements. Complete
details on the optimization analysis are found on Chapter 3. Once the optimization analysis is finished, the
design will be analyzed using ANSYS.
Along with the different other analysis performed, one very important factor to be considered and
that needed to be determined early on are the materials that the different parts will be made of. Not only does
the material selection affect the mechanical and structural characteristics of the design, but it also plays a big
role on the costs of the system. Because of this, several material options were considered for each of the
different parts and the best choices selected. In the case of the Star-shaped adapter, it was decided that the
best choice was Lexan and thus, the optimization was performed according to Lexan’s properties. Material
selection is discussed in greater detail on Section 5.3.
14
2.2 Design Selection
From the any ideas resulting from brainstorming, the best possible ideas were taken and
developed into more solid concepts. The result of this was that two ideas came out as good candidates
for the final design and to choose between them, Pugh’s Matrix Method was used. The two ideas,
along with the evaluation categories and the actual Pugh Matrix are presented in the next sections.
2.2.1 First Idea: Off-Center Rotating Star Adapter
The first proposed idea consists of a rotating “star-shaped” adapter mounted on an axis parallel
to the axis of rotation of the power tool. The star will hold a particular toolbit in each of its points. To
switch between bits, the user first disengages the current toolbit by moving the star forward, then
rotates the star until the new, desired toolbit is aligned with the power tool’s tip, and finally, moves the
star back to its original position to engage the new toolbit with the tool’s rotating tip. The adapter is
held in place via an adjustable clamp that can be adapted to fit a variety of rotary tools.
Figure 2.1 - Preliminary Concept for Idea 1
15
2.2.2 Second Idea: Centered Rotating Star Adapter
The second proposed idea also consists of a “star-shaped” adapter with the different toolbits
on the star’s tips, but now, the center of the star is aligned with the axis of rotation of the tool, and
instead of rotating the star to the desired bit, each of the star’s points have a hinge mechanism
allowing the toolbits to rotate towards the center of the star (which is the same as the tool’s rotating
tip) and engages into place for use. To switch between bits, the currently installed bit is rotated back
and the next one is rotated in.
Figure 2.2 - Preliminary Concept for Idea 2
2.2.3 Selected Design
To select between the different ideas a Pugh’s Matrix analysis was performed. The ideas were
evaluated according to seven different categories. These categories are:
•
Production Cost – This based on the approximate, expected cost of manufacturing one
unit. It’s an important category since we are aiming for a predetermined price range.
•
Weight – This is the weight of the complete adapter system. Weight is a very
important factor since the lighter the adapter, the less it affects or interferes with the
standard operation of the rotary tool.
16
•
Ease of use – The adapter must be easy to use and provide for a straightforward
operation.
•
"Universality" – The same adapter should be able to fit a wide variety of rotary tools,
instead of requiring a different adapter for each different rotary tool kind or model.
•
Appearance – This refers to the physical appearance of the system. Not a lot of
consideration was given to this particular category, but still, since this is a consumer
product, it needs to be considered.
•
Visibility – The adapter should not interfere, or the least possible, with the actual
visibility that the rotary tool provides the user.
•
Toolbit Switching Time – This category refers to the expected time it would take to
change from one toolbit to another using the adapter. Since this is the main purpose of
the adapter, it was given the most consideration out of all the categories.
Based on these categories, the first Idea, the Off-Center Rotating Star Adapter design was
selected as the best choice between the two ideas. The actual analysis used to select the best idea is
presented in the following table.
Table 2.1 - Pugh's Matrix Analysis
Designs
Criteria
% Scale
Idea 1
Idea 2
Score
Weighted
Score
Weighted
Production Cost (1 Unit)
15
0
0
-1
-15
Weight
15
0
0
0
0
Ease of use
15
0
0
1
15
"Universality"
12.5
0
0
-1
-12.5
Appearance
7.5
0
0
1
7.5
Visibility
10
0
0
1
10
Toolbit Switching Time
25
0
0
-1
-25
100
0
0
0
-20
17
2.3 Methodology
To carry out this type of project, it is necessary to go through a series of steps in order to achieve our
goal. Our first step was to research and study our target market to properly identify some of the needs of the
customer. During this step we spent time looking for the necessities of the workers in a machine shop. After
identifying their needs, the next step was to perform a general brainstorming of ideas to determine which of
those needs we could satisfy and how could we do it.
As part of the brainstorming, we as a group started bringing up ideas to satisfy the needs of those
customers. At this point of the project a lot of ideas show up, and after some general analysis and
consideration, the RTSA was chosen as the concept to be designed. A new brainstorming was carried out to
further develop the concepts for the RTSA and several different ideas were obtained, out of which only two
were taken into final consideration as best choice candidates (see Figures 2.1 and 2.2).
To select between the two final ideas for the design, Pugh’s Method was used, using the categories
described in the previous section. The selected idea was the first one and with the actual design idea
selected, the next stage was selecting a particular part from that design to be optimized and further analyzed.
The Star-shaped adapter was selected as the part to be optimized because it is the main
component of the system. We used the Lagrange Multiplier Method to solve the optimization of
our part, more specifically, minimize the adapter’s size/weight. This type of analysis was done
by hand and is discussed in more detail in the next chapter.
The other type of analysis that will be used to achieve our goals is Finite Element Method
(FEM) which is the method used to construct the approximations needed in a variation or
weighted-residual approximation of the solution of a problem over each element as defined on
the class textbook. This analysis is not yet completed.
Once all the analysis is completed and we are sure that our design is capable of being
manufactured, we may pass to our next and final step, which is the construction of a prototype of
our design. On the following figures you can appreciate some of the machines that could be used
for the creation of our design.
18
Figure 2.3 - Welding Machine
Figure 2.4 - Electric Cut-off Saw
Figure 2.5 - Industrial Milling Machine
19
Chapter 3.
Optimization
3.1 Design Problem
The design of any product involves a series of steps and processes, including different
analysis. The number and depth of the analysis depend on what properties or parameters need to be
verified (like static or dynamic analysis), depending the conditions the product will be subjected to,
and its intended used. For intended use we mean loads, impacts, pressure or concentration of stresses
for which the piece or component is designed. One very critical specification of our product is that its
weight should not affect the rotary device usage, by adding unnecessary weight to the front end
making it hard for the user to manage it. For this we have selected a specific component to perform a
optimization analysis, to minimize its weight.
It is the intention of this section to provide you with the information of the optimization
process that was performed on our RTSA (Figure 3). The RTSA is composed of two main
components, which are: (1) the star-shaped toolbit holder, and (2) supporting clamp that will attach the
toolbit holder to the drill. For purpose of this report, we have decided to minimize the star-shape
adapter (Figure 3.2) weight, through a Lagrange Optimization Method. This method will enable us to
find the optimum dimensions for this adapter, for which we can achieve the minimum weight possible
without compromising certain desired properties. For this specific optimization, we focused on
eliminating the probability of the product failing by bending at any of the 6 extensions of the star
shape component. For this we included a constraint in our problem to account for that requirement.
It is very important to clarify that this analysis was done for a specific case that include
some fixed values and specifications, and the values obtained by this optimization will only hold
true for the ratios and values chosen as constant. We fixed certain dimensions that were desired
for this specific design, in order for this component to work properly with the rotary device for
20
which it is intended. The area “AT” of the star component of the RTSA has a number of
dimensions which have fixed ratios between them. Some of these values were chosen for
convenience reasons and were fixed as constants and others were specified in terms of ratios of
other dimensions. The equation of the area of the star-shape component for the design will be
described in detailed in Section 3.4 together with the dimension constraints used to arrive at the
optimum point.
Figure 3.1 - Star-Shaped Adapter
Load
P
Failure line
Figure 3.2 - Tool bit holder
L
Figure 3.3 - Bending Moment at the star extension
21
3.2 Design Variables:
The design variables for our problem are some of the dimensions of the volume of the Star-shape
component. The rest of the dimensions are fixed as specified values chosen for other purposes.
The design variables include:
X1=b
( 1)
X2=t
( 2)
X3=L
( 3)
X4=l
( 4)
( 5)
X5=a
( 6)
X6=h
Where we can put them together in vector form:
( 7)
X = {X 1 X 2 X 3 X 4 X 5 X 6 }
T
The dimensions of the star that are held constant for this specific design are the
following:
d= 0.5 in
R= 3 in
(Standard hole for the toolbit holder insert)
( 8)
( 9)
(Star external radius)
Parameters that are important in this analysis:
A. Parameters
ft
(Acceleration of Gravity)
s2
lb ⋅ ft
gc= 32.174
(Gravitational Constant)
lbf ⋅ s 2
SF=2
(Safety factor used)
g= 32.174
22
( 10 )
( 11 )
( 12 )
B. Material Properties
lb
ρ Lexan = 0.04405 3
ft
σy
= 9.35 E 3
Lexan
P = 30lb
lbf
in 2
( 13 )
(Density of Lexan)
(Yield strength of Lexan)
(Load applied at corner of extension)
Figure 3.4 - Design Variables Diagram
23
( 14 )
( 15 )
b
L
L
Figure 3.5 - Variation of Design Variables to fit constraints
3.3 Objective Function
The product is intended to be used as a front adapter of a standard rotary device. The star-shape
component needs to be as light as possible, and for that its weight needs to be optimized. To minimize
the weight (W) of the star-shape component of the RTSA we have the following equations.
Where the objective function is:
W = mg/gc = ρgV = ρg(ATt)
( 16 )
f(x) = (ATt)
( 17 )
AT = 6 (Atriangle + Ascquare + Aarc) – 7Acircle
( 18 )
Atriangle = ½ Ab = ½(b/2)(h) = ¼ bh = ¼ X1X6
( 19 )
Ascquare = Ab = (b)(l) = bl = X1X4
( 20 )
Aarc = ½ Acircle =
1 ⎛π d 2
⎜
2 ⎜⎝ 4
⎛π d2
7Acircle = 7⎜⎜
⎝ 4
⎞ 7
⎟⎟ = π d 2
⎠ 4
(
⎞ 1 ⎛ π b2 ⎞ 1
1
⎟⎟ = ⎜⎜
⎟⎟ = π b 2 = π X 12
8
⎠ 2⎝ 4 ⎠ 8
(
)
)
(
)
( 21 )
( 22 )
24
3.4 Constraints
There are certain specified ratios that are fixed between the dimensions:
1. Diameter “d” of the hole of each of the six extensions which has a value of 0.5 inches.
This is a standard value for the piece toolbit holder to fit inside the hole. It is also the
same value of the diameter of the center of the star.
2. Radius “R” of the star is fixed to a value of 3 inches, to provide an external overall
dimension of 6 inches of diameter.
A. Inequality Constraints:
Bending Moment:
This constraint is included here to make sure the dimensions b, t and L
arrived at through the optimization process meet the requirements of
providing for the piece not to exceed the allowable stress (which is a ratio
of the yield strength of the material to a safety factor SF equal to 2)
through bending due to an applied load P at the end of any of the six
extensions of the star-shape components.
σ − σ a ≤ 0 Where σ =
g1 ( X ) =
σ
6 PL
; and σ a = y
2
bt
SF
6 PL σ y
−
= (6SFPL − σ y bt 2 ) ≤ 0
2
bt
SF
g1 ( X ) = SF 6 PX 3 − σ y X 1 X ≤ 0
2
2
( 22 )
( 23 )
( 24 )
Thickness range:
For a desired possible range of the value for thickness “t” tmin < t < tMax
for which we get the following
25
g 2 ( X ) = t min − t ≤ 0 ⇒ g 2 ( X ) = t min − X 2 ≤ 0
( 25 )
g 3 ( X ) = t − t max ≤ 0 ⇒ g 3 ( X ) = X 2 − t max ≤ 0
( 26 )
Width of each of the 6 extremities
Where we want b to be at least 3 time the dimension of d
b ≥ 3d
( 27 )
g 4 ( X ) = 3d − b ≤ 0 ⇒ g 4 ( X ) = 3d − X 1 ≤ 0
( 28 )
Length of each of the extremities
Where it has to bigger than b
g5 ( X ) = L ≥ b ⇒ g5 ( X ) = b − L ≤ 0 ⇒ g5 ( X ) = X1 − X 3 ≤ 0
( 29 )
B. Equality Constraints:
Length = “L”
( 30 )
L = R – bcos(30)
h1(X) = L – R + bcos(30) = 0
h1(X) = X3 – R + X1cos(30) = 0
Length = “l”
26
( 31 )
( 32 )
l = L – b/2
( 33 )
h2(X) = l – L + b/2
( 34 )
h2(X) = X4 – X3 + X1/2
( 35 )
Dimension = “a”
b−d
⇒ h3 ( X ) = 2a − b + d = 0
2
h3 ( X ) = 2 X 5 − X 1 + d = 0
a=
( 36 )
( 37 )
Dimension = “h”
h = bcos(30)
( 38 )
h4(X) = h – bcos(30)
( 39 )
h4(X) = X6 – X1cos(30) = 0
( 40 )
27
3.5 Method of Lagrange Multipliers
General Form
ζ = Lagrangian
(42 )
4
4
i =1
k =1
ζ ( X ) = f ( X ) + ∑ λi g i + ∑ µ k hk
(43 )
For our specific case
ζ (b,t,L,l,a,h) =
⎡ ⎛⎛ 1 ⎞
⎛ 1 2 ⎞⎞ 7 2 ⎤
2
⎢6⎜ ⎜ 4 bh ⎟ + (bl ) + ⎜ 8 πb ⎟ ⎟ − 4 πd ⎥t + λ1 (SFGPL − σ y bt ) + λ2 (tmin − t )
⎠
⎝
⎠⎠
⎣ ⎝⎝
⎦
b⎞
⎛
+ λ3 (t − tmax ) + λ4 (3d − b1 ) + λ5 (b1 − L ) + µ1 (L − R + b cos(30) ) + µ2 ⎜ l − L + ⎟
2⎠
⎝
+ µ3 (2a − b + d ) + µ4 (h − b cos(30) )
(44 )
Written in Term of Design variables {X}
ζ (X ) =
Where X = {X 1 X 2 X 3 X 4 X 5 X 6 }
(
)
⎡ ⎛⎛ 1
⎤
⎞
⎛1
2 ⎞⎞
2
⎢6⎜ ⎜ X 1 X 6 ⎟ + ( X 1 X 4 ) + ⎜ πX 1 ⎟ ⎟ − 1.374⎥ X 2 + λ1 SF 6 PX 3 − σ y X 1 X 2 + λ2 (t min − X 2 )
4
8
⎝
⎠
⎝
⎠
⎠
⎣ ⎝
⎦
X ⎞
⎛
+ λ3 ( X 2 − t max ) + λ4 (3d − X 1 ) + λ5 ( X 1 − X 3 ) + µ1 ( X 3 − R + X 1 cos(30) ) + µ 2 ⎜ X 4 − X 3 + 1 ⎟
2 ⎠
⎝
(
)
(
)
+ µ 3 2 X 5 − X 1 + d + µ 4 X 6 − X 1 cos(30)
(45 )
Stationary Point of Lagrangian
∇ζ = 0
(46 )
28
⎧ ∂ζ ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂ζ ⎪
⎪ ∂X 2 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂X 3 ⎪
⎪ ∂ζ ⎪
⎪ ∂X ⎪
⎪ ∂ζ 4 ⎪
⎪
⎪
⎪ ∂X 5 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂X 6 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ1 ⎪
⎪ ∂ζ ⎪
⎪
⎪
∇ζ = 0 = ⎨
⎬
⎪ ∂λ2 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ3 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂λ4 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ ⎪
⎪ 5⎪
⎪ ∂ζ ⎪
⎪ ∂µ1 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂µ 2 ⎪
⎪ ∂ζ ⎪
⎪ ∂µ ⎪
⎪ 3⎪
⎪ ∂ζ ⎪
⎪⎩ ∂µ 4 ⎪⎭
Then with (I) λi, µk can be acquired
To get possible solution Points:
design variables X
Using the second set of equations we acquired the design optimum point:
29
(47 )
X 1 = 1.5
X 2 = 0.2088
X 3 = 1.7009
X 4 = 0.95
X 5 = 0.5
X 6 = 1.299
Verify K-Tucker
For design point:
X = {1.5 0.2088 1.7009 0.95 0.5 1.299 }
*
(48 )
T
I. Condition : Check All inequality and equality constraints for feasibility
g1
X =X*
≤ 0 = (SF )(6)(P )( X 3 ) − σ Y X 1 X 22 ≤ 0
(2)(6)(30)(1.7009 ) − 9.35 × 10 −3 (1.5)(0.2088)2 = 0
g2
X =X *
≤ 0 = t min − X 2 ≤ 0
X =X*
X =X *
(50 )
(52 )
≤ 0 = X 2 − t max ≤ 0
(53 )
0.20894 − 0.375 = −0.16605 ≤ 0 OK
g4
OK
(51 )
0.125 − 0.20894 = −0.08395 ≤ 0 OK
g3
(49 )
(54 )
≤ 0 = 3d − X 1 ≤ 0
(55 )
3(0.5) − 1.5 = 0 ≤ 0 OK
(56 )
30
g5
X =X *
≤ 0 = X1 − X 3 ≤ 0
(57 )
1.5 − 1.7009 = −0.2009 ≤ 0 OK
(58 )
h1 X = X * = 0 = X 3 − R + X 1 cos(30)
(59 )
= 1.7009 − 3 + 1.5 cos(30)
1.7009 − 3 + 1.5 cos(30)
= −0.00006 ≈ 0 OK
h2
X =X*
= 0 = X4 − X3 +
(60 )
X1
2
(61 )
= 0.95 − 1.7009 +
1.5
2
(62 )
= −0.00009 ≈ 0 OK
h3
X =X *
= 0 = 2X 5 − X1 + d
(63 )
= 2( X 5 ) − X 1 + d
2(0.5) − 1.5 + 0.5 = 0 OK
h4
X =X*
(64 )
= 0 = X 6 − X 1 cos(30)
(65 )
(1.299) - 1.5cos(30)
= -0.00004 ≈ 0 OK
(66 )
31
II. Condition: check Lagrange Multipliers λ1 inequality constraints are none
negative nonnegative
Inequality Constraint #1
λ1 g1 X = X * = 0 Since g1 = 0; then λ1 > 0
(67 )
λ1 = 0.00026 > 0 OK
Inequality Constraint #2
λ2 g 2 X = X * = 0 Since g 2 = −0.08395; then λ 2 = 0
(68 )
λ 2 = 0 OK
Inequality Constraint #3
λ3 g 3 X = X * = 0 Since g 3 = −0.16605; then λ3 = 0
λ3 = 0 OK
(69 )
Inequality Constraint #4
λ4 g 4 X = X * = 0 Since g 4 = 0; then λ4 > 0
λ4 = 1.01 > 0 OK
(70 )
Inequality Constraint #5
λ5 g 5 X = X * = 0 Since g 5 = −0.2009; then λ5 = 0
λ5 = 0 OK
32
(71 )
III. Condition: Gradient of the function and constraints of lagrange are equal to
zero
5
4
i
k
∇f + ∑ λi g i ∇ + ∑ µ k hk ∇ = 0
(72 )
⎧1 ⎫
⎧− 1⎫
⎧− 407.04 ⎫
⎧ 2.786 ⎫
⎧0.866⎫
⎪2 ⎪
⎪ − 5856.8 ⎪
⎪ 15.374 ⎪
⎪ 0 ⎪
⎪0⎪
⎪0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪ ⎪
⎪ 360 ⎪
⎪ 0 ⎪
⎪ 1 ⎪
⎪0⎪
⎪0 ⎪
⎬ + (1.01)⎨ ⎬ + (− 1.499 )⎨
⎬ + (0.00026)⎨
⎨
⎬ + (− 1.592 )⎨ ⎬
0
⎪
⎪
⎪ 1.5923 ⎪
⎪1 ⎪
⎪ 0 ⎪
⎪1⎪
⎪
⎪
⎪ 0 ⎪
⎪ ⎪
⎪
⎪
⎪
⎪
0
0
0
⎪
⎪
⎪
⎪
⎪0 ⎪
⎪
⎪
⎪ ⎪
0
.
18291
0
0
0
⎭
⎩
⎭
⎩
⎭
⎩
⎩ ⎭
⎪⎩ 0 ⎪⎭
⎧− 0.866⎫ ⎧− 0.3465⎫ ⎧0⎫
⎪ 0 ⎪ ⎪ − 0.386 ⎪ ⎪0⎪
⎪ ⎪ ⎪
⎪ ⎪
⎪
⎪ ⎪0⎪
⎪ 0 ⎪ ⎪ 0
+ (− 0.183)⎨
⎬≈⎨ ⎬
⎬=⎨
0
0
⎪ ⎪0⎪
⎪ ⎪
⎪
⎪ ⎪0⎪
⎪ 0 ⎪ ⎪ 0
⎪ ⎪ ⎪
⎪ ⎪
⎪
⎭ ⎩0⎭
⎩ 1 ⎭ ⎩ 0
(73 )
Now using the design variables, inserting the values in the equation for total area, and
Calculating the objective function of volume;
(74 )
W = mg/gc = ρgV = ρg(ATt)
f(x) = (ATt)= Volume = 4.583 in 3
(75 )
Weight = ρgV = 0.2088lbf
(76 )
33
3.6 Discussion
The design variables dimensions that were achieved provide values that comply with our
desires: (1) keeping the external dimensions of the star inside the desired 6 in diameter, and still
manage to get appropriate values for the other parameters; (2) comply with all our constraints, to make
sure it will not fail by bending under stated conditions.
This optimization process brought values of the dimensions for the area of the star, and
thickness, with which we can determine volume and thus weight. The values of the area AT are locked
with one another, and any change in one value, will have a direct change in our other parameters. In
order for us to comply with values that wouldn’t fail by bending, we found a value of thickness
0.2088in. For construction purposes we have set this number to the nearest value of commercially
available plates of Lexan, which is 0.25in. This value will still work with our constraints, because is
greater than the minimum t required.
Throughout this process, we can arrive at various other possible solutions that were not good
candidates, because even though the values were positive, some were not logical for construction
purposes. The other point didn’t have the bending constraint active, so it had dimensions that could
make the star, but not guaranteeing it wouldn’t break by bending under the conditions specified. The
other point provided values that violated some of the constraints, so they were not good candidates.
Finally it is important to clarify that this process and values are only valid for the ratios and
constants specified. If they were to change the same process would be repeated, but with the desired
constraints, that can be added.
34
Chapter 4.
Finite Element Analysis
4.1 Introduction
Part of these project objectives, aside from optimizing the weight of the most important
component (the star-shape component), is to perform a structural analysis to verify that our
selected dimensions, a For this purposes we have selected ANSYS, to perform a Finite Element
Analysis(FEA) of our most relevant component, which is the Star-shape . This FEA includes the
three main general stages that any computer aided design program for FEA has: Pre-processing;
Processing and Post-processing. Following, the main parts of FEA of the star-shape component
will be described, and explained. For more details, a step- by-step guide of this FEA has been
included in the Appendix C.
4.2 Geometry
Once the component to be analyzed has been selected, it needs to be described in terms
that the program can understand, and interpret. This requires the definition of the material
properties, and the geometry of the component evaluated. In ANSYS, you need to draw the
geometry using a coordinate system, which the program calls “key points”.
For the purpose of our FEA analysis, we focused our interest in acquiring structural
analysis of our component, when subjected to specified loads, and restrictions. Specifically we
were after deflection and stresses results due to a point load one of the extremities of the starshape component, and this way verify that the design variables obtained in the Lagrange
Optimization Method comply with the conditions the component will be subjected to. Because
we have modeled the as having one point load, found at one extremity to evaluate bending (to
model the case of the unit falling to the floor together with the rotary device, on one corner), we
modeled the geometry of one extremities of the star, because we have symmetry of shape. For
35
our specific geometry we use geometric shape areas to draw the geometry with keypoints
(coordinates) to develop the extremity component. After the geometry’s front area was drawn (2D), then the individual areas can be selected and united, using “Glue” command. Then, they can
be extruded to form the desired one-volume (3-D) figure (see Figure 4.1).
Figure 4.1 - 2D Area to be extruded
4.3 Mesh Creation
For this particular FEA, we choose to divide our principal geometry, using a very fine mesh,
to attain a high level of results. If this particular analysis was to be performed on a high scale manner,
it would become too costly in time and resources, but for this particular study, the resources, and
equipment were available to perform a very specific and accurate FEA., and the analysis wouldn’t
take large amounts of disk space, which is also a main reason to diminish the element division just to
the needed level of study. For our Analysis we used the smart meshing tool, but with a “coarse” of
“1”, meaning, it would provide us very minuscule element divisions, that approximate our solution to
a very high degree. From a consumer point of view, this represents a higher degree of guarantee. See
Figure 4.2 for more clarification.
36
Figure 4.2 - ANSYS Geometry and Mesh Plot
For the element divisions, we chose a solid element, to represent our component behavior.
From the solids available we choose, Solid 187. This element has plasticity, hyperelasticity, creep,
stress stiffening, large deflection, and large strain capabilities. It also has mixed formulation capability
for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible
hyperelastic materials. Pressures may be input as surface loads on the element faces, which makes it
suitable for our analysis. (See Appendix D for more detailed information.)
4.3 Loads and Boundary Conditions
Once the main element geometry has been defined subdivided into sub-elements, using the
meshing mentioned in the previous section, the loads and boundary conditions are specified. This
includes defining the restrictions of the figure, to simulate the real conditions of that specific
component in the real life. This permits the evaluation of resulting stresses and deflections due to a
condition applied with the specified restricted ends.
37
For our analysis, we wanted to simulate the effect of having a point load of 30lbs on one
extremity of the star. We established earlier in this report during the optimization process that the
rotary device to which this RTSA will be adapted to should weigh between 8-15 lbs. assuming the
worst case. When we modeled one segment of the star, we need to model it as a cantilever element,
which has the beginning restricted. The point load is applied at the very end, at a distance L of 1.7in
from the origin, which is established at the constrained end, in one corner. See Figure 4.3 observe the
component restrictions and loads applied.
Applied Force
Constrained Face
Figure 4.3 - Applying Load and Restrictions
38
4.5 Analysis
From the ANSYS analysis, we can appreciate that the design can handle the type and
magnitude of forces that it was subjected to. In general, Lexan proved to be a very good material
choice since it allowed the part to be very light while at the same time, comply with the strength
requirements.
We also learned that it is very important for the outcome of the results how the problem
is approached and the different analysis choices that are made. From the type of element that is
selected, to how the geometry is drawn or the constraints that are established. In the end, all these
parameters were selected to suit our specific model and the model ran very reliably.
The stress results show a stress concentration point at the location were the force was
applied which is expected. The stress then is distributed along the piece very uniformly until it
rises again at the point were the model is constrained, this is the area where the real part would
meet with the rest of the piece. The maximum stresses experienced by the part are below the
ultimate stress that the material (Lexan) can handle.
Figure 4.4 - ANSYS von Mises Stress Plot
The deflection results show a maximum deflection of the piece of only 0.15 inches,
which is an acceptable value and should not interfere with the performance of the adapter.
39
Figure 4.5 - ANSYS Deflection Plots
40
4.6 Additional Analysis (COSMOSWorks)
In addition to the ANSYS analysis, the model was also analyzed using the
COSMOSWorks application from the SolidWorks 2005 design software. This analysis was used
as a comparison to the ANSYS results, and not surprisingly, both were very similar. The model
used for the analysis was of the complete star-shaped adapter, restraint in the center hole and
with a force of 30lbs. applied to one of the star’s edges.
The following plot shows the mesh that the program created to analyze the model, as well
as the location of the applied force (pink arrow) and the constrained elements (green arrows).
Also in the figure is a plot of the Safety Factor in the model, which is calculated using the von
Mises stress on the element in relation to the material’s ultimate strength.
Applied Force
Constrained
Elements
Figure 4.6 – COSMOSWorks Meshing and Safety Factor Plot
The following plots show the stress results and deflection, both of which were also within
acceptable parameters and very close to the ANSYS results. As expected, the maximum stresses
are experienced in the area were the star point meets the rest of the structure, since this is the area
were the moment caused by the force is greatest. Still, the stresses here are well below the limits
of the design (Max. von Mises = 12,300psi). In terms of the deflection, the maximum deflection
occurs at the point were the force is applied and is equal to 0.31in, which in the case of our
design, is an acceptable value that should not affect the performance of the adapter.
41
Figure 4.7 – COSMOSWorks von Mises Stress Plot
Figure 4.8 – COSMOSWorks Deflection Plot
42
Chapter 5.
Results and Discussion
5.1 Present Design
The following set of figures show how the design of the RTSA would look like according to the
results of the analysis performed. The dimensions were determined from the optimization through the
Lagrange Multipliers Method and the materials decisions were based mostly on the cost analysis. The
ANSYS analysis helped prove that the design decisions made were correct, as the results show that the
stresses and were below the limits of our design.
Figure 5.1 - RTSA - Isometric View
Figure 5.2 - RTSA - Front View
43
Figure 5.3 - RTSA - Side View
Figure 5.4 - RTSA - Top View
44
Figure 5.5 - RTSA - Engineering Drawing
45
5.2 Discussion
Our inspiration to create this design was to solve a need in the automotive industry. We decide to
work with rotary devices because we observed a distinctive preoccupation in many customers about how
tedious were the action that they had to constantly change the toolbits when they were using a drill. Because
of this we decide to create the RTSA so the workers could end a task in the minimum amount of time
without the preoccupation of change the toolbits each time they used a different tool. The recollected
information led us to develop a set of desired specifications and criteria for the RTSA. After evaluating all
the ideas we end with the two best possible designs. Then we went through a process of choosing the best
idea for our design and at the same time that it can be feasible.
The Pugh’s Method was the method used to determine which idea we were going to use in our
design. After the method was performed, we got that the first idea was better because it domains the second
one in almost all the categories we decide to use. Even though the second idea looks very appealing to the
eye its manufacturing cost, its universality, and its toolbit switching time, which were the ones with most
weigh in the Pugh’s Method, were below the first idea and this affected in great scale the final selection: the
firs idea.
The optimization process included in this report provides good design dimensions for the star-shape
component of the RTSA, from a manufacturing point of view, after adjusting the value of the thickness to
the next Lexan plate available commercially. It provides a safety factor of two, which will ensure even more
that the piece will not fail by bending of the star-shape extremities.
5.3 Cost Analysis
As part of the project a cost analysis was performed to determine the approximate costs
involved in the productions of one unit. Since we are trying to minimize the weight of our adapter
but at the same time, while at the same time keeping costs down, several material options were
considered and compared for the different components of the adapter.
46
Table 5.1 - Star-Shaped Adapter Material Comparison
Material
Thickness
(in)
Dimensions
(in)
Weight
(lb/ft2)
Price
($)
Polycarbonate
.25
12 X 12
1.5624
14.07
Hot roll Mild Steel A36
.25
12 X 12
10.188
21.27
Stainless Plate 316 Annealed
.25
12 X 12
10.332
79.25
The proposed materials for Star-shaped adapter are shown in Table 2. These materials all
fit the specifications of our design and will need to be further evaluated taking under
consideration another variables that could affect our design. The materials under consideration
are: Plastic Sheet Polycarbonate (Lexan), Hot-rolled Mild Steel, and Stainless Steel. The table
shows that all the materials were chosen with the same thickness and the same dimensions so we
could make an appropriate comparison. Since we are looking for the best choice that fulfills all
of our needs we can say that the best choice for our star is the Plastic Polycarbonate because it is
the lightest material of all and it is also the most affordable. Another advantage of this material is
that it is clear and this will help the user by not interfering with their visibility while using the
adapter. The mechanical properties of Lexan also make it a very good choice because they
provided a unique combination of high impact strength, flame retardancy, and thermoformability
that makes it ideally suited for this application. The other two materials may also satisfy our
needs, but their prices are too high in comparison with the polycarbonate and they don’t provide
for an uninterrupted field of vision as Lexan does.
47
Table 5.2 - Adapter Support Material Comparison
Material
OD / ID
(in)
Length
(ft)
Weight
(lb/ft)
Price
($)
Welded Stainless Tube 316
.75 / .62
1
.4818
4.00
Alloy Steel Tube 4130
.75 / .62
1
.4818
7.65
Copper Tube 122 Hard Drawn
.75 / .62
1
.5422
6.50
After choosing the material for the Star-shaped adapter, the material that needs to be
selected is the one for the support clamp that holds the Star-shaped adapter in place. For this part,
three different materials were considered. These materials are: Welded Stainless Tube 316, Alloy
Steel Tube 4130, and Copper Tube 122 Hard Drawn. The principal characteristic we were
looking for was the materials resistance to bending. Of the three materials, the stainless steel
came out as the best choice for our application. This was because it possesses a very high
strength and is a non-corrosive. This is very important because of the environment in which this
type of tool is exposed. Also of all the materials, stainless steel is one of the lightest per lineal
foot and is the cheapest one. The copper is a good material but is corrosive and not as strong, and
the price of the steel tube is higher in comparison with the stainless steel.
Table 5.3 - Expected Production Costs
Application
Quantity
Price
Polycarbonate
1 sheet
$14.07
Stainless Steel
1 feet
$4.00
Machining
Renting Equipment
$25.00
Labor
1 person @ $7/hr
$14.00
Miscellaneous
Nuts, bolts, and others
$6.00
Total
1 adapter
$63.07
48
The previous table shows the cost for the production of just one adapter. Most of the
prices and costs were approximate values of the expected costs and should be further analyzed
before actual productions of the unit. Because this table shows the cost of just one adapter
produced by hand it may be a lot higher than the actual mass-produced cost of the adapter.
Assuming that we were to produce a total of 100 adapters per week by working eight hours a
day, it means that we would need five workers to achieve the goal. This means it will be a total
of $4,432/wk that leads to a total of $230,464/yr. Obviously, these prices will change if we
produce the adapter in mass because the material would be cheaper and is recommended to buy
the machines instead of renting them. We can also create a system in which it can be created
automatically and not by hand and this will save a lot of money.
49
Chapter 6.
Final Remarks
6.1 Conclusion
The existence of indispensable repetitive process that takes place on a daily basis on
production lines in the Automotive Industry creates a need and consequently an excellent profitable
market for efficient time reducing type of tools. This opportunity provides a great timing to capitalize
from these growing market, by introducing the innovative RTSA unit, which as demonstrated
throughout this report, represents for the automotive industry, an extraordinary alternative to
conventional tool-handling, in a cost effective and affordable manner.
The RTSA proves to be a robust design, excellent for the harsh repeatedly machine shop
conditions, while providing a more straight forward route, with its capacity to hold close, accessible
and redy to switch several toolbits at a time. This way reducing the time of the overall in-line process,
which to any high scale producing corporation, translates to millions of dollars saved due to time
reduction, as well as thousands more in extra revenue resulting from higher efficiency of individual
tasks, and thus higher gross overall production. This being one of the many endless reasons why the
RTSA is the way to convert this great opportunity into a profitable solution.
50
6.2 Recommendations
The RTSA design is an excellent alternative to stated need improving efficiency at automotive
line processes requiring repeated processes. In the occasion that this design was implemented to be
produced in a mass scale production, it is recommended that specific analysis is performed to the other
remaining components, but adjusting the analysis conditions to not only the intended use, but also the
production conditions, which may include thermal, depending on the manufacturing processes of the
parts.
Also, it is advise that an ergonomics study of be performed to adapt the existing features of the
design to one which complies with the time and for of usage of the tools for which this is intended.
51
References
1. Junvinall, Robert C. and Kurt M. Marshek. Fundamentals of Machine Components Design. 3rd ed. New
York: John Wiley and Sons, 2000.
2. WEB: http://www.plunkettresearch.com/automobile/automobile_statistics2.htm
3. WEB: http://www.dewalt.com
4. WEB: http//www.sears.com
5. WEB: http://www.makita.com
6. WEB: Goyal, Vijay K. http://www.me.uprm.edu/vgoyal, University of Puerto Rico at Mayaguez,
Mayaguez, PR.
52
Appendix A.
Submitted Proposal
53
Proposal
To:
Dr. Vijay K. Goyal
From: Borrero Valentín, Susana E. (802-00-0837); Diana Vélez, Roberto (802-00-1925);
Rivera Maldonado, Rodolfo (802-00-6025)
Date:
February 7, 2005
Re:
Engineering Design Project Proposal (INME4058: Section 126)
Dear Dr. Goyal:
Greetings. The intention of this document is to propose the design and optimization of the Rapid Toolbit Switching
Adapter (RTSA) for rotary tools. This device is a disk-like adapter for hand-operated rotary devices that will hold,
close and handy, a set of chosen toolbits at a time, allowing for quick and easy switching between them, thus reducing
the time required for a set of tasks. Enclosed is a 10-page brief description of the project.
With over 5 million automobiles produced each year in the U.S. alone, plus many millions more already on the
streets requiring periodic repair and services, a system that could reduce the time, effort, and consequently money
spent in the hand tasks involved in the assembly and repair of such, will have a significant impact on the
Manufacturing Industry, and more specifically in the Automotive sector. These tasks involve a considerable number
of different tools, and toolbits, making the proper access to them a time challenge. The RTSA, a light disk-like
adapter, will eliminate that challenge and the problem of lost tool-bits, by keeping the operator’s chosen set of bits
together at his fingertips in a circular array around the rotary device front end, making his work a lot easier in less
steps.
Our main goal is to deliver the optimized design of the RTSA at a cost that should not exceed $60, to make it an
affordable and attractive solution to our costumer. This will involve a thorough design evaluation which will include a
complete static, dynamic and finite element analysis to achieve the maximum design performance as well as safety
standards associated. Constraints such as weight, attachment and removal of device, visibility, ergonomics and others
will also be taken into consideration.
We appreciate your time and consideration in this project. If we can be of any further help, please do not hesitate to
contact us.
Regards,
Susana Borrero Valentín
Roberto Diana Vélez
54
Rodolfo Rivera Maldonado
Design of a Rapid Toolbit
Switching Adapter for Rotary Tools
Susana Borrero Valentín,
Roberto Diana Vélez and Rodolfo Rivera Maldonado
In any high scale line production, every extra second per task becomes a bottle neck for the
next, causing delays down the line, and subsequently lost money to the business. In the Automobile
Industry, where constant repetitive tasks takes place along the line, success depends on the efficiency
of performing each task in the least amount of steps possible. The problem is that rotary devices, such
as drills, which are one of the most employed tools on assembly lines, don’t provide for the complete
task to be performed in a none-stop single step process. Instead, the time is lost in moving to the side
to look for the next toolbit required, detaching the one on the tool, attaching the next one, moving back
to the operating position to finally performing the drilling activity. Based on this need to increase line
efficiency, the engineering team has come up with the idea of the Rapid Toolbit Switching Adapter
(RTSA), which enables for a reduction in steps, time and effort, by providing quick accessibility and
placement of toolbits by holding them in a disk-like adapter that is placed on the front end of the
rotary device, for a quick switch between bits.
The RTSA is disk-like toolbit holder for regular rotary devices. The disk has a star-shape form
that can fit up to 8 toolbits at its corners, and functions by one hand rotation of the disk about its axis
to click the desired bit in place. It has an adjustable ring clamp for attaching it to the different rotary
devices available. To function, it will not consume any energy from the main device, because its
revolving mechanism will relay on a gear-type mechanism for proper toolbit alignment with the
rotating tip of the main device. The cost of this design should be within the $60 to $80 range, and a
Manufacturers Suggested Retail Price (MSRP) of around $100. The user should be able to perform a
set of related tasks (requiring different toolbits), in an easier and time reduced manner, with the same
tool, without the need to stop activity or move from operating position to find the needed tool.
An evaluation of technology available (toolbelts, tool cabinets, ceiling tool racks, toolbit kits)
show that even though some of these available solutions solve part of the time related problems, they
still present inconveniences to the operator, because of extra man power required for effective usage
of far tool racks for example; space constraints limits the use of racks which are too big; accessibility
from operating position, and weight limitations of tool carriers. But most important, none of them
provide for an extremely quick switch between toolbits, mounted on the tool. The RTSA will satisfy
the need of user friendly, step reducing tools that can increase productivity in any manufacturing line.
In order to achieve the high level design and optimized mechanism, a complete static,
dynamic and finite element analysis will be performed, and important constraints such as safety, ease
of use, weight, ergonomics, visibility and others will be considered.
55
Introduction
For the hundreds of tasks in the automotive industry that require the use of rotary tools,
whether it is for an assembly line of the millions of vehicles produced annually; for the industrial
machine shops that repair the many other that are on the streets, or simply for personal use, the reality
is that even though there are many rotary tools available for the different tasks, the problem of
dispersion of toolbits, at-hand type of accessibility, quick and easy switching between toolbits on a
single tool is a common problem to all.
From the existing devices available, let’s take for example the tool holder waist strap. This
particular tool carrier permits the operator to hold the tools and toolbits on him, but still requires him
to 1) take the current toolbit from the rotary device and put it back in the strap, 2) take the next toolbit,
which requires him to take his eyes away from the job to look at the strap, 3) place it on the rotary
device tip, and finally 4) perform task, which is still a long process. Other available tool carriers work
properly in keeping tools in place, avoiding misplacement, as for example, toolbit kits (see figure 2 on
Appendix C). Still, they don’t solve the specific need of having them ready and mounted in the rotary
tool itself for quick switching. The RTSA, will provide for both of these needs while minimizing the
steps required to perform a given task, by holding the desired set of toolbits mounted and ready for use
in the tool itself.
The focus of the design team will be developing an adapter for currently existing rotary
devices such as drills and electric or pneumatic wrenches. This adapter will employ a rotating starshaped disk to hold up a set number of toolbits in its corners, allowing for quick switching between
the toolbits. The RTSA disk will function in such a way that it will require only a hand rotation of the
disk, and its revolving mechanism about its own axis will allow the desired toolbit to auto-click in
place. The operator can choose how many toolbits and which types to accommodate in the disk at any
given time. The disk will have a number of holes that accept a variety of standard toolbit types and
sizes, for an even higher user convenience.
This venture will be achieved at a cost effective manufacturing cost, but without trading off
the top of the line reliability, robustness, or product high quality expected.
Problem description
After studying the Manufacturing Industry, specifically the Automotive, we evaluated
different machine shops, and found there was a common need to increase the mechanic’s efficiency
when performing a repetitive type of task. When further observing these tasks, required in processes
such as assembly of parts, we found that rotary devices are the most frequently used, quantifying for
most of the repetitive activities that take place in automotive machine shops and industrial assembly
lines. One way the efficiency of such processes can be increased is by reducing the amount of steps
required and thus, the time a mechanic spends changing toolbits when using rotary tools, which is a
very common and inevitable time-consuming process as already mentioned.
56
A brief analysis was performed to better understand what could be the needs and most
common complaints of mechanics and technicians that use this type of tools. Users of this type of
tools tend to need the same group of toolbits for a specific task over and over again, providing for an
opportunity to optimize the steps taken to perform such a task. The main problem pointed was the
need to change tools or constantly reach out for the correct toolbit to change it and then perform the
action. Second was the fact that in a machine shop, assembly line or personal workshop area, it is
often hard to keep all tools handy and ready for the specific task. Often, tools get misplaced and lost
because they are on the floor or laying around. A toolbit holder that provides the user the ability to see
the set of toolbits in a disk while using them, while having them mounted and ready for quick
placement to the rotary tool, would reduce the problems mentioned above, as well as keep the user
focused on the task without distractions, which is the mayor cause of injuries in the work place.
Given the existence of a wide variety of hand-operated rotary tools in the market (see figure 4
of Appendix C), but no actual device available to provide for rapid toolbit switching, by developing an
adapter which will hold the chosen set of bits handy and ready for quick switching, mounted on the
tool itself, we can achieve this. The solution involves the ability to insert a preset number of toolbits
on the corners a star-shaped disk, which has specially designed holes to accept different sizes and
types of toolbits. These specially designed holes will hold the bits in place, but will permit the toolbit
to rotate inside the disk without abrasion of the toolbit or disk.
In order for this device to be effective and practical it should comply with the following
specifications: (a) light weight; (b) easy to install/remove; (c) fits a wide variety of rotary tools
(universal adapter); (d) enables quick tool bit switching; (e) accepts standard tool bits including
sockets, drill bits, screwdriver bits, etc.; (f) able to fit between 4-8 different tool bits at a time; (g) does
not interfere with the typical operation or capabilities of the rotary tool and (h) provides a user-friendly
type of tool manageability.
Major Assumptions
The user already has both, an appropriate tool to which the adapter will be installed, and the
necessary toolbits which fit the adapter. The task to be performed is adequate for the adapter’s capabilities,
meaning that it involves repetitive processes and the use of different tools/toolbits to fasten, unfasten, drill,
etc. User-experience with the regular operation of the tool is expected. The adapter won’t rob the tool of any
power or performance it currently has. The toolbits holes found in the disk will permit the free rotation of the
toolbits placed in them, without abrasion of the toolbit or disk, without interfering with the rotary device
intended performance. The weight of the adapter and toolbits together won’t affect the task to be performed
by the operator, thus it will be considered to make sure it won’t exceed an appropriate value.
57
Expected Outcomes
The goal of this project is to design an adapter that will provide the user of a rotary tool the ability to
quickly and effortlessly swap between a number of toolbits. The adapter needs to be universal, fitting a wide
variety of rotary tools without any additional parts. Its cost should be within the $60-$80 range. It will be
lightweight and shall not interfere with the typical operation or capabilities of the tool and it must work
with standard toolbits already available in the market. This adapter will decrease the amount of time needed
for performing certain tasks and should help improve the overall efficiency of the processes in a reliable
robust way.
Budget Summary
A good quality heavy-duty rotary tool’s price, like a drill, electric or pneumatic wrench, usually
fluctuates in the $100-$200 range. Based on the price of the tools the adapter will be used on, we are aiming
for it to be priced within the $60-$80 range. We expect most expenses to go toward the cost of the materials
and the manufacturing, since we want a high-quality, reliable, heavy-duty product that can stand up to
constant use and abuse. Final product price will depend on the material selection and manufacturing
processes selected, which in turn, will be determined after a final design is selected and a comprehensive
engineering analysis is performed.
Table 2 - Preliminary Costs
Item Description
Cost
Disk and Clamp Materials
$10 - $15
Toolbit Holder Materials
$15 - $20
Machining and Labor
$20 - $25
Misc. Expenses
$15 - $20
Total
$60 - $80
58
Conclusions
Given the proper evaluation that was performed on the Industry to understand the problem of
efficiency found in the automotive sector, researching the market for what is available and what is lacking
from the existent solutions, and understanding the needs and requirements of the customer, the design
proposed by the engineering team meets and surpasses all the aspects that were brought for consideration.
Under the specifications and performance level defined by the costumer’s need, the RTSA complies with
being an excellent and right-on target solution by reducing the amount of steps, time, money, effort and
space required to perform the common set of repetitive tasks. The static, dynamic, and ANSYS analysis will
ensure only the optimal design is produced from this process, guarantying that it will not only solve the
problem, but in the most efficient way possible.
The Rapid Toolbit Switching Adapter will revolutionize the Automotive Industry as you know it.
This holistic design promises to deliver the effectiveness and results needed to boost efficiency up and
beyond. The design will deliver a robust and cost effective solution to time consuming multi-step processes
required by tasks involving hand-operated rotary tools. This device guarantees the elimination of the bulk
part of your mechanic work and effort, as well as the lost and misplaced tools dilemma found in machine
shops. With this simple to use adapter you will make repetitive-task involve in the use of these rotary
devices a reliable and most important uninterrupted simpler process, thus no more bottle necks due to this
problems in your continuous assembly line once and for all.
References
1. Junvinall, Robert C. and Kurt M. Marshek. Fundamentals of Machine Components Design. 3rd ed.
New York: John Wiley and Sons, 2000.
2. WEB: http://www.plunkettresearch.com/automobile/automobile_statistics2.htm
3. WEB: http://www.dewalt.com
4. WEB: http//www.sears.com
5. WEB: http://www.makita.com
6. WEB: Goyal, Vijay K. http://www.me.uprm.edu/vgoyal, University of Puerto Rico at Mayaguez,
Mayaguez, PR.
Appendix A. Qualifications
Miss. Susana Borrero is currently a 5th year mechanical engineering student at the University of
Puerto Rico – Mayagüez Campus. She has worked on process lines and has experience designing
packaging equipment for production plants as part of her Coop experience with Kraft Foods in New Jersey.
Also worked with the developing of new technology for P&G for two terms in Cincinnati, OH, and can
provide some insight on Quality and efficiency standards required by their high speed production facilities.
59
Mr. Roberto Diana is currently a 5th year mechanical engineering student at the University of Puerto
Rico – Mayagüez Campus. He has participated in the design and development of Energy Systems for
thermodynamic applications. Mr. Diana has successfully completed design courses such as Machine
Element Design, Mechanism Design, and Project Management courses, which have familiarized him with
design steps and concepts.
Mr. Rodolfo Rivera is currently a 5th year mechanical engineering student at the University of
Puerto Rico – Mayagüez Campus. He has worked as an Engineering Coop at GE Aircraft Engines in
Cincinnati, Ohio for two terms. He has also been part of RUM’s Rumblebots fighting-robot program/course
and has plenty of personal experience working with power tools.
Appendix B. Task Breakdown
The following table describes the expected task/time distribution for the project. It encompasses a
preliminary table illustrating anticipated tasks and other tasks could be added, removed or modified if
necessary as the project progresses.
The project overall duration is estimated for 3 month and a half. As currently planned, the final
design selection should be done by March 4, 2005, giving way then the Materials, Cost, Static/Dynamic and
Mechanism analysis methods can be performed in time for the Project Part 1 Report which is due on April
20, 2005(. The final step of the analysis, the ANSYS analysis, will be performed next . The results are
scheduled to be presented in the Project Part 2 Report and the Project Presentation, scheduled for May 12
and May 14, 2005, respectively.
Table 3 - Gantt Chart
(Legend: SB = Susana Borrero , RB = Roberto Diana , RR = Rodolfo Rivera)
60
Appendix C. Figures
Figure 6 - Standard Tool belt
Figure 7 - Drill Bit Set
Figure 9 - Examples of various types of rotary tools
Figure 10 – Example of various standard toolbits
61
Figure 8 - Tool Cabinet
Individual Rotating
Toolbit Holders
Figure 11 – Sketch of proposed Adapter Design
62
Appendix B.
Method of Lagrange Multipliers
II.
Objective Function:
To minimize the weigh (W):
Where:
W = mg = ρgV = ρgATt
ATt = 6 (Atriangle + Ascquare + Aarc) – 7 Acircle
Atriangle = ½ Ab = ½(b/2)(h) = ¼ bh = ¼ X1X6
Ascquare = Ab = (b)(l) = bl = X1X4
Aarc = ½ Acircle =
1 ⎛π d 2
⎜
2 ⎜⎝ 4
⎛π d2
7Acircle = 7⎜⎜
⎝ 4
⎞ 7
⎟⎟ = π d 2
⎠ 4
(
⎞ 1 ⎛ π b2 ⎞ 1
1
⎟⎟ = ⎜⎜
⎟⎟ = π b 2 = π X 12
8
⎠ 2⎝ 4 ⎠ 8
(
)
(
)
)
III. Design Variables
X1=b; X2=t; X3=L; X4=l; X5=a; X6=h
X = {X 1 X 2 X 3 X 4 X 5 X 6 }
T
IV. Constraints:
A. Inequality Constraints: g1 ( X ) = SF 6 PX 3 − σ y X 1 X 22 ≤ 0
63
1) Bending
σ − σ a ≤ 0 Where σ =
g1 ( X ) =
σy
GPL
; and σ a =
2
SF
bt
GPL σ y
−
= ( SFGPL − σ y bt 2 ) ≤ 0
2
SF
bt
2) For a range of desired value for thickness “t” tmin < t < tMax
g 2 ( X ) = t min − t ≤ 0 ⇒ g 2 ( X ) = t min − X 2 ≤ 0
3)
g 3 ( X ) = t − t max ≤ 0 ⇒ g 3 ( X ) = X 2 − t max ≤ 0
4)
b ≥ 3d
g 4 ( X ) = 3d − b ≤ 0 ⇒ g 4 ( X ) = 3d − X 1 ≤ 0
5)
g5 ( X ) = L ≥ b ⇒ g5 ( X ) = b − L ≤ 0 ⇒ g5 ( X ) = X1 − X 3 ≤ 0
B. Equality Constraints:
1)
Length = “L”
L = R – bcos(30)
h1(X) = L – R + bcos(30) = 0
h1(X) = X3 – R + X1cos(30) = 0
2)
Length = “L”
l = L – b/2
h2(X) = l – L + b/2
h2(X) = X4 – X3 + X1/2
3) Dimension = “a”
b−d
⇒ h3 ( X ) = 2a − b + d = 0 ⇒ h3 ( X ) = 2 X 5 − X 1 + d = 0
2
4) Dimension = “h”
a=
64
h = bcos(30)
h4(X) = h – bcos(30)
h4(X) = X6 – X1cos(30) = 0
V. Lagrangian
ζ = Lagrangian
A.
General Form
4
4
i =1
k =1
ζ ( X ) = f ( X ) + ∑ λi g i + ∑ µ k hk
B.
For our specific case
ζ (b,t,L,l,a,h) =
⎡ ⎛⎛ 1 ⎞
⎛ 1 2 ⎞⎞ 7 2 ⎤
2
⎢G⎜⎜ ⎜ bh ⎟ + (bl ) + ⎜ πb ⎟ ⎟⎟ − πd ⎥t + λ1 SFGPL − σ y bt + λ 2 (t min − t ) + λ3 (t − t max )
⎝8
⎠⎠ 4
⎣ ⎝⎝ 4 ⎠
⎦
b⎞
⎛
+ λ 4 (3d − b1 ) + λ5 (b1 − L ) + µ1 (L − R + b cos(30) ) + µ 2 ⎜ l − L + ⎟ + µ 3 (2a − b + d ) + µ 4 (h − b cos(30) )
2⎠
⎝
(
)
Written in Term of Design variables {X}
ζ (X ) =
Where X = {X 1 X 2 X 3 X 4 X 5 X 6 }
(
)
⎡ ⎛⎛ 1
⎤
⎞
⎛1
2 ⎞⎞
2
⎢6⎜⎜ ⎜ X 1 X 6 ⎟ + ( X 1 X 4 ) + ⎜ πX 1 ⎟ ⎟⎟ − 1.374⎥ X 2 + λ1 SF 6 PX 3 − σ y X 1 X 2 + λ 2 (t min − X 2 ) + λ3 ( X 2 − t max )
⎠
⎝8
⎠⎠
⎣ ⎝⎝ 4
⎦
X ⎞
⎛
+ λ 4 (3d − X 1 ) + λ5 ( X 1 − X 3 ) + µ1 ( X 3 − R + X 1 cos(30) ) + µ 2 ⎜ X 4 − X 3 + 1 ⎟ + µ 3 (2 X 5 − X 1 + d )
2 ⎠
⎝
+ µ 4 ( X 6 − X 1 cos(30) )
V. Stationary Point of Lagrangian
∇ζ = 0
65
⎧ ∂ζ ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂ζ ⎪
⎪ ∂X 2 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂X 3 ⎪
⎪ ∂ζ ⎪
⎪ ∂X ⎪
⎪ ∂ζ4 ⎪
⎪
⎪
⎪ ∂X 5 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂X 6 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ1 ⎪
⎪ ∂ζ ⎪
⎪
⎪
∇ζ = 0 = ⎨
⎬
⎪ ∂λ2 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ3 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂λ4 ⎪
⎪ ∂ζ ⎪
⎪ ∂λ ⎪
⎪ 5⎪
⎪ ∂ζ ⎪
⎪ ∂µ1 ⎪
⎪ ∂ζ ⎪
⎪
⎪
⎪ ∂µ 2 ⎪
⎪ ∂ζ ⎪
⎪ ∂µ ⎪
⎪ 3⎪
⎪ ∂ζ ⎪
⎪⎩ ∂µ 4 ⎪⎭
1)
∂ζ
∂X 1
Then with (I) λi, µk can be acquired
To get possible solution Points
⎛ ⎡⎛ 1
⎞
⎛ ⎛1⎞
⎞⎤
⎞
: ⎜⎜ 6⎢⎜ X 6 ⎟ + X 4 + ⎜ 2⎜ ⎟πX 1 ⎟⎥ − 1.374 ⎟⎟ X 2 + λ4 (− 1) + λ5 (1) + λ1 (− σ y X 22 )
⎠
⎝ ⎝8⎠
⎠⎦
⎝ ⎣⎝ 4
⎠
⎛1⎞
+ µ1 (cos(30) ) + µ 2 ⎜ ⎟ + µ3 (− 1) + µ4 (− cos(30) )
⎝2⎠
66
2)
∂ζ
:
∂X 2
1
⎛1
⎞
6⎜ X 1 X 6 + X 1 X 4 + πX 12 ⎟ − 1.374 + λ1 (− 2σ y X 1 X 2 ) + λ 2 (− 1) + λ3 (1)
8
⎝4
⎠
3)
∂ζ
:
∂X 3
λ1 (SF 6 P ) + λ5 (− 1) + µ1 (1) + µ 2 (1)
4)
∂ζ
:
∂X 4
[6( X 1 ) − 1.374]X 2 + µ 2 (1)
5)
∂ζ
:
∂X 5
µ 3 (2)
6)
∂ζ
:
∂X 6
⎡ ⎛1 ⎞
⎤
⎢6⎜ 4 X 1 ⎟ − 1.374⎥ X 2 + µ 4 (− 1)
⎠
⎣ ⎝
⎦
7)
∂ζ
:
∂λ1
(SF 6PX
8)
3
− σ y X 1 X 22 )
∂ζ
:
∂λ2
67
(tmin − X 2 )
9)
∂ζ
:
∂λ3
( X 2 − tmax )
10)
∂ζ
:
∂λ 4
(3d − X 1 )
11)
∂ζ
:
∂λ5
(X 1 − X 3 )
12)
∂ζ
:
∂µ1
( X 3 − R + X 1 cos(30) )
13)
∂ζ
:
∂µ 2
X ⎞
⎛
⎜ X4 − X3 + 1 ⎟
2 ⎠
⎝
14)
∂ζ
:
∂µ 3
(2 X 5 − X 1 + d )
15)
∂ζ
:
∂µ 4
( X 6 − X 1 cos(30) )
68
Derivatives of the Inequality constraints, which are used for solving for the Lagrange multipliers:
1)
⎞
∂ζ ⎛⎜ ⎡⎛ 1
⎛ ⎛1⎞
⎞⎤
⎞
: ⎜ 6⎢⎜ X 6 ⎟ + X 4 + ⎜ 2⎜ ⎟πX 1 ⎟⎥ − 1.374 ⎟⎟ X 2 + λ4 (− 1) + λ5 (1) + λ1 (− σ y X 22 )
∂X 1 ⎝ ⎣⎝ 4
⎠
⎝ ⎝8⎠
⎠⎦
⎠
⎛1⎞
+ µ1 (cos(30) ) + µ 2 ⎜ ⎟ + µ3 (− 1) + µ4 (− cos(30) )
⎝2⎠
2)
∂ζ
:
∂X 2
1
⎛1
⎞
6⎜ X 1 X 6 + X 1 X 4 + πX 12 ⎟ − 1.374 + λ1 (− 2σ y X 1 X 2 ) + λ 2 (− 1) + λ3 (1)
8
⎝4
⎠
3)
∂ζ
:
∂X 3
λ1 (SF 6 P ) + λ5 (− 1) + µ1 (1) + µ 2 (1)
4)
∂ζ
:
∂X 4
[6( X 1 ) − 1.374]X 2 + µ 2 (1)
5)
∂ζ
:
∂X 5
µ 3 (2)
6)
∂ζ
:
∂X 6
⎡ ⎛1 ⎞
⎤
⎢6⎜ 4 X 1 ⎟ − 1.374⎥ X 2 + µ 4 (− 1)
⎠
⎣ ⎝
⎦
69
These derivatives are used to solve for the possible optimum point, or design variables X
7)
∂ζ
:
∂λ1
(SF 6PX
8)
3
− σ y X 1 X 22
)
∂ζ
:
∂λ 2
(tmin − X 2 )
9)
∂ζ
:
∂λ3
( X 2 − t max )
10)
∂ζ
:
∂λ4
(3d − X 1 )
11)
∂ζ
:
∂λ5
(X 1 − X 3 )
12)
∂ζ
:
∂µ1
( X 3 − R + X 1 cos(30) )
13)
∂ζ
:
∂µ 2
70
X ⎞
⎛
⎜ X4 − X3 + 1 ⎟
2 ⎠
⎝
14)
∂ζ
:
∂µ 3
(2 X 5 − X 1 + d )
15)
∂ζ
:
∂µ 4
( X 6 − X 1 cos(30) )
VI.
Now using the derivatives above to solve for the design variables:
X3 – R + X1 cos(30) = 0
3d – X1 = 0
X1 = 3d
X1 = 1.5
Substitute (X1) in (1)
X3 = R – (X1) cos(30)
= 3 – (1.5) cos(30)
X3 = 1.7009
X 22 =
6(SF )(P )( X 3 ) 6(2 )(30 )(1.700 )
612
=
=
3
σ y (X 1 )
9.35 x10 (1.5) 14025
X 22 = 0.0436
X 2 = 0.20889318
Which is
t min < X 2 < t max
71
X1
= 0.95
2
X − d 1.5 − 0.5
X5 = 1
=
= 0.5
2
2
X 6 = X 1 cos(30) = 1.299
X4 = X3 −
Possible optimum points
X 1 = 1.5
X 2 = 0.2088
X 3 = 1.7009
X 4 = 0.95
X 5 = 0.5
X 6 = 1.299
VII.
Calculate Lagrange Multiplier Values
µ3 = 0
µ4 = -[6(0.25(1.5) – 1.374](0.2089) = -0.183
µ2 = -[6(1.5) – 1.374](0.2089) = -1.5931
λ2, λ3, λ5 → inactive, not used to arrive at points
1
⎛⎛ 1 ⎞
2⎞
6⎜ ⎜ ⎟(1.5)(1.2999 ) + (1.5)(0.95) + π (1.5) ⎟ − 1.374 + λ1 (2 )(9.35 X 103 )(1.5)(0.2088)
8
⎝⎝ 4 ⎠
⎠
λ1 = 0.00026
λ1 (SF (6)(P )) + µ1 (− 1) + µ 2 (1) = 0
µ1 = −(− 1.5931) − 0.000256(2)(6)(30)
µ1 = 1.4999
72
⎡⎡ ⎛
⎤
2
⎞⎤
2
3
⎢ ⎢6⎜ (0.25)(1.299 ) + 0.95 + π (1.5)⎟⎥ − 1.374⎥ (0.2089 ) + (0.00286 ) − 9.35 × 10 (0.2088)
8
⎠⎦
⎣⎣ ⎝
⎦
(
)
⎛1⎞
+ (0.8505)(cos(30) ) + (− 1.8801)⎜ ⎟ + (− 0.4698)(cos(30) ) = λ 4
⎝2⎠
2.7874 + -1.167 + 0.7365 – 0.94005 – 0.94005 – 0.40686
λ4 = 1.01
λ1 = 0.00026
λ2 = 0
λ3 = 0
λ4 = 1.01
λ5 = 0
µ1 = −1.4999
µ 2 = −1.5931
µ3 = 0
µ 4 = −0.183
VIII.
Verify K-Tucker X = {1.5 0.2088 1.7009 0.95 0.5 1.299 }
*
T
I. Condition: Check All inequality and equality constraints for feasibility
g1
X =X*
≤ 0 = (SF )(6)(P )( X 3 ) − σ Y X 1 X 22 ≤ 0
(2)(6)(30)(1.7009 ) − 9.35 × 10 −3 (1.5)(0.2088)2 = 0
g2
X =X *
≤ 0 = t min − X 2 ≤ 0
0.125 − 0.20894 = −0.08395 ≤ 0 OK
73
OK
g3
X =X*
≤ 0 = X 2 − t max ≤ 0
0.20894 − 0.375 = −0.16605 ≤ 0 OK
g4
X =X *
≤ 0 = 3d − X 1 ≤ 0
3(0.5) − 1.5 = 0 ≤ 0 OK
g5 X = X * ≤ 0 = X 1 − X 3 ≤ 0
1.5 − 1.7009 = −0.2009 ≤ 0 OK
h1 X = X * = 0 = X 3 − R + X 1 cos(30)
= 1.7009 − 3 + 1.5 cos(30)
1.7009 − 3 + 1.5 cos(30)
= −0.00006 ≈ 0 OK
h2
X =X*
= 0 = X4 − X3 +
= 0.95 − 1.7009 +
X1
2
1.5
2
= −0.00009 ≈ 0 OK
74
h3
X =X *
= 0 = 2X 5 − X1 + d
= 2( X 5 ) − X 1 + d
2(0.5) − 1.5 + 0.5 = 0 OK
h4
X =X *
= 0 = X 6 − X 1 cos(30)
(1.299) - 1.5cos(30)
= -0.00004 ≈ 0 OK
IV.Condition: check Lagrange Multipliers λ1 inequality constraints are none negative nonnegative
Inequality Constraint #1
λ1 g1 X = X * = 0 Since g1 = 0; then λ1 > 0
λ1 = 0.00026 > 0 OK
Inequality Constraint #2
λ2 g 2 X = X * = 0 Since g 2 = −0.08395; then λ2 = 0
λ2 = 0 OK
Inequality Constraint #3
λ3 g 3 X = X * = 0 Since g 3 = −0.16605; then λ3 = 0
λ3 = 0 OK
Inequality Constraint #4
λ4 g 4 X = X * = 0 Since g 4 = 0; then λ4 > 0
λ4 = 1.01 > 0 OK
Inequality Constraint #5
λ5 g 5 X = X * = 0 Since g 5 = −0.2009; then λ5 = 0
λ5 = 0 OK
75
V. Condition: Gradient of the function and constraints of lagrange are equal to zero
5
4
i
k
∇f + ∑ λi g i ∇ + ∑ µ k hk ∇ = 0
⎧ ∂f ⎫
⎪ ∂X ⎪
2
⎤ ⎫
⎞
⎪ 1 ⎪ ⎧⎡ ⎛ 1
⎪ ∂f ⎪ ⎪ ⎢6⎜⎝ 4 X 6 + X 4 + 8 (17 )X 1 ⎟⎠ − 1.374⎥ X 2 ⎪ ⎧ 2.786 ⎫
⎦ ⎪
⎪ ∂X 2 ⎪ ⎪ ⎣
⎪
⎪
1
1
⎪ ∂f ⎪ ⎪ 6⎛⎜ X + X X + πX 2 ⎞⎟ − 1.374 ⎪ ⎪ 15.374 ⎪
6
1
4
1
⎪
⎪ ⎪
⎪
8
⎝4
⎠
⎪ ∂X 3 ⎪ ⎪
⎪ ⎪ 0 ⎪
=⎨
∇f = ⎨
0
⎬
⎬
⎬=⎨
∂f
1.592 ⎪
⎪ ⎪
⎪
⎪
⎪
[6 X 1 − 1.374]X 2
⎪ ∂X 4 ⎪ ⎪
⎪ ⎪ 0 ⎪
0
⎪ ∂f ⎪ ⎪
⎪
⎪ ⎪
⎪ ∂X ⎪ ⎪
⎪ ⎩0.18291⎭
⎛ 1
⎞
⎜ 6 X 1 − 1.374 ⎟ X 2
⎪ 5⎪ ⎪
⎪
⎝ 4
⎠
⎭
⎩
f
∂
⎪
⎪
⎪⎩ ∂X 6 ⎪⎭
76
⎧ ∂g1 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂g1 ⎪ ⎧ (− σ X 2 ) ⎫
2
Y
⎪ ∂X 2 ⎪ ⎪
⎪
⎪ ∂g1 ⎪ ⎪(− 2σ Y X 1 X 2 )⎪
⎪
⎪
⎪ ∂X ⎪ ⎪⎪ (SF (6)(P )) ⎪⎪
∇g1 = ⎨ 3 ⎬ = ⎨
⎬
∂g
(0)
⎪ 1⎪ ⎪
⎪
⎪ ∂X 4 ⎪ ⎪
⎪
(0)
⎪ ∂g1 ⎪ ⎪
⎪
⎪⎭
(0)
⎪
⎪ ⎪⎩
⎪ ∂X 5 ⎪
⎪ ∂g1 ⎪
⎪⎩ ∂X 6 ⎪⎭
⎧ ∂g 2 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂g 2 ⎪ ⎧ 0 ⎫
⎪ ∂X 2 ⎪ ⎪ ⎪
⎪ ∂g 2 ⎪ ⎪− 1⎪
⎪
⎪
⎪ ∂X ⎪ ⎪ 0 ⎪
∇g 2 = ⎨ 3 ⎬ = ⎨ ⎬
∂g
⎪ 2⎪ ⎪0⎪
⎪ ∂X 4 ⎪ ⎪ 0 ⎪
⎪ ∂g 2 ⎪ ⎪ ⎪
⎪ ∂X ⎪ ⎩ 0 ⎭
⎪ 5⎪
⎪ ∂g 2 ⎪
⎪⎩ ∂X 6 ⎪⎭
⎧ ∂g 3 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂g 3 ⎪ ⎧0⎫
⎪ ∂X 2 ⎪ ⎪ ⎪
⎪ ∂g 3 ⎪ ⎪1⎪
⎪
⎪
⎪ ∂X ⎪ ⎪0⎪
∇g 3 = ⎨ 3 ⎬ = ⎨ ⎬
∂g
⎪ 3 ⎪ ⎪0⎪
⎪ ∂X 4 ⎪ ⎪0⎪
⎪ ∂g 3 ⎪ ⎪ ⎪
⎪ ∂ ⎪ ⎩0⎭
⎪ X5 ⎪
⎪ ∂g 3 ⎪
⎪⎩ ∂X 6 ⎪⎭
77
⎧ ∂g 4 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂g 4 ⎪ ⎧− 1⎫
⎪ ∂X 2 ⎪ ⎪ ⎪
⎪ ∂g 4 ⎪ ⎪ 0 ⎪
⎪
⎪
⎪ ∂X ⎪ ⎪ 0 ⎪
∇g 4 = ⎨ 3 ⎬ = ⎨ ⎬
∂g
⎪ 4⎪ ⎪1⎪
⎪ ∂X 4 ⎪ ⎪ 0 ⎪
⎪ ∂g 4 ⎪ ⎪ ⎪
⎪
⎪ ⎩0⎭
⎪ ∂X 5 ⎪
⎪ ∂g 4 ⎪
⎪⎩ ∂X 6 ⎪⎭
⎧ ∂g 5 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂g 5 ⎪ ⎧ 1 ⎫
⎪ ∂X 2 ⎪
⎪ ∂g 5 ⎪ ⎪ 0 ⎪
⎪
⎪ ⎪ ⎪
⎪ ∂X ⎪ ⎪− 1⎪
∇g 5 = ⎨ 3 ⎬ = ⎨ ⎬
∂g
⎪ 5⎪ ⎪0⎪
⎪ ∂X 4 ⎪ ⎪ 0 ⎪
⎪ ∂g 5 ⎪ ⎪ ⎪
⎪
⎪ ⎩0⎭
∂
X
5
⎪
⎪
⎪ ∂g 5 ⎪
⎪⎩ ∂X 6 ⎪⎭
∇hi stationary point of equality constraint ( i. 1-4)
78
⎧ ∂ h1
⎪ ∂X
1
⎪
∂
h
1
⎪
⎪ ∂X 2
⎪ ∂ h1
⎪
⎪ ∂X 3
∇ h1 = ⎨
∂ h1
⎪
⎪ ∂X 4
⎪ ∂ h1
⎪
⎪ ∂X 5
⎪ ∂ h1
⎪⎩ ∂ X 6
⎫
⎪
⎪
⎪ ⎧ cos( 30 ) ⎫
⎪ ⎪
⎪
0
⎪ ⎪
⎪
⎪ ⎪
⎪
1
⎪
⎬
⎬ = ⎨
0
⎪ ⎪
⎪
⎪ ⎪
⎪
0
⎪ ⎪
⎪
0
⎭
⎪ ⎩
⎪
⎪
⎪⎭
⎧ ∂h2 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂h2 ⎪ ⎧ 1 ⎫
⎪ ∂X 2 ⎪ ⎪ 2 ⎪
⎪ ∂h2 ⎪ ⎪ 0 ⎪
⎪ ⎪ ⎪
⎪
⎪ ∂X 3 ⎪ ⎪ 0 ⎪
=
∇h2 = ⎨
∂h ⎬ ⎨ ⎬
⎪ 2 ⎪ ⎪1 ⎪
⎪ ∂X 4 ⎪ ⎪ ⎪
⎪ ∂h2 ⎪ ⎪ 0 ⎪
⎪ ∂X ⎪ ⎪⎩ 0 ⎪⎭
⎪ 5⎪
⎪ ∂h2 ⎪
⎪⎩ ∂X 6 ⎪⎭
79
⎧ ∂h3 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂h3 ⎪ ⎧− 1⎫
⎪ ∂X 2 ⎪ ⎪ ⎪
⎪ ∂h3 ⎪ ⎪ 0 ⎪
⎪
⎪
⎪ ∂X ⎪ ⎪ 0 ⎪
∇h3 = ⎨ 3 ⎬ = ⎨ ⎬
∂h
⎪ 3⎪ ⎪0⎪
⎪ ∂X 4 ⎪ ⎪ 2 ⎪
⎪ ∂h3 ⎪ ⎪ ⎪
⎪∂ ⎪ ⎩ 0 ⎭
⎪ X5 ⎪
⎪ ∂h3 ⎪
⎪⎩ ∂X 6 ⎪⎭
⎧ ∂h4 ⎫
⎪ ∂X ⎪
⎪ 1⎪
⎪ ∂h4 ⎪ ⎧− cos(30) ⎫
⎪ ∂X 2 ⎪ ⎪
⎪
0
⎪ ∂h4 ⎪ ⎪
⎪
⎪
⎪
0
⎪
⎪ ∂X 3 ⎪ ⎪
=⎨
∇h4 = ⎨
⎬
⎬
∂h
0
⎪ 4⎪ ⎪
⎪
⎪ ∂X 4 ⎪ ⎪
⎪
0
⎪ ∂h4 ⎪ ⎪
⎪
1
⎭
⎪ ∂X ⎪ ⎩
⎪ 5⎪
⎪ ∂h4 ⎪
⎪⎩ ∂X 6 ⎪⎭
80
⎧1 ⎫
⎧ 2.786 ⎫
⎧− 407.04 ⎫
⎧0.866⎫
⎧− 1⎫
⎪2 ⎪
⎪ 15.374 ⎪
⎪ − 5856.8 ⎪
⎪ 0 ⎪
⎪0⎪
⎪0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪ ⎪
⎪ 0 ⎪
⎪ 360 ⎪
⎪ 1 ⎪
⎪0⎪
⎪0 ⎪
⎨
⎬ + (0.00026)⎨
⎬ + (1.01)⎨ ⎬ + (− 1.499 )⎨
⎬ + (− 1.592 )⎨ ⎬
0
⎪ 1.5923 ⎪
⎪
⎪
⎪ 0 ⎪
⎪1 ⎪
⎪1⎪
⎪ 0 ⎪
⎪
⎪
⎪ 0 ⎪
⎪ ⎪
⎪0⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪0 ⎪
⎪ ⎪
0
0
.
18291
0
0
⎩
⎭
⎩
⎭
⎩
⎭
⎩ ⎭
⎪⎩ 0 ⎪⎭
⎧− 0.866⎫ ⎧− 0.3465⎫ ⎧0⎫
⎪ 0 ⎪ ⎪ − 0.386 ⎪ ⎪0⎪
⎪ ⎪ ⎪
⎪ ⎪
⎪
⎪ ⎪0⎪
⎪ 0 ⎪ ⎪ 0
+ (− 0.183)⎨
⎬≈⎨ ⎬
⎬=⎨
⎪ ⎪0⎪
⎪ 0 ⎪ ⎪ 0
⎪ ⎪0⎪
⎪ 0 ⎪ ⎪ 0
⎪ ⎪ ⎪
⎪ ⎪
⎪
⎭ ⎩0⎭
⎩ 1 ⎭ ⎩ 0
For this reason, it is indeed a good candidate.
81
Appendix C.
ANSYS Tutorial
Step by Step Procedure
Defining Materials Properties
In ANSYS main menu:
Preprocessor<Material Properties<Material Models
Under “Material Models Available”
Structural<Linear<Elastic<Isotropic
EX=325000
PRXY=0.38
OK
82
83
Under “Material Models Available”
Material Density
0.00003<OK
84
Close “Material Properties Menu”
Defining Element Type
Preprocessor<Element Type<Add/Edit/Delete
Under “Element Type”
Add
Under “Library of Element Types”< Under “Structural Mass”
Solid< choose Tet 10node 187<OK
85
Close “Element Type”
86
Constructing Geometry
Constructing Arc
First construct a rectangle with the following dimensions Width=1.5 and Height=0.95
Preprocessor<Modeling<Create<Areas<Rectangle<by 2 corners
Under “Rectangle by 2 corners”
Enter the following numbers
WP X = 0
WP Y = 0
Width = 1.5
Height = 0.95
87
OK
88
Construct a circle with the following dimensions r = 0.75 with center (0.75, 0.95, 0)
Preprocessor<Modeling<Create<Areas<Circle<Solid Circle
Under “Solid Circular Area”
Enter the following numbers
WP X = 0.75
WP Y = 0.95
Radius = 0.75
89
OK
Subtract Areas
Preprocessor<Modeling<Operate<Booleans<Subtract<Areas
90
91
Select the area that you want to keep
92
OK
93
Select the area that you want to remove
94
OK
95
Constructing Rectangle
Preprocessor<Modeling<Create<Area<Rectangle<By 2 Corners
Under “Rectangle by 2 Corners”
Enter the following numbers
WP X=0
WP Y=0
Width=1.5
Height=0.95
96
OK
97
Add both areas (A1+A2)
98
Preprocessor<Modeling<Operate<Booleans<Add<Areas
Under “Add Areas”
Pick All
99
Extrude the area
Preprocessor<Modeling<Operate<Extrude<Areas<Along Normal
100
Select the area that you want to extrude
101
OK
Under “Extrude Area along Normal”
Enter the following number
Distance Length of extrusion = 0.2088
102
OK
Glue all segments of the extruded area
Preprocessor<Modeling<Operate<Boolean<Glue<Areas
103
Pick all
104
Constructing Hole
Create a cylinder with the dimensions of the hole and with center (0.75, 0.95) and subtract the
cylinder.
Preprocessor<Modeling<Create<Volume<Cylinder<Solid Cylinder
Under” Solid Cylinder”
Enter the following numbers
WP X = 0.75
WP Y = 0.95
Radius =0.25
Depth =0.2088
105
OK
106
Subtract the cylinder
Preprocessor<Modeling<Operate<Booleans<Subtract<Volume
107
Select the volume you want to keep
108
OK
109
Select the volume you want to remove.
110
OK
111
To allow easy selection of the face that will be constrained, rotate -900 the volume in the X
axis rotation.
In the utility Menu
Plot Ctrls<Pan Zoom Rotate
112
Under the “Pan Zoom Rotate” tool bar
Press Bot
113
Close the “Pan-Zoom-Rotate”
In the Utility Menu
Plot Ctrls<Numbering
114
Turn on the area numbers
115
Boundary Conditions and Loads
Preprocessor<Load<Define Loads<Apply<Structural<Displacement<On Areas
116
Under “Apply U, ROT on Areas”
Enter the number of the area that you want to select
117
Press enter
118
OK
119
Under” Apply U,ROT on Areas”
All DOF
In “Value Displacement value” enter 0
120
OK
121
Change bottom view to isometric view, to allow for easy placement of the point load at the
unrestricted end.
In the utility Menu
Plot Ctrls<Pan Zoom Rotate
122
Under the “Pan Zoom Rotate” tool bar
Press Iso
123
Close the “Pan-Zoom-Rotate”
In the Utility Menu
Plot Ctrls<Numbering
124
Turn on the keypoint numbers
125
OK
126
Boundary Conditions and Loads
Preprocessor<Load<Define Loads<Apply<Structural<Force/Moment<On Keypoints
127
Select key point number 7
128
OK
Under “Apply F/M on KPs”
In “Direction of Force/mom” Select FZ
In “Value Force/Moment value” enter -30
129
OK
130
Meshing Definition
Preprocessor<Meshing<Meshing Attributes<Picked Volumes
131
Pick All
132
OK
133
Preprocessor<Meshing<Mesh tool
134
Under “MeshTool”
Select Smart Size and move the Fine Course slider to 1
135
Under “MeshTool”
Select Mesh
136
Select Pick All
137
138
SOLVE
Solution<Analysis Type<New Analysis
Under “New Analys”
Select Static
139
OK
140
Solution<Solve<Current Ls
141
OK
142
Plot Results
Displacement
General Postproc<Plot Results<Contour plot<Nodal Solution
Under “Contour Nodal Solution Data”
Displacement vector sum
143
OK
144
145
General Postproc<Plot Results<Contour plot<Nodal Solution
Under “Contour Nodal Solution Data”
Von Misses Stress
146
OK
147
148
149
Appendix D.
Complementary Information
A. ANSYS Log File
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!
RTSA
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!
!===================================
! DEFFINING PARAMMETERS
!===================================
! LOADING CONSTANTS
!------------------FORCE=30
!
! MATERIAL PROPERTIES FOR Lexan
!----------------------------------YOUNG=325000
POISSON=0.38
DENSITY=0.00003
!
!==================================
! PRE PROCESSING
!==================================
!
/PREP7
!
! ELEMENT TYPE & MATERIAL DEFINITION
! -----------------------------------!
! ELEMENT TYPE
! -----------ET,1,SOLID187
! MATERIAL
! --------MP,EX,1,YOUNG
MP,PRXY,1,POISSON
MP,DENS,1,DENSITY
!
! CONSTRUCTING GEOMETRY
!----------------------! Constructing Arc
!----------------------BLC4,0,0,1.5,.95
CYL4,.75,.95,.75
ASBA,
2,
1
! Constructing Rectangle
!----------------------BLC4,0,0,1.5,.95
! BOOLEAN OPERATIONS
!-------------------AADD,all
!
Extrude
!t=THICKNESS
!----------------------VOFFST,2,.2088
! BOOLEAN OPERATIONS
!--------------------
150
AGLUE,all
!
Iso Positioning
!----------------------/ANG, 1
/VIEW, 1 ,1,1,1
/ANG, 1
/REP,FAST
! Constructing Hole
!----------------------CYL4,.75,.95,.25, , , ,.2088
VSBV,
1,
2
!
!
!
! ==============================
! BOUNDARY CONDITIONS AND LOADS
! ==============================
FLST,2,1,5,ORDE,1
FITEM,2,6
/GO
DA,P51X,ALL,0
FLST,2,1,3,ORDE,1
FITEM,2,7
/REPLOT,RESIZE
/GO
FK,P51X,FZ,-FORCE
! MESHING DEFINITIONS
! --------------------CM,_Y,VOLU
VSEL, , , ,
3
CM,_Y1,VOLU
CMSEL,S,_Y
CMSEL,S,_Y1
VATT,
1, ,
1,
0
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
CM,_Y,VOLU
VSEL, , , ,
3
CM,_Y1,VOLU
CHKMSH,'VOLU'
CMSEL,S,_Y
VMESH,_Y1
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!
! ==============================
!
SOLVE
! ==============================
FINISH
/SOL
ANTYPE,0
/STATUS,SOLU
SOLVE
FINISH
151
B. SOLID187: 3-D 10-Node Tetrahedral Structural Solid
MP ME ST PP ED
Element Description
SOLID187 element is a higher order 3-D, 10-node element. SOLID187 has a quadratic
displacement behavior and is well suited to modeling irregular meshes (such as those produced
from various CAD/CAM systems).
The element is defined by ten nodes having three degrees of freedom at each node: translations
in the nodal x, y, and z directions. The element has plasticity, hyperelasticity, creep, stress
stiffening, large deflection, and large strain capabilities. It also has mixed formulation capability
for simulating deformations of nearly incompressible elastoplastic materials, and fully
incompressible hyperelastic materials. See the ANSYS, Inc. Theory Reference for more details
about this element.
Figure 187.1. SOLID187 3-D 10-Node Tetrahedral Structural Solid
Input Data
The geometry, node locations, and the coordinate system for this element are shown in
SOLID187.
In addition to the nodes, the element input data includes the orthotropic or anisotropic material
properties. Orthotropic and anisotropic material directions correspond to the element
coordinate directions. The element coordinate system orientation is as described in Linear
Material Properties.
Element loads are described in Node and Element Loads. Pressures may be input as surface
loads on the element faces as shown by the circled numbers on SOLID187. Positive pressures
152
act into the element. Temperatures may be input as element body loads at the nodes. The node
I temperature T(I) defaults to TUNIF. If all other temperatures are unspecified, they default to
T(I). If all corner node temperatures are specified, each midside node temperature defaults to
the average temperature of its adjacent corner nodes. For any other input temperature pattern,
unspecified temperatures default to TUNIF.
As described in Coordinate Systems, you can use ESYS to orient the material properties and
strain/stress output. For large deflection analyses (NLGEOM,ON), use KEYOPT(4) to choose
output that follows the material coordinate system or the global coordinate system.
KEYOPT(6) = 1 or 2 sets the element for using mixed formulation. For details on the use of
mixed formulation, see Applications of Mixed U-P Formulations in the ANSYS Elements
Reference.
KEYOPT(10) = 1 is used to read initial stress data from a user subroutine. For details about user
subroutines, see the ANSYS Guide to User Programmable Features.
The effects of pressure load stiffness are automatically included for this element. If an
unsymmetric matrix is needed for pressure load stiffness effects, use NROPT,UNSYM.
The next table summarizes the element input. Element Input gives a general description of
element input.
SOLID187 Input Summary
Element Name
SOLID187
Nodes
I, J, K, L, M, N, O, P, Q, R
Degrees of Freedom
UX, UY, UZ
Real Constants
None
Material Properties
EX, EY, EZ, ALPX, ALPY, ALPZ,
(PRXY, PRYZ, PRXZ or NUXY, NUYZ, NUXZ),
DENS, GXY, GYZ, GXZ, DAMP
Surface Loads
Pressures -face 1 (J-I-K), face 2 (I-J-L), face 3 (J-K-L), face 4 (K-I-L)
Body Loads
Temperatures -T(I), T(J), T(K), T(L), T(M), T(N), T(O), T(P), T(Q), T(R)
Special Features
Plasticity, Hyperelasticity, Viscoplasticity, Creep, Swelling, Stress stiffening (available
with NLGEOM,ON only), Large deflection, Large strain, Initial stress import, Birth and
death. Supports the following types of data tables associated with the TB command:
ANEL, BISO, MISO, NLISO, BKIN, MKIN, KINH, CHABOCHE, HILL, RATE, CREEP, HYPER, and
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USER.
Note
See the ANSYS, Inc. Theory Reference for details on the material models.
KEYOPT(4)
0 -Use
1 -Use
KEYOPT(6)
0 -Use
1 -Use
2 -Use
the global coordinate system for element output.
element or ESYS coordinate system for element output.
pure displacement formulation (default)
mixed formulation, hydrostatic pressure is constant in an element.
mixed formulation, hydrostatic pressure is interpolated linearly in an element.
Note
If mixed formulation is needed, KEYOPT(6) = 1 is recommended for hyperelastic
materials, and KEYOPT(6) = 2 is recommended for nearly incompressible
elastoplastic materials.
KEYOPT(10)
0 -No user subroutine to provide initial stresses (default).
1 -Read initial stress data from user subroutine USTRESS
Note
See the ANSYS Guide to User Programmable Features for user written
subroutines.
Output Data
The solution output associated with the element is in two forms:
•
•
nodal displacements included in the overall nodal solution
additional element output as shown in Element Output Defintions
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Student Vitae
The group was composed of three students, these are:
Name: Borrero Valentín, Susana
Discipline: Mechanical Engineering
Parent Institution: University of Puerto Rico – Mayagüez Campus
Expected date of graduation: December, 2005
E-mail address: [email protected]
Name: Rivera Maldonado, Rodolfo
Discipline: Mechanical Engineering
Parent Institution: University of Puerto Rico – Mayagüez Campus
Expected date of graduation: December, 2005
E-mail address: [email protected]
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Evaluation Sheet
Overall
_________/5
Chapter 1. Premilinary Remarks
_________/5
Chapter 2. Design Selection
_________/10
Chapter 3. Optimization
_________/10
Chapter 4. Finite Element Analysis
_________/10
Chapter 5. Results and Discussion
_________/10
Chapter 6. Final Remarks
_________/10
References
_________/5
Appendix, Figs and Tables
_________/5
Turned in on time
TOTAL
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