1

KINEMATIC AND DYNAMIC BEHAVIOR OF A WEARING JOINT IN A CRANK-SLIDER MECHANISM

By

NATHAN A. MAUNTLER

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

2

2009 Nathan A. Mauntler

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To my wonderful wife Nicole

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ACKNOWLEDGMENTS

I would first like to thank my advisors Tony Schmitz and Greg Sawyer for getting me into,

keeping me in, and getting me out of grad school. I would also like to thank Nam Ho Kim and

Elif Akcali for serving on my graduate committee. Saad Mukras deserves recognition for his fine

work on the Coupled Evolution Wear Model. I would like to thank the members of the Tribology

Laboratory and the Machine Tool Research Center at the University of Florida for their support.

Specifically, Jim Keith, Andres Duarte, and Hyo Soo Kim deserve recognition for their

contributions to this project.

I would like to thank John Deere and the National Science Foundation for their financial

and technical support. In particular I would like to thank Michael Messman (now of Clemson

Universitys i-CAR center) and Ryan Blodig of the Advanced Vehicle Development Lab for

providing, and then patiently explaining, the shear beam load cell used in this project.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS ...............................................................................................................4

LIST OF TABLES ...........................................................................................................................7

LIST OF FIGURES .........................................................................................................................8

ABSTRACT ...................................................................................................................................13

CHAPTER

1 INTRODUCTION AND SCOPE ...........................................................................................14

Introduction and Motivation ...................................................................................................14 Scope of the Project ................................................................................................................14

2 PREDICTED BEHAVIOR OF THE MECHANISM AND JOINT UNDER STUDY .........16

Idealized Kinematics and Dynamics of a Crank-Slider Mechanism ......................................16 Mechanism Kinematics ...................................................................................................16 Modeling Ideal Joint Forces ............................................................................................19 Sensitivity of the Joint Dynamics to Mechanism Parameters .........................................21

Characterizing Error Motions of the Revolute Joint ...............................................................22 Modeling Compliance in a Revolute Joint .............................................................................23 Modeling Mechanism Mechanics with Consideration of Non-Ideal Joints ...........................26 Predicting Wear in the Joint of Interest ..................................................................................27

Friction and Wear Behavior of PTFE under Dry Sliding Conditions .............................27 Modeling Joint Wear in Mechanisms ..............................................................................28

3 TEST BED DESCRIPTION AND EXPERIMENTAL PROCEDURES ..............................39

Test Apparatus ........................................................................................................................39 Crank-Slider Mechanism .................................................................................................39 Manipulating Joint Forces ...............................................................................................40 Instrumentation ................................................................................................................41

Spindle Encoder .......................................................................................................41 Pin Load Cell ............................................................................................................41 Capacitance Probes ..................................................................................................46

Operation of the Test Bed ................................................................................................47 Sample Preparation and Characterization ................................................................47 Mechanism Preparation ............................................................................................48 Operation and Data Collection .................................................................................49

4 RESULTS FROM A SPRING DOMINATED WEAR TEST ...............................................61

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Joint Dynamics .......................................................................................................................61 Wear Characterization ............................................................................................................62 Comparison between Experimental and Modeled Wear Results ...........................................63

5 ADDITIONAL INSTRUMENTATION OF THE TEST BED ..............................................70

Direct Measurement of Spring Forces Using a Uniaxial Load Cell .......................................70 Direct Measurement of Stage Position Using a Linear Displacement Measuring

Interferometer ......................................................................................................................70

6 COUPLED JOINT WEAR AND DYNAMICS UNDER INERTIAL LOADING................74

Crank-Slider Test Bed Results................................................................................................74 Ex-Situ Wear Measurements ...................................................................................................76 Discussion ...............................................................................................................................76

7 SENSITIVITY OF THE DYNAMIC SYSTEM TO CHANGES IN THE MECHANISM AND JOINT PARAMETERS .......................................................................88

Repeatability of Dynamic Test Results ..................................................................................88 Sensitivity to Changes in the Mechanism Operating Conditions ...........................................89

Influence of Varying the Crank Speed ............................................................................89 Influence of Increasing the Stage Mass ...........................................................................90 Influence of Increasing Spring Rate ................................................................................92

Short Dynamic Tests Conducted Using a Previously Worn Bushing ....................................93 Discussion ...............................................................................................................................94

8 CONCLUDING REMARKS................................................................................................114

APPENDIX

A MATLAB SIMULATION OF AN IDEALIZED CRANK-SLIDER MECHANISM .........116

B MATLAB SIMULATION OF A SIMPLE LINE CONTACT MODEL .............................119

C MATLAB SIMULATION OF SECONDARY CAPACITANCE PROBE SENSITIVITY ......................................................................................................................120

D INVESTIGATING THE PLANAR MECHANISM ASSUMPTION ..................................122

Motivation .............................................................................................................................122 Measurement and Analysis ...................................................................................................122 CMM Measurement Results .................................................................................................123 Discussion .............................................................................................................................124

LIST OF REFERENCES .............................................................................................................129

BIOGRAPHICAL SKETCH .......................................................................................................132

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LIST OF TABLES

Table page 3-1 Crank-slider mechanism parameters. .................................................................................30

3-2 Contact model simulation parameters ................................................................................37

4-1 Wear test parameters ..........................................................................................................65

4-2 Experimental and model predictions of bushing wear amount for initial wear test. .........69

7-1 Test conditions of five short dynamic repeatability tests...................................................95

D-1 Pin axis vector components as imported into Geomagic Studio software. ......................127

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LIST OF FIGURES

Figure page 2-1 The crank-slider is modeled as a planar mechanism. ........................................................29

2-2 The follower angle is defined as the angle between the negative X and follower axes. ...30

2-4 Stage velocity as a function of crank angle at a spindle speed of 30 rpm. ........................31

2-5 Stage acceleration with respect to crank angle. .................................................................32

2-6 Sliding velocity at the joint of interest is plotted versus crank angle for a constant spindle speed of 30 rpm and a pin-bushing joint diameter of 19.05 mm...........................32

2-7 Free body diagrams used in the dynamic analysis. ............................................................33

2-8 Predicted cyclic joint force magnitude profiles at varying crank speeds. ..........................33

2-9 Predicted joint force magnitude cyclic profile at varying spring rates. .............................34

2-10 Sensitivity of the joint force profile to changes in the stage mass. ....................................35

2-11 Sensitivity of the predicted joint force profile to changes in the Y-direction location of the stage center of mass. ................................................................................................35

2-12 Static cylinder contact model. ............................................................................................36

2-13 Contact model assumes flat deformation profile and semi-elliptical pressure profile.......36

2-14 The contact model may be modified to accommodate internal line contacts by multiplying the inner radius of the bushing by (-1). ..........................................................37

2-15 Results from the contact simulation indicate contact pressures less than 1.5 MPa, contact widths less than 12 mm, penetration values less than 30 mm, and contact stiffness values on the order of 2.1x106 N/m. ....................................................................38

3-1 The crank-slider bed used to perform experimental tests. .................................................50

3-2 Kinematic components of the crank-slider mechanism. ....................................................51

3-3 The cyclic joint load profile may be adjusted through the addition of stage mass or by adding springs between the slide stage and table. .........................................................51

3-4 A hollow rotary encoder clamped to the spindle provides crank position and speed information. ........................................................................................................................52

3-5 Custom steel pin load cell. .................................................................................................52

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3-6 Each load cell channel is constructed from two strain gage shear rosettes totaling four gages. ..........................................................................................................................53

3-7 The two shear rosettes associated with a single channel are mounted on diametrically opposite sides of the necked portion of the pin..................................................................53

3-8 Wheatstone bridge circuit diagram for a single pin force channel. ...................................54

3-9 Two strain gages oriented symmetrically about an axis of interest combine to form a shear rosette. ......................................................................................................................54

3-10 General transverse shear loading of the pin transducer. ....................................................55

3-11 General bending moment loading of the pin transducer. ...................................................55

3-13 Axial loading of the pin transducer. ...................................................................................56

3-14 Pin load cell channel coordinates. ......................................................................................56

3-15 Forces measured by the pin load cell can be transformed into a global coordinate system using the measured crank angle . .........................................................................56

3-16 Global coordinate forces may be further transformed to a coordinate system fixed to the bushing and follower link using the calculated follower angle. ..................................57

3-17 Capacitance probes measure the location of the pin from fixed locations on the follower link. ......................................................................................................................57

3-18 The capacitance probe in this picture is sensitive not only to pin motion along its axis, but also secondarily in the orthogonal direction........................................................58

3-19 Estimated cross sensitivity error as a function of off-axis motion of the pin. ...................58

3-20 Geometry of the test bushings............................................................................................59

3-21 The bushing center is defined as the intersection of two lines drawn between opposing sets of wear collection grooves. .........................................................................59

3-22 Preparation of the test bed is performed in steps. ..............................................................60

4-1 Joint force magnitude predictions and measured data. ......................................................65

4-2 CEWM simulation of the effect of stage friction on the dynamics at the joint of interest. ...............................................................................................................................66

4-3 Discrepancies between the predicted and measured global force component profiles were likely due to an offset between the follower-stage revolute joint and the spring load point of application. ...................................................................................................66

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4-4 Capacitance probe measurements of the pin displacement relative to the bushing center showed similar vibration content as the pin force measurements. ..........................67

4-5 Evolution of the cyclic pin path relative to the bushing center with increasing cycle number indicated a change in the bushing shape in the first 1000 cycles. ........................67

4-6 Contact pressure evolution predicted by the CEWM. .......................................................68

4-7 Bushing wear scar mapped by plotting the change in bushing radius as calculated from a center created by the intersection of lines connecting opposing wear collection grooves. .............................................................................................................69

5-2 A laser interferometer is used to measure the position of the stage as a function of time. ...................................................................................................................................72

5-3 The stage location is defined relative to the maximum extension of the mechanism and as the minimum distance measured by the interferometer. ......................73

6-1 The evolution of high frequency dynamics with wear can be observed from single cycle joint force magnitude plots. ......................................................................................78

6-2 Capacitance probe data from the wear test shows an evolution of both the bushing shape and high frequency motion of the pin corresponding to high frequency load dynamics. ...........................................................................................................................79

6-3 Higher frequency dynamic content appears to evolve only in the X-direction of the bushing coordinate system. ................................................................................................80

6-4 Capacitance probe output as a function of crank angle. ....................................................81

6-5 Small changes in the stage path profile are observed by removing the gross motion profile from the interferometer output. ..............................................................................82

6-6 Fast Fourier transform (FFT) of bushing-coordinate force results. ...................................83

6-7 Capacitance probe FFT results...........................................................................................84

6-8 Interferometer FFT results. ................................................................................................84

6-9 Change in radius as a function of bushing angle as measured by the three axis CMM.....85

6-10 Capacitance probe measurements from two short tests intended to challenge the assumption of a rigid pin....................................................................................................85

6-11 Comparison between measured and modeled bushing force profiles. ...............................86

6-12 Due to incorrect zeroing of the pin load cell during the wear test, a separate short dynamic test was run under identical operating conditions. ..............................................87

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7-1 Pin force magnitude profiles from a series of five repeat dynamic tests. ..........................96

7-2 Capacitance probe measurements from a series of five repeat dynamic tests. ..................97

7-3 Joint force magnitude single cycle profile at varying crank speeds. .................................98

7-4 Global coordinate force components at varying crank speeds...........................................99

7-5 Single cycle capacitance probe readings at varying crank speed. ...................................100

7-6 Single cycle joint force magnitude plots with changing stage mass. ...............................101

7-7 Capacitance probe measurements from a series of tests varying the mass of the dovetail slide stage. ..........................................................................................................102

7-8 Force magnitude data from a series of short dynamic tests in which varying amounts of mass were set on the pillow blocks in order to increase stage mass. ..........................103

7-9 Global force components subject to varying stage mass. ................................................104

7-10 Changes in the capacitance probe pin displacement measurements were difficult to discern as the stage mass was increased. .........................................................................105

7-11 Force magnitude at the joint of interest subject to varying stage mass and using a worn bushing. ...................................................................................................................106

7-12 Spring rates were found to be linear and with comparable preloads. ..............................107

7-13 Contact force magnitude subject to increasing spring load. ............................................108

7-14 Capacitance probe output with increasing spring load. ...................................................109

7-15 Global force components subject to increasing spring load. ...........................................110

7-16 Joint force magnitude measurements from five repeat dynamic tests using a worn bushing. ............................................................................................................................111

D-1 The FARO ARM articulating CMM was bolted to the test bed optical table to collect point cloud data. ...............................................................................................................125

D-2 Point clouds on the pin surfaces were carried out by bringing the CMM probe tip into contact with the pin, depressing the probe trigger, and moving the probe tip over as much of the exposed pin surface as could be reached without losing contact. ................125

D-3 Point clouds were imported into Geomagic Studio 9 software for analysis. ...................126

D-4 The imported point clouds were then fit with least-square cylinders. .............................126

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D-5 The joint component angular misalignment that will force the pin into contact with the edges of the bushing may be estimated as the ratio of the joint clearance c to the bushing width wb. .............................................................................................................128

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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

KINEMATIC AND DYNAMIC BEHAVIOR OF A WEARING JOINT IN A CRANK-SLIDER

MECHANISM

By

Nathan A. Mauntler

August 2009 Chair: Tony L. Schmitz Cochair: W. Gregory Sawyer Major: Mechanical Engineering

When modeling joint wear behavior, it is often necessary to consider the coupled evolution

of wear geometry and contact load conditions. In this way, a mechanical designer is afforded a

means of predicting not only length of life, but performance degradation over the life cycle.

Unfortunately, such models are often computationally expensive and require simplifying

assumptions regarding the behavior of the rest of the mechanism, making experimental

validation a challenge. This dissertation describes the design and evaluation of an instrumented

crank-slider mechanism with a single wearing bushing. In the construction of this device, care

was taken to isolate friction, wear, and error motions to the joint of interest: in this case, the

revolute joint connecting the crank and follower arms. Experimental results are presented on the

coupled evolution of joint wear and machine kinematics and dynamics. Comparisons are drawn

between experimental results and predictions made by a simple, idealized dynamic model as well

as a contemporary coupled wear model.

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CHAPTER 1 INTRODUCTION AND SCOPE

Introduction and Motivation

Software currently being developed at the University of Florida is being used to

incorporate wear models into dynamics simulations and finite element analyses. This Coupled

Evolution Wear Model (CEWM) uses the finite element method and known material wear rates

to iteratively update contact pressures and geometry. Ultimately this software package will be

made available to designers in order to predict the influence of wear on the performance of

moving parts over the course of their useful lives. The CEWM is currently being applied to a

variety of complex kinematic scenarios. One such application is a crank slider mechanism, where

the integrity of the joint between the crank and connector rod can influence the ability to

accurately control the location of the slide. In order to validate the CEWM in this scenario, a

crank slider test bed is being developed in this work in order to complete controlled wear tests at

a single revolute joint which connects the crank and follower links.

Scope of the Project

The scope of this project is to design, build, and evaluate the crank slider test bed to be

used for comparison against the CEWM software predictions. The test bed evaluation will be

comprised of: 1) identification of the error motions caused by manufacturing and assembly

imperfections and wear in the joint under test; 2) identification of the system dynamics from the

same sources; and 3) evaluation of the actual wear amount and wear profile for the joint under

test. The test bed is based on the classic crank slider geometry, but the primary design

consideration is isolating all wear to a single joint, the revolute joint which connects the crank

and follower links. Naturally, in order to isolate wear to this joint, design constraints include

minimizing friction end error motions in all other joints.

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The crank-follower revolute joint is comprised of a steel pin clamped in the crank link and

a bushing clamped in the follower link. Orthogonally-mounted capacitance probes are used to

observe the evolution of the path of the steel pin axis relative to the bushing centerline as the

bushing material wears. Simultaneously, the dynamic forces imparted on the bushing are

monitored via a load cell built into the steel pin. This load cell is composed of two full-bridge

strain gage arrays which monitor the transverse shear load in the pin while cancelling axial,

bending, and torsion strain signals. An encoder enables the evolving force and wear data to be

plotted as a function of the crank position. Furthermore, an air bearing slide, thrust air bushings,

and a high-stiffness precision spindle limit friction and wear to the pin-bushing joint as much as

possible. This isolation helps to idealize the motion of the mechanism and reduce any

confounding influences on the measured dynamics at the joint of interest.

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CHAPTER 2 PREDICTED BEHAVIOR OF THE MECHANISM AND JOINT UNDER STUDY

The goal of this project is to create a test bed with which revolute joint mechanics and

wear can be studied in the context of a working mechanism. This chapter provides a brief

background into the three constituent fields of study: mechanism kinematics and dynamics,

revolute joint behavior, and joint wear modeling.

Idealized Kinematics and Dynamics of a Crank-Slider Mechanism

The first step in predicting the behavior of the test bed is to perform an idealized

Newtonian mechanical analysis. In this section, all joints are assumed to be planar, frictionless,

and have a single degree of freedom. Additionally, all components are assumed to be rigid.

Finally, the angular velocity of the crank link is assumed to be constant. A MATLAB program

file used to demonstrate the idealized kinematics and dynamics of a crank-slider is included in

Appendix A.

Mechanism Kinematics

Kinematic analysis of an idealized crank-slider mechanism is a common exercise

performed in many introductory mechanics courses. The following analysis is based on the

analytical methods discussed by Shigley and Uicker [1] and Wilson and Sadler [2]. A simple

crank-slider mechanism under inertial and spring loading is shown in Figure 2-1. This

mechanism has four bodies, including ground. In this analysis, the spring is considered solely as

a load source, rather than a separate body. For this idealized system, the crank angle , is

sufficient to specify the mechanism orientation. The mass centers of links 1 and 2 (the crank and

follower) are located at the midpoint between their respective revolute joints. The center of mass

of link 3 is located as shown in Figure 2-1B. Nominal dimensional and mass parameters for this

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crank-slider mechanism are listed in Table 2-1. These parameters are representative of the test

bed design described in Chapter 4.

Kinematic analysis of the mechanism begins by describing the positions of the centers of

mass of each body as a function of the crank angle. To carry out this step, it is convenient to

define the follower angle as the angle between the negative X-axis and the axis of the follower

link (Figure 2-2). The value of the follower angle with respect to the crank angle is calculated

using Equation 2-1. The X- and Y-axis coordinates of the link centers of mass, {xi,yi

3x

} for i = 1 to

3, may then be calculated using Equations 2-2 through 2-7. The X-axis stage location is

shown as a function of crank angle in Figure 2-3.

1 1

2

sin sin( )LL

=

(2-1)

11 cos( )2

Lx = (2-2)

11 sin( )2

Ly = (2-3)

22 1 cos( ) cos( )2

Lx L = + (2-4)

22 sin( )2

Ly = (2-5)

3 1 2cos( ) cos( )x L L = + (2-6)

3 3y L= (2-7)

The link center of mass velocity equations are then obtained by calculating the time

derivatives of Equations 2-2 through 2-7. It should be noted that the crank speed is assumed to

be constant. Additionally, since the value of 3y is a constant, the slide velocity has no

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component in the Y-direction. The velocity equations for the crank-slider links are shown in

Equations 2-8 through 2-13. The stage velocity is plotted with respect to the crank angle in

Figure 2-4 where the rotational velocity of the spindle and crank is 30 rpm.

( )

( )

1

2

2

1

2

cos

1 sin

LL

LL

=

(2-8)

11 sin( )2

Lx = (2-9)

11 cos( )2

Ly = (2-10)

22 1 sin( ) sin( )2

Lx L = (2-11)

22 cos( )2

Ly =

(2-12)

3 1 2sin( ) sin( )x L L = (2-13)

The link accelerations are then obtained by differentiating the velocity equations with

respect to time (Equations 2-14 through 2-19). These equations are considerably simplified due

to the assumption of constant crank velocity. The stage acceleration is plotted versus crank angle

in Figure 2-5 using a spindle speed of 30 rpm.

( )

( )( )

( )

( )

221

221

22

1

2212

2 21

2

cos1 sin

1 sinsin

1 sin

LLL

L LLL

L LL

=

(2-14)

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211 cos( )2

Lx = (2-15)

211 sin( )2

Ly = (2-16)

( ) ( )2 222 1 cos( ) cos( ) sin( )2Lx L = + (2-17

( )222 sin( ) cos( )2Ly = + (2-18)

( ) ( )2 23 1 2cos( ) cos( ) sin( )x L L = + (2-19)

Since the joint under investigation is the revolute joint between the crank and follower

links, the angular and sliding velocities between the joints pin and bushing are also relevant. In

order to calculate these velocities, it is convenient to define the angle as the angle (in radians)

between the crank and follower links (Equation 2-20). Differentiating this equation with respect

to time yields the joint angular velocity (Equation 2-21). Finally, given the nominal diameter of

the pin dj, the sliding velocity of the joint vj

=

may be calculated using Equation 2-22. The revolute

joint sliding velocity is plotted as a function of crank angle using a crank velocity of 30 rpm and

joint diameter of 19.05 mm in Figure 2-6.

(2-20)

= (2-21)

2j

j

dv = (2-22)

Modeling Ideal Joint Forces

A model of the ideal joint forces is formed from the free body diagrams shown in Figure 2-

7. Again, for the purposes of this idealized analysis, all joints are assumed to be frictionless, all

kinematic terms and inertial terms are assumed known, and the input crank angular velocity is

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assumed to be constant. In this discussion, the nomenclature Fij indicates the force F on body i

due to contact with body j. Inertial effects are effectively treated as loads according to

DAlemberts principle of inertial forces [1].

The spring load Fspr on the slide is calculated as the as the sum of a spring preload Fs0 and

the product of the linear spring rate ks and the extension of the spring from the pre-loaded

configuration (Equation 2-23). Considering the free body diagram of the stage (Figure 2-7A) and

summing moments about point a, the value of M34

0 1 2 3( )spr s sF F k L L x= + +

is calculated by Equation 2-24.

(2-23)

34 3 3 3 spr sprM m x L F y= + (2-24)

The contact forces at the joint of interest may then be obtained from the free body diagram

of the follower link and slide together. The value of Fx21 is obtained by summing forces in the X-

direction (Equation 2-25). This term is the contact force in the global X-direction on the bushing.

The global Y-direction force on the bushing, Fy21, is then obtained by summing moments about

the center of mass of the slide (Equation 2-26). In order to write Equation 2-26 compactly, Fy21 is

separated into external and inertial terms Fe (Equation 2-27) and Fi

21 2 2 3 3x sprF m x m x F= +

(Equation 2-27). Here g is

acceleration due to gravity. Contact forces experienced by the pin are equal and opposite to those

on the bushing.

(2-25)

21y e iF F F= + (2-26)

( )34 3 21 2 32

(2 )cos( )

spr spr xe

M F y y F y yF

L +

= (2-27)

( ) ( ) ( )( ) ( )( )2 22 2 2 3 2 2 3 2 2 3 2 2 3 22 cos( )

i

m x y y I m x x y y m g y x xF

L

+ + + + =

(2-28)

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Sensitivity of the Joint Dynamics to Mechanism Parameters

In this section, a series of simulations are performed in order to illustrate the sensitivity of

the ideal model to changes in several mechanism parameters. In each case, all parameters except

the single variable of interest are held constant. Additionally, unless otherwise specified, the link

length L3 and the spring load offset yspr

In Figure 2-9, the joint force magnitude is plotted versus crank angle under varying spring

rates. Simulations are performed with spring rate values of 0 N/m, 200 N/m, and 400 N/m. The

spring offset y

are assumed to be zero. Simulation conditions listed in

this section are indicative of conditions that may be achieved using the experimental apparatus

described in Chapter 3.

In Figure 2-8, the cyclic force magnitude profile is plotted at increasing crank speeds.

Simulations are performed at 15 rpm, 30 rpm, 45 rpm, and 60 rpm. As the crank speed increases,

acceleration of the stage mass is increased, which affects the shape of the force profile.

Additionally, larger variations in the force magnitude are seen with increasing crank speed. It

should be noted that under the range of crank speeds used in this group of simulations, the force

magnitude does not approach a value of zero at any point in the crank cycle. A condition of zero

force magnitude is an indication that the inertia of the stage overtakes the driving velocity of the

crank. If clearance exists between the revolute joint components, such a condition will result in

the loss of contact and subsequent impact at another location on the joint perimeter.

spr is set to 0 mm in Figure 9A and 55 mm in Figure 9B. In each case, the spring

preload is set to 0 N. It can be seen that at large spring rate values, the spring force can be made

to dominate the load profile. This can be advantageous in the initial stages of validating a

dynamic model as spring rates may be simpler to predict than inertial properties. While spring

forces and the location where the spring load is applied are easily measured, it can be difficult to

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accurately determine the center of mass of an assembly built from various materials and intricate

shapes.

The sensitivity of the crank slider dynamic system to changes in inertial properties is

illustrated in Figures 2-10 and 2-11. In Figure 2-10 a series of simulations are performed with the

stage mass set to 9 kg, 13 kg, and 17 kg. In Figure 2-11, the vertical distance from the follower-

stage revolute joint to the stage center of mass (L3

Characterizing Error Motions of the Revolute Joint

) is increased from 0 mm to 60 mm in 20 mm

increments. In both cases, changing the inertial characteristics can affect the shape and severity

of the force magnitude profile, although not to the same degree as varying the spring rate.

As noted, the previous analysis assumes that the kinematic and dynamic relationships

between links are governed by ideal revolute joints. In other words, each revolute joint is

constrained to a single rotational degree of freedom with its axis perpendicular to the page.

Additionally, the joint cannot support any motion about its axis. In reality, each revolute joint is

comprised of (at least) two separate bodies, each with its own nominal axis of rotation. In-plane

and out-of-plane errors in the physical joint are dependent both on flaws in the components as

well as misalignments between them. Much of the work surrounding the definition and

measurement of axis of rotation errors has arisen from the machining community, where the

accuracy of a feature being produced is heavily dependent on the quality of spindle motion

produced in a turning or milling operation.

Bryan et al [3] suggested that the five remaining degrees of freedom other than the

intended one be grouped into three categories: angular, axial, and pure radial. These three

categories were envisioned as motions of an imaginary centerline of a cylinder positioned in

space, nominally oriented in the direction of the spindle axis. Bryan later formalized these

motions as radial motion, axial motion, face motion, and tilt motion [4]. The American National

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Standard (which Bryan largely contributed to) for specifying these parameters is ANSI/ASME

B89.3.4M-1985 [5]. Here, axial motion at the centerline was differentiated from face motion at

some radial location from the nominal center. Radial motion was defined as normal to the

nominal rotation centerline, while tilt motion was defined as pitching and yawing of the axis of

rotation. These definitions were intended to replace the less-favored terms such as run-out and

face run-out, measurements of which could be influenced by multiple error motions as described

above.

These motion categories were then considered in terms of sensitive directions with

respect to a given machining operation and workpiece surface. Bryan defined the sensitive

direction as parallel to a line perpendicular to the ideal generated workpiece surface through the

instantaneous point of machining or gauging. Insensitive directions were defined as being

perpendicular to the sensitive direction.

In the context of this project, Bryans error motion definitions are most clearly applicable

to the crank spindle axis. However, a similar approach might be applied to the joint of interest by

considering the centerline of the ground steel pin as the nominal joint axis and treating the

relative location of the bushing centerline axis as an error motion. Furthermore, the notion of in-

plane error motions being sensitive directions and out-of-plane error motions being insensitive

directions is relevant to this study. Here, radial and tilt errors throughout the mechanism are

likely to be more pertinent than axial motions.

Modeling Compliance in a Revolute Joint

Error motions may be due to manufacturing defects, but can also be caused by the

compliance of the joint components. In order to simulate compliance in the revolute joint

discussed in this paper, a two-dimensional line contact model for cylinders is used [6]. In this

model, cylinder a is brought into contact with cylinder b with normal force Fn (Figure 2-12).

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Contact conditions are simulated using a composite radius R and composite Youngs modulus E

1' 1 1

a b

RR R

= +

(Equations 2-29 and 2-30). Here, the symbol represents the Poissons ratio for a given material.

One assumption of this model is that contact deformations result in a flat deformed face on each

body (Figure 2-13). While this is not an accurate assumption when the two bodies have

substantially different geometric and material properties, it serves adequately as a first order

estimate of contact stiffness. In order to accommodate the geometry of a revolute joint with this

model, the value of the internal radius of body b is made negative (Figure 2-14)

(2-29)

2 2' 1 1a b

a b

EE E

= +

(2-30)

The width of the deformed region w is calculated using Equation 2-31 from the contact

normal load, composite radius and modulus, and the contact length Lc

1/2'

'4n

c

F RwL E

=

. The maximum contact

pressure value is calculated using Equation 2-32. Due to the assumed flatness of the contact

region and the circular cross section of the bodies in contact, the pressure profile across the

contact is semi-elliptical with the distance s from the contact center (Equation 3-33).

Deformation at the contact may then be calculated using Equation 2-34. In the revolute joint

under consideration, the Youngs modulus of the steel pin is approximately three orders of

magnitude greater than that of the polymer bushing. Because of this, the joint deformation may

be thought of as a penetration of the pin into the bushing. The contact stiffness is then calculated

using Equation 3-35.

(2-31)

25

1/2'

max 'n

c

F EPL R

=

(2-32)

2

max2( ) 1 sP s Pw

=

(2-33)

2 22 1 8 1 81 1ln ln2 2

n a a b b

c a b

F R RL E w E w

= + + +

(2-34)

nci

FK

= (2-35)

Contact parameters representative of the joint under study in this work are listed in Table

3-2. Simulation results from the contact model are shown in Figure 2-15. This particular contact

model predicts the penetration depth of the pin into the bushing to be less than 30 m for normal

loads under 150 N. Interestingly, the contact stiffness Kci is predicted to be relatively insensitive

to normal load and on the order of 2.2x10-6 N/m. The MATLAB code used to perform these

calculations is included in Appendix B.

It should be noted that this model has several limitations. As previously discussed, the

assumption of a flat contact region may be inappropriate depending on the mismatch of

component material properties. Additionally, this model does not account for frictional shear

effects that would be present during revolute joint motion. In fact, the model is purely static and

does not consider dynamic concerns such as viscoelasticity. However, it is a simple and

analytical model that could be easily employed by the mechanical designer. Like the idealized

kinematic and dynamic model presented in the previous section, this contact model is intended to

provide a readily available basis for comparison with measured results rather than a fundamental

prediction of the joints compliant behavior.

26

Modeling Mechanism Mechanics with Consideration of Non-Ideal Joints

If the assumption of ideal revolute joint behavior breaks down, another approach must be

implemented to define the relative position of connected links. In other words, more general

formulations of the equations of motion and constraints are required. Such formulations are

common in computer-aided kinematics and dynamics simulations. One common approach as

described by Haug and Nikravesh [7, 8] is to define a system of equations which algebraically

define the driving (D) and kinematic (K

( , )0

( , )

K

D

q t

q t

= =

) system constraints. For example, a driving constraint

would confine the crank to rotate with a constant angular velocity, while a kinematic constraint

might confine the slide to travel in a horizontal line. In this method, mechanism constraints are

defined as functions of component Cartesian coordinate locations (q) and time (t) (Equation 2-

36). Constraint equations may be formulated to reflect rigid physical boundaries, external force

constraints, intermittent contact, or even elastic and viscoelastic contact models [7, 9-12].

(2-36)

The constraint equations are then incorporated as bounds into the equations of motion as

described by Equation 2-36. Here, M

is the mass matrix, q

is the Jacobian, is the vector of

Lagrange multipliers, extF

is the vector of externally applied forces, is a collection of

acceleration constraints (Equation 2-38). The constants and are numerical stabilization

parameters that help prevent constraint violation during numerical integration [13]. Dynamic

simulations are performed by specifying a set of initial kinematic conditions, then solving the set

of differential equations described by Equation 2-37. While a detailed description of these

numerical techniques is beyond the scope of this study, it is mentioned since it forms the basis of

the kinematic and dynamic modeling portions of the Coupled Evolution Wear Model (CEWM).

27

Dynamic and wear predictions of the CEWM are compared with experimental results in Chapter

4. For an in-depth description of the application of these methods to the crank-slider mechanism

discussed in this paper, the interested reader is referred to Mukras [14].

20 2

Textq

q

FM q

=

(2-37)

( ) 2q q qt ttqq q q q = =

(2-38)

Predicting Wear in the Joint of Interest

Previous sections of this chapter described methods for the inclusion of non-ideal joint

conditions such as compliance and irregular geometry into dynamic mechanism analysis. While

conditions such as manufacturing defects and compliance in the joint components can and do

contribute to mechanism errors, these pre-existing conditions may be outweighed by the effects

of wear as a mechanisms useful life is consumed. This section discusses general wear modeling,

wear mechanisms of polytetrafluoroethylene (PTFE), and wear modeling in the context of the

mechanism under study.

Wear losses are typically quantified in terms of a generalized wear rate, k, as described by

the Archard wear model (Equation 2-39) [15]. In this equation, V is the volume of material worn

away at normal load Fn

n

VkF d

=

after a sliding distance d. As Archard stipulates, k is not an intrinsic

material property and is dependent not only on the contact pair, but also on the experimental

conditions. Still, it serves as a useful comparative and modeling tool.

(2-39)

Friction and Wear Behavior of PTFE under Dry Sliding Conditions

The contact pair used in this study is a PTFE (wearing) bushing mated with a (non-

wearing) steel shaft. PTFE is commonly chosen as a solid lubricant in contact with steel due to

28

its low friction, stability at elevated temperatures, and chemically inert behavior. When running

on hard, smooth surfaces such as glass or polished steel, PTFE has a tendency to form thin

transfer films which may bond with the surface under contact conditions, effectively forming a

self-mated contact from a chemical standpoint [16, 17]. However, the tribological properties of

PTFE are not straightforward. Like many thermoplastics, PTFE exhibits some viscoelastic

tendencies [18]. Generally speaking, the friction and wear behavior of PTFE on steel may be

influenced by sliding speed, normal load, ambient temperature, and the direction of sliding

relative to a pre-existing transfer film [14-21]. At sliding speeds greater than 10 mm/s, PTFE

may be expected to exhibit wear rates ranging from 10-3 to 10-5 mm3/Nm [16] with values on the

order of 5x10-4 to 8x10-4

Modeling Joint Wear in Mechanisms

being commonly reported [22-25]. Friction coefficient values are

typically less than 0.2 [26]. While the relatively high wear rates exhibited by PTFE pose a

challenge to designers seeking long component lives, this property provides an advantage in this

case since wear tests can be performed more quickly than with low wearing materials.

Published or otherwise known wear rates may be applied by a designer in a predictive

fashion by removing material from a modeled component as a function of dynamics, kinematics,

and cycles completed. However, simply extrapolating a cyclic wear volume or depth over many

cycles may prove inaccurate as geometric shape change in the joint may alter joint pressure

profiles or even the dynamic behavior of the overall mechanism, especially in cases of severe

wear [27-29].

Often, contact conditions and geometry are iteratively updated in order to provide a more

realistic simulation of wear evolution [27-33]. Such is the case with the CEWM [14]. In this

analysis approach, kinematic and dynamic numerical simulations, finite element analysis contact

pressure simulations, and wear geometry are iteratively updated. However, due to the high

29

computational costs of the dynamic and finite element simulations, the wear geometry and

contact conditions are not iterated on a cycle-by-cycle basis. Instead, single cycle wear results

are extrapolated with stability feedback. The details of this technique are beyond the scope of

this study.

A

B

Figure 2-1. The crank-slider is modeled as a planar mechanism. A) This mechanism has four bodies: the crank link, the follower link, the slide, and ground. B) Global coordinate system, link lengths of the four bodies, and load point of the spring.

30

Table 3-1. Crank-slider mechanism parameters. Parameter Symbol Value Units Crank length L1 0.0762 m Follower length L2 0.2032 m Slide center of mass offset L3 0 m Spring load offset yspr 0 m Crank mass m1 0.4 kg Crank mass moment of inertia I1 2.0x10-4 kg-m2 Follower mass m2 0.8 kg Follower mass moment of inertia I2 5.5x10-4 kg-m2 Slide mass m3 9 kg

Figure 2-2. The follower angle is defined as the angle between the negative X and follower axes.

31

0 2 4 60

0.05

0.1

0.15

0.2

0.25

crank angle (rad)

X-a

xis s

lide

posi

tion

x 3 (m

)

Figure 2-3. Stage location as a function of crank angle.

0 2 4 6

-0.2

-0.1

0

0.1

0.2

0.3

crank angle (rad)

slid

e ve

loci

ty (m

/s)

Figure 2-4. Stage velocity as a function of crank angle at a spindle speed of 30 rpm.

32

0 2 4 6-1.5

-1

-0.5

0

0.5

1

1.5

crank angle (rad)

stag

e ac

cele

ratio

n (m

/s2)

Figure 2-5. Stage acceleration with respect to crank angle.

0 2 4 6-0.05

-0.04

-0.03

-0.02

-0.01

0

crank angle (rad)

join

t slid

ing

velo

city

(m/s

)

Figure 2-6. Sliding velocity at the joint of interest is plotted versus crank angle for a constant

spindle speed of 30 rpm and a pin-bushing joint diameter of 19.05 mm.

33

A B Figure 2-7. Free body diagrams used in the dynamic analysis. A) Free body diagram of the stage.

B) Free body diagram of the stage and follower link.

0 1 2 3 4 5 60

10

20

30

40

crank angle (rad)

join

t for

ce m

agni

tude

(N)

15 rpm30 rpm45 rpm60 rpm

Figure 2-8. Predicted cyclic joint force magnitude profiles at varying crank speeds.

34

A

0 1 2 3 4 5 60

10

20

30

40

50

crank angle (rad)

join

t for

ce m

agni

tude

(N)

ks = 0 N/m

ks = 200 N/m

ks = 400 N/m

B

0 1 2 3 4 5 60

10

20

30

40

50

60

crank angle (rad)

join

t for

ce m

agni

tude

(N)

ks = 0 N/m

ks = 200 N/m

ks = 400 N/m

Figure 2-9. Predicted joint force magnitude cyclic profile at varying spring rates. It can be seen

that at higher spring rates, the spring force quickly overwhelms the effects of the system inertia. A) Spring offset yspr of 0 mm. B) Spring offset yspr of 55 mm.

35

0 1 2 3 4 5 60

5

10

15

crank angle (rad)

join

t for

ce m

agni

tude

(N)

m3 = 8 kg

m3 = 13 kg

m3 = 17 kg

Figure 2-10. Sensitivity of the joint force profile to changes in the stage mass.

0 1 2 3 4 5 60

5

10

15

20

crank angle (rad)

join

t for

ce m

agni

tude

(N)

L3 = 0 mm

L3 = 20 mm

L3 = 40 mm

L3 = 60 mm

Figure 2-11. Sensitivity of the predicted joint force profile to changes in the Y-direction location

of the stage center of mass.

36

Figure 2-12. Static cylinder contact model.

Figure 2-13. Contact model assumes flat deformation profile and semi-elliptical pressure profile.

37

Figure 2-14. The contact model may be modified to accommodate internal line contacts by

multiplying the inner radius of the bushing by (-1).

Table 3-2. Contact model simulation parameters Parameter Symbol Value Units Body a radius Ra 9.48 mm Body b radius Rb -9.55 mm Body a Young's modulus Ea 210 GPa Body b Young's modulus Eb 542 MPa Body a Poisson's ratio a 0.3 Body b Poisson's ratio b 0.45 Contact load Fn 0-150 N

38

A 0 50 100 150

0

0.5

1

Contact Normal Force Fn (N)

Max

imum

Con

tact

Pre

ssur

e, P

max

(MPa

)

B0 50 100 150

0

2

4

6

8

10

Contact Normal Force Fn (N)

Con

tact

Wid

th, w

(mm

)

C 0 50 100 150

0

5

10

15

20

25

Contact Normal Force Fn (N)

Con

tact

Pen

etra

tion,

(

m)

D 0 50 100 150

1.5

2

2.5 x 106

Contact Normal Force Fn (N)

Con

tact

Stif

fnes

s, K

ci (N

/m)

Figure 2-15. Results from the contact simulation indicate contact pressures less than 1.5 MPa,

contact widths less than 12 mm, penetration values less than 30 mm, and contact stiffness values on the order of 2.1x106

N/m.

39

CHAPTER 3 TEST BED DESCRIPTION AND EXPERIMENTAL PROCEDURES

This chapter describes the equipment and experimental techniques used in the study. The

test bed design and evaluation reinforce the purpose of this study, specifically, to characterize the

behavior of a real revolute joint and to examine how that behavior changes as the joint wears.

Test Apparatus

The crank-slider test apparatus used in the study is shown in Figure 3-1. The design

philosophy was to attempt to isolate friction, wear, and error motions exclusively to the joint

under consideration. In order to minimize confounding dynamic contributions from the other

components in the mechanism, every attempt was made to idealize all other joints in the

mechanism.

Crank-Slider Mechanism

The kinematic components of the crank-slider can be grouped into one of three categories:

power train components intended to supply a constant input crank velocity constituent components of the joint under study components intended to transfer reciprocating motion from the joint of interest to the slide.

Power train components include an electric motor, a gear reducer, a timing belt, a block

spindle, a flywheel, and the crank link (Figure 3-2). The 560 W (0.75 HP) DC electric motor

with attached gear reducer drives the block spindle through a timing belt. A 76.2 mm (3 in.) long

aluminum crank link is clamped to the spindle shaft at one end. The mass moment of inertia of

the spindle is increased by the addition of a 19.27 kg flywheel. This steel wheel has a mass

moment of inertia of 0.161 kg-m2 and helps maintain a constant crank speed by helping to

overcome torque fluctuations caused by the oscillating stage mass.

40

The revolute joint under study consists of a 19.00 mm diameter steel pin and a polymer test

bushing. The pin is clamped in the crank link at one end and is free to rotate, subject to sliding

friction, within the test bushing. The test bushing is clamped in the follower link.

As the crank link turns, it drives the 203.2 mm (8 in.) long follower link which drives the

linear slide. Friction, error motions, and wear in these components are reduced through the use of

porous carbon air bearings for the revolute joint between the follower link and the slide stage, as

well as prismatic joint for the linear slide. The revolute joint is comprised of a 31.75 mm (1.25

in.) diameter stainless steel ground pin which rotates within a pair of New Way Air Bearings

model C303202 thrust air bushings supplied with compressed air at 0.55 MPa (80 psi). While no

data is available from the supplier regarding the radial or thrust stiffness of this particular model,

similar 38.1 mm diameter air bushings also made by New Way have a radial stiffness of 72

N/m when supplied at 0.41 MPa (55 psi). These air bushings are supported by aluminum pillow

blocks which are bolted to the slide stage. The prismatic joint is a New Way Air Bearings model

S40-06150-095457 dovetail slide with 304.8 mm (12 in.) of travel. This slide can exhibit joint

stiffness of approximately 100-300 N/m depending on the load direction and air supply

pressure.

Manipulating Joint Forces

The contact and friction loads experienced by the joint under study can be manipulated in

two ways. First, up to 9.6 kg of mass (madd) in the form of steel weights can be bolted to the

dovetail slide stage. The corresponding inertial force is dependent on the stage acceleration,

which is a function of the stage location and the crank velocity. Additionally, single or dual coil

tension springs may be attached in parallel between the stage and the table. With the assumption

that the springs have no significant strain rate dependency, the spring force is nominally a

function of stage position. Springs used in this study have a nominal spring rate of ks = 220 N/m.

41

Instrumentation

While the kinematic and dynamic behavior of the crank-slider can be theoretically

estimated, it is important that the mechanisms behavior be verified experimentally. Indeed,

comparison between theoretical and actual performance is a primary motivation for this work. As

such, the test bed has been instrumented to measure forces and displacements at key locations.

All signals are read as analog voltages by a National Instruments 16-bit PCI-series data

acquisition card. Data acquisition is controlled by a program written using National Instruments

LabVIEW 7.1 software.

Spindle Encoder

For the crank-slider, the angular location of the crank specifies the orientation of the rest of

the mechanism. Often, it is useful to plot force, error motions, or link locations as a function of

the crank angle. For this reason, the angular location of the crank is measured using a BEI Model

HS35 hollow shaft encoder clamped to the spindle axis as shown in Figure 3-4. The 3600 count-

per-revolution digital signal generated by the encoder is then converted to an absolute analog

output using a BEI M-series digital-to-analog module.

Pin Load Cell

Contact forces at the joint of interest are measured using a load cell built onto a necked

portion of the joints hollow steel pin (Figure 3-5). This load cell, provided by Deere &

Companys Advanced Vehicle Development group, uses two full-bridge strain gage circuits to

measure the transverse shear two orthogonal channels, Xp and Yp, with 1 N resolution. Each

full-bridge channel is constructed from two Vishay Micromeasurements EA-06-062TV-350 90

degree shear rosettes (Figure 3-6). Signals are transmitted from the load cell through a Michigan

Scientific S-series 10 circuit slip ring to a National Instruments SG-024 modular amplifier/signal

conditioner and 16 bit data acquisition card.

42

The physical arrangement of the strain gages and the circuitry of each Wheatstone bridge

channel make them sensitive to transverse shear and insensitive to other loading conditions.

Additionally, each full bridge is automatically compensated. In order to demonstrate these

conditions, the physical layout and circuitry of a single channel (Xp) are shown in Figures 3-7

and 3-8, respectively. When any load is applied to the pin transducer, the change in resistance

(R) of each gage is equal to the product of the strain in that gage (), the nominal gage

resistance (R), and the gage factor (Sg

gR S R =

) [34]. This relationship is expressed mathematically in

Equation 3-1. This linear relationship holds for strain values less than 3% for the gages used in

this study.

(3-1)

When two gages are arranged at symmetric angles to a given axis, their measured strains may be

used to calculate the shear strain between the axis of symmetry and a second, in-plane,

orthogonal axis (Figure 3-9) [35]. This relationship is governed by Equation 3-2. Here, XZ

represents the shear strain in the X-Z plane, 1 and 2

1 2

sin(2 )XZ

=

represent the tensile strains in gages 1 and

2, respectively, and represents the half-angle of symmetry about the Z-axis. The half-angle

value for the shear rosettes used in this study is /2 radians, making the denominator of Equation

3-2 equal to one.

(3-2)

When two rosettes are wired into a full bridge, the measured bridge output voltage Vout is a

function of the source voltage Vs and the gage resistances R1, R2, R3, and R4 (Eq. 3-3). Applying

mechanical load to the pin changes the resistance of each gage according to Equation 3-1.

Assuming that the nominal resistance of each gage is R, the resulting change in the bridge

voltage is described by Equation 3-4. This formulation assumes that second order effects are

43

neglected ( 0i jR R for any two g ages i and j). With the understanding that resistance change

and gage strain are proportional, Equation 3-4 may be rewritten in terms of strains (Equation 3-

5). Here, XZ,a and XZ,b

( )( )1 3 2 4

1 2 3 4out s

R R R RV VR R R R

= + +

are the X-Z plane shear strains at each respective shear rosette. For the

purposes of this discussion, the rosette constructed from gages 1 and 2 is designated rosette a

while the rosette constructed from rosettes 3 and 4 is designated rosette b.

(3-3)

( )1 2 3 44s

outVV R R R RR

= + (3-4)

( ) ( )1 2 3 4 , ,4 4s s

out XZ a XZ bV VVR R

+ = + (3-5)

The bridge voltage response for a given channel under a particular loading condition can

then be calculated by obtaining the state of stress and strain at each gage and calculating each

gages change in resistance. The following paragraphs describe the pin response to transverse

shear, bending moment, torsion, axial, and thermal loading conditions. For this study, the pin

material (steel) is assumed to be isotropic. The stress-strain relationships may then be described

as in Equation 3-6. In Equation 3-6, E and represent the Youngs modulus and Poisson ratio of

the pin material.

44

( )

( )

( )

1 0 0 0

1 0 0 0

1 0 0 0

2 10 0 0 0 0

2 10 0 0 0 0

2 10 0 0 0 0

X X

Y Y

Z Z

XY XY

XZ XZ

YZ YZ

E E E

E E E

E E E

E

E

E

= + +

+

(3-6)

A general transverse shear loading condition is shown in Figure 3-10. Note that gages 3

and 4 lie on the far side of the pin and are thus not visible in this figure. For simplification, only

a portion of the pin is shown. Under this loading condition, shear stress (and therefore shear

strain) in the X-Z plane is proportional to the transverse shear load Fx and insensitive to Fy.

Since both rosettes are centered on the neutral axis relative to the Fx

( ), ,4out XZ a XZ bVsV

R = +

load, the shear strains

registered by each are equal and at a maximum. Therefore, Equation 3-5 may be simplified to

Equation 3-7.

( 3-7)

A general bending moment loading condition is shown in Figure 3-11. First consider the

effects of moment My. Both shear rosettes lie centered on the neutral plane with respect to this

moment. Because of this, gages 1 and 4 are under tension while gages 2 and 3 are under

compression. Due to this symmetry, the strain in gages 1 and 4 are equal and opposite to the

strain in gages 2 and 3. Considering Equation 3-5, this causes the net change in the bridge

voltage output to be zero. A similar consideration of the effects from Mx shows that the strain in

45

gages 1 and 2 are equal and opposite to the strain in gages 3 and 4. Again, the net bridge voltage

is not affected. In either case, the pin load cell is insensitive to bending moments.

A general torsion load condition is shown in Figure 3-12. Since the two rosettes point in

anti-parallel directions, their respective shear strain measurements are equal and opposite.

Subsequently, the shear strain terms in Equation 3-5 cancel and the bridge voltage is not

affected.

A general axial loading condition is shown in Figure 3-13. In this scenario, the tensile

stress in the Z-direction is proportional to the applied load Fz

, ,

, ,

cos( ) sin( )sin( ) cos( )

X g X p

Y g Y p

F FF F

=

while all other stresses are zero. All

four gages are under equal tension, and the bridge voltage is unchanged.

Furthermore, each channel is temperature compensated as long as the temperature changes

in each gage are equal. Whether uniform ohmic (resistive) heating increases the resistance of

each gage or thermal expansion of the pin occurs, the change in resistance of each gage will be

equal and the change in bridge voltage will be zero.

It should be noted that the two channels of the pin load cell measure forces with respect to

a coordinate system fixed to the pin and crank link (Figure 3-14). It is often convenient to

consider contact forces in a global, or world, coordinate system (Figure 3-15) or even a

coordinate system attached to the bushing and follower link (Figure 3-16). Pin forces can be

converted to global forces using the coordinate transformation shown in Equation 3-8. Contact

forces imparted on the bushing may be expressed in a coordinate system fixed to the follower

link by Equation 3-9.

(3-8)

46

,,

,,

cos( ) sin( )sin( ) cos( )

X gX b

Y gY b

FFFF

=

(3-9)

It must be noted that a substantial portion of the pin mass extends beyond the necked

sensor region. The inertial and gravitational effects of this mass (mpin,e = 0.2 kg) are not

insignificant and while they affect load cell measurements, they do not affect the contact force at

the joint of interest. Because of this, the global X- and Y- direction affects inertial loads Fpinx and

Fpiny

( )2, 1 cos( )pinx pin eF m L = are calculated and removed according to Equation 3-10 and 3-11.

(3-10)

( )2, 1 sin( )piny pin eF m L g = (3-11)

Capacitance Probes

Relative motion between the pin and bushing is measured using capacitance probes (Figure

3-17). Two Lion Precision model C23-B probes measure the pin location from fixed,

orthogonally-mounted locations on the follower link. These probes have a range of 1250 m and

a resolution of 40 nm. The sensing are on this probe model has a diameter of 3.2 mm. It should

be noted that these probes are typically calibrated using a flat target, rather than the cylindrical

target used in this study. To remedy this, Lion Precision recommends reducing the calibration

coefficient, in V/m, by 3%.

Additionally, a rounded target surface causes the probes to be somewhat sensitive to off-

axis motion. The output of each capacitance probe is proportional to the average distance from

the average distance from the sensing area to the target (Figure 3-18). As the target moves

orthogonally to the direction a probe is intended to measure, the perspective of the probe changes

from state i to state ii. As the pin moves, the average distance from the probe sensing area to

47

the target surface is affected. While this cross sensitivity is difficult to predict and remove, it

is expected to result in less than 20 m of error in the capacitance probe reading (Figure 3-19).

Since the capacitance probes only offer a two-dimensional perspective, they offer an

incomplete characterization of the error motions at the joint of interest. The capacitance probes

offer no information as to angular misalignment between the revolute joint components. While

the addition of extra probes along the pin axis could add such information, more probes were not

available. Instead, angular misalignments between the mechanism axes are characterized in

discrete static conditions in Appendix D.

Operation of the Test Bed

A typical procedure for operating the crank-slider test bed is carried out as described in the

following paragraphs.

Sample Preparation and Characterization

Bushing samples are machined from 38.1 mm diameter polymer stock using a table-top

computer-numerically controlled (CNC) mill. The bushing inner diameter is nominally 19.05

mm (0.75 in.), but may be adjusted depending on the desired test conditions. After the inner

diameter of the bushing is machined, four grooves (2.38 mm radius) are milled into the inner

circumference (Figure 3-20). This provides a location for debris accumulation during testing.

Following machining, polymer burrs are manually removed. The sample is then washed in water

and isopropyl alcohol and allowed to dry.

Following preparation, the bushing inner diameter is measured using a coordinate

measuring machine (CMM). The CMM used in this study is a Brown and Sharpe PFX 3-axis

computer controlled machine outfitted with a MIP-20 digital touch trigger probe and 3 mm

diameter ruby probe tip. First, the edge of one of the four wear collection grooves is marked with

a permanent marker. This is done so that the bushing may be measured in the same orientation

48

following wear testing. The bushing is then lightly clamped in v-blocks to the CMM table for

measurement. Alignment of the bushing coordinate system is performed by first measuring each

of the four wear-collection grooves and determining their radial centers. The center of the

bushing bore is then defined as the intersection of the lines connecting each pair of opposing

wear groove centers (Figure 3-21). Alignment is performed in this way since the wear collection

grooves are not drastically affected as the bushing wears. The remaining portion of the bushing

which will contact the pin is then measured as four quadrants, with 50 touch-points to a quadrant.

This procedure is then repeated at the conclusion of a wear test in order to quantitatively

demonstrate how the bushing shape evolves.

While CMM results are useful to track shape changes in the bushing shape following a

test, the amount of worn material can be more accurately measured by massing the sample. Mass

measurements are taken prior to and following wear tests using a Mettler-Toledo AX205

analytical microbalance with 10 g resolution.

Mechanism Preparation

Once the sample bushing has been prepared, the crank-slider mechanism is assembled.

First, the bushing is clamped in the follower arm, aligned as shown in Figure 3-22. The pin is

then inserted through the bushing and clamped in the crank arm. If the kinematic components

have been disassembled since the previous test, some component fasteners may be loosened by

1/8 turn and the mechanism run at less than 10 rpm. This allows the mechanism to shift into a

nominally aligned state. For example, if the stage has been disassembled, the cap screws

fastening the pillow blocks to the stage may be loosened and allowed to align. If either the

dovetail slide or the spindle has been removed from the table, typically only the spindle fasteners

are loosened.

49

Operation and Data Collection

Once the test bed has been assembled, the motor is turned on and set to the desired speed.

At this point in the procedure, the spindle speed is measured using a non-contact hand held

tachometer with 1 rpm resolution. The test bed is capable of running at speeds from less than 5

rpm to greater than 45 rpm. Once the speed has been set, the motor is switched off.

The data acquisition program is then started. This program records data in two ways. First,

for each complete rotation of the crank, the root mean square, maximum, and minimum of each

data channel is recorded and appended to one spreadsheet. Additionally, entire cycles of data

from each channel can be periodically saved to separate spreadsheets.

Once the file paths are set up, calibration coefficients are verified and each channel is

zeroed. Load cell signals are zeroed by subtracting the no-load voltage from each channel. The

capacitance probes are zeroed using the probe amplifier.

Once all channels are properly initialized, the stage spring assembly is connected and the

motor is started. After the motor runs up to speed, data collection is initiated. It should be noted

that the number of cycles completed on the test bushing as the mechanism is aligned and the

speed is set is relatively low. Typically, less than 25 cycles are completed prior to the start of

data collection, while a wear test generally requires over 25,000 cycles.

50

A

B Figure 3-1. The crank-slider bed used to perform experimental tests. A) Line drawing of the test

bed showing components. B) Photograph of the actual test bed.

51

Figure 3-2. Kinematic components of the crank-slider mechanism.

Figure 3-3. The cyclic joint load profile may be adjusted through the addition of stage mass or by

adding springs between the slide stage and table.

52

Figure 3-4. A hollow rotary encoder clamped to the spindle provides crank position and speed

information.

Figure 3-5. Custom steel pin load cell.

53

Figure 3-6. Each load cell channel is constructed from two strain gage shear rosettes totaling four gages.

Figure 3-7. The two shear rosettes associated with a single channel are mounted on diametrically opposite sides of the necked portion of the pin. Each rosette is aligned parallel to the pin cylinder axis and anti-parallel with the other rosette.

54

Figure 3-8. Wheatstone bridge circuit diagram for a single pin force channel. Gages are numbered as pictured in Figure 3-7.

Figure 3-9. Two strain gages oriented symmetrically about an axis of interest combine to form a shear rosette.

55

Figure 3-10. General transverse shear loading of the pin transducer.

Figure 3-11. General bending moment loading of the pin transducer.

Figure 3-12. General torsion loading of the pin transducer.

56

Figure 3-13. Axial loading of the pin transducer.

Figure 3-14. Pin load cell channel coordinates.

Figure 3-15. Forces measured by the pin load cell can be transformed into a global coordinate system using the measured crank angle .

57

Figure 3-16. Global coordinate forces may be further transformed to a coordinate system fixed to the bushing and follower link using the calculated follower angle.

Figure 3-17. Capacitance probes measure the location of the pin from fixed locations on the

follower link.

58

Figure 3-18. The capacitance probe in this picture is sensitive not only to pin motion along its axis, but also secondarily in the orthogonal direction. As the pin moves at a right angle to the intended sensitive direction, the average distance from the probe sensing face to the pin surface changes.

Figure 3-19. Estimated cross sensitivity error as a function of off-axis motion of the pin. The MATLAB code used to generate this figure is provided in Appendix C.

59

Figure 3-20. Geometry of the test bushings. A) Nominal dimensions of the bushing B) Four 2.38

mm radius grooves provide an outlet for wear debris that might otherwise artificially reduce wear.

Figure 3-21. The bushing center is defined as the intersection of two lines drawn between opposing sets of wear collection grooves.

60

Figure 3-22. Preparation of the test bed is performed in steps. A) The bushing is clamped in the follower link. B) The pin is inserted through the bushing and clamped in the crank link. C) The pin Y-direction channel is zeroed in an orientation not subject to gravitational loading. D) The pin X-direction channel is zeroed. E) The capacitance probe channels are tared such that both channels record zero volts when the pin and bushing centers are aligned. F) The crank speed is set using a non-contact tachometer.

61

CHAPTER 4 RESULTS FROM A SPRING DOMINATED WEAR TEST

An initial wear test was completed using test parameters selected to make the spring force,

rather than inertial effects, the dominant influence in the load profile (Table 4-1). This was done

to enable two simplifying assumptions to be made regarding the contact forces. First, larger

spring forces reduce shifting of the contact location relative to the bushing as the crank turns.

Second, the force generated by extending the spring by a 150 mm stroke is relatively insensitive

to joint wear values of less than a millimeter. An initial diametric clearance of 108 m was used

to provide a tight fit while still allowing relative rotation subject only to friction. Following

experimental testing, results were compared with dynamic and wear predictions made by the

Coupled Evolution Wear Model (CEWM). For this test, the CEWM dynamic model was capable

of accounting for joint component compliance and shifting contact location on the bushing

surface.

Joint Dynamics

Joint force data measured by the pin load cell showed little evidence of evolving over the

course of the test (Figure 4-1). This made sense given the limited wear amount for the test. The

maximum wear scar depth was limited to less than one millimeter by the capacitance probe

range. This amount of wear did not significantly affect the spring load profile. High frequency

dynamic signals were witnessed throughout the test. The amplitude of these vibrations was

highest where the slide reversed direction, i.e., approaching crank angles of rad and 2 rad. At

no point did the joint force magnitude approach a value of 0 N. This was an indication that the

pin and bushing never lost contact.

Of particular interest was the sudden drop in joint force magnitude just prior to a crank

angle of rad followed by a period of decaying high frequency vibration as well as the dynamic

62

spikes in the vicinity of 6 rad. One possible explanation for this event was the presence of

friction between the dovetail slide components. A series of CEWM dynamic simulations with

increasing stage friction was performed (Figure 4-2). The CEWM predicted similar dynamic

disturbances at higher values of the friction coefficient, although at crank angles lagging behind

the experimental data. High coefficient of friction values at the dovetail slide may have been

caused by abnormally low air pressure or abnormally high moments on the stage causing

starvation of the air bearings.

Dynamic results from the CEWM simulation were generally comparable in shape and

magnitude to the experimental data. However, the simulation was not able to capture the high

frequency content. When comparing the force components, it was noted that the predicted global

Y-direction forces (FYg) were somewhat lower in magnitude that the measured results (Figure 4-

3). This may have been the result of an assumption in the CEWM simulation which did not

account for a vertical offset between the follower-stage revolute joint and the spring force point

of application (yspr).

Similar vibration was noted in the capacitance probe channels, with data from cycle 1

shown in Figure 4-4. By comparing the relative magnitudes of the joint force and displacement

high frequency content, the stiffness of the contact was estimated to be between 4x106 N/m and

8x106

Wear Characterization

N/m. This was somewhat higher than the value predicted by the line contact model

described in Chapter 2. This may have been due to viscoelasticity of the bushing material.

As described in Chapter 3, bushing wear was measured in three ways. First, evolution of

the pin cyclic path relative to the bushing was measured in situ by the capacitance probes.

Second, a coordinate measuring machine was used to directly measure the profile of the bushing

63

both before and following the wear test. Third, mass loss and, indirectly, volume loss was

measured using an analytical balance.

Unlike the joint force profile, which remained relatively unchanged over the course of the

wear test, capacitance probe data indicated an evolution in the shape of the bushing in the first

thousand cycles (Figure 4-4). After this, the cyclic path of the pin remained relatively unchanged

while the bushing continued to wear.

While no experimental measurement of contact pressure could be made, evolution of the

bushing shape was echoed by changes in the contact pressure as predicted by finite element

analysis, see Figure 4-6. This figure indicates that, initially, the contact was limited to a single

lobe between wear collection channels. However, as the pin wore into the bushing, the contact

wrap angle approached rad.

Comparison between Experimental and Modeled Wear Results

Despite differences in the predicted and measured joint force profiles, the CEWM was able

to largely capture the shape and magnitude of the bushing wear scar as measured by the

coordinate measuring machine (Figure 4-7). When considering Figure 4-7, the profiles of the

bushing radius change are very similar in both shape and magnitude although a small angular

misalignment was observed between the measured and modeled wear scar. This is believed to be

a result of the differences in the predicted load profile and the gross measured joint force profile.

Additionally, a deviation between the predicted and measured profile can be seen in the vicinity

of the bushing angle = 3/2 rad. This may have been the result of a build-up of wear debris.

This would have been caused by a wear collection groove being filled to capacity, or debris may

simply not have been swept out of the contact.

Wear mass and volume losses predicted by the CEWM also correlated well with

experimental values (

64

vary by as much as an order of magnitude (Table 4-2). In the case of the model, wear volume

was predicted and mass loss calculated. Experimentally, the mass loss was measured directly and

volume loss obtained by dividing by density. In either case, a density of 2.2 g/cm3 was assumed.

65

Table 4-1. Wear test parameters Parameter Value Crank speed (rpm) 30 Initial diametric clearance (m) 108 Spring rate (N/m) 525 Spring preload (N) 52 Slide mass (kg) 8.5 Cycles run 21,400

A0 2 4 6

0

50

100

150

200

crank angle (rad)

join

t for

ces (

N)

measuredCEWM

B0 2 4 6

0

50

100

150

200

crank