ANALYSIS AND DESIGN OF PLANAR MULTIBODY...
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1 ANALYSIS AND DESIGN OF PLANAR MULTIBODY SYSTEMS WITH REVOLUTE JOINT WEAR By SAAD M. MUKRAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009
Transcript of ANALYSIS AND DESIGN OF PLANAR MULTIBODY...
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ANALYSIS AND DESIGN OF PLANAR MULTIBODY SYSTEMS WITH REVOLUTE JOINT WEAR
By
SAAD M. MUKRAS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2009
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© 2009 Saad M. Mukras
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To my parents, Mohamed Mukras and Bauwa Mukras, to my siblings AbduRahman, Suleiman, and Mariam, to my wife Amina and finally to my son Talha
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ACKNOWLEDGMENTS
I express my humility and utmost gratitude to Allah for his blessings in my life. Verily no
success would have been achieved without his grace and mercy. I would next like to thank my
parents for their support in all aspects of my life. I owe them much more than I can ever give
back. Next I thank my wife for her unwavering support, her encouragement and for the patience
that she has shown as I pursue my studies. I would like to acknowledge Dr Nam-Ho Kim, my
adviser, for the support that he has provided. Because of his advice and challenges, I have
matured as a student and as a researcher. I would like to thank my graduate committee members,
Dr Tony L. Schmitz, Dr W.G. Sawyer, Dr B.J. Fregly and Dr Jörg Peters for their assistance and
guidance during my Ph.D. pursuit. I would also like to acknowledge the assistance that I have
received from my colleagues, friends and members of the university staff. Indeed, it would be
negligent not mention the support that I have received from the members of the Masaajid in
Gainesville who have enabled me to feel at home while away from home.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS ...............................................................................................................4
LIST OF TABLES...........................................................................................................................8
LIST OF FIGURES .......................................................................................................................10
NOMENCLATURE ......................................................................................................................13
ABSTRACT...................................................................................................................................14
CHAPTER
1 INTRODUCTION ..................................................................................................................16
Motivation...............................................................................................................................16 Background.............................................................................................................................17 Objectives ...............................................................................................................................20
2 DYNAMIC ANALYSIS OF RIGID MULTIBODY SYSTEMS ..........................................22
Introduction.............................................................................................................................22 Kinematics Analysis ...............................................................................................................23
Kinematic Constraints .....................................................................................................24 Examples of Kinematic Constraints ................................................................................26
Revolute joint. ..........................................................................................................26 Translational joint. ...................................................................................................27
Example: Kinematic Analysis of a Slider-Crank Mechanism................................................28 Dynamic Analysis...................................................................................................................29
Direct Integration.............................................................................................................31 Constraint Stabilization Method......................................................................................32 Generalized Coordinate Partitioning Method..................................................................33 Hybrid Constraint Stabilization-Generalized Coordinate Partitioning Method ..............34 Modified Lagrangian Formulation ..................................................................................34
Example: Dynamic Analysis of a Slider-Crank Mechanism..................................................34 Summary and Discussion .......................................................................................................35
3 DYNAMICS OF MULTIBODY SYSTEMS WITH IMPERFECT REVOLUTE JOINTS...................................................................................................................................44
Introduction.............................................................................................................................44 General Imperfect Revolute Joint ...........................................................................................46
Contact-Impact Force Model...........................................................................................46 Modeling the General Imperfect Joint Model .................................................................51
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Example: Slider-Crank Mechanism with Joint Clearance Between the Crank and the Connecting Rod (General Imperfect Joint)............................................................54
Summary and Discussion .......................................................................................................56
4 WEAR-PREDICTION METHODOLOGY ...........................................................................62
Introduction.............................................................................................................................62 Wear Model ............................................................................................................................64 Wear Simulation Procedure....................................................................................................67
Computation of Contact Pressure....................................................................................67 Determining the Wear .....................................................................................................68 Geometry Update Procedure ...........................................................................................69 Boundary Displacement ..................................................................................................72
Reducing Computational Costs ..............................................................................................73 Adaptive Extrapolation Scheme .............................................................................................76 Experimental Validation.........................................................................................................77 Summary and Discussion .......................................................................................................78
5 INTEGRATED MODEL: SYSTEM DYNAMICS AND WEAR PREDICTION ................88
Introduction.............................................................................................................................88 Dynamic Analysis...................................................................................................................88 Wear Analysis.........................................................................................................................89 Demonstration of the Integration Process...............................................................................90 Summary and Conclusions .....................................................................................................92
6 INTERGRATED MODEL: SYSTEM DYNAMICS AND WEAR PREDICTION USING THE ELASTIC FOUNDATION MODEL ...............................................................97
Introduction.............................................................................................................................97 Elastic Foundation Model.......................................................................................................97 Analysis of Multibody Systems with Joint Wear Using the EFM........................................100
Modeling the Non-Ideal Revolute Joint Using the EFM...............................................100 Integrated Model: System Dynamics and Wear Prediction Using EFM.......................103
Summary and Conclusion.....................................................................................................106
7 EXPERIMENTAL VALIDATION OF THE INTEGRATED MODELs............................113
Introduction...........................................................................................................................113 Experiments for Model Validation .......................................................................................113 Summary and Conclusions ...................................................................................................116
8 DESIGN OF A MULTIBODY SYSTEM FOR REDUCED JOINT WEAR MAINTENANCE COSTS....................................................................................................124
Introduction...........................................................................................................................124 Design Example: Design of a Slider-Crank for Reduced Maintenance Cost .......................125
Problem Definition ........................................................................................................125
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Analysis of the Slider-Crank with Multiple Joints Wearing .........................................126 Solution of Optimization Problem.................................................................................127
Summary and Concluding Remarks .....................................................................................130
9 SUMMARY AND FUTURE WORK ..................................................................................137
Summary and Discussion .....................................................................................................137 Future Work..........................................................................................................................140
LIST OF REFERENCES.............................................................................................................142
BIOGRAPHICAL SKETCH .......................................................................................................151
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LIST OF TABLES
Table page 2-1 Dimension and mass parameter for slider-crank mechanism.................................................37
3-1 Dimension and mass parameter for slider-crank mechanism.................................................58
3-2 Material properties for the joint components .........................................................................58
3-3 Parameters for the force model...............................................................................................58
4-1 Wear test information for the pin-pivot assembly ..................................................................81
4-2 Simulation parameters for the pin-pivot simulation test ........................................................81
4-3 Comparison of results form the simulation and Expt. wear tests. ..........................................81
5-1 Dimension and mass properties of the slide-crank mechanism..............................................93
5-2 Properties of the pin and bushing ...........................................................................................93
5-3 Test and simulation parameters ..............................................................................................93
6-1 Dimension and mass parameter for slider-crank mechanism...............................................107
6-2 Material properties for the joint components .......................................................................107
6-3 Wear simulation parameters .................................................................................................107
6-4 Comparison of results from prediction based on FEM and EFM.........................................107
7-1 Dimension and mass properties of the slide-crank mechanism............................................119
7-2 Properties of the pin and bushing .........................................................................................119
7-3 Test and simulation parameters ............................................................................................119
7-4 Comparison of wear results for FEM and EFM models (21,400 crank cycles) ...................119
7-5 Comparison of wear results between test and FEM model (21,400 crank cycles)...............119
7-6 Comparison of wear results between test and EFM model (21,400 crank cycles)...............120
8-1 Dimension and mass parameter for slider-crank mechanism...............................................132
8-2 Material properties for the joint components .......................................................................132
8-3 Parameter and design space specifications for the optimization ..........................................132
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8-4 Solution of optimization problem (Eq. (8-1)).......................................................................132
8-5 Comparison of results between surrogate and high-fidelity model......................................132
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LIST OF FIGURES
Figure page 1-1 Examples of Multibody systems.............................................................................................21
2-1 Revolute joint between bodies i and j.....................................................................................38
2-2 Translational joint between body i and j ................................................................................38
2-3 A slider-crank mechanism......................................................................................................39
2-4 Disassembled slider-crank mechanism...................................................................................39
2-5 Results from the kinematic analysis (crank)...........................................................................40
2-6 Results from the kinematic analysis (slider)...........................................................................41
2-7 Direct integration procedure for the Differential Algebraic Equation ...................................42
2-8 Effect stabilization parameters ( and ) on the second kinematic constraint ( 2K ) ...........42
2-9 Results from the dynamic analysis .........................................................................................43
2-10 Joint reaction force between the crank and the connecting rod............................................43
3-1 A revolute joint with clearance...............................................................................................59
3-2 Kelvin-Voigt viscoelastic model ............................................................................................59
3-3 Contact-force model with hysteresis damping (by Lankarani & Nikravesh).........................59
3-4 General imperfect joint...........................................................................................................60
3-5 Penetration during contact between the pin and the bushing .................................................60
3-6 Slider-crank mechanisms with joint clearance .......................................................................61
3-7 Comparison of reaction between the ideal and the non-ideal joints.......................................61
4-1 Pin-pivot assembly .................................................................................................................82
4-2 Pin-pivot finite element model ...............................................................................................82
4-3 Wear simulation flow chart for the ‘step update’ procedure ..................................................83
4-4 A three-node contact element used to represent the contact surface......................................83
4-5 Geometry updates process ......................................................................................................84
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4-6 Geometry updates process using the boundary displacement approach.................................84
4-7 Contact pressure distribution on a pin-pivot assembly...........................................................84
4-8 Extrapolation history for a pin-pivot assembly ......................................................................85
4-9 Cumulative maximum wear on pin and pivot ........................................................................85
4-10 Evolution of the contact pressure for nine intermediate cycles............................................86
5-1 Integration of wear analysis into system dynamics analysis ..................................................94
5-2 Slider-crank mechanism with a wearing joint between the crank and the connecting rod ....94
5-3 Initial joint reaction force for joint with clearance 0.0005mm...............................................95
5-4 Locus of contact point ‘C’ for a complete crank cycle...........................................................95
5-5 Wear on bushing as a function of the bushing angular coordinate after 5,000 cycles ...........96
6-1 Elastic foundation model ......................................................................................................108
6-2 Procedure to determine the contact pressure using the EFM ...............................................108
6-3 General imperfect joint.........................................................................................................109
6-4 Penetration during contact between the pin and the bushing ...............................................109
6-5 Slider-crank mechanisms with joint clearance .....................................................................109
6-6 Comparison of reaction force between the ideal joint model and the EFM.........................110
6-7 Locus of contact point ‘C’ for a complete crank cycle.........................................................110
6-8 Comparison of reaction force between ideal joint model EFM............................................111
6-9 Comparison of joint wear between the EFM and FEM after 5,000 cycles ..........................112
7-1 Experimental slider crank mechanism..................................................................................121
7-2 Comparison of the initial joint reaction force between the two models and the Expt..........121
7-4 Comparison of the wear prediction between the models......................................................122
7-5 Comparison of the wear profile for the models and the experiment ....................................123
8-1 Backhoe system with three main revolute joints ..................................................................133
8-2 Slider-crank mechanism with two wearing joints wearing ..................................................133
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8-3 Joint reaction force for Joint1 and Joint 2 ............................................................................134
8-4 Incremental sliding angle Joint 1 and Joint 2 .......................................................................134
8-5 Locus of the center of the contact region .............................................................................135
8-6 DOE used to construct surrogate for cyc1 and cyc2 ..............................................................135
8-7 Response generated using the surrogate for cyc1..................................................................136
8-8 Response generated using the surrogate for cyc2..................................................................136
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NOMENCLATURE
A Contact area
EA Extrapolation factor
Penetration
re Coefficient of restitution
E Young’s modulus
WE Composite elastic modulus (Elastic foundation model)
NF Normal force in the contact interface
h Wear depth
k Dimensioned wear coefficient
K Non-dimensional wear coefficient
L Thickness of elastic layer (Elastic foundation model)
λ Vector of Lagrange multipliers
M Mass matrix
p Contact pressure
q Position vector
AQ Vector of applied loads
s Sliding distance
t Time
Poisons ratio
Φ Constraint vector
qΦ Jacobian matrix
Crank velocity
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
ANALYSIS AND DESIGN OF PLANAR MULTIBODY SYSTEMS WITH REVOLUTE
JOINT WEAR
By
Saad M. Mukras
August 2009 Chair: Nam Ho Kim Major: Mechanical Engineering
Wear prediction on the components of a mechanical system without considering the
system as a whole will, in most cases, lead to inaccurate predictions. This is because the wear is
directly affected by the system dynamics which evolves simultaneously with the wear. In
addition, the contact condition (regions of contact for the wearing bodies) also depends on the
system dynamics and, in most cases, can only be determined in a multibody dynamics
framework.
In this work, a procedure to analyze planar multibody systems in which wear is present at
one or more revolute joints is presented. The analysis involves modeling multibody systems with
revolute joints that consist of clearance. Wear can then be incorporated into the system dynamic
analysis by allowing the size and shape of the clearance to evolve as dictated by wear. An
iterative wear prediction procedure based on the Archard’s wear model is used to compute the
wear as a function of the evolving dynamics and tribological data. In this framework, two
procedures for the analysis of planar multibody systems with joint wear are developed. In the
first procedure contact force at the concerned joint is determined using a contact force law and
the wear prediction is based on the finite element method. In the second procedure, contact force
determination and the wear prediction are based on the elastic foundation model.
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The two procedures are validated by comparing the wear predictions with wear on an
experimental slider-crank mechanism. The experimental slider-crank is also used as a reference
to assess the performance of the two models. It turns out that the procedure based on the finite
element method provides reasonably accurate predictions for both wear profile and wear
volume/mass whereas the procedure based on the elastic foundation model provides reasonably
accurate estimates on the wear volume/mass, is computationally faster but provides progressively
poor estimates on the wear profile.
Finally an example is presented to illustrate an application of the procedures. In the
example, a slide-crank mechanism is designed so that the maximum allowable wear depth at two
joints occurs simultaneously.
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CHAPTER 1 INTRODUCTION
Motivation
In this work, a study of the analysis of planar multibody systems with wear at a single or
multiple joints is presented. This analysis will involve the dynamic analysis of multibody system
coupled with a wear prediction procedure to estimate the wear that the wearing joints. The
motivation behind this work is based on the need to understand the behavior of such systems
under the influence of wear. Informed and therefore improved design of these systems is thus
enabled.
Figure 1-1 shows two examples of multibody/mechanical systems namely a backhoe
system and a flat engine. The two systems have different functionalities but are, otherwise,
similar in the sense that they are both made of multiple components connected by joints. Due to
the contact and relative motion of the joint components it is inevitable that wear will occur at the
joints of both systems. Despite this fact, knowledge of how wear affects the system dynamics
and how the system dynamics affect the wear can empower the designer to develop better design.
For instance, in the flat engine shown in Figure 1-1, it is known from the analysis of a similar
system (in this work) that the revolute joint between the crank and the connecting rod will
experience the most wear. This is especially true if the reaction force at this joint and that of
other joints are similar (which is the case since the resultant reaction force on the piston due to
compression in the cylinder will dominate inertial forces). One possible way to reduce the wear
is to increase the joint diameter; this will have the effect of reducing the maximum contact
pressure and thus reducing the maximum wear depth. On the contrary increasing the diameter
will have the opposing effect or increasing the wear since the incremental sliding distance is
proportional to the diameter. Thus the decision to increase or decrease the diameter will depend
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on which of the two quantities (the contact pressure or the sliding distance) is more dominant.
Arriving at this decision will require an analysis that couples both the system dynamics and wear
prediction which will be the discussion of this work.
Another example can be drawn from the backhoe system shown in Figure 1-1. Being able
to perform multibody analysis on this system while accounting for wear at the joints, can allow
the engineers to correctly predict wear at the joints and thereafter issue appropriate warranties or
prepare appropriate maintenance schedules. Prediction of wear at the joints is however not
trivial. Wear prediction at component level may be inaccurate since these procedures do not
account for how the overall system will affect a particular joint. For instance, it will not be a
simple task to determine the regions in a joint that will be in contact and thus wear without
considering the entire system. In addition wear that occurs at the joint may further affect the
subsequent regions of contact. Once again it is clear that to determine this information it is
necessary to perform an analysis in which the system dynamics is coupled with a wear prediction
procedure.
Background
As was mentioned earlier, the analysis of multibody systems with joint wear will be the
focus of this work. While this work is not unique, more research is still required to properly
address this area. Most of the work related to this topic has been aimed at modeling the effect of
joint clearance on multibody systems. Early studies, in this area, focused on simple models to
obtain insight into the behavior of systems with joint clearances. Dubousky et al [1,2] developed
a contact impact pair model to study the elastic joint with clearance. In their model, joint
elasticity and damping were modeled via springs and a viscous coefficient. The study served to
represent the complexity of the joint clearance using a simple model. Dubowsky and Gardner [3]
later extended this work to include flexible mechanism as well as multiple clearance connections.
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Earles et al [4] proposed to model joint clearance using a mass-less rigid link whose length
was equal to the clearance size. The components of the joint were thus assumed to be in contact
at all times. Wu et al [5] later used the model to predict contact loss between the joint
components for planar mechanism. The concept of a mass-less link was also used by Furuhashi
et al [6-9] to study the dynamics of a four-bar linkage with clearance. Once again the joint
components were assumed to be in contact at all times.
In later year, more complex models were developed to study the effect of clearance on
system dynamics. Farahanchi et al [10] modeled joint clearance by considering three
configurations of the joint components i.e. 1) free-flight motion; when the components are not in
contact 2) impact condition; when the components establish contact and 3) sliding condition;
when the components are in contact and relative motion. In their model the reaction force at the
joint (when the joint components are in contact) was determined assuming no clearance was
present. They used a slider-crank mechanism to demonstrate the procedure and studied the effect
of clearance size, friction, crank speed and impact parameters. Rhee et al [11] also used the three
modes of motion to model the joint clearance. They used an approach similar to that of
Farahanchi [10] to determine the reaction force during the sliding motion. They studied the
response of a four-bar mechanism with a revolute joint clearance.
The three mode approach was also used by Khulief et al [12] to model the clearance at the
joint. In their approach, termed as discontinuous method, the analysis (integration of the
equations of motion) is divided into two parts namely; pre and post-collision during which
momentum balance is performed to determine the post-collision velocities. Velocities in the
analysis are then updated and the analysis is resumed.
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Ravn [13], once again, utilized the three-mode approach to model the joint clear. However,
in his approach, the reaction force during the impact and the sliding mode is computed using a
contact force model. The analysis in this case has been termed continuous since integration of the
equations of motion is not halted as in the case of the discontinuous method. A number of
researchers [14-18] have since used this technique to model as well as study the effect of
clearance in the joints of multibody systems.
The literature that has been presented in the preceding paragraphs was primarily directed at
modeling multibody systems with joint clearance that remains constant. The concern of this
work, however, is in studying the case in which the clearance changes in shape and size due to
wear. The majority of wear prediction procedures that are available generally take into account
the changes in the contact pressure distribution due to the geometry evolution caused by wear.
This is however at a component level rather than at the system level where the effect of the
system dynamics is accounted for in the wear prediction. Recently, Flores [19] presented a
procedure to model multibody systems with joint wear. The procedure involves the analysis of
multibody system in the manner similar to the three-mode approach. Based on the contact force
at the concerned joint, wear is estimated for the joint. While this work is in line with the basic
theme of this dissertation details of the work differ substantially. These include: 1) the manner in
which the system dynamics is integrated with the wear prediction, 2) computation of quantities
required for wear analysis such as i) calculation of the sliding distance, ii) prediction of the
contact surface, and iii) calculation of the contact pressure, 3) the procedure in which the
geometry is updated to reflect the wear, 4) the procedure in which the analysis is accelerated in
order to reduce computational expense, 5) the manner in which the contact force is computed,
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and 6) experimental validation of the procedures. These differences have significant
consequences in the performance of the procedures presented.
Objectives
In line with the motivation, four objectives to this work are listed as follows:
To develop a procedure that can be used to analyze multibody systems with joint wear.
To demonstrate the need of a multibody framework in predicting wear at the joints of mechanical systems.
To validate the procedure through experiments.
To present an example that will illustrate the use of the procedure as a design tool.
Since the subject of multibody system analysis is extensive, this work will be limited to
analysis of planar systems. Furthermore the components in the multibody systems will be
assumed to be rigid. While in real system wear is expected to occur in all components that are in
contact and in relative motion, this work will only consider wear at the revolute joints. It is,
however, possible to extend this work to cover wear in other types of joints.
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Figure 1-1. Examples of Multibody systems.
Revolute joints
Flat Engine
Piston Connecting-rod Crank
Revolute joints
Backhoe System
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CHAPTER 2 DYNAMIC ANALYSIS OF RIGID MULTIBODY SYSTEMS
This Chapter presents a discussion on the dynamic analysis of multibody systems.
Formulation and solution techniques of the equations of motion for in plane, rigid, multibody
systems are presented. The intent of the information in this Chapter is to acquaint the reader with
the subject of multibody dynamics and lay the foundation for the subsequent topic of analyzing
multibody systems with joint wear. A more in depth discussion of the subject can be accessed
from the literature [20-24].
Introduction
A multibody system refers to a system consisting of a set of interconnected rigid or flexible
bodies that undergo large displacements and rotations. Many mechanical systems such as
vehicles, robots, etc are in fact multibody systems. They consist of several interconnected bodies
that may be loaded at various locations and whose motions are restricted to achieve a desired
function. The analysis of such systems to determine their response to applied loads is termed as
dynamic analysis of multibody systems. Depending on the degree of complexity of the system
being modeled, the formulations of the equations of motion can be quite simple or complicated.
The most general example of such a system would consist of a set of rigid and/or flexible
components subjected to various loading conditions (forces, contact force, torque etc) and whose
motions are spatial. In reality, however, appropriate assumptions can be made so as to manage
the complexity of the analysis for a corresponding system. For instance, a systems whose
motions are primarily confined to a single plane may be modeled as planar instead spatial and
thus substantially reducing the modeling efforts. Another example would be to model a system
with rigid bodies instead of flexible bodies if the deformations of the bodies are reasonable
small.
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The two simplifications previously mentioned are probably the most important
consideration when analyzing a multibody system. In this work only systems with components
whose motions are restricted within a plane are considered. In this Chapter, dynamic analysis of
systems with rigid bodies (referred to as rigid multibody dynamics) will be discussed.
As was mentioned earlier, dynamic analysis of a multibody system is usually conducted in
order to determine the response of a system to applied loads. Thus the joint reactions forces or
the motion (position, velocity and acceleration) of the components of the system are typical
quantities of interest. In cases in which only the motion of the system is require, it may be
possible to simply perform a kinematic analysis. A discussion of dynamics analysis will thus be
preceded by a brief outline of the kinematic analysis of a mechanical system.
Kinematics Analysis
Kinematic analysis involves estimating the position, velocity and acceleration of the
components of a multibody system without regard to the forces that produce the motion. The
analysis efforts are therefore less than that of the dynamic analysis.
The position and orientation of all bodies in a multibody system can uniquely be specified
by a set of generalized coordinates which may vary in time if the system is in motion. In this
text, the generalized coordinates are expressed as a column vector, 1 2, , ,T
ncq q qq , where nc
is the number of generalized coordinates. Specification of the system components for a planar
system is achieved by fixing an x’-y’ reference frame on each body in the system. Body i can
then be located by specifying the global coordinates of the origin of the corresponding reference
frame ,T
i ix yr and the relative angle between the reference frame and the global frame i .
The location of a body in the system is thus specified as , ,T
i ix y q . The generalized
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coordinates can then be expressed as 1 2 1 2, , , , , , , , , , ,
TT T T TT T Tnb nb
x y x y x y q q q q
where nb is the number of bodies in the multibody system. It is thus clear that a planar system
with nb bodies will have nc = 3nb generalized coordinates.
Kinematic Constraints
The bodies in a multibody system are interconnected by joints which impose condition on
the relative motion of the bodies. Consequently, the generalized coordinates are usually not
independent. When these conditions are expressed as algebraic equations in terms of the
generalized coordinates, they are referred to as holonomic kinematic constraints. The kinematic
constraints for a system with nh constraints can be expressed as
1 2, , ,TK K K K
nh Φ q q q q 0 . (2-1)
Furthermore if the constraints depend on time t they are referred to as time-dependent constraints
and are expressed as
,K t Φ q 0 (2-2)
If t does not appear explicitly in Eq. (2-2) then the constraints are called stationary constraints.
Other general constraints that contain inequalities and/or depend on system coordinates as well
as velocities are said to be nonholonomic constraints. In this work only holonomic kinematic
constraints are considered and are referred to simply as constraints.
If the number of generalized coordinates exceed the number of constraints ( nc nh ), then
the constraint equations cannot be solved to uniquely determine the position (q ) of the
components. The system in this case is said to have nc nh degrees-of-freedom (DOF). In order
to determine the motion of the system one may specify a total of DOF nc nh additional
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constraints. These additional constraints are termed as driving constraints and are shown in Eq.
(2-3).
,D t Φ q 0 (2-3)
If the kinematic as well as the driving constraints are independent the complete set of constraints
(shown in Eq.(2-4)) will consist of nc (where nc nh DOF ) independent constraint equations
which can simultaneously be solved to uniquely determine the position ( tq ) of the
components of the system at any time.
,,
,
K
D
tt
t
Φ qΦ q 0
Φ q (2-4)
Alternatively, instead of specifying the driving constraints, one may specify appropriate loads on
the system. In this case, tq is the solution of a set of differential equations (the differential
equations of motion). This will be the subject of the dynamic analysis.
The expression for the velocity q as well as the acceleration q of the system components
can be obtained by differentiating the constraint vector (Eq. (2-4)). Using the chain rule
expression (2-4) is differentiated once with respect to time to yield the following velocity
equation,
t qΦ q Φ , (2-5)
where qΦ is the Jacobian of the constraint vector. The velocity q can then be determined if the
Jacobian is nonsingular. Equation (2-5) can further be differentiated to yield the expression for
the acceleration. The result is shown in Eq. (2-6). Similar to the velocity equation, if the Jacobian
is nonsingular the acceleration can also be determined.
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Determination of the position, velocity and acceleration completes the kinematic analysis.
It is worth noting that instead of supplying the driving constraints, appropriate driving forces or
torques can be applied. In this case a kinematic analysis cannot be performed (since the numbers
of unknowns exceed the number constraint equation), and instead a dynamic analysis will have
to be performed.
2
2
t tt
t tt
q q qq
q q qq
Φ Φ q q Φ q Φ q Φ 0
Φ q Φ q q Φ q Φ γ
(2-6)
Examples of Kinematic Constraints
Various types of kinematic constraints can be formulated in order to achieve desired
relative motion between components in a multibody system. In what follows, two kinds of
constraints, the revolute joint and translational joint, will be presented in order to demonstrate the
formulation procedure. The two constraints have been specifically selected as they will also be
used in the later sections of this work.
Revolute joint. The revolute joint imposes a constraint on the relative translation between
two bodies i and j. However, it permits relative rotation between the bodies about a point P that
is common to both bodies. An illustration of the revolute joint is shown in Figure 2-1. The
constraint equation that describes the revolute joint can be formulated by ensuring that point P
on body i and point P on body j coincide at all times. This constraint can be expressed in vector
form as
( , )r i ji i i j j j Φ r A s r A s 0 , (2-7)
where ir and jr are position vectors in the global coordinate system (x-y) that describe the
location of the origin of the body-fixed coordinates (xi-yi and xj-yj). The vectors is and js are
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position vectors in the body-fixed coordinate systems that locate the point P. iA and jA are
transformation matrices that transform vectors in body-fixed coordinates systems into vectors in
the global system.
Translational joint. The translational joint, contrary to the revolute joint, allows relative
translation between two bodies but prevents relative rotation between the bodies. For instance,
Figure 2-2 shows two bodies, i and j, which are connected by a translation joint. Body i translates
relative to body j along the axis A-B. Relative rotation between the two bodies is however
constrained.
With reference to Figure 2-2, this joint can be modeled by requiring that the vector ih on
body i remain collinear to vector jh on body j. The two will be collinear if ih is parallel to jh
and ih is parallel to ijl , where ijl is a vector that connects a point on vector ih and another on
vector ih . The constraint can be expressed mathematically as
( , ) i ijt i j
i j
h lΦ 0
h h
. (2-8)
It should be noted ijl in Eq. (2-8) can assume a value of zero when the two points in vector ih
and ih coincide. This occurrence essentially means that the two vectors, ih and jh , touch each
other and the second constraint ( i j h h 0 ) will ensure that the vectors are parallel.
The two joints discussed above are simple but probably the most widely used constraints in
planar multibody systems. The interested reader is referred the works of Nikravesh [20] and
Haug [21], for a comprehensive discussion on constraint formulation.
28
Example: Kinematic Analysis of a Slider-Crank Mechanism
In order to demonstrate the procedure of constraint formulation and kinematic analysis, a
study will be done on a slider-crank mechanism. The slider-crank is selected for this task since it
is a simple mechanism with all the relevant features necessary for the demonstration.
Furthermore the slider-crank will be used to facilitate the central idea in this work.
The slider-crank mechanism, shown in Figure 2-3, consists of three rigid bodies (crank,
connecting rod, and slider). In Figure 2-4, the disassembled components of the mechanism are
shown in the global axis. Each component can translate and rotate in the plane. The mechanism
is modeled by imposing constraints on the motion of the components as described in the previous
section. The constraints corresponding to the slider-crank mechanism, shown in Figure 2-4,
consist of nine nonlinear simultaneous equations expressed explicitly as
1 1
1 1 1
2 1 1 2 2
2 1 1 2 2
3 1 1 2 2
3 1 1 2 2
3
3
1
cos 0
sin 0
2 cos cos 0
2 sin sin 0
2 cos 2 cos 0
2 sin 2 sin 0
0
0
0
x l
y l
x l l
y l l
x l l
y l l
y
t
Φ . (2-9)
The first two constraints in Eq. (2-9) confine point P1 (on the crank) to the origin and
describe a revolute joint between the origin/ground and the crank. The next two constraints also
describe a revolute joint between the crank and the connecting rod. They ensure that points P2
(on the crank) and P3 (on the connecting rod) coincide at all times. This condition is synonymous
to an ideal joint and later will be relaxed when modeling the imperfect joint. The fifth and sixth
constraints in Eq. (2-9) represent a perfect revolute joint between the connecting rod and the
29
slider. The next two constraints ensure that the slider remain on the x-axis without rotation.
These two constraints represent the translational joint. The final constraint is the driving
constraint which specifies the motion of the crank. For the current case a constant angular
velocity is imposed on the crank.
It can be seen from the set of simultaneous equations above, that the number of equations
is exactly equal to the number of unknowns. The unknowns are the locations of the center of
masses of the slider-crank components, denoted as 1 1 1 2 2 2 3 3 3, , , , , , , ,T
x y x y x y q . Kinematic
analysis is performed by simultaneously solving Eq. (2-9) to determine q and solving Eq. (2-5)
and (2-6) to determine the velocities q and accelerations q of the system.
A kinematic analysis is performed for the slider-crank mechanism with crank velocity of π
rad/sec and dimension parameters as shown in Table 2-1. Figure 2-5 shows the position velocity
and acceleration at the crank mass of center from the kinematic analysis. The corresponding
results for the slider are shown in Figure 2-6. Since the motion of the slider is restricted along the
global x-axis (i.e. translation along the x-axis), the vertical position, velocity as well as
acceleration of its mass center should all be zero. This is confirmed from the plots in Figure 2-6.
Dynamic Analysis
As was mentioned earlier, dynamic analysis is the study of motion and the forces that
cause it. With reference to this definition, our primary interest is to develop and solve the
equation of motion for the concerned system. Two approaches can be used to formulate the
equations of motion namely: the Newton-Euler formulation and the Lagrangian formulation. The
Newton-Euler formulation involves the application of the law of motion which involves applied
loads and reactions, whereas in the Lagrangian formulation the system dynamics is described in
30
terms of work and energy. Both approaches lead to equivalent results. A more comprehensive
discussion of the approaches can be found in the works of Shabana [22] and Schiehlen [24].
The equations of motion for a constrained rigid multibody system are stated here without
derivation due to the length and complexity of their derivations. The interested reader is referred
to the work of Haug [21] for a detailed derivation of these equations using the Lagrangian
approach. The equation can be expressed as
T AqMq +Φ λ Q ’ (2-10)
where M is the mass matrix consisting of masses and moments of inertia for the system
components. q and TqΦ are the acceleration vectors and Jacobian of the constraints (as
previously defined) respectively. λ is a vector of Lagrange multipliers and AQ is a vector of
applied load. It is worth noting that in this work the body-fixed coordinate system is selected to
be at the center of mass (COM) of the corresponding body which significantly simplifies the
general form of the equations of motion. In particular, the mass matrix becomes diagonal. The
product of the Jacobian and the vector of Lagrange multipliers ( TqΦ λ ) is in fact the vector of
reaction forces. This is the second term on the left hand side of Eq. (2-10). Thus for an
unconstrained system the vector or Lagrange multipliers are zero and this term would disappear.
Equations (2-10) and (2-6) can be combined to result in a mixed system of differential
algebraic equations also known as the differential algebraic equation of motion (DAE). The
equations are expressed as
T A
q
q
M Φ q Q
Φ 0 λ γ
’ (2-11)
31
where 2 t tt q q qqγ Φ q Φ q q Φ q Φ . For a meaningful system the coefficient matrix of Eq.
(2-11) must be nonsingular. This is guaranteed if the mass matrix M is positive definite and the
Jacobian matrix qΦ is full row rank (or constraints are independent).
The process of determining the system dynamics involves solving the DAE. In addition to
the DAE Eq. (2-11) both the position and velocity equations must be satisfied, that is
, t Φ q 0 , (2-12)
t
t
q
q
Φ Φ q Φ 0
Φ q Φ
(2-13)
In the next section a procedure termed as direct integration will be used to demonstrate how to
solve the DAE (Eq. (2-11)).
Direct Integration
For a proper set of initial conditions (initial position and velocity conditions), the system of
equations (Eq.(2-11)) can simultaneously be solved for a unique solution. The system of
equations for a particular time instant consist of a set of linear equations that can easily be solved
to determine the acceleration vector q and vector of Lagrange multipliers λ (which give the
joint reactions). For that time instant, the velocity q and position q vectors can be determined
by integrating the acceleration vector q using methods such as Runge-Kutta, Adams-Bashforth,
and Adams-Moulton predictor-corrector methods [25]. The process is then repeated for the next
time instant with the new values of position velocity and acceleration. This procedure is referred
to as direct integration and is illustrated in Figure 2-7.
Although the direct integration procedure is straight forward, there is no guarantee that,
after several iterations, the position and velocity equations will be satisfied. This is especially
32
true because not all the components of q and q are independent. In order to combat this
problem a number of algorithms have been developed. Some of these include the Constraint
Stabilization Method, Generalized Coordinate Partitioning Method, Hybrid Constraint
Stabilization- Generalized Coordinate Partitioning Method, and the Modified Lagrangian
Formulation. A brief discussion of these techniques will follow in the next sections.
Constraint Stabilization Method
The Constraint Stabilization Method, due to Baumgarte [26], is probably the most
attractive procedure due to its simplicity both in concept and implementation. Baumgarte noted
that the differential equations Φ 0 can be unstable. Baumgarte, however, observed that if the
acceleration equation, repeated here as
qΦ Φ q γ 0 , (2-14)
was modified to
22 Φ Φ Φ 0 , (2-15)
such that 0 and 0 , it became stable. It is clear from Eq. (2-15) that since the
acceleration equation is zero ( Φ 0 ) both the position and velocity equations are approximately
zero i.e. Φ Φ 0 . Equation (2-15) can explicitly be written as
22 t q qΦ q γ Φ q Φ Φ . (2-16)
Both Eq. (2-14) and (2-15) are equivalent analytically however with regards to stability the term
22 Φ Φ in Eq. (2-15) serves as an error control so that both the position and velocity
constraints will eventually be satisfied.
33
The DAE can then be modified by replacing the acceleration equations with the modified
equivalent. The resulting DAE is shown in Eq. (2-17). The system dynamics can then be
determined by a similar procedure as outlined for the direct integration method.
22
AT
t
q
q q
QM Φ q
Φ 0 λ γ Φ q Φ Φ
. (2-17)
The choices for and are important in the solution process, however, no generally
accepted procedure appears to have been adopted in the selection of these parameters. Baumgarte
[26] used a trial and error procedure through numerical experiments to determine reasonable
values for the parameters. Lin et al [27,28] suggested that the appropriate values of the
parameters were dependent on the type of integration method employed. They discussed ways of
determining appropriate values of and for the Runge-Kutta method [27], the Adams-
Bashforth and the Adams-Moulton predictor-corrector methods [28]. Nikravesh [20] suggested
that for most practical problems, values between 1 and 10 will suffice for both parameters. It was
also suggested by Wittenburg [23] that selecting will stabilize the response more quickly.
Generalized Coordinate Partitioning Method
The Generalized Coordinate Partitioning Method is a more rigorous technique that has
been found to have good error control. The constraints (position and velocity equations) are
satisfied to the precision set by the user. The method involves partitioning the generalized
coordinates into dependent and independent variables. Equation (2-11) can then be solved to
determine the independent accelerations (together with the dependent accelerations) and the
Lagrange multipliers λ . The independent accelerations are integrated for independent velocities
and positions. The kinematic constraint can then be solved to determine the dependent positions
(using techniques such as the Newton-Raphson method). Finally the dependent velocity is
34
determined by solving the partitioned velocity equation. More details of this technique can be
found in the works of Haug [21] and Wehage et al [29].
Hybrid Constraint Stabilization-Generalized Coordinate Partitioning Method
As the name suggests, the Hybrid Constraint Stabilization-Generalized Coordinate
Partitioning Method is a hybrid between the two techniques. The idea behind the method is to
derive benefits from the two techniques namely: to maintain good error control of the
partitioning method and to retain the computational speeds of the constraint stabilization method.
A complete study of the method is available in the works of Park [30].
Modified Lagrangian Formulation
The approach used in the modified Lagrangian formulation method to remedy the
constraint violations is quite different from the three procedures just described. In this procedure
the constraint violations are catered for using a penalty approach. The constraint condition is
inserted into the Lagrange equations by means of a penalty formulation rather than appending
them as was the case in the non-modified formulation (Eq.(2-11)). The benefit of this technique
is that no new unknowns are introduced which is contrary to the regular Lagrange formulation
where the new unknowns are the Lagrange multipliers. The details of the formulations have been
presented by Bayo et al [31].
Four procedures to solve the equations of motions have been presented. All these
techniques have been shown to produce acceptable results. In this work, however, the Constraint
Stabilization Method has been adopted because of its simplicity in implementation.
Example: Dynamic Analysis of a Slider-Crank Mechanism
A slider-crank mechanism, which was introduced earlier, will be used to demonstrate the
dynamic analysis procedure. A diagram of the slider-crank mechanism is shown in Figure 2-3.
35
The kinematic constraint equation for the slider-crank are identical to those used in the
previous example (Eq. (2-9)). In this case, the crank is constrained (driving constraint) to rotate
at a constant angular velocity of π rad/sec. The dimensions and mass properties for the
mechanism are shown in Table 2-1. The stabilization parameters used for this analysis were
250 . The differential equations of motion can then be assembled as described in Eq.
(2-17). For this case, the only applied load is the gravitational force. The equations of motion
may be solved through the procedure outlined in Figure 2-7 in conjunction with the constraint
stabilization method.
In Figure 2-8, the value of the second kinematic constraint (constraint corresponding to the
joint between the crank and the ground) is plotted for different constraint stabilization
parameters. It is clear that when no constraint stabilization parameters are used (i.e. direct
integration method) the constraint is violated and the violation increases as the integration
progresses. A value of 250 was found to provide reasonably stability to the constraint.
The motion (position, velocity and acceleration) of the center of mass of the crank,
obtained from the dynamic analysis, is plotted in Figure 2-9. The results shown in the figure
show close resembles to those obtained from the kinematic analysis in Figure 2-5. In addition to
the motion, the joint reaction forces at the various revolute joints can be determined. As was
mentioned earlier, this information can not be obtained by performing a kinematic analysis. The
reaction force at the revolute joint between the crank and connecting rod is shown in Figure 2-
10.
Summary and Discussion
In this Chapter the procedure to perform kinematic and dynamic analysis for a planar rigid
multibody system was presented. The kinematic analysis involves simultaneously solving a set
36
of nonlinear kinematic constraints equations. The analysis provides information regarding the
motion of the bodies in the system; i.e., the positions, velocities and accelerations of the bodies.
In order to determine the reaction forces, a dynamic analysis is required. In the dynamic analysis,
a differential algebraic equation of motion needs to be assembled. The equations of motion can
once again be solved to determine the motion as well as reaction forces of the system. In the
solution process it is necessary to integrate the accelerations in order to determine both the
velocity and the position. It has, however, been noted that a direct integration procedure will lead
to violations in the kinematic constraints. Various techniques, such as the constraint stabilization,
generalized coordinate partitioning and the modified lagrangian formulations, have been
proposed to mitigate or eliminate the constraint violation problem. In this work the constraint
stabilization method was used. The kinematic and dynamic analysis procedures were
demonstrated using the slider-crank mechanism.
37
Table 2-1. Dimension and mass parameter for slider-crank mechanism Length (m) Mass (kg) Moment of inertia (kg-m2) crank 1.00 10.00 45.00connecting rod 1.75 15.00 35.00slider -- 30.00 --
38
Figure 2-1. Revolute joint between bodies i and j. The joint imposes a constraint on the relative translation between the bodies.
Figure 2-2. Translational joint between body i and j. The joint imposes a constraint on the relative rotation between the bodies.
y
x
P
yi
xi
i
yj
xjj
ri rj
si sj
y
x
hi
Pi
ih
Pj
hi
lij
j
i
yi xi
yi xi
B
A
39
Figure 2-3. A slider-crank mechanism
Figure 2-4. Disassembled slider-crank mechanism
Connecting rod
Crank
Slider
GroundGround
y3
x3
x1 y1 y2
x2
y
x
y
x
P1
y1 x1
P2
P3
y2
x2
P4
y3
x3
P5
Crank
Connecting rod
Slider mass length crank m1 l1 connecting rod m2 l2 slider m3 l3
40
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.5
-1
-0.5
0
0.5
1
1.5
2
Time (sec)
Po
sitio
n (
m)
Position of Mass Center for the Crank
x1
y1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-3
-2
-1
0
1
2
3
4
5
6
Time (sec)
Ve
loci
ty (
m/s
)
Velocity of Mass Center for the Crank
dx1/dt
dy1/dt
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15
-10
-5
0
5
10
15
20
Time (sec)
Acc
ele
ratio
n (
m/s
2 )
Acceleration of Mass Center for the Crank
d2x1/dt2
d2y1/dt2
Figure 2-5. Results from the kinematic analysis. Position velocity and acceleration of the crank mass center.
41
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3
4
5
6
7
8
Time (sec)
Po
sitio
n (
m)
Position of Mass Center for the Slider
x1
y1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8
-6
-4
-2
0
2
4
6
8
10
12
14
Time (sec)
Ve
loci
ty (
m/s
)
Velocity of Mass Center for the Slider
dx1/dt
dy1/dt
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-40
-30
-20
-10
0
10
20
30
Time (sec)
Acc
ele
ratio
n (
m/s
2 )
Acceleration of Mass Center for the Slider
d2x1/dt2
d2y1/dt2
Figure 2-6. Results from the kinematic analysis Position velocity and acceleration of the slider mass center.
42
Figure 2-7. Direct integration procedure for the Differential Algebraic Equation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Time (sec)
2n
d C
on
stra
int
Constraint
=0 =0
=5 =5
=100 =100
Figure 2-8. Effect stabilization parameters ( and ) on the second kinematic constraint ( 2K ).
Initial conditions
0t , 0q , 0q
Assemble M , qΦ , AQ , γ
Solve linear DAE T A
q
q
M Φ q Q
Φ 0 λ γ
for q λ
Assemble vel. & acc. Vector
t
qg
q
Integrate for pos. and vel.
integratet t t
qg g
q
Time increment t t t
endt t
43
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.5
-1
-0.5
0
0.5
1
1.5
2
time (sec)
Po
sitio
n (
m)
Position of Mass Center for the Crank
x1
y1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-3
-2
-1
0
1
2
3
4
5
6
Time (sec)
Ve
loci
ty (
m/s
)
Velocity of Mass Center for the Crank
dx1/dt
dy1/dt
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15
-10
-5
0
5
10
15
20
time (sec)
Acc
ele
ratio
n (
m/s
2 )
Acceleration of Mass Center for the Crank
d2x1/dt2
d2y1/dt2
Figure 2-9. Results from the dynamic analysis. Position velocity and acceleration of the crank mass center.
0 1 2 3 4 5 6-1500
-1000
-500
0
500
1000
1500
2000
2500
Crank Position (rad)
Join
t Re
act
ion
Fo
rce
(N
)
Joint Reaction Force (crank and connecting rod)
Fx
Fy
Figure 2-10. Joint reaction force between the crank and the connecting rod.
44
CHAPTER 3 DYNAMICS OF MULTIBODY SYSTEMS WITH IMPERFECT REVOLUTE JOINTS
In this Chapter a discussion on how to model the dynamics of multibody systems with
imperfect revolute joints will be presented. In particular constraints will be developed to model
the imperfect joint.
Introduction
In the previous Chapter a discussion on the procedure to model multibody systems was
presented, based on kinematic constraint formulations for revolute and translational joints. The
formulations assume that clearances at these joints are nonexistent. However in real mechanical
systems, joint clearances do in fact exist. The clearances are due to manufacturing constraints as
well as the occurrence of wear. In the latter case the two components of the joint, are in contact
and in relative motion and as a result wear occurs so that the clearance size increases with time.
Joint clearances have been noted to affect the performance and service life of mechanical
systems. This may be attributed to the increased vibration, excessive wear and dynamic force
amplification as discussed by Dubowsky [32]. The clearances may also cause uncertainty in the
orientations and positions of mechanisms that are designed to have precise motions such as
manipulators [33]. Due to the significance of the problem, numerous studies have been
conducted with the goal of understanding the response of these systems in the presence of joint
clearances [1-11,13-18,32-41]. These studies have evolved from the analysis of less complex
planar multibody systems [1-11,13-18,32,33,35,36,38,40,41] to more complex spatial systems
[34,37,39] as well as from rigid multibody analysis [4-9,14,16-18,32,33,38,40] to flexible
multibody analysis [3,34,36,37,42]. The studies have demonstrated that the presence of
clearances alter the response of the system appreciably.
45
Figure 3-1 shows a diagram of a revolute joint with a joint clearance. The joint consists of
a pin and a bushing (also referred to as a journal and a bearing, respectively). The pin and the
bushing are attached to either of the two bodies that share the revolute joint. This joint differs
from the ideal joint, discussed in Chapter 2, in that the pin is free to move within the inner
perimeter of the bushing as dictated by the dynamics of the system. Thus the centers of the
components of the joint do not necessarily coincide. The ideal revolute joint was, however,
modeled by assuming that the centers of the joint components coincided at all times.
Furthermore, the ideal revolute joint was simply considered to be a single point (axis) about
which relative rotation between the two bodies occurs. The actual physical joint was not modeled
Two main approaches have been used to model the clearances in multibody systems. In
one approach, the components of the joint are assumed to be in contact at all times, that is, the
pin and the bushing are in continuous contact. Furuhashi et al [6-9] modeled the clearance as a
fixed distance between the two centers of the joint components. The fixed distance was
interpreted as a mass-less link that introduced an extra degree-of-freedom. This technique has
also been used by several other researchers [33, 38] to investigate the effect of joint clearance on
the motion and dynamics of mechanical systems.
In a different approach the components of the revolute joint are not assumed to be in
continuous contact. Rather, three different configurations of the joint components are considered.
In one of configurations, the pin is assumed to be in free flight [17]. This is when the pin is
completely detached from the bushing and moves freely within the bushing depending on the
system dynamics. In another configuration, the pin establishes contact with the bushing through
an impact motion, and in the final mode the pin moves while in contact with the bushing. The
latter configuration has been referred to as following motion [17]. This approach appears to be
46
the most realistic approach available. This latter approach will be used to model the non-ideal
joint and subsequently be integrated with a wear prediction procedure in order to determine the
wear evolution in a joint. In this work the later approach will be termed as the general imperfect
joint.
General Imperfect Revolute Joint
The general imperfect revolute joint closely models the actual joint in which the pin and
bushing can assume three different configurations; i.e., freefall, impact or the following motion.
In the real joint the motion of the pin is restricted within the bushing. Thus the pin moves freely
within the inner perimeter of bushing until contact is established. When contact/impact occurs,
both bodies deform locally and a reaction force is developed in the contact region. The reaction
force has the effect of restricting the pin within the bushing. This behavior is modeled in the
general imperfect joint model by imposing a force constraint. A force, determined by a contact
law, is applied every time contact is established between the two joint components. The effect of
the force is that the pin is prevented from leaving the bushing inner perimeter. In what follows a
discussion of the contact-impact force model will be presented thereafter followed by a
discussion on modeling the general imperfect joint.
Contact-Impact Force Model
A number of papers dealing with the subject including contact and impact of bodies in
multibody systems have been authored. Two different approaches with regard to the treatment of
this problem have been reported. These approaches are referred to as discontinuous and
continuous. In the discontinuous approach, discussed by Khulief et al [12], the collision is
assumed to be instantaneous. The dynamic analysis during this period is decomposed into two
parts namely; pre and post-collision. The analysis is conducted as usual until impact/contact is
detected. Momentum balance is then performed to determine the post-collision velocities. The
47
velocities in the analysis are then updated and the analysis is then resumed and performed as
usual. Due to the momentum balance (and velocity updates), discontinuities are observed in the
velocities. Although the method has been found to be efficient, the assumption of instantaneous
impact becomes invalid if the duration of contact in the collision is large. This would limit the
use of the method since the system configuration would have changed appreciably over that
duration [43].
In the continuous approach, a continuous force is used to describe the contact-impact force
that results from the interaction of the two bodies. During the period of contact (including the
impact), a force normal to the plane of contact is developed. The scenario is accounted for in the
dynamic analysis by including the contact forces in the Differential Algebraic Equations of
motion (as applied forces). The forces need to be updated at each time step, during the contact, to
reflect the changing system configuration. In order to apply this method, it is necessary to know
how the forces vary during the impact-contact period. Two main contact laws have been used in
determining these forces in multibody systems, which are the Kelvin-Voigt viscoelastic model
applied by Kulief et al [44] and a contact force model with hysteresis damping applied by
Lankarani et al [43], Ravn [13,15] and Flores et al [14,16-18].
The Kelvin-Voigt viscoelastic model is a linear contact force model in which a parallel
spring-damper assembly is used. The spring represents the elasticity of the joint components,
whereas the damper accounts for the energy dissipated in the process of collision. Energy
dissipation is simulated by multiplying the rebound force by a coefficient of restitution. The
model, as reported by Ravn [13], is expressed as,
Nr
K loadingF
K e unloading
(3-1)
48
where K , and re are the elastic stiffness, deformation/penetration and coefficient of
restitution, respectively. As an example, the contact force for this model is plotted in Figure 3-
1(b). The force is plotted for the penetration shown in Figure 3-2(a) with a stiffness value of
65 10 N/m. The plot of contact force verses the relative penetration is shown in Figure 3-2(c). It
can be seen that during the rebound (unloading) the contact force is less than that in the
compression. The dissipated kinetic energy is also shown as the enclosed area.
Although the Kelvin-Voigt model can provide continuous contact force as well as accounts
for energy dissipation during contact, it suffers from the difficulty of determining the elastic
stiffness K . In addition, the linear relationship between the penetration and the contact force is a
major simplification since the contact force would depend on other factors in addition to the
material such as the geometry of the bodies in contact.
The contact force model with hysteresis damping for multibody systems was first applied
by Lankarani et al [43]. The model is an extension of the Hertz contact theory for spherical
bodies in contact. It is based on the idea of dissipation of energy in terms of internal energy.
According to Lankarani et al [43], the continuous contact force can be decomposed into an
elastic and a damping term. The elastic term represents elasticity whereas the damping term is
concerned with the dissipated energy during collision. The model can be expressed as,
nNF K D (3-2)
where K , D , and are the stiffness coefficient, damping coefficient the relative penetration
and the relative penetration velocity. n is an exponent whose value is taken to be 1.5n . Unlike
the Kelvin-Voigt contact force model, the stiffness coefficient K is determined based on the
geometry and material properties of the bodies in contact. The stiffness expression for colliding
spheres i and j is given as,
49
1
24
3i j
i ji j
R RK
R Rh h
(3-3)
In Eq. (3-3) Ri and Rj are the radii of the spheres. The variables hi and hj are material properties
which are dependent on the Young’s modulus E and Poisson’s ratio and can be computed as,
21
, mm
m
h m i jE
(3-4)
Hunt et al [45] proposed an expression for the damping coefficient written as,
nD (3-5)
where is referred to as a hysteresis damping factor. Considering the kinetic energy of the
system before and after the collision, Lankarani et al [43] determined an expression for the
hysteresis damping factor. The expression which depends on the stiffness K , the initial
penetration velocity ( ) and a coefficient of restitution re is written as ,
23 1
4
rK e
(3-6)
The contact force model with hysteresis can then be written as,
2
1.53 1
14
r
N
eF K
(3-7)
To demonstrate the model, the contact force is plotted in Figure 3-3(b) of the penetration shown
in Figure 3-3(a) with a stiffness value of 65 10 N/m1.5. The plot of contact force verses the
relative penetration is shown in Figure 3-3c. Once again it can be seen that during the rebound
phase the contact force is less than that in the compression, indicating that energy has been lost.
The dissipated kinetic energy is shown as the enclosed area.
50
It should be noted that the contact force with hysteresis model (Eq. (3-7)) was derived for
contact of spherical bodies. However, the concerned in this work is the contact in a planar
revolute joint which corresponds to cylindrical joint components. Contact-Impact models for
cylindrical components have been proposed but these modes are quite difficult to use. For
instance Dubowsky [1], proposed a contact force model for cylindrical bodies that can be
expressed as,
3
= ln 1i ji j
N
n i j i j
R R aF
a F R R
(3-8)
where a is the length of the shorter cylinder, Ri and Rj are the radii of the cylinders, and other
term are as previously defined. It is clear from this model that determining the contact force NF
is complicated, requiring some iterative procedures such as the Newton-Rahpson method.
Despite the difficulty of determining the contact force using Eq. (3-8), it has been shown [17]
that there are no considerable gains in using the cylindrical model instead of the spherical contact
model (eq. (3-7)) with the cylinder radii used in place of the sphere radii. Furthermore the
cylindrical models do not account for dissipated energy. Thus in this work the contact model
with hysteresis damping is used to determine the continuous contact force in the revolute joint
during collision of the components (i.e. pin and bushing).
In addition to the contact force, the modeling of the imperfect joint can be enhanced by
considering the friction as a result of the contact between the joint components. In this work,
Coulomb friction model has been adopted. However, other models can also be used. The model,
whose line of action is perpendicular to that of the contact force, is expressed as
f k NF F . (3-9)
51
In this expression NF is the contact force determined by Eq. (3-7) and k is the coefficient of
kinetic friction and can be determined through experiments as described by Schmitz et al [46].
Modeling the General Imperfect Joint Model
A discussion of how to determine the contact forces between the joint components of the
general imperfect revolute joints was presented. In what follows, modeling of the imperfect joint
will be presented. The discussion here follows closely the work of Ravn [13] and Flores [17].
Consider a revolute joint with clearance c (general imperfect joint) as shown Figure 3-1.
The joint consist of two components namely; a pin and a bushing. For the sake of generality it is
assumed that the pin and the bushing are rigidly attached to bodies i and j. The two bodies,
referenced in the global coordinates, are shown in Figure 3-4. Body coordinates xi - yi and xj - yj
are fixed to the center of masses of the bodies A and E respectively. The body-fixed coordinates
are oriented at angles i and j relative to the global horizontal axis.
When contact between the two bodies is established, a contact force (described by Eq.
(3-7)) is generated for the entire contact period. Furthermore if there is relative motion between
the joint components, a kinetic friction (described by Eq.(3-9)) is experience. In Figure 3-4, the
point of contact C is defined as the center of contact region between the pin and the bushing.
This point can be located using the eccentric vector e which is a vector connecting the bushing
center D and the pin center B. At the time of contact the eccentric vector will point in the
direction of the contact. This vector is expressed as,
i i i j j je r A s r A s (3-10)
where ir and jr are vectors linking the global origin and the center or masses of the bodies. is
and js are vectors in the local coordinate system that link the center of masses to the pin and
52
bushing centers respectively. iA and jA are transformation matrices that transform a vector
from the local coordinate system to the global system. In this particular case they transformation
vectors is and js into their global equivalent. The location of contact point C with respect to the
pin and bushing can then be expressed as,
,Cm m m m mR m i j r r A s n . (3-11)
In Eq. (3-11) Rm are the pin and bushing radii and n is a unit vector in the direction of e and is
written as,
ee
en
e. (3-12)
The penetration between the pin and bushing during the contact is computed as the
different between the eccentric and the clearance as shown in Eq. (3-13). When the pin is not in
contact with the bushing the eccentric is smaller than the clearance, and the penetration has a
negative value. When the penetration has a value equal or greater than zero, contact is
established. Thus when is greater than zero a contact force is applied between the bodies. The
contact force vanishes when is equal to or less than zero. These configurations are depicted in
Figure 3-5.
e c (3-13)
In determining the contact force (using Eq. (3-7)) the relative penetration velocity is also
required. This will essentially be the difference between the velocities of the contact point. The
velocity of each contact point is obtained as,
, ,m
Cm m m m m mR m i j r r Α s n . (3-14)
53
The relative velocities in the normal and tangential direction can then be computes as shown in
(3-15). In Eq. (3-15), n is the unit tangent vector defined as ˆ n k n and k̂ is the unit vector
in the global z-coordinate. In Eq. (3-15), the normal velocity nv is also the relative penetration
velocity and is positive during the penetration period and negative during the rebound period.
C Cn i j
C Ct i j
v
v
r r n
r r n
. (3-15)
Once the relative penetration velocity has been computed, the contact and friction forces
can be computed and assembled into the DAE to determine the dynamics of the system. It is
noted that since the body-fixed coordinates were fixed at the center of masses of bodies i and j,
the forces must also be applied at these locations rather than at the points of contact. Thus the
transfer of the loads to the mass centers will result in an addition moment in each body. The
force and moment equation for body acting at the center of mass for body i is given as,
i N
Ci i i i
F F F
m F r r, (3-16)
where
N N
k N
F
F
F n
F n. (3-17)
The corresponding loads for body j are
j N
Cj j j j
F F F
m F r r. (3-18)
Once the forces (contact and friction forces) for the non-ideal revolute joint are known, the
description of the joint is complete. It should be noted that no kinematic constraint was
54
introduced while describing the non-ideal joint. Instead a force constraint was used in the
description. As a result a multibody system with this type of non-ideal revolute joint will have 2
additional degrees-of-freedom which are catered for by the force constraint. Thus a kinematic
analysis on a multibody system with a general imperfect joint (to determine the motion of the
system) is not possible. Instead the motion and the dynamics of the system must be determined
through the dynamic analysis. A demonstration of incorporating the general imperfect joint into a
multibody system for a dynamic analysis will be presented in the following section.
Example: Slider-Crank Mechanism with Joint Clearance Between the Crank and the Connecting Rod (General Imperfect Joint)
The use of the general imperfect joint model will be demonstrated by modeling a slider-crank
mechanism that has a non-ideal joint.
Figure 3-6 shows a diagram of the slider-crank mechanism which consists of four
components (ground, crank, connecting rod and slider). The components are connected to each
other by three revolute joints and a translational joint. For this example, the revolute joint
between the crank and the connecting rod is modeled as a general imperfect joint. The dimension
and mass properties for the mechanism are shown in Table 3-1. Also, the radii and material
properties for the joint components (the pin and bushing) are shown in Table 3-2. For the contact
force model, a value of 0.13 and 0.8 were used for the friction coefficient and the coefficient of
restitution, respectively. These are summarized in Table 3-3. It is assumed in this analysis that
the pin and the bushing are rigidly attached to the crank and the connecting rod, respectively.
The crank is assumed to rotate at a constant angular velocity of 30 rpm ( rad/sec).
The kinematic constraint equations for this mechanism can be obtained using the
procedures described in the previous Chapter. For this mechanism the constraint equations are as
shown in Eq. (3-19). In Eq. (3-19) the first two rows describe the revolute joint between the
55
crank and the ground, whereas the third and fourth describe the revolute joint between the
connecting rod and the slider. The fifth and sixth rows model the translational motion of the
slider. The last row is the driving constraint that specifies the constant angular velocity of the
crank.
1 1 1
1 1 1
2 3 2 2
2 3 2 2
3
3
1
cos 0
sin 0
cos 0
sin 0
0
0
0
x l
y l
x x l
y y l
y
t
Φ . (3-19)
It is noted that no constraint for the joint with clearance appears in Eq.(3-19). Instead a
force constraint as previously described is used to restrict the motions of the crank relative to the
connecting rod. The differential algebraic equations of motion (Eq. (2-17)) repeated here as
22
AT
t
q
q q
QM Φ q
Φ 0 λ γ Φ q Φ Φ
’ (3-20)
can be assembled and solved (using techniques discussed in Chapter 2) to determined the
dynamics of the system. A value of 250 is used for both stabilization parameters and .
Figure 3-7 shows representative results from the dynamic analysis of the mechanism. In
the figure a comparison of the joint reaction force between the ideal and the non-ideal joints is
shown for four clearance sizes. In the first diagram the clearance size is 0.5 m. For this case the
two plots almost overlay each other as would be expected since the clearance is effectively zero.
It should be mentioned that a value of zero for the clearance would lead to numerical difficulties
since the eccentric vector will assume a value of zero (see Eq. (3-12)). As the clearance size is
increased to 5mm, the change in the system dynamics becomes quite evident. The curve of the
56
reaction force is seen to evolve from a smooth one to one characterized by peak forces that
increase in size with the clearance. In addition, the general magnitude of the force is also seen to
be higher for the case of the non-ideal joint.
The location of the peak contact force can be explained by noting that they occur right after
the minimum contact force (reaction force) between the joint components. This minimum
corresponds to the slider-crank configuration when contact between the components is also
minimal. Thereafter the contact and thus the reaction force between the pin and the bushing
increases rapidly. For the case of large clearance the pin has more freedom to move within the
bushing and will therefore develop larger contact force peaks as the contact increases. It should
be mentioned that the slider-crank configuration when there is minimal contact between the joint
components is also a candidate for contact loss between the components. This is especially true
when the clearance size is large. The last diagram in Figure 3-7 shows the contact force between
the pin and the bushing when the clearance is quite large (5mm). The diagram shows that contact
between the two components is briefly lost when the slider-crank assumes the concerned
configuration. Since the clearance is larger, the peak contact force that results after contact is
reestablished is also higher than was previously observed.
Summary and Discussion
In this Chapter, a non-ideal revolute joint model that closely represents a real joint was
presented. The model, termed as the general imperfect joint, assumes that the joint components
(pin and bushing) locally deform/penetrate when they are in contact. The contact between the
joint components generates a contact force that is determined by a modified form of the Hertz
contact law. This force prevents further penetration and restricts the motion of the pin within the
inner perimeter of the bushing. Thus the imperfect joint is modeled using force constraints and
not the kinematic constraint that were used in the previous Chapter.
57
The developed joint model was used to model a non-ideal joint in a slider-crank
mechanism. In the mechanism, a clearance was present in the revolute joint between the crank
and the connecting rod. A dynamic analysis for this mechanism showed that the dynamics of the
system is altered when the clearance is present at the joint. As the clearance is increased, force
peaks are observed in addition to an increase in the force magnitude. These observations can be
used to infer that wear at the joints also alters the system dynamics in a similar manner.
58
Table 3-1. Dimension and mass parameter for slider-crank mechanism Length (mm) Mass (g) Moment of inertia (kg-m2) Crank 1.00 10.00 45.00Connecting Rod 1.75 15.00 35.00Slider -- 30.00 --
Table 3-2. Material properties for the joint components Pin Bushing Young’s modulus 0.29 0.38Poisson ratio 206.8 GPa 0.5 GPaRadius 20 mm 20.0005, 20.05, 20.5, 25 mm
Table 3-3. Parameters for the force model Parameter Value Friction coefficient (pin & Bushing) 0.13Coefficient of restitution ( re ) 0.8
59
Figure 3-1. A revolute joint with clearance
0.331 0.4330
0.1155
Time
m
m
0.331 0.4330
600
Time
For
ce N
0 0.1155
600
mmF
orce
N
Figure 3-2. Kelvin-Voigt viscoelastic model. a) Relative penetration; b) Contact force during contact; c) Force verses the relative penetration during contact.
0.331 0.4330
0.1155
Time
m
m
0.331 0.4330
600
Time
For
ce N
0 0.1155
600
mm
For
ce N
Figure 3-3. Contact-force model with hysteresis damping (by Lankarani & Nikravesh). a) Relative penetration; b) Contact force during contact; c) Force verses the relative penetration during contact.
Clearance - c
Bushing
Pin
(a) (c) (b)
(a) (c) (b)
60
Figure 3-4. General imperfect joint.
Figure 3-5. Penetration during contact between the pin and the bushing.
0e c
c
0e c
0e c
ePin
Bushing e
O
y
ri
Si
Sj
rj
A
B
E
C
D
j
i
yi xi
yj
xj
x
e
y1 x1
y2
x2 Body-iBody-j
61
Figure 3-6. Slider-crank mechanisms with joint clearance.
8 10 12 14 16 180
1000
2000
3000
4000
5000Reaction Force (clearance = 0.0005mm)
Crack Angle (rad)
Fo
rce
(N
)
Non-Ideal JointIdeal Joint
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 0.05mm)
Non-Ideal JointIdeal Joint
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 0.5mm)
Non-Ideal JointIdeal Joint
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 5mm)
Non-Ideal JointIdeal Joint
Figure 3-7. Comparison of reaction between the ideal and the non-ideal joints for various joint clearance.
Crank
Connecting rod
Slider
Joint clearance
GroundGround
y3
x3
x1 y1 y2
x2
y
x
62
CHAPTER 4 WEAR-PREDICTION METHODOLOGY
The procedure to predict wear occurring at the interface of bodies in contact and in relative
motion will be presented in this Chapter. Emphasis is placed on ways to reduce computational
costs while ensuring accuracy and stability in the predictions.
Introduction
When two bodies are in contact and are in relative motion, with respect to each other, wear
is expected to develop on the regions of contact [47]. Similarly, in the case of a revolute joint
wear is expected to occur since the two components of the joint (pin and bushing) are in contact
and revolve relative to each other. Intuitively, it is safe to claim that the amount of wear at such a
joint is affected by the type of material (which the joint components are made of), the relative
sliding distance between the two components and the operating conditions. Here, the operating
conditions refers to the amount of reaction force developed at the joint and the condition of the
joint which could be dry, lubricated, or contaminated with impurities.
An enormous amount of effort has been placed into developing techniques to predict wear.
A general trend that has emerged is the use of Archard’s wear model in an iterative procedure.
Archard’s model is a linear wear model that estimates wear based on information about the
contact condition (contact pressure and sliding distance) and Tribological data that reflects
information about the materials in contact and the operating conditions. Thus, wear procedures
typically involve an iterative process in which incremental wear is estimated at each iteration
(based on the wear model) and accumulated up to the desired number of iterations.
In earlier prediction procedures, the linear model was employed to estimate worn geometry
based on initial contact conditions. The procedures assumed that the geometry and thus contact
pressure did not evolve with wear. Thus only a single iteration was required in which linear
63
extrapolations were applied to determine the final geometry i.e. the geometry that would result
after several thousand iterations. This procedure has been found to produce erroneous results
[48,49]. The primary reason for the inaccuracy is that the evolution of the geometry was
neglected.
In later procedures, wear predictions have been based on evolving contact conditions. The
procedures allow the contact geometry to vary gradual and thus resulting in an iterative
procedure in which the contact pressure and the sliding distance are computed at each iteration.
The geometry is also updated at each iteration to reflect the worn material and thus geometry
evolution. Depending on the effectiveness of the geometry update and the accuracy of the
contact pressure calculations, the iterative procedures have been found to yield useful results [50-
55]. This has allowed the procedure to be used in various application such as in the prediction of
gear wear [56-58], cam-follower wear [59-61], knee joint wear [62], and hip joint wear [63,64].
Despite the reported success, the iterative procedure has been found to be quite
computationally expensive. This is primarily due to the iterative process that is required to
capture the evolving geometry. Several ideas have been implemented in an attempt to reduce
computational costs associated with the wear simulation process. Põdra et al. [52] attempted to
minimize the computational cost by using the Winkler or elastic foundation model to determine
the contact pressure distribution. The Winkler model was used as an alternative to the more
expensive but relatively accurate FEM. Although the method was found to be less expensive it
can be argued that the benefit of using the more accurate results from the finite element
technique outweigh the gains in computational efficiency when complicated geometries are
considered. Põdra et al. [52] also employed a scaling approach to tackle the problem of
computational costs. In this approach the incremental wear at any particular cycle of the
64
simulation was scaled based on a predefined maximum allowable wear increment. The scaling
factor was obtained as a ratio between the maximum allowable wear increment and the current
maximum wear increment (maximum wear increment of entire geometry). They found that this
procedure was more computationally effective. Kim et al. [50] used a constant extrapolation
technique to reduce the computational costs for the wear problem. In their technique one finite
element analysis was made to represent a number of wear cycles. Through this procedure, they
were able to reduce the total number of analyses needed to estimate the final wear profile. A
similar procedure was employed by McColl et al. [65] as well as Dickrell et al. [66]. In another
paper [67], the computational costs of simulating a pin on a rotating disc was reduced by
approximating the state of strain on the center of the wear track as plain strain. A less costly two-
dimensional idealization was then used in place of the more expensive three-dimensional
problem.
In what follows the widely used iterative wear prediction procedure based on the Archard’s
wear model will be presented. The procedure is then enhanced using an adaptive extrapolation
scheme so as to reduce computational costs. The adaptive extrapolation scheme is an
improvement to the constant extrapolation scheme that was used by Kim et al. [50]. A pin-pivot
joint (a type of revolute joint), shown in Figure 4-1, will be used to demonstrate the procedure.
Wear predictions on the pin-pivot assembly will be conducted and validated with experiments.
Wear Model
As was mentioned the Archard’s wear law [68] forms the basis for the wear-prediction
methodology. In that model, first published by Holm [69], the worn out volume, during the
process of wear, is considered to be proportional to the normal load. The model is express
mathematically as follows:
65
NFVK
s H (4-1)
where V is the volume lost, s the relative sliding distance between the two bodies in contact, K
is the dimensionless wear coefficient, H is the Brinell hardness of the softer material, and NF is
a normal force. Since the wear depth is the quantity of interest, as opposed to the volume lost,
Eq. (4-1) is usually written in the following form:
N
hAkF
s (4-2)
where h is the wear depth and A is the contact area such that the volume becomes V hA . The
non-dimensioned wear coefficient K and the hardness are bundled up into a single dimensioned
wear coefficient k (Pa-1). It should be noted that the wear coefficient k is not an intrinsic
material property but is also dependent on the operating condition. Equation (4-2) can further be
simplified by noting that the contact pressure may be expressed with the relation Np F A so
that the wear model is expressed as
h
kps (4-3)
The wear process is generally considered to be a dynamic process (rate of change of the
wear depth with respect to sliding distance) so that the first order differential form of Eq. (4-3)
can be expressed as:
( )dh
kp sds
(4-4)
where the sliding distance is considered as the time in the dynamic process, and the contact
pressure is a function of the sliding distance.
A numerical solution for the wear depth may be obtained by estimating the derivative in
Eq. (4-4) with a finite divide difference to yield the depth as follows:
66
1j j j jh h kp s (4-5)
In Eq. (4-5), jh refers to the wear depth at the thj iteration while 1jh represents the wear depth
at the previous iteration. The last term of Eq. (4-5) is the incremental wear depth which is a
function of the contact pressure and incremental sliding distance ( js ) at the corresponding
iteration.
If information about the wear coefficient k , the contact pressure jp and the sliding
distance js is available at all iterations ( j ), the wear depth on a contact interface for a specified
sliding distance s can be estimated using Eq. (4-5). Here the sliding distance is an accumulation
of the incremental sliding distance for all iterations ( _n iter ) and is expressed as
_
1
n iter
jj
s s
. (4-6)
The contact pressure ( p ) may be obtained using numerical methods. Various methods
have been employed in computing the contact pressure and sliding distance. These include the
finite element method (FEM) [50,51,53,54,65,67,70-72], boundary element method [73],
Winkler model [52,56,74], Hertz contact model [57,58] each having its pros and cons with
regard to accuracy and computational expense. In this work the FEM is used.
The value of k for a specific operating condition and given pair of materials may be
obtained by experiments [50, 75, 76]. It is worth noting that measured values of wear coefficients
usually have large scatter and may affect wear predictions significantly. Care should thus be
taken in obtaining these values. Uncertainty analysis for measured values of wear coefficients,
such as those presented by Schmitz et al. [75], may be of considerable benefit.
67
Since all the necessary components of Eq. (4-5) can be determined, the simulations to
predict wear can be conducted. A discussion on the wear simulation procedure will be presented
as follow.
Wear Simulation Procedure
A number of papers that demonstrate the implementation of Eq. (4-5) in predicting wear,
have been published [50-52,54,56,65,77]. Although the details of the various procedures differ,
three main steps are common to all of them. These include the following:
Computation of the contact pressure resulting from the contact of bodies.
Determination of the incremental wear amount based on the wear model.
Update of geometry to reflect the wear amount and to provide the new geometry for the next iteration.
A brief discussion of these steps will follow. To facilitate the discussion, a pin-pivot joint that
experiences wear at the contact interface will be used. The pin-pivot assembly is shown in Figure
4-1. It consists of a pin that oscillates inside a pivot. The objective would thus be to determine
the amount of wear that occurs on both the pin and the pivot after several thousand pin
oscillations.
Computation of Contact Pressure
The contact pressure, as previously mentioned, can be determined in a variety of ways. In
this work, however, commercial finite element software, ANSYS, which is well adopted for this
type of problem, is used.
Figure 4-1 shows the diagram of the pin-pivot assembly. The corresponding finite element
model (a discretization of the pin-pivot domain) is shown in Figure 4-2. Eight-node quadrilateral
elements are used to model the pin and the pivot whereas three-node contact elements are used to
represent the contact surface. The contact elements are generated on the outer and inner surfaces
68
of the pin and pivot, respectively, and provide information about the contact between the two
bodies. The use of the eight-node quadrilateral and three-node contact elements is not a
limitation; other elements may also be used depending on the application and accuracy desired.
Since the pin-pivot model is discretized, the contact pressure on the contact surface (both
pin and bushing contact surface) can be obtained at discrete locations, that is, at the nodal
locations (or integration points). Details of how to perform the contact analysis can be found in
the ANSYS user’s manual.
Determining the Wear
In addition to the spatial discretization, the motion of the pin is also discretized in order to
specify the incremental sliding distance ( s ) (temporal discretization). A pin cycle which
involves oscillation of the pin from one extreme to the other is discretized into several steps. The
discretization is such that each step corresponds to a specific pin angle between the two extremes
from which the incremental sliding distance can be calculated.
At each step a finite element analysis is performed to determine the contact pressure over
the contact region. The wear depth during any cycle and at any location on the contact surface
can then be determined by Eq. (4-7) which is a modification of Eq.(4-5) to include the
discretization of the cycles.
, , , 1, ,n i j n i j i n ih h kp s . (4-7)
In Eq.(4-7), n refers to the contact surface node numbers which may or may not establish
contact with the opposing surface. The subscript i and j indicate the current step and cycle,
respectively. All other terms are as defined previously.
The simulation procedure will involve determining the incremental wear depth ( ,i n ikp s ) at
a particular step for the entire contact region. The geometry is then updated to reflect the wear
69
amount. The pin is then incrementally rotated to the next step and the corresponding incremental
wear depth is computed. The geometry is once again updated. This is repeated until the cycle is
completed. The next cycle iteration is started and the procedure repeated to the desired number
of cycles. The simulation procedure is summarized in a flow chart shown in Figure 4-3.
Geometry Update Procedure
The process of geometry update is necessary in order to account for material loss through
the process of wear. Indeed material removal changes the contact surface and causes a
redistribution of the contact pressure. These changes can only be captured if the surface is altered
through a geometry update. The geometry update procedure involves two steps. These steps are
outlined below:
Determination of the normal direction (vector) of the contact surface at the location of each surface node (contact node).
Shift the position of the surface nodes in the direction of the normal vector by an amount equal to the wear increment for each iteration.
The contact surface is made up of a collection of contact elements. In the present case each
contact element has three contact nodes. One such element is shown in Figure 4-4. Any location
on the contact surface can be determined by an interpolating its coordinates from the surrounding
nodal locations (surrounding contact nodes). Thus the contact surface can be expressed using
nodal locations and a set of interpolation functions known as the shape functions. This
information can be exploited to derive the expression for the normal direction of the contact
surface at each node.
The shape functions for the contact element shown in Figure 4-4 may be written as:
70
1
2
3
11
21 1
11
2
N t t
N t t
N t t
(4-8)
where t is the local coordinate parameter. The surface of an element can then be described in
terms of the nodal coordinates and as a function of the local coordinate. The expression for the
surface is given in Eq. (4-9).
1
1
1 2 3 2
1 2 3 2
3
3
0 0 0
0 0 0
x
y
N N N xx
N N N yy
x
y
(4-9)
where kx and ky are the coordinates of node k ( 1, 2,3k )for the element of interest. Thus a
value of 1t will supply the location of the first node 1 1( , )x y and a value of 1t will supply
the location of the third node 3 3( , )x y whereas the second node is determined by 0t . Thus if
the nodal locations of the contact nodes are known, any location of the contact surface is
completely defined by expression in Eq. (4-9).
If the vector tangent to the surface (contact element surface) is denoted as tv then its value
for the element can be obtained as follows:
0x y
t t
tv i j k (4-10)
where the partial differentials are given in Eq. (4-11) or (4-12).
71
1
131 2
2
231 2
3
3
0 0 0
0 0 0
x
yNN Nxxt t t tyy NN N
t t t t x
y
(4-11)
or
3
1
3
1
rr
r
rr
r
Nxx
t t
Nyy
t t
(4-12)
The vector normal ( nv ) to the surface can be expressed as a cross product of the tangent
vector ( tv ) and the vector perpendicular to the plane of the surface ( 0,0,1pv ). This cross
product is expressed in Eq.(4-13) where n denotes the contact node number.
,,
,
nn
n
t p
n
t p
v vv
v v (4-13)
The resulting unit normal vector then appears as follows:
, 2 2n
y xt t
x yt t
n
i jv (4-14)
or,
, _ , _ ,n norm x n norm y nv v nv i j (4-15)
where _norm xv and _norm yv are the components of the vector normal to the surface.
Once the contact pressure distribution and normal vectors at all the nodes on the surface
have been determined, the geometry can be updated. The update is done by moving the surface
72
nodes in the direction of the unit normal vector. The coordinate of the new node position at any
step of any cycle can be written as follows:
, , , 1, _ ,
,, , , 1, _ ,
n i j n i j norm x n
n in i j n i j norm x n
x x vkp s
y y v
(4-16)
The process of the geometry update is shown in Figure 4-5. In this diagram the wear depth
is grossly exaggerated to illustrate the concept. The procedure for the geometry update has been
used successfully in the wear-simulation process.
Boundary Displacement
The update procedure discussed in the previous section involved moving only the nodes at
the boundary to reflect the wear. If the amount of wear is large, the elements associated with the
boundary nodes will become distorted and the results from the finite element analysis will
become questionable. It is thus necessary to implement some strategy that will prevent this
distortion. On possible solution is applying a procedure, termed as boundary displacement. The
procedure, borrowed from structural shape optimization [78], involves repositioning the internal
nodes in addition to the boundary nodes. This ensures that the mesh regularity is maintained.
Figure 4-6 illustrates how the internal node is repositioned in order to reflect the wear.
The process of repositioning the internal nodes in this approach can be broken down into
three steps. The steps are outlined as follows:
1. Perform wear analysis and geometry update as has been discussed in the two previous sections. This is done until the amount of wear is equal to or exceeds a specified value. The wear analysis is then halted and the boundary displacement needs to be performed.
2. Using the initial geometry (before wear), perform a static finite element analysis using the accumulated wear depth as the displacement boundary condition.
3. Save the displacement of all the nodes. Using this configuration of nodes as the updated geometry, proceed with the wear analysis (in step 1).
73
This procedure has previously been used in wear analysis [50] and has been found prevent mesh
distortion. In addition it allows for fine mesh to be used, allowing for more accurate finite
element results.
Reducing Computational Costs
The procedure discussed in the previous section provides a way to simulate the wear
resulting from bodies in contact and in relative motion. However, the process can be quite
expensive. For instance, if one desires to simulate 100,000 cycles for a case in which each cycle
is discretized into 10 steps then 1,000,000 finite element analyses (nonlinear) as well as
geometry updates would be required. Clearly this may not be practical and the need for
techniques to reduce the computational cost becomes immediately apparent. One procedure that
has previously [50,65,66] been employed to reduce the computational costs is the use constant
extrapolations. Extrapolations have been used in various forms with the goal of reducing
computational costs. In this work an extrapolation factor ( A ) is used to project the wear depth at
a particular cycle to that of several hundreds of cycles. Essentially, the extrapolation is the total
number of cycles for which extrapolation is desired. Thus according to this definition, the
extrapolation factor can only take on positive integers values.
Equation (4-7), repeated here as
, , , 1, ,n i j n i j i n ih h kp s , (4-17)
can be modified slightly in order to incorporate an extrapolation factor. It is first noted that the
first term on the right hand side (R.H.S.) of Eq. (4-17) refers to the cumulative wear depth from
previous cycles whereas the last term refers to the incremental wear depth at the current step and
cycle. As a way to minimize computational costs, it is assumed that the next “ A ” cycles (as
many cycles as the value of the extrapolation) will have the same amount of wear depth as that of
74
the current step and cycle. The total incremental wear depth for those many cycles may then be
obtained by multiplying the last term of Eq. (4-17) with the extrapolation factor. The resulting
expression is shown in the following equation:
, , , 1, ,n i j A n i j i n ih h kAp s (4-18)
Utilizing the same concept, a new expression can be written to describe the position of the
contact nodes for the geometry update process. This expression is as follows:
, , , 1, _ ,
,, , , 1, _ ,
n i j A n i j norm x n
n in i j A n i j norm x n
x x vkAp s
y y v
(4-19)
As may be expected, the level of accuracy of the wear simulation is reduced when
extrapolations are used. This is directly related to the assumption that the same value of
incremental wear depth is maintained for several cycles. In reality the geometry would
continuously evolve resulting in a continuous redistribution of the contact pressure. The evolving
contact pressure would thus dictate a continuous change in the incremental wear depth at each
cycle. Intuitively if the extrapolation sizes are small the corresponding inaccuracy will also be
small.
Use of extrapolations may also cause problems in simulation stability. Here stability is
defined with regard to the contact pressure distribution and hence the wear profile. An ideally
stable wear simulation would be defined as one in which the contact pressure distribution
remained smooth (with no sharp or sudden changes in the distribution) for the entire duration of
the simulation. It is however unlikely to have smooth pressure distribution throughout the
simulation process. As a result a more relaxed definition of stability is adopted where by sudden
changes in the pressure distribution are allowed to occur. In Figure 4-10a, the contact pressure in
the contact interface of the pin-pivot assembly is seen to vary smoothly over the contact region
75
except for small peaks at the contact edges. The peaks are attributed to the transition from a
region of contact to a region of no contact. This transition occurs at a point which can not be
represented by a discrete model. The result is that there is an abrupt change in the surface
curvature which causes high contact pressure. If such contact pressure distribution is maintained
through out the simulation, the simulation can be referred to as stable. On the contrary, the
diagram in Figure 4-10b is representative of contact pressure distribution that would constitute
an unstable wear simulation.
When very large extrapolation sizes are used, wavy pressure distributions (Figure 4-10b)
are observed and the simulation becomes unstable. The shift to instability due to the use of large
extrapolation sizes can be explained as follows. The contact pressure distribution (obtained from
the finite element analysis) is generally not perfectly smooth (e.g. the pressure peaks at the
contact edges). This may be due to the discretization error stemming form the finite element
analysis. The use of an extrapolation factor magnifies these imperfections so that when the
geometry is updated the contact surface smoothness is reduced. If large extrapolation sizes are
used, the regions that experience high contact pressure in a particular step of the simulation are
worn out excessively so that in the following step these regions experience little or no contact.
On the other hand, the regions that did not experience high contact pressure will be worn out less
and thus will experience greater contact pressure in the next step. This behavior will repeat in
subsequent steps causing the surface to become increasingly unsmooth. The simulation will then
become unstable. If, however, smaller extrapolation sizes are used the wearing process acts as an
optimizer to smoothen the surface but computational costs increase. This raises the question as to
whether an optimum extrapolation size exists. In an effort to determine an optimum extrapolation
76
size, the adaptive extrapolation scheme was developed. This scheme will be discussed in the
following section.
Adaptive Extrapolation Scheme
The adaptive extrapolation technique is an idea proposed as an alternative to the constant
extrapolation scheme. The idea behind it is to seek for the largest extrapolation size while
maintaining a state of stability (smooth pressure distribution) throughout the simulation process.
The scheme is a three-step process. The first part of the scheme involves the selection of an
initial extrapolation size ( 0A ). The selection is based on experience, however it has been
observed that as longs as the initial value is not too large the wear prediction is not affected. This
is because the adaptive scheme will automatically adjust to the appropriate value. In the second
part of the adaptive extrapolation scheme, a stability check is performed. The stability check
involves monitoring the contact pressure distribution within an element for all elements on the
contact surface. This essentially translates to monitoring the local pressure variation. If the
contact pressure difference within an element is found to exceed a stated critical pressure
difference critp then a state of instability is noted and vice versa. In the final step of the adaptive
scheme, the extrapolation size is altered based on the result of the stability check. That is, the
extrapolation size is increased for the stable case and decreased for the unstable case. This
process can be summarized as follows:
1 inc ele crit
1 dec ele crit
if
if j
jj
A A p pA
A A p p
(4-20)
It must be mentioned that in order to maintain consistency in the geometry update as well
as in the ‘bookkeeping’ of the number of simulated cycles, a single extrapolation size must be
maintained through out a cycle. That is, every step in the discretized cycle will have the same
77
extrapolation size while different cycles may have different extrapolation sizes. Figure 4-8 shows
a graph of the extrapolation history for the oscillating pin-pivot assembly. From the graph, it can
be seen that the extrapolation took on a conservative initial value of about 3900 and increased
steadily up to the 12th cycle (actual computer cycles not considering the extrapolations).
Thereafter the extrapolation size oscillated about a mean of 6000.
Experimental Validation
Probably the most convincing way to validate the results of a simulation is to compare
them against those from an actual experiment. In this work the simulation procedure is validated
by comparing simulation results to results from a wear tests. The wear test consisted of a fixed
steel pin inside an un-lubricated oscillating steel pivot as shown in Figure 4-1.
The pivot was set to oscillate with amplitude of 30 and was loaded in the direction of its
shoulder. The resulting pressure at the cross-sectional of the pivot was 60MPa. The load was
kept approximately constant throughout the test. A total number of 408,000 cycles were
completed during the test to yield a maximum wear depth of about 2mm. It should be noted, for
the sake of comparison, that the definition of the test cycles is different from that of the
simulation cycles. Here a test cycle is defined as a complete rotation from one extreme to the
other and then back to the starting position (in this case -30 to 30 and back to -30) where as in the
simulation a cycle is considered to be a rotation from one extreme to the other(i.e. case -30 to 30).
The test information is summarized in Table 4-1.
Simulation test were performed with the pin oscillation amplitude and pivot loading
identical to that of the actual wear test. The simulation test was run for 100,000 cycles
(considering the extrapolation). A wear coefficient of 51.0 10 mm3/Nm (typical on un-
lubricated steel on steel contact) was used. This value is obtained from pin-on-disk tests results
78
reported by Kim et al. [50].The simulation test parameters are summarized in Table 4-2. In
Figure 4-9, the history of wear for the pin and pivot nodes that experienced the most wear is
shown. From the figure, a transient and steady state wear regime can be identified as discussed
by Yang et al. [67]. The transient wear regime corresponds to the beginning of the simulation. As
the simulation progresses the geometry and the pressure distribution evolve and the wear
transitions to a steady state regime. The evolution of the contact pressure is depicted in Error!
Reference source not found. where contact pressures of nine intermediate cycles are plotted.
Within the steady state wear regime, the wear is approximately linear with respect to the
cycles as can be seen in Figure 4-9. This information may be exploited to approximate the wear
after 408,000 cycles (since the analysis was only done to 100,000 cycles). Through linear
extrapolation, a value of 1.867mm was predicted as the maximum wear depth on the pin.
Although this value underestimates the wear depth it provides useful information about the wear.
The variation of the extrapolation size is depicted in Figure 4-11.
Summary and Discussion
In this Chapter a methodology to predict wear in a revolute joint was presented. The
methodology is built upon a widely used iterative wear prediction procedure that is based on the
linear Archard’s wear model. In the procedure, incremental wear is estimated base on the contact
pressure resulting from the contact of the joint components, the incremental sliding distance and
the wear coefficient (which reflects both the material which the joint components are made of
and the operating condition). The geometry is then updated so as to reflect the evolution of the
surface and thus to account for changing contact conditions. This process is then iterated and the
incremental wear is accumulated up to the desired number cycles.
To ensure that the computational costs are reduced, extrapolations are used. In the past
extrapolations have been used to project the wear depth at a particular cycle to that of several
79
hundreds of cycles and thus reducing computational costs. The difficulty associated with this
approach is in determining an appropriate extrapolation size. This is because large extrapolation
sizes would cause the geometry to distort after several iterations while small iterations would
result in suboptimal use of resources. One of the contributions in this work was the development
of the adaptive extrapolation scheme, which optimizes the selection of the extrapolation size.
The adaptive scheme was developed to be used with the FEM. In the scheme, the extrapolation
size is continuously increased until the geometry begins to distort (or more specifically the mesh
begins to distort). Distortion is monitored locally by comparing the pressure difference within an
element, for all the surface elements in the contact region, against a predefined critical. When the
element pressure difference exceeds the predefined critical, distortion is noted and the
extrapolation size is reduced.
A validation of the wear procedure was also conducted. The validation is done by
comparing the results from the simulation to that of an experimental counterpart. The wear
occurring at the contact interface of pin-pivot assembly was simulated. The predicted wear depth
deviated from the actual experimental wear depth by approximately 7%. Even though this
deviation appears to be large the predicted results is able to give a good insight into the wear
occurring at the interface. Indeed like any other approximation technique, errors are inherent. A
number of factors contribute to this discrepancy including the wear model, which is not an exact
representation of wear and the finite element analysis, which is an approximation technique.
Another contributor is the wear coefficient. The wear coefficient is obtained experimentally and
as was mentioned has a large scatter. Errors in the wear coefficient considerably affect the results
of the simulation. For instance, using a wear coefficient of 51.2 10 mm3/Nm resulted in
predicted wear depth of 2.028mm. The new wear coefficient, which is still within the range of
80
scatter according to Kim et al. [50], has a deviation of about 1.4% from the experimental value.
This is indeed a large improvement from the previous predictions. It is thus concluded that even
though the procedure does not accurately predict the wear the results obtained are of the correct
order of magnitude and can be used for preliminary design.
81
Table 4-1. Wear test information for the pin-pivot assembly
Test Parameters Values
Oscillation amplitude ±30
Load (cross-sectional pressure) 60MPa
Test condition Un-lubricated steel on steel
Total cycles 408,000
Max wear depth on pin ~2.00mm
Table 4-2. Simulation parameters for the pin-pivot simulation test
Simulation Parameters Value
Oscillation amplitude ±30
Load (cross-sectional pressure) 60MPa
Wear coefficient ( k ) 51.0 10 mm3/Nm
Total cycles 100,000
Steps per cycle 10
Table 4-3. Comparison of results form the simulation and Expt. wear tests
Max. wear depth (pin) (mm)
Simulation time (min.)
Actual test 2.000 --
Step update 1.867 206
82
Figure 4-1. Pin-pivot assembly.
Figure 4-2. Pin-pivot finite element model.
83
Figure 4-3. Wear simulation flow chart for the ‘step update’ procedure.
Figure 4-4. A three-node contact element used to represent the contact surface.
Input Model
Solve Contact problem (Contact Pressure)
Application of Wear Rule 1) Determine wear amount 2) Determine new surface
loc.
Update Model
Total Cycles?
End of Simulation
Cycle Count
Step count
Total Steps/cycle
?
84
Figure 4-5. Geometry updates process.
Figure 4-6. Geometry updates process using the boundary displacement approach.
Figure 4-7. Contact pressure distribution on a pin-pivot assembly. a) The case of a stable wear simulation. b) The case of an unstable wear simulation.
ba
Initial Geometry Updated Geometry
Old surface
New surface
85
0 10 20 30 400
1000
2000
3000
4000
5000
6000
7000
Cycles
Ext
rapo
latio
nExtrapolation Vs Cycles
Figure 4-8. Extrapolation history for a pin-pivot assembly.
0 25 50 75 1000
0.06
0.12
0.18
0.24
Cycles x 1000
Wea
r D
epth
(m
m)
Wear on Pin & Pivot
PinPivot
Figure 4-9. Cumulative maximum wear on pin and pivot.
86
Figure 4-10. Evolution of the contact pressure for nine intermediate cycles.
87
0 5 10 15 200
1000
2000
3000
4000
5000
6000
7000
Cycles
Ext
rapo
latio
nExtrapolation Vs Cycles
Figure 4-11. Extrapolation history plot for the step updating simulation procedure.
88
CHAPTER 5 INTEGRATED MODEL: SYSTEM DYNAMICS AND WEAR PREDICTION
The analysis of multibody systems with joint wear will be presented in this Chapter. As
before, the analysis will be restricted to rigid multibody systems and only wear in the revolute
joints will be considered.
Introduction
The procedure for analyzing multibody systems with joint clearance was presented in
Chapter 3. The procedure enables information about the non-ideal joint to be extracted. This
information includes contact location, incremental sliding distance and joint reaction forces
which are critical for wear prediction. Furthermore the procedure is able to capture any changes
in the system dynamics due to changes in the joint dimensions. A wear prediction procedure to
estimate the wear occurring in a revolute joint was also presented in Chapter 4. The wear
prediction procedure can be integrated into the dynamic analysis of a multibody system in order
to gain insight into the overall performance of a system when wear is present at one or more of
its revolute joints. The integrated model is composed of two parts namely; dynamic analysis and
wear analysis. The model is discussed in the following subsections.
Dynamic Analysis
In the first part of the integration process a dynamic analysis is performed to determine the
joint reaction force and the incremental sliding distance. These are the two quantities required
(from the dynamic analysis) to perform the wear analysis. The analysis is done for a complete
cycle and the reaction force as well as the incremental sliding distance is obtained at each time
increment of the discretized range. The reaction force at the non-ideal joint is determined by the
contact force model. Thus for a non-ideal joint b, the contact and friction force at time increment
ti is expressed as:
89
, , ,
, , ,
i i i
i i i
bN t N t t
bt k N t t
F
F
F n
F n (5-1)
where NF is the contact force and n is a unit normal vector pointing in the direction of contact.
NF and n are describe in Eq. (3-7) and (3-12) respectively. The values of NF and n are
computed during the integration of the equations of motion. It is worth noting that in the case of
the ideal joints, the reaction force is determined by the product of the Jacobian and the Lagrange
multiplier. For instance, the reaction force for an ideal joint k can be obtained as shown in the
following expression:
j
b T bj r F Φ λ (5-2)
where bjF is the reaction force at the revolute joint k corresponding to body j,
j
TrΦ and bλ are the
Jacobian sub-matrix and the Lagrange multipliers corresponding to the concerned revolute joint.
The incremental sliding distance is also obtained at each increment and described as:
1i i it j t ts R
(5-3)
where jR is the bushing radius and it
is the angle difference (in radians) between the local x-
axes of the two bodies i and j that share a revolute joint at a current time whereas 1it
corresponds to the difference at a previous time. The value of t is obtained as i j .
Wear Analysis
The second part of the integration process involves a wear analysis. The amount of wear is
determined at each increment based on the reaction force and sliding distance from the previous
dynamic analysis. The reaction force at each time increment is used to determine the contact
pressure (through a finite element analysis) between the joint components (pin and bushing).
Incremental wear can then be computed and the geometry is updated (see Chapter 4).
90
The wear prescribed by geometry update increases the clearance. The inner perimeter of
the bushing is altered from its circular shape to one dictated by the wear. This implies that the
value of c in Eq. (3-13) is no longer a constant value. Instead c depends on the location of
contact C (as defined in Figure 3-4 and in Eq.(3-11)) and must also to be updated during the
dynamic analysis. The corresponding update for c is give by
ic R Cb jr - r (5-4)
where iR is the original pin radius, and Cjr and br are the position vector of the contact point C
and the bushing center, respectively. The last term on the left hand side of Eq. (5-4) ( Cb jr - r ) is
the distance between the bushing center and the contact location C. This is the quantity that will
vary as the bushing is worn out.
After the pin and bushing geometry is updated and the clearance size is adjusted to reflect
the wear, another dynamic analysis is performed. The wear is then computed based on the result
of the new dynamic analysis. The geometry and the clearance size are once again updated. This
process is iterated to the desired number of cycles. The process is summarized in the flowchart
shown in Figure 5-1.
Demonstration of the Integration Process
In this section the integrated model will be used to predict the wear occurring at the joint of
a slider-crank mechanism. The slider-crank mechanism, encountered in Chapter 3, will be used
to facilitate the demonstration. A diagram of the slider-crank is shown in Figure 5-2. The
dimensions and mass parameters for the slider-crank are shown in Table 5-1andTable 5-2.
The demonstration involves determining the joint wear after several thousand revolutions
of the crank. For this mechanism, wear is assumed to occur at the joint between the crank and the
connecting rod. All other joints are assumed to have no wear. The joint of interest consists of a
91
pin that is attached to the crank and a bushing attached to the connecting rod. To simplify the
analysis, the pin is assumed to be made of hardened steel while the bushing is made of poly-
tetra-fluoro-ethylene (PTFE). This allows the pin to be modeled as a rigid body and the pin as a
deformable body. In addition, the interaction of steel and PTFE results in a very low wear rate
for the steel and a high wear rate for PTFE. Wear on the pin is thus negligible and can be
disregarded for low number of cycles. The concern is therefore to account for the wear on the
PTFE bushing while performing the dynamic analysis.
A value of 5.05x10-4 mm3/Nm, based on wear tests by Schmitz et al [75], was used for
PTFE bushing wear rate. In addition, the friction was considered at the joint and a value of 0.13
was used for the friction coefficient (Schmitz et al [46]). For the demonstration, the crank was
operated at 30 rpm for 5,000 crank cycles. A summary of these simulation parameters is shown
in Table 5-1
Representative results of the initial dynamic analysis for the configuration are shown in
Figure 5-3 and Figure 5-4. A plot of the reaction force at the joint of interest is shown Figure 5-3.
The initial clearance for this joint was 0.005mm as listed in Table 5-2. In Figure 5-4 the locus of
the center of the contact region in the dynamic analysis is shown. Thus each point on the graph
corresponds to the center of the contact region. The center of the contact region was defined as
the contact point ‘C’ in Figure 3-4.
From Figure 5-4b it can be seen that the location of contact point ‘C’ is concentrated on the
left and the right side of the bushing inner diameter. This corresponds to bushing angular
coordinates of approximately 0 and π radians as defined in Figure 5-4c. This indicates that these
two regions will experience more wear that other regions.
92
The wear result from the integrated model after 5,000 crank cycles is shown in Figure 5-5.
The figure shows a plot of the wear on the bushing as a function of the bushing angular
coordinate. As was expected, two peaks are observed to occur at 0 and π radians. These peaks
correspond to the location where there was a concentration of the contact point ‘C’ during the
dynamic analysis (see Figure 5-4). The reason for the two peaks can be explained by considering
the motion of the crank during a particular cycle. In the first half of the cycle, the crank tends to
pull both the connecting-rod and the slider whereas in the second half of the cycle the crank
pushes the rod and slider. This results in the pin establishing contact with the bushing with the
center of contact at approximately π radians during the first half cycle and then establishing
contact on the opposite side (0 radians) during the second half cycle.
Summary and Conclusions
In this Chapter a procedure to analyze multibody systems with revolute joint was
presented. The procedure involves an alternation between dynamic analysis of the system and
wear prediction of the concerned joint. The alternating analyses are iterated to the desired
number of cycles. Integration is achieved by updating the joint geometry based on the results of
the wear analysis. The changes in the joint geometry are captured in successive dynamic
analyses and thus the need for the alternation. The procedure is capable of determining the
potential regions of contact and is able to quantify wear in those regions. The analysis of a slider
crank mechanism with wear occurring at one of its joints was presented to demonstrate the use of
this procedure.
93
Table 5-1. Dimension and mass properties of the slide-crank mechanism Length (m) Mass (kg) Inertia x10-6 (kg.m2)
Crank 1.00 10.00 45.00Connecting rod 1.75 15.00 35.00Slider -- 30.00 --
Table 5-2. Properties of the pin and bushing Pin Bushing Initial radius 20 mm 20.005mDepth -- 12.8mmPoisson ratio 0.29 0.38Young’s Modulus 206.8 GPa 0.5 GPa
Table 5-3. Test and simulation parameters Parameter Value Crank velocity 30 RPM (π rad/sec)Crank cycles 5,000 cyclesFriction coefficient [46] 0.13Wear coefficient [75] 5.05x10-4 mm3/Nm
94
Figure 5-1. Integration of wear analysis into system dynamics analysis
Figure 5-2. Slider-crank mechanism with a wearing joint between the crank and the connecting
rod.
Crank
Connecting rod
Slider
Wearing Joint
GroundGround
Input Geometry
Wear Analysis
FEA
Wear Rule
Geometry Update
Dynamic Analysis
Input Mechanism Parameters/Geometry
Assemble
M , qΦ , AQ , γ
Solve DAE for q and λ
Integrate q for
q and q
Reaction Force
Sliding Distance
95
7 8 9 10 11 120
1000
2000
3000
4000
5000F
orc
e (
N)
Reaction Force (Contact Force Model)
Crank Position (rad)
Figure 5-3. Initial joint reaction force for joint with clearance 0.0005mm.
-30 -15 0 15 30-30
-15
0
15
30Locus of the Center of Region Contact (Pin)
x-coordinate
y-co
ord
ina
te
-30 -15 0 15 30 -30
-15
0
15
30Locus of the Center of Region Contact (Bushing)
x-coordinate (mm)
y-co
ord
ina
te (
mm
)
Figure 5-4. Locus of contact point ‘C’ for a complete crank cycle. a) Locus for the pin. b) Locus for the bushing. c) Definition of the angular bushing coordinate.
a b c
x
y
θ
96
Figure 5-5. Wear on bushing as a function of the bushing angular coordinate after 5,000 cycles.
-4 -2 0 2 40
0.1
0.2
0.3
0.4
0.5Wear on Bushing
Bushing Angular Coordinate (rad) ()
We
ar
on
Bu
shin
g (
mm
)
FEM
θ
97
CHAPTER 6 INTERGRATED MODEL: SYSTEM DYNAMICS AND WEAR PREDICTION USING THE
ELASTIC FOUNDATION MODEL
The use of the Elastic Foundation model in the analysis of planar multibody systems with
joint wear will be presented in this Chapter.
Introduction
In Chapter 4, a procedure to predict wear occurring at the interface of bodies in contact and
relative motion was presented. The procedure required determination of the contact pressure
resulting from the contact of the bodies. This was achieved using the FEM (FEM). An alternative
technique is the elastic foundation model (EFM) also known as the Winkler Surface model. This
model has previously been used in wear prediction procedures. Flodin and Andersson [56]
simulated mild wear in spur gears using the EFM to compute contact pressure. Podra and
Andersson [53] used the EFM to simulate the wear for a sliding sphere on flat and cylinder on
flat. They compared the wear results with result for wear simulations conducted with FEM. They
reported reasonable agreements.
In addition to computing the contact pressure the EFM may also be to compute the joint
reaction force in a multibody dynamics framework. This approach was employed by Bei and
Fregly [79] to perform multibody dynamic simulation of knee contact. In this Chapter the use of
the elastic foundation model in the analysis of multibody system will be presented.
Elastic Foundation Model
In the EFM the contact surface is modeled as a set of springs (spread over the contact
surface). The model is derived from plane strain elastic theory where an elastic layer of known
thickness is bounded onto a rigid surface [80]. The springs represent the elastic layer and the
thickness of the layer is composed of the thickness of one or both bodies (depending on whether
one of the bodies is defined as rigid). The deformation on the elastic layer is produced by a rigid
98
indenter when a force Fin is applied on the indenter. The contact surface of the indenter takes into
account the shape of the two bodies [80]. A figure of the EFM is shown in Figure 6-1.
The EFM assumes that the springs are independent from each other and thus the shear
force between them is neglected. The consequence of this assumption is that the model does not
account for how pressure applied at one location affects the deformation at other location. This is
contrary to what is experience in elastic contact where the displacement at one location is a
function of the pressure applied at other locations. Although this simplifying assumption violates
the very nature of contact problems, some benefits can be derived from its use. In particular, the
simplified model results in reduced pressure computational costs, facilitated analysis of
conformal geometry, layered contact and nonlinear materials [79]. In addition, when the EFM is
used in the analysis of multibody systems with joint wear, the joint reaction forces required for
dynamic analysis can be recovered as a byproduct of the contact pressure. The need for a contact
force model, therefore, disappears. This additional advantage will become clear when the
discussion of the use of the EFM in multibody analysis is presented.
The contact pressure for any spring (spring i ) in the elastic foundation can be calculated
from [56]
Wi i
i
Ep
L (6-1)
where ip is the contact pressure, WE is the elastic modulus for the elastic layer, iL is the
thickness of the elastic layer and i is the deformation of the spring. When both bodies are
deformable WE is a composite of the elastic modulus and Poisson ratio of the two bodies. The
procedure to determine the composite modulus is discussed by Podra [52] and in more detail by
99
Johnson [80]. For the purpose of illustration, it is assumed that only one of the bodies in contact
is deformable. For this case, a common expression for WE is give by [79, 81-88]
1
1 1 2W
EE
(6-2)
where E and are the elastic modulus and Poisson ratio of the deformable body, respectively.
The contact pressure for the spring i can then be determined from
1
1 1 2i
ii
Ep
L
. (6-3)
The total load supported by the elastic layer can then be computed as
Load i iF p A , (6-4)
where iA is the area of each spring element. It should be noted that total load supported by the
elastic layer must equal the load applied on the indenter. Thus the following equation must hold:
In LoadF F (6-5)
If the shape of the rigid indenter and the deformation of a particular spring are known, the
deformation at other springs can be determined. Equation (6-3) can then be used to determine
contact pressure. It is, however, not typical that the deformation in springs is known. Rather,
what is common is that the indenter force (shown as Fin in Figure 6-1) which causes the
deformation is known. The task is then to determine the contact pressure that is caused by the
force. This requires an iterative procedure that is outlined as follows:
1. Guess an initial deformation for one of spring.
2. Based on the shape of the indenter and the initial guess, determine the deformation for the other springs.
3. Compute the contact pressure using Eq. (6-3).
4. Find the sum of the load ( LoadF ) supported by the elastic layer using Eq. (6-4).
100
5. Check if Eq. (6-5) is satisfied within some specified tolerance. If it is, stop the iteration otherwise repeat part 1-4 until convergence is achieved.
The iteration procedure is summarized in Figure 6-2.
Analysis of Multibody Systems with Joint Wear Using the EFM
In this section, a discussion of how the EFM can be used in the analysis of multibody
systems with imperfect revolute joints will be presented. While the framework for this analysis,
presented in Chapter 5, remains the same, the manner in which the revolute joint is modeled will
be different. In addition the wear prediction procedure will involve the EFM rather than the
FEM.
Modeling the Non-Ideal Revolute Joint Using the EFM
The imperfect revolute joint was defined as consisting of two components, a pin and a
bushing, that are rigidly attached to bodies i and j which share the joint. A realization of the
revolute joint with the two bodies, referenced in the global coordinates, is shown in Figure 3-1.
The imperfect revolute joint was modeled so that the pin and the bushing centers do not
necessarily coincide during the motion of the mechanism. The pin is allowed to move within the
bushing inner perimeter. Whenever contact is established, a contact force, determined by
Eq.(3-7), is applied so as to restrict the pin within the bushing.
Instead of using the contact force model, the EFM can be used to determine the appropriate
force required to restrict the pin within the bushing inner perimeter. The procedure will first
require determination of the contact pressure resulting from the contact between the pin and
bushing using Eq.(6-3). In this equation, the material properties ( E and ) and the thickness of
the elastic layer ( iL ) will generally be known from the problem definition. The deformation,
however, ( i ) is dependent on the system dynamics. Referring to Figure 6-3, the deformation
may be computed as:
101
e c (6-6)
where e is the eccentricity and is defined as the distance between the pin and the bushing, and c
is the initial clearance in the joint. When the pin is not in contact with the bushing the
eccentricity is smaller than the clearance, and the penetration has a negative value. When the
penetration has a value equal or greater than zero, contact is established. Thus when is greater
than zero a contact pressure can be computed using Eq.(6-3). The contact pressure vanishes
when is equal to or less than zero. These configurations are depicted in Figure 6-4.
The eccentricity can be determined from the eccentric vector which is a vector that points
in the direction potential contact. As previously discussed in Chapter 3, the vector is calculated
from
i i i j j je r A s r A s , (6-7)
where ir and jr are vectors linking the global origin and the center of masses of the bodies. is
and js are vectors in the local coordinate system that link the center of masses to the pin and
bushing centers respectively. iA and jA are transformation matrices that transform a vector
from the local coordinate system to the global system. The eccentricity is then give as the
magnitude of the eccentric vector defined as
e e . (6-8)
Once the contact pressure has been evaluated, the contact force can then be computed from Eq.
(6-4). It is also possible to compute the friction force using the Coulomb friction model discussed
in Chapter 3. Both the contact and the friction forces may then be assembled into the differential
algebraic equations of motion which can then be solved (as discussed in Chapter 2) to determine
the system dynamics. It should be noted that since the body-fixed coordinates were fixed at the
102
center of masses of bodies i and j, the forces must also be applied at these locations rather than at
the points of contact. Thus the transfer of the loads to the mass centers will result in an additional
moment in each body. The appropriate transformations are expressed in Eqs. (3-16), (3-17) and
(3-18).
Example: dynamic analysis of a slider-crank with an imperfect joint using EFM. The
slider-crank mechanism encountered in Chapters 2 and 3 will be used to demonstrate the use of
the EFM in analysis of multibody system with imperfect joints. A diagram of the slider-crank
mechanism is shown in Figure 6-5. The revolute joint between the crank and the connecting rod
is modeled as an imperfect joint. All other joints, including the translational joint (slider), are
modeled as ideal. The dimension and mass properties for the mechanism are shown in Table 6-1.
For this example, it is assumed that the pin is made of steel, while the bushing is made of PTFE.
The material properties as well as the radii of the pin and bushing are listed in Table 6-2. The
crank is constrained to rotate at a constant angular velocity of 30 rpm (π rad/sec).
The kinematic constraint equations for this mechanism are identical to the one discussed in
Chapter 4 and are repeated in Eq. (6-9) for convenience. It is worth noting that only driving
constraint and the constraint equations representing the ideal joints appear in the set of kinematic
constraint equations. The dynamics of the slider crank mechanism can then be determined by
assembling and solving the differential algebraic equations of motion.
1 1 1
1 1 1
2 3 2 2
2 3 2 2
3
3
1
cos 0
sin 0
cos 0
sin 0
0
0
0
x l
y l
x x l
y y l
y
t
Φ . (6-9)
103
Representative results of the dynamic analysis are shown in Figure 6-6, Figure 6-7 and
Figure 6-8. In Figure 6-6 the joint reaction force (between the crank and the connecting rod) for
the EFM based model is compared to the joint force from the ideal joint model. For the non-ideal
model a clearance of 0.0005 mm was used. Since this clearance is quite small, it is expected that
the response of the EFM based model should closely resemble those from the ideal model as is
observed in Figure 6-6. In Figure 6-7, the locus of the center of the contact region is plotted for
both the pin and the bushing. From this plot it is seen that the pin experiences contact on only
one side whereas the bushing has a concentration of the contact points at two locations. This
behavior was observed in the case of the previous analysis procedure based on the Hertzian
contact model (see Figure 5-4). Comparison of the predicted force between the ideal model and
the EFM base model for various joint clearances is shown in Figure 6-8. It can be seen from the
plots that as the clearance increases, the reaction force begins to exhibit some oscillations. The
amplitude and frequency predicted by the EFM model differ from those predicted by contact
force model (see Figure 3-7). However the trend, in terms of the location of the oscillation is
quite similar.
Integrated Model: System Dynamics and Wear Prediction Using EFM
It was seen, in the previous section, that in order to perform dynamic analysis of a system
with imperfect revolute joints using the EFM, it was first necessary to determine the contact
pressure at the revolute joint. Contact pressure information is thus available after every dynamic
analysis and can be used for wear prediction using the procedures discussed in Chapter 4. The
integration of the wear prediction procedure into the dynamic analysis is achieved by updating
the bushing geometry after every wear analysis1. Since the inner perimeter of the bushing is
1 For simplicity it has been assumed that one of the joint components (pin) is rigid and thus does not wear.
104
altered from its circular shape to one dictated by the wear, the value of c in Eq. (6-6) is no
longer a constant value but instead depends on the location of contact C (as defined in Figure 6-3
and in Eq.(3-11)). Thus, the value of c must also be updated in progressive dynamic analyses.
The expression used to update the clearance c was discussed in Chapter 3 and is repeated here
for convenience as:
ic R Cb jr - r (6-10)
where iR is the original pin radius, and Cjr and br are the position vector of the contact point C
and the bushing center, respectively. The first term on the left hand side of Eq. (6-10) ( Cb jr - r )
is the distance between the bushing center and the contact location C. This is the quantity that
will vary as the bushing is worn out.
Example: analysis of a slider-crank with joint wear using EFM. Wear prediction at a
joint of the slider-crank mechanism will be used to demonstrated integrated model based on
EFM. For purposes of comparison with integrated model based on FEM, simulation parameters
are selected similar to the example discussed in Chapter 5. The slider-crank mechanism is shown
in Figure 6-5 and its dimensions and mass properties are listed in Table 6-1. The pin and the
bushing are assumed to be made of steel and PTFE respectively. The material properties of the
pin and bushing are listed in Table 6-2 and their radii and bushing depth are listed in Table 6-3.
The crank is constrained to revolve at a constant angular of 30 rpm for 5,000 cycles.
A plot of the locus of the center of the contact region (contact point ‘C’ as defined in
Figure 6-3) for the initial dynamics is shown in Figure 6-7. It can be seen that for the pin (Figure
6-7a), the contact point ‘C’ is concentrated on the left side whereas for the bushing (Figure 6-7b)
the contact point ‘C’ is concentrated on both the left and the right side of the bushing inner
diameter. This corresponds to bushing angular coordinates of approximately 0 and π radians as
105
defined in Figure 6-7c. This indicates that these two regions will experience more wear than
other regions. The same behavior was observed in the integrated model based on the FEM (see
Figure 5-4).
In Figure 6-9 and Table 6-4, the wear predicted by the integrated model base on the EFM
after 5,000 crank cycles is compared to the wear predicted by the FEM based model. In the
figure, wear is plotted as a function of the bushing angular coordinate. As was expected, two
peaks are observed to occur at approximately 0 and π radians. The maximum wear depth
predicted by the FEM based model is seen to be greater than that of FEM based model. On the
contrary, the EFM based model predicted more wear in the region where the FEM based model
experienced the least amount wear. This difference in wear profile is directly linked to the
assumption in the EFM that the individual springs are independent of each other and lateral
effects are neglected.
Another interesting observation from wear results, in Figure 6-9 and Table 6-4, is that
while the maximum wear depth predicted by the two models differ, the wear profile for the
models is such that the worn volumes predicted by the two models are approximately equal. This
equality is a manifestation of the equality in the force as shown in Figure 6-6. It should be
mention however that as the wear increases in the joint the force predictions for the two models
differ slightly. As a result the wear volume will also differ.
Also in Table 6-4 the computation time for the two models is compared. The EFM has a
shorter computational time. This is because of two reasons: 1) the EFM is quite simple because
the springs in the elastic layer are assumed to be independent, and 2) in the case of EFM based
model only one set of analyses for each cycle is required to simultaneously determine both the
contact force and the contact pressure during the dynamic analysis, whereas in the case of the
106
FEM based model the contact force is determine in the dynamic analysis which in turn is used to
determine the contact pressure.
Summary and Conclusion
In this Chapter the procedure to determine contact pressure and joint reaction force of a
multibody system using the elastic foundation model was discussed. Wear prediction was then
introduced into the dynamic analysis to allow for the analysis of the system with joint wear. A
slider-crank mechanism was used to demonstrate the integrated model as well as for comparison
with the FEM based model. The force prediction of the two models was identical for near zero
clearance but differed in magnitude as the clearance was increased. This difference is attributed
to the different techniques employed in the models used to compute the force. It was seen that
the maximum wear depth and the wear profile predicted by the two models differed. However,
the wear volume predictions from the two models were similar. The computation time for the
EFM based integration model was found to be shorter than that of the FEM based model.
While it is well known that the FEM is more superior to the EFM, the use of the EFM may
be justified because of computational speed. The EFM based integrated model may be used to
provides some qualitative information about the system. However results from the model should
be interpreted with caution to avoid erroneous conclusions.
107
Table 6-1. Dimension and mass parameter for slider-crank mechanism Length (mm) Mass (g) Moment of inertia (kg-m2) Crank 1.00 10.00 45.00Connecting Rod 1.75 15.00 35.00slider -- 30.00 --
Table 6-2. Material properties for the joint components Pin Bushing Young’s modulus 0.29 0.38Poisson ratio 206.8 GPa 0.5 GPaRadius 20 mm 20.00, 20.05, 20.5, 25 mmDepth -- 12.8 mmFriction coefficient (pin & Bushing) [46] 0.13
Table 6-3. Wear simulation parameters Parameter Value Crank speed 30 rpmCrank cycles 5,000 cyclesBushing inner diameter 20.005 mmPin diameter 20.000 mmBushing depth 12.8 mmWear coefficient [75] 5.05x10-4 mm3/Nm
Table 6-4. Comparison of results from prediction based on FEM and EFM FEM EFM Error Maximum wear depth 0.34 mm 0.29mm 13.0%Wear Volume 231.42 mm3 241.76 mm3 4.3%Computation Time 162 min 92 min
108
Figure 6-1. Elastic foundation model.
Figure 6-2. Procedure to determine the contact pressure using the EFM.
L Elastic Layer
Rigid Surface
δi
FIn Rigid Indenter
Spring i
End of Simulation
Initial guess for δ1
Determine δi based shape of indenter and δ1
Calculate Contact Pressure pi
Find Fload
Load i iF p A
Update value of δ1
Fload=Fin ?
109
Figure 6-3. General imperfect joint.
Figure 6-4. Penetration during contact between the pin and the bushing.
Figure 6-5. Slider-crank mechanisms with joint clearance.
0e c
c
0e c
0e c
ePin
Bushing e
Crank
Connecting rod
Slider
Joint clearance
GroundGround
y3
x3
x1 y1
y2
x2
y
x
O
y
ri
Si
Sj
rj
A
B
E
C
D
j
i
yi xi
yj
xj
x
e
y1 x1
y2
x2 Body-iBody-j
110
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)Reaction Force (clearance = 0.0005mm)
Elastic Foundation ModelContact Model
Figure 6-6. Comparison of reaction force between the ideal joint model and the EFM.
-30 -20 -10 0 10 20 30-30
-20
-10
0
10
20
30Locus of the Center of Region Contact (Pin)
x-coordinate
y-co
ord
ina
te
-30 -20 -10 0 10 20 30
-30
-20
-10
0
10
20
30Locus of the Center of Region Contact (Bushing)
x-coordinate (mm)
y-co
ord
ina
te (
mm
)
Figure 6-7. Locus of contact point ‘C’ for a complete crank cycle. a) Locus for the pin. b) Locus for the bushing. c) Definition of the angular bushing coordinate.
a b c
x
y
θ
111
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)Reaction Force (clearance = 0.0005mm)
Elastic Foundation ModelIdeal Model
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 0.05mm)
Elastic Foundation ModelIdeal Model
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 0.5mm)
Elastic Foundation ModelIdeal Model
8 10 12 14 16 180
1000
2000
3000
4000
5000
Crack Angle (rad)
Fo
rce
(N
)
Reaction Force (clearance = 5mm)
Elastic Foundation ModelIdeal Model
Figure 6-8. Comparison of reaction force between ideal joint model EFM.
112
-4 -2 0 2 40
0.1
0.2
0.3
0.4
0.5Comparison of Wear on Bushing Using FEM and EFM
Bushing Angular Coordinate (rad) ()
We
ar
on
Bu
shin
g (
mm
)
Finite Element MethodElastic Foundation Model
Figure 6-9. Comparison of joint wear between the EFM and FEM after 5,000 cycles.
113
CHAPTER 7 EXPERIMENTAL VALIDATION OF THE INTEGRATED MODELS
Introduction
Two procedures to analyze multibody systems with joint wear were presented in Chapters
5 and 6. In addition predictions from the two procedures were compared in Chapter 6. Before
these models can be utilized for practical purposes, their validity should be assessed. In this
Chapter, the integrated models will be validated through experiments.
Experiments for Model Validation
For the purpose of validation, an experimental slider-crank mechanism was built. The
mechanism was built at the Tribology Laboratory in the University of Florida. Likewise, all the
wear tests were conducted at this laboratory. A diagram of the slider-crank mechanism, built for
this purpose, is shown in Figure 7-1.
The focus of the experimental test is to determine the wear that occurs at the joint between
the crank and the connecting rod after several thousand revolutions of the crank. This joint
essentially consists of a pin that is attached to the crank and a bushing attached to the connecting
rod. The pin is made of hardened steel and is assumed to be hard enough so that no appreciable
wear occurs on its surface. The bushing on the other hand is made of poly-tetra-fluoro-ethylene
(PTFE) which is soft and will experience considerable wear. A spring is attached to the slider
which serves as a means to increase the joint reaction force and hence accelerates the wear
occurring at the joint. The spring also ensures that contact between the pin and the bushing is
maintained even in the presence of excessive wear. The slider-crank mechanism was built so as
to minimize friction and wear (to a negligible amount) at all joints except at the joint of interest
(joint between crank and connecting rod). This is achieved by building the joint between
connecting rod and the slider with a thrust air bearing and using a dovetail air bearing slider.
114
Details of the construction of the mechanism can be found in the works of Mauntler et al [89,
90]. The dimensions and mass properties for the experimental slider-crank are shown in Table 7-
1. The dimensions, the material properties, the fiction coefficient ant the wear coefficient of the
joint components (bushing and pin) are listed in Table 7-2.
The validation involves comparing the wear on the bushing from the experiment to the
wear prediction by the two models (model based on FEM and EFM). For the validation, both the
maximum wear depth and the wear profile (location of wear) will be compared for the two
models. The comparison will thus give an indication of the performance of each model with
respect to each other and with respect to the experiment.
In the experimental test, the slider-crank mechanism was operated for 21,400 cycles. The
crank was constrained to revolve at a constant velocity of 30 rpm. A spring with spring constant
of 525 N/m was used. The test and simulation parameters are summarized in Table 7-3.
Figure 7-2 and Figure 7-3 show representative results from the initial system dynamics.
Three plots of the joint reaction force from 1) the experiment 2) the model based on the EFM and
3) the model based on the contact force model are shown in Figure 7-2. The two models, which
are identical in this case, predict the joint reaction force reasonably well over the entire crank
cycle except at about π radians. At this location, the measured force exhibits high frequency
oscillation for a short duration. The location of these oscillations corresponds to one half of the
crank rotation when the slider changes direction. It is the belief of the author that these higher
order dynamics is a result of the change in the direction of the slider which most likely involves a
slight rotation of the slider and thus a moment of brief impact with the sliding rail. It should be
mention that, although the magnitude of these oscillations is large, there effects on the wear
115
prediction is quite small. This is because the corresponding incremental sliding distance is also
quite small. Thus according to equation (4-5), repeated here for convenience as
1j j j jh h kp s , (7-1)
the value of pj will be larger because of the large amplitude oscillations but the value of js will
be quite small since the oscillation occur over a short duration. The result is that the incremental
wear depth, for this region, will not vary significantly from the incremental wear depth of its
neighboring regions.
In Figure 7-3 the locus of the center of the region of contact are plotted for the pin and
bushing. Figure 7-3(a) shows the locus of points when the model based in the EFM was used. It
is seen that the entire pin surface will experience contact. On the other had the center of the
region of contact on the bushing is concentrated on the left side of the bushing. This means that
only one side of the bushing will experience wear and that the maximum wear will occur where
there is a concentration of the center of the region of contact. The concentration of the contact
point in this location is reasonable because the spring that restrict the motion of the pin relative
to the bushing. Figure 7-3(b) corresponds to the plot of the locus or the center of the region of
contact when the contact force model is used. It is clear that the two models give approximately
identical predictions.
The wear predicted by the FEM and the EFM models are compare in Figure 7-4 and Table
7-4. In Figure 7-4, it can be seen that while the FEM base model predicts a larger maximum wear
depth, the EFM has a wider base. The wider base means that a wider region in the bushing
surface is worn out. An interesting observation is that while the wear depth for the two models
differs, their wear profile is such that the worn volume is equal for the two models. This equality
116
is a manifestation of the equality in the force as seen was in Figure 7-2. This behavior was
identified and explained in the previous chapter.
The computation time for the wear prediction based on the FEM and the EFM models is
compared in Table 7-4. As was seen in the previous chapter the EFM model has a faster
computation time.
The wear results from the experiment and the simulation results from the two models are
compared in Figure 7-5, Table 7-5 and Table 7-6. From Figure 7-5(a) and Table 7-5, it can be
seen that the maximum wear depth, the wear profile and the wear volume from the experiment
have accurately been predicted by the FEA based model. There is however a discrepancy
between 4.5<θ <6.3 that is attributed to the measurement of the wear on the bushing in the
experiment.
In the case of the EFM model, the wear profile and the maximum wear depth are
incorrectly predicted as shown in Figure 7-5(b) and Table 7-6. The location of maximum wear
and the wear volume are however correctly predicted as expected.
Summary and Conclusions
The objective of this Chapter was to validate the procedures for analysis of multibody
systems with joint wear. The two procedures were presented in Chapters 5 and 6. For the
validation, results from an experimental slider-crank mechanism were compared to results from
the two models. In addition the performance of the two models was analyzed by comparing
results between models and against the experimental results. The focus of the comparison was
the joint force, contact location and wear on the bushing at the joint between the crank and the
connecting-rod. All other joints were assumed to be free of wear and free friction as necessary;
provisions were made for these assumptions.
117
For the initial dynamics the two models provided reasonably accurate prediction of the
contact force. The two models produced identical prediction for the location of maximum wear.
Through the experiment, this location was later verified to be correct. Although this location was
correctly predicted by both models, only the FEM based model gave an accurate prediction of
the wear profile and maximum wear depth. Prediction from the FEM based model differed by
6.7% from the experiment, whereas the prediction from the EFM model differed by 12.1% from
the experiment. It should, however, be noted while the predictions on the wear profile differed,
the wear volume predictions of the two models were identical and reasonably close to the
experimental wear volume (8.2 %).
Despite the poor prediction on the wear profile and maximum wear depth, the EFM based
model had a shorter computation time. The experiment took about 12 hours (excluding
construction and setup time) and FEM based model took 11 hours while the EFM based model
took 7 hours to complete. The speed of the EFM based model is associated with the assumption
that the springs in the elastic foundation model are independent.
From the comparison between the results of the two models and between the results of the
two models and the experiments the following conclusions can be made: 1) the two procedures
(FEM and EFM based procedures) can accurately predict the contact force, the contact locations
and the wear volume, 2) FEM based procedure is a better predictor of maximum wear depth and
the wear profile than the EFM based procedure, and 3) EFM model is computationally faster
than the FEM based model.
It can be concluded that the FEM based procedure will be a better procedure in the analysis
of multibody systems with joint wear when the computational costs in not of concern. On the
other hand if the cost is of concern and only qualitative information about the system is needed
118
then the EFM based procedure will be the suitable choice. Other scenarios will require a
compromise on either accuracy or computational costs.
119
Table 7-1. Dimension and mass properties of the slide-crank mechanism Length (m) Mass (kg) Inertia x10-6 (kg.m2)
Crank 0.0381 0.4045 204.0Connecting rod 0.1016 0.8175 5500.0Slider - 8.5000 --
Table 7-2. Properties of the pin and bushing Pin Bushing Bushing Inner radius -- 9.533 mmOuter radius 9.500 15.875mmDepth -- 13.100mmPoisson ratio 0.29 0.38Density 7.8 g/cm3 2.25g/cm3
Young’s Modulus 206.8 MPa 0.500 MPaFriction coefficient (steel & PTFE) [46] 0.13Wear coefficient (steel & PTFE) [75] 5.05x10-4 mm3/Nm
Table 7-3. Test and simulation parameters Properties Value Crank velocity 30 RPMCrank cycles 21400Spring constant 525.4 N/m
Table 7-4. Comparison of wear results for FEM and EFM models (21,400 crank cycles)
Table 7-5. Comparison of wear results between test and FEM model (21,400 crank cycles)
FEM EFM Difference Wear Volume 106.71 mm3 106.68 mm3 0.02%Max wear depth 0.4779 mm 0.4263 mm 10.70%Computation time 11hrs 7hrs 4 hrs
Experimental Simulation (FEM) Difference Worn mass 0.2616 g 0.2401 g 8.2%Wear Volume 116.27 mm3 106.71 mm3 8.2%Max wear depth 0.4850 0.4779 mm 1.5%
120
Table 7-6. Comparison of wear results between test and EFM model (21,400 crank cycles)
Experimental Simulation (EFM) Difference Worn mass 0.2616 g 0.2400 g 8.2%Wear Volume 116.27 mm3 106.68 mm3 8.2%Max wear depth 0.4850 0.4263 mm 12.1%
121
Figure 7-1. Experimental slider crank mechanism
0 1 2 3 4 5 60
50
100
150
200
Crank Position (rad)
Join
t Fo
rce
(N
)
Comparison Joint Force Between Models and Expt
EFMContact ModelExperiment
Figure 7-2. Comparison of the initial joint reaction force between the two models and the experiment.
122
Figure 7-3. Locus of the center of the contact region. a) Prediction based the elastic foundation model. b) Prediction based on the contact force model.
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5Comparison of Wear on Bushing Using FEM and EFM
Bushing Angular Coordinate (rad) ()
We
ar
on
Bu
shin
g (
mm
)
FEMEFM
Figure 7-4. Comparison of the wear prediction between the models.
a
-10 -5 0 5 10
-10
-5
0
5
10
Locus of the Center of Region Contact (EFM)
x-coordinate (mm)
y-co
ord
ina
te (
mm
)
PinBushing
b
-10 -5 0 5 10
-10
-5
0
5
10
Center of Region Contact (Contact Force Model)
x-coordinate (mm)y-
coo
rdin
ate
(m
m)
PinBushing
x
y
θ
123
Figure 7-5. Comparison of the wear profile for the models and the experiment. a) Comparison between experiment and FEM. b) Comparison between experiment and EFM.
a
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5Comparison of Wear (Expt and FEM)
Bushing Angular Coordinate (rad) ()
We
ar
on
Bu
shin
g (
mm
)
ExptFEM
b
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5Comparison of Wear (Expt and EFM)
Bushing Angular Coordinate (rad) ()
We
ar
on
Bu
shin
g (
mm
)
ExptEFM
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CHAPTER 8 DESIGN OF A MULTIBODY SYSTEM FOR REDUCED JOINT WEAR MAINTENANCE
COSTS
In Chapters 5 and 6, the procedure to analyze multibody systems with joint wear was
presented. The methodology in the analysis discussed in Chapter 5 was based on the contact
force law and the finite element method (FEM), whereas the procedure presented in Chapter 6
was based on the elastic foundation model (EFM). These procedures were later validated against
experiments in Chapter 7 and their performances are evaluated. In this Chapter, application of
the FEM based procedure in the design of a mechanical system will be presented.
Introduction
Most mechanical/multibody systems consist of multiple joints that connect the system
components. For example, the backhoe system shown in Figure 8-1 consists of three main
revolute joints which connect the bucket to the dipper, the dipper to the boom and the boom to
the main body, respectively. For this particular system wear is expected to occur at all the joint.
If one cycle is defined as scooping and dropping off the dirt using the bucket then the wear depth
at any of the joints is a function of 1) the number of cycles of operation, 2) the joint reaction
forces, 3) the material used to construct the joint components (pin and bushing), 4) the size of the
joint components (diameter and depth of the joint components) and 5) the operating conditions at
the joint. In conventional design of such systems, it is unlikely that the maximum allowed wear
at all the joints will simultaneously occur at the same number of operation cycles. As a
consequence, when the maximum allowable wear occurs on one of the joints the use of the
system will have to be halted and the joint has to be repaired or replaced. The system can then be
operated until the maximum allowable wear occurs on the other joint. The system operation is
once again halted and the second joint is repaired or replaced. In the case of the backhoe system,
repair of all the main joints would require that the system be out of operation in three occasions
125
and thus leading to high maintenance cost and loss of revenue (since the equipment is out of
service). Another alternative is that whenever one joint reaches the maximum allowable depth,
the system operation is halted and all the joints are replace. For purposes of illustration it
assumed that this is not the case.
One design requirement could be to design the joints of the backhoe system such that the
maximum allowable wear on all the joints occur at the same (or approximately the same) number
of operation cycles. The result of this design is that the maintenance cost and loss of revenue is
reduced since the system will be out of operation only once as opposed to three times. This
concept will be illustrated using a slider-crank mechanism with wear occurring at two joints.
Design Example: Design of a Slider-Crank for Reduced Maintenance Cost
The slider-crank mechanism used in Chapters 2, 3, 5 and 6 will be used to illustrate the
design example. In this slider-crank, however, wear is allowed to occur at two joint namely; the
joint between the crank and the connecting-rod which will be referred to as Joint 1 and the joint
between the connecting-rod and the slider referred to as Joint 2. A diagram of the slider-crank is
shown in Figure 8-2 and its dimensions are shown in Table 8-1. Similar to the previous
examples, it is assumed that the two joints consist of a steel pin and PTFE bushing and thus wear
on the pin is negligible when compared to the wear on the bushing. The material properties
including the friction and wear coefficients for the joint components are listed in Table 8-2.
Problem Definition
The design problem can be stated as follows: given that the crank is operated at 30rpm,
design the bushing (inner diameter and depth) for the two joints so that the maximum allowable
wear at the two joints is attained simultaneously (at the same number of cycles of operation).
This problem can be stated mathematically as follows:
126
2
1 2 1 2 1 1 1 2 2 2
1 2
1 2
Minimize : , , , , ,
Such that : ,
,LB UB
LB UB
f d d h h cyc d h cyc d h
d d d d
h h h h
(8-1)
where cyc1 and cyc2 are the number of operation cycles required for wear to accumulate to the
maximum allowable amount for Joint1 and Joint2, respectively, and the objective function f is
the square of the difference between cyc1 and cyc2. In Eq.(8-1), the design variables are the
dimensions of the bushing; i.e., the diameter and the depth of the bushings (d1 d2 h1 h2). The
variables are constrained between their corresponding upper (dUB hUB) and lower bounds (dLB
hLB).
The solution of the design problem will be decomposed into two parts. In the first part, the
analysis of the slider-crank with two joints wearing will be discussed. This will involve the
integration of the dynamic analysis of the mechanism and the wear analysis at the two joints. The
second part will involve the solution of the optimization problem stated in Eq.(8-1). As will be
seen later, the objective function is quite expensive to evaluate and thus a surrogate based
optimization technique will be employed.
Analysis of the Slider-Crank with Multiple Joints Wearing
The procedures presented for the analysis of multibody systems with joint wear consisted
of two parts, namely; 1) dynamic analysis that accounts for joint clearance and 2) wear analysis
to predict the amount of wear based on the preceding dynamic analysis. The dynamic analysis
involves assembling the differential algebraic equations (DAE) of motion in which the system
kinematic constraints and applied loads should be specified. For this particular example the
kinematic constraints can be expressed as shown in Eq.(8-2). In Eq.(8-2) the generalized
coordinates are similar to those described in Figure 3-6. It is worth mentioning that only the
driving constraint, the crank’s constraint to the ground, and the rotational constraint of the slider
127
appear in Eq. (8-2). The kinematic constraints that would normally have represented Joint1 and
Joint2 (for perfect joints) are now replaced with force constraints described in Eq.(3-16).
1 1 1
1 1 1
3
3
1
cos 0
sin 0
= 0
0
0
x l
y l
y
t
Φ (8-2)
The kinematic constraint and the force constraints can be assembled into the DAE which in
turn can be solved to reveal the system dynamics. Representative results from the dynamic
analysis of this system are shown in Figure 8-3, Figure 8-4 and Figure 8-5. For purposes of
illustration and comparison, bushing diameters of 40mm and clearances of 0.0005mm was used
(for both joints) to generate these results. In Figure 8-3 the reaction forces for Joint1 and Joint2
have been plotted. The force response for Joint1 is identical to what was obtained for the
dynamics based on the contact force model and the EFM based model (see Figure 6-6). This was
expected since the clearances in the two joints are small enough so that the systems are also
identical. In Figure 8-4, a plot of the incremental sliding angle is shown for both the joints. This
is different from the incremental siding distance which would be a function of the bushing/pin
radius. It should be pointed out that for a larger radius the incremental sliding distance would
increase and so would the wear. The locus of the center of the contact region (for one cycle) is
shown in Figure 8-5. For the same reasons, these results are once again identical to those
observed in Figure 5-4 for the contact force based model and Figure 6-7 for the EFM based
model.
Solution of Optimization Problem
Upon careful examination of Eq.(8-1), one will realize that the objective function is
extremely expensive to evaluate. This is because determining a single value of cyc1 or cyc2 for
128
any combination of the joint diameters and depth (d1 d2 h1 h2) requires a complete wear analysis
involving several thousand cycles. Since numerous evaluation of the objective function is
required during optimization, the current implicit form of the objective function will render the
optimization problem unreasonably costly.
Instead of conducting the optimization using the current form of the objective function,
also called the high fidelity model/function, a surrogate based optimization approach [91-96] can
be employed. This approach entails constructing surrogate models, such as the response surface
approximations [97-99], support vector regression [100 101], and kriging [102 103], using data
obtained from the high-fidelity models. The surrogates replace the high-fidelity objective
functions (and constraints) and thus offer fast approximations of the objective functions (and
constraints) at other locations in the design space. The consequence is that the speed of
optimization is substantially increased but at the cost of accuracy. There are however procedures
that can be used to improve the surrogate based optimization results such as the trust region
approach in which construction of the surrogate is focused in regions of possible optima based
on previous optimization results [96 104].
The process of drawing data from the high-fidelity models to construct the surrogate model
is referred to as the design of experiments (DOE). It should be mentioned that the choice of DOE
will have a great influence on the quality of the surrogate model. The reader is referred to the
works by Simpson et al [105], Giunta et al [106] and Goel [107] for a detailed discussion on the
subject.
Despite the various possible surrogate models available, for this example, the objective
function will be replaced by a response surface approximation. Furthermore the design points are
selected by a space filling technique called Latin Hypercube Sampling (LHS) [108]. For
129
convenience the surrogate toolbox developed by Viana [109] was used to generate the DOE as
well as the surrogate models.
For the current slider-crank problem the maximum allowable wear depth for both joints
was set at 2mm and the design range for the bushing diameters (for the two joints) was set
between 20mm and 70mm, whereas the range for the bushing depth was set between 10 mm and
50mm. These specifications are summarized in Table 8-3. The design space is defined as any
combination of the two variables with the two ranges. A plot of the design points, generated
using LHS is shown in Figure 8-6. In this plot the design space is boxlike with both variables
(diameter and depth) normalized between 0 and 1. It is emphasized that this DOE was used to
construct the surrogate models for cyc1 and cyc2 in Eq.(8-1).
In Figure 8-7 and Figure 8-8, the response of cyc1 and cyc2 generated using the constructed
surrogate models for cyc1 and cyc2 are shown. It can be inferred from both plots that as the
bushing depth increases, cyc1 and cyc2 also increase. This is reasonable since a longer bushing
depth would mean that the contact pressure responsible for wear is distributed over a longer
length. As a result, the rate of change of the wear depth would reduce and cyc1 and cyc2 would
increase. It can also be inferred from both figures that as the diameter increases, cyc1 and cyc2
decreases. This observation seems counterintuitive since a larger diameter would generally mean
that the resulting contact pressure is distributed over a greater perimeter and thus dictating that
rate of change of the wear depth would reduce causing cyc1 and cyc2 to increase. In reality
however, an increase in the diameter will cause the incremental siding distance to increase which
will in turn cause the rate of wear depth to increase and cyc1 and cyc2 to decrease. The later
behavior is however not as strong as the former. It is therefore true to state that the bushing depth
has a dominating effect on the value of cyc1 and cyc2.
130
With the surrogates for cyc1 and cyc2 available, the optimization problem in Eq.(8-1) can
be solved. For this problem a standard optimizer in the Matlab software was used to solve the
problem. The results from the optimization is shown in Table 8-4. The combination of bushing
dimensions (diameter and depth) for both joints necessary to minimize the difference between
cyc1 and cyc2 are shown in the table. The difference from the optimization is zero cycles which
means that (according to the surrogate models) the maximum allowable wear depth in both joints
will be achieved simultaneously. It is however expected that the results are not dead accurate
since they were obtained using surrogate models which provide approximates for the high-
fidelity model. Listed in Table 8-5, are the results generated using the high-fidelity model for the
combination of bushing dimension at the optimum solution. For the optimum configuration the
high-fidelity model predicted a difference of 4000 cycles (3.3%) between cyc1 and cyc2. The
ideal condition would require that the difference be equal to zero. Also the high-fidelity model
predicted a difference of 2165 cycles and 1835 cycles from the surrogate for cyc1 and cyc2
respectively. These differences are attributed to the approximate nature of the surrogate model.
These results can however be improved by using the trust region approach [96,104]
Summary and Concluding Remarks
In this Chapter an illustration of the application of the procedure to analyze multibody
systems with joint wear was presented. The application involved the design of a slider-crank
mechanism so as to allow the maximum allowable wear depth at two joints to be attained
simultaneously. This was restated as an optimization problem that would involve minimizing the
difference in the number of cycles required to attain the maximum wear depth for both cycles.
The solution was obtained by minimizing a surrogate model that was constructed in place of the
actual objective function.
131
The application demonstrates an example of how the procedures presented can be use to
improve the design of planar multibody system with joint wear. Other examples include
minimizing joint wear by changing the system mass and dimensions, reducing out of plane
motion cause by the wear by redesigning the system, predicting amount of wear at the joints of
system for scheduling maintenance or issuing warranties etc. The procedures may therefore be a
useful tool for the designer.
132
Table 8-1. Dimension and mass parameter for slider-crank mechanism Length (mm) Mass (g) Moment of inertia (kg-m2) Crank 1.00 10.00 45.00Connecting Rod 1.75 15.00 35.00slider -- 30.00 --
Table 8-2. Material properties for the joint components Pin Bushing Young’s modulus 0.29 0.38Poisson ratio 206.8 GPa 0.5 GPaWear coefficient (pin & bushing[75] 5.05x10-4 mm3/NmFriction coefficient (pin & bushing) [46] 0.13
Table 8-3. Parameter and design space specifications for the optimization Joint 1 Joint2 Maximum allowable wear depth 2 mm 2 mmBushing diameter design range 20 mm ≤ d1 ≤ 70 mm 20 mm ≤ d2 ≤ 70 mmBushing depth design range 10 mm ≤ h1 ≤ 50 mm 10 mm ≤ h2 ≤ 50 mm
Table 8-4. Solution of optimization problem (Eq. (8-1)) Value Value of objective ( f ) 0 cycles
Bushing diameter Joint 1 ( 1d ) 46.58 mm
Bushing diameter Joint 2 ( 2d ) 48.60 mm
Bushing depth Joint 1 ( 1h ) 37.25 mm
Bushing depth Joint 2 ( 2h ) 14.00 mm
Table 8-5. Comparison of results between surrogate and high-fidelity model cyc1 cyc2 Difference Surrogate 119435 119435 0 (0%)High-fidelity (actual wear simulation) 121600 117600 4000 (3.3%)Difference 2165 (1.8%) 1835 (1.5%) --
133
Figure 8-1. Backhoe system with three main revolute joints.
Figure 8-2. Slider-crank mechanism with two wearing joints wearing.
Crank
Connecting rod Wearing Joint 1
Ground Ground
SliderWearing Joint 2
Joint 3 Joint 1
Joint 2
Dipper
Boom
Bucket
134
0 1 2 3 4 5 60
1000
2000
3000
4000
5000
Crank Position (rad)
Fo
rce
(N
)Joint Force for Joint 1 & Joint 2
Joint 1Joint2
Figure 8-3. Joint reaction force for Joint1 and Joint 2
0 1 2 3 4 5 60
0.005
0.01
0.015
0.02
0.025
0.03
Crank Position (rad)
Incr
em
en
tal S
lidin
g A
ng
le (
rad
)
Incremental Sliding Angle
Joint 1Joint 2
Figure 8-4. Incremental sliding angle Joint 1 and Joint 2.
135
-30 -20 -10 0 10 20 30-30
-20
-10
0
10
20
30
x-coordinate (mm)
y-co
ord
ina
te (
mm
)Locus of the Center of the Contact Region (Joint1)
PinBushing
-30 -20 -10 0 10 20 30
-30
-20
-10
0
10
20
30
x-coordinate (mm)y-
coo
rdin
ate
(m
m)
Locus of the Center of the Contact Region (Joint2)
PinBushing
Figure 8-5. Locus of the center of the contact region. a) Locus for Joint 1 b) Locus for Joint 2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Diameter
De
pth
LHS (Normalized Design Space)
Figure 8-6. DOE used to construct surrogate for cyc1 and cyc2.
136
0
0.5
1
0
0.5
10
1
2
3
4
x 105
Diameter
Number of Cycles to Reach Max. Allowable Wear (Joint 1)
Depth
Cyc
les
Figure 8-7. Response generated using the surrogate for cyc1.
0
0.5
1
0
0.5
10
2
4
6
8
x 105
Diameter
Number of Cycles to Reach Max. Allowable Wear (Joint 2)
Depth
Cyc
les
Figure 8-8. Response generated using the surrogate for cyc2.
137
CHAPTER 9 SUMMARY AND FUTURE WORK
In this Chapter, a summary of the main ideas in this work will be presented. In addition,
suggestions for future research will be discussed.
Summary and Discussion
Analysis of planar multibody systems with revolute joint wear was the central theme of
this work. Two procedures to analyzing such systems were developed. In the first procedure the
analysis is based on a contact forces law, used to predict the joint reaction forces, and the finite
element method employed for the purpose of wear prediction. In the second procedure the elastic
foundation model is used to determine the contact forces as well as for the wear prediction. The
basic framework for theses procedures consists of three main parts. These are; 1) modeling a
non-ideal revolute joint that would replicate the behavior of a joint with clearance, 2) developing
an efficient procedure to predict wear and 3) integrating the wear procedure into the dynamic
analysis of multibody systems. The three parts will briefly be summarized in the context of the
two procedures.
For the two procedures, a non-ideal revolute joint (planar) was modeled using a procedure
that closely resembles a real non-ideal joint. Unlike the ideal joint which uses kinematic
constraints to restrict the motion of the joint components, the non-ideal joint uses force
constraints to guide the motion of the joint components. The procedure assumes that the joint
components can exhibit three possible configurations, namely: 1) free-flight, 2) impact and 3)
following motion. In the case of the free-flight, no contact occurs between the components. This
configuration is modeled by requiring that the contact/reaction force be zero. Thus no restriction
is imposed on the motion of the joint components. For the impact and following motion, contact
138
between the joint components is established. Contact force is thus developed based on the
amount of penetration experienced during contact.
In the first procedure, the contact force is computed using a modified form of the Hertz
contact law, whereas in the second procedure the contact force is determined using the elastic
foundation model. In the latter case, the contact pressure between the joint components is first
determined, and thereafter, the resultant contact force can be evaluated. In both procedures,
however, the magnitude of the force is a function of penetration between the joint components.
To enforce the joint constraint, the contact force is applied on the components to prevent
further penetration. Due to this contact force, the motion of the components is restricted. The
implementation of this joint was illustrated with the aid of a slider-crank mechanism with a non-
ideal revolute joint between the crank and the connecting rod. Using the mechanism, it was
shown that the system dynamics is altered as the joint clearance size varies. The two procedures
showed reasonable agreement in the force magnitude for a small clearance but appreciable
disagreement in the magnitude for relatively large clearances. However, the trend in the force
distribution for the force distribution remained the same.
A methodology to predict wear that is built upon a widely used iterative prediction
procedure was discussed. In the iterative procedure, the wear occurring at the contact interface
between two bodies that are in relative motion is determined based on Archard’s wear model.
Incremental wear is determined at each iteration and accumulated up to the desired number of
cycles. In every iteration (or several iterations) the geometry is updated to reflect the worn
material. In the first procedure (for analysis of multibody systems with joint wear) the finite
element method is used to determine the contact pressure necessary for wear calculation,
whereas in the second procedure the contact pressure is readily available from the analysis used
139
to determine the contact force. In order to reduce computational costs, an adaptive extrapolation
scheme is used. The two main contributions in the wear prediction procedures include: 1)
development of an adaptive extrapolation scheme in which a systematic approach in selecting the
extrapolation was proposed and 2) the development of a updating procedure that maintains
smooth boundary and ensures regularity of the finite element mesh at the boundary (in the case
of the FEM based procedure).
Finally the analysis of multibody systems with joint wear is completed by integrating the
wear prediction procedures into the dynamic analysis. In the integration process, the dynamic
analysis provides information about the system dynamic for multibody systems with joint
clearance. This information includes, contact force (and contact pressure for the second
procedure), location of contact and the incremental siding distance. On the other hand, the wear
prediction determines the amount of wear at the joints based on the result of the system dynamics
and the contact geometry is updated after every cycle to reflect the worn out material. The
changes in the geometry due to the wear are accounted for in the dynamic analysis by updating
the clearance. In this case the clearance is a vector that is dependent on the bushing angular
coordinate. Thus any change in the system dynamics (contact force and contact locations) due to
the wear is captured in the integration process.
The use of the integrated model for the two procedures was demonstrated using a slider-
crank mechanism in which the joint between the crank and the connecting rod was allowed to
wear. It was found that the two procedures predicted different wear profiles and different
maximum wear depth but the same location of the maximum wear and the same wear volume.
The difference in the wear profiles is attributed to the simplification in the EFM where contact
pressure is determined without considering how displacements at one contact region affects the
140
displacement at other regions (independent springs). On the other hand the equality of the wear
volume for the two models is a manifestation of the equality in the force predicted by the two
models.
Experiments were conducted to validate the two procedures and to assess the performance
between the two procedures. An experimental slider-crank mechanism was used to facilitate the
validation. The validation revealed that both procedures predicted reasonably accurate wear
volumes and location of maximum wear. With regard to the wear profile and maximum wear
depth, the FEM based procedure gave more accurate predictions than the EFM based procedure.
On the other hand the EFM based procedure had a much faster computation time than the FEM
based procedure. The conclusion that can be drawn from the validation is that the FEM
procedure will be a better procedure in the analysis of multibody systems with joint wear when
the computational costs in not of concern. On the other hand if the cost is of concern and only
qualitative information about the system is needed then the EFM based procedure will be the
suitable choice. Other scenarios will require a compromise on either accuracy or computational
costs.
Future Work
This work has primarily focused on the analysis of systems for which the joints are non-
lubricated. When considering real systems, this condition can be considered to be among the
extreme cases since in most real systems some type of lubrication is used. Thus, this works will
serve as a starting or as a reference point for other realistic cases. Naturally, the lubricated joint
should be the next case considered. In this case the lubricant is expected to affect the system
dynamics. The dynamic analysis of multibody systems with joint lubricant has already been
explored [14 15 18, 111,112]. It is also expected that the lubricant will affect the wear
mechanism and thus the wear rate.
141
Another area interest that will follow on from the lubricated joint is the lubricated joint
with trapped impurities. This case is probably the most realistic case of all three mentioned. In
most joint encountered, the joint is lubricated. However, while the system is in operation,
impurities such as sand may become trapped between the joint components together with the
lubricants. The presence of the impurities in the joint will most certainly have an effect on the
wear and possible an effect on the system dynamics (could serve to increase the friction or act as
an additional lubricant).
The final suggestion for future research is the analysis of spatial multibody systems with
joint wear. This can be coupled with joint lubricants with impurities to make it the most general
and realistic case.
142
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BIOGRAPHICAL SKETCH
Saad Mukras was born in Nairobi, Kenya. He was raised in Nairobi and partially in
Gaborone, Botswana, where he completed his secondary education. He then joined University of
Botswana and then transferred to Embry Riddle Aeronautical University in Daytona Beach,
Florida. There, he studied aircraft engineering technology and received his bachelor’s degree in
2003. In 2004 he joined the University of Florida to pursue a master’s and later a doctorate
degree in mechanical engineering. He worked under the supervision of Dr. Nam-Ho Kim,
completing several research projects, earning his masters degree in 2006 and doctorate in 2009.
