THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY ÇAĞIL MERVE TANIK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING JANUARY 2014
135
Embed
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF MIDDLE EAST TECHNICAL UNIVERSITY
BY
ÇAĞIL MERVE TANIK
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
JANUARY 2014
Approval of the thesis:
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT
submitted by ÇAĞIL MERVE TANIK in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences _____________________ Prof. Dr. Süha Oral Head of Department, Mechanical Engineering _____________________ Prof. Dr. F. Suat Kadıoğlu Supervisor, Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Co-Supervisor, Mechanical Engineering Dept., HU _____________________ Examining Committee Members: Prof. Dr. Orhan Yıldırım Mechanical Engineering Dept., METU _____________________ Prof. Dr. F. Suat Kadıoğlu Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Mechanical Engineering Dept., HU _____________________ Prof. Dr. Metin Akkök Mechanical Engineering Dept., METU _____________________ Assist. Prof. Dr. Ergin Tönük Mechanical Engineering Dept., METU _____________________
Date: 30.01.2014
iv
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced
all material and results that are not original to this work.
Name, Last name : Çağıl Merve TANIK
Signature :
v
ABSTRACT
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT
UNIVERSAL JOINT
Tanık, Çağıl Merve
M.S., Department of Mechanical Engineering
Supervisor: Prof. Dr. Fevzi Suat Kadıoğlu
Co-Supervisor: Assoc. Prof. Dr. Volkan Parlaktaş
February 2014, 115 pages
In this study, a compliant version of the cardan universal joint whose compliant parts
are made of blue polished spring steel is considered. The original design consist of
two identical parts assembled at right angles with respect to each other. Identical
parts can be produced from planar materials; thus, it has the advantage of easiness in
manufacturing. As a design example, two mechanisms are dimensioned with
different plate thicknesses. The resultant stresses at flexural hinges of these samples
are determined via analytical and finite element analysis method. Torque capacity of
these mechanisms are determined. Also fatigue analysis of these mechanisms are
performed. Further, one of these samples is manufactured and operated under three
different conditions. It is verified that results of experiments are consistent with
theoretical approaches.
Keywords: Compliant Mechanisms, Universal Joint, Finite Element Analysis,
Fatigue Analysis
vi
ÖZ
ESNEK KARDAN MAFSALININ TEORİK VE DENEYSEL ANALİZLERİ
Tanık, Çağıl Merve
Yüksek Lisans, Makine Mühendisliği Bölümü
Tez Yöneticisi: Prof. Dr. Fevzi Suat Kadıoğlu
Ortak Tez Yöneticisi: Doç. Dr. Volkan Parlaktaş
February 2014, 115 sayfa
Bu çalışmada esnek kısımları yay çeliğinden yapılan kardan milinin esnek bir
versiyonu incelenmiştir. Özgün tasarım birbirlerine dik olarak monte edilmiş iki
özdeş parçadan oluşmaktadır. Özdeş parçalar düzlemsel malzemelerden imal
edilebilir, bu nedenle mekanizmanın üretimi kolaydır ve bu önemli bir avantajdır. Bu
çalışmada iki farklı plaka kalınlığında mekanizma tasarım örnekleri
boyutlandırılmıştır. Bu örneklerin bükülen mafsallarındaki gerilmeler analitik
metotlar ve sonlu elemanlar analizi ile incelenmiştir. Böylece mekanizmaların tork
kapasiteleri belirlenmiştir. Ayrıca mekanizmaların yorulma analizleri de yapılmıştır.
Yapılan analitik ve sayısal çalışmaları doğrulamak amacıyla, tasarlanan örneklerin
bir tanesi üretilmiş ve üç farklı koşulda çalıştırılmıştır. Bu deneylerin sonuçlarının
kuramsal yaklaşımlar ile tutarlı olduğu gözlemlenmiştir.
Table 5.15 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.75 mm) ......................................................................................................... 90
Table 5.16 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.5 mm) ........................................................................................................... 91
Table 5.17 Maximum and Minimum Normal Stress Values for Combined Loading 94
Table 6. 1 Experiments for Different Conditions .................................................... 101
xiii
LIST OF FIGURES
FIGURES
Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş
V. (2011)) ..................................................................................................................... 2
Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012)) ..... 4
Figure 1.3 Flow Chart of the Stress Analysis .............................................................. 6
Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b) ..................... 10
Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body
Model with a Torsional Spring .................................................................................. 12
Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model .............................. 13
Figure 2.4 Plot of Characteristic Radius Factor, �, versus n...................................... 14
Figure 2.5 Flexible Beam with a Moment at the Free End ........................................ 17
Figure 2.6 Component Characteristics of Links ........................................................ 18
Figure 2.7 Component Characteristics of Segments .................................................. 18
Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams ...................... 19
Figure 2. 9 An Example of a Compliant Mechanism Diagram ................................. 20
Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid
Body Models .............................................................................................................. 22
Figure 5.59 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�ℎ = 0.5 mm) ..................................................................................................................... 91
Figure 5.60 Von-Mises Stress Distribution, Most Critical Point on the Flexural
Hinge, its Infinitesimal Cube and ��� Stress Distribution ........................................ 92
Figure 5.61 Infinitesimal Cube of Critical Point ....................................................... 93
Figure 5.62. Normal Stress Variations for Different Shaft Angles............................ 94
Figure 5.63 Performance Graph of the Compliant Universal Joint ........................... 95
Figure 6.1 Assembly of the Prototype ....................................................................... 97
Figure 6.2 Components of the Prototype ................................................................... 98
Figure 6. 3 Different Ways of the Connections ......................................................... 99
Figure 6.4 Components of the Experimental Setup ................................................. 100
Figure 6.5 A Failed Compliant Universal Joint ....................................................... 102
Figure 6.6 Shaft Angle vs. Torque Output and Experimentally Verified Data ....... 102
Figure A.1 Properties of AISI 1080 ......................................................................... 109
Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness ......... 110
xvii
Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness ............ 111
Figure C.1Specifications of UCP 200 Bearing ........................................................ 112
Figure D.1 Specifications of the DC Electric Motor................................................ 114
Figure E.1 Isometric View of the Fatigue Test Setup .............................................. 115
xviii
LIST OF SYMBOLS
� Horizontal component of free end, length of rigid link, angular acceleration,
" Horizontal coefficient of applied force, factor of safety
xix
"# Horizontal component of force
# Vertical component of force
$ Notch sensitivity
% Radius
& Contact surface of compliant universal joint
'( Endurance limit
'() Unmodified endurance limit
'*� Ultimate tensile strength
'+ Yield strength
, Torque
�- Thickness of flexural hinge
�. Thickness of compliant universal joint
/ Width of flexural hinge
01 Initial position vector
23 Engineering normal strain for fracture
4 Pseudo rigid body angle, angular deflection
� Characteristic radius factor, angle of rotation for shafts
��5 Pseudo rigid body constant
6 Deflection
7 Bend angle, shaft angle
8 Angle
� Normal stress
9 Angular velocity
: Direction angle of force
1
CHAPTER 1
INTRODUCTION
1.1 Literature Review
Universal joints are common mechanical devices which are used for transmitting
rotary motion between misaligned intersecting shafts. That is, a universal joint is a
joint or coupling that is capable of transmitting rotary motion from one shaft to
another which are not in line with each other. Classical analysis of the universal joint
involves the determination of angular displacements, velocities, accelerations and
torque ratios of the shafts.
In this study a novel universal joint design is proposed. A compliant universal joint
whose compliant parts are made of blue polished spring steel is taken into
consideration. This chapter presents a survey of the related literature for compliant
mechanisms and universal joint mechanisms.
In literature universal joint became an attractive topic because of its capabilities of
easy mounting, resisting high loads and commercial availability. Early articles on
universal joints made of rigid links address various aspects of these mechanisms.
Basically a universal joint is a spherical four bar linkage. In literature there are lots
of studies about this type of mechanism addressing its analysis, synthesis,
applications and type determination. For example Mohan et al. (1973) introduced
closed form synthesis of a spatial function generation mechanism which consists of a
spherical four bar linkage. Freudenstein (1965) proposed a new type of a spherical
2
mechanism. Yang (1965) worked on static force and torque analysis of a spherical
four bar mechanism. Dynamic analysis of a universal joint and its manufacturing
tolerances are introduced by Chen and Freudenstein (1986). Freudenstein and Macey
(1990) worked on the inertia torques of the Hooke joint. Moment transmission by a
universal joint is studied by Porat (1980). Homokinetic joint allows to transmit
power through a variable angle, at constant rotational speed. For a double cardan
homokinetic joint Wagner and Cooney (1979) developed a new approach to increase
its dynamic mechanical efficiency.
Universal joint has the advantage of easiness in manufacturing. On the other hand
traditional universal joints consist of many parts which are assembled and therefore
manufacturing tolerances on these parts must be complied with. Tolerances of a
universal joint are studied by Fischer and Freudenstein (1984).
Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş
V. (2011))
Compliant mechanisms are flexible mechanisms, which gain some or all of their
motion through the deflection of members. They can be fully or partially compliant.
Generally compliant mechanisms have lower number of parts which reduce
manufacturing and assembly time. Some of them may even be made of a single
3
piece. They are lighter and they have fewer number of movable joints, which cause
wear and need lubrication. The main disadvantage of compliant mechanisms is that,
their analyzes and design is difficult to accomplish. The pseudo rigid body model is
used to simplify the analysis and design of compliant mechanisms. In Figure 1.1 first
compliant spatial four bar mechanism is shown which is designed by Tanık and
Parlaktaş (2011).
Salamon (1989) introduced a methodology which uses a pseudo rigid body model of
the compliant mechanisms with compliance modelled as torsional and linear springs.
Howell and Midha (1994) and (1998) used closed form elliptic integral solutions to
develop deflection approximations for an initially straight flexible segment subjected
to bending.
A spherical four bar mechanism which is a special case of spatial four bar
mechanism that possesses out of plane motions is studied by Tanık and Parlaktaş
(2012). Another spatial four link mechanism studied by Parlaktaş and Tanık (2011)
is the compliant spatial slider crank mechanism.
In the literature a compliant universal joint is previously considered by Trease et al.
(2005) who proposed a design for a compliant universal joint. There is also a
prototype of a compliant universal joint in the library of Cornell University that can
be found in http://kmoddl.library.cornell.edu website. In that prototype the compliant
section is made of leather.
Recently, Tanık and Parlaktaş (2012) proposed a new design for a compliant cardan
universal joint which is shown in Figure 1.2. The design consists of two identical
parts assembled at right angles with respect to each other. In that study, dimensions
of the mechanism are designed in order to satisfy the Cardan joint theory and to
avoid an undesired contact between the identical parts for proper functioning of the
mechanism. This prototype is made of polypropylene and manufactured and
operated under specified loading conditions to verify the theoretical approaches.
4
Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))
1.2 Objective and Scope of the Thesis
The successful implementation of a compliant universal joint in real life applications
depends not only on its kinematic design but its strength as well. The purpose of this
thesis is therefore to analyze the stresses and fatigue strength of a compliant
universal joint, whose flexible parts are made of blue polished spring steel by
analytical, numerical and experimental methods. Hence theoretical approaches will
be experimentally verified. To the best of Author's knowledge there are not any
studies in the literature which address the strength issues of compliant universal
joints except the study of Tanık and Parlaktaş (2012) , where only a preliminary
5
finite element analysis has been done to determine the torque capacity of the
mechanism.
Here, a design is proposed according to the dimensional constraints that satisfy the
theory of universal joints and thereby avoid undesired contact between the parts. The
stress analysis is done analytically and numerically by finite element method using
ANSYS software. Fatigue analysis is done and it is experimentally verified with the
prototypes that are manufactured. The flow chart of the strength analysis can be seen
in Figure 1.3 schematically.
Outline of the thesis is as follows. In Chapter 2, a review of compliant mechanisms
is done. Then pseudo rigid body model is briefly explained and deflection and stress
equations are derived. Rigid body replacement synthesis is presented. The kinematic
equations of the universal joint are given in Chapter 3. In Chapter 4 dimensioning of
the mechanism and selection of the material is covered. In Chapter 5 the analytical
and numerical solution procedure and approaches are discussed. Experimental setup
is introduced in Chapter 6. Finally, the results are discussed in Chapter 7.
Figure 1.3
Determine stresses under static loading
Perform stress analysis using FEA
Evaluate fatigue life of the mechanism
Manufacture a real model
Compare the fatigue life estimates with
Discuss the results by various methods
and explain discrepancies if any
6
3 Flow Chart of the Stress Analysis
Determine stresses under static loading
analytically
Perform stress analysis using FEA
Evaluate fatigue life of the mechanism
by analytical methods
Manufacture a real model
Compare the fatigue life estimates with
experiments
Discuss the results by various methods
and explain discrepancies if any
Conclusions
7
CHAPTER 2
COMPLIANT MECHANISMS
2.1 Introduction to Compliant Mechanisms
According to Shigley and Uicker (1980) a mechanism is a mechanical device used to
transfer or transform motion, force, or energy. Rigid-link mechanisms gain their
mobility from the movable joints.
Compliant mechanisms are flexible mechanisms, that gain some or all of their
motion through the deflection of flexible members rather than movable joints.
Compliant mechanisms can be fully compliant or partially compliant. Fully and
partially compliant mechanism definitions are given by Howell (2001). Fully
compliant mechanisms obtain all their motion from the deflection of compliant
members and partially compliant mechanisms contain one or more traditional
kinematic pairs along with compliant members.
Required input output relationship is obtained by the combination of the rigid and
compliant parts or fully compliant elements. The strength of the deflecting members
limits deflection of compliant link therefore a compliant link cannot produce a
continuous rotational motion.
The advantages of compliant mechanisms can be divided into two subgroups: cost
reduction and increased performance. Compliant mechanisms require fewer parts to
accomplish a certain task. A reduction in the number of parts reduces manufacturing
and assembly time, and cost. Some compliant mechanisms can be manufactured as a
8
single piece by injection molding process. Compliant mechanisms also have fewer
movable joints. That results in reduced wear and need for lubrication.
Using compliant mechanisms reduces the number of movable joints which increases
mechanism precision since backlash may be reduced or eliminated. Vibration and
noise caused by the revolute and sliding joints of rigid-body mechanisms may also
be reduced by using compliant mechanisms.
In compliant mechanisms energy is stored in the form of strain energy in the flexible
members. This property can be an advantage for some cases and a disadvantage for
some other cases. As an advantage the stored or transformed energy can be released
at a later time or in a different manner. A bow and arrow system is a good example.
All of the energy is not transferred, but some is stored in the mechanism.
Compliance becomes a disadvantage if function of a mechanism is to transfer energy
from input to output.
A major disadvantage of compliant mechanisms is the lack of knowledge regarding
analysis and synthesis methods for such mechanisms and the requirement to
determine the deflections of flexible members. Therefore analysis and design of
compliant mechanisms has difficulties compared to conventional mechanisms.
Fatigue analysis is another vital issue. Some compliant members are loaded in a
cyclic manner. To perform prescribed functions it is important to design those
compliant members with sufficient fatigue life.
Compliant links that remain under stress for long periods of time or subject to high
temperatures may experience stress relaxation and creep.
2.2 Pseudo-Rigid-Body Model
The purpose of the pseudo-rigid-body model is modelling the deflection of flexible
members by using rigid-body components which have equivalent force-deflection
9
characteristics. This method of modelling allows well-known rigid-body analysis
methods to be used in the analysis of compliant mechanisms. Howell (2001) says
that the pseudo-rigid-body model is a bridge that connects rigid-body mechanism
theory and compliant mechanism theory.
Salamon (1989) introduced a methodology for compliant mechanism design that
used a pseudo-rigid-body model of the compliant mechanism with compliance
modelled by torsional and linear springs. These models are much easier to analyze
than idealized models that require finite element or elliptic integral solutions. The
most important attribute of the pseudo-rigid-body model is that it significantly
simplifies the design process.
Closed-formed elliptic-integral solutions are used by Howell and Midha (1994 and
1998) to develop deflection approximations for an initially straight, flexible segment
with linear material properties.
2.2.1 Small-Length Flexural Pivots
An important component that exist in compliant mechanisms is the so called ''small
length flexural pivot''. The beam shown in Figure 2.1 has two segments. The small
segment is shorter and more flexible than the large segment. This small segment is
called small-length-flexural pivot. Usually large segment is at least 10 times larger
than small segment. The large segment is also much stiffer.
� ≫ � (2.1)
(��)< ≫ (��)= (2.2)
10
Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b)
For the flexible segment with end moment loading the deflection equations are
derived by Howell (2001) as follows:
8> = ��� (2.3)
6+� = 1 − �A&8>8> (2.4)
6B� = 1 − &C"8>8> (2.5)
These equations could be used to model small-length flexural pivots with pseudo-
rigid-body model. Figure 2.1 shows a member and its pseudo-rigid-body model . The
model consist of two rigid equal links, connected by a characteristic pivot.
Characteristic pivot represents the displacement and torsional spring models the
beam stiffness or resistance to deflection. This model gives an accurate solution for
the deflection path of the beam end for a given end load. The percentage error
between this model and the closed-form elliptic integral solutions is 0.5 for large
11
deflections. The angle of pseudo rigid link is the pseudo rigid body angle, 4, that is
equal to the beam end angle for small-length flexural pivots.
Θ= 8>(small-length flexural pivots) (2.6)
A torsional spring with spring constant � is used to model the beam's resistance to
deflection. The required torque to deflect the torsional spring at an angle 4 is
, = �Θ (2.7)
From the elementary beam theory the spring constant � could be found. For a beam
with an end moment, the end angle is
8> = �(��)= (2.8)
Since = , and 4 = 8>, the spring constant can be found as
� = (��)=� (2.9)
This model is more accurate for bending dominant cases than transverse and axial
loading dominant cases.
2.2.2 Active and Passive Forces
Figure 2.2 shows a cantilever beam with a force at the free end. The force, F, must
be defined by its magnitude and direction. The direction may be defined by the angle : or by the horizontal and vertical components of the force. In Figure 2.2 horizontal
component is shown as "# and the vertical component as #. F is a nonfollower
12
force which means it remains at the same angle regardless of the deflection of the
beam. The nonfollower force's magnitude and direction is
� = #D"E + 1 (2.10)
: = atan 1−" (2.11)
Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body
Model with a Torsional Spring
The force can also be resolved into its normal and tangential components. The
tangential component to the path which is also normal to the pseudo-rigid-link, ��,
causes a moment at the torsional spring
, = ��(� + �2) (2.12)
The tangential component, which causes the deflection of the pseudo-rigid-link, is
called an active force. The normal component is called a passive force and it has no
contribution to the deflection of the beam.
13
When the deflection changes the active and passive components change because F is
a non-follower force. Then the active force is,
�� = �&C"(: − 4) (2. 13)
2.2.3 Cantilever Beam with a Force at the Free End
In Figure 2.3 the flexible beam with a force applied at its free end is shown. If
deflections are large, the linear beam deflection equations may not give accurate
solutions. To perform the analysis, elliptic integral solutions or nonlinear finite
element analysis could be used. Instead of these methods, pseudo-rigid-body model
which is a simpler but accurate method of analysis may also be used.
Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model
14
The location of the characteristic pivot is expressed in terms of the characteristic
radius factor, �, which represents the fraction of the beam length at which the pivot
is located. Once � is determined the deflection path may be parameterized in terms
of 4, the pseudo-rigid-body angle.
The characteristic radius factor is a function of ", that is horizontal coefficient of the
applied force. Howell (2001) introduced a formulation of � in terms of " as:
�ΘTU( = V �Θ�"WXWYV �"WXWY �A% g "e = 5"E = 10S (2.20)
�ΘTU( = 2.61 (2.21)
considering the loads are in a range of 63h < : < 135h or −0.5 < " < 1.0,
�ΘTU( = 2.65 A% �Θ ≅ i� (2.22)
The torque applied at the pin joint is,
, = �4 (2.23)
where � is the torsional spring constant [!/%��] and 4 is the angular deflection.
Torsional spring constant is defined by Howell (2001) as,
� ≅ i�E ���
(2.24)
where ��Θ is pseudo-rigid body constant.
For different values of n, value of the coefficients are shown in Table 2.1,
Table 2.1 The Coefficients of the Compliant Mechanisms
n m nopq(m) rs ts nopq(ts)
0 0.8517 64.3o 1.2385 2.677 58.5o
2 0.8276 109.0o 1.2511 2.597 69.0o
5 0.8192 121.0o 1.2557 2.562 67.5o
-0.5 0.8612 47.7o 1.2358 2.693 44.4o
-3 0.8669 16.0o 1.2119 2.688 12.9o
17
2.2.4 Moment at the Free End
The flexible beam with an end moment at its free end is shown in Figure 2.5.
Figure 2.5 Flexible Beam with a Moment at the Free End
The coordinates of the free end, maximum normal stress value and angle of the
beam end are determined by Howell (2001) as,
� = �[1 − 0.7346(1 − �A&4)] (2.25)
� = 0.7346�&C"4 (2.26)
8> = 1.51644 (2.27)
�fTB = >�� (2.28)
2.3 Nomenclature
In rigid-body mechanisms motion is transferred or transformed by rigid links and
traditional joints. However the working principle of the compliant mechanisms are
different. The deflection of the flexible members gives the motion. Therefore
identification of the compliant mechanism's parts are more difficult than the rigid
body mechanisms.
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
of one or more kinematic pairs.
Figure 2.6
For a rigid-body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
dependent on link geometry and the forces. As seen in Figure
rigid or compliant and a compliant link can be simple or compound.
Figure 2.7 Component Characteristics of Segments
Rigid Link
18
of the compliant mechanism's parts are more difficult than the rigid
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
of one or more kinematic pairs.
Component Characteristics of Links
body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
dependent on link geometry and the forces. As seen in Figure 2.6 the links may be
rigid or compliant and a compliant link can be simple or compound.
Component Characteristics of Segments
Link
Rigid Link Compliant Link
Simple Compound
Segment
Rigid Compliant
of the compliant mechanism's parts are more difficult than the rigid-
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
the links may be
19
When a compliant link is analyzed, it could be observed that the compliant segment
may be composed of rigid or compliant segments. Material discontinuities or
geometric changes are often the starting points of a new segment. Segments can be
either rigid or compliant as shown in Figure 2.7.
2.4 Diagrams
Skeleton diagrams are used to represent rigid-body mechanisms easily. Similar
diagrams are also used for compliant mechanisms. Symbols which represent joints
and segments are shown in Figure 2.8.
Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams
By using symbols an example of a compliant mechanism diagram is shown in Figure
2.9. The mechanism has two compliant links. First link is compliant and it is a
Rigid Segment
�
Axially Compliant
Segment
Pin or Revolute
Joint Flexural Pivot
Slider (Prismatic)
Joint
or
Fixed Connection
or
20
composition of a compliant and a rigid segment. Second link is also compliant and
has 3 segments that are rigid and compliant. There are rigid traditional kinematic
pairs therefore it is a partially compliant mechanism.
Figure 2. 9 An Example of a Compliant Mechanism Diagram
2.5 Pseudo-Rigid-Body Replacement
In a compliant mechanism analysis or synthesis, sometimes transformation of the
mechanisms is required. This transformation can be compliant mechanism to rigid-
body mechanism or vice versa. In compliant mechanism analysis, a pseudo rigid
segment 1(rigid)
link 1
(compliant)
segment 2 (compliant)
segment 5(rigid)
link 2
segment 4 (compliant)
6 variable
21
body model is obtained from the compliant mechanism. However, in rigid body
replacement synthesis, a pseudo rigid model is equivalent to a rigid-body mechanism
model and the resulting mechanism is determined from these models.
In rigid-body mechanisms the distance between the joints is the kinematically most
important parameter. During all the replacement processes the joints must be fixed.
Joint is on the midpoint of the compliant segment in small length flexural hinges if
transformation is compliant mechanism to rigid-body mechanism or vice versa.
In flexible beams there is a relationship between the rigid and compliant link which
is defined by characteristic radius factor, �.
� = �� (2.29)
Where � is length of the rigid link and � is the length of the compliant link. The
characteristic radius factor value according to " values can be seen in Table 2.1.
Another important issue during pseudo-rigid-body replacement is taking the torque
,which comes from the deflection of complaint link, into account. Therefore torsional
springs must be attached to the mechanism.
In Figure 2.10 a four bar and slider crank mechanism's pseudo-rigid-body
replacements are shown. In both mechanisms the black mechanism represents
compliant mechanism and grey one is its pseudo-rigid-body model.
22
Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid
Body Models
2.6 Material Considerations
For a beam that is shown in Figure 2.11 the deflection at the free end is,
6 = 4��`��ℎ` (2.30)
23
Figure 2.11 Flexible Cantilever Beam
The maximum stress occurs at the fixed end and equals to,
�fTB = 6���ℎE (2.31)
The failure occurs when the �fTB equals to the yield strength, '+, then,
'+ = 6���ℎE (2.32)
From Equation 2.32 � can be expressed in terms of yield strength as,
� = '+�ℎE6� (2.33)
24
Substituting Equation 2.33 into Equation 2.30 results in the maximum deflection, 6fTB (assuming geometrically linear load vs. deflection relationship is valid),
6fTB = 23 '+� �Eℎ (2.34)
According to maximum deflection equation, ratio of the strength to modulus of
elasticity shows us how much does a beam deflect without permanent deformation.
Thus the material with the highest value of '+/� will allow larger deflection. In
Table 2.2 ratio of yield strength to modulus of elasticity for several materials are
shown.
Table 2.2 Ratio of Yield Strength to Young's Modulus for Some Materials
Material u (vwp) xy (zwp) (xy/u) × {|||
Steel (1010 hot rolled) 207 179 0.87
Steel (4140 Q&T@400) 207 1641 7.9
Blue Polished Spring Steel
(AISI 1080)
200 880 4.4
Aluminium (1100 annealed) 71.7 34 0.48
Aluminium (7075 heat treated) 71.7 503 7.0
Titanium (Ti-13 heat treated) 114 1170 10
Polyethylene (HDPE) 1.4 28 20
Polypropylene 1.4 34 25
As seen in Table 2.2 elastic modulus does not change much with addition of alloying
elements or heat treatments. However yield strength value could be increased by heat
treatment that also makes the material more brittle.
Polypropylene has a very high yield strength to young's modulus ratio which allows
large deflections. It is available in the market, inexpensive, very ductile, easy to
process and has a low density. It can yield thousands of time without fracturing.
25
CHAPTER 3
CONVENTIONAL UNIVERSAL JOINT
3.1 Introduction to Cardan Universal Joint
A universal joint is a joint or coupling that is capable of transmitting rotary motion
from one shaft to another which are not in line with each other. It consists of a pair
of hinges, oriented at 90o to each other that are connected by a cross shaft as seen in
Figure 3.1. Kinematically a universal joint is equivalent to a slotted sphere type of a
joint which has two degrees of rotational freedom.
Figure 3.1 Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))
7
26
3.2 Kinematic Analysis of the Cardan Universal Joint
Even when the input shaft rotates at a constant speed, output shaft could rotate at a
variable speed. For this reason the universal joint suffers from vibration and wear.
The speed of the output shaft varies and this variation depends on the configuration
of the joint that is specified by the variables given below,
� �e, the angle of rotation for axle 1
� �E, the angle of rotation for axle 2
� 7, the bend angle of the joint or the input shaft angle of the joint
Figure 3.2 Variables of Cardan Universal Joint
These variables can be seen in Figure 3.2. The red plane and axle 1 are perpendicular
to each other and axle 2 is always perpendicular to the blue plane. These planes are
at an angle 7. �e and �E are the angular displacement of each axle. �e, �E are the
angles between the 01e and 01E and initial positions along the 0 and � axes. The 01e
and 01E vectors are fixed with the cross shaft that connects the two axles therefore
they remain perpendicular to each other.
01E �1
01
01e
�e
�E
7
27
01e draws the border of the red plane and related to �eby,
q}{ = [�A&�e, &C"�e, 0] (3.1)
01E draws the border of the blue plane and is the result of the unit vector 01 = [1,0,0] being rotated through Euler angles [i/2, 7, 0],
q}~ = [−�A&�&C"�e, �A&�E, &C"�&C"�E] (3.2)
The 01e and 01E vectors are fixed with the cross shaft therefore they must remain at
right angles,
q}{. q}~ = 0 (3.3)
Thus the equation relating the rotations of axles is,
tan�e = cos7 tan�E (3.4)
and the solution for �Ein terms of �eand 7 is,
�E = ���"2(&C"�e, �A&7 �A&�e) (3.5)
The angles �e and �E are the functions of the time. Differentating the rotation angles
with respect to time gives us the angular velocities of the axles,
9e = ��e/�� (3.6)
9E = ��E/�� (3.7)
The relationship between the angular velocities of axle 1 and 2 is,
28
9E = 9e�A&71 − &C"7E�A&�eE (3.8)
The relationship between the angular accelerations can be derived by differentiating
According to the three constraint equations given above, the dimensions of the
design are determined. Two different designs are proposed where thicknesses of the
small length flexural hinges are 0.5 and 0.75 mm, respectively in the experiments
performed in this study, the geometry with 0.5 mm thickness is used thus the
detailed design of this geometry will be explained.
Free parameters are chosen as follows: � =20 mm, �- =0.5 mm, � =15 mm and �. =5.68 mm. For the ease of manufacturing three plates will be assembled therefore �. is the summation of blue polished spring steel, two steel plates and tolerances of
the plates. According to the Equation 4.1, 4.2 and 4.5 the dimensions are determined
36
as, � =25 mm, � =5.68 mm and the minimum value of � =50.34 mm. All of the
dimensions of the design can be seen in Appendix B. The isometric view can be seen
in Figure 4.5 and deflected position of the mechanism for different input shaft angles
is shown in Figure 4.6.
Figure 4.5 Isometric View of the Design
Flexural Hinges
Shaft 1
Shaft 2
37
Figure 4.6 For a 10° Shaft Angle and Different Input Shaft Angles Views of the
Mechanism
38
THEORETICAL ANALYSIS
5.1 Stress Analysis of the
Preliminary stress analyses of the design are
then finite element analysis is u
ANSYS 13.0 Workbench software is used.
Firstly, both of the geometries are sketched with Catia V5.
analyses are performed
the analysis can be seen in Figure
following parts.
Importing the CAD
model of the design into the ANSYS 13.0
Programme's design
modeller module
Meshing the model
Defining the boundary
conditions and loadings
39
CHAPTER 5
CAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT
Stress Analysis of the Compliant Universal Joint with FEA
stress analyses of the design are performed by analytical methods and
hen finite element analysis is used for comparison. For finite element analysis
ANSYS 13.0 Workbench software is used.
both of the geometries are sketched with Catia V5. Then
performed for different shaft angles and torque values. The flow chart of
the analysis can be seen in Figure 5.1. Details of the analysis will be explained in the
Figure 5.1 Flow Chart of the Analysis
Importing the CAD
model of the design into the ANSYS 13.0
Programme's design
modeller module
Pre processing of the
model (defining rigid and flexible parts, simplfying
the geometry etc.)
Importing the model to
the analyser module
Assigning the material
properties to the parts
Defining the contacts
and joints between partsMeshing the model
Defining the boundary
conditions and loadings
Analysis settings and
solving the problem
Obtaning the results and
making comments
COMPLIANT UNIVERSAL JOINT
Compliant Universal Joint with FEA
by analytical methods and
sed for comparison. For finite element analysis
hen, finite element
angles and torque values. The flow chart of
etails of the analysis will be explained in the
Importing the model to
the analyser module
Assigning the material
properties to the parts
Obtaning the results and
making comments
40
The steps shown in Figure 5.1 will be explained schematically for ANSYS 13.0
Workbench.
Defining the type of the analysis (Modal, Static, Transient Structural, etc.)
Entering the engineering data for AISI 1080
41
By using ANSYS Workbench Design Modeller prepare the model so that it is ready
for the analysis (Deleting the idle parts, improving the surfaces, etc.)
Analysis part
42
Defining the behaviors of the parts (Flexible or rigid)
Assigning the material properties that are defined in Engineering Data
43
Creating new coordinate systems to identify motions or forces
Defining contacts and joints between parts
44
Meshing the parts
Defining the analysis settings
45
Defining the boundary conditions and loadings
46
Getting the solutions and making comments
5.1.1 Boundary Conditions and Meshing
The spring steel universal joint is divided into subgroups whose characteristics will
be different in the analysis. Small length flexural hinges are the most critical parts of
the design therefore these hinges are meshed finely which means the number of
elements for unit area has the highest value compared to the other parts. And the
other parts are meshed coarsely which behave as relatively rigid and not as much
critical as flexible hinges. Hence the analysis took less time with a good accuracy of
results. The parts of the joint could be seen in Figure 5.2.
47
Figure 5.2 Definitions of the Parts
The connections of the parts are shown in Figure 5.3. Shaft 1 and clutch 1 are
modelled as a unique part by using Form New Part command. And also compliant
joint 1 is bonded to clutch 1 and body 1. Bonding creates a multi point constraint so
that the bonded surfaces behave like a single surface. Same procedure is done for the
other compliant joints. The connections and the parts that have been formed, are
shown in Figure 5.3.
Figure 5.3 Connections and Newly Formed Parts
As seen in Figure 5.4 the model is divided into two main parts whose meshes are
finer than other part. Part A is meshed with body sizing of 0.65 mm and part B is
48
meshed with 2 mm hex-dominant method which uses an unstructured meshing
approach to generate a quad-dominant surface mesh and then fill it with a hex-
dominant mesh. The total number of the elements is 19053 and there are 27428
nodes for the design with 0.75 mm thickness. The model with 0.5 mm thickness has
21705 elements 31896 nodes.
Figure 5.4 The Meshing of the Model
5.2 Static Analysis for Deflection-Only Case
Firstly the bending capacity of the compliant universal joint without loading of a
torque should be identified. In the design thicker parts are modelled as rigid and are
not bent compared to the flexural hinges.
For different shaft angles stress values are calculated analytically and numerically.
The analysis of the design with thickness of 0.75 mm is done for shaft angles
between 1° to 13°. Same procedure is done for the design with 0.5 mm thickness for
shaft angles 1° to 18°.
49
Analytical calculations that can be seen in Table 5.1 are done with the procedure
given below,
Width, thickness and length of the hinge are,
/ = 20 × 10d` (5.1)
�- = 0.75 × 10d` (5.2)
� = 20 × 10d` (5.3)
Second moment of area is,
� = /�-̀12 = 7.031 × 10de`a (5.4)
Modulus of elasticity for blue polished spring steel is,
� = 200 × 10�#� (5.5)
Taking shaft angle, 7, in radians, moment and maximum stress values could be
found with the equations given below,
= ��7� (5.6)
�fTB = �-/2� (5.7)
50
Figure 5.5 Representation of the Boundary Condition and Shaft Angle
For the numerical solution the model which is shown in Figure 5.5 is used. One end
is fixed and the other end is bent from analysis settings by inserting supports, fixed
support and remote displacement, in FEA. Remote displacement command is used
by rotating the shaft in the desired direction and for the desired shaft angle. The
equivalent stress values are obtained. Numerical solutions can be seen in Table 5.1
for the material with thickness of 0.75 mm and in Table 5.2 for the other model for
shaft angles as limited by yielding. The average of the chosen data is calculated.
During the analysis for the angles between 1° to 4° small deflection analysis is used
and for the other angles large deflection analysis is chosen from analysis settings.
Stress concentration is a highly localized effect. In some cases the reason may be
surface scratches. Engineering normal strain for fracture, 23, is an important
parameter for defining a material as ductile or brittle. If 23 ≥ 0.05, material is
ductile, otherwise material is brittle. In ductile materials the stress concentration
factor is not usually applied to predict the critical stress, because plastic strain in the
region of the stress is localized and has a strengthening effect. In other words, if the
material is ductile and the load is static, load may cause yielding in the critical
loading near the notch. This yielding can involve strain hardening of the material and
increases yield strength at the small critical notch location. In brittle materials the
geometric stress concentration factor, ��, is applied to the nominal stress.
51
AISI 1080 material has an elongation of 10-14 % which means 23 changes between
0.1 to 0.14. Therefore blue polished spring steel is a ductile material. Considering
that loads are static and the material is ductile, this part can withstand the loads with
no general yielding. Budynass and Nisbett (2011) points that in these cases the
designer sets the geometric (theoretical) stress concentration factor, ��, to unity.
Therefore, in this study all the following static analyses are performed by
Howell, L.L., Midha, A., "Parametric deflection approximation for end-loaded,
large-deflection beams in compliant mechanisms", Journal of Mechanical Design
117, p156-165, 1988.
http://kmoddl.library.cornell.edu/model.php?m¼500, Number 099 in the Clark
collection
L.L. Howell, A. Midha, "A Method for The Design of Compliant Mechanism with
Small-Length Flexurl Pivots", Journal of Mechanical Design 116, p280-290, 1994.
Mohan, R. A. V., Sandor, G. N., Kohli, D., and Soni, A. H., 1973, “Closed Form
Synthesis of Spatial Function Generating Mechanism for the Maximum Number of
Precision Points,” ASME J. Eng. Ind., 95, pp. 725–736.
Norton R. L., 2000, Machine Design, 2nd Ed., Prentice Hall, Upper Saddle River,
NJ.
Parlaktas, V., and Tanik, E., 2011, “Partially Compliant Spatial Slider-Crank (RSSP)
Mechanism,” Mech. Mach. Theory, 46(11), pp. 1707–1718.
Porat, I., 1980, “Moment Transmission by a Universal Joint,” Mech. Mach. Theory,
15(4), pp. 245–254.
Salamon, B.A. "Mechanical Advantage Aspects in Compliant Mechanisms Design",
M.S. Thesis, Purdue University, 1989.
Savage, M., and Hall, J., 1970, “Unique Descriptions of All Spherical Four-Bar
Linkages,” ASME J. Eng. Ind., 92, pp. 559–563.
Shigley, J. E. and Uicker, J. J.., Theory of Machines and Mechanisms, McGraw-Hill, New York, 1980, p. 262. Shigley, J. E., and Mischke, C. R., 1996, Standard Handbook of Machine Design,
2nd Ed., McGraw-Hill, New York.
Shigley, J. E., and Mischke, C. R., 2001, Mechanical Engineering Design, 6th Ed.,
McGraw-Hill, New York.
107
Tanik, E. and Parlaktas, V., 2011, “Compliant Cardan Universal Joint,” ASME J.
Mech. Des., 134 (2012) 021011
Tanik, E., and Parlaktas, V., 2011, “A New Type of Compliant Spatial Four- Bar
(RSSR) Mechanism,” Mech. Mach. Theory, 46(5), pp. 593–606.
Trease, B., Moon, Y., and Kota, S., 2005, “Design of Large-Displacement Compliant
Joints,” ASME J. Mech. D., 127, pp. 788–798.
Wagner, E. R., and Cooney, C. E., 1979, “Cardan or Hooke Universal Joint”
Universal Joint and Driveshaft Design Manual, Society of Automotive
Engineers,Warrendale, PA, pp. 39–75.
Yang, A. T., 1965, “Static Force and Torque Analysis of Spherical Four-Bar
Mechanisms,” ASME J. Eng. Ind., 87, pp. 221–227.
108
109
APPENDIX A
PROPERTIES OF AISI 1080
Figure A.1 Properties of AISI 1080
110
APPENDIX B
TECHNICAL DRAWINGS
Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness
111
Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness
112
APPENDIX C
SPECIFICIONS OF BEARINGS
Figure C.1Specifications of UCP 200 Bearing
113
APPENDIX D
SPECIFICIONS OF DC ELECTRIC MOTOR
114
Figure D.1 Specifications of the DC Electric Motor