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Kinematic and Dynamic Behavior of a Wearing Joint in a Crank-Slider Mechanism

Permanent Link: http://ufdc.ufl.edu/UFE0024958/00001

Material Information

Title: Kinematic and Dynamic Behavior of a Wearing Joint in a Crank-Slider Mechanism
Physical Description: 1 online resource (132 p.)
Language: english
Creator: Mauntler, Nathan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamics, tribology, wear
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nathan Mauntler.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.
Local: Co-adviser: Sawyer, Wallace G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024958:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024958/00001

Material Information

Title: Kinematic and Dynamic Behavior of a Wearing Joint in a Crank-Slider Mechanism
Physical Description: 1 online resource (132 p.)
Language: english
Creator: Mauntler, Nathan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dynamics, tribology, wear
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nathan Mauntler.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.
Local: Co-adviser: Sawyer, Wallace G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024958:00001


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1 KINEMATIC AND DYNAMIC BEHAVIOR OF A WEARING JOINT IN A CRANKSLIDER MECHANISM By NATHAN A. MAUNTLER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009

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2 2009 Nathan A. Mauntler

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3 To my wonderful wife Nicole

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4 ACKNOWLEDGMENTS I would first like to thank my advisors Tony Schmitz and Greg Saw yer 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 pr oject. 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 De velopment Lab for providing, and then patiently explaining, the shear beam load cell used in this project.

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5 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 NonIdeal 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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6 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 Dis placement 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 T HE DYNAMIC SYSTEM TO CHANGES IN THE MECHANISM AND JOINT PARAMETERS .......................................................................88 Repeatability of Dynamic Test Results ..................................................................................88 Sensitivity to Ch anges 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 CRANKSLIDER MECHANISM .........116 B MATLAB SIMULATION OF A SIMPLE LINE CONTACT MODEL .............................119 C MATLAB SIMULATION OF SECONDARY CAPACITANCE PROB E 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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7 LIST OF TABLES Table page 31 Crank slider mechanism parameters. .................................................................................30 32 Contact model simulation parameters ................................................................................37 41 Wear test parameters ..........................................................................................................65 42 Experimental and model predictions of bushing wear amount for initial wear test. .........69 71 Test conditions of five short dynam ic repeatability tests ...................................................95 D 1 Pin axis vector components as imported into Geomagic Studio software. ......................127

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8 LIST OF FIGURES Figure page 21 The crank slider is modeled as a planar mechanism. ........................................................29 22 The follower angle is defined as the angle between the negative X and follower ax es. ...30 24 Stage velocity as a function of crank angle at a spindle speed of 30 rpm. ........................31 25 Stage acceleration with respect to c rank angle. .................................................................32 26 Sliding velocity at the joint of interest is plotted versus crank angle for a constant spindle speed of 30 rpm and a pinbushing joint diameter of 19.05 mm. ..........................32 27 Free body diagrams used in the dynamic analysis. ............................................................33 28 Predicted cyclic joint force magnitude profiles at varying crank speeds. ..........................33 29 Predicted joint force magnitude cyclic profile at varying spring rates. .............................34 210 Sensitivity of the joint force profile to changes in the stage mass. ....................................35 211 Sensitivity of the predicted joint force profile to changes in the Y direction location of the stage center of mass. ................................................................................................35 212 Static cylinder contact model. ............................................................................................36 213 Contact model assumes flat deformation profile and semi elliptical pressure profile. ......36 214 The contact model may be modified to accommodate internal line contacts by multiplying the inner radius of the bushing by ( 1). ..........................................................37 215 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 31 The crank slider bed used to perform experimental tests. .................................................50 32 Kinematic components of the crank slider mechanism. ....................................................51 33 The cycli c joint load profile may be adjusted through the addition of stage mass or by adding springs between the slide stage and table. .........................................................51 34 A hollow rotary encoder clamped to the spindle provides c rank position and speed information. ........................................................................................................................52 35 Custom steel pin load cell. .................................................................................................52

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9 36 Each load cell channel is constructed from two st rain gage shear rosettes totaling four gages. ..........................................................................................................................53 37 The two shear rosettes associated with a single channel are mounted on diametrically opposite sides of the necked portion of the pin. .................................................................53 38 Wheatstone bridge circuit diagram for a single pin force channel. ...................................54 39 Two strain gages oriented symmetrically about an axis of interest combine to form a shear rosette. ......................................................................................................................54 310 General transverse shear loading of the pin transducer. ....................................................55 311 General be nding moment loading of the pin transducer. ...................................................55 313 Axial loading of the pin transducer. ...................................................................................56 314 Pin load cell channel coordinat es. ......................................................................................56 315 Forces measured by the pin load cell can be transformed into a global coordinate .........................................................................56 316 Global coordinate forces may be further transformed to a coordinate system fixed to the bushing and follower link using the calculated follower angle. ..................................57 317 Capacitance probes measure the location of the pin from fixed locations on the follower link. ......................................................................................................................57 318 The capacitance probe in this picture is sensitive not only to pin motion along its axis, but also secondaril y in the orthogonal direction........................................................58 319 Estimated cross sensitivity error as a function of off axis motion of the pin. ...................58 320 Ge ometry of the test bushings. ...........................................................................................59 321 The bushing center is defined as the intersection of two lines drawn between opposing sets of wear collection grooves. .........................................................................59 322 Preparation of the test bed is performed in steps. ..............................................................60 41 Joint force magnitude predictions and measured data. ......................................................65 42 CEWM simulation of the effect of stage friction on the dynamics at the joint of interest. ...............................................................................................................................66 43 Discrepancies between the predicted and measured global force compone nt 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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10 44 Capacitance probe measurements of the pin displacement relat ive to the bushing center showed similar vibration content as the pin force measurements. ..........................67 45 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 46 Contact pressure evolution predicted by the CEWM. .......................................................68 47 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 52 A laser i nterferometer is used to measure the position of the stage as a function of time. ...................................................................................................................................72 53 mechanism and as the minimum distance measured by the interferometer. ......................73 61 The evolution of high frequency dynamics with wear can be observed from single cycle joint force magnitude plots. ......................................................................................78 62 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 63 Higher frequency dynamic content appears to evolve only in the X direction of the bushing coordinate system. ................................................................................................80 64 Capacitance probe output as a function of crank angle. ....................................................81 65 Small changes in the stage path profile are observed by removing the gross motion profile from the interferometer output. ..............................................................................82 66 Fast Fourier transform (FFT) of b ushing coordinate force results. ...................................83 67 Capacitance probe FFT results. ..........................................................................................84 68 In terferometer FFT results. ................................................................................................84 69 Change in radius as a function of bushing angle as measured by the three axis CMM. ....85 610 Capa citance probe measurements from two short tests intended to challenge the assumption of a rigid pin ....................................................................................................85 611 Comparison between measured and modeled bushing force profiles. ...............................86 612 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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11 71 Pin force magnitude profiles from a series of five repeat dynamic tests. ..........................96 72 Capacitance probe measurements from a series of five repeat dynamic tests. ..................97 73 Joint force magnitude single cycle profile at varying crank speeds .................................98 74 Global coordinate force components at varying crank speeds. ..........................................99 75 Single cycle capacitance probe readings at varying crank speed. ...................................100 76 Single cycle joint force magnitude plots with changing stage mass. ...............................101 77 Capacitance probe measurements from a series of tests varying the mass of the dovetail slide stage. ..........................................................................................................102 78 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 79 Global force components subject to varying stage mass ................................................104 710 Changes in the capacitance probe pin displacement measurements were difficult to discern as the stage mass was increased. .........................................................................105 711 Force magnitude at the joint of interest subject to varying stage mass and using a worn bushing. ...................................................................................................................106 712 Spring rates were found to be linear and with comparabl e preloads. ..............................107 713 Contact force magnitude subject to increasing spring load. ............................................108 714 Capacitance probe output with increasin g spring load. ...................................................109 715 Global force components subject to increasing spring load. ...........................................110 716 Joint force magnitude measurements f rom 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 re ached without losing contact. ................125 D 3 Point clouds were imported into Geomagic Studio 9 software for analysis. ...................126 D 4 The impor ted point clouds were then fit with least square cylinders. .............................126

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12 D 5 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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13 Abstract of Dissertation Pres ented 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 CRANKSLIDER 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 mechanic al 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 frictio n, 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 dra wn between experimental results and predictions made by a simple, idealized dynamic model as well as a contemporary coupled wear model.

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14 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 geomet ry. 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 kinemat ic 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 isol ating 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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15 The crank follower re volute 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 l oad 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 highstiffness precision s pindle 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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16 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 Newton ian mechanical analysis. In this section, all joints are assumed to be planar, frictionless, and have a single degree of freedom. Additionally, all component s 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 21. 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 21B. Nominal dimensional and mass parameters for this

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17 crank slider mechanis m 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 ca rry 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 21. 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 22 through 27. The X axis stage location is shown as a function of crank angle in Figure 23. 1 1 2sinsin() L L ( 21) 1 1cos() 2 L x (2 2) 1 1sin() 2 L y (2 3) 2 21cos()cos() 2 L xL (2 4) 2 2sin() 2 L y (2 5) 312cos()cos() xLL (2 6) 33yL (2 7) The link center of mass velocity equations are then obtained by calculating the time derivative s of Equations 22 through 2 7. It should be noted that the crank speed is as sumed to be constant. Additionally, since the value of 3y is a constant, the slide velocity has no

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18 component in the Y direction. The velocity equations for the crankslider links are shown in Equations 28 through 2 13. The stage v elocity 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 2cos 1sin L L L L (2 8) 1 1sin() 2 L x (2 9) 1 1cos() 2 L y (2 10) 2 21sin()sin() 2 L xL (2 11) 2 2cos() 2 L y (2 12) 312sin()sin() xLL (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. 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2cos 1sin 1sin sin 1sin L L L L L L L L L L (2 14)

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19 2 1 1cos() 2 L x (2 15) 2 1 1sin() 2 L y (2 16) 22 2 21cos()cos()sin() 2 L xL (2 17 22 2sin()cos() 2 L y (2 18) 22 31 2cos()cos()sin() xLL (2 19) Since the joint under investigation is the revolute joint betwee n 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) betw een the crank and follower links (Equation 220). 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 222. Th e 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 26. (2 20) (2 21) 2 j jd v (2 22) Modeling Ideal Joint Forces A model of the ideal joint forces is formed from the free body diagrams shown in Figure 27. 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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20 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 ar e effectively treated as loads according to DAlemberts principle o f 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 sprin g from the pre loaded configuration (Equation 223). Considering the free body diagram of the stage (Figure 2 7A) and summing moments about point a, the value of M34 0123()sprssFFkLLx is calculated by Equation 2 24. (2 23) 34333 sprsprMmxLFy (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 Fx 21 is obtained by summing forces in the X direction (Equation 225). This term is the contact force in the global X direction on the bushing. The global Y direction force on the bushing, Fy 21, is then obtained by summing moments about the center of mass of the slide (Equation 226). In order to write Equation 226 compactly, Fy 21 is sep arated into external and inertial terms Fe (Equation 227) and Fi 212233 x sprFmxmxF (Equation 227). Here g is acceleration due to gravity. Contact forces experienced by the pin are equal and opposite to those on the bushing. (2 25) 21 yeiFFF (2 26) 34 32123 2(2) cos()sprspr x eMFyyFyy F L (2 27) 22 22232232232232 2cos()imxyyImxxyymgyxx F L (2 28)

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21 Sensitivity of the Joint Dynamics to Mechanism Parameters In this section, a series of simulations are performed in order to illustrate the se nsitivity of the ideal model to changes in several m echanism 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 ysprIn 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 sp eeds. 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 indicat ion 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 5 5 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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22 accurately determine the center of mass of an assembly built from various materials and intricate shape s The sensitivity of the crank slider dynamic system to changes in inertial properties is illustrated in Figures 210 and 211. 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 211, t he vertical distance from the follower stage revolute joint to the stage center of mass ( L3Characterizing Error Motions of the Revolute Joint ) is increased from 0 mm to 60 mm in 20 mm increments. In both cases, changi ng 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 joint s. 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 le ast) two separate bodies, each with its own nominal axis of rotation. Inplane 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 th at 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, nominall y 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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23 Standard ( w hich Bryan largely contributed to) for specifying these parameters is A NSI/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 p itching and yawing of the axis of rotation. These definitions were intended to replace the less favored terms such as runout and face run out, measurements of which could be influenced by multiple error motions as described above. These motion categori es 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 in stantaneous 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. Howe ver, 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 twodimensional 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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24 Contact conditions are simulated using a composite radius R and composite Youngs modulus E 1 '11abR RR (Equations 229 and 230 ) 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 neg ative (Figure 2 14) (2 29) 22 '11ab abE EE (2 30) The width of the deformed region w is calculated using Equation 231 from the contact normal load, composite radius and modulus, and the contact lengt h Lc 1/2 '4n cFR w LE T he 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 dista nce s from the contact center (Equation 333). Deformation at the contact may then be calculated using Equation 234. 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 335. (2 31)

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25 1/2 max n cFE P LR (2 32) 2 max2 ()1 s PsP w (2 33) 2221818 11 ln ln 22naabb ca bFRR LEwEw (2 34) n ciF K (2 35) Contact parameters representative of the joint under study in this work are listed in Table 32. Simulation results from the contact model are shown in Figure 2 15. This particular contact loads under 150 N. Interestingly, the contact stiffness Kci is predicted to be relatively insensitive to normal load and on the order of 2.2x106 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 co mponent 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. Howe ver, 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 com parison with measured results rather than a fundamental prediction of the joints compliant behavior.

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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 co mmon 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 Dqt qt ) 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 236). Constraint equations may be formulated to reflect rigid physical boundaries, external force constraints, intermittent contact, or even elasti c and visco elastic contact models [ 7, 912]. ( 236) The con straint equations are then incorporated as bounds into the equations of motion as described by Equation 236. Here, M is the mass ma trix, q is the Jacobian, is the vector of Lagrange multipliers, extF is the vector of externally applied forces, is a collection of acceleration constrain ts (Equation 238). 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 237. 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 )

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27 Dyn amic and wear predictions of the CEWM are compared with experimental results in Chapter 4. For an indepth description of the application of these methods to the crankslider mechanism discussed in this paper, the interested reader is referred to Mukras [ 14]. 20 2T ext q qF M q ( 237) ()2q q qttt qqqqq ( 238) Predicting Wear in the Joint of Interest Previous sections of this chapter d escribed methods for the inclusion of nonideal joint conditions such as compliance and ir regular 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 m echanisms useful life is consumed. This section discusses general wear modeling, wear mechanisms of polytetrafluoroe thylene (PTFE), and wear modeling in the context of the mechanism under study. Wear losses are typically quantified in terms of a generali zed wear rate, k as described by the Archard wear model (Equation 239) [ 15]. In this equation, V is the volume of material worn away at normal load Fn nV k Fd 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. ( 239) 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 (nonwearing) steel shaft. PTFE is commonly chosen as a solid lubricant in contact with steel due to

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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]. Howev er, the tribological properties of PTFE are not straightforward. Like many thermoplastics, PTFE exhibits some viscoelastic tendencies [1 8]. 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 preexisting transfer film [ 1421]. At sliding speeds greater than 10 mm/s, PTFE may be expected to exhibit wear rates ranging from 103 to 105 mm3/Nm [ 16] with values on the order of 5 x104 to 8 x 104 Modeling Joint Wear in Mechanisms being commonly reported [ 2225]. Friction coefficient values are typically less than 0.2 [ 26]. While the relatively high we ar 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 modele d 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 j oint pressure profiles or even the dynamic behavior of the overall mechanism, especially in cases of severe wear [ 2729]. Often, contact conditions and geometry are iteratively updated in order to provide a more realistic simulation of wear evolution [ 2733]. 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

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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 21. The crank slider is modeled as a planar mechanism A) This mechanism has four bodies: t he 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.

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30 Table 3 1. Cr ank slider mechanism parameters. Parameter Symbol Value Units Crank length L 1 0.0762 m Follower length L 2 0.2032 m Slide center of mass offset L 3 0 m Spring load offset y spr 0 m Crank mass m 1 0.4 kg Crank mass moment of inertia I 1 2.0 x10 4 kg m 2 Follower mass m 2 0.8 kg Follower mass moment of inertia I 2 5.5 x10 4 kg m 2 Slide mass m 3 9 kg Figure 2 2. The follower angle is defined as the angle between the negative X and follower axes.

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31 0 2 4 6 0 0.05 0.1 0.15 0.2 0.25 crank angle (rad)X-axis slide position x3 (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)slide velocity (m/s) Figure 2 4. Stage velocity as a function of crank angle at a spindle speed of 30 rpm.

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32 0 2 4 6 -1.5 -1 -0.5 0 0.5 1 1.5 crank angle (rad)stage acceleration (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)joint sliding velocity (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 pinbushing joint diameter of 19.05 mm.

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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 6 0 10 20 30 40 crank angle (rad)joint force magnitude (N) 15 rpm 30 rpm 45 rpm 60 rpm Figure 2 8. Predicted cyclic joint force magnitude profiles at varying crank speeds.

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34 A 0 1 2 3 4 5 6 0 10 20 30 40 50 crank angle (rad)joint force magnitude (N) ks = 0 N/m ks = 200 N/m ks = 400 N/m B 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)joint force magnitude (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 sys tem inertia. A) Spring offset yspr of 0 mm. B) Spring offset yspr of 55 mm.

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35 0 1 2 3 4 5 6 0 5 10 15 crank angle (rad)joint force magnitude (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 6 0 5 10 15 20 crank angle (rad)joint force magnitude (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.

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36 Figure 212. Static cylinder contact model Figure 2 13. Contact model assumes flat deformation profile and semi elliptical pressure profile.

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37 Figure 2 14. The contact model may be modified to accommodate int ernal 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 R a 9.48 mm Body b radius R b 9.55 mm Body a Young's modulus E a 210 GPa Body b Young's modulus E b 542 MPa Body a Poisson's ratio a 0.3 Body b Poisson's ratio b 0.45 Contact load F n 0 150 N

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38 A 0 50 100 150 0 0.5 1 Contact Normal Force Fn (N)Maximum Contact Pressure, Pm a x (MPa) B 0 50 100 150 0 2 4 6 8 10 Contact Normal Force Fn (N)Contact Width, w (mm) C 0 50 100 150 0 5 10 15 20 25 Contact Normal Force Fn (N)Contact Penetration, ( m) D 0 50 100 150 1.5 2 2.5 x 106 Contact Normal Force Fn (N)Contact Stiffness, Kc i (N/m) Figure 2 15. Results from the contact simulation indicate contact pressures less than 1.5 M P a, contact widths less than 12 mm, penetration values less than 30 mm, and contact stiffness values on the order of 2.1x106 N/m.

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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 i s shown in Figure 31. 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 c onst ituent components of the joint under study c omponents 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 t he 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.

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40 The revo lute 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 fol lower 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 be tween 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 thr ust 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 stiffn ess of 72 blocks which are bolted to the slide stage. The prismatic joint is a New Way Air Bearings model S4006150095457 dovetail slide with 304.8 mm (12 in.) of travel. This slide can exhibit joint stiffness of approximately 100pressure. Manipulating Joint Forces The contact and friction loads experienced by the joint under study can be manipulated in two w ays. 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. Addit ionally, 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 = 22 0 N/m.

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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, com parison 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 crankslider, 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 show n in Figure 34. The 3600 count per revolution digital signal generated by the encoder is then converted to an absolute analog output using a BEI M series digitalto 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 35). 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 06062TV 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.

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42 The physical arrangement of the strain gages and the circuitry of each Wheatstone bridge channel make them sensitive to transverse sh ear 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 38, respectivel y. When any load is applied to the pin transducer, the change in resistance ( resistance ( R ), and the gage factor ( Sg gRSR ) [ 34] This relationship is expressed mathematically in Equation 31. 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) [3 5]. This relationship is governed by Equation 32. Here, XZ represents the shear strain in the X Z plane, 1 and 2 12sin(2)XZ represent the tensile strains in gages 1 and angle of symmetry about the Z axis. The half angle dians, making the denominator of Equation 32 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. 33). 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 34. This formulation assumes that second order effects are

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43 neglected ( 0ijRR for any two g ages i and j ). With the understanding that resistance change and gage strain are proportional, Equation 34 may be rewritten in terms of strains (E quation 35). Here, XZ,a and XZ,b 1324 1234 out sRRRR VV RRRR 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 rose tte b. (3 3) 12344s outV VRRRR R (3 4) 1234 ,,44ss out XZaXZbVV V RR (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 (s teel) is assumed to be isotropic. The stress strain relationships may then be described as in Equation 36. In Equation 36, E and represent the Youngs modulus and Poisson ratio of the pin material.

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44 1 000 1 000 1 000 21 000 00 21 0000 0 21 00000 XX YY ZZ XY XY XZ XZ YZ YZEEE EEE EEE E E E (3 6) A general transv erse shear loading condition is shown in Figure 310. 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 XZaXZbVs V R load, the shear strains registered by each are equal and at a maximum. The refore, Equation 35 may be simplified to Equation 37. ( 3 7) A general bending moment loading condition is shown in Figure 311. First consider the effects of moment My. Both shear rosettes lie centered on the neutral p lane 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 35, 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

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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 312. 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 35 cancel and the bridge voltage is not affected. A general axial loading condition is shown in Figure 313. In this scenario, the tensile stress in the Z direction is proportional to the applied load Fz ,, ,, cos()sin() sin()cos()Xg Xp Yg Yp FF FF w hile 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 315) 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 38. Contact forces imparted on the bushing may be expressed in a coordinate system fixed to the follower link by Equation 39. (3 8)

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46 ,cos()sin() sin()cos()Xg Xb Yg YbF F F F (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 ,1cos()pinxpineFmL are calculated and removed according to Equati on 310 and 311. (3 10) 2 ,1sin()pinypineFmLg (3 11) Capacitance Probes Relative motion between the pin and bushing is measured using capacitance probes (Figure 317). Two Lion Precision model C23B probes m easure the pin location from fixed, orthogonally 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 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 i ntended 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

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47 the target surface 19). Since the capacitance probes only offer a two dimensional perspective, the y 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 locat ion 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 coordina te measuring machine (CMM). The CMM used in this study is a Brown and Sharpe PFX 3axis 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

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48 following wear testing. The bushing is then lightly clamped in vblocks to the CMM table for measurement. Alignment of the bushing coordinate s ystem 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 (Figur e 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 touchpoints 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 ma terial can be more accurately measured by massing the sample. Mass measurements are taken prior to and following wear tests using a Mettler Toledo AX205 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 322. The pin is then inserted through the bushing and clamped in the crank arm. If the kinematic components have been disassemb led 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.

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49 Operation and Data Collection Once the test bed h as 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 tha n 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.

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50 A B Figure 3 1. The crankslider bed used to perform experimental tests. A) Line drawing of the test bed showing components. B) Photograph of the actual test bed.

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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.

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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.

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53 Figure 3 6. Each load cell channel is constructed from two strain gage shear rosettes totaling four gages. Figure 37. 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.

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54 Figure 3 8. Wheatstone bridge circuit diagram for a single pin force channel. Gages are numbered as pictured in Figure 37. Figure 3 9. Two strain gages oriented symmetrically about an axis of interest combine to form a shear rosette.

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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.

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56 Figure 3 13. Axial loading of the pin transducer. Figure 3 14. Pin load cell channe l coordinates. Figure 3 15. Forces measured by the pin load cell can be transformed into a global coordinate

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57 Figure 316. Global coordinate forces may be further transformed to a coordinate system fixed to the b ushing 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.

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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 sens itivity error as a function of off axis motion of the pin. The MATLAB code used to generate this figure is provided in Appendix C.

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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 321. The bushing center is defined as the intersection of two lines drawn between opposing sets of wear collection grooves.

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60 Figure 3 22. Preparation of the test bed is perf ormed 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 directio n 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.

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61 CHAPTER 4 RESULTS FROM A SPRIN G 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 41). This was done to enable two simplifying assumptions to be made regar ding 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 va lues o f less than a millimeter. An initial diametric clearance of 108 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 41). This ma de sense given the limited wear amount for the test. T he 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. H igh frequency dynamic signals were witnessed throughout the test The amplitude of these vibrations was highest 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 riod of decaying high frequency vibration as well as the dynamic

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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 increasi ng 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 s lide may have been caused b y 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. How ever, 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, wi th data from cycle 1 shown in Figure 44. 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 8x106Wear Characterization N/m. This was somewhat higher tha n 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

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63 both before and following the wear test. Third, mass loss and, indirectly, v olume 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 cyc les (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 ang le appr 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 coordina te measuring machine (Figure 4 7 ). When considering Figure 47, 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 be lieved 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 the result of a buildup 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 (<8% disagreement), especially when it was considered that wear rates may

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64 vary by as much as an order of magnitude (Table 42). 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.

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65 Table 4 1. Wear test parameters Parameter Value Crank speed (rpm) 30 Initial diametr ic clear 108 Spring rate (N/m) 525 Spring preload (N) 52 Slide mass (kg) 8.5 Cycles run 21,400 A 0 2 4 6 0 50 100 150 200 crank angle (rad)joint forces (N) measured CEWM B 0 2 4 6 0 50 100 150 200 crank angle (rad) y joint forces (N) measured CEWM Figure 4 1. Joint force magnitude predictions and measured dat a. A) Cycle 1. B) Cycle 20,500.

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66 0 2 4 6 50 100 150 crank angle (rad) y ,joint force magnitude (N) = 0 = 0.01 = 0.1 Figure 4 2. CEWM simulation of the effect of sta ge friction on the dynamics at the joint of interest. 0 1 2 3 4 5 6 -200 -150 -100 -50 0 50 100 150 200 crank angle (rad) y joint forces (N) measured FX g measured FY g CEWM FX g CEWM FY g Figure 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. Experimental data is from cycle 1.

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67 0 2 4 6 -20 0 20 40 60 crank angle (rad)cap probe measurement ( m) x (probe 1) y (probe 2) Figure 4 4. Capacitance probe measurements of the pin displacement relative to the bushing center showed similar vibration content as the pin force measurements. Figure 4 5. Evolution of the cyclic pin pat h relative to the bushing center with increasing cycle number indicated a change in the bushing shape in the first 1000 cycles.

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68 A B Figure 4 6. Contact pressure evolution predicted by the CEWM. A) Maximum contact pressure at cycle 1. B) Maximum contact pressure at cycle 21400

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69 Figure 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. Radius change is plotted as a function of bushing angular coordinate Table 4 2. Experimental and model predictions of bushing wear amount for initial wear test. Experimental CEWM Mass loss (g) 0.1714 0.1589 Volume loss (mm 3 ) 78 72

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70 CHAPTER 5 ADDITIONAL INSTRUMENTATION OF THE TEST B E D The test bed was designed in order to monitor evolving conditions at the joint of interest. I n order to validate, or conversely to challenge, several simplifying assumptions made regarding parameters which affect the load profile in the revolute joint s everal changes were made to the test bed following the spring loaded wear test. First, the test bed was moved from a granite table set on a steel frame to a more stable location on an optical table. Second a uniaxial load cell was added in order to quanti fy tensile forces in the springs. Finally a laser interferometer assembly was added to directly measure the location of the sliding stage. Direct Measurement of Spring Forces Using a Uniaxial Load Cell Following the initial wear test, s pring force values were measured directly using a uniaxial load cell rather than estimating their values based on the stage location and assumed spring rate. A Honeywell Sensotec Model 31 load cell with 445 N (100 lbfDirect Measurement of Stage Position Using a Linear Displacement Measu ring Interferometer ) capacity was used for this application (Figure 51). This load cell was mounted to the optical table rather than the slide as not to confound the spring force signal with the inertial load that would be present if fixed to the moving stage. Typically, the designed purpose of a crank slider mechanism is to accurately prescribe reciprocating motion of the slide from the rotation of the crank. Because of this, one way in which the performance of the mechanism can be judged is the fidelity with which the path of the slide emulates the ideal reciprocating path described by Equation 2 6. As such, the laser interferometer setup shown in Figure 52 was added in order to track the position of the dovetail slide stage. In this setup, light emitted from the two frequency laser head is split into two paths. One frequency follows a fixed reference path, while the other frequency travels to and is

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71 reflected back from a target located on the sliding stage. The two light signals are recombi ned, and the interference between them is used to identify the stage displacement. Collecting data from the interferometer requires a specialized data acquisition (DAQ) card, and requires its own computer. Because of this, the measured stage position are not directly distance measured by the interferometer during a single crank cycle is used as a reference location and assumed to correspond to a crank angle of 0 radia a given crank angle is defined from this datum as shown in Figure 53.

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72 Figure 5 1. Spring forces are measured using a uniaxial load cell. Figure 5 2. A laser interferometer is used to measure the position of the s tage as a function of time.

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73 Figure 5 mechanism and as the minimum distance measured by the interferometer. This minimum is assumed to occur when the crank angle is equal to 0 r adians.

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74 CHAPTER 6 COUPLED JOINT WEAR AND DYNAMICS UNDER INERTIAL LOADING In the wear test described in Chapter 4, an attempt was made to simplify contact conditions and wear progression at the joint of interest Large, predictable spring loads were use d to constrain the contact location and overwhelm load contributions of the oscillating stage mass. This provided an initial verification of the CEWM under relatively simple conditions. Conversely, the wear test described in this chapter was intended to hi ghlight nonideal dynamics contributed by the joint of interest as well as to investigate evidence of coupled evolution in the joint dynamics and bushing wear. In order to allow free vibration of the pin within the bushing, no springs were used in this we ar test No extra mass was added to the slide stage, making the slide mass 8.5 kg. The crank speed was set to 30 rpm for a total of 420,000 cycles The PTFE test bushing was initially 19.066 mm in diameter, 0.022 mm out of round, and 13.1 mm thick. Given t he pin diameter of 18.958 mm, the initial clearance was 108 um. At the start of the test, the bushing mass was 13.1566 g. Data from the crank slider test bed and the interferometer was acquired at 2 kHz. While buffer limitations in the interferometer data acquisition system allowed only a single cycle of data to be collected at once, 20,000 samples per channel (five cycles) were read to the crank slider acquisition buffer at each acquisition. As of the time of this writing, no predictions from the CEWM wer e available for this wear test. Crank Slider Test Bed Results Results from this wear test indicated the evolution of high frequency dynamic content as the bushing wore. These dynamics were in signals measured by both the pin load cell (Figure 61) and the capacitance probes (Figure 6 2). In particular, changes in the single cycle pin path

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75 within the bushing as measured by the capacitance probes suggested a change in the bushing shape as well as size. When the force components were transposed into a bushing fixed coordinate system as described in Chapter 3, the high frequency dynamic content was observed in the bushing X direction (XB) direction only (Figure 6 4). However, when the results from the individual capacitance probe channels were plotted as a func tion of crank angle (Figure 6 5), Some vibration was observed in the YB direction as well. Since the XB direction is more closely aligned with the global X direction and therefore the degree of freedom of the slide prismatic axis, the XB direction was like ly more susceptible to vibratory error motions. Any smaller magnitude YB direction vibration that may have been present was likely drowned out by the approximately 2 N magnitude noise in the pin load cell. With the addition of the linear interferometer to the test bed, the effects of error motions at the joint of interest on the motion of the slide could be observed. Single cycle slide motion profiles are plotted in Figure 65 along with the ideal slide position x3In order to investigate the frequency content of the joint vibrations, fast Fourier transforms (FFT) were performed on the bushing force, capacitance probe, and interferometer data at cycle 10 and cycle 419,000. Magnitude plots from these transforms were then plotted in Figures 66 through 68. Disregarding the gross force signals, vibration content in the force signal wa s obtained using the kinematic simulation de scribed in Chapter 2. In order to discern micrometer scale changes in the slide motion, the gross motion profile was removed by subtracting the ideal slide position from the measured interferometer data. In this way, the evolution of higher frequency conte nt could be observed in the slide motion as well. Interestingly, the average error was seen to decrease over the course of the wear test.

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76 initially present at approximately 10 Hz (Figure 4 6). As the bushing wore, content was observed to develop between 20 Hz and 50 Hz in the FXbEx Situ Wear Measurements signal by cycle 419,000. The 10 Hz content did not show up in the capacitance probes at any cycle while content developed up to approximately 30 Hz as the bushing wore (Figure 4 7) While the cumulative vibration content seemed to shift towards higher frequencies as the bushing wore, these changes were difficult to quantify (Figure 48) Following the wear test, the final mass of the bushing was measured to be 13.1233 g for a cumulative loss of 0.0334 g. The worn bushing was measured to be 19.129 mm in diameter and 8 Changes in the bushing radius are plotted as a function of the bushing angular coordinate in Figure 6 9. spots on the bushing being subjected to higher contact pressures and wearing preferentially. This reduction in form error may also explain the reduction in stage position error shown in Figure 6 5. Discussion One concern that was raised following this wear test was whether the vibrations measured by the capacitance probes were substantially affected by compliance in the pin transducer. In order to test the rigid pin hypothesis, two short dynamic tests were run using the worn bushing. In the first test, the usual pin load cell was used, while in the second test, the load cell was replaced with a ground steel pin. Capacitance probe measurements from these short tests were t hen plotted for comparison (Figure 6 10) The magnitude of the measured pin vibration appeared reduced for the case of the solid steel pin. However, a third dynamic test run with the load cell again in place showed even lower vibration magnitudes than the solid pin (Figure 6-

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77 11C ). This suggested that differences between the vibrations measured in each test were more likely a result of relative misalignment of joint components between tests. A further concern following the wear test was the shape cyclic forc e profiles as measured by the pin. Prior to the test, the loa d cell pin had been incorrectly tared before being clamped in the crank arm. The effects of clamping the pin in the crank arm were sufficient to result in poor agreement with the dynamics predict ed by the ideal model described in Chapter 2 (Figure 6 11) However, it was believed that the development of high the high frequency content as the bushing wore was a real phenomenon. In order to validate this assertion as well as to demonstrate the actual dynamic profile, a short dynamic test was run with the worn bushing and the load cell zeroed appropriately (Figure 612). Agreement between this second set of dynamics and the ideal joint model was improved compared to the wear test measured dynamics.

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78 A 0 1 2 3 4 5 6 0 5 10 15 20 25 y crank angle (rad) joint force (global coordinates) (N) B 0 1 2 3 4 5 6 0 5 10 15 20 25 crank angle (rad)joint force (global coordinates) (N) C 0 1 2 3 4 5 6 0 5 10 15 20 25 crank angle (rad)joint force (global coordinates) (N) D 0 1 2 3 4 5 6 0 5 10 15 20 25 crank angle (rad)joint force (global coordinates) (N) Figure 6 1. The evolution of high frequency dynamics with wear can be observed from single cycle joint force magnitude plots. A) Cycle 10 B) Cycle 100,000 C) Cycle 200,000 D) Cycle 419,000.

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79 A -80 -60 -40 -20 0 20 -50 0 50 100 150 200 250 300 Pin center position (X)(um)Pin center position (Y)(um) = 0 rad = /2 rad = 3 /2 rad = rad B -80 -60 -40 -20 0 20 -50 0 50 100 150 200 250 300 y Pin center position (X)(um)Pin center position (Y)(um) C -80 -60 -40 -20 0 20 -50 0 50 100 150 200 250 300 Pin center position (X)(um)Pin center position (Y)(um) D -80 -60 -40 -20 0 20 -50 0 50 100 150 200 250 300 y Pin center position (X)(um)Pin center position (Y)(um) Figure 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. A) Cycle 10. B) Cycle 100,000. C) Cycle 200,000. D) Cycle 419,000.

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80 A 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)force on bushing (follower coords) (N) FB x FB y B 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)force on bushing (follower coords) (N) C 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)force on bushing (follower coords) (N) D 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)force on bushing (follower coords) (N) Figure 6 3. Higher frequency dynami c content appears to evolve only in the X direction of the bushing coordinate system. A) Cycle 10 contact force components. B) Cycle 100,000 force components. C) Cycle 200,000 force components. D).Cycle 419,000 force components.

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81 A 0 2 4 6 -100 0 100 200 300 crank angle (rad)capacitance probe output ( m) probe 1 probe 2 B 0 2 4 6 -100 0 100 200 300 y crank angle (rad)capacitance probe output ( m) probe 1 probe 2 C 0 2 4 6 -100 0 100 200 300 crank angle (rad)capacitance probe output ( m) probe 1 probe 2 D 0 2 4 6 -100 0 100 200 300 y crank angle (rad)capacitance probe output ( m) probe 1 probe 2 Figure 6 4. Capa citance probe output as a function of crank angle. A) Cycle 10. B) Cycle 100,000. C) Cycle 200,000. D) Cycle 419,000.

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82 A 0 2 4 6 -150 -100 -50 0 crank anglestage position x3 (mm) theoretical measured B 0 2 4 6 -150 -100 -50 0 crank anglestage position x3 (mm) theoretical measured C 0 1 2 3 4 5 6 0 0.5 1 crank anglestage position error (mm) D 0 1 2 3 4 5 6 0 0.5 1 crank anglestage position error (mm) Figure 6 5. Small changes in the stage path profile are observed by removing the gross motion profile from the interferometer output. Over the course of the test, the average cyclic stage position error is seen to decrease while higher frequency dynamics are seen to evolve. A) Theoretical and measured stage position profile at cycle 10. B) Theoretical and measured stage positi on profile at cycle 419,000. C) Stage position error profile at cycle 10. D) Stage position error profile at cycle 419,000.

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83 A 0 10 20 30 40 50 0 1 2 3 4 5 frequency (Hz)bushing force magnitude |Fx B| (N) B 0 10 20 30 40 50 0 1 2 3 4 5 frequency (Hz)bushing force magnitude |Fx B| (N) C 0 10 20 30 40 50 0 1 2 3 4 5 frequency (Hz)bushing force magnitude |Fy B| (N) D 0 10 20 30 40 50 0 1 2 3 4 5 frequency (Hz)bushing force magnitude |Fy B| (N)y Figure 6 6. Fast Fourier t ransform (FFT) of bushing coordinate force results. A) Bushing X direction force profile at cy cle 10. B) Bushing X direction contact force profile at cycle 419,000. C) Bushing Y direction contact force profile at cycle 10. D) Bushing Y direction contact force profile at cycle 419,000.

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84 A 0 10 20 30 40 50 0 10 20 30 40 50 frequency (Hz)cap. probe 1 magnitude ( m)y B 0 10 20 30 40 50 0 10 20 30 40 50 frequency (Hz)cap. probe 1 magnitude ( m)y C 0 10 20 30 40 50 0 10 20 30 40 50 frequency (Hz)cap. probe 2 magnitude ( m) D 0 10 20 30 40 50 0 10 20 30 40 50 frequency (Hz)cap. probe 2 ( m) Figure 6 7. Capacitance probe FFT results. A) Probe 1 cycle 10. B) Probe 1 cycle 419,000. C) Probe 2 cycle 10. D) Probe 2 Cycle 419,000. A 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 frequency (Hz)stage position error magnitude (mm) B 0 10 20 30 40 50 0 0.05 0.1 0.15 0.2 frequency (Hz)stage position error magnitude (mm) Figure 6 8. Interferometer FFT results A) Cycle 10. B) cycle 419,000

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85 0 2 4 6 -50 0 50 100 150 bushing angle (rad)radius change R ( m) Figure 6 9. Change in radius as a function of bushing angle as measured by the three axis C MM. A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)capacitance probe output ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)capacitance probe output ( m) Xb Yb Figure 6 10. Capacitance probe measurements from two short tests intended to c halleng e the assumption of a rigid pin. A) Sensor pin. B) Solid ground pin.

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86 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)force on bushing (follower coords) (N) FX b meas. FY b meas. FX b mod FY b mod Figure 6 11. Comparison between measured and modeled bushing force profiles. Ex periment al data is from cycle 1

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87 A 0 1 2 3 4 5 6 0 5 10 15 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)bushing contact force (N) xm e a s ym e a s xt h e o yt h e o C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) XB YB Figure 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 Results from this test reflected predictions from an ideal dyna mic model. A) Measured and predicted joint force magnitude. B) Measured and predicted global force components. C) Pin displacement as measured by the capacitance probes.

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88 CHAPTER 7 SENSITIVITY OF THE D YNAMIC SYSTEM TO CHANGES IN THE MECHANIS M AND JOINT P ARAMETERS Following the second wear test, a series of short tests were conducted in order to further explore the dynamics at the joint of interest. The lengths of these tests were held to less than 100 cycles in order to limit wear effects. The goal of the se short tests was to demonstrate the sensitivity of the joint force and displacement measurements to a variety of factors associated with the joint of interest and the mechanism as a whole J oint forces measured during these tests were compared with predi ctions made by the simple dynamic model presented in Chapter 2 in order to highlight deviations from ideal dynamics For all of the tests described in this chapter, data acquisition was performed at 8 kHz. It should be noted that the same bushing was used for all tests in this section and that no change in the bushing diameter or roundness error was detected by the coordinate measuring machine Repeatability of Dynamic Test Results In order to investigate the sensitivity of the dynamic profile to disassemb ly and reassembly, a series of five repeat tests were performed under nominally identical conditions. Experimental parameters used in these tests were comparable to those used in the wear test described in Chapter 6 (Table 71). The disassembly and reassem bly procedures described below were equivalent to the steps that would ha ve to be taken in order to add mass the stage bracket. First, the load cell pin was removed from the crank link and test bushing. This effectively separated the crank and follower lin ks. The PTFE test bushing was then removed from the follower link. The screws connecting the thrust air bushing pillow blocks to the stage bracket were then removed, allowing the pillow blocks, air bushings, follower stage joint pin, and follower link to be removed from the stage together. While these items could be removed as one unit, they were no longer fixed together. The steel stage bracket was then removed from the dovetail slide stage.

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89 The crank slider test bed was then reassembled in reverse order. During assembly, the screws holding the pillow blocks to the stage bracket were not tightened. Five to ten revolutions of the crank were then completed at 10 rpm to allow the bushing and follower link to align with the stage. The pillow block screws were then tightened, the speed set to 30 rpm, and data acquisition was initiated Single cycle force magnitude plots from the five tests showed reasonable agreement with the dynamic model (Figure 7 1). Based on the relatively large bushing clearance, high frequenc y dynamics were expected in each of the force profiles. However, extraneous vibration was only witnessed in the force and capacitance probe data from the second trial (Figure 7 1B and Figure 72B). This result suggested that in most cases, residual misa lignment between the pin and bushing axes prevented the joint components from losing contact A series of measurements of the misalignment between the mechanism revolute joints is included in Appendix D which suggested that for even modest misalignments be tween the pin and bushing, the pin is likely in contact with at least the edges of the bushing at all times. With the exception of the high frequency content from trial 2, very little variability between the force measurements was seen. Capacitance probe p rofile Sensitivity to Changes in the Mechanism Operating Conditions A series of short tests were conducted in which the sensitivity of the joint dynamics to the mechanism operating parameters was investigated and compared with the ideal dynamic model. In this section, tests were completed while varying crank speed, stage mass, and spring rate. Unless otherwise note d, the conditions of these tests were identical to those listed in Table 71. Influence of Varying th e Crank Speed Short dynamic test s were conducted at crank speeds of {15.2, 30.0, 50.3, and 60.0} rpm. At no time during th ese test s was the mechanism disassembled. Joint force measurements from these tests are displayed in Figures 73 and 74. Capacitance probe readings for these tests are

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90 displayed in Figure 75. The simple dynamic model was most effective at capturing the joint force profile at a crank speed of 30.0 rpm. High frequency dynamics were observed in force and capacitance probe data at 15.2 rpm 50.3 rpm, and 60.0 rpm The frequency of vibration in the 15.2 rpm force data was difficult to characterize due to a low signal to noise ratio. One possible explanation for the high frequency content presence during a low kinetic energy stage was that th e system inertia may have been low enough for the pin to ride over small features on the bushing surface rather than penetrate into them. Stick slip may have also contributed to these dynamics. High frequency dynamics at higher crank speeds were more likel y a result of natural frequencies associated with the system inertia and joint stiffness being excited by the oscillating stage mass and exacerbated by the joint clearance. One interesting feature that was noted in the force data was a spike that was obser ved to grow in magnitude and shift from a crank angle of approximately 0.4 rad to 0.7 rad as the crank speed was increased. Similarity with the global X direction force profile suggest s that the XbInfluence of Increasing the Stage Mass displacement of the pin measured by the capacitance probes was related to joint compliance. A series of short dynamic tests were conducted in order to investigate the effects of added stage mass on the joint performance. The first set of varying mass experiments was conducte d by bolting additional weights under the stage bracket as shown in Figure 3 3. Data was collected at stage mass values of 8.5 kg (no mass added to the stage), 12.8 kg (a single steel weight bolted to one side the stage), and 17.1 kg (steel weights bolted to either side of the stage). For the single mass case, a repeat test was performed with the mass bolted to the opposite side of the stage to investigate the effects of out of plane moments. Force magnitude results from the bolted mass tests are provided in Figure 7 6. Pin displacement results are shown in Figure 7 7. While some vibration was observed in all tests, the

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91 most substantial high frequency content was observed when a single mass was bolted to either the far side (closest to the DC motor) or near side of the slide bracket. Additionally, sharp jumps in the Xb direction capacitance probe measurements were observed in the 8.5 kg and 12.8 kg rad rad It was likely that this phenomenon resulted from the inertia of the follower link and stage overcoming the driving speed of the crank at the joint of interest. Following these experiments, there was concern that the severe vibrations in the 12.8 kg stage mass tests was influenced by out of plane moments caused by the inertial imbalance of the stage. Further concern was raised that imbalance of the stage mass could result in undesired contact between the mechanism air bearings and guideways. A second set of experiments was then conducted in which masses were simply placed on top of the thrust air bearing pil low blocks therefore reducing the out of plane mass moment of inertia. In this set of experiments, it was not necessary to stop or disassemble the rig between tests. Force data from these tests shows reasonable agreement with the ideal dynamic model descr ibed in Chapter 2 (Figure 7 8 and Figure 7 9). Dynamic spikes at crank angles of {0.4, 2.3, 4.1, and from 56} rad were seen to grow in magnitude with increasing stage mass. These spikes were visible only in the global X direction. No discernable difference was observed in the force magnitude profiles when a single 4.3 mass was moved from one pillow block to the other. Furthermore, c hanges in the capacitance probe measurements with increasing mass were difficult to distinguish for these tests (Figure 7 10). Interestingly, the shift in the stage center of mass caused by moving the location of the added weights did not appear to greatly affect the joint force profile.

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92 A third set of experiments was conducted in which the bushing was replaced with the worn bushing from the Chapter 6 wear test. Again, the stage mass was increased by setting steel weights on top of the pillow blocks. The high frequency dynamics associated with the large clearance and small roundness form error of the worn bushing did not appear to increase in severity as the stage mass was increased (Figure 7 11) Influence of Increasing Spring Rate A series of joint dynamic tests was conducted in order to investigate the effects of increasing spring rate on the joint dynamics. Experiments were car ried out with zero, one and two springs added to the test bed The nominal stiffness of each spring was 220 N/m. The measured spring forces are shown as a function of spring extension in Figure 712. Unfortunately, when this series of tests was conducted, the interferometer was not functioning and the spring extension had to be calculated using the ideal kinematic model described in Chapter 2. This was an acceptable substitution for this application as interferometer results from Chapter 6 indicated that the model accurately predicted stage position to better than 1 mm. A linear least square fit was applied to Figure 7 12 in order to calculate the actual spring rates. The spring rate for a single spring was found to be 236 N/m while the total spring rate for a two spring array was found to be 469 N/m. In each case the spring preload was approximately 46 N. Joint force magnitude profiles from these tests were noisier than previous short tests (Figure 7 13). This was caused by two failing soldered connections between the strain gage bridge outputs and the slip ring. Shortly after these tests were conducted, these connections separated and were repaired. Nonetheless, vibration at approximately 8 Hz was observed in both instances where springs were used. Since si milar force perturbations were not evident in the spring load profiles, these vibrations were thought to be related to a damped natural frequency associated with the pin bushing stiffness and the mechanism inertia. Similar vibrations were,

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93 however, observe d in the capacitance probe measurements (Figure 7 14). Furthermore, the joint force drop that occurred in the springd was not observed in these experiments. Measured global force components compared reasonably well w ith the ideal dynamic model (Figure 7 15). This was somewhat surprising since the vertical offset between the follower stage revolute joint and the point of spring load application ( ysprShort Dynamic Tests Conducted Using a Previously Worn Bushing ) was set to zero. It is not clear why the force profiles measured dur ing these tests did not reflect the true offset which was on the order of 65 mm. Short tests conducted with increasing stage mass suggested that high frequency vibration (at greater than 20 Hz) was accommodated to a greater degree with the worn bushing described in Chapter 6 than with a freshly machined bushing. Over the course of the wear test, form errors in the bushing profile would have worn preferentially due to higher contact pressures. Ad ditionally, misalignment between the joint components may have led to higher contact pressures at the bushing edges. This would have lead to a worn shape that was able to accommodate misalignment with free motion. In order to test these theories, further s hort dynamic tests were performed using the worn bushing. Five repeated short dynamic tests were conducted using the worn bushing. These tests were conducted using a crank speed of 30 rpm, a stage mass of 8.5 kg, and with no added springs. Following each t est, the crank slider was disassembled and reassembled. Each disassembly included removal of the pin and bushing as well as removal of the stage mounting plate from the dovetail slide. Joint force measurements and capacitance probe measurements from these five repeat tests are shown in Figures 7 16 and 717, respectively. All trials showed evidence of high frequency dynamics to one degree or another. This validated the theory that the

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94 shape of the wear scar in the bushing served to reduce sensitivity to mis alignment. In other words, the bushing had worn to accommodate the driving path of the pin and to allow a more free transfer of motion. Following the repeatability tests, additional short experiments were conducted using the worn bushing in order to invest igate the sensitivity of the high frequency dy namics to increasing crank speed Speed sensitivity trials were conducted at crank speeds of {15, 30, 45, and 60} rpm (Figure 7 18) As expected, high frequency vibration content was observed to grow in magnitude with the increased kinetic energy provided by higher speeds. Discussion Several conclusions were drawn from the experiments conducted in this chapter. First, the presence or absence of high frequency dynamics in the unworn bushing appeared to be depende nt on how well the pin and bushing are aligned at the start of the test. In most cases, misalignment was sufficient for contact with the bushing edges and roundness form error to prevent high frequency (2050 Hz) vibration. Increasing the crank speed and t he stage mass led to corresponding increases in the kinetic energy of the mechanism. This in turn increased the severity of nonideal dynamics in the 5 10 Hz range, but not in the 2050 Hz range in the cases where a freshly machined bushing was used. When a worn bushing was used, alignment and form errors were not sufficient to prevent high frequency (2050 Hz) vibration. Finally, the test bed as constructed was found to be relatively insensitive to offsets between the follower stage revolute joint axis and the stage center of mass as well or the line of action of the spring force.

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95 Table 7 1. Test conditions of five short dynamic repeatability tests Test Condition Value Units Stage mass 8.5 kg Crank speed 30 rpm Spring rate 0 N/m Bushing diameter 19.1 23 mm Bushing roundness form error 26 Pin load cell diameter 18.958 mm Nominal joint clearance 165

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96 A 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical C 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical D 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical E 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical Figure 7 1. Pin force magnitude profiles from a series of five repeat dynamic tests. A) Trial 1. B) Trial 2. C) Trial 3. D) Trial 4. E) Trial 5.

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97 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb D 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb E 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb Figure 7 2. Capacitance probe measurements from a series of five repeat dynamic tests. A) Trial 1. B) Trial 2. C) Trial 3. D) Trial 4. E) Trial 5.

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98 A 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) p measured theoretical B 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) p measured theoretical C 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) p measured theoretical D 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) p measured theoretical Figure 7 3. Joint force magnitude single cycle profile at varying c rank speeds. A) 15.2 rpm. B) 30.0 rpm. C) 50.3 rpm. D) 60.0 rpm

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99 A 0 1 2 3 4 5 6 -40 -20 0 20 40 60 crank angle (rad)global contact force (N) x meas y meas xtheo y theo B 0 1 2 3 4 5 6 -40 -20 0 20 40 60 crank angle (rad)global contact force (N) x meas y meas xtheo y theo C 0 1 2 3 4 5 6 -40 -20 0 20 40 60 crank angle (rad)global contact force (N) x meas y meas xtheo y theo D 0 1 2 3 4 5 6 -40 -20 0 20 40 60 crank angle (rad)global contact force (N) x meas y meas xtheo y theo Figure 7 4. Global coordinate force components at varying crank speeds. A) 15.2 rpm. B) 30.0 rpm. C) 50.3 rpm. D) 60.0 rpm

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100 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) p Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb D 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) p Xb Yb Figure 7 5. Single cycle capacitance p robe readings at varying crank speed. A) 15.2 rpm. B) 30.0 rpm. C) 50.3 rpm. D) 60.0 rpm

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101 A 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical C 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical D 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical Figure 7 6. Single cycle joint force magnitude plots with changing stage mass. A) No mass added to the stage. B) A single 4.3 kg mass is attached to the far side of the stage. C) A single 4.3 kg mass is added to the near side of the stage. D) Two masses for a total of 9 kg are added to the stage.

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102 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb D 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb Figure 7 7. Capacitance probe measurements from a series of tests varying the mass of the dovetai l slide stage. A) No mass added to the stage. B) A single 4.3 kg mass is attached to the far side of the stage. C) A single 4.3 kg mass is added to the near side of the stage. D) Two masses for a total of 9 kg are added to the stage.

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103 A 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical C 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical D 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) measured theoretical Figure 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. A) No added mass. B) 4.3 kg of added on the far pillow block. C) 4.3 kg of mass set on the near pil low block. D) 8.6 kg of added mass.

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104 A 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)global contact force (N) FX g meas FY g meas FX g model FY g model B 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad) global contact force (N) C 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad) global contact force (N) D 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad) global contact force (N) Figure 7 9. Global force components subject to varying stage mass A) No added mass. B) 4.3 kg of added on the far pillow block. C) 4.3 kg of mass set on the near pillow block. D) 8.6 kg of added mass.

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105 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb D 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb Figure 7 10. Changes in the capacitance probe pin displacement measurements were difficult to discern as the stage mass was increased. A) No added stage mass. B) 4.3 kg of mass set on the far pillow block. C) 4.3 kg of mass set on the near pillow block. D) 8.6 kg of mass set on the pillow blocks.

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106 A 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad) pin force magnitude (N) B 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) C 0 1 2 3 4 5 6 0 5 10 15 20 25 30 crank angle (rad)pin force magnitude (N) Figure 7 11. Force magnitude at the joint of interest subject to varying stage mass and using a worn bushing. A) No mass added to the stage. B) 4.3 kg of mass set on the far pillow block. C) 8.6 kg of mass set on the pillow blocks.

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107 A 0 0.05 0.1 0.15 0 20 40 60 80 100 120 spring extension (m) spring force (N) y = 235.7*x + 46.46 measured spring force linear regression B 0 0.05 0.1 0.15 0 20 40 60 80 100 120 spring extension (m) spring force (N) y = 468.9*x + 46.08 measured spring force linear regression Figure 7 12. Spring rates were found to be linear and with comparable preloads. A) Single spring. B) Two springs.

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108 A 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured modeled B 0 2 4 6 0 20 40 60 80 100 120 crank angle (rad) pin force magnitude (N) C 0 2 4 6 0 20 40 60 80 100 120 crank angle (rad) pin force magnitude (N) Figure 7 13. Contact force magnitude subject to increasing spring load. A) No spri ngs added. B) Single spring with spring rate 235.7 N/m and 46.5 N preload. C) Two spring array with combined spring rate of 468.9 N/m and 46.1 N preload.

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109 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position ( m) Xb Yb Figure 7 14. Capacitance probe output with increasing spring load A ) No added springs. B) Single spring with spring rate 235.7 N/m and 46.5 N preload. C ) Two spring array with combined spring rate of 468.9 N/m and 46.1 N preload.

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110 A 0 1 2 3 4 5 6 -20 -10 0 10 20 crank angle (rad)global contact force (N) pg FX g meas FY g meas FX g model FY g model B 0 2 4 6 -50 0 50 100 crank angle (rad)global contact force (N) C 0 2 4 6 -50 0 50 100 crank angle (rad)global contact force (N) Figure 7 15. Global force components subject to increasing spring load. A) No springs added. B) Sing le spring with spring rate 235.7 N/m and 46.5 N preload. C) Two spring array with combined spring rate of 468.9 N/m and 46.1 N preload.

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111 A 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical C 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical D 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical E 0 1 2 3 4 5 6 0 5 10 15 20 crank angle (rad)pin force magnitude (N) measured theoretical Figure 7 16. Joint force magnitude measurements from five repeat dynamic tests using a worn bushing. Hi gh frequency dynamics were more prevalent than with an unworn bushing of similar clearance. A) Trial 1. B) Trial 2. C) Trial 3. D) Trial 4. E) Trial 5.

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112 A 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) Xb Yb B 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) Xb Yb C 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) Xb Yb D 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) Xb Yb E 0 2 4 6 -200 -100 0 100 200 crank angle (rad)pin center position (um) Xb Yb Figure 7 17. Capacitance probe measurements from a series of five short dynamic tests u sing a worn bushing. A) Trial 1. B) Trial 2. C) Trial 3. D) Trial 4. E) Trial 5.

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113 A 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) measured theoretical B 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) measured theoretical C 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) measured theoretical D 0 1 2 3 4 5 6 0 10 20 30 40 50 60 crank angle (rad)pin force magnitude (N) p measured theoretical Figure 7 18. Joint force magnitude results at varying crank speeds using a worn bushing. A) 15.1 rpm. B) 30.1 rpm C) 44.8 rpm. D) 59.7 rpm.

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114 CHAPTER 8 CONCLUDING REMARKS An instrumented crank slider test bed has been constructed to satisfy two goals: 1) to generate experimental data for comparison with dynamic and wear models, and 2) to highlight the dynamic contributions of a single nonideal and wearing revo lute joint to the mechanism performance. The non ideal joint under consideration in this study was a pinbushing revolute joint in dry sliding and located between the mechanism crank and follower arms. While the joint under study was instrumented to observe the evolution of dynamics and wear, the remainder of the crank slider mechanism was intended to contribute as little error motion, friction, and wear as possible. In fulfillment of the project goals, two wear tests were conducted. In the first test, large spring forces dominated the load profile providing accelerated wear while simplifying the kinematic and dynamic evolution. Results from this test were compared with the results of a Coupled Evolution Wear Model (CEWM). While differences between the predi cted and measured joint forces were observed, the wear predictions of the CEWM proved to be accurate in magnitude. Additionally, results from this wear test highlighted the dynamic effects of friction in the mechanisms sliding prismatic joint. A second we ar test was conducted in which the inertia of the sliding stage dominated the load profile. Joint force measurements from this test were compared with an ideal dynamic model in order to highlight nonideal force behavior. As the bushing wore, the simultane ous evolution of high frequency dynamics and vibration of the pin was observed. These dynamics increased in magnitude as the joint clearance increased and the roundness error of the bushing decreased. Furthermore, the vibrations at the joint under study we re found to a ffect the motion of the mechanisms prismatic joint.

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115 In order to further explore the contributions of the single real joint as well as to generate experimental data for future comparison with comprehensive dynamic models, a series of short nonwearing tests were conducted. These experiments were designed to investigate the sensitivity of the behavior of the joint of interest to changes in the mechanism parameters. Increasing the kinetic energy of the mechanism through increases in crank speed or inertia was found to exacerbate nonideal dynamics associated with joint compliance and clearance. Conversely, increased spring loading acted to constrain the contact and therefore did not further excite vibration in the joint. A series of repeated dyn amic tests following disassembly and reassembly of the mechanism indicated that high frequency dynamics were not always observed prior to bushing wear, even with substantial joint clearance. It was postulated that roundness form error of the joint components in conjunction with angular misalignment of the pin and bushing maintained contact and therefore reduced vibration. This theory was supported by a series of measurements of misalignment between the mechanism joint axes.

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116 APPENDIX A MATLAB SIMULATION OF AN IDEALIZED CRANKSLIDER MECHANISM %This program simulates the kinematics and dynamics for an idealized crank %slider mechanism % %N. Mauntler % %Bodies: %1 crank link %2 follower link %3 stage %4 ground % clear all % close all clc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mechanism Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %gravitational constant g = 9.81; %m/s %speed input w = pi; %crank speed (radians/s) %geometric parameters dj = 19.00/1000; %pinbushing joint diameter l1 = 3*25.4/1000; %crank length (3 inches) (m) l2 = 8*25.4/1000; %follower length (8 inches) (m) l3 = 0/1000; %vert. dist. b/w follower revolute and slide axes (m) yspr = 0/1000; %vert. dist. b/w follower revolute and spring (m) m1 = 0.4; %kg, measured using digital scale I1 = 200/1000^2; %kg*m^2, calculated using SolidWorks m2 = 0.812; %kg, measured using digital scale I2 = 5500/1000^2; %kg*m^2, calculated using SolidWorks m3 = 8.5; %kg % m3 = 9+4.324; %kg % m3 = 9+4.324+4.305; %kg %spring values Fs0 = 0; %Spring preload, N k = 0; %Linear spring rate, N/m k2 = 0; %nonlinear spring rate, N/m^2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Kinematics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Create a time vector

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117 dt = 2/1000; %s time = dt:dt:2; %s %Assuming a constant crank velocity, create a vector of crank angle theta theta = w*time; %From the crank angle, calculate the follower angle phi phi = asin(l1/l2*sin(theta)); %Calculate the center of mass locations for bodies 1, 2, and 3 (m) x1 = l1/2 cos(theta); y1 = l1/2 sin(theta); xpin = l1*cos(theta); ypin = l1*sin(theta); x2 = 2*x1 + l2/2*cos(phi); y2 = l2/2*sin(phi); x3 = l1*cos(theta) + l2*cos(phi); y3 = l3; %Calculate the angle kappa between the crank and follower links kappa = 180 theta phi; %Calculate the center of mass velocities (m/2) dphi = (l1/l2*cos(theta)*w) ./ sqrt(1(l1/l2*sin(theta)).^2); dx1 = l1/2 sin(theta)*w; dy1 = l1/2 cos(theta)*w; dx2 = l1*sin(theta)*w l2/2*sin(phi).*dphi; dy2 = l2/2*cos(phi).*dphi; dx3 = l1*sin(theta)*w l2*sin(phi).*dphi; dy3 = 0; dxpin = l1*sin(theta)*w; dypin = l1*cos(theta)*w; %Calculate the rotational velocity and sliding velocity at the joint of %interest dkappa = w dphi; vj = dj/2*dkappa; s = sum(vj*dt) %Calculate the center of mass accelerations A = (l1/l2*sin(theta)).^2; B = sqrt(1A); C = (l1/l2*cos(theta)).^2; ddphi=l1/l2*w^2*sin(theta).*(BC./B)./(1A); ddx1 = l1/2 cos(theta)*w^2; ddy1 = l1/2 sin(theta)*w^2;

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118 ddx2 = l2*(cos(theta)*w^2) l2/2*(cos(phi).*dphi.^2 + sin(phi).*ddphi); ddy2 = l2*(sin(phi).*dphi.^2 + cos(phi).*ddphi); ddx3 = l1*cos(theta)*w^2 l2*(cos(phi).*dphi.^2 + sin(phi).*ddphi); ddy3 = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Dynamics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Spring force xspr = x3 + l1+l2; %spring extension from start point Fspr = Fs0 + k*xspr + k2*xspr.^2; % figure(4) % subplot(2,1,1) % plot(time,xspr,'.k') % xlabel('time (s)') % ylabel('delta xstate') % subplot(2,1,2) % plot(time,Fspr,'.k') % xlabel('time (s)') % ylabel('spring force') %The following are obtained by a free body diagram of the slider stage %neglecting friction in the slide M34 = m3*ddx3*l3 + Fspr*yspr; %The following are obtained by a free body diagram of the follower and %slider together. %Fx21,Fy21 = forces on the bushing %Fy12,Fy12 = forces on the pin Fx21 = m2*ddx2 + m3*ddx3 Fspr; Fe = (M34 + Fspr.*(yspry3) Fx21.*(2*y2y3)) ./ (l2.*cos(phi)); Fi = (m2*ddx2.*(y2y2) (I2+m2*((x3x2).^2+(y2y3).^2)).*ddphi +m2*(g+ddy2).*(x3x2)) ./ (l2.*cos(phi)); Fy21 = Fe + Fi; % Fy21 = (M30 + Fspr.*(ysprl3) + m2*ddx2.*(y2+l3) + m2*(g+ddy2).*(x3x2) Fx21.*(2*y2+l3))./(x3x2.*cos(phi)); %Calculate the forces on the pin in global coordinates Fpinx = Fx21; Fpiny = Fy21; Fmag = sqrt(Fpinx.^2 + Fpiny.^2); %Pin forces on bushing in bushing coordinates Fbbx = (Fpinx.*cos(phi) Fpiny.*sin(phi)); Fbby = (Fpiny.*cos(phi) + Fpinx.*sin(phi));

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119 APPENDIX B MATLAB SIMULATION OF A SIMPLE LINE CONTAC T MODEL %This program calculates the max contact pressure, contact depth, %stiffness, etc of a pin in contact with an annular bushing. The model is %twodimensional and takes into consideration only static, normal loading. % %All input units should be SI (Fn in N and Lc in m) function [pmax, delta,Kci,w,E,R] = cylcontmod(Fn,Lc) %Pin properties and geometry Ra = 18.958/2/1000; %Radius, m Ea = 210E9; %Young's Modulus, Pa (210 GPa) nua = 0.3; %Poisson's ratio %Bushing properties and geometry Rb = 19.1/2/1000; %Radius, m Eb = 542E6; %Pa nub = 0.45; %Poisson's ratio %Calculate effective radius R = 1/(1/Ra + 1/Rb); %Calculate effective contact modulus E E = 1/((1nua^2)/Ea + (1nub^2)/Eb); %Calculate contact width w (m) w = 4*sqrt(Fn/Lc*R/pi/E); %Top out the contact width at half the outer circumference if w > pi*abs(Rb) w = pi*abs(Rb); end %Calculate the maximum contact pressure (Pa) pmax = sqrt(Fn/Lc*E/pi/R); %Calculate the pressure P (Pa) distribution as a function of s (m) s = w/2:w/1000:w/2; P = pmax*sqrt(1(2*s/w)); %Calculate the penetration, m delta = 2*Fn/Lc/pi*((1nua^2)/Ea*(log(8*Ra/w)+0.5)+(1nub^2)/Eb*(log(8*Rb/w)+0.5)); %Calculate the instantaneous contact stiffness kci, N/m Kci = Fn/delta;

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120 APPENDIX C MATLAB SIMULATION OF SECONDARY CAPACITANC E PROBE SENSITIVITY %This function calculates what the capacitance probe should output when the %pin is located at particular location from the intersection of the cap %probe paths (origin). All units are in mm. % % %Assumptions: %1) The cap probe signal is zeroed when the pin center is %coincident with the origin. % %2) The capacitance probes are perfectly orthogonal with respect to their %sensing faces located 10.40 mm from the origin and 0.875 mm from the closest %point on the pin face(in the middle of the cap probe range) % %3) The pin is exactly 19.00 mm in diameter and is perfectly round % % %SYNTAX: % [Probe1Output, Probe2Output] = CapProbeOutput(PinCenter) % clear all % close all % clc function [Probe1Output, Probe2Output] = CapProbeOuput(PinCenter) %Origin%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Define the origin, the point where theoretically, the measurement paths of %the capacitance probes meet O = [0,0]; %Pin%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Locate the center of the pin % PinCenter = [0,0]; %mm PinR = 19.00/2; %mm Theta = pi/4:pi/10000:3*pi/4; %Radians %Define the border of a perfectly circular pin PinX = PinR.*cos(Theta)+PinCenter(1); PinY = PinR.*sin(Theta)+PinCenter(2); %Cap Probe Faces%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Define the locations of the capacitance probe measurement faces. Locate %them orthoganally at a distance of 10.40 mm from the center. Cap1x = 10.40; %mm Cap1ymax = 1.6; %mm Cap1ymin = 1.6; %mm %Vector of y values of cap probe 1 face

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121 Cap1yvis = 1.6:3.2/1000:1.6; i1 = 1:length(Cap1yvis); Cap1xvis(i1) = Cap1x; Cap2y = 10.40; %mm %Vector of y values of cap probe 1 face Cap2xvis = 1.6:3.2/1000:1.6; Cap2yvis(i1) = Cap2y; %Calculate the average distance measured by each Cap Probe%%%%%%%%%%%%%%%%% %Find indices where the cap probe paths intersect the pin and calculate the %pin output %For Cap probe 1 n1 = find(PinY > 1.6 & PinY < 1.6); Probe1PinInt = PinX(n1); %Take the mean x value to be read by probe 1 MeanPinPosX = mean(Probe1PinInt); %Calculate the output from Probe 1. Note that the probe should be "zeroed" %to when the pin center is at 0,0. Probe1Output = Cap1x MeanPinPosX 0.945250318838903; n2 = find(PinX > 1.6 & PinX < 1.6); Probe2PinInt = PinY(n2); %Take the mean y value to be read by probe 2 MeanPinPosY = mean(Probe2PinInt); %Calculate the output from Probe 2. Note that the probe should be "zeroed" %to when the pin center is at 0,0. Probe2Output = Cap2y MeanPinPosY 0.945250318838903;

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122 APPENDIX D INVESTIGATING THE PL ANAR MECHANISM ASSUM PTION Motivation One of the primary goals of the experimental test bed was to make the crank slider operate as a planar mechanism as much as possible. As s uch, error motions at the joint of interest were measured only in a twodimensional framework using capacitance probes. In order to quantify out of plane misalignments of the test bed, the pin axis of each revolute joint w as measured using a FARO ARM model IND 03 articulating coordinate measuring machine (CMM) outfitted with a scanning probe (Figure D 1). Measurement and Analysis Measurement of the revolute joint axes was accomplished by collecting point clouds of data from the cylindrical surfaces of the joint pins. In the case of the joint of interest, a ground steel pin with a diameter of 18. 97 mm was used instead of the pin load cell in order to provide a more uniform, less obstructed target for the CMM. The three pins measured in this study were the gr ound pin clamped in the spindle collet, the ground pin at the joint of interest, and the ground pin between the follower link and stage. Since point clouds could not be collected while the mechanism was in motion, static measurement of the pins was carried out with the crank arm at angles of { 2} rad The set of four static measurements was then repeated. Measurements at a given static location were carried out by first setting the crank to the appropriate position, then cutting off the air supply to the crankslider air beari ngs. This served to effectively lock the mechanism in place. Point clouds of the pin surfaces were then collected by placing the CMM probe tip in contact with a given pin surface and depressing the probe trigger (Figure D 2). The probe tip was then scanned over as much of the exposed pin surface as

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123 could be covered without losing contact. If multiple portions of a given pin were exposed, multiple point clouds were required. Following data collection, the point clouds were imported into Geomagic Studio 9 (Re search Triangle Park, NC) analysis software (Figure D 3). Point clouds were imported using the CMM coordinate system. While this coordinate system was constant from one measurement to the next, it had no defined relationship to crank slider coordinate syst em. Least square cylinders were then fit to the point clouds and the cylinder axis vectors were recorded (Figure D 4). The angle ax between two given vectors v1 and v2 1 12 12cosaxvv vv is calculated using Equation D 1. Ideally, all three axes are parallel for the planar crank slider mechanism. Were this actually the case, the calculated angles would be zero. (D 1) CMM Measurement Results The measured pin axis direction cosines are displayed in Table D 1. In this table, the angle cosine c omponents a, b, and c correspond to the CMM coordinate system X, Y, and Z axes, respectively. It should be noted that over the course of the trials, measurement of the spindle pin was the least repeatable. The standard deviation of the spindle pin axis as measured by the C MM was found to be 0.012 rad (0.70 deg). However, it is likely that this was due to uncertainty in identifying the cylinder axis rather than the axis of rotation errors associated with the spindle and collet. Th is is because the spindle pi n had only a limited surface available to be measured when compared with the crank follower and follower stage axis pins. The angular axis of rotation errors associated with the crank follower and follower slide revolute joints, however, were more relevant to this work. Over the course of the two crank revolutions and eight measurements, the standard deviation of the crank follower axis angle was

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124 6.2x104 rad (0.035 deg ). The standard deviation of the follower stage pin axis was found to be 4.0x104Discussion rad (0. 023 deg ). The calculated angular misalignment between the crank slider axis pins was t ypically less than 0.013 rad (0.74 deg ) (Table D 2). the spindle pin and the crank follower pin (at the joint of interest) angle between the spindle pin and the follower between the angle between the crank follower pin and the follower stage pin. The larger standard deviations in the calculat ed values of determining the spindle pin axis. While the measurements described in this appendix gave some indication of the mechanism joints in the assembled test bed, they did not directly measure the misalignment between the pin and the bushing. A simple model (Figure D 5) of the pin bushing cross section shows that for a given bushing width of wb bc w and a joint clearance c required to force the pin into contact with the bushing edges may be calculated using Equation D 2 (small angle approximation) As a representative example, for a liberal bushing clearance of ). The implication is that the average misalignment measured in this appendix is sufficient to prevent the pin from losing contact with the bushing under most circumstances. Roundness error in the bushing internal geometry may also help to prevent the pin from losing contact. (D 2)

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

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126 Figure D 3. Point clouds were imported into Geomagic Studio 9 software for analysis. Figur e D 4. The imported point clouds were then fit with least square cylinders.

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127 Table D 1. Pin axis vector components as imported into Geomagic Studio software. The axis components a, b, and c are relative to the CMM coordinate system. Crank angles are in radians. Trial Crank Angle Spindle Pin Axis Crank/Follower Pin Axis Follower/Stage Pin Axis a b c a b c a b c 1 0 0.900 0.226 0.372 0.895 0.220 0.387 0.899 0.224 0.377 1 /2 0.898 0.214 0.385 0.896 0.220 0.386 0.899 0.225 0.377 1 0.898 0. 207 0.389 0.896 0.220 0.386 0.899 0.224 0.377 1 0.897 0.217 0.384 0.895 0.221 0.387 0.898 0.225 0.377 2 0 0.901 0.218 0.376 0.895 0.222 0.387 0.898 0.225 0.377 2 0.892 0.217 0.395 0.896 0.220 0.386 0.898 0.225 0.377 2 0.883 0.243 0.402 0.895 0.221 0.387 0.898 0.225 0.377 2 0.900 0.220 0.377 0.895 0.221 0.387 0.898 0.225 0.378 Average 0.896 0.220 0.385 0.895 0.221 0.387 0.898 0.225 0.377 Stdev 0.006 0.011 0.010 0.001 0.001 0.001 0.001 0.000 0.000 Table D 2. Re lative angles between the revolute joint pin axes. Crank angles are in radians. Trial Crank Angle Angle Angle Degrees Radians Degrees Radians Degrees Radians 1 0 0.97 1.7x102 0.31 5.4 x103 0.66 1.1 x102 1 0.37 6.4 x10 3 0.78 1.4 x10 2 0.61 1.1 x10 2 1 0.77 1.3 x10 2 1.19 2.1 x10 2 0.59 1.0 x10 2 1 0.31 5.4 x1003 0.61 1.1 x102 0.64 1.1 x102 2 0 0.75 1.3 x102 0.44 7.6 x103 0.62 1.1 x102 2 0.59 1.0 x10 2 1.18 2.1 x10 2 0.60 1.0 x10 2 2 1.67 2.9 x10 2 1.96 3.4 x10 2 0.64 1.1 x10 2 2 0.64 1.1 x102 0.31 5.5 x103 0.59 1.0 x102 Average 0.76 1.3 x102 0.85 1.5 x102 0.62 1.1 x102 Stdev 0.43 7.5 x10 3 0.57 9.9 x10 3 0.03 4.4 x10 4

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128 Figure D 5. T he joint component angular misalignment he 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 (small angle approximation)

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129 LIST OF REFERENCES 1. J.E. Shigley and J.J. Uicker, Jr. Theory of m achines and m echanisms, 2nd2. C.E. Wilson and J.P. Sadler. Kinematics and d ynamics of m achinery, 3 Edi tion. Schaum Division, McGraw Hill, Inc, NY (1995) rd3. J.B. Bryan, R. Clouser, and E. Holland. Spindle a ccuracy. American Machinist, 612 (1967) 149164. Edit ion. Pearson Education, Inc, NJ (2003) 4. J.B. Bryan, P. Vanherck. Unification of t erminology c oncerning the e rror m otion of a xes of r otation. Annals of the CIRP Annual Meeting, Freudenstadt, West Germany, August ( 1975 ) 5. ANSI/ASME 89.3.4M 1985. Axes of Rotation: Methods for s pecifying and t esting (1985) 6. R.D. Arnell, P.B. Davies, J. Halling, T.L. Whomes. Tribology: principles and design Applications. Springer, NY (1991). 7. E.J. Haug. Computer a ided kinematics and d ynamics of m echanical s ystems. Allyn and Bacon, MA (1989) 8. P.E. Nikravesh, Comput er a ided a nalysis of m echanical s ystems. Prentice Hall, NJ. (1988) 9. H.M. Lankarani, P.E. Nikravesh. A Contact f orce m odel w ith h ysteresis damping for impact a nalysis of m ultibody s ystems. Journal of Mechanical Design 112 (1990) 369376. 10. H.M. Lankarani. A Poisson based f ormulation for f rictional i mpact a nalysis of m ultibody m echanical s ystems with open or c losed kinematic c hains. Journal of Mechanical Design 122 (2000) 489497. 11. P. Ravn. A c ontinuous a nalysis m ethod for planar m ultibody s ystems with j oint c learance. Multibody System Dynamics 2 (1998) 124. 12. P. Flores and J. Ambrsio. Revolute j oints with c learance in m ultibody s ystems. Computers & Structures 82 (2004) 13591369. 13. J. Baumgarte. Stabilization of c onstraints and i ntegrals of m otion in d ynamical s yst ems. Computer Methods in Applied Mechanics and Engineering 1 (1972) 1 16. 14. S. M. Mukras. Numerical m odeling of w ear for bodies in oscillatory c ontact. Master s thesis, University of Florida, FL (2006) 15. Archard, J. and Hirst, W. The w ear of m etals under unlubricated c onditions. (1956) 16. K.R. Makinson and D. Tabor. The f riction and t ransfer of polytetrafluoroethylene. Physics and Chemistry of Solids 281 (1964) 4961.

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130 17. S.K. Biswas and K. Vijayan. Friction and w ear of PTFE a r eview. Wear 158 (1992) 193211. 18. C.M Pooley and D. Tabor. Friction and m olecular s tructure: t he behaviour of s ome t hermoplastics. Surface Physics 329 (1972) 251274. 19. Y.M. Pleskachevsky and V.A. Smurugov. Thermal f luctuations at PTFE f riction and t ransfer. Wear 209 (1997) 123127. 20. D.L. Burris. E ffects of nanoparticles on the w ear r esistance of polytetrafluoroethylene. Doctoral dissertation, University of Florida (2007) 21. I. Jang, D.L. Burris, P.L. Dickrell, P.R. Barry, C. Santos, S.S. Perry, S.R. Phillpot, S.B. Sinnott, W.G. Sawyer. Sliding orienta tion e ffects on the t ribological p roperties of polytetrafluoroethylene. Journal of Applied Physics 102 (2007). 22. T.A. Blanchet and F.E. Kennedy. Sliding w ear m echanism of polytetrafluroethylene (PTFE) and PTFE Composites. Wear 152 (1992) 229243. 23. D. Gong, Q. Xue, H. Wang. Study of the wear of filled polytetrafluroethylene, Wear 134 (1989) 283295. 24. H. Unal, A. Mimaroglu, U. Kadioglu, H. Ekiz. Sliding f riction and w ear behaviour of polytetrafluoroethylene and its c omposites under dry c onditions. Materials and D esign 25 (2004) 239245. 25. W.G. Sawyer, K.D. Freudenberg, P. Bhimaraj, L.S. Schadler. A s tudy on the f riction and w ear b ehavior of PTFE f illed with a lumina nanoparticles. Wear 254 (2003) 573580. 26. T.L. Schmitz, J.E. Action, J.C. Ziegert, W.G. Sawyer. Wear r at e uncertainty a nalysis. Journal of Tribology 126 (2004) 802808. 27. T.A. Blanchet. The i nteraction of w ear and d ynamics of a s imple m echanism. Journal of Tribology 119 (1997) 597599. 28. W.G. Sawyer. Wear predictions for a s imple c am i ncluding the c oupled e volut ion of w ear and l oad. Lubrication Engineering (2003) 31 36. 29. D.J. Dickrell III, D.B. Dooner and W.G. Sawyer. The e volution of g eometry for a w earing c ircular c a m: a nalytical and c omputer s imulation with c omparison to e xperiment. Journal of Tribology 125 (2003) 187192. 30. B. Anders and J. Hugnell, S. Bjrklund, S. Andersson. Simulation of the m ild w ear in a c am f ollower c ontact with f ollower r otation. Wear 199 (1996) 202210. 31. P. Pdra and S. Andersson. Wear s imulation with the W inkler s urface m odel. Wear 207 (1 997) 7985.

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BIOGRAPHICAL SKETCH Nate Mauntler was born in 1981, the second of four sons of John and Margaret Mauntler of Troy, Ohio. He graduated from Troy High School in June of 1999. Mr. Mauntler then left Ohio for the University of Florida, where he obtained Bachelor of Science i n 2004. He immediately entered graduate school and in 2006 completed a Master of Science in m echanical e ngineering with a thesis titled Dimensional Study of an Interference Fit Allograft. In February of 2009, Mr. Mauntler married Nicole Emily Parker in G ainesville, Florida. Following the completion of his doctoral degree, Mr. Mauntler plans to begin work as a civilian engineer with the United States Navy.