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:

_______________________________________________

Jeffrey LaPierre

May 30, 2008

Approved:

________________________________________________

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

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.

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.

Contents

1. Introduction..………………………………………………………………...1

2. Project Scope…………………………………………………....…..……….2

2.1 Goal Statement………………………………………….……....2

2.2 Project Objective……………………………………….....…….2

2.3 Approach………………………………………………….…….2

3. Background Study…………………………………….…………….…...5

3.1 Tension Members……………………………………….………5

3.2 Timing Belts……………………………………………….…....5

3.3 Flat Belts ………………………………………………….……7

3.4 Pre-Stretched Wire Rope………………………………….……7

3.5 Metal Drive Tapes………………………………….…………..8

3.6 Literature Review……………………………………………..10

4. Conceptual Design……………………………………………………..13

5. Preliminary Modeling………………………………………………….18

6. Refined Design………………………………………………………...27

7. Analysis………………………………………………………………...30

8. Fabrication……………………………………………………………...44

9. Experimentation………………………………………………………..50

9.1 Data Collection……………………………………………….50

9.2 Experimental Results……………………………….…….......52

9.3 Experimental vs. Simulated Results……………………….....54

9.4 Iteration of Simulation Parameters………...............................56

9.5 Follower vs. End Effector………………………………….…61

10. Conclusion……………………………………………………………75

11. Recommendations……………………………………………………77

References………………………………………………………………..79

Bibliography……………………………………………………………...81

Appendix A………………………………………………………………82

Appendix B………………………………………………………...…….85

Appendix C………………………………………………………………88

Appendix D………………………………………………………………95

Appendix E……………………………………………………………..100

Appendix F……………………………………………………………..103

Appendix G…………………………………………………………….114

1

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.

2

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.

3

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.

4

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.

5

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

6

semi-precision linear position is required (+/- 0.005 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.

7

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

8

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 +/-

0.0005 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

9

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.

10

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.

11

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.

12

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.

13

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.

108.7 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

14

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 11.140 inch long follower

arm was determined to be 76,800 deg/ sec2. 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/sec2. Through the use of Newton’s second law, F = ma,

15

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.

16

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:

17

Segment 1: Rise 0.5 inches in 50° with 4-5-6-7 polynomial displacement

Segment 2: Dwell for 40°

Segment 3: Fall 0.5 inches in 50° with 3-4-5 polynomial displacement

Segment 4: Dwell for 40°

Segment 5: Rise 0.5 inches in 50° with 3-4-5 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/sec2

and the resulting tangential acceleration due to

the 13.50 inch arc of the follower arm was determined to be 17,945 in/sec2

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.

18

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

].

σtotal = σwork + σbending (5.1)

σwork = τ / (w x t) (5.2)

σbending = (E x t) / (1- u2)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

19

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

].

Ntape = (1/3 x Sy) / σtotal (5.4)

Where:

Sy = 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 1095

20

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.

Cload = 1 (Axial Loading)

Cload = 0.70 (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.

A95 = 0.05 (thickness) x (width) (5.5)

A95 = 0.05 (0.010 inches) x (2.00 inches) = 0.001

dequiv = (A95 / 0.0766)0.5

(5.6)

dequiv = (0.001 / 0.0766)0.5

= 0.114

Csize = 1 (Where dequiv < 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.

21

Table 1-1A:

Surface Finish A (kpsi) b (kpsi)

Ground 1.34 -0.085

Machined or Cold Rolled 2.7 -0.265

Hot-Rolled 14.4 -0.718

As-Forged 39.9 -0.995

Csurf = A (Sut)b

(5.7)

Csurf = 1.34 (173,500 psi) -0.085

= 0.481

The temperature correction factor was based on the following criteria and the fact

that the machine will be operated at room temperature.

Ctemp= 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 % Creliab

50 1.000

90 0.897

99 0.814

99.9 0.753

99.99 0.702

99.999 0.659

Creliab = 0.897

22

Application of the correction factors can be seen in equation 5.8 as follows.

Se = Cload Csize Csurf Ctemp Creliab 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

1.615 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

1.50

1.52

1.54

1.56

1.58

1.60

1.62

1.64

0.01

0

0.01

2

0.01

3

0.01

4

0.01

5

0.01

6

0.01

7

0.01

8

0.01

9

0.02

0

0.02

1

0.02

2

0.02

3

0.02

4

0.02

5

0.02

6

0.02

7

0.02

8

0.02

9

0.03

0

0.03

1

0.03

2

0.03

3

0.03

4

0.03

5

Tape Thickness (inches)

Nta

pe

Figure 5-1: Plot of tape safety factor vs. tape thickness

23

The problem with using a tape thickness falling within this ideal range (.025 -

.030 inches), is that the pulley must be between 15.00 and 18.75 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 6.250 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

24

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

25

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 0.625 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 0.625 inches

Bearing Length 1.500 inches

Shaft Speed 210 rpm

26

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 15840 in this

case, eliminating the possibility of most plastic bearings. A bronze 954 alloy

bearing with a 125000 PV rating was found to be more than sufficient.

27

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.

28

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.

29

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.

30

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.

meff = Izz / r2

(7.1)

Where:

Izz = Mass moment of inertia of follower arm about pivot point

r = Radius from pivot point to tape attachment point

31

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 (Izz) 0.6687 blob-in2

Radius (r) 13.50 inches

Effective Mass (meff) 0.00367 blobs*

The effective mass of each component due to the lever ratio of the follower arm

was then determined using equation (7.2) below.

meff = m (r1/r2)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.

Norton16

.

32

Table 7-2

Component Mass (blobs) Effective Mass at Follower

(blobs)

Follower Arm 0.00367 0.015831

Follower Roller 0.0008 0.0008

Spring Pivot Block 0.00147 0.00147

Spring Clamp Plate 0.001728 0.005648

Tape Termination 0.00093 0.004012

Tape Termination Clamp Plate 0.000155 0.000669

Force Transducer 0.00016 0.00016

Tape Termination Yoke 0.00104 0.00104

Shoulder Bolt & Nut 0.00055 0.00237

Point Mass of Pulley 0.00203 0.00875

Oscillating Mass & Hardware 0.0033 0.01406

Metal Drive Tape 0.00044 0.001898

Total 0.0116 0.0567

The spring rate of the 0.010 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 0.020 in2

Length of Tape 31.50 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 in2)( 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

33

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 (r1/r2)2

(7.4)

keff = (18,700 lb/in) (13.50 / 6.50) 2

= 80,500 lb/in

Where:

r1 = Distance between follower arm pivot point and tape attachment.

r2 = Distance follower arm pivot to roller follower.

Table 7-4

Tape Spring Constant 18,700 lb/in

r1 13.50 inches

r2 6.50 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

34

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, two-

degree 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.

35

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 = m1

Fc (t) – Fd – Fs – k2(z-x)-c2( ) = m

Fc (t) – c1 - k1z – k2 (z-x) – c2 ( - ) = m

Fc (t) – c1 – k1z – k2z + k2x – c2 + c2 = m

Fc (t) = m1 + c1 + k1z + k2z – k2x + c2 – c2

Fc (t) = m1 + (c1 + c2) + (k1 + k2)z – k2x – c2

Where Fc (t) = 0 :

m1 = -(c1 + c2) – (k1 + k2)z + k2x + c2

36

= -(c1 + c2) – (k1 + k2)z + k2 x + c2 (7.5)

m1 m1 m1 m1

Derivation of Mass 2 Equation:

Σ F = m2

m2 = k2(z-x) + c2( - ) – k3 x – c3

m2 = k2z - k2x + c2 - c2 – k3 x – c3

m2 = k2z – (k2 + k3) – (c2 + c3) + c2

= k2z – (k2 + k3)x – (c2 + c3) + c2 (7.6)

m2 m2 m2 m2

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/Second2

x = Displacement of Mass 2 (Oscillating Mass) in Inches

37

= Velocity of Mass 2 in Inches/Second

= Acceleration of Mass 2 in Inches/Second2

Through the use of four dummy variables, the following state space equations

were obtained:

Where: y 1 = x y2 = y3 = s y4 =

4 = -(c1 + c2) y4 – (k1 + k2) y3 + (k2) y1 + c2 (y2) (7.7)

m1 m1 m1 m1

3 = y4 (7.8)

2 = (k2) y3 – (k2 + k3) y1 -(c2 + c3) y2 + (c2) y4 (7.9)

m2 m2 m2 m2

1 = y2 (7.10)

In order to solve these state space equations (7.7 - 7.10) 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.

38

The damping coefficients (c1, c2, and c3) were also calculated in the Matlab

model using the equations 7.11 and 7.12.

(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/sec2 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/sec2. Plots of the

39

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.

40

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.

41

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.

42

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 0.0035 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 7-9.

43

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.

44

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 8-3.

45

Figure 8-1: Rear view of cam test bed showing completed fixture mounted on

test bed and metal tape routed around flat pulley.

46

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.

47

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

test bed.

48

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.

49

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.

50

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 end-

termination 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 9-1.

51

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

Point A

Point C

Point B

52

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

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

0

0.0

2

0.0

3

0.0

5

0.0

7

0.0

9

0.1

0.1

2

0.1

4

0.1

5

0.1

7

0.1

9

0.2

1

0.2

2

0.2

4

0.2

6

0.2

7

0.2

9

0.3

1

0.3

2

0.3

4

0.3

6

0.3

8

0.3

9

0.4

1

0.4

3

0.4

4

0.4

6

0.4

8

0.5

Time (s)

Acce

lera

tio

n (

in/s

ec^2

)

Figure 9-2: Plot of experimental follower acceleration (point A) over

approximately two cam revolutions at 250 rpm.

53

End Effector Acceleration at 250 RPM

-6000

-4000

-2000

0

2000

4000

6000

0

0.0

2

0.0

3

0.0

5

0.0

7

0.0

9

0.1

0.1

2

0.1

4

0.1

5

0.1

7

0.1

9

0.2

1

0.2

2

0.2

4

0.2

6

0.2

7

0.2

9

0.3

1

0.3

2

0.3

4

0.3

6

0.3

8

0.3

9

0.4

1

0.4

3

0.4

4

0.4

6

0.4

8

0.5

Time (s)

Acce

lera

tio

n (

in/s

ec^2

)

Figure 9-3: Plot of experimental end effector acceleration (point B) over

approximately two cam revolutions at 250 rpm.

Tape Tension at 250 RPM

0

20

40

60

80

100

120

140

160

0

0.0

2

0.0

3

0.0

5

0.0

7

0.0

8

0.1

0.1

2

0.1

3

0.1

5

0.1

7

0.1

8

0.2

0.2

2

0.2

3

0.2

5

0.2

7

0.2

8

0.3

0.3

2

0.3

3

0.3

5

0.3

7

0.3

8

0.4

0.4

2

0.4

3

0.4

5

0.4

6

0.4

8

0.5

Time (s)

Te

nsio

n (

lbf)

Figure 9-4: Plot of experimental tension in tape (point C) over approximately two

cam revolutions at 250 rpm.

54

Torque at 250 RPM

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

0.0

0

0.0

2

0.0

3

0.0

5

0.0

7

0.0

9

0.1

0

0.1

2

0.1

4

0.1

5

0.1

7

0.1

9

0.2

1

0.2

2

0.2

4

0.2

6

0.2

7

0.2

9

0.3

1

0.3

2

0.3

4

0.3

6

0.3

8

0.3

9

0.4

1

0.4

3

0.4

4

0.4

6

0.4

8

0.5

0

Time (s)

To

rqu

e (

in-lb

)

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 9-6.

55

Simulated vs. Experimental End Effector Acceleration

-6000

-4000

-2000

0

2000

4000

6000

0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 0.220 0.240

Time (sec)

Acce

lera

tio

n (

in/s

ec^2

)

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

56

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 0.001 inch

resolution was placed against the follower arm to measure the deflection. The

57

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 9-1.

58

Table 9-1:

Parameter Value

Mass 1 0.032 blobs

Mass 2 0.025 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 z2 0.01

Damping Ratio z3 0.05

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 9-11.

Simulated vs. Experimental End Effector Acceleration

-6000

-4000

-2000

0

2000

4000

6000

0.000 0.050 0.100 0.150 0.200 0.250

Time (sec)

Acce

lera

tio

n (

in/s

ec^2

)

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.

59

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

0

20

40

60

80

100

120

140

160

0.00 0.05 0.10 0.15 0.20 0.25

Time (sec)

Te

nsio

n (

lbf)

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.

60

Simulated vs. Experimental Camshaft Torque

-600

-400

-200

0

200

400

600

800

0 0.05 0.1 0.15 0.2 0.25

Time (sec)

To

rqu

e (

in-lb

)

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 9-

12.

61

Position Error z-x

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

Time (sec)

Dis

pla

ce

me

nt

z-x

(in

ch

es)

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.

62

Follower vs. End Effector Acceleration

-50

-40

-30

-20

-10

0

10

20

30

40

Time (s)

Acce

lera

tio

n (

g)

Follower Accel End Effector Accel

A

B

C

D

E

F

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

63

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

0

0.05

0.1

0.15

0.2

0.25

0 160 320 480 640 800

Frequency (Hz)

Figure 9-13: Plot of experimental follower acceleration linear spectrum (point

A).

64

End Effector Acceleration Linear Spectrum

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 160 320 480 640 800

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

65

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 9-15.

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

Point D

66

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

67

The results from the three hammer tests performed are shown in Figures 9-16

through 9-24.

Follower Acceleration Linear Spectrum Out

0

0.05

0.1

0.15

0.2

0.25

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-16: Plot of follower acceleration linear spectrum output from dynamic

testing (accelerometer at point D).

68

Follower FRF

0

0.2

0.4

0.6

0.8

1

1.2

0 160 320 480 640 800 960 1120 1280 1440 1600

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.

69

Follower Acceleration Linear Spectrum In

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-18: Plot of deconvolved follower acceleration linear spectrum (point D).

70

Tape Termination Acceleration Linear Spectrum Out

0

0.05

0.1

0.15

0.2

0.25

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-19: Plot of tape termination acceleration linear spectrum output from

dynamic testing (point A).

Tape Termination Acceleration FRF

0

0.5

1

1.5

2

2.5

3

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-20: Plot of tape termination acceleration FRF obtained from hammer

test (point A).

71

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-21: Plot of deconvolved tape termination acceleration linear spectrum

(point A).

72

End Effector Accel. Linear Spectrum Out

0

0.05

0.1

0.15

0.2

0.25

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-22: Plot of end effector acceleration linear spectrum output from

dynamic testing (point B).

End Effector Acceleration FRF

0

0.5

1

1.5

2

2.5

3

3.5

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-23: Plot of end effector acceleration FRF obtained from hammer test

(point B).

73

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

0

0.2

0.4

0.6

0.8

1

1.2

0 160 320 480 640 800 960 1120 1280 1440 1600

Frequency Hz

Figure 9-24: Plot of deconvolved end effector acceleration linear spectrum

(point B).

74

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.

75

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.

76

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 tension-

member 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.

77

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

78

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 0.006 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 tension-

member 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.

79

References

1 “Modular Linear Actuator.” Nook Industries. 7 Sept. 2007

<www.precisionactuator.com>.

2 “No-Stretch Timing belts.” Fenner Precision. 14 Sept 2007.

<http://www.fennerprecision.com/flatbelts.php>.

3 United States Department of Defense. “Wire Rope Assemblies: Aircraft Proof

Testing and Prestretching of.” MIL-DTL-5688E. 1999.

4 Cable Stretch, Pre-Stretching & Proof Testing.” The Cable Connection. 6 Sept.

2007. <http://www.thecableconnection.com>.

5 Wire Rope Technical Board. Wire Rope User’s Manual. Virginia: Alexandria,

2005.

6 Wire Rope Technical Board. Wire Rope Inspection Guidelines. Virginia:

Alexandria, 2004.

7 Hasengawa, Junzo, Susumu, Kawabata, and Kono, Kiichi. “Flexible Type

Rapier Loom.” US Patent 4344466. 17 Aug. 1982.

8 Mass, William. “The Decline of a Technological Leader: Capability, Strategy,

and Shuttless Weaving, 1945-1974,” University of Lowell, Lowell

Massachusetts.

9 Norton, R.L. Cam Design and Manufacturing Handbook. New York: Industrial

Press, 2002.

10

Norton, R. L. Machine Design - An Integrated Approach. Third Edition. New

Jersey: Prentice Hall, 2006.

11

G. Dalpiaz, A. Rivola, “A kineto-elastodynamic model of a mechanism for

automatic machine”, Department of Mechanical Engineering. University of

Bologna, Bologna, Italy.

12

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.

13

"Metal Belt Design Guide." Belt Technologies. 2007. 9 Dec. 2007

<http://www.belttechnologies.com>

80

14

"Metal Belt Design Guide." Belt Technologies. 2007. 9 Dec. 2007

<http://www.belttechnologies.com>

15

Dowling, N.E. Mechanical Behavior of Materials. New Jersey: Prentice Hall,

1993.

16

Norton, R. L. Machine Design - An Integrated Approach. Third Edition. New

Jersey: Prentice Hall, 2006.

17

Koster, M. P. Vibrations of Cam Mechanisms. Phillips Technical Library Series,

London: Macmillan Press Ltd.

81

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 117. 1995:150-155.

“Flat Belts.” Fenner Precision. 14 Sept. 2007 <http://www.fennerprecision.com>.

Killion, Christopher, Spangler, Joseph, and Van Sant, Glen. “Precision Cable Drive.”

US Patent 6503163. 7 Jan. 2003.

Mallard, Robert G. “Loom Raper Drive Mechanism.” US Patent 4243076. 6 Jan.

1981.

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, 2004.

82

Appendix A

Dynacam Model of Project Sponsor’s

Rise-Fall-Dwell Cam

83

84

85

Appendix B

Dynacam Model of Test Bed Four Dwell Cam

86

87

88

Appendix C

TK Solver Model

89

Rule

CrossSectionalArea = TapeThickness * TapeWidth

PulleyOD = ( 625 * TapeThickness)

BendingStress = (Young * TapeThickness) / (( 1 - Poisson ^ 2 ) * ( PulleyOD))

WorkingStress = (Tension) / (TapeThickness * TapeWidth)

TotalStress = ( WorkingStress + BendingStress )

TotalStrength = (1 / 3) * YieldStrength

AllowableTens = 0.25 * UTS

Ntape = TotalStrength / TotalStress

MassTape = DensityTape * TapeThickness * TapeWidth

PulleyRimWidth = (TapeWidth + .120)

PulleyRimID = (PulleyOD) - (PulleyRimThickness * 2)

PulleyHubWidth = (TapeWidth + .140)

VolumePulleyRim = ((.25 * Pi() * (PulleyOD)^2) - (.25 * Pi() * (PulleyRimID)^2)) * (PulleyRimWidth)

VolumePulleyWeb = ((.25 * Pi() * (PulleyRimID)^2) - (.25 * Pi() * (PulleyBore)^2)) * (WebThickness)

VolumePulleyHub = ((.25 * Pi() * (PulleyHubOD)^2) - (.25 * Pi() * (PulleyBore)^2)) * (PulleyHubWidth - WebThickness)

PulleyVolume = (VolumePulleyRim) + (VolumePulleyWeb) + (VolumePulleyHub)

90

MassPulley = (DensityPulley) * (PulleyVolume)

MassPulleyRim = ((VolumePulleyRim) * (DensityPulley) ) / (386.4)

MassPulleyWeb = ((VolumePulleyWeb) * (DensityPulley)) / (386.4)

MassPulleyHub = ((VolumePulleyHub) * (DensityPulley)) / (386.4)

MassPulleyBlobs = (MassPulleyRim + MassPulleyWeb + MassPulleyHub)

InertiaRim = ((MassPulleyRim) * ((((PulleyOD)/2)^2) + (((PulleyRimID)/2)^2))) / (2)

InertiaWeb = ((MassPulleyWeb) * ((((PulleyRimID)/2)^2) + (((PulleyBore)/2)^2))) / (2)

InertiaHub = ((MassPulleyHub) * ((((PulleyHubOD)/2)^2) + (((PulleyBore)/2)^2))) / (2)

InertiaPulleyTotal = (InertiaRim) + (InertiaWeb) + (InertiaHub)

ωFollowerArm = (FollowerArmDeg)/(180) * (pi)

TapeVelocity = (FollowerArm) * (FollowerArmRadius)

Pulley = (TapeVelocity) / ((PulleyOD)/2)

pi = 3.14159

AngularPulleyDispDegrees = (ToolingStroke / PulleyCircumference) * (360)

AngularPulleyDisp = (AngularPulleyDispDegrees / 180) * (pi)

PulleyCircumference = (PulleyOD) * (pi)

αFollowerArm = ((αFollowerArmDeg) / (180)) * (pi)

91

TapeAccel_Inches = (ᆯllowerArm) * (FollowerArmRadius)

TapeAccel = ((TapeAccel_Inches) * (2.54)) / (100)

TensionNewtons = (OscillatingToolingForceNewtons) + (ForcePulleyNewtons) + (SpringForceCompressedNewtons)

Tension = ((TensionNewtons) / (4.448))

KspringMetric = ((OscillatingToolingForceNewtons) + (ForcePulleyNewtons)) / (ToolingStroke)

TorquePulley = (InertiaPulleyTotal) * (ᐵlley)

αPulley = (TapeAccel_Inches) / ((PulleyOD)/2)

ForcePulley = (TorquePulley) / ((PulleyOD) / (2))

ForcePulleyNewtons = (ForcePulley) * (4.448)

OscillatingToolingForceNewtons = (ToolingMass) * (TapeAccel)

OscillatingToolingForce = (OscillatingToolingForceNewtons) / (4.448)

SpringForceCompressedNewtons = (KspringMetric) * (ToolingStroke)

Kspring = (KspringMetric) / (175.126)

SpringForceCompressed = (SpringForceCompressedNewtons) / (4.448)

92

Input Name Output Unit Comment

13.500 FollowerArmRadius in

549.500 ωFollowerArmDeg deg

ωFollowerArm 9.591 rad/sec

TapeVelocity 129.473 in/sec

αFollowerArmDeg deg/sec^2

αFollowerArm 1,340.412 rad/sec^2

TapeAccel_Inches 18,095.558 in/sec^2

TapeAccel 459.627 m/sec^2

TensionNewtons 1,259.488 N

1.000 ToolingMass kg

OscillatingToolingForce 103.333 lb

OscillatingToolingForceNewtons 459.627 N

Pi 3.142

Tension 283. lb

280,000. UTS

2.6E7 Young

.285 Poisson

2.6E5 YieldStrength

.010 TapeThickness in

2.000 TapeWidth in

CrossSectionalArea .02000 in^2

BendingStress 45,278.

WorkingStress 14,158.

TotalStress 59,436.

TotalStrength 86,667.

AllowableTens 70,000.000

Ntape 1.458

8.030 DensityTape g/cc

93

MassTape .003 kg

PulleyOD 6.250 in

.150 PulleyRimThickness

PulleyRimID 5.950 in

PulleyRimWidth 2.120 in

.200 WebThickness in

.750 PulleyHubOD in

.500 PulleyBore in

PulleyHubWidth 2.140 in

PulleyCircumference 19.635 in

VolumePulleyRim 6.094 in^3

VolumePulleyWeb 5.522 in^3

VolumePulleyHub .476 in^3

.098 DensityPulley lb/in^3

PulleyVolume 12.092 in^3

MassPulley 1.185 lb

MassPulleyRim .002 blob

MassPulleyWeb .001 blob

MassPulleyHub 1.21E-4 blob

MassPulleyBlobs .003 blob

InertiaRim .014 lb-in-sec^2

InertiaWeb .006 lb-in-sec^2

InertiaHub 1.23E-5 lb-in-sec^2

InertiaPulleyTotal .021 lb-in-sec^2

AngularPulleyDispDegrees 28.877 deg

AngularPulleyDisp .504 rad

ωPulley 41.431 rad/sec

αPulley 5,790.579 rad/sec^2

TorquePulley 119.518 in-lb

94

ForcePulleyNewtons 170.117 N

ForcePulley 38.246 lb

1.575 ToolingStroke in

KspringMetric 399.837 N/m

Kspring 2.283 lb/in

SpringForceCompressedNewtons 629.744 N

SpringForceCompressed 141.579 lb

95

Appendix D

Matlab Code for Dynamic Simulation of Tension-Member

Cam Follower Train

96

clear; close; clc; global m1 m2 k1 k2 k3 c1 c2 c3 f1 f2 t1 t2

%% Import Cam Data thData=dlmread('Dynacam Model at 258 RPM_6-2-08.dat', '', 6, 0); angle=thData(:,1); z=thData(:,2); zdot=thData(:,3); zdotdot=thData(:,4);

%% Import Dynamic Response from Dynacam % and use it to validate the Matlab results % dynacamData=dlmread('DynamicResponse.dat', '', 6, 0); % xDynacam=dynacamData(:,3); % xdotDynacam=dynacamData(:,4); % z_xDynacam=dynacamData(:,2);

%% System Parameters m1=.032;

k1=288;

fp1=174;

m2=.025;

k2=80522;

fp2=58;

k3=323;

zeta1=0.01;

zeta2=0.05;

zeta3=0.01;

c1=(2*(m1)*(sqrt((k1+k2)/(m1))))*(zeta1);

c2=(2*(m2)*(sqrt((k2+k3)/(m2))))*(zeta2);

c3=(2*(m2)*(sqrt((k2+k3)/(m2))))*(zeta3);

%% Force Known Variables % f=(m1*zdotdot)+(c1+c2)*zdot+(k1+k2)*z; f=(c2)*zdot+(k2)*z;

%% Initial Conditions

97

x1(1)=0; x1dot(1)=0; x1dotdot(1)=0 wCam=250; %rpm dt=(1/(wCam/60))/1440; % sec t=[0:1:1440]*dt; % time for a rotation

N=length(t);

%% Solve ODE nip=3; for i=1:N-1 y0=[x1(i); x1dot(i);]; %

fc(i)=(fp1)+(fp2)+(m1*zdotdot(i))+(c1+c2)*zdot(i)+(k1+k2)*z(i)-

(k2*x1(i))-(c2*x1dot(i)); ft(i)=(fp2)+(z(i)-(x1(i)))*(k2); tc(i)=(fc(i)*(x1dot(i)))/((wCam)*(.104719)); %

t1=t(i); t2=t(i+1); tspan = (t1:(t2-t1)/nip:t2); fi=f(i); f1=f(i); f2=f(i+1); % for better accuracy

options = odeset('RelTol',1e-6, 'AbsTol',1e-9);

[t_y,y] = ode45('ysystem_Main',tspan,y0, options);

% [t_y,y] = ode45('ysystem_Main',tspan,y0); % [t_y,y] = ode23s('ysystem_Main',tspan,y0); x1(i+1)=y(nip+1,1); % x2(i+1)=y(nip+1,3);

x1dot(i+1)=y(nip+1,2); % x2dot(i+1)=y(nip+1,4);

% x1dotdot(i+1)=y(nip+1,3);

x1dotdot(i)=((k2)/(m2))*(z(i))-((k2+k3)/(m2))*(x1(i))-

((c2+c3)/(m2))*(x1dot(i))+((c2)/(m2))*(zdot(i));

end

fc(N)=(m1*zdotdot(N))+(c1+c2)*zdot(N)+(k1+k2)*z(N)-(k2*x1(N))-

(c2*x1dot(N)); ft(N)=(z(N)-x1(N))*(k2); tc(N)=(fc(N)*(x1dot(N)))/((wCam)*(.104719));

x1dotdot(N)=((k2)/(m2))*(z(N))-((k2+k3)/(m2))*(x1(N))-

((c2+c3)/(m2))*(x1dot(N))+((c2)/(m2))*(zdot(N));

98

%% Plot results figure hold on; plot(t, z-x1', 'r'); grid; ylabel('z-x(Inches)'); xlabel('Time(Seconds)') % legend('dynacam', 'Matlab')

figure plot(t, fc) grid; ylabel('Force (Pounds)'); xlabel('Time(Seconds)') legend('Cam Force')

figure plot(t, ft) grid; ylabel('Force (Pounds)'); xlabel('Time (Seconds)') legend('Tension in Tape')

figure plot(t, tc) grid; ylabel('Torque (In-Lb)'); xlabel('Time(Seconds)') legend('Camshaft Torque')

figure subplot(3,1,1); hold on; plot(t, x1); grid; ylabel('x(Inches)'); xlabel('Time(Seconds)') legend('Displacement x') % legend('dynacam', 'Matlab')

subplot(3,1,2); hold on; plot(t, x1dot, 'r'); grid; ylabel('xdot(In/Sec)'); xlabel('Time(Seconds)') legend('xdot') % legend('dynacam', 'Matlab')

subplot(3,1,3); plot(t, x1dotdot) grid; ylabel('xdotdot (In/Sec^2)'); xlabel('Time(Seconds)') legend('xdotdot')

figure hold on; plot(t, x1); grid; ylabel('x(Inches)'); xlabel('Time(Seconds)') legend('Displacement x') % legend('dynacam', 'Matlab')

figure

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hold on; plot(t, x1dot, 'r'); grid; ylabel('xdot(In/Sec)'); xlabel('Time(Seconds)') legend('xdot') % legend('dynacam', 'Matlab')

figure plot(t, x1dotdot) grid; ylabel('xdotdot (In/Sec^2)'); xlabel('Time(Seconds)') legend('xdotdot')

figure plot(t, zdotdot) grid; ylabel('zdotdot (In/Sec^2)'); xlabel('Time(Seconds)') legend('zdotdot')

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Appendix E

Discussion of ODE Solver Selection

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Matlab offers several different versions of both fixed step and adaptive step

ordinary differential equation (ODE) solvers. For this research only adaptive

step solvers were considered as they yield results of far superior accuracy due

to the fact that they interpolate between each input data point. The problem

with using a fixed step size solver is that it is possible to lose points where the

signal frequency is greater than the frequency of the solver. Adaptive step

solvers use a large step size where there is low frequency data and a small step

size when there is high frequency data. Descriptions of the Matlab adaptive

step algorithms that were considered for this problem are as follows:

The ode45 solver is the most common and it is based on an explicit Runge-

Kutta (4,5) formula known as the Dormand-Prince pair. The ode45 is a one

step solver as it only needs one step immediately preceding the time point

(y(tn-1) ) in order to calculate y(tn).

The ode23 solver uses an explicit Runge-Kutta (2,3) pair of Bogacki and

Shampine. Like the ode45, the ode23 solver is a one step solver and only

requires one preceding time step. The advantage of this solver is that it is

more efficient than the ode45 method, however, it sacrifices the accuracy of

the results.

The ode113 solver, unlike the two previously described is a multistep solver.

Instead of simply taking the previous time step and interpolating, this solver

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uses several of the preceding time point to compute the current solution. The

ode113 solver also implements a variable order Adams-Bashforth-Moulton

PECE method. Again, like the ode23 method this is computationally more

efficient than the ode45 solver, and although more accurate than the ode23 it

is not as accurate as the ode45 method.

All three of these adaptive step algorithms interpolate by first taking a step,

then estimating the error at the step, determining if the value is greater than or

less than the tolerance, then the step size is adjusted accordingly.

For this research, the ode45 algorithm was selected due to the fact it would

yield the most accurate results despite the increased calculation run time.

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Appendix F

Drawings for Tension Member Test Fixture

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105

106

107

108

109

110

111

112

113

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Appendix G

Data Collected from Direct Measurement of Metal Tape

Stiffness

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Trial Force in Tape

(lbf)

Displacement

(inches)

Force

Displacement

1 39.8 0.005 7960.0

2 25.78 0.007 3682.9

3 36.02 0.006 6003.3

4 44.023 0.007 6289.0

5 42.49 0.007 6070.0

Appendix Table 1C: Data obtained from direct measurement of steel tape

stiffness.

Based on the data that was obtained in table 1C the average stiffness of the

metal tape was calculated to be 6001 lb/in. The effective stiffness at the

follower roller was then determined to be 29,863 lb/in through the application

of the lever ratio described in section 7.