A Novel Tension-Member Follower Train for a Generic Cam-Driven Mechanism

Transcription

1 A Novel Tension-Member Follower Train for a Generic Cam-Driven Mechanism A Thesis Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial requirement for the Degree of Master of Science In Mechanical Engineering By: Approved: Jeffrey LaPierre May 30, 2008 Professor Robert L. Norton, Advisor Professor Holly K. Ault, Thesis Committee Member Professor James D.Van de Ven, Thesis Committee Member Professor Cosme Furlong, Graduate Committee Member

2 ABSTRACT Many assembly machines for consumer products suffer from the fact that the mechanisms used to impart the necessary assembly motions to the product are orders of magnitude more massive than the product payloads that they carry. This characteristic subsequently limits the operating speed of the machine. If the follower train could be made less massive without sacrificing accuracy and control, it would therefore allow higher speeds. It is well-known that structures that carry only tensile loads can be much less massive than those that must also carry compressive loads. This concept is demonstrated in many structures, such as the suspension bridge. This master s project set out to investigate the feasibility of a tension-member follower train for a generic cam-driven pick and place mechanism. This system was first dynamically simulated using a computer model, and then tested by constructing a proof of concept prototype. A cam-driven, low-mass tension member (in this case a spring steel strip over pulleys) under spring preload was used to replace the bellcranks and connecting rods typical of a conventional follower train. The system was determined to be feasible and will allow for increased operating speeds at potentially lower costs as an additional benefit.

3 ACKNOWLEDGEMENTS I would like to thank Professor Norton for his guidance throughout the course of my research. I would also like to thank Tim Sweet for presenting me with the concept that became the basis of my research and the Gillette Company for sponsoring this project.

4 Contents 1. Introduction Project Scope Goal Statement Project Objective Approach Background Study Tension Members Timing Belts Flat Belts Pre-Stretched Wire Rope Metal Drive Tapes Literature Review Conceptual Design Preliminary Modeling Refined Design Analysis Fabrication Experimentation Data Collection.50

5 9.2 Experimental Results Experimental vs. Simulated Results Iteration of Simulation Parameters Follower vs. End Effector Conclusion Recommendations 77 References..79 Bibliography...81 Appendix A 82 Appendix B Appendix C 88 Appendix D 95 Appendix E..100 Appendix F..103 Appendix G.114

6 1. INTRODUCTION Cam actuated mechanisms are common in pick and place assembly stations where simultaneous assembly motions must be kept in synchronization from station to station. In order to maintain synchronization, these mechanisms are typically driven via a common central cam shaft having multiple cams. Due to the location of this common shaft it is necessary to have a relatively extensive linkage comprised of numerous bellcranks, rocker arms, and connecting rods to transmit motion to the tooling. In many cases the mass of these components can far exceed the mass of the tooling that they actuate. In the case of a force closed cam system a preload device such as a spring must be used in order to compensate for the inertia of the follower train and maintain contact between the cam and the follower. In order to increase the speed of the system, the preload force must also increase which in turn imparts greater force on the follower train and associated parts. Many of these parts must become more massive in order to withstand the increased force resulting in a dog chasing its tail scenario. Additionally, the friction in all the moving parts increases in proportion to the force, and therefore the motor required to drive the system must also increase in size. For these reasons it is advantageous to design cam follower train mechanisms to have as little mass as possible. 1

7 2. PROJECT SCOPE 2.1 Goal Statement Research and test the feasibility of a low-mass, tension-member cam follower train in a high speed application. 2.2 Project Objective The objective of this project was to design a tension member follower train that would be capable of oscillating a 1 kg mass with a 40mm stroke at more than 200 cycles per minute. The follower train had to also maintain accuracy and repeatability with respect to the placement of the mass and have high cycle life. A prototype was constructed to study the dynamic characteristics of the follower train at 250 cycles per minute. 2.3 Approach The research and development of this mechanism took place as follows: Research: Based on the parameters described in the project objective, potential tension members were sought and researched. After compiling data on all applicable tension members, the member showing the most promise was selected for further analysis. 2

8 Conceptual Design: A conventional cam follower train typical of the sponsor s application was reverse engineered in order to determine system parameters. Using these figures, a test fixture incorporating both a pulley and an oscillating dummy mass to represent the tooling was then developed. This fixture was designed in such a way that it could be installed on a special cam dynamics testing machine. Preliminary Modeling: A mathematical model was created using TK solver to assist in the optimization of the pulley and tension member. This step was necessary in order to determine whether the tension member selection was still viable and to eliminate two of the unknowns, the pulley and the tension member, thereby allowing further analysis. Refined Design: The conceptual design was further refined in a Solidworks solid model of the test fixture and detailed drawings of each component were produced. The solid model also served to verify the mass and moment of inertia values of the preliminary model. Analysis: A dynamic mathematical model representative of the finalized design was then created in Matlab in order to understand the influence different parameters had on the system. This analysis allowed the determination of certain unknown system parameters such as the required spring constants and preloads and the resultant cam shaft torque. 3

9 Fabrication: Various components of the test fixture were either purchased or, in the case of machined parts, manufactured to print in house and assembled. Experimentation: After completing the installation of the test fixture on the test bed data was obtained from various forms of instrumentation and compared to results of the dynamic mathematical model. 4

10 3. Background study Research conducted early on in this project was aimed at the study of specific tension members used to transmit linear motion in similar applications. The remainder of background research focused on previous research in the modeling of tension members and cam follower trains. 3.1 Tension Members Tension members have several advantages over other means of transmitting linear motion. Most importantly, tension members can be much lighter than members which experience compressive loading due to the fact that the cross sectional geometry is irrelevant. With tension members only the cross sectional area and the material which it is comprised of limits the strength. Another benefit is that tension members have the flexibility to be routed around pulleys in order to transmit motion to remote locations. This enables the elimination of multiple pin joints inherent in linkages which can result in loss of precision and require constant maintenance. Opposed to complex rigid members found in conventional linkages, tension members have very simple geometry and can therefore be manufactured at significantly reduced cost. 3.2 Timing Belts Timing belts are used quite often as linear motion transmission devices. They can be commonly found in office equipment, robotic arms, and machine tools where 5

11 semi-precision linear position is required (+/ inches). Some of the key advantages to timing belts are their low mass, small bend radius, and low wear on pulleys. The fact that they are equipped with teeth over their entire length enables synchronization through intermediate pulleys as well as the ends when used as a drive tape. Although timing belts can be used in as drive tapes in free end applications it is more common to see them as a continuous belt as shown in Figure 3-1 below. Figure 3-1: Typical application of timing belt used to transmit linear motion Source: Nook Industries [ 1 ] The problem with the use of timing belts in free end applications that are subjected to dynamic loading is failure at the end attachments. This problem stems from the fact that the strength of timing belts is due to internal cords made from Kevlar, Hypalon, or steel wire which act as the actual tension member[ 2 ]. Because these cords are embedded within the belt covering it means that all force must be transmitted to them from the end attachment through the polymer belt covering. Over time, the end attachment eventually strips the covering off the inner cords resulting in failure of the tension member. 6

12 3.3 Flat Belts Flat belts share many of the same characteristics as timing belts when used in linear positioning. The main advantage synthetic flat belts have over timing belts is that they can be purchased with a woven outer covering. This outer covering is what gives the belt strength and it can readily be attached to an end termination. Although this is an improvement over timing belts, under cyclical dynamic loading there is still a potential for failure at the end terminations due to pull out. 3.4 Pre-Stretched Wire Rope Pre-stretched wire rope is used extensively in controls hence the name aircraft cable and in various linear motion devices such as copiers, printers, scanners etc. Pre-stretched wire rope is manufactured from wire rope by subjecting it to repeated tensile loading of approximately 75% of the cable tensile strength in accordance with MIL-C-5688 [ 3 ]. The intent of this process is to eliminate constructional stretch in the structure of the cable due to movement of the individual strands as they close around the inner core of the cable under tension. Unfortunately, the amount of constructional stretch in a cable is somewhat unpredictable. Although most of the constructional stretch in the cable is removed in the process described above, it can, in some cases change over the life of the cable [ 4 ]. This is especially true in cases with varying loads. Another problem inherent with wire rope is bending fatigue in the internal wires which can eventually result in failure [ 5 ]. This is a problem which is difficult to 7

13 detect and therefore requires a predetermined service life dependant on the application [ 6 ]. One advantage to wire rope that is not possible with the other potential tension members previously discussed is the capability to route around pulleys in three dimensions. Because of the round cross section of wire rope many complex direction changes can be made possible. However, due to the physical structure of wire rope, the outer surface has less than ideal wear properties. It is difficult to find a pulley material that will have good wear compatibility characteristics with wire rope. 3.5 Metal Drive Tapes Metal drive tapes are yet another means of linear motion transmission that was explored. They are used in many of the same applications as timing belts and flat belts, however they are better suited to free end applications. Metal drive tapes offer low mass and very low stretch due to the high stiffness of metals, specifically steel. Position repeatability of these systems can be as good as +/ inches. Metal drive tapes can be made from a multitude of metals including, but not limited to, Inconel, titanium, 301 high yield stainless steel, and carbon spring steel. Both the carbon spring steel and high yield stainless steel are the most common choices due to their high tensile strengths of 347,000 psi and 280,000 psi respectively. Because of the high strength these steels, a relatively small cross section is needed to transmit rather high loads. The ability to maintain a thin cross section allows metal drive tapes to have the capability of being routed 8

14 around reasonably small pulleys, in addition to having a high strength to weight ratio. Infinite life is attainable through selection of proper tape thickness and pulley sizing. One application that has successfully demonstrated the fatigue life and other benefits of flexible metal drive tapes is the shuttle-less loom developed by the Draper Corporation in the 1940 s. This loom was revolutionary in that it eliminated the use of traditional shuttles that carried the thread back and forth across the loom s weft in the production of woven textile materials. The traditional shuttles were comprised of heavy blocks of hardwood bound with steel points, severely limiting the operating speed of the loom. The shuttle-less loom replaced the shuttle with a flexible metal tape often referred to as a rapier [ 7 ]. The rapier was stored on a reel on one side of the loom and in operation would extend across the width of fabric transporting thread to the opposite side where it would detach and return to grab another loop of thread. This process would then repeat millions of cycles per month. In this example, the implementation of a metal tape allowed the production output of the loom to increase by as much as 300 percent [ 8 ]. The characteristics of metal tapes have also made them quite popular in the field of robotics where tension members are commonly used to actuate arms and end effectors. Many of these applications involve relatively high intermittent loading similar to that of a follower train. 9

15 3.6 Literature Review Extensive research was conducted in order to understand different techniques used to model cam follower systems and to determine what others had experienced with similar tension member systems. This research included, but was not limited to the reading of Cam and Design and Manufacturing Handbook [ 9 ] and Design of Machinery [ 10 ] by R. L. Norton. Both books describe the modeling of various styles of cam linkages in great detail. G. Dalpiaz and A. Rivola [ 11 ] studied the modeling of a high performance automatic packaging machine that utilized a cam actuated arm connected via a timing belt. This mechanism similar to the one in question is depicted in the diagram labeled Figure 3-2. Figure 3-2: Schematic of Dalpiaz and Rivola experimental machine Source: A kineto-elastodynamic model of a mechanism for automatic machine. 10

16 In their case, the timing belt used to transmit oscillating motion from the cam follower to the rocker arm was used continuously around two pulleys as opposed to the more troublesome free end situation which is more prone to failure. Dalpiaz and Rivola describe the kineto-elastodynamic analysis of their machine using the lumped parameter method. They used a 5 degree of freedom model to describe the torsional elements in each of the machine s sub-systems as denoted by the numbered balloons in the above diagram. The parameters for the moments of inertia and stiffness of the each element of the system were calculated based on the dimensions of the links with exception of the timing belt between the cam follower and rocker which was obtained empirically. Viscous dampers corresponding to each tension member were included in the model to take into account both structural and coulomb damping. The damping coefficient for each of the viscous dampers was then estimated based on the stiffness of the member it corresponded with. After completing their theoretical model, they compared the numerical results to experimental data collected from an accelerometer mounted on the oscillating rocker arm as shown in Figure 3-2. Although initial results of the model resembled the experimental data, damping values were adjusted in order to achieve a closer correlation. In conclusion, Dalpiaz and Rivola found that their 5 degree of freedom, lumped parameter model was capable of accurately predicting the dynamics of the machine that was the basis of their research. 11

17 In 1999 Xiang-Rong Xu, Won-Jee Chung, and Young-Hyu Choi [ 12 ] set out to develop a new method for the dynamic modeling of robots with flexible links, specifically those utilizing revolute joints and open loop mechanisms. They first explain both the Rayleigh-Ritz method and the finite element method commonly used to develop a kineto-elastodynamic model. The Rayleigh-Ritz method assumes that a link is a continuous body, and only one link is assumed to be elastic. The finite element method is used to first divide the link into finite elements then, derive a system of equations which ultimately results in the dynamic analysis of the system. Furthermore, there are two variations of the finite element method, the lumped parameter method and the distributed parameter method. Although the distributed parameter method is computationally more efficient than the former method because it eliminates the selection of element types and model shape function of the displacement, it is limited to closed chain systems. Xiang-Rong Xu, Won-Jee Chung, and Young-Hyu Choi developed a series of motion equations that can be used to model elastic open loop systems. They also validated their new method through comparisons to more time consuming traditional methods. 12

18 4. Conceptual Design: Following the research of various tension members, it was determined that the metal drive tape was best suited for this application. However, before proceeding to design it was necessary to first determine the approximate loading the tension member would be subjected to in a typical operation. This would verify that the use of a metal drive tape was feasible. The project sponsor supplied the solid model shown in Figure 4-1 in addition to the following system parameters: Rise/Fall in 120 Deg. Dwell for 240 Deg mm (4.28 in) Prime Radius 1 Kg (2.204 lbs.) Oscillating mass 40 mm (1.57 in) Stroke (mass at end of follower arm) 400 cycles per minute 13

19 Figure 4-1: Cam follower system typically in use at sponsor s operation Using the parameters provided along with the solid model, the cam was recreated in program Dynacam using a polynomial rise-fall-dwell function in order to determine the peak acceleration. The parameters entered into Dynacam in addition to the resulting position (s), velocity (v), acceleration (a), and jerk (j) can be seen in Appendix A. The peak acceleration of the inch long follower arm was determined to be 76,800 deg/ sec 2. Due to the arc of the follower arm, the tangential acceleration at the tape attachment point on the follower arm was calculated to be 15,000 in/sec 2. Through the use of Newton s second law, F = ma, 14

20 the resulting force due to the oscillating mass was determined to be approximately 85 lbf, which was well within the range of a metal belt. After determining that the metal belt was indeed a feasible tension member, a means of testing this element was devised. A test fixture equipped with a sliding mass (representative of the mass of the project sponsor s tooling) and a metal drive tape was designed. The fixture was designed in such a way that the metal tape could be connected to the follower arm on a special cam dynamics testing machine (Figure 4-2) located in the Vibrations Laboratory at the Worcester Polytechnic Institute. Figure 4-2: Cam dynamics test machine prior to installation of tension-member test apparatus. 15

21 The design of this fixture incorporated a pulley that the drive tape would be routed around 180 degrees in order to study the effect of bending on the tape. A compression spring at the sliding mass would preload the metal tape in tension against the pulley. This preliminary design is depicted in the Figure 4-3 below. Figure 4-3: Preliminary design of tension member test fixture. The cam dynamics testing machine is fitted split cams to facilitate the installation and removal of different cams in order to simulate different situations. Due to the physical constraints of the machine (limited swing radius) in addition to the costliness associated with the machining of a custom plate cam, a four dwell cam that had previously been used for another experiment was selected. The parameters of the four dwell cam used for this experimentation are as follows: 16

22 Segment 1: Rise 0.5 inches in 50 with polynomial displacement Segment 2: Dwell for 40 Segment 3: Fall 0.5 inches in 50 with polynomial displacement Segment 4: Dwell for 40 Segment 5: Rise 0.5 inches in 50 with modified trapezoidal acceleration Segment 6: Dwell for 40 Segment 7: Fall 0.5 inches in 50 with modified sine acceleration Segment 8: Dwell for 40 Using Dynacam, the peak angular acceleration of the follower arm at 400 rpm was found to be 76,163 deg/sec 2 and the resulting tangential acceleration due to the inch arc of the follower arm was determined to be 17,945 in/sec 2 at the point of tape attachment. The Dynacam program parameters in addition to the resulting SVAJ plots can be seen in Appendix B. The force resulting from the oscillating mass at this acceleration was then calculated to be approximately 100 lbf, which was still within the reach of a metal tape. Clearly, one can see that the four dwell cam that was selected will result in a more than adequate simulation of the forces that the mass and tension member would undergo with the cam program presented by the project sponsor at 400 RPM. 17

23 5. Preliminary Modeling At this point it was necessary to establish what the diameter of the pulley would need to be in order to maintain a reasonable stress in the metal tape. The goal was to make the pulley as small as possible in order to minimize its moment of inertia which would add effective mass to the system. According to metal belt manufacturer design guidelines, it is recommended that a pulley diameter be at least 625 times greater than the thickness of the belt to achieve infinite life expectancy. The manufacturer also states that the total stress of a metal belt or tape (equation 5.1) not exceed one third the belt material s yield strength [ 13 ]. σ total = σ work + σ bending (5.1) σ work = τ / (w x t) (5.2) σ bending = (E x t) / (1- u 2 )D (5.3) Where: τ = Tension in Belt w = Tape Width t = Tape Thickness E = Young s Modulus of Elasticity u = Poisson s Ratio D = Pulley Diameter 18

24 Using these equations, a mathematical model was developed in TK Solver to allow various parameters such as the tape thickness, tape width, and pulley diameter to be easily optimized in order to achieve minimal bending stress (Appendix C). Parameters were also added to this model to account for the force due to the oscillating point mass of the pulley, and the force due to the spring that would preload the tape in tension. The force of this spring would have to counteract the force due to the inertia of the pulley, assuming that there would be no slippage between the tape and pulley. Based on the availability of belt material and constraints in the mechanism it was decided that a two-inch-wide AISI 1095 steel belt would be most appropriate. In order to determine the ideal thickness of the drive tape, the safety factor of the tape was calculated for various thicknesses from outputs of the model. The equation used to compute the safety factor (5.4) was based on experimentation performed by a metal belt manufacturer [ 14 ]. N tape = (1/3 x S y ) / σ total (5.4) Where: S y = Yield Strength of Tape Material This equation was found to be rather conservative based on the fact that in order to achieve infinite fatigue life for steels having a tensile strength greater than 200,000 psi, the endurance strength is 100,000 psi. The tensile strength for

25 steel hardened to 60 Rc. was determined to be approximately 347,000 psi [ 15 ], meaning that it would have a 100,000 psi uncorrected endurance limit (Se ). The following correction factors were then calculated and applied to the endurance limit to take into account for physical differences between the standard fatigue test specimen and the metal tape. The loading correction factor was based on the fact that the tape is subjected to both bending and axial loading. C load = 1 C load = 0.70 (Axial Loading) (Bending) In order to determine the size correction factor the Kuguel method was used where the equivalent diameter of the tape was found using equation 5.5 and 5.6. A 95 = 0.05 (thickness) x (width) (5.5) A 95 = 0.05 (0.010 inches) x (2.00 inches) = d equiv = (A95 / ) 0.5 (5.6) d equiv = (0.001 / ) 0.5 = C size = 1 (Where d equiv < 0.3 inches) The correction factor for the ground surface of the steel tape was determined using equation 5.7 and the values in Table 1-1A. 20

26 Table 1-1A: Surface Finish A (kpsi) b (kpsi) Ground Machined or Cold Rolled Hot-Rolled As-Forged C surf = A (S ut ) b (5.7) C surf = 1.34 (173,500 psi) = The temperature correction factor was based on the following criteria and the fact that the machine will be operated at room temperature. C temp = 1 (Where temperature < (840 F) The reliability correction factor was based on Table 1-2A and the fact that 90 percent reliability was desired. Table 1-1A: Reliability % C reliab C reliab =

27 Application of the correction factors can be seen in equation 5.8 as follows. Se = C load C size C surf C temp C reliab Se (5.8) Se = (0.70)(1)(0.481)(1)(0.897)(100,000) = 30,202 psi The corrected endurance limit for the metal tape is 30,202 psi. Based on these calculations it was determined that a maximum safety factor of could be obtained with a tape thickness between.025 and.030 inches. It is interesting to note that the safety factor remained essentially constant over this range of tape thicknesses as depicted in Figure 5-1. Tape Safety Factor vs. Tape Thickness N tape Tape Thickness (inches) Figure 5-1: Plot of tape safety factor vs. tape thickness

28 The problem with using a tape thickness falling within this ideal range ( inches), is that the pulley must be between and inches in diameter due to the bending stress. Not only would a pulley of this size be impractical for this application, but the resultant moment of inertia would be too large. For this reason it was decided that a 1.5 safety factor attained through the use of a.010 inch thick tape would be adequate. This reduction in tape thickness would mean that the pulley could be as small as inches in diameter, and the resultant force in the tape due to the effective mass of the pulley would be cut in half. The next step was to optimize the geometry of the pulley for low mass moment of inertia about the axis of rotation. Unlike traditional flat belts which are made out of more compliant materials such as leather or woven synthetics, metal belts can not be kept on track through the use of crowned pulleys. The tracking of metal belts and drive tapes must be influenced solely by the precise alignment and parallelism of the pulley axes with respect to one another. The peripheral surface of the pulley must be kept perfectly flat and concentric with the center axis. In extreme cases where the distance between end attachment points is great, flanged pulleys may be used to force belt tracking. This technique is not recommended for situations such as this one, where pulleys are located close to the end attachments and belt tension is high as flanges will cause rapid tape and pulley wear. Fortunately, both of these factors will simplify the pulley design, manufacture, and reduce its moment of inertia. For strength and manufacturing 23

29 purposes it was determined that an aluminum pulley having an I-beam cross section would be most practical. A solid model of the pulley was then created in Solidworks, and optimized to reduce the mass moment of inertia about its pivot. The resulting pulley is shown in Figure 5-2. Figure 5-2: Final design of aluminum flat pulley With the design of the pulley finalized, the moment of inertia of the pulley was entered into the first TK Solver model to determine the total tension in the tape. According to the model, the total tension in the tape due to the oscillating mass 24

30 and pulley, and the force of the spring used to counteract these forces would be approximately 270 lbf. After determining the design of the pulley and the estimated load that it must carry, the axle and bearings for the pivot could be sized. Plain bearings were selected based on the fact that bearings which utilize rolling elements are known to introduce vibrations to the system. A hardened and ground inch diameter dowel pin was selected for the axle as it would ensure minimal deflection under these loads over such a short span. The surface finish and hardness of this axle would also have ideal bearing compatibility with common bearing materials. In order to determine the bearings that would be needed to support the load at the projected speed, the bearing pressure in psi and velocity in feet per minute at the bearing interface were calculated using equations 5.9 and 5.10 and the parameters listed in Table 5-1. P = (Bearing Load) / (Shaft Dia. x Bearing Length) (5.9) V = (Shaft RPM) x (.262) x (Shaft Dia.) (5.10) Table 5-1: Bearing Load 270 lbf. Shaft Diameter inches Bearing Length inches Shaft Speed 210 rpm 25

31 The resulting bearing pressure and velocity was found to be 288 psi (1.5 inch long journal) and 55 fpm respectively. These numbers were then multiplied together to obtain the PV value. The PV rating is a number which bearing manufacturers use to rate various bearing materials in order to determine if a certain material will be suitable for a given application. The PV value was calculated to be in this case, eliminating the possibility of most plastic bearings. A bronze 954 alloy bearing with a PV rating was found to be more than sufficient. 26

32 6. Refined Design: The conceptual design described in section four was then refined based on the calculations made in the previous section. A three dimensional solid model of the existing cam test bed and the new test fixture was constructed in Solidworks. A view of the resulting model can be seen in Figures 6-1 and 6-2. Figure 6-1: Solid model of assembled test fixture (rear view) showing the oscillating mass and preload spring. 27

33 Figure 6-2: Solid model of assembled test fixture (side view) showing the drive tape and follower arm attachment point. The side plates used to support the pulley pivot were constructed of.625 inch 6061 T-6 aluminum to insure stiffness. A THK linear ball bearing slide was selected to guide the oscillating mass vertically, in-line and tangent with the pulley. Clamps were designed to attach the metal tape at both the follower arm and at the oscillating mass. This clamp style of attachment was chosen in order to minimize the tendency for fatigue that would be inherent with other means. 28

34 Provisions were made at the follower end clamp for both an inline piezoelectric force transducer and an accelerometer. Provisions for an accelerometer were also made at the oscillating mass, enabling comparisons to be made between the two points. A mount for the preload spring was located above the oscillating mass and equipped with a hollow jack screw to facilitate installation. The entire fixture was designed so that it could be easily removed from the machine and would not affect the use of the machine for the experimentation for which it was designed. 29

35 7. Analysis: Although the basic design of the machine had been established, a few questions were left unanswered. How stiff does the spring need to be at the oscillating mass? How stiff does the follower arm return spring need to be? What are the preload requirements of both springs? Given the fact that the cam test bed was originally designed to operate at 120 rpm, would the motor have enough power to operate this system at 400 rpm? Were the assumptions made in the preliminary analysis correct? The solution to answering these questions was to develop a kineto-elastodynamic, two-mass, two-degree of freedom computer model. The first step toward creating this model was to determine the effective mass of each component in the follower train at the follower roller. This was accomplished by first obtaining the mass of each component in the solid model and the mass moment of inertia of the follower arm about its pivot using the mass properties calculator in Solidworks. The effective point mass of the follower arm was then found by applying equation 7.1. m eff = I zz / r 2 (7.1) Where: I zz = Mass moment of inertia of follower arm about pivot point r = Radius from pivot point to tape attachment point 30

36 The mass moment of inertia of the arm in addition to the resulting effective mass at the radius (r) from the follower arm pivot can be seen in Table 7-1. Table 7-1: Mass Moment of Inertia (I zz ) blob-in 2 Radius (r) inches Effective Mass (m eff ) blobs* The effective mass of each component due to the lever ratio of the follower arm was then determined using equation (7.2) below. m eff = m (r 1 /r 2 ) 2 (7.2) Where: r1 = The distance from the follower arm pivot point to the mass in question. r2 = The distance from the follower arm pivot point to the roller follower. The mass of each of the follower train components in addition to the resulting effective mass at the follower roller can be seen in Table 7-2. * A blob represents the inch pound system unit for mass as defined by Robert L. Norton

37 Table 7-2 Component Mass (blobs) Effective Mass at Follower (blobs) Follower Arm Follower Roller Spring Pivot Block Spring Clamp Plate Tape Termination Tape Termination Clamp Plate Force Transducer Tape Termination Yoke Shoulder Bolt & Nut Point Mass of Pulley Oscillating Mass & Hardware Metal Drive Tape Total The spring rate of the thick x 2.00 wide metal tape was then determined to be 18,700 lb/in using the parameters in Table 7-3 and equation 7.3. Table 7-3 Cross Sectional Area of Tape in 2 Length of Tape inches Young s Modulus of Steel 30,000,000 psi Tape Spring Rate 18,700 lb/in K = (AE / L) (7.3) K = ((0.020 in 2 )( 30,000,000 psi)) / (31.50) = 18,700 lb/in Where: K = Spring Constant A = Cross Sectional Area of Tape E = Young s Modulus of Elasticity 32

38 L = Length of Tape The effective stiffness of the tape at the roller follower due to the lever ratio of the follower arm was then found to be 80,500 lb/in, using equation 7.4 and the parameters in Table 7-4. keff = k (r 1 /r 2 ) 2 (7.4) keff = (18,700 lb/in) (13.50 / 6.50) 2 = 80,500 lb/in Where: r 1 = Distance between follower arm pivot point and tape attachment. r 2 = Distance follower arm pivot to roller follower. Table 7-4 Tape Spring Constant 18,700 lb/in r inches r inches Effective Stiffness 80,500 lb/in The next step toward the computer model was to develop a lumped mass model that was representative of the system. Based on analysis, it was determined that there were essentially two sub-systems that interact with one another dynamically. One sub-system was the top half, consisting of the oscillating mass, pulley, and preload spring. The bottom sub-system was comprised of the remaining parts 33

39 such as the follower arm, follower return spring and hardware, tape end attachment clamps, etc. For this reason, it was decided that a two-mass, twodegree of freedom model would best represent the situation. This model is depicted in Figure 7-1. Figure 7-1: Lumped mass model of system ( s = z when cam is in contact with follower roller). In this model, mass 2 (0.025 blobs) and mass 1 (0.032 blobs) represent the top and bottom sub-systems respectively. The follower return spring is represented by k1, the steel tape by k2 (80,522 lb/in), and the oscillating mass preload spring by k3. The system damping due to the damping of the springs and coulomb friction in the various pivots is represented by c1, c2, and c3. The position of mass 1 and mass 2 are represented by z and x respectively, while s represents the displacement of the cam. 34

40 From the lumped mass model free body diagrams representing both masses were created in Figure 7-2. Figure 7-2: Free body diagrams of mass 1 and mass 2 The free body diagrams show the direction of the forces due to the springs and dampers. These diagrams were the basis for the following differential equations: Derivation of Mass 1 Equation: Σ F = m 1 F c (t) F d F s k 2 (z-x)-c 2 ( ) = m F c (t) c 1 - k 1 z k 2 (z-x) c 2 ( - ) = m F c (t) c 1 k 1 z k 2 z + k 2 x c 2 + c 2 = m F c (t) = m 1 + c 1 + k 1 z + k 2 z k 2 x + c 2 c 2 F c (t) = m 1 + (c 1 + c 2 ) + (k 1 + k 2 )z k 2 x c 2 Where F c (t) = 0 : m 1 = -(c 1 + c 2 ) (k 1 + k 2 )z + k 2 x + c 2 35

41 = -(c 1 + c 2 ) (k 1 + k 2 )z + k 2 x + c 2 (7.5) m 1 m 1 m 1 m 1 Derivation of Mass 2 Equation: Σ F = m 2 m 2 = k 2 (z-x) + c 2 ( - ) k 3 x c 3 m 2 = k 2 z - k 2 x + c 2 - c 2 k 3 x c 3 m 2 = k 2 z (k 2 + k 3 ) (c 2 + c 3 ) + c 2 = k 2 z (k 2 + k 3 )x (c 2 + c 3 ) + c 2 (7.6) m 2 m 2 m 2 m 2 Notation Key: c = Damping Coefficient k = Spring Constant Fc = Force of Cam on Follower Fs = Force of Spring on Follower Fd = Force of Damper on Follower m = Mass of Moving Elements t = Time in Seconds s = Rise of Cam z = Displacement of Mass 1 in Inches = Velocity of Mass 1 in Inches/Second = Acceleration of Mass 1 in Inches/Second 2 x = Displacement of Mass 2 (Oscillating Mass) in Inches 36

42 = Velocity of Mass 2 in Inches/Second = Acceleration of Mass 2 in Inches/Second 2 Through the use of four dummy variables, the following state space equations were obtained: Where: y 1 = x y 2 = y 3 = s y 4 = 4 = -(c 1 + c 2 ) y 4 (k 1 + k 2 ) y 3 + (k 2 ) y 1 + c 2 (y 2 ) (7.7) m 1 m 1 m 1 m 1 3 = y 4 (7.8) 2 = (k 2 ) y 3 (k 2 + k 3 ) y 1 -(c 2 + c 3 ) y 2 + (c 2 ) y 4 (7.9) m 2 m 2 m 2 m 2 1 = y 2 (7.10) In order to solve these state space equations ( ) for a numerical solution, a model utilizing an adaptive step Runge-Kutta method was developed in Matlab (Appendix D). A discussion of the specific ordinary differential equation (ODE) solver that was selected for problem is located in Appendix E. This model would not only allow the determination of the required spring constants and preloads, but also the resulting cam shaft torque and position error of the oscillating mass. Additionally, the model enabled the verification of earlier calculations for the tension in the tape. 37

43 The damping coefficients (c1, c2, and c3) were also calculated in the Matlab model using the equations 7.11 and (7.11) (7.12) Where: ζ = Damping Ratio (typically less than 0.1 according to Koster 17 ) Iterative simulations were performed using the completed Matlab model to determine the stiffness requirements for both the follower return and oscillating mass preload springs in order to prevent follower jump and maintain tension in the metal tape at the target speed of 400 rpm. Unfortunately, the camshaft torque resulting from the necessary springs at this speed would far exceed the available torque of the cam test bed drive motor. For this reason the peak operating cam speed was scaled down to 250 rpm. At this speed, the peak acceleration of the oscillating mass would be about 7000 in/sec 2 vs. the 15,000 in/sec 2 that the mass would see in the project sponsor s intended application. Ultimately, the tension in the tape would be reduced from the projected 620 lbf in the sponsor s application to 290 lbf in the experiment on the cam test bed at 7000 in/sec 2. Plots of the 38

44 displacement, velocity, and acceleration of the system at 250 rpm can be seen in Figures 7-3 through 7-5. Figure 7-3: Plot of simulated cam follower displacement over one revolution of cam. 39

45 Figure 7-4: Plot of simulated cam follower velocity over one revolution of cam. Figure 7-5: Plot of simulated end effector acceleration over one cam revolution. According to the model, at 250 rpm the follower return spring would need to have an effective spring stiffness of 288 lb/in at the follower roller and would need to have a preload of 140 lbf in order to prevent follower jump. It was also determined that the spring maintaining tension in the steel tape would need to have an effective spring rate of 323 lb/in and a preload of 50 lbf. The resulting force on the cam and tension in the metal tape at 250 rpm is depicted in Figures 7-6 and 7-7 respectively. 40

46 Figure 7-6: Plot of simulated cam follower force over one cam revolution. Figure 7-7: Plot of simulated tension in metal tape over one cam revolution. 41

47 The position error between the two masses can be seen in the plot of z-x Figure 7-8 below. Figure 7-8: Plot of simulated position error between mass one and mass two. One can observe that the total simulated position error of the top mass with respect to the bottom mass is approximately inches. This position error is due to the axial deflection of the tape, which although rather stiff at 80,522 lb/in due to the lever ratio, still has some measurable deflection even with a peak load of a little more than 285 lbf. Overall, this position error is minimal in comparison to conventional linkages which would have clearances in multiple pin joints in addition to deflection in its members. The torque imposed on the camshaft due to the follower return and oscillating mass preload springs can be seen in Figure

48 Figure 7-9: Plot of simulated camshaft torque over one revolution of cam. As one can see from the plot, the peak torque was approximately 500 in/lb neglecting friction in the cam shaft bearings and roller follower. The cam dynamics test bed is equipped with a three horsepower electric motor with a full load torque rating of 132 in-lb. This motor drives the cam shaft of the machine via belt drive with a reduction ratio of 5.7:1, meaning the peak available torque at the cam shaft is roughly 750 in-lb. Based on this information the motor would have ample power to drive the machine at 250 rpm with the parameters used in the model. 43

49 8. Fabrication: After determining the values for the unknown parameters, and confirming that the design was feasible in the mathematical model, it was time to build the test fixture to enable the collection of experimental data. The first step was to obtain all necessary purchased parts and materials from various vendors. Using equation 7.4 the required spring constants were determined for both the oscillating mass preload spring and the follower arm return spring. A die spring having a 75 lb/in spring rate and a 4 inch free length was selected to fulfill the 323 lb/in effective spring rate that was determined necessary in order to preload the oscillating mass and tape in tension. Likewise, an extension spring having a 88 lb/in spring rate with a built in 75 lb preload was chosen to meet the 288 lb/in effective spring rate requirement for the follower arm return spring. Machined parts such as the main support plates, pulley, and tape end attachments were all made to prints located in Appendix F. After assembling, the completed fixture was then aligned with the follower arm on the test bed and mounted to the plate above the camshaft using four 3/8-16 UNC socket head cap screws. The force transducer and accelerometers were then installed and wired into the Dytran current power source. Photos of the completed test fixture installed on the test bed can be seen in Figures 8-1, 8-2, and

50 Figure 8-1: Rear view of cam test bed showing completed fixture mounted on test bed and metal tape routed around flat pulley. 45

51 Figure 8-2: This rear view of cam test bed depicts the tape end clamp at the follower arm, equipped with force transducer and accelerometer. 46

52 Figure 8-3: Front view of test fixture showing support plates and connection to test bed. 47

53 Figure 8-4: Front view of test fixture showing oscillating mass mounted to the linear slide along with the tape preload spring between the two pulley support plates. One can also see the accelerometer located to the left of the preload spring that will be used to monitor the acceleration of the oscillating mass. In the process of mounting the test fixture, it became apparent how critical the alignment between the oscillating mass, pulley, and follower end clamp must be. 48

54 If this alignment is off, or the shaft that the pulley rides on is not perfectly parallel with the camshaft, it causes the belt to pull to one side and the opposite side to lift up off the pulley. In this example, the fixture was carefully adjusted prior to mounting to keep misalignment to a minimum. However, in an industrial application, parallelism between pulley axis and the camshaft axis must be taken seriously. Furthermore, it may be advantageous to develop an end attachment with a pivot to allow for rotational compliance in the belt. The fixture was found to be operational at the desired speed of 250 rpm without any sign of separation between the cam and cam follower. One problem that was discovered was that after running for prolonged periods of time the machine would develop a severe vibration. After extensive trouble shooting it was determined that the cam test bed s original follower arm had excessive play in its pivot joint allowing the arm to rub against the side of the cam and the split in the cam was hitting the follower arm. This play was due to the fact that the pivot bearings were located too close to one another to compensate for the clearance in the bearings. Although the clearance in the bearings was only a few thousandths of an inch, the length of the follower arm magnified this allowable movement to an unacceptable degree. This problem was remedied by adjusting the follower arm pivot mount. 49

55 9. Experimentation: The objective of experimentation was to compare real life data obtained through instrumentation on the test fixture with the results of the computer model in Matlab. This validation of the computer model would allow future designs for use in an industrial setting to be accurately modeled to determine whether such a system would be viable. 9.1 Data Collection: Two Dytran 3145A 50g accelerometers were installed, one at the tape endtermination at the follower arm (point A) and the other at the oscillating mass (point B) in order to monitor the input and output accelerations of the system. The tape end-termination at the follower arm was also equipped with an inline Dytran 1051V4 500lbf force transducer (point C) to measure the dynamic tension in the metal tape. The locations of this instrumentation can be seen in Figure

56 Point B Point A Point C Figure 9-1: Location of follower accelerometer at point A, end effector accelerometer at point B, and tape end-termination force transducer at point C respectively. In addition to instrumentation on the fixture, the cam dynamics test bed itself was equipped with a torque transducer between the flywheel driven by the motor and the cam to measure the camshaft torque, and a rotary encoder that was used as a time trigger. Data was collected from the various instrumentation using an 51

57 Acceleration (in/sec^2) Agilent Technologies / HP 36070A Dynamic Signal Analyzer. This analyzer is equipped with four channels, samples at 256 khz, and has resolutions of 100, 200, 400, and 800 lines in the frequency domain and 256, 512, 1024, 2048 points in the time domain, respectively. The analyzer also has the capability of storing data on a 3.5 inch floppy disk for later analysis. 9.2 Experimental Results: A series of tests were performed at 250 rpm and data were obtained for acceleration of the follower arm, acceleration of the end effector, tension in the tape, and camshaft torque. The data obtained can be seen in the plots shown in Figures 9-2 through 9-5. Follower Acceleration at 250 RPM Time (s) Figure 9-2: Plot of experimental follower acceleration (point A) over approximately two cam revolutions at 250 rpm. 52

58 Tension (lbf) Acceleration (in/sec^2) End Effector Acceleration at 250 RPM Time (s) Figure 9-3: Plot of experimental end effector acceleration (point B) over approximately two cam revolutions at 250 rpm. Tape Tension at 250 RPM Time (s) Figure 9-4: Plot of experimental tension in tape (point C) over approximately two cam revolutions at 250 rpm. 53

59 Torque (in-lb) Torque at 250 RPM Time (s) Figure 9-5: Plot of experimental camshaft torque over approximately two cam revolutions at 250 rpm. 9.3 Experimental vs. Simulated Results: The experimental results were then superimposed over the results from the Matlab simulation in order to test the validity of the model. This comparison of the theoretical and experimental results can be seen in Figure

60 Acceleration (in/sec^2) Simulated vs. Experimental End Effector Acceleration Time (sec) Simulated End Effector Acceleration Experimental End Effector Acceleration Figure 9-6: Plot of simulated and experimental end effector acceleration (point B) over one cam revolution at 250 rpm. In Figure 9-6 it can be observed that although the phasing of the simulated acceleration is somewhat similar to that of the experimental acceleration, the magnitude is much less in the simulation. Furthermore, the peaks in the simulated acceleration lack the valleys that are displayed in the experimental data. Due to these differences in the data, it was determined that some iteration of the input parameters in the model was required in order to obtain data that was representative of the experiment. Despite the differences between the theoretical results from the Matlab simulation and the experimental results, the model proved to be a valuable asset. The mathematical simulation allowed the approximation of several unknown design 55

61 parameters such as the spring constants, spring preloads, and resulting camshaft torque. A mathematical model utilizing the same techniques could easily be used to aid in the design of a similar tension-member mechanism in an industrial application. 9.4 Iteration of Simulation Parameters: In an attempt to further validate the simulation various input parameters such as the damping ratios, metal tape stiffness, and the distribution of mass between mass 1 and mass 2 were adjusted. Through this iterative process it became apparent that the stiffness of the steel drive tape had the most significant effect on the valleys in the peaks between the dwells of the acceleration plot. As the stiffness of the metal tape approached 20,000 lb/in, the valleys in the simulated data resembled the valleys evident in the experimental results to a greater degree. For this reason, the strip stiffness that had been calculated to be 80,500 lb/in became suspect. In order to verify the stiffness of the steel strip the following test was performed to obtain a direct measurement of the deflection in the tape, follower arm, and associated hardware for a given load. The oscillating mass was held at the bottom of its stroke using a block placed between the mass and the bracket for the preload screw. The in-line force transducer was then connected to the dynamic analyzer to measure the tension in the tape and a dial test indicator with inch resolution was placed against the follower arm to measure the deflection. The 56

62 flywheel connected to the main drive shaft of the cam test machine was then rotated until an appreciable displacement was measured on the dial indicator. This displacement and the force measurement from the dynamic analyzer was recorded and used to compute the stiffness of the strip by dividing force by displacement. This test was performed several times in order to attain a range of data which was then averaged. Data from this testing can be found in Appendix G. The resulting effective stiffness was found to be 29,863 lb/in which confirmed the suspicion that the strip and end-terminations were not as stiff as had been calculated. This experimentally obtained value for stiffness also includes the stiffness of the follower arm which was not included in the original calculations. The simulation in Matlab was then run using this value for the tape stiffness and the remaining values for the damping ratio and mass distribution were adjusted to obtain graphical outputs that were more representative of the experimental data. Through this iteration, it was eventually determined that the original assumptions for the system s mass distribution were likely correct. These simulation input parameters are shown in Table

63 Acceleration (in/sec^2) Table 9-1: Parameter Value Mass blobs Mass blobs Effective Follower Return Spring Stiffness (k1) 323 lb/in Effective Follower Return Spring Preload 140 lbf Effective Tape Stiffness (k2) 29,863 lb/in Effective Tape Preload Spring Stiffness (k3) 288 lb/in Effective Tape Preload 50 lbf Damping Ratio z1 0.1 Damping Ratio z Damping Ratio z The following plots of acceleration (point B), camshaft torque, and tape tension (point C) were found to best represent the experimental results and can be seen in figures 9-9 through Simulated vs. Experimental End Effector Acceleration Time (sec) Simulated End Effector Acceleration Experimental End Effector Acceleration Figure 9-9: Plot of simulated end effector acceleration vs. experimental end effector acceleration (point B) over one cam revolutions at 250 rpm. 58

64 Tension (lbf) One can observe that the valleys that were absent in the original simulation are now present with a lower tape stiffness. The ringing in the dwells is also more realistic in comparison to the original simulation which had little ringing and virtually no taper in magnitude. Simulated vs. Experimental Tape Tension Time (sec) Simulated Tape Tension Experimental Tape Tension Figure 9-10: Plot of simulated tape tension vs. experimental tape tension (point C) over one cam revolutions at 250 rpm. Again the simulated tension in the tape (Figure 9-10) is similar to that of the experimental data with the corrected tape stiffness. The vibrations in both the high and low dwells are much more evident than with the previous simulation which utilized the theoretical tape stiffness. 59

65 Torque (in-lb) Simulated vs. Experimental Camshaft Torque Time (sec) Experimental Camshaft Torque Simulated Camshaft Torque Figure 9-11: Plot of simulated camshaft torque vs. experimental camshaft torque over one cam revolutions at 250 rpm. Although there is some oscillation in the camshaft torque (Figure 9-11) in addition to some phasing issues one can see there is some resemblance between the simulated and experimental data. The most probable cause for this torsional oscillation is unstable camshaft speed resulting in varying momentum of the flywheel. Although the test fixture was not equipped with sufficient instrumentation to verify deviation between follower displacement (z) and oscillating mass displacement (x), the experimentally obtained tape stiffness was re-entered into the Matlab model. Results of the simulated position error can be seen in Figure

66 Displacement z-x (inches) Position Error z-x Time (sec) Figure 9-12: Plot of position error between the follower and oscillating mass over one cam revolution at 250 rpm. As one might expect, the position error of the system is roughly double with the tape stiffness at 29,863 lb/in than it was at 80,000 lb/in. However, this is still comparable to and perhaps better in some cases than a conventional follower train. It is believed the decrease in tape stiffness is not due to the strip itself, but the associated hardware by which it is connected at either end. It could also be affected by the compliance of the pulley mounting system which was not included in the simulation. 9.5 Follower vs. End Effector: In Figure 9-13 the experimental follower acceleration plot was superimposed over the experimental end effector acceleration plot. 61

67 Acceleration (g) Follower vs. End Effector Acceleration B D E A C F -50 Time (s) Follower Accel End Effector Accel Figure 9-13: Plot of experimental follower acceleration vs. experimental end effector acceleration over two cam revolutions at 250 rpm. One can observe that the acceleration is nearly identical between the driver and the driven elements of the system. In fact, the only significant difference between the follower and end effector acceleration appears to be the magnitude of the vibration caused by the split in the cam which occurs in the middle of every other dwell shown as points A, B, D, and E. This high-frequency vibration caused by the split in the cam appears to have been filtered by the tension member follower train. Points C and F also show a slightly higher acceleration at the follower than at the end effector. These two points occur at the same location on the cam, and it is believed to have been caused by some other undetermined source of noise in the system. The RMS averages of both sets of data was calculated using the dynamic signal analyzer and despite the earlier visual observations that the 62

68 vibrations were lower at the end effector than at the follower arm, it was discovered that there was actually a 4 percent increase in RMS average vibration from the follower arm acceleration to the end effector acceleration. This observation of the system s vibration damping aspects prompted further investigation. The linear spectrum of the experimental follower acceleration and the experimental end effector acceleration can be seen in Figures 9-13 and 9-14 respectively. Follower Acceleration Linear Spectrum Frequency (Hz) Figure 9-13: Plot of experimental follower acceleration linear spectrum (point A). 63

69 End Effector Acceleration Linear Spectrum Frequency (Hz) Figure 9-14: Plot of experimental end effector acceleration linear spectrum (point B). The dynamic signal analyzer was used to calculate the RMS average for the acceleration linear spectra of both the follower and the end effector. In this case, the RMS average at the end effector was 3.8 percent greater than the RMS average at the follower arm. This correlates with the 4 percent difference in the time data. The fact that the acceleration linear spectrum RMS increase was slightly less than the time acceleration RMS increase is to be expected because the linear spectrum data in the frequency domain omits all data above 1600 Hz. Impulse hammer tests were performed at both the follower arm and oscillating mass at the end effector to study the dynamic response of each element. With the 64

70 machine at a standstill, a Dytran 5850A impulse hammer equipped with a 100mv/lbf force transducer was used to excite the system. The resulting response was then measured with the accelerometers mounted on the follower tape termination (point A) and end effector (point B). In the case of the follower arm which was not equipped with a permanently mounted accelerometer, an accelerometer was temporarily mounted with beeswax at point D in Figure Point D Figure 9-15: Location of accelerometer mounted to top surface of follower arm. The dynamic analyzer recorded both the force input from the hammer and the response signal from the accelerometer and was used to calculate the resulting 65

71 frequency response function (FRF). The FRF is the quotient of the Fourier transforms of the system input and output functions as shown in equation 9.1. H(f) = O (f) / I (f) (9.1) Where: O = Response Acceleration from Hammer Test I = Input Force from Hammer Test The FRF is valuable as it allows the prediction of the systems behavior in response to any input. Likewise, if the response of the system is known it is possible to determine the input function through deconvolution. The output of the system is the convolution of the input with the FRF. The output function and the FRF can each be measured independently as shown above. Deconvolution of the FRF from the output gives the spectrum of the input function and allows observations to be made with respect to how elements of the system modify the system input which in this case is the cam. The acceleration linear spectrum that was obtained during dynamic testing (system output) was used to deconvolve the dynamic input of the system as shown in equation 9.2. Acceleration Linear Spectrum = System Input Spectrum (9.2) FRF 66

72 The results from the three hammer tests performed are shown in Figures 9-16 through Follower Acceleration Linear Spectrum Out Frequency Hz Figure 9-16: Plot of follower acceleration linear spectrum output from dynamic testing (accelerometer at point D). 67

73 Follower FRF Frequency Hz Figure 9-17: Plot of follower acceleration FRF obtained from hammer test (accelerometer at point D). One can observe that the FRF of the follower arm displays natural frequencies at 202, 228, 242, 270, 320, 416, and 518. The evidence of sharp narrow peaks is typical of systems having low structural damping. 68

74 Follower Acceleration Linear Spectrum In Frequency Hz Figure 9-18: Plot of deconvolved follower acceleration linear spectrum (point D). 69

75 Tape Termination Acceleration Linear Spectrum Out Frequency Hz Figure 9-19: Plot of tape termination acceleration linear spectrum output from dynamic testing (point A). Tape Termination Acceleration FRF Frequency Hz Figure 9-20: Plot of tape termination acceleration FRF obtained from hammer test (point A). 70

76 The FRF of the tape termination (point A) exhibits multiple natural frequencies that span the bandwidth of 1600 Hz. This is expected as the tape termination hammer test was performed with other system elements assembled. The peaks appear to be wider than the FRF of the follower arm (Figure 9-17) suggesting that the system has more structural damping when assembled under preload. Tape Termination Acceleration Linear Spectrum In Frequency Hz Figure 9-21: Plot of deconvolved tape termination acceleration linear spectrum (point A). 71

77 End Effector Accel. Linear Spectrum Out Frequency Hz Figure 9-22: Plot of end effector acceleration linear spectrum output from dynamic testing (point B). 3.5 End Effector Acceleration FRF Frequency Hz Figure 9-23: Plot of end effector acceleration FRF obtained from hammer test (point B). 72

78 Again, the FRF of the end effector (Figure 9-23) exhibits natural frequencies across the bandwidth as would be expected due to the additional components. The peaks have become even broader and rounder in comparison to those of the tape termination FRF (Figure 9-20), further supporting that there has been an increase in structural damping with the addition of the metal tape. End Effector Accel. Linear Spectrum In Frequency Hz Figure 9-24: Plot of deconvolved end effector acceleration linear spectrum (point B). 73

79 The relatively low increase in vibration between the follower arm and end effector is a very unusual characteristic of this system in comparison to traditional follower trains which typically exhibit much more vibration at the output than at the input. The comparatively low increase in vibration in this system is believed to be due to both the elimination of pivot points which are abundant in conventional follower trains, and the internal damping characteristics of the system. With conventional systems, the necessary clearance at each pivot to allow for free motion creates significant noise as the forces vary throughout the cycle of the cam. In tension member systems not only are many of the pivots eliminated, but the few that do exist are always under unidirectional loading, therefore removing any backlash. It is believed that most of the system damping is due to the pulley which serves as damper in two ways. The pulley separates the bottom half of the follower train consisting of the system input from the cam profile from the top half which is comprised of the oscillating mass. Essentially, the pulley limits the vibration transmitted through the metal tape, just as a fret limits the vibration of a guitar string. The second way the pulley acts as a damper is through the coulomb friction inherent in its bearing. Although this is a pivot point, it is always under load due to the tape preload spring and the follower return spring. The vibration damping characteristics of this system are an important added benefit in comparison to traditional follower trains which would not exhibit this notable vibration absorption quality. There are many mechanisms used to perform delicate, vibration sensitive assembly operations that would benefit from the application of a tension member cam follower train. 74

80 10. Conclusion: The tension member cam follower train was found to be a viable alternative to conventional cam follower trains. The use of tension members allows follower trains to be less massive, more precise, and less costly to produce. Additionally the vibration damping characteristics of the tension member allows greater control of the tooling at the end effector for critical operations at high speed. Despite the fact that the testing performed in this research was at accelerations less than the target goal, the tension member apparatus was not the limiting factor. There is little doubt that with a properly designed test bed, the target could easily be attained. The kineto-elasto dynamic method that was chosen to model the dynamics of this system was also found to be acceptable but could be improved with better estimates of element compliance and additional degrees of freedom. The method presented here could be used to model potential applications and serve as a valuable starting point for the design of a tension member follower train. In addition to the dynamic behavior of tension member follower trains, a great deal was learned about the mechanics of such a mechanism. 75

81 The tape end attachments could be improved by adding a pivot that would eliminate twist in the tape and allow the tape to conform to the pulley in the event of misalignment among the follower, pulley(s), and end effector. The most significant finding from this investigation is the fact that the tensionmember system appears to provide more internal damping between the cam follower and end effector than a conventional, multi-link mechanism. This is most likely due to the friction in the bearing of the pulley, as a result of the high preload necessitated in the metal tape to keep it in tension. This damping resulted in attenuation at the end effector of high frequency vibration present in the cam follower arm. This is the opposite of a conventional follower train as the intermediate links typically increase vibration at the end effector to a greater degree than the 4 percent seen here. 76

82 11. Recommendations: Although a great deal of information regarding tension member follower trains was obtained in this investigation, some recommendations for further study are as follows: Improve Dynamic Model: An improved dynamic model of the system could include such parameters as compliance of the pulley mounting, necessary for allowing overtravel when the end effector stroke is limited by hard stops. Achieve Target Speed: Both the limited torque output of the dynamics testing machine drive motor and the machine s follower arm were the limiting factors in this research. Due to the excessive clearance in the follower arm pivot bearings, it was almost impossible to keep the follower arm from coming into contact with the cam at speeds greater than 250 rpm. For these reasons alone, the peak cam speed of 400 rpm and the resulting tape accelerations and forces were not reached. The machine could, however be fitted with a larger motor and a new follower arm. The follower arm could be redesigned with the roller follower located in the center of the arm so as not to create a couple and with pivot bearings further apart to minimize the movement of the follower arm from side to side. Redesign End Attachment: As was described in the conclusion, the tape end attachment could be redesigned with a swivel that would eliminate the tendency 77

83 for the tape to twist. This would provide for some unintentional misalignment in the system and possibly relax manufacturing tolerances. Investigate End Effector Position Error: The position error of the system that was projected in the mathematical model could be confirmed with the addition of another LVDT located at the oscillating mass. Due to the fact that this deviation is believed to be as small as inches, the pivot points on the existing LVDT located on the follower arm would need to be redesigned in order to have meaningful results. Investigate Partial Pulley: In applications where the pulleys used in a tensionmember follower train rotate less than 360 degrees over the stroke of the cam follower, it may be possible to further reduce the rotating moment of inertia thru the use of a partial pulley. This partial pulley would have the unused section of the rim removed and the drive tape would be attached to the remaining section of the rim via pin or clamp to prevent independent movement of the tape and pulley. This reduction of rotating moment of inertia would become more important as the number of pulleys in the system increase resulting in accumulated effective mass at the follower. 78

84 References Modular Linear Actuator. Nook Industries. 7 Sept < No-Stretch Timing belts. Fenner Precision. 14 Sept < United States Department of Defense. Wire Rope Assemblies: Aircraft Proof Testing and Prestretching of. MIL-DTL-5688E Cable Stretch, Pre-Stretching & Proof Testing. The Cable Connection. 6 Sept < Wire Rope Technical Board. Wire Rope User s Manual. Virginia: Alexandria, Wire Rope Technical Board. Wire Rope Inspection Guidelines. Virginia: Alexandria, Hasengawa, Junzo, Susumu, Kawabata, and Kono, Kiichi. Flexible Type Rapier Loom. US Patent Aug Mass, William. The Decline of a Technological Leader: Capability, Strategy, and Shuttless Weaving, , University of Lowell, Lowell Massachusetts. Norton, R.L. Cam Design and Manufacturing Handbook. New York: Industrial Press, Norton, R. L. Machine Design - An Integrated Approach. Third Edition. New Jersey: Prentice Hall, G. Dalpiaz, A. Rivola, A kineto-elastodynamic model of a mechanism for automatic machine, Department of Mechanical Engineering. University of Bologna, Bologna, Italy. Young-Hyu Choi, Won-Jee Chung, Xiang-Rong Xu, Modeling of Kineto- Elastodynamics of Robots with Flexible Links, Department of Mechanical Design and Manufacturing, Chang-Won National University, Changwon, Korea. "Metal Belt Design Guide." Belt Technologies Dec < 79

85 "Metal Belt Design Guide." Belt Technologies Dec < Dowling, N.E. Mechanical Behavior of Materials. New Jersey: Prentice Hall, Norton, R. L. Machine Design - An Integrated Approach. Third Edition. New Jersey: Prentice Hall, Koster, M. P. Vibrations of Cam Mechanisms. Phillips Technical Library Series, London: Macmillan Press Ltd. 80

86 Bibliography Dresner, T.L. and Barkan, P., New Methods for the Dynamic Analysis of Flexible Single-Input and Multi-Input Cam-Follower Systems. Journal of Mechanical Design Transactions of the ASME volume : Flat Belts. Fenner Precision. 14 Sept < Killion, Christopher, Spangler, Joseph, and Van Sant, Glen. Precision Cable Drive. US Patent Jan Mallard, Robert G. Loom Raper Drive Mechanism. US Patent Jan Nayfeh, A. Samir and Varanasi, K. Kripa. Damping of Drive Resonances in Belt- Driven Motion Systems Using Low-Wave-Speed Media. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts. Norton, R. L. Design of Machinery: An Introduction to Synthesis and Analysis of Mechanisms and Machines. New York: Mc Graw-Hill,

87 Appendix A Dynacam Model of Project Sponsor s Rise-Fall-Dwell Cam 82

88 83

89 84

90 Appendix B Dynacam Model of Test Bed Four Dwell Cam 85

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92 87