martin finalreport

8/12/2019 Martin FinalReport

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Multibody Dynamic Analysis of a Planetary Belt-Coupled Linear

Mechanism using RecurDyn

by

Kyle Martin

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING

Major Subject: MECHANICAL ENGINEERING

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

August, 2012


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Copyright 2012

by

Kyle Martin

All Rights Reserved


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CONTENTS

LIST OF TABLES ............................................................................................................ iv

LIST OF FIGURES ........................................................................................................... v

LIST OF SYMBOLS ........................................................................................................ vi

ACKNOWLEDGMENT ................................................................................................. vii

ABSTRACT ................................................................................................................... viii

1. Introduction.................................................................................................................. 1

1.1 Kinematic Interests............................................................................................. 1

1.2 Slider-Crank Mechanism ................................................................................... 1

1.3 Mechanism Kinematics ...................................................................................... 2

1.4 Mechanism Dynamics ........................................................................................ 4

1.5 Previous Analyses .............................................................................................. 6

2. Methodology ................................................................................................................ 7

2.1 Timing Belt Properties ....................................................................................... 7

2.2 Sprocket Geometry............................................................................................. 8

2.3 Mechanism Assembly and Constraints .............................................................. 9

2.4 Mechanism Analysis Steps .............................................................................. 11

3. Results........................................................................................................................ 14

3.1 Analysis versus hand solution .......................................................................... 14

3.2 Effect of timing belt tooth form ....................................................................... 18

3.3 Mechanism Behavior with Different External Loads ...................................... 21

4. Conclusions................................................................................................................ 25

5. References .................................................................................................................. 27


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iv

LIST OF TABLES

Table 1Synchronous Belts Modeled .............................................................................. 7

Table 2Timing Belt Sprocket Parameters Modeled ....................................................... 9

Table 3Summary of Connections in Mechanism......................................................... 11


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v

LIST OF FIGURES

Figure 1Typical Slider-Crank Mechanism .................................................................... 1

Figure 2Generic 3 Body Linkage .................................................................................. 2

Figure 32D Representation of Mechanism .................................................................... 3Figure 4Forces and Moments on Link 3 ........................................................................ 4

Figure 5Forces and Moments on the Planet Sprocket ................................................... 5

Figure 6Forces and Moments Acting on Link 2 ............................................................ 6

Figure 7Typical Timing Belt Geometry ........................................................................ 7

Figure 8Typical Timing Belt Sprocket Geometry Parameters ...................................... 8

Figure 9Isometric View of the Mechanism as Assembled .......................................... 10

Figure 10Motion Profile of Node B............................................................................. 14

Figure 11Lag of Link Arm Relative to Drive Arm Position ....................................... 15

Figure 12Tension of Tight Side of Timing Belt Relative to Drive Arm Position ....... 16

Figure 13Sun Sprocket Reaction Torque during One Cycle ....................................... 17

Figure 14Torque Input to the Mechanism ................................................................... 17

Figure 15Backlash in Different Tooth Profiles ........................................................... 19

Figure 16Drive Synchronization - Selected Profiles ................................................... 20

Figure 17Torque Transmitted Through Torque Shaft - Selected Profiles ................... 21

Figure 18Arm Synchronization for Studied Loads ...................................................... 22

Figure 19Mechanism Performance Metrics for Studied Loads ................................... 23

Figure 20Torque Shaft Torque for Studied Loads ....................................................... 24


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LIST OF SYMBOLS

Symbol Units Description

mm Length of carrier arm mm Length of link arm

Degrees Angle of carrier arm relative to ground Degrees Angle of link arm relative to ground Teeth Tooth count of Sun Sprocket

Teeth Tooth count of Planet sprocket Nmm Reaction torque of planet sprocket Nmm Driving torque of mechanism

N External load applied to mechanism N Reaction of carrier arm on link arm N Tight-side tension in timing belt N Reaction of carrier arm on planet sprocket


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ACKNOWLEDGMENT

I owe a debt of gratitude to Victoria Thai for providing the necessary motivation and

encouragement to compel me to complete this project. I would also like to thank

Ernesto Gutierrez-Miravete for his support on this project. Finally, I would also like tothank my family for providing lifelong support and encouragement, without which I

would not be where I am today.


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viii

ABSTRACT

This document presents a study of the dynamics of a mechanism used to convert

continuous rotational input motion into reciprocating straight line output motion. This is

accomplished using a planetary arrangement of a timing belt drive and one additionallink body. The dynamics of the mechanism are studied to understand the influence of

specific elements of the mechanism and external loading.

The derivation of the ideal case is presented and used as a basis for comparison with

the simulation model created in RecurDyn. The procedure for constructing the

analytical model using RecurDyn is also detailed. Specifically, the following aspects are

considered in the analysis: the ideal case versus the baseline simulation, the influence of

the timing belt tooth profile and the effect of the magnitude of the external load on

mechanism performance. Further possible studies are proposed as well as conclusions

based on the overall modeling process and the outcome of the analyses.


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1. Introduction1.1 Kinematic Interests

The conversion of motion between rotation and translation has been a popular topic

in kinematics for some time. This is due in large part to its widespread use in converting

rotating sources of power into linear motion or vice versa. A large majority of devices

that use a rotary motor to convert electricity into motion will use some form of

kinematics to generate the desired output motion.

A second popular topic in the study of kinematics is the generation of straight line

motion using combinations of links and joints. Numerous linkages have been developed

to generate a straight line based on specific kinematic constructions. Some mechanisms

will generate an approximate straight line in theory, such as the Watt linkage whileothers will generate a theoretically exact straight line, such as the Peaucellier

mechanism.1

1.2 Slider-Crank MechanismTraditionally, conversion from rotation to linear motion has been accomplished with

a slider-crank mechanism as shown inFigure 1

2

O

3

A

Figure 1 Typical Slider-Crank Mechanism

The slider-crank mechanism consists of two link elements; link 2 and 3. The

uncoupled end of link 3, node A is also guided along a straight line by some external

means such as rollers within a track, a piston within a cylinder or some sort of linear

bearing. In practice, link 3 is made longer than link 2 to avoid any toggling of the

linkage. In this form, the linkage has a single degree of freedom. Motion is input as

1(Mabie & Reinholtz, 1987)


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rotation of link 2 about point O and the corresponding output is the displacement of node

A along its guided path. While this mechanism is simple, the performance of the

mechanism is degraded due to the external guidance. The resulting reaction force acting

on the guidance introduces friction opposite the motion. In addition, the motion profile

of node A can likely be undesirable due to the kinematics of the mechanism.

1.3 Mechanism KinematicsFigure 2 shows a simple 2-D linkage where link 2 is connected to ground at node O

with a revolute joint, and link 3 is connected to link 2 at node A. The kinematics of any

point in the linkage can be solved with simple trigonometry.2

y

x

2

O

A

3

2131

B

Figure 2 Generic 3 Body Linkage

The displacement of node B from the origin in the X and Y direction can be found

with equations 1 and 2

[1] [2]

The velocity of node B relative to the origin can be found by taking the time

derivative of the above equations. The acceleration can also be found by again taking

the time derivative of the velocity equations. As can be seen from the above equations,

the linkage has two degrees of freedom as the motion of node B is a function of angles

and . Since the objective of the linkage is to transform a single input motion,rotation of link 2, to a single output motion, translation of node B, the mechanism must

be further constrained to a single degree of freedom before it can perform its function.

2(Mabie & Reinholtz, 1987)


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To reduce the linkage to a single degree of freedom, link 3 is coupled to a timing

sprocket (body 4) at node A. A second timing sprocket is located centered about the

origin and coupled to ground. Those sprockets are then coupled with a timing belt and

the resulting mechanism is shown inFigure 3

2

O

B21

31

A

4

3

Figure 3 2D Representation of Mechanism

With the addition of these elements, the relative rotation of link 2 and link 3 is now

given by the following relation

[3]

If the sprockets are chosen such that

, the relation between the two angles

becomes

[4]

Finally, if the mechanism was constructed such that link 2 and link 3 are equal length

and , the position of node B is given by [5] [6]

As is an even function and is an odd function, the equations further simplify inthis case to


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[7] [8]

As can be seen from the above equations, rotational input motion, , results in atranslational output motion,

, in the X direction only. Consequently, the velocity

and acceleration of node B are solely in the X direction and can be found by taking the

first and second time derivative of equation 7

[9]

[10]

1.4 Mechanism DynamicsAs the purpose of the mechanism being considered is to perform work along the

translation axis, it is necessary to include an external force acting upon the linkage. In

the case of the mechanism of interest, the force acts on node B in the X direction, as

shown inFigure 4

B

31

A

3

Fext

F32

M34

Figure 4 Forces and Moments on Link 3

The hand calculations will assume that the linkage is mass-less and without friction.The applied external force,is resisted by a force acting on element 3 from element 2,and a moment acting on element 3 from the planet sprocket,

[11] [12]


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The planet sprocket sees the equal and opposite moment acting on it from link 4,

, which is resisted by tension in the timing belt, . The timing belt is connected tothe sun sprocket and ensures that there is no relative slip of the two sprockets due to its

positive engagement. As the timing belt is a tension member, it cannot develop any

compressive loads. Neglecting any pre-tension of the timing belt and assuming isalways positive, then will only act on one side of the timing belt span so as to resistthe imparted moment. The other span will be un-tensioned, and is shown in gray in

Figure 5.

A

M43

F41

F41

rp,planet

42

F42

41

Figure 5 Forces and Moments on the Planet Sprocket

The tension in the timing belt acts at a radius from the center of the sprocket defined

as the pitch radius of the planet sprocket, which is a function of the sprocketpitch and the number of teeth on the sprocket,

[13]

The magnitude of the tension in the timing belt acting opposite can be found [14]

The angle at which the tension acts is determined by the distance between the two

sprockets and their respective pitch counts

[15]

The angle of tension vector relative to ground can then be found

[16]


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The X and Y components of this tension are resisted by the X and Y components of .Finally, by analyzing the forces and moments acting on link 2 it is then possible to

determine the input torque, necessary to overcome the externally applied forces asshown inFigure 6.

2

O

21

A

F24

F23

Tin

F21

Figure 6 Forces and Moments Acting on Link 2

The input torque is found as the opposite of the sum of the moments resulting from and

[17]

1.5 Previous AnalysesThis specific mechanism described above has been analyzed in the past, with more

advanced techniques, accounting for the elasticity of the timing belt and using

differential equations.3 This analysis was exhaustive within the assumptions considered,

however one of those assumptions was to neglect the non-ideal nature of the contact

between the timing belt teeth and the sprockets of the mechanism. Previous efforts have

demonstrated that RecurDyn is well suited for the analysis of both multi-body

mechanisms, as well as mechanisms that employ timing belts and sprockets.4 In this

particular analysis, RecurDyn is employed specifically for its ability to accurately model

the interface between the timing belt and sprocket.

3(Al-Dwairi & Al-Lubani, 2007)4(Zhanguo, Jiangming, & Jiaxing, 2010)


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2. Methodology2.1 Timing Belt Properties

In order to model the system in RecurDyn, it was necessary to accurately define the

geometry and properties of the timing belt used to synchronize rotation of the planet

sprocket about the sun sprocket. The timing belts and sprockets used in the analysis

were standard, commercially available designs. Figure 7 shows the typical timing belt

geometry parameters used to define a belt in RecurDyn and Table 1 lists the values of

those dimensions for the different timing belt profiles considered during the analysis.

A A

Hr H

s a

Ra

Rr

SPb

Figure 7 Typical Timing Belt Geometry

Table 1 Synchronous Belts Modeled5

Symbol T20 AT20 XH Units

A 20 25 20

Hr 5 5 6.35 Mm

Hs 8.0 8 11.2 Mm

Pb 20 20 22.225 Mm

S 10.15 15.1 12.57 Mm

Rr 0.8 2.5 1.6 Mm

Ra 0.8 1.6 1.2 Mm

a 1.5 1.2 1.4 Mm

W 50 Mm

CSP 37410 60692 37410 N/mmCp 93525 151732 84161 N/mm

In addition to the geometry, it was also necessary to define the mechanical behavior

of each belt. Table 1 lists the tensile stiffness properties of each timing belt. The

5(SIT Spa, 2003), (Gates Mectrol GmbH, 2008)


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specific stiffness of the T profile belts were found from a commercial catalog6, while the

AT, X and XH properties were found in a separate catalog7. All belts were modeled

using the stiffness values for steel cords. Specific stiffness, CSPis defined as the tensile

spring constant of a unit long, unit wide segment of belt8. This number is typically only

valid for belts being used inside of their working tension range as belts typically

exhibit non linear stress-strain behavior at extremely high or low tensions. As the

tensions observed in simulations were generally within the working range of the belts,

a linear elastic behavior assumption was reasonable. It was necessary to define the

stiffness of each tooth element , according to the following [18]

This was necessary because RecurDyn constructs a timing belt group by discretizing

a complete belt into individual tooth segments, each with a length equal one pitch of the

profile.

2.2 Sprocket GeometrySimilarly, it was necessary to define the sprocket geometry according to accepted

standards and in a form suitable for input to RecurDyn. Figure 8 shows the relevant

standard dimensions used to define a timing belt pulley in RecurDyn.

Bg

Do

R2 Hr

R1

A

Figure 8 Typical Timing Belt Sprocket Geometry Parameters

6(F.N. Sheppard & Co., 2000)7(Gates Mectrol GmbH, 2008)8(Gates Mectrol, Inc., 2006)


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Table 2 lists parameters used for each different timing belt profile. Timing belts and

sprockets are intended to be used as a matched set so all analyses were done with

properly matched sprockets. The T20 belt profile did have two possible tooth forms: the

standard tooth form is known as T20, while the reduced backlash form is referred to as

T20SE. Both were analyzed and the effects are examined later in this document.

Table 2 Timing Belt Sprocket Parameters Modeled

Symbol T20 T20SE AT20 XH Units

A 25 25 20

Hr 6.0 5.2 4.65 6.88 mm

Bg 7.0 6.8 10.51 7.59 mm

R1 0.8 1.5 2.01 mm

R2 1.2 2.5 1.93 mm

Do, planet 111.75 111.7 117.47 mm

Do, sun 226.35 226.3 237.74 mm

Nplanet 18 18 17 -

Nsun 36 36 34 -

2.3 Mechanism Assembly and ConstraintsWith the timing belt and sprocket geometry defined, constructing the mechanism

from these components along with other standard kinematic elements was the next step.

The mechanism was constructed from eight individual bodies, a timing belt group and

nine joints in RecurDyn, the individual bodies of the mechanism are arranged in three

planar groups. The timing belt, (1) sun sprocket, (2) tensioner roller, (3) tensioner shaft

and (4) planet sprockets are all in the sprocketplanes. The (5) drive arm is the only

item solely in the drive planeand the (6) link arm is the only body solely in the link

plane. The (7) torque shaft passes through all three planes; it serves to connect the link

arm to the planet sprocket. The drive arm acts as the carrier for the torque shaft,

providing axial and radial support, while allowing tangential rotation.

In the sprocket plane, the first body of the assembly was the sun sprocket which was

fixed to the ground body in all six degrees of freedom, as it is fixed in the practical

application. The planet sprocket was assembled at a radial distance from the axis of the

sun sprocket equal to the drive arm length,. The torque shaft was assembled coaxialwith the planet sprocket and was rigidly connected to the planet sprocket. The tensioner

shaft was then located such that it would be at the approximate middle of the slack side


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span of the timing belt. The tensioner roller was assembled coaxially with the tensioner

shaft in the sprocket plane using a revolute joint, representing the tensioner being fixed

to the shaft with rolling bearings. Lastly, the timing belt group was built around these

three bodies to complete the system.Figure 9 shows the completed assembly.

Figure 9 Isometric View of the Mechanism as Assembled

In the drive plane, the drive arm was represented by link element of length andlocated with the primary axis of the link passing through the sun sprocket axis and the

secondary axis passing through the planet sprocket axis. The arm was fixed to ground

using a revolute joint about the primary axis. The tensioner shaft described above was

6.) Link Arm

(Link 3)

7.) Torque Shaft

5.) Drive Arm

(Link 2)

2.) Tensioner Roller

1.) Sun Sprocket

4.) Planet Sprocket

3.) Tensioner Shaft

B

O

A

Tension Sensor Location


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fixed to the drive arm using a translational joint perpendicular to the length of the drive

arm, which represents the direction in which the tensioner assembly moves to apply pre-

load to the timing belt. The torque shaft also described above was fixed to the secondary

axis of the drive arm using a revolute joint.

And finally in the link plane, the link arm was represented by link element of length

, it was assembled with the primary axis aligned with the secondary axis of the drivearm and the secondary axis aligned with the primary axis of the drive arm. The arm was

rigidly connected to the torque shaft. Table 3 summarizes the connections of the

mechanism, as well as the directions in which the components are either constrained

(), or free to move ().

Table 3 Summary of Connections in Mechanism

Connection Body Fixed to Joint Target Csys

X Y Z

1 Drive Arm Ground Revolute

2 Tension Roller Tension Shaft Revolute

3 Tension Shaft Drive Arm Translational

4 Link Arm Torque Shaft Revolute 5 Sun Sprocket Ground Fixed

6 Torque Shaft Drive Arm Revolute

7 Planet Sprocket Torque Shaft Fixed

2.4 Mechanism Analysis StepsIn order to develop a stable solution, it was necessary to run the model in two steps.

The purpose of the first step was to bring the mechanism to equilibrium and establish

proper alignment of the components once the timing belt was pre-tensioned and seated

on each sprocket. The second step of the analysis was to simulate regular operation of

the mechanism where the external load was applied and the drive arm was rotated.

The first step of the analysis was important for several reasons. First, it allowed allcomponents to settle in their positions as gravity is abruptly applied at time t=0 in

RecurDyn. Second, during assembly the timing belt must be constructed from an integer

number of teeth elements at a pitch as defined inTable 2,constraining the pitch length of

the overall belt group to increments equal to one pitch length of the belt. At the same


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time, the required pitch length of the sprockets changed based on the dimensions of the

sprocket teeth.

The analysis had to be conducted with the distance between the sun and planet

sprockets held constant throughout all cases of the study. Therefore, it was not possible

to get an exact match between the pitch length of the belt group and required pitch

length on the sprockets, while different teeth dimensions on both the sprocket and timing

belt were studied. Finding a one-size-fits-all center dimension to accommodate all

possible pitch lengths was not possible in this case. For similar reasons, as well as to aid

in assembly of the mechanism, a tensioner roller is used in the practical application of

the mechanism to take up the excessive pitch length of the belt and keep the teeth of the

belt properly engaged in the groves. In the first step of the analysis, a force was applied

to the tensioner shaft along the direction of the translational joint, fixing it to the drive

arm. This force acted through the revolute joint between the tensioner shaft and the

tensioner roller and finally onto the timing belt group. As the sun sprocket was fixed in

all directions, while the planet sprocket was free to rotate about the Z-axis, the link arm

would become misaligned from its starting position relative to the drive arm as a result

of this application of tension.

While applying this tension it was also necessary to apply a fixed external load at

node B. Where the tensioner roller force was used to seat the teeth in the grooves, this

second force was used to impart a torque on the planet sprocket and bring the flanks of

the timing belt teeth into contact with the flanks of the sprockets. The load was equal in

magnitude and direction to the load expected for the particular input angle of arm. Now

that the teeth were properly seated as expected, the final element of the first step of the

analysis was to realign the link arm to the drive arm before the mechanism was run in

step 2.

In step 2, relative rotation between the link arm and the torque shaft was prescribed

as zero in the input motion field for the revolute joint, fixing these 2 bodies together in

all directions. An external force was then applied to the link arm. The force was defined

as a fixed magnitude, acting in a direction along the X-axis, opposite to the X direction

component of the motion of node B. The drive arm was then rotated about the revolute

joint passing through the central axis of the sun sprocket. The rotation of the drive arm


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was prescribed at the joint as a gradual acceleration up to a constant speed. The portion

of the analysis with constant speed rotation of the input arm was the main focus in

analyzing the results and drawing conclusions.


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3. Results3.1 Analysis versus hand solution

For the baseline result, the T20 profile timing belt was analyzed with the standard

T20 sprocket profile. The magnitude of the external applied force was 200N and

pretension of the timing belt was fixed at 1250N. All joints were modeled without

friction.

As can be seen in Figure 10,aspects of the simulation results vary from the ideal

case of the hand calculations presented earlier. The significant difference is in the Y

direction displacement of node B. The simple hand calculation predicts no vertical

displacement while the model shows a small displacement of node B in the Y direction.

As the simulation includes many more factors than the ideal case presented in the handcalculations, this result is not entirely surprising.

Figure 10 Motion Profile of Node B

Looking further at the two link elements of the mechanism it becomes evident that

the synchronization is certainly not ideal between the two of them. The shape of the plot

in Figure 11 suggests that there are multiple components contributing to this

misalignment.

-5

-4

-3

-2

-1

0

1

2

-550 -450 -350 -250 -150 -50 50 150 250 350 450 550

NodeBYDisplace

ment[mm]

Node B X Displacement [mm]

Displacement of Node B Over One Cycle

Simulation Results

Hand Calculations


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Figure 11 Lag of Link Arm Relative to Drive Arm Position

The small periodic element of the lag is due to the meshing of the timing belt teeth

onto the sprockets. The red vertical lines in Figure 11 are spaced at 10 degree

increments and correspond with peak of each of these small variations. The 10 degree

increment is a result of the number of teeth on the sun sprocket. In this case 36 teeth are

engaged over 360 degrees of input motion, resulting in one mesh event every 10 degrees.

The large periodic element of the misalignment is due to the elastic behavior of the

timing belt, as its tension changes with position of the mechanism as is shown inFigure

12. Finally, there is an offset error of the alignment, where the absolute misalignment

ranges between -1 and 0.5 degrees. This is likely attributable to a static misalignment of

approximately 0.25 degrees.

Figure 12 shows the tension of the timing belt on the tight side, the side which

tension is developed in reaction to the applied load as the mechanism operates. The

location within the span where the tension is measured can be seen in the Figure 9.

Again, the engagement of each tooth of the timing belt is evident in this plot, with 36

distinct peaks showing up, corresponding to the number of teeth engaged over one

revolution of the mechanism. The peaks appear to shift the tension above the values

determined in the hand calculation. This would suggest that the tension peaks are

-1.5

-1

-0.5

0

0.5

1

0 90 180 270 360

LinkArmL

ag[degree

s]

Drive Arm Position [degrees]

Link Arm Lag vs. Drive Arm Position

Simulation Results

Hand Calculations


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additive to the theoretical tension that is required to maintain the kinematics of the

mechanism.

Figure 12 Tension of Tight Side of Timing Belt Relative to Drive Arm Position

A further area of interest is in the torque transmitted through the planet sprocket as

well as the torque required to be input to the mechanism to generate the constant speed

motion at the specified conditions.Figure 13 shows the torque transmitted through the

planet sprocket, which is a direct result of the tight side tension acting on the sprocket

through the pitch radius of the sprocket. Like the previous plots, the effects of the

engagement of each of the timing belt teeth can be readily observed while the overall

plot closely resembles the hand calculation in magnitude and shape. However, unlike

the previous plots, the variation in the torque begins to display random non-periodic

noise, which can be seen specifically around drive arm positions of 75 and 135

degrees.

0

500

1000

1500

2000

2500

0 90 180 270 360

TimingBeltTension[N]

Drive Arm Position [deg]

Timing Belt Tension vs. Drive Arm Position

Simulation ResultsHand Calculations


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Figure 13 Sun Sprocket Reaction Torque during One Cycle

Coming to the torque input in Figure 14,we see similar noise as with the sprocket

torque inFigure 13,however there are many more locations of the drive arm showing

this noise.

Figure 14 Torque Input to the Mechanism

-80000

-70000

-60000

-50000

-40000

-30000

-20000

-10000

0

10000

0 90 180 270 360PlanetSprocketReactionTorque[Nmm]


Planet Sprocket Reaction Torque vs. Drive Arm Position

Simulation Results

Hand Calculations

-100000

-50000

0

50000

100000

150000

200000

250000

0 90 180 270 360

In

putDrivingTorque[Nmm]


Input Torque vs. Drive Arm Position

Simulation ResultsHand Calculations


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The amplitude of the noise element appears greater while the shape and magnitude of the

overall plot agree with the hand calculation results. The source of this noise is not clear;

it was examined as other suspect parameters were varied but none of the parameters

varied was found to control it.

3.2 Effect of timing belt tooth formThere are many potential reasons for discrepancies between the hand calculation

and the results obtained from the simulation. One particular difference is the

characterization of the timing belt. In the hand calculation, the timing belt is assumed to

be inelastic, frictionless and without back lash, while in the simulation all of these things

are modeled to yield the final results. In the real implementation of this mechanism, all

of these things are present, so the simulation results will be closer to reality than the

hand calculations.

The teeth of timing belts are, generally speaking, smaller than the grooves on the

sprocket in which they sit. This is necessary for proper functioning of the system since

there should be no interference between the tip of the belt tooth and the flank of the

sprocket teeth as the belt comes engaged, conforming to the sprocket.

These things were considered when formulating standards for tooth profiles. Backlash is

the direct consequence of the fact that the fit of the timing belt tooth and the sprocket

groove cannot be line-to-line. The amount of displacement to move contact from one

flank to the other flank of the mating pair is the direct realization of backlash.

Timing belt designers have tried various steps to reduce the amount of backlash for

a given tooth profile standard. In the case of the T20 profile, the SE profile is used on

timing sprockets to reduce the amount of backlash in the belt/sprocket system. Figure 15

shows the difference between the normal metric T timing profile and the SE version of

the profile. The sprocket dimensions are modified to reduce the amount of backlash in

the teeth.


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Figure 15 Backlash in Different Tooth Profiles9

As part of the analysis, the geometry of the timing sprocket grooves were

constructed according to the T20SE profile, while all other simulation parameters were

held constant. In order to quantify the amount of backlash in the mechanism, the relative

position of the carrier arm (link 2) was compared to the position of the link arm (link 3)

throughout one revolution of the mechanism. As can be seen in Figure 16,the T20SE

profile did demonstrate better synchronization than the standard T20 profile, but not by alarge margin. Other profiles were also analyzed, including the AT20 profile and the XH

profile. The AT profile was developed as a refinement of the T profile specifically to

reduce backlash and allow use of a stiffer tension member for the same pitch.10

Referring again toFigure 16,the AT profile does improve upon the performance of both

the standard T profile, as well as the SE version of the T profile. The XH profile was

also analyzed and exhibited better alignment than both variations of the T profile as well.

9(ContiTech Antriebssysteme GmbH, 2008)10(ContiTech Antriebssysteme GmbH, 2008)


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Figure 16 Drive Synchronization - Selected Profiles

Other results from the simulation were essentially unaffected by changes to the belt

and sprocket geometry. Looking at the torque transmitted through the timing shaft to

maintain alignment of the mechanism, shown inFigure 17,there is little variation in the

average torque transmitted. The comparison was based on a 10-point rolling average as

shown, since the noise element of each simulation result made for a congested figure.

Each of the tooth profiles seemed to exhibit similar amplitude of noise based on the

scatter of the data points around the moving average lines. This similarity in basic

performance is not surprising since the fundamental operation of the mechanism remains

unchanged.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 90 180 270 360

LinkArmL

ag[degrees]


Link Arm Lag vs. Drive Arm PositionSelected Tooth Forms

AT20

XH

T20

T20SE


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21

Figure 17 Torque Transmitted Through Torque Shaft - Selected Profiles

3.3 Mechanism Behavior with Different External LoadsHaving demonstrated that modeling the mechanism with RecurDyn could produce

accurate results when compared to the hand calculations with the same mechanism

geometry and external load; the next step was to examine the effect of the magnitude of

the external load on the performance of the mechanism. For this analysis a T20 timing

belt was used with sprockets constructed according to the T20SE tooth form. All other

simulation variables were kept constant.

As with the previous study of different timing belt tooth shapes, the synchronization

accuracy of the timing belt was examined to understand how well the timing belt system

would respond to different external loads. Figure 18 shows the synchronization of the

mechanism with forces applied at node B ranging from zero load, up to 750N.

-10000

0

10000

20000

30000

40000

50000

60000

70000

80000

0 90 180 270 360

TorqueShaftTorque[Nmm

]


Torque Shaft Torque for Selected Teeth Profiles10 Point Moving Average

AT20

XH

T20

T20SE


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Figure 18 Arm Synchronization for Studied Loads

The tooth mesh events can be seen distinctly in each case, occurring with the same

frequency in all cases. This is expected since the number of teeth on the sprockets and

rotational speed of the input arm was constant in all cases. Overall, increasing the load

appears to produce a uniform change in the synchronization of the mechanism. The link

arm lags the driver arm throughout the full range of motion when no external load is

applied. However, the link arm begins to lead the driving arm as load is applied. In all

cases, the mechanism is well synchronized when the input arm is at 90 and 270 degrees,

while the synchronization is worst at 0, 180 and 360 degrees.

Another way to examine the synchronization is to compare the root mean squared

(RMS) value of the link arm synchronization over a full revolution of the input. The

RMS value of the Y displacement of Node B was also calculated and examined as a

second metric of mechanism performance. With both metrics, a smaller value indicatedbetter performance in generating a straight line. Figure 19 shows the comparison of

these metrics for various external loads.

-1.5

-1

-0.5

0

0.5

1

1.5

0 90 180 270 360

LinkArmL

ag[degrees]


Link Arm Lag vs. Drive Arm PositionVaried Loading

P=0

P=100

P=150

P=200

P=500

P=750


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Figure 19 Mechanism Performance Metrics for Studied Loads

In general, it seems that the higher the RMS value of link arm lag, the higher the

RMS value of the Y displacement of node B. Interestingly, the 150N external load case

appeared to perform the best from the Y displacement standpoint. Whereas the 100N

case was slightly better from the synchronization standpoint. The no-load case

performed significantly worse than these two cases, on par with the 500N case for

synchronization and worse than the 750N case for Y displacement. The no-load case is

possibly due to the lack of a persistent torque on the timing belt mechanism. In the

previous analysis of tooth forms, Figure 17 showed that the torque transmitted through

the torque shaft did vary over one cycle of the mechanism. However, the torque was

always positive. As discussed previously, a torque was necessary to bring the flanks of

the timing belt teeth into contact with the flanks of the teeth in the timing sprockets.

Figure 20 shows that in the no-load case, the torque through the timing belt wasoscillating between 10,000 Nmm and -20,000 Nmm while an external load of just 100N

was enough to bring the torque up to a positive value through the complete cycle.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 100 200 300 400 500 600 700 800

YDisplacementofNodeB[mmRMS]

LinkArmL

ag[DegRMS]

External Force Applied [N]

Mechanism Performance MetricsVaried Loading

Link Arm Lag [Deg RMS]

Y Displacement of Node B [mm RMS]


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Figure 20 Torque Shaft Torque for Studied Loads

The degradation in performance as the external load was increased was expected

since it would require a greater torque to be transmitted by the timing belt system to

maintain alignment of the mechanism, as can be seen in Figure 20. This increased

torque led to increased belt tension, which increased elongation of the belt and adversely

affected the synchronization of the system. Along with timing belt tension, system

variables such as input torque, torque shaft revolute joint reaction forces and sun

sprocket reaction moment all increased proportionally to the magnitude of the applied

external load. This was demonstrated by the stability of the subject mechanism over a

range of input loads.

-50000

0

50000

100000

150000

200000

250000

0 90 180 270 360

TorqueShaftTorque[Nmm

]


Torque Shaft Torque vs. Drive Arm PositionVaried loading

P=0

P=100

P=150

P=200

P=500

P=750


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4. ConclusionsThe results attained from the RecurDyn simulations compare favorably to those

determined with the hand calculations. Furthermore, the performance of the mechanism

responded proportionally and repeatably to the magnitude of the external load applied.While corresponding measurements from the physical model of the mechanism

were not available from comparison, general observations of the operation, particularly

with regard to the motion of node B, show similar characteristics as those in the

RecurDyn model. A potential future extension of this analysis could include taking

physical measurements of performance metrics of the actual mechanism and comparing

them to the RecurDyn model.

The RecurDyn simulation software was very well suited for this task. The

RecurDyn software is extremely robust and powerful, providing detailed output and also

allowing great flexibility in formulating the model. The built-in timing belt modeling

functionality made constructing the mechanism relatively straight forward. Once the

mechanism was constructed it was a bit more challenging to configure the model to

deliver a reasonable result. While simple bodies like arms and shafts were easily

modeled, the more complex bodies required research and experimentation to determine

reasonable characteristic values.

A prime example of this difficultly was in characterizing the timing belt tensile

stiffness. When creating a new timing belt system in RecurDyn the belt stiffness and

damping properties are assigned default values, rather than prompting for entry by the

user. The tensile stiffness default value is 100N/mm, which is two orders of magnitude

less than the published values for the timing belts being modeled. During the initial

modeling phase of this project where feasibility was assessed, this default value caused

instability in solving the simulation. The input for tensile stiffness was found after

further research in the RecurDyn software. Once rectified, similar challenges with

damping and roller compliance were found and addressed to achieve a more stable and

accurate solution.

Based on the trends observed in the studies presented in this paper it would be

possible to achieve further improvement in the performance of the mechanism. This can

be done by improving one or more of the attributes of the timing belt. A belt with higher


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26

tensile stiffness would decrease the elongation of the timing belt that results from the

external load and therefore maintain better synchronization accuracy. This could be

achieved with either a stiffer tension member, or simply a wider belt. Using a newer

involute in place timing belt profile, such as the STD profile11

, in place of the trapezoidal

profile would also improve synchronization accuracy due to reduced backlash.

Even without the improvements suggested above the subject mechanism appears to

perform very well in translating node B in an approximate straight line without any

external guidance. In the case of the varied external loading, the RMS Y direction

displacement of Node B was found to range from a minimum of 0.2mm to a maximum

of 1mm over the analyzed load range of no load to 750N. Considering the total X

direction travel of Node B was 1070mm for each of the cases analyzed, this is excellent

performance.

11(Perneder & Osborne, 2012)


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5. References1.) Al-Dwairi, A. F., & Al-Lubani, S. E. (2007). Modeling and Dynamic Analysis of a

Planetary Mechanism with an Elastic Belt. Mechanism and Machine Theory, 39, 343-

355.

2.) ContiTech Antriebssysteme GmbH. (2008, December). Conti Sychroflex Timing

Belts: Overall Catalog.Retrieved May 2, 2012, from ContiTech Antriebssysteme GmbH

Web site: http://www.contitech.de/pages/produkte/antriebsriemen/antrieb-

industrie/download/td_synchroflex_en.pdf

3.) F.N. Sheppard & Co. (2000). Belt Design Catalog. Erlanger, Kentucky.

4.) Gates Mectrol GmbH. (2008, May). Catalogue; Polyurethane Timing Belts.

Retrieved May 12, 2012, from Gates Mectrol Corporation Web site:

http://www.gatesmectrol.com/mectrol/downloads/download_common.cfm?file=GatesM

ectrol_Belt_Pulley_br_5_08.pdf&folder=brochure

5.) Gates Mectrol, Inc. (2006, October). Timing Belt Theory. Retrieved May 4, 2012,

from Gates Mectrol Corporation Web site:

http://www.gatesmectrol.com/mectrol/downloads/download_common.cfm?file=Belt_Th

eory06sm.pdf&folder=brochure

6.) Mabie, H. H., & Reinholtz, C. F. (1987). Mechanisms and Dynamics of Machinery.

New York: John Riley and Sons.

7.)Perneder, R., & Osborne, I. (2012). Handbook Timing Belts. Berlin Heidelberg:

Springer-Verlag.

8.) SIT Spa. (2003, September). Trasmissioni a cinghia dentata. Retrieved May 14,

2012, from SIT Spa Web site: http://www.sitspa.it/94.pdf


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