DYNAMIC MODELING OF HIGH PRECISION SERVO SYSTEMS WITH GEARBACKLASH

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

ZAFER YUMRUKÇAL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

MECHANICAL ENGINEERING

AUGUST 2013

Approval of the thesis:

DYNAMIC MODELING OF HIGH PRECISION SERVO SYSTEMS WITH GEARBACKLASH

submitted by ZAFER YUMRUKÇAL in partial fulfillment of the requirements for the degreeof Master of Science in Mechanical Engineering Department, Middle East TechnicalUniversity by,

Prof. Dr. Canan ÖZGENDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Süha ORALHead of Department, Mechanical Engineering

Prof. Dr. Eres SÖYLEMEZSupervisor, Mechanical Engineering Department, METU

Examining Committee Members:

Prof. Dr. Y. Samim ÜNLÜSOYMechanical Engineering Department, METU

Prof. Dr. Eres SÖYLEMEZMechanical Engineering Department, METU

Prof. Dr. Kemal IDERMechanical Engineering Department, METU

Assist. Prof. Dr. Yigit YAZICIOGLUMechanical Engineering Department, METU

Dr. Murat GÜLTEKINDefense Systems Technologies Division, ASELSAN

Date:

I hereby declare that all information in this document has been obtained and presentedin accordance with academic rules and ethical conduct. I also declare that, as requiredby these rules and conduct, I have fully cited and referenced all material and results thatare not original to this work.

Name, Last Name: ZAFER YUMRUKÇAL

Signature :

iv

ABSTRACT

DYNAMIC MODELING OF HIGH PRECISION SERVO SYSTEMS WITH GEARBACKLASH

Yumrukçal, Zafer

M.S., Department of Mechanical Engineering

Supervisor : Prof. Dr. Eres SÖYLEMEZ

August 2013, 70 pages

Although the mechanical solutions, like spring preloaded hinge mechanism, compensate forbacklash in servo drive systems with high precision positioning requirement, these solutionsmay not be preferred because of their drawbacks like high friction forces/torques and dimin-ished life cycles for mating gears. Besides, the use regular gear meshes could be enforced byany reason, like cost efficiency or simplicity of the solution. On such occasions, to maintainperformance requirements like high positioning accuracy, alternative control methods can beused with regular gear meshes. For the purpose of developing a controller, it is important todevelop a model for the physical system. This thesis work aims to fulfill the requirement fora dynamic model of a servo system with gear backlash which is to be used as a base for thedevelopment of a controller.

In the scope of this thesis work, firstly, a literature survey has been conducted to examine theapproaches for identifying and modeling a dynamic system with gear backlash. Followingthe literature survey, a real servo system with gear backlash has been identified and a dynamicmodel of the system has been developed. Simulation results from the system model comparedwith the test results obtained from physical system and validity of servo system model withvarying backlash limits has been presented.

Keywords: Gear, Backlash, Varying Backlash, Dynamic System Model, System Identification

v

ÖZ

DISLI BOSLUGUNA SAHIP YÜKSEK HASSASIYETLI SERVO SISTEMLERINDINAMIK MODELLENMESI

Yumrukçal, Zafer

Yüksek Lisans, Makina Mühendisligi Bölümü

Tez Yöneticisi : Prof. Dr. Eres SÖYLEMEZ

Agustos 2013 , 70 sayfa

Hassas konumlama gereksinimi bulunduran servo sistemlerde disli boslugu her ne kadar yaylıbaskı mekanizması gibi mekanik önlemlerle giderilebilse de bu tekniklerin yüksek sürtünmekuvvetleri/torkları ve düsük disli ömürleri gibi sakıncaları bulundugundan tercih edilmeyebi-lirler. Dahası, çözümün basitligi ya da maliyet avantajı gibi sebeplerle disli bosluguna sahipsıradan disli çiftlerinin kullanımı zorunlu kılınmıs olabilir. Bu gibi durumlarda disli boslu-gunun, konumlama hassasiyetini düsürmesi gibi olumsuz etkilerinden kurtulmak için sıradandisli çiftleri kullanımı ile alternatif kontrol çözümlerinin uygunlanması mümkündür. Bu tezçalısması kontrolcu tasarım çalısmalarına temel olusturacak nitelikte, disli bosluguna sahipdinamik servo sistem modeli gereksinimini karsılamayı amaçlamaktadır.

Tez çalısması kapsamında, ilk asamada, disli bosluklu sistemlerin tanımlanması ve model-lenmesi hakkında literatürde yer alan yaklasımlar incelenmistir. Literatür taramasını takibendisli bosluguna sahip gerçek bir servo sistem tanımlanmıs ve sisteme ait dinamik model olus-turulmustur. Çalısma sonucunda sunulan model ile elde edilen simulasyon çıktısı veri, fizikselsistem üzerinde yapılan test sonuçları ile kıyaslanarak disli boslugu limitleri degisken servosistem modelinin geçerliligi gösterilmistir.

Anahtar Kelimeler: Disli, Bosluk, Degisken Disli Boslugu, Dinamik Sistem Modeli, Sistem

Tanımlama

vi

To those who can smile expecting nothing in return. . .

vii

ACKNOWLEDGMENTS

I am grateful to my supervisor, Prof. Eres SÖYLEMEZ, for his aid and guidance through-

out my study. I am happy for having the chance to work with him and to study under his

exceptional supervision.

I would like to express my gratitude to Evrim Onur ARI, Murat GÜLTEKIN, Umut BATU

and Mustafa Burak GÜRCAN for sharing their experience & vast knowledge and helping me

through tough times; to Bülent BILGIN and all my other administrators for granting me such

a valuable opportunity to proceed in my academic career; to ASELSAN A.S for providing all

hardware/software requirements of my study.

I would like to appreciate the help of my young yet, skillful and capable colleagues in Un-

manned Systems Department in ASELSAN and thank them for their sincere friendship.

I acknowledge TUBITAK for financially supporting me during my study.

For each and every success in my carrier, I am indebted to my parents, Sadıka and Bülent, to

my lovely sister, Orkide, and to my grandparents, Nihat and Sevim. I am thankful to them for

being my source of power and will throughout my life. Special thanks go to my dearest, Ayla,

for believing in my success more than I did, for convincing me that completing this thesis

work is something I can achieve and for giving her heartwarming and everlasting support

generously. I appreciate my cousin, Özcan, for his long lasting fellowship and great times we

had together.

viii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 LITERATURE SURVEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Single Degree of Freedom Gear Mesh Model . . . . . . . . . . . . . 3

2.2 Modeling of Backlash Phenomenon . . . . . . . . . . . . . . . . . . 5

2.2.1 Backlash as a Simple Deadband . . . . . . . . . . . . . . 5

2.2.2 Backlash with Impact Dynamics . . . . . . . . . . . . . . 7

2.2.2.1 Analysis on One Dimensional Model . . . . . 7

2.2.2.2 Impact Modeling in Terms of Energy Loss . . 9

2.3 Discussion on Gear Mesh Stiffness . . . . . . . . . . . . . . . . . . 14

2.4 System Identification in Frequency Domain . . . . . . . . . . . . . 16

3 EXPERIMENTAL SETUP AND MEASURING METHODS . . . . . . . . . 21

3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Calibration for Measurements . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Torque Applied . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Calibration of Input and Output Rotational Speed . . . . . 23

3.2.3 Calibration of Input and Output Angular Position . . . . . 25

3.3 Test Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Externally Exciting the System . . . . . . . . . . . . . . . 26

3.3.2 Driving the System with Sine Wave Torque Input . . . . . 26

3.3.3 Driving the System with Square Wave Torque Input . . . . 27

3.3.4 Driving the System with Constant Speed Actuation . . . . 27

ix

4 MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Basics for Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Discussion on Stiffness of the System . . . . . . . . . . . . . . . . . 30

4.3 Discussion on Backlash of the Gearbox . . . . . . . . . . . . . . . . 31

4.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.5.1 Drive System . . . . . . . . . . . . . . . . . . . . . . . . 33

4.5.2 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5.3 Gearbox Reduction . . . . . . . . . . . . . . . . . . . . . 34

4.5.4 Gear Mesh Dynamics . . . . . . . . . . . . . . . . . . . . 35

4.6 Development of Model . . . . . . . . . . . . . . . . . . . . . . . . 35

4.6.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.6.2 Model with Constant Backlash . . . . . . . . . . . . . . . 35

4.6.3 Model with Varying Backlash . . . . . . . . . . . . . . . 36

5 PARAMETER IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Identification of Drive System Parameters . . . . . . . . . . . . . . 37

5.1.1 Motor Viscous and Coulomb Friction . . . . . . . . . . . 37

5.1.2 Motor Inertia . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Identification of Load Parameters . . . . . . . . . . . . . . . . . . . 41

5.2.1 Load Bearing Viscous and Coulomb Friction . . . . . . . 41

5.2.2 Load Inertia . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Identification of Gear Mesh Parameters . . . . . . . . . . . . . . . . 42

5.3.1 Identification of Mechanical Clearance . . . . . . . . . . . 42

5.3.2 Identification of Stiffness . . . . . . . . . . . . . . . . . . 43

5.3.3 Identification of Damping . . . . . . . . . . . . . . . . . 45

5.4 Parameters Determined for Experimental Setup Configurations . . . 47

6 COMPARISON OF SIMULATION AND TEST RESULTS . . . . . . . . . . 51

6.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Model with Constant Backlash . . . . . . . . . . . . . . . . . . . . 51

6.2.1 Discussion on Varying Reduction Ratio . . . . . . . . . . 53

6.3 Model with Varying Backlash . . . . . . . . . . . . . . . . . . . . . 57

7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

APPENDICES

x

A EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B MEASUREMENTS FOR IDENTIFICATION . . . . . . . . . . . . . . . . . 69

xi

LIST OF TABLES

TABLES

Table 3.1 General properties of experimental setup . . . . . . . . . . . . . . . . . . . 23

Table 5.1 Stiffness estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Table 5.2 Estimated parameters for test configurations . . . . . . . . . . . . . . . . . 47

Table B.1 Inertia values of driving system calculated for different excitations . . . . . 69

Table B.2 Inertia values of total system calculated for different excitations . . . . . . . 69

Table B.3 Friction measurement for load side . . . . . . . . . . . . . . . . . . . . . . 70

xii

LIST OF FIGURES

FIGURES

Figure 1.1 Anti-backlash pinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Figure 1.2 Preloaded anti-backlash gear assembly . . . . . . . . . . . . . . . . . . . 2

Figure 2.1 Dynamic model of a gear pair . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 2.2 Equivalent SDOF model of a gear pair . . . . . . . . . . . . . . . . . . . 5

Figure 2.3 Backlash model without any impact . . . . . . . . . . . . . . . . . . . . . 6

Figure 2.4 Backlash model with impact . . . . . . . . . . . . . . . . . . . . . . . . . 6

Figure 2.5 Simple mass-spring-damper system with deadzone . . . . . . . . . . . . . 7

Figure 2.6 Non-linear spring model contact force . . . . . . . . . . . . . . . . . . . . 10

Figure 2.7 Contact Force versus Penetration Depth . . . . . . . . . . . . . . . . . . . 11

Figure 2.8 Contact force variation during impact . . . . . . . . . . . . . . . . . . . . 12

Figure 2.9 Foundation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 2.10 Two masses connected with a spring . . . . . . . . . . . . . . . . . . . . . 17

Figure 2.11 Frequency response from Motor Torque to Motor Speed . . . . . . . . . . 18

Figure 2.12 Frequency response from Motor Torque to Load Speed . . . . . . . . . . . 19

Figure 3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 3.2 Sample loading case for gyro and resolver speed calibration . . . . . . . . 24

Figure 3.3 Difference between gyro speed and resolver speed . . . . . . . . . . . . . 24

Figure 3.4 Difference between gyro speed and resolver speed after calibration . . . . 25

Figure 3.5 Profile used to calibrate input and output position . . . . . . . . . . . . . . 26

Figure 3.6 Difference between encoder reading and load position after calibration . . 27

Figure 4.1 Schematic for experimental setup . . . . . . . . . . . . . . . . . . . . . . 30

Figure 4.2 Frequency response for driver assembly . . . . . . . . . . . . . . . . . . . 31

Figure 4.3 Overall look of the system model . . . . . . . . . . . . . . . . . . . . . . 33

xiii

Figure 5.1 Motor friction for different rotational speeds . . . . . . . . . . . . . . . . 38

Figure 5.2 Friction for positive rotational speeds . . . . . . . . . . . . . . . . . . . . 38

Figure 5.3 Friction for negative rotational speeds . . . . . . . . . . . . . . . . . . . . 39

Figure 5.4 Sine wave form angular speed of the system and input motor torque . . . . 40

Figure 5.5 Frequency content of torque and resolver speed signals . . . . . . . . . . 41

Figure 5.6 Total Friction vs Load Speed . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 5.7 Load position and motor position data during external excitation . . . . . . 43

Figure 5.8 Frequency response from Motor Torque to Motor Speed . . . . . . . . . . 43

Figure 5.9 Inertia information generated from frequency response data . . . . . . . . 44

Figure 5.10 Transient vibration after gear mesh engaged . . . . . . . . . . . . . . . . . 46

Figure 5.11 Input and output position differences calibrated . . . . . . . . . . . . . . . 48

Figure 5.12 Input and output position differences calibrated with modified method . . . 48

Figure 5.13 Frequency responses for different configurations . . . . . . . . . . . . . . 49

Figure 6.1 Test data and simple model output, motor speed comparison, step input . . 52

Figure 6.2 Test data and simple model output, motor speed comparison, sine input . . 52

Figure 6.3 Simple model speed difference comparison . . . . . . . . . . . . . . . . . 53

Figure 6.4 Simple model position difference comparison . . . . . . . . . . . . . . . . 54

Figure 6.5 Constant backlash model position difference, square wave input . . . . . . 54

Figure 6.6 Constant backlash model position difference, step input . . . . . . . . . . 55

Figure 6.7 Constant backlash model position difference, sine wave input . . . . . . . 55

Figure 6.8 Constant backlash model position difference, square wave input . . . . . . 56

Figure 6.9 Test data vs varying backlash model output before calibration . . . . . . . 57

Figure 6.10 Test data vs varying backlash model output after calibration . . . . . . . . 58

Figure A.1 Spring preloaded anti-backlash mechanism . . . . . . . . . . . . . . . . . 65

Figure A.2 Spring preloaded anti-backlash mechanism hinge position . . . . . . . . . 66

Figure A.3 Spring preloaded anti-backlash mechanism 3D wireframe top view . . . . 67

xiv

CHAPTER 1

INTRODUCTION

In most of the modernization projects of out-dated weapon systems, the turret is upgradedby an automatic electrical drive system for precise targeting. In such a case, original drivesystems of the turret usually has backlash at undesired levels for an automated high precisionturret system.

To reduce the unwanted amount of backlash in the original system, there are some mechan-ical solutions that can be employed. One of these methods involves usage of anti-backlashpinion as referred in literature [1–3] and presented in Fig.1.1. It is basically two of the samegear which are attached to each other in the center and preloaded against each other withsome compliant element. This solution is mostly limited to applications with low torque re-quirement since high preload may reduce service life drastically, or it may not be possible topreload the component as it is required because of the physical restrictions.

Figure 1.1: Anti-backlash pinion (adapted from [2])

Another method is to use a driving gear which is preloaded against the driven gear. It isachieved either by directly preloading the driving gear itself by means of applying preloadingforce or by using idler gear(s) between the driving pinion and driven gear to eliminate back-

1

lash [4, 5](see Fig.1.2). Mentioned methods involve preloading in some way, which causeshigher resistance torques/forces, lower life cycles and loss of efficiency which may get worsedepending upon amount of preloading required by design and they require mechanical modi-fications to the original system as well.

Figure 1.2: Preloaded anti-backlash gear assembly (adapted from [4])

Although there are effective backlash compensation solutions which can be achieved by me-chanical modifications as mentioned; in a modernization project, it is essential to keep thesystem in its original configuration as much as possible and avoid any structural modificationunless it is highly necessary, while providing desired functionality imposed within the require-ments of the project. At this stage, control methods to compensate for undesired contributionsfrom backlash can be considered as alternative solutions as suggested in [6–8]. Related con-trol methods basically make use of predictors, i.e system models, together with self-learningalgorithms for dynamically changing system parameters and they do only require the use ofstandard gear pairs in their simplest form. To develop such controllers, as a preliminary work,whole system with backlash component should be well-defined and accurately modeled.

With this motivation, in this thesis work, it is aimed to identify the system parameters exper-imentally and develop a mathematical model by using these system parameters which willclosely approximate the actual system. A literature survey on modeling and identificationmethods has been presented as the first part of the study. The experimental setup, which isthe reference system for modeling, has been introduced and calibration methods that havebeen employed are also mentioned. Approaches for modeling the experimental setup andidentifying the system parameters have been presented. Finally, experimental data has beencompared with simulation data and validity of the second order system model with varyingbacklash limits has been shown.

2

CHAPTER 2

LITERATURE SURVEY

Gear backlash, or gear play, existed in the world of mechanics in very day geared systemsare started to be used. In fact, it is the outcome of the gap between parts and referred asthe the play between a mating pair of gear teeth [9]. To sustain a smooth operation andavoid jamming, overloading and overheating, gear backlash should exist in gear applications.Excessive backlash, on the other hand, causes unwanted impact and noise. Kinematically,backlash is a simple deadband for motion transmission as defined in [10]. But for transientregime where gear teeth get in contact with each other, vibrations show up. This transientregime is where gear teeth impact phenomenon occurs. Disregarding the effect of impact wasquite logical when there were mostly big machines operating in one direction of rotation.

With the growing demand for high precision machinery, impact occurring because of backlashhas become an important topic. Even watches, which are high precision mechanisms but havesmall inertia which leads to negligible shock forces, have been observed to employ somespring preloaded mechanisms which are used to compensate for possible backlash and avoidimpact impact of the gear teeth. Applications in robotics community and defense industryare also considered to be high precision. Such applications require positioning high inertiapayloads with tight positioning tolerances. This requirement makes inclusion of backlashimpact consideration in system models essential.

Following survey covers approaches that exist in literature related to modeling of gear pairwith backlash. Basic parameter identification methods for a second order system are alsodiscussed.

2.1 Single Degree of Freedom Gear Mesh Model

In literature, the most basic model is considered to be single degree of freedom (SDOF) modelwhich only considers torsional stiffness of gears. In [11], author considers mesh stiffness asthe only energy storing element thus neglecting deformations of all other elements like shaftsand bearings. It also neglects shock that can be caused by backlash since backlash has notbeen included. This approach is useful in order to uncouple the vibration caused by gearmesh from other sources of vibrations as stated in [12]. Mesh stiffness, mesh damping and

3

excitation due to gear errors are the most crucial elements to be included into SDOF model.

Figure 2.1: Dynamic model of a gear pair (from [12])

Using notation in Fig.2.1 simple equation of motion is given as follows,

I1θ1 +R1cm(R1θ1−R2θ2)−R1c1e1−R2c2e2 +R1km(R1θ1−R2θ2)−R1k1e1−R2k2e2 = T1

(2.1)

I2θ2 +R2cm(R2θ2−R1θ1)+R2c1e1 +R2c2e2 +R2km(R2θ2−R1θ1)+R2k1e1 +R2k2e2 = T2

(2.2)where

Ii Mass moment of inertia of gear i

θi Angular displacement of gear i

Ri Radius of base circle of gear i

cm Viscous damping coefficient of the gear mesh (total).

ci Viscous damping coefficient of ith tooth pair in mesh

km Stiffness of the gear mesh (total).

ki Stiffness of the ith teeth pair in mesh

ei Displacement excitation representing the relative gear errors of the ith meshing teethpair

Ti Torque applied on ith gear

Introducing variable x,

x = R1θ1−R2θ2 (2.3)

Eq. (2.1) and (2.2) are reduced to the following,

4

mex+ cmx− c1e1− c2e2 + kmx− k1e1− k2e2 =W0 (2.4)

where equivalent mass me and static load W0 are defined as follows,

me =I1I2

I1R22 + I2R2

1(2.5)

W0 =T1

R1=

T2

R2(2.6)

In [12], single degree of freedom translational system model has been given as an equivalentrepresentation of a SDOF model for a gear pair.

Figure 2.2: Equivalent SDOF model of a gear pair (from [12])

e is given as relative gear errors of the meshing teeth. Gear or transmission error is usually de-fined as tangential linear displacement at the contact. There are numerous ways of measuringor numerically estimating e [11,13,14]. Vibrations of input and output gears can be observedto find gear transmission error [14].

Transmission error is an important topic and in order to reduce noise and sustain the smooth-ness of the drive. It is also important when study involves steady state response of gear pairunder operation [11].

2.2 Modeling of Backlash Phenomenon

2.2.1 Backlash as a Simple Deadband

Simple SDOF model introduced assumes that contact occurs in all cases.But, in reality,change of direction of forcing would cause backlash phenomenon. Backlash is basically asource of mechanical hysteresis for gear transmission as defined in [10]. It is also underlinedthat there are two main features of backlash phenomenon. Those are hysteresis and impactphenomenon. Backlash is well-defined when compared to friction which is a more compli-cated phenomenon. It can be modeled by using a simple deadband (see Fig.2.3)and deadband

5

characteristic can be extended to include impact dynamics (see Fig.2.4). Most basic backlashmodel is said to be a mere deadband and reflect backlash and gear mesh stiffness characteris-tics.

Figure 2.3: Backlash model without any impact (from [10])

Figure 2.4: Backlash model with impact (from [10])

To model backlash phenomenon dynamically, one should consider discontinuity at transitionstate. For that purpose, three modes of operations can be introduced [15]. These modes arenamely,

• Contact occurring in CW direction of motion, driving force transmitted,

• Contact occurring in CCW direction of motion, driving force transmitted,

• No contact occurring, no force is transmitted.

6

Equations (2.1) and (2.2) can be adapted to three different cases simply by allowing forcetransmission or zero force transmission depending upon the contact case. In literature, thereare detailed analyses to simulate 3 different modes mentioned above by employing SDOFgear pair model [16].

2.2.2 Backlash with Impact Dynamics

Impact occurring between hard materials tends to have transition vibrations which have bothlow and high frequency components [10]. Vibrations of lower frequencies, which can alsobe referred as structural vibrations, show themselves as a positioning error occurring in tran-sition state. On the other hand, higher frequency components, can be identified by acousticnoise. Assumption of purely elastic collusion inherently ignores high frequency componentswhich contribute to energy loss. However, energy loss caused by high frequency vibrationsis negligible compared to energy loss due to low frequency vibrations. In terms of damping,material damping is considered as the mere source during impact.

2.2.2.1 Analysis on One Dimensional Model

For impact case, simple one dimensional mass-spring-damper model has been examined.

Figure 2.5: Simple mass-spring-damper system with translational deadzone(from [10])

Mx(t)+Cx(t)+Kx(t) = f (t) (2.7)

where,

M ,

[m1 00 m2

]C , c

[+1 −1−1 +1

]K , k

[+1 −1−1 +1

]

f (t) =

[f1(t)f2(t)

]x(t) =

[x1(t)x2(t)

]

7

Simple system given in Fig.2.5 can be underdamped, overdamped or critically-damped . Im-pact phenomenon caused by backlash should be interpreted as underdamped. Because twobody will not be moving together after first contact but instead there will be vibration whichtakes few cycles to damp. With the assumption of underdamped system, series of manipula-tions are made to obtain well-known position and velocity profiles [10]. Following formulashave been obtained for position and velocities.

x1(t) = γ11 + γ12t + e−ζω0t [γ21sinωdt + γ22cosωdt]

x2(t) = γ11 + γ12t− m2

m1[γ21sinωdt + γ22cosωdt]

(2.8)

x1(t) = γ12 + e−ζω0t [−(γ22ωd + γ21ζω0)sinωdt− (γ21ωd + γ22ζω0)cosωdt]

x2(t) = γ12−m2

m1e−ζω0t [−(γ22ωd + γ21ζω0)sinωdt− (γ21ωd + γ22ζω0)cosωdt]

(2.9)

where

w0 ,

√km

ζ ,cc0

=c

2mω0ωd , ω0

√1−ζ2 (2.10)

and γ’s are constants to be determined according to initial conditions as follows,

γ11 =m1x10 +m2x20

m1 +m2γ22 =

mm2

δx0

γ12 =m1x10 +m2x20

m1 +m2γ21 =

mm2ωd

(δ0x+ζω0δx0)

(2.11)

where

δx0 , x10− x20

These manipulations result in fundamentals of free vibrations. Further manipulations aremade to obtain position and velocity profiles for an assumed relative velocity of vc yields,

δx(t) =vc

ωde−ζω0t(−ζω0sinωdt +ωdcosωdt) (2.12)

Above equation shows that when there is a relative velocity between two bodies before theyengage (i.e impact), consecutive vibrations following the impact phenomenon has no contri-bution from rigid body modes and they only reflect transient vibrations. Impact force at themoment of collusion can be calculated as it is mentioned in [10] as follows,

8

∆ f (t), cδx(t)+ kδx(t); (2.13)

Since δx is zero, impact force becomes,

∆ f (tc = 0), f0 = cvc (2.14)

Equation (2.14) is the result of simplest approach to model impact force and energy loss. Insome of the studies damping constant taken to be a linear multiple of mesh stiffness [11].

2.2.2.2 Approaches for Impact Modeling in Terms of Energy Loss

Impact phenomenon is also a trend topic in civil engineering and there are numerous studiesto model and identify energy damped in a structure for certain conditions such as earthquakesand pounding phenomenon as defined in civil engineering.

In [17], five different approaches have been examined and evaluated for modeling purpose.First approach is so called “Stereomechanical model” which makes use of principle of con-servation of momentum. For energy loss, coefficient of restitution has been taken into consid-eration and following relations have been obtained.

v′1 = v1− (1+ e)m2(v1− v2)

m1 +m2(2.15)

v′2 = v2− (1+ e)m1(v1− v2)

m1 +m2(2.16)

Second approach is “Linear spring model” which assumes that impact occurs between twobodies between which only stiffness exists. This approach is very primitive since no energyloss has been modeled. It is easy to conclude that this approach overestimates the displace-ment, accelerations (thus velocities) and time to reach to steady state.

Third approach is Kelvin model which is the basic model to consider damping element sameas equation (2.13). In this model relative position between two colliding bodies have beentaken into consideration for elastic deformation, whereas a damping element has been in-cluded as the multiplier of relative velocity. Model can be realized as follows,

Fc = kk(u1−u2−gp)+ ck(u1− u2)); u1−u2−gp ≥ 0 (2.17)

Fc = 0; u1−u2−gp < 0 (2.18)

9

where,

u1 absolute position of the first body,

u2 absolute position of the second body,

gp clearance between two bodies,

u1 absolute velocity of the first body

u2 absolute velocity of the second body

Fourth approach is making use of non-linear spring stiffness and called Hertz model. Similarto second approach this model does not include any damping element. Mentioned non-linearbehavior has been plotted as follows,

Figure 2.6: Change in contact force when using a Non-linear spring stiffness (from [17])

Additionally, force representation for the mentioned model is given as follows,

Fc = kh(u1−u2−gp)n; u1−u2−gp ≥ 0

Fc = 0; u1−u2−gp < 0

In this representation n is typically taken to be 3/2. As the applied force (or penetration intomaterial) increases, contact surface increases which increases stiffness value. In other words,instead of constant stiffness value, stiffness is being represented as a function of deflection.

Fifth approach is simply the extension of fourth approach by the inclusion of nonlineardamper. Making use of previously defined variables,

Fc = kh(u1−u2−gp)n + ch(u1− u2); u1−u2−gp ≥ 0

Fc = 0; u1−u2−gp < 0

10

where

ch damping as a function of relative displacement, δ, ch = ζδn,

ζ damping constant,

δ relative displacement, u1−u2−gp,

Similarly n is to be selected as 3/2 as it is suggested in [17]. Making use of coefficientof restitution, author derived the following expression which relates damping coefficient todamping ratio,

ζ =3kh(1− e2)

4(v1− v2)(2.19)

In [18] similar approach has been used to model impact when robot arm/leg collides with amassive object (e.g. ground).

Further study has been conducted to observe effect of coefficient of restitution, damping,spring constant and mass on penetration depth, contact force and penetration velocity. One ofthe parametric study conducted explains that as coefficient of restitution increases for sameimpact velocity, depth of penetration and final velocity (by definition) decreases. In Fig.2.7,for very small value of e, system simply acts as if it only consists of a spring element, wherecoefficient of restitution e = 1−αvi and vi is the impact velocity.

Figure 2.7: Contact Force versus Penetration Depth for e values of 0.01 0.2 0.4 0.6 (inner toouter loop) (from [18])

Fig.2.8 is crucial for understanding the approach of modeling for impact. As the contact oc-curs, contact force starts to build up and goes to zero as two bodies separate. As stiffnessincreases maximum force increases but time between initial contact and separation decreases.

11

Areas under curves are equal as damping is constant. But as stiffness increases, forces occur-ring become more impulsive.

Figure 2.8: Contact force versus time for (bottom to top) k=10,30,50 and 70 kN/m duringimpact (from [18])

“Foundation model” can also be used for stiffness and damping modeling [18]. Main propertyof related model is that so called foundation has distributed spring and damping elements.And there is a spherical or cylindrical mass that collides with the foundation with an initialvelocity. Making use of basic definitions like compliance, strain and rate of strain, followingdefinition is determined for damping force,

Fd = 2πR(

η

h

)( g1

g1 +g2

)δδ (2.20)

Disregarding the material and geometrical parameters, this definition tells that damping forceis a function of strain. Thus, previously defined relations, which include damping force as afunction of deformation together with rate of deformation, valid and accurate considering socalled “Foundation Model”.

In [18], there is comparison in terms of energy loss due to impact when different modelsare used. Energy loss is obtained as a quantity related to cube of maximum deformation δ

for foundation model and nonlinear contact model. Models are reported to yield consistentresults compared to real life applications. On the other hand, when modeling approach islinear contact modeling, energy loss is related to square of maximum deformation which isstated to be less accurate.

All mentioned approaches assume there is an energy loss not only during approach but alsoduring restitution. Considering the Hertz approach, author divided impact phenomenon intotwo stages [19]. One is defined as approach period and the other is defined as restitutionperiod. Main difference between those two stages is that approach period contains damping

12

Figure 2.9: Foundation model (a) rigid sphere penetrating an elastic foundation and (b) dis-tributed viscoelastic foundation model element (from [18])

term whereas restitution period does not. Following definitions are force expressions formentioned two phases,

F = βδ3/2 + cδ for δ > 0 (approach period of collision)

F = βδ3/2 for δ≤ 0 (restitution period of collision)

where

δ relative position between two masses

δ relative velocity between two masses

β stiffness matrix

m1 first mass

m2 second mass

Referencing [20], Jankowski quoted the following relation for damping [19],

c = 2ξ

√β

√δ

m1m2

m1 +m2(2.21)

13

Study explained in [19] mainly aims to relate coefficient of restitution to damping ratio usingenergy method. With the assumption of elastic collision, following relation has been obtainedfor restitution period after collusion. In model, no damping is included during restitutionperiod and 2.22 is obtained for restitution phase(δ < 0),

δ =−

√(δ f

)2− 4

5β(m1 +m2)

m1m2δ5/2 (2.22)

where

δ f final relative velocity between two masses

To obtain the relation regarding to approach period (δ > 0) further manipulation is required.First method is to divide both sides of eq. (2.22) with coefficient of restitution, assuming thatmanipulated formula concerns not only δ = 0 but whole range of δ > 0. This approach yieldsfollowing relation,

ξ =

√5

2π

1− e2

e(2.23)

Another method explained is based on the assumption of linear decrease of approach velocity(damping is applied as a multiple of velocity itself). This leads to the following relation,

ξ =9√

52

1− e2

e(e(9π−16)+16)(2.24)

Equations 2.23 and 2.24 have been compared with numerical solutions and 2.24 is reportedto be the closest analytical solution compared to numerical results. This relation can be usedto determine damping coefficient experimentally.

2.3 Discussion on Gear Mesh Stiffness

During operation of a gear pair, number of gear teeth in contact varies with time. And thisvariation affects mesh stiffness. It is suggested to define mesh stiffness with respect to numberof gear teeth pairs in contact. Thus, mesh stiffness can be defined as function of relativeangular position of meshing gears [16].

There are linear [20–23] and non-linear [11,24,25] time-varying stiffness models in literature.Linear model simply assumes that meshing stiffness does not change unless number of gearteeth in contact changes. Mesh stiffness switches from one case to the other. Those cases aresimply single pair of teeth meshing case and two pairs of teeth meshing case [11].

14

One way of evaluating stiffness parameter is making use of the same approach used for twoisotropic spheres colliding. This assumptions yields the following relation,

kh =4

3π(h1 +h2)

[R1R2

R1 +R2

]1/2

(2.25)

where R1 and R2 are radii of colliding spheres and

hi =1− v2

i

πEii = 1,2 (2.26)

in which v & E are Poisson’s ratio and modulus of elasticity respectively. For determiningequivalent radii for different bodies colliding, following relation has been suggested.

Ri = 3

√3mi

4πρi = 1,2 (2.27)

In [24], it is stated that both Hertzian deformation and tooth bending deflection contributesto the elastic deformation of the tooth. Thus, both should be considered separately and thensuperposed to find total mesh stiffness. Hertzian stiffness per unit face width is defined asfollows,

kh =πE

4(1− v2)

where v and E are Poisson’s ratio and modulus of elasticity respectively. Single gear toothbending stiffness per unit face width loaded at position r can be approximated as follows,

ki(r) = (A0 +A1Xi)+(A2 +A3Xi)

[r−Ri

(1+Xi)m]

]

A0 = 3.867+1.612Ni−0.02916N2i +0.0001553N3

i

A1 = 17.06+0.7289Ni−0.01728N2i +0.0000999N3

i

A2 = 2.637−1.222Ni +0.02217N2i −0.0001179N3

i

A3 =−6.330−1.033Ni +0.02068N2i −0.0001130N3

i

where Ni is the number of teeth and Xi is the addendum modification coefficient. If gearscontact each other at two points, namely points A and B following defines the total meshstiffness for points A and B.

1ka

=1

k1(r1A)+

1k2(r2A)

+1kh

ka

F=

k1(r1A)k2(r2B)kh

k1(r1A)k2(r2A)+ k1(r1A)kh + k2(r2A)kh(2.28)

15

1kb

=1

k1(r2B)+

1k2(r2B)

+1kh

kb

F=

k1(r1B)k2(r2B)kh

k1(r1B)k2(r2A)+ k1(r1A)kh + k2(r2A)kh(2.29)

where,

ka single tooth pair stiffness for gear mesh at mating point A in N/mm

kb single tooth pair stiffness for gear mesh at mating point B in N/mm

F width of the spur gear pair in mm

r1A radial distance from the center of first gear to point A, at which contact occurs, in mm

R radius of pitch circle in mm

m module of gears in mm

Defining geometrical properties of the gear pair and kinematically solving their motion, onecan approximate contact positions A and B at a given time. Formulation and a script forfinding the contact point for a gear pair depending on various parameters like center distance,addendum modification etc. are already provided in Appendix of [25].

Another mesh stiffness definition for a gear pair is given as follows [11],

k(t) = km

[(−1.8(εT )2 t2 +

1.8εT

t +0.55]/(0.85ε) (2.30)

where T is the meshing period (T = 2/pi/ω) and ε is the contact ratio. Int this approach km

is defined as,

km = 0.85εkmax (2.31)

kmax in eq. (2.31) can be calculated using maximum stiffness of one pair of teeth defined inISO/DIN 6336-1.2 (1990). A discussion related to calculation is given in [26] as suggestedin [11].

2.4 System Identification in Frequency Domain

For identification of drive systems, like experimental setup used in this study, methods thatmake use of frequency response of the systems are commonly used in literature [10, 27–30].

16

[27–29] considers the system illustrated in Fig.2.10 which is a system consisting of twomasses connected with a compliant element, similar to the experimental setup explained insection 3.1.

Eq.(2.32) is the transfer function which can be driven from equations of motion (which arerotational version of equations (2.7) without output forcing f2(t)) for input of motor torqueand output of motor speed [29]

Figure 2.10: Two masses connected with a spring [29]

TM Motor Input TorqueJM Motor InertiaKs Stiffness of the connecting compliant elementJL Motor Inertiabs Damping of the connecting compliant element (not shown)

ΘM(s)T (s)

=JLs2 +bss+Ks

s(JL + JM)(JLJM

JL + JMs2 +bss+Ks)

(2.32)

Plotting the frequency response of the system defined with transfer function (2.32), one caneasily see that there is one anti-resonance and one resonance in Fig.2.11. In [27], frequenciesof resonance and anti-resonance have been driven and defined as follows,

FR =1

2π

√Ks

JpFAR =

12π

√Ks

JL(2.33)

where Jp is the equivalent inertia defined as,

Jp =JLJM

JL + JM(2.34)

Considering the anti-resonance phenomenon, it can be interpreted as load side is connectedto ground with a compliant element. To make the analogy clear, mentioned anti-resonanceoccurrence zone is where motor movement is mostly impaired but load continuous to oscillatein accordance with provided torque excitation. For reference, transfer function (eq.(2.35)) andbode plot (Fig.2.12) from input torque to load speed have been presented. Load side speedhas no sign of anti-resonance.

17

Figure 2.11: Frequency response from Motor Torque to Motor Speed (adapted from [29])

ΘL(s)T (s)

=bss+Ks

s(JL + JM)(JLJM

JL + JMs2 +bss+Ks)

(2.35)

Additionally, it has been suggested that one may extract information for both load inertia andmotor inertia from given frequency response Fig.2.11. For lower frequency excitation inputtorque and motor (input) observation result can be interpreted as follows. For lower frequencyexcitations whole system (both of the inertias) tend to move together like a single inertia i.eit does rigid body motion. As frequency of excitation increases effects of existing compliantelement start to show up. So trend of magnitude in Fig.2.11 upto 1−2rad/s (lower frequencyof excitation) yields the information of total inertia of the system [27,29]. Converting magni-tude back from dB and curve fitting to mentioned zone on Fig.2.11 gives the equivalent inertiavalue reduced to the motor side.

As frequency of excitation increases further and further, motor fails to excite the load sidebecause of the compliant element which connects input and output. Magnitude values forabove frequencies 30− 40rad/s in Fig.2.11 can be used to obtain input side (motor) inertia[27, 29].

For the compared quantities speed and applied torque, it is expected to have -90◦ phase onbode plot when only an inertia excited. It should be noted that phases for both 0 < f < 2rad/sand 30< f < 100rad/s zones are around -90◦ implying excitation of inertia is dominant aboveall other effects.

18

Figure 2.12: Frequency response from Motor Torque to Load Speed

19

20

CHAPTER 3

EXPERIMENTAL SETUP AND MEASURING METHODS

In this study, to simulate a servo system with gear backlash, a mathematical model is used.In order to verify findings from the model, an experimental setup is essential and simulationresults should be compared with outcomes of the practical application. To make a goodapproximation to the main source of the problem, which is a turret design with gear pairbacklash, the experimental setup should consist of the following components essentially.

• A motor,

• An inertial load,

• A gear pair with constant backlash,

An experimental setup, which consists of the components mentioned above, has been es-tablished. In this chapter, general properties of the experimental setup has been presentedtogether with the measuring methods and calibration techniques that have been used.

3.1 Experimental Setup

Experimental setup, as it is shown in Fig.3.1, consists of the following,

• A stand,

• A ring gear mounted on the stand to support the rotating load and transmit the drivingtorque,

• A platform mounted on the ring gear,

• An encoder -with anti-backlash pinion- mounted on the platform,

• Servo motor bracket -with spring preloaded mechanism- mounted on the platform,

• A Servo motor and a gearbox which are mounted on the bracket,

• A Servo driver.

21

Figure 3.1: Experimental setup

More detailed information related to experimental setup is given in Appendix A. Generalproperties of test setup can be summarized as in Table 3.1.

Backlash and inertia of the experimental setup can be set as desired. Inertia is set to maximumto observe transient shocks more distinctively. Although the configuration with high inertiahas been used for simulation, there are mainly two configurations of experimental setup asfollows,

First Configuration Minimum Inertia, Moderate Backlash1

Second Configuration Maximum Inertia, Moderate Backlash1

1 Denoted level of backlash is moderate compared to achievable backlash limits of test setup

22

Table3.1: General properties of experimental setup

Property Unit Value

Max. Motor Torque Nm 22,5

Max. Load Speed deg/s 110

Total Reduction Ratio - 110

Input Position Measuring Accuracy deg 0.0055

Output Position Measuring Accuracy deg 0.0004

Input Speed Measuring Accuracy deg/s 0.015

Output Speed Measuring Accuracy deg/s 0.015

Load Inertia kgm2 40 to 110 approx.

Size mm 1300x1300x1000

3.2 Calibration for Measurements

3.2.1 Torque Applied

Torque measurement is done using well calibrated current transducer. Servo motor and servodriver have been tested together under dynamic loading conditions to make sure that the cur-rent value read can be correlated to servo motor output shaft torque.

Servo motor is separately tested and relation between motor current input and motor outputshaft torque has been identified. For all torque demands in working range (0-22,5 Nm), servomotor delivers physical torque output with a linear correlation between current measured andtorque applied.

3.2.2 Calibration of Input and Output Rotational Speed

For measuring input side (servo motor shaft) rotational speed, a resolver; for measuring outputside rotational speed a gyro have been used. Even though reduction ratios are well-defined,small errors that may prove to be significant, should be compensated making use of well-defined cases. One of these cases is continuous rotation to one direction where input andoutput speed should match.

It is known that if the gears are in contact and they are rotating in the same direction of steadystate with a non-varying forcing, it is expected to have all oscillatory behavior damped andhave both meshing gears rotating with same linear speed at contact point. Making use of thisinformation one can suggest to set input and output difference to zero to calibrate input andoutput speed.

Under constant forcing, sensor data (as they are given in Fig.3.2 and Fig.3.3) have beenrecorded and used to calibrate input and output speed accurately. Difference between load

23

speed and reduced servo motor shaft speed has been recalculated making use of slope of fit-ted curve in Fig.3.3. During this calculation gyro data is assumed to be the actual load sidespeed and it reflects load side speed accurately.

Figure 3.2: Sample loading case for gyro and resolver speed calibration

Figure 3.3: Difference between gyro speed and resolver speed(reduced to load side)

After calibration operation, variation of load speed, input side shaft speed and differencebetween those two quantities is given in Fig.3.4. This calibrated input-output speed differencedata is to be used to compare measurements from real system and findings from mathematicalmodel.

24

Figure 3.4: Difference between gyro speed and resolver speed (reduced to load side) aftercalibration together with loading condition and actual speed data from gyro and resolver

3.2.3 Calibration of Input and Output Angular Position

First step to calibrate input and output positions would be to get an absolute sensing devicefor driving or driven side in the system. Encoder has a solid connection. So it is rathereasy to tune the encoder readings precisely to be sure that it surely reflects the load position.Encoder has been calibrated to get exactly 360 degrees reading for one full revolution. It isdone by rotating the load multiple times in one direction and by checking the consistency ofthe readings. Then one can proceed with the assumption that encoder measures the absoluteposition of the load.

Similar to method followed for input and output speeds in section 3.2.2, input and output po-sitions of the system are calibrated to obtain consistent results from separate sensors. Selectedprofile for calibration is given in Fig.3.5. It should be noted that applied torque is relativelysmall and almost constant. It also allows movement of the system to sweep whole range of ro-tation. As constant forcing assumption satisfied and enough number of samples are obtained,one can proceed to fit a curve to difference in readings of input and output position sensors asa function of input position, similar to what is done in Fig.3.3.

Fig.3.6 reflects the situation after calibration. Position information should have an initial con-dition or a reference value. One should impose an initial condition for reduced motor shaftposition data to match with load side position data obtained from encoder. The first method

25

that would appear in mind would be taking encoder reading from stationary system as refer-ence. In that case, reference value will be a random position in backlash zone. This will causeto have non-symmetric backlash values for different directions of rotation. Non-symmetricbacklash values should be compensated during simulation by selecting proper backlash limits.

Figure 3.5: Profile used to calibrate input and output position

3.3 Test Runs

For identification and model verification purposes, a number of tests have been determined.Each test step explained, have been repeated for high inertia and low inertia load cases. Thesetests can be listed as follows,

3.3.1 Externally Exciting the System

Rotating platform has been excited externally by means of applying torque to load by hand,while motor brake is active (system movement hindered from input side). By doing so, it isexpected to gather information about the backlash of the drive system. Detailed informationrelated to backlash measurement is given in section 5.3.1.

3.3.2 Driving the System with Sine Wave Torque Input

For identification of system parameters, sine sweep is a method commonly employed in lit-erature. For identification purpose, system has been excited with sine wave torque input atdifferent frequencies.

26

Figure 3.6: Difference between encoder reading and load position reduced from servo motorshaft position after calibration together with loading condition and actual position data fromencoder and resolver (unwrapped)

For frequency sweep tests, frequency range from 0.25Hz to 100Hz with step size of 0.25Hzhas been used for excitation and system has been exposed to each excitation for 10 cycles.

Additionally sine wave torque input excitation with a known bias has been applied to ensurethat the movement is in one direction. By this way, it has been expected to eliminate highlynon-linear effect of static friction observed during zero velocity crossings.

3.3.3 Driving the System with Square Wave Torque Input

Square wave excitation test is basically done for providing a reference output for developedmathematical model. Square wave excitations with different amplitudes and periods havebeen practiced together with some biased profiles.

3.3.4 Driving the System with Constant Speed Actuation

Experimental setup has the ability to trace a provided load speed demand which requiressimple tuning depending on the load. Using this ability, load has been driven with differentconstant speeds. Constant speed motion at different speeds provides information about fric-tion characteristics of the system. Method employed has been explained in sections 5.1.1 and5.2.1.

27

28

CHAPTER 4

MODELING

4.1 Basics for Modeling

Experimental setup mainly consist of the following with their respective simplifications inmodeling aspect,

• Servo motor which is modeled as an inertia,damping and Coulomb friction.

• Gearbox which is modeled as a combination of inertia, stiffness, damping and Coulombfriction.

• Spur gear mesh which is modeled as a combination of stiffness, damping. Additionallyforce transmission over gear pair is modeled as a function of input and output positionsfor gear pair to simulate backlash.

• Load side bearing modeled as a combination of damping and Coulomb friction.

• Load is modeled as an inertia.

Each of the required parameter has been calculated with units of applied torque (reduced toload side) Nm, load position rad, load speed rad/s and load acceleration rad/s2. This enablesthe usage of the test data directly collected from the experimental setup. For the purpose ofdetermining motor parameters, motor speed has been reduced to load side and related motorparameters have been calculated in terms of load side units.

Coulomb friction related items can be extended to have a "rate of force applied" dependencyfor static case. There are numerous works in literature that have emphasis on dependency ofbreak-away friction on forcing rate [31]. In the scope of thesis work, both damping constantand coulomb friction parameters are only taken to be direction of motion dependent but po-sition independent. Justification of constant friction over the working range can be explainedwith loads of driven and driver sides on bearings being constant.

Inertia information has been mainly determined by exciting the system harmonically andexamining the system response. For the driving and the driven sides those are taken to be

29

constant for the same test run which is making use of same servo motor and same load inertia.

A single stiffness and a single damping element has been assumed which is connecting servomotor and load. This can be taken as combined stiffness of gear mesh and gearbox and gearmesh damping itself. This approach has been mentioned with details in section 4.2.

Figure 4.1: Schematic for experimental setup

4.2 Discussion on Stiffness of the System

Considering servo motor itself with gearbox attached to it, subsystem has been modeled asa inertia with resistance elements. On the other hand, motor itself carries a pinion attachedto the output of the gearbox. Thus, as it is illustrated in Fig.2.12, driver assembly can beconsidered as a two mass system. In that case, inertial components of two mass model arenamely motor inertia and output pinion inertia.

To justify the assumption of single mass, single stiffness for driver assembly, it is logical tocheck frequency response of respective sub-component only. If driver assembly was a sub-component like in Fig.2.12, frequency response of the assembly from motor torque to motorspeed would be like in Fig.2.11 for the frequency of range up to 100Hz. But frequency sweeptest data for driver assembly can be interpreted as in Fig.4.2. No observable resonance oranti-resonance exist in Fig.4.2 unlike in Fig.2.11. Furthermore, trend in the graph can be in-terpreted as inertia and damping components are dominant. Because decay rate of magnitude

30

in logarithmic scale is almost constant related to inertia. And the phase1 is higher than −90◦

which is the theoretical value for purely inertial loads.

Figure 4.2: Frequency response from Motor Torque to Motor Speed for driver assembly alone

Therefore it can be concluded that small mass (pinion) attached to the complaint element(gearbox) can be neglected. When contact occurs at the limits of backlash zone, gear pairwill deform together with complaint element of driver assembly. In fact, if pinion mass isneglected (as it is recently justified for the experimental setup), gear mesh in contact case isthe only case where complaint element of driver assembly deforms. Thus, it is also logical tocombine driver assembly stiffness with gear mesh stiffness and assume these two componentsas one compliant element.

4.3 Discussion on Backlash of the Gearbox

In order to determine the backlash of the gearbox for the purpose of comparison with totalbacklash of the system, gear pair backlash is set to the lowest value possible by engagingthe driver pinion to the ring gear as close as the configuration permits. And measurementmethod explained in section 5.3.1 has been repeated for the configuration with lowest gearpair backlash possible. Mean value for total backlash (backlash of the gearbox and spurgear pair) has been found to be approximately 0.004◦ for experimental setup configurationwith lowest achievable value of gear pair backlash. It is about %10 of the total backlash ofthe configuration used during tests. Gearbox backlash catalog value, on the other hand, is0.0015◦ which is even smaller than %10 of the total backlash of the test configuration.

1 Damping has positive contribution to increase in phase [32]

31

4.4 Equations of Motion

Modeling of experimental setup, which is illustrated schematically in Fig.4.1, has been basedon single degree of freedom gear mesh model explained in section 2.1.

Equations of motion for the system shown in Fig.4.1 can be written as follows,

I1θ1 + c1θ1 + keq(θ1−θ2)+ ceq(θ1− θ2)+Tf s1 = Tin (4.1)

I2θ2 + c2θ2 + keq(θ2−θ1)+ ceq(θ2− θ1)+Tf s2 = 0 (4.2)

where

Tin is torque applied by drive systemI1 is driver inertia (motor and gearbox combined)c1 is viscous damping constant for driver(motor and gearbox combined)

keq is total stiffness for drive line (gearbox and gear mesh stiffness combined)ceq is gear mesh damping constant

Tf s1 is Coulomb friction for drive systemTf s2 is Coulomb friction for load bearing

As it can be concluded equations (4.1) and (4.2) are stated for the case when gear mesh isoperating without any backlash. When backlash phenomenon occurs, drive system separatesfrom the load. This case can be expressed with the following equations,

I1θ1 + c1θ1 +Tf s1 = Tin (4.3)

I2θ2 + c2θ2 +Tf s2 = 0 (4.4)

which are simply removal of terms related to gear mesh. Having equations of motion readyfor contact and no-contact cases, conditions should be defined for respective situations. Whencontact occurs, relative position between input and output is at or beyond boundary which isdefined by backlash value i.e if |(θ2−θ1)| ≥ b. Then drive system torque is being transmittedthrough gear pair.Assumptions made for modeling are as follows,

• Inertias of pinions are negligible (any resonance/anti-resonance that would be observedin eq.(4.3) due to hanging inertia after gearbox stiffness is disgarded) as justified insection 4.2.

• Gearbox considered to have very small backlash compared to backlash in gear pair andbacklash of the system is modeled as a single stage backlash. Discussion on gearboxbacklash is given in section 4.3.

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• Connections between

– Load and Gear Pair

– Gear box and Gear Pair

are both considered to be much stiffer than the rest of the drive line. Any resonance/anti-resonance caused by these connecting compliant elements would be beyond frequencyrange observed, i.e. 0 < f < 100 Hz

• Static friction is only dependent of the static load that bearings carry. Static frictionobserved in system does not depend on force appliance rate nor load position.

• Drive line is rigid enough so that only rigid body motion is observed for low frequencyoscillations (like ≤ 1Hz).

4.5 Implementation

Overall look of the model is as given in Fig.4.3. Base elements that form the system modelare as follows.

Figure 4.3: Overall look of the system model prepared in Matlab R© Simulink R© Environment

4.5.1 Drive System

Drive system component is built to model inertia and resistive elements of drive assembly.Inputs for the block are applied motor torque and torque transmitted to gear pair (a natural

33

feedback) whereas outputs are motor speed and motor position.

In terms of friction modeling, a constant resistance plus a viscous element has been usedwhich changes its direction of appliance with respect to direction of motor speed. Whenparameters obtained from experimental setup for resistive elements are examined, there is aconsiderable change for both viscous damping constant and coulomb friction parameter whendirection of rotation changes. In accordance with the fact, two different damping constantsand two different coulomb friction components each for having in relation with a direction ofrotation, have been included into model.

Since test data will be used as the reference torque input, noise in reference signal is almostunavoidable. It has no disadvantage to have noise in torque data in transient regime but whenthe system is at rest and torque input oscillates around zero, model tends to make unnecessarynumber of zero crossing checks, which slows down the model drastically. Torque input data inrange of ±0.75 Nm, which is about 1.5 times of the amplitude of the noise in original signal,have been considered to be zero. By this way, total time to run the simulation is enhancedto ten to fifteen times compared to simulation without dead-zone application depending uponthe test data provided.

A similar dead zone has been implemented for simulation speed data which is fed to frictionblock. Motor speeds less than 4×10−5deg/s are considered to be zero (which is practicallyzero) to relief calculation load created by zero checks in friction block.

4.5.2 Load

Pretty similar to drive system, load system has been modeled as a combination of inertia andresistive elements with same approach explained in section 4.5.1. Input for the block is torquetransmitted from gear pair (which drives the load) and outputs are load speed & load position.

4.5.3 Gearbox Reduction

For gearbox reduction, two blocks, namely Gearbox Torque Transmission and Gearbox Kine-matics, have been included into model in which torque or position data is simply multipliedby a scalar depending upon reduction ratio.

Reduction ratio for the model is taken to be one. Since all parameters are defined with unitsof input torque reduced to load side in Nm, load position in rad, load speed in rad/s and loadacceleration in rad/s

2, requirement for inclusion of a reduction ratio has been eliminated.

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4.5.4 Gear Mesh Dynamics

Gear Mesh Dynamics block is basically the part of the model where gear mesh compliance,gear mesh damping and gear backlash have been implemented.

This block compares load position and reduced motor position and decides if gear pair is inbacklash zone or not. If absolute value of the difference between input and output positionis greater or equal to defined backlash parameter, it generates torque information which istransmitted from driver side to load side, using the gear pair, as defined in eq. (4.3) and (4.4).If it decides the gear pair is in backlash zone, transmitted torque becomes equal to zero whichmeans load is decoupled from driver, as defined in eq. (4.1) and (4.2).

To satisfy the initial condition for backlash, it should be possible to set starting position dif-ference for the system. Therefore for each simulation data, time response of experimentalsetup should be examined and a proper offset value should be defined for backlash parameter.This functionality has been provided by defining positive and negative backlash values for thesystem.

4.6 Development of Model

4.6.1 Basic Model

Basic model consists of

• driver inertia and friction

• load inertia and friction

• a damper and a spring connecting input and output without any clearance

With this model, it is aimed to distinguish the effect of backlash when introduced.

Tuned parameters are checked on this model and determined if related parameters needs fur-ther tuning.

4.6.2 Model with Constant Backlash

In addition to items given in section 4.6.1, this model has backlash dynamics. As it is men-tioned in section 4.5.4, model accepts two backlash parameters one denoting backlash innegative direction and the other one denoting backlash in positive direction. By this way, ithas been intended to set positive and negative backlash parameters accordingly to match withthe initial condition in test data.

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4.6.3 Model with Varying Backlash

Experimental setup makes use of a ring gear which has number of teeth above 280 whilepinion is smaller with much less number of teeth of 15. So it is practically impossible to havea constant center distance all over the working range.

As a matter of fact, it is obvious that center distance thus reduction ratio, backlash etc. varieswith position as it is stated in section 5.4.

To provide this functionality, deadband dynamics defined in section 4.5.4 has been modifiedto accept dynamically changing upper and lower limit.

Since model position output may have some deviation in terms of distance traveled, positionrecorded during test is used as reference input for data in Fig.5.11 in simulation environment.For a known load position, position difference variation determined while moving with posi-tive constant speed is fed as the upper limit for deadband, while position difference variationdetermined for negative constant speed is used as the lower limit.

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CHAPTER 5

PARAMETER IDENTIFICATION

In this identification study, there are three main components to identify. These are,

• Drive system (motor including gear box)

• Gear Pair

• Load

Measurements are made to obtain necessary parameters experimentally to be used in model-ing. Parameters that are determined are as follows,

• Inertia and friction for drive system,

• Inertia and friction for load

• Stiffness, damping and backlash for gear mesh,

Following part explains the approaches for determining required parameters of experimentalsetup. Presented sample application of each method is valid for configuration 1 i.e experi-mental setup configuration with low inertia.

5.1 Identification of Drive System Parameters

For identification of motor parameters output shaft of motor gearbox assembly has been sep-arated from load side during tests. A series of tests have been conducted to determine param-eters required to make drive system model complete.

5.1.1 Motor Viscous and Coulomb Friction

Motor shaft has been rotated with different constant rotational speeds, torque values havebeen recorded.

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Mean values of applied torques for different rotational speeds are plotted in Fig.5.1 . If nega-tive and positive speed zones is considered separately, linear curves can be fit for each. By thisway, friction characteristic has been modeled as a constant term plus additional term whichvaries with speed. Constant term is directly related to Coulomb friction. And speed dependentterm is simply the viscous friction.

Figure 5.1: Motor friction for different rotational speeds

Figure 5.2: Friction for positive rotational speeds (fitted curve Tf = 13.7θ+63.0)

Considering curves fit in Fig.5.2 and Fig.5.3, total friction for drive system can be defined asfollows,

• For positive load speeds Tf = 13.7θ+63.0

• For negative load speeds Tf = 11.5θ−62.7

where θ is in rad/s and Tf is in Nm.

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Figure 5.3: Friction for negative rotational speeds (fitted curve Tf = 11.5θ−62.7)

5.1.2 Motor Inertia

Considering the motor assembly,

Tmot = Imot × θmot + cmot × θmot +Tf smot(5.1)

where Tf smotis coulomb friction defined for driving system. For different excitation frequen-

cies and amplitudes system has been excited to trace speed command input of sine wave withan offset. Results given in Fig.5.4. To avoid stiction phenomenon system has been excited sothat it always has positive speed away from low speed zones i.e. highly non-linear region ofstiction.

Since torque signal in Fig.5.4 is noisy, a sine wave function should be fit to data or frequencycontent of the signal should be examined. In this study, frequency content has been exam-ined. Using Fast Fourier Transform algorithm, frequency content of torque data has beendetermined. Analysis has been repeated by subtracting mean values (coulomb friction fortorque data, non-oscilotory component for speed data) for both torque and speed information.They are theoretically 0 Hz components, thus they have no effect on higher frequency excita-tions as it can be seen in Fig.5.5. Peak at approximately 1 Hz tells that excitation with torquehaving 1 Hz sine wave form and 18.0 Nm amplitude, causes motor to rotate with a speedhaving 1 Hz sine wave form and 21.34 deg/s amplitude as it is plotted in Fig.5.5.

Since both speed data and viscous damping coefficient are known, viscous damping term ineq. (5.1) is subtracted from left hand side, i.e. torque input data which is given in Fig.5.4. Eq.(5.1) becomes,

Tmotmod = Imot × θmot (5.2)

Where Tmotmod stands for motor input torque data corrected by subtracting viscous dampingtorque cmot × θmot .

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To make use of eq. (5.2) acceleration definition should be converted to known quantity, angu-lar speed.

θmot(t) = θ∗motsin(wt−π/2) = θ

∗motcos(wt)

Differentiating angular speed considering excitation at a single frequency,

θmot(t) =−θ∗mot ×wsin(wt)

Now substituting above definition into eq. (5.2)

T ∗motmod sin(wt) =−Imot θ∗motwsin(wt) (5.3)

One can obtain following relation for inertia, using eq. (5.3),

Imot =−T ∗motmod

w× θ∗mot(5.4)

Using collected and analyzed data and eq. (5.4), one can find motor inertia. If w is in rad/s,θ∗mot is in rad/s and T ∗motmod is in Nm ,inertia unit is kgm2.

Figure 5.4: Sine wave form angular speed of the system and input motor torque

For multiple speed and frequency values, test has been repeated. Results are as given in TableB.1. Ignoring the outliers from Table B.1, mean value for motor inertia comes out to be7kgm2. It should be noted that speed data is the motor speed reduced to load side.

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Figure 5.5: Frequency content of torque and resolver speed signals

5.2 Identification of Load Parameters

5.2.1 Load Bearing Viscous and Coulomb Friction

Similar to the procedure followed in section 5.1.1 load has been rotated with different constantspeeds using servo motor which is engaged to the load. Measurement results are given inTable B.3 and in Fig.5.6. It should be noted that these results are for complete system, i.e. itcontains friction values for both driving system and load bearing.

Figure 5.6: Total Friction vs Load Speed

Total friction for test system can be defined as follows,

For positive load speeds Tf = 18.0θ+79.4

For negative load speeds Tf = 21.7θ−77.9

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where θ is in rad/s and Tf is in Nm.

5.2.2 Load Inertia

Drive system is now engaged to system. And the system is driven with sinusoidal speeddemand pretty similar to what is done in section 5.1.2.

Ignoring the outliers from Table B.2, mean value for total inertia comes out to be 38.8kgm2

which is close to approximate inertia of the experimental setup, 40kgm2.

5.3 Identification of Gear Mesh Parameters

5.3.1 Identification of Mechanical Clearance

For backlash measurement load side has been externally excited while driver brake is hin-dering the motion in the drive system. It is expected to determine mechanical clearance orbacklash in the system by observing load position.

When system is externally excited, it is unavoidable to observe non-moving motor shaft dur-ing tests since brake may allow little amount of movement. Considering brake slip case, loadposition data have been used in combination with the driver position data. Encoder positionvariation where motor shaft does not move, considered to be in backlash zone and recordedas mechanical clearance in the drive-line.

As input and output engage, it is possible to observe bit-wise movement on motor positiondata since brake may slip slightly. Similarly, during disengagement of input and output, motorshaft may also move slightly. This should be interpreted as spring-back on servo drive side.Re-coupling of load side with drive side will show itself as a comparably sharp rise/decay inmotor position data.

An example evaluation is presented in Fig.5.7. In time interval from 80.386s to 80.542s,motor position is not changing considerably, other than 3 bit of change which is assumedto be slight spring back. At 80.542s motor position comparably changes, there it is said tobe motor and load coupled again. Now the position change read in load position between80.386s to 80.542s is considered as backlash which is 0.027◦ as measured from load side.

Previously mentioned method is somehow direct and trustworthy. But as backlash varieswith respect to input or output position; a test made for whole range of operation, like it ismentioned in section 5.4, can be used for determining backlash. It is possible to comparedirectly measured backlash at a known load position with the one obtained by using methodexplained in section 5.4.

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Figure 5.7: Load position and motor position data during external excitation of load

5.3.2 Identification of Stiffness

Even though inertia parameters are estimated in previous section by using low frequencyexcitation experiment as explained in section 2.4, extending the previous method to higherfrequencies would be a good practice in terms of double checking the parameters.

For that purpose frequency response of the system has been determined by exciting the systemwith sine sweep torque input. Result is given in Fig.5.8

Figure 5.8: Frequency response from Motor Torque to Motor Speed

Making use of eq.(5.4) and data in Fig.5.8, Fig.5.9 has been generated over whole range of

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frequency sweep. For low frequency and high frequency excitations (which are away fromresonance and anti-resonance zones), calculated inertia values are almost constant. Extractingdata from Fig.5.9, in reference to section 2.4,

• For frequency range 0.5 < f < 2Hz calculated mean inertia value 48.0 kgm2 which istotal inertia of the system

• For frequency range 40 < f < 60Hz calculated mean inertia value 6.2 kgm2 which ismotor inertia

Figure 5.9: Inertia information generated from frequency response data

For identifying overall stiffness, frequencies of resonance and anti-resonance are required.From Fig.5.8, frequency of anti-resonance is 15.25Hz , frequency of resonance is 38.75Hz.Considering eqs. (2.33), FR, FAR, JM and JL are all-known. FR and FAR may be consideredto be accurate information. Therefore ratio of FR to FAR can be used to find the ratio JL/JM.Determining the ratio between FR and FAR,

FR

FAR=

√JL

Jp

FR

FAR=

√JM + JL

JM

(5.5)

FRFAR

ratio is equal to 2.54. Thus, JM+JLJM

is 6.46. It comes out that JLJM

is 5.46. Here usingpreviously found inertia values for load and motor will yield JL

JM= 7.11 which is significantly

different than the ratio derived by using frequencies of resonance and anti-resonance. Thereare three approaches to handle the situation,

1. Assume measured total inertia and frequencies of resonance and anti-resonance arecorrect, estimate a new motor inertia

2. Assume measured motor inertia and frequencies of resonance and anti-resonance arecorrect, estimate a new total inertia

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3. Assume measured motor inertia and load inertia are correct, estimate frequencies ofresonance and anti-resonance

For a quick evaluation Table 5.1 has been prepared. Each row is the application of the alterna-tives listed above to overcome over-defined inertia ratio issue. To keep the system response ofmodeled system and the real system as close as possible, 1st alternative is the best option sincesystem response (in terms of resonance and anti-resonance frequencies) is going to match andtotal inertia of the system won’t be changing significantly from experimentally determinedvalues. In relation with 1st solution alternative, first row (for configuration 1) and fourth row(for configuration 2) of Table 5.1, reflect a deviation in estimated motor inertia about 5% oftotal estimated inertia of the system.

Table5.1: Stiffness estimation

Config. FR FAR Jm Jl Ks11 Jp Ks2

2 FR3 FAR

4

Hz Hz kgm2 kgm2 103Nm/rad kgm2 103Nm/rad Hz Hz

136 14.5 8.4 40.5 336 6.9 355 35.0 14.9

36 14.5 6.0 29.2 242 5.0 256 35.0 14.9

36 14.5 6.0 42.9 356 5.3 270 41.3 12.6

237.75 10.5 8.9 105.8 460 8.2 460 37.8 10.5

37.75 10.5 6.3 74.9 326 5.8 326 37.8 10.5

37.75 10.5 6.3 108.4 472 5.9 334 44.9 8.8

Assuming the total inertia found 48.0 kgm2 in section 5.3.3 is true, JM comes out to be 8.4kgm2. Using these inertia values, Ks comes out to be 355.103Nm/rad with reference to firstrow of Table 5.1. When system parameters JM,JL and Ks are selected as in the first row ofTable 5.1 calculated FR becomes 35Hz (which was 36Hz experimentally) and calculated FAR

becomes 14.9Hz (which was 14.5Hz experimentally).

5.3.3 Identification of Damping

To estimate the equivalent mesh damping ceq, system response to a step input has been ex-amined. Considering eq. (2.12), it is possible to fit a curve to peaks of damped oscillationas given in Fig.5.10 and determine −ζω0 term experimentally. By making use of −ζω0,damping can be estimated. Equality for curve fitting can be defined as eq. (5.6).

y = a.eτ f it x+b + c (5.6)

1 Calculated by using FAR relation in eq. (2.33)2 Calculated by using FR relation in eq. (2.33)3 Calculated by using Ks14 Calculated by using Ks2

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Figure 5.10: Transient vibration just after gear mesh engaged when step torque input applied

where y is input-output speed difference in rad/s, x is time in s, τ f it is the constant to bedetermined for evaluation of damping and a,b,c are unused constants that are included tocompensate for offsets/gains, such as initial condition for time, that are of no interest.

After determination of term τ f it , following equations can be used to obtain ceq in eq. (4.1)and (4.2),

τ f it =−ζω0 =−ωdζ√1−ζ2

(5.7)

A =τ f it

−ωd=

ζ√1−ζ2

(5.8)

ζ =

√A2

1+A2 (5.9)

ceq = ζcc (5.10)

where ωd is damped natural frequency which can be determined from frequency of oscillationof transient vibration in Fig.5.10 or from the frequency response of the system. cc is thecritical damping coefficient which is defined as 2

√keqJp using Jp defined in eq. (2.34).

Measurement made for experimental setup yielded the result 515Nm/rad/s which is thentuned to 460Nm/rad/s for mathematical model for better performance of the model.

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5.4 Parameters Determined for Experimental Setup Configurations

As it is mentioned in section 3.1, two different configurations of test setup have been exam-ined. Determined parameters for each configuration are given in Table 5.2.

Approach presented in section 3.2.2 for input and output speed calibration works quiet satis-fying. However method explained in section 3.2.3 for calibration of input and output positionsneeds a modification. Method tells to use a constant speed movement in one direction whichhas a relatively long travel. But the problem with one direction movement is that deflectioncaused by applied torque remains uncompensated. So motor position becomes calibrated forapplied torque during movement in one direction. As it can be seen in Fig.5.11 position dif-ference varies more for motion in positive direction compared to motion in negative direction.

Luckily resistance to rotation does not vary much with the direction of rotation when deflec-tion reflected to motor shaft is considered. One may assume deflection for case 1 (constantspeed movement in positive direction) and for case 2 (constant speed movement in negativedirection) are the same. With this assumption, better correction constants can be defined byusing mean values of calibration constants for case 1 and case 2. When done so, result is asgiven in Fig.5.12. Variation in difference between input and output positions do match betterfor case 1 and case 2.

Table5.2: Estimated parameters for test configurations

Config.

ViscousDamping Coulomb

TotalInertia

MotorInertia Stiffness

Equiv. MeshDamping Backlash

Nm/(rad/s) Nm kgm2 kgm2 103Nm/rad Nm/rad/s deg

θ > 0 θ < 0 θ > 0 θ < 0

Motor Only 13.7 11.5 63.0 -62.7 - 7.0 - - -

1 18.0 21.6 79.4 -77.9 48.9 6.0 355 515 0.0253

2 21.3 23.0 79.8 -76.0 114.6 6.3 460 483 0.0307

Collected data for backlash shows a variation up to 15% with respect to load position. Tointegrate experimental data to dynamic model with constant backlash, mean value has beencalculated. With a weight function, domination of highly sampled backlash measurementpositions has been eliminated.

Sources of mentioned variation are mainly varying center distance of gears and inaccuracy intooth profile manufacturing. To model the deviation in backlash, it is possible to use well-calibrated position difference variation like the one given in Fig.5.12. It should be noted thata defined load position does not have a unique matching motor shaft position. In other words,after full revolution of load, backlash can not be guaranteed to be the same value as onerevolution before. Therefore, variation measurement should be done in whole position rangeof operation.

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Figure 5.11: Input and output position differences calibrated with method explained in section3.2.3

Figure 5.12: Input and output position differences calibrated with modified method

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Frequency response results for two different configurations of experimental setup are given inFig.5.13. As it is expected, anti-resonance frequency is lower for second configuration whichis mainly because of the increased load inertia. Frequency of oscillation, however, does notchange considerably because it is dependent on stiffness of the system and Jp which is definedby eq. (2.34). When motor inertia is constant and very small compared to load inertia, Jp doesnot change much with the change in load inertia.

Figure 5.13: Frequency responses for different configurations

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50

CHAPTER 6

COMPARISON OF SIMULATION AND TEST RESULTS

In this chapter, simulation results are presented and compared with experimental data. Testdata has been collected for 2nd configuration of experimental setup and mathematical modelhas been tuned accordingly. There are three mathematical models having different levels ofcomplexity in terms of backlash modeling.

6.1 Basic Model

Basic model was intended to build for checking overall response of the system without anybacklash dynamics implemented. Evaluation of system parameter estimation and approachesfor modeling of elements like friction and inertia are made with this model.

Basic model yields a satisfying result considering the response measured from driver sidespeed sensor which is given in Fig.6.1. On the other hand, input and output speed differencegiven in Fig.6.3 has significant difference for first peak observed just after the backlash impact.Model output peak value is almost half of the peak value of measured data. Since backlashhas not been introduced, it is irrelevant to expect a peak denoting a gear impact. But transientvibrations after impact is quite satisfying. This evaluation shows that discussion related tomesh stiffness and damping in section 4.2 is valid. In terms of position difference betweeninput and output which is given in Fig.6.4 there is no sign of backlash in simulation data asexpected. Additionally results related to excitation in sine wave form presented in Fig.6.2 forreference.

6.2 Model with Constant Backlash

Although backlash varies with load position, to use mean value calculated over the workingrange for backlash is a way employed for simulating the system. When done so, peaks withlow amplitudes observed in Fig.6.6 are replaced with finely matching speed profile.

Discontinuities in position difference graphs are also tracing the form in reference data quite

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Figure 6.1: Test data and simple model output, motor speed comparison, step input

Figure 6.2: Test data and simple model output, motor speed comparison, sine input

satisfactorily. Position difference vs time is given in Fig.6.7 and Fig.6.8.

Deviation in input-output position difference data in Fig.6.7 does exist. Because the variationof backlash limits with respect to load position is neglected. Consecutive gear mesh impactscauses input and output position difference data to settle at different levels. However un-compensated variation along contact zone in Fig.6.7 could be caused because of the limitedquality of calibration. As reduction ratio changes -which is caused by variation in center dis-tance for both driver and load side position sensor- position difference data from simulationfails to trace experimental data.

Considering Fig.6.8, position difference result obtained from simulation matches with testresults quite satisfactorily except transient zone. Transient response however fails to tracedamped peak at corners and impact shows itself as high peaks in position difference data atimpact points. This could be caused by underestimation of impact damping. As mentionedbefore impact modeled with a single damping, but it hasn’t been handled separately. Speeddifference data obtained from simulation, on the other hand, seems to match with test resultsas it is in the case presented in Fig.6.6.

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Figure 6.3: Test data and simple model output comparison, input & output speed difference,step input

6.2.1 Discussion on Varying Reduction Ratio

As it is observed in Fig.6.5, after impact at gear pair, measured position difference data tendsto deviate from what is estimated in simulation depending on application direction of force.Direct conclusion would be that this deviation is mainly because of the scaling error for inputsensor. This scaling error can be caused by the application of non-varying reduction ratiowhich is precisely tuned for minimal forcing1. First item appear in mind related to reduc-tion ratio change under forcing is the structural compliance. As the applied force increases,radial component of contact force pushes the driving pinion away from the driven ring gearwhich causes an increase in center distance, thus increase in reduction. In accordance, minorincrease in reduction causes underestimation of position data which is derived using the inputposition and a constant scale. Thus, inaccuracy in input and output position difference databecomes unavoidable with constant reduction ratio.

1 Method used for tuning is explained in section 3.2.3

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Figure 6.4: Test data and simple model output comparison: input & output position difference,sine input

Figure 6.5: Test data and constant backlash model output comparison: input & output positiondifference, square wave input

54

Figure 6.6: Test data and constant backlash model output comparison, input & output speeddifference, step input

Figure 6.7: Test data and constant backlash model output comparison, input & output positiondifference, sine wave input

55

Figure 6.8: Test data and constant backlash model output comparison, input & output positiondifference, square wave input

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6.3 Model with Varying Backlash

Model explained in section 4.6.3, has been evaluated in this part. Aim of using this modelis to match model and simulation data with higher accuracy when position differences arecompared.

Practically, upper and lower limits for clearance are shaped by using data from a look up table.This look up table is basically the variation data obtained in section 5.4 which is presented inFig.5.12. When this data directly integrated into model, there appeared an inconsistency inspeed and position difference estimation as it is presented in Fig.6.9. It is obvious that usedbacklash in model is higher than the one observed in experimental setup. Ratio of backlashvalues in test and simulation data consistently yield almost same result for different sam-ples. This consistently obtained ratio has been used to scale down backlash limits given inFig.5.12. For accurate estimation of position difference between input and output, developedmodel can be used considering the result indicated in Fig.6.10. It should be noted that testdata in Fig.6.10 has been detrended piece-wisely to eliminate slope in input and output po-sition difference data possibly caused by constant reduction ratio assumption. By this way,contribution of varying backlash can be observed solely. A detailed discussion on piece-wiseslope that has been observed in position difference data has been presented in section 6.2.1.

Considering Fig.6.10 while simulation data with constant backlash tends to follow a lineartrend, simulation data with varying backlash which makes use of imperfections measuredfrom system, successfully traces test data.

Figure 6.9: Test data and varying backlash model output comparison before calibration

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Figure 6.10: Test data, constant backlash model output and varying backlash model outputcomparison after calibration

58

CHAPTER 7

CONCLUSION

In this thesis work, main focus was to adapt identification and modeling techniques for highprecision servo systems with gear backlash. There are numerous references in literature ex-plaining basics of identification and modeling for such systems. But detailed studies relatedto practical application of system identification and fine calibration of the sensors are rare.Every identification method employed in this thesis work is experiment based directly. Ad-ditionally, fine tuning methods of input and output position/speed sensors are also discussedin the scope of this study. Following the fine tuning procedure, reference data have been col-lected from experimental setup for different conditions of excitation. Simulation environmentallows the use of actual torque input employed to excite the experimental setup during tests.Thus, developed second order system model with varying backlash limits could have beenused to generate simulation results to be compared with the test results.

For development of a valid model, a simple second order model has been considered as thebase model. Final model has been obtained by including backlash dynamics and increasingcomplexity of the basic model in a controlled manner. During this development phase, ithas been concluded that even without backlash consideration, second order model is able topredict transient vibrations (excluding the backlash impact zone) provided that system pa-rameters like stiffness and damping are estimated accurately. On the other hand, introducingconstant backlash limits into model is just enough to successfully estimate input-output speeddifference before impact.

Final model with varying backlash limits successfully predicts variation of both position andspeed, and thus can be used to achieve better accuracy in terms of input and output positiondifference estimation. Success of the second order system model with varying backlash limitsalso proves that experimental method used to determine the varying upper and lower backlashlimits is valid.

The system model with varying backlash limits explained in this study can be further extendedto increase its accuracy in terms of input-output position and speed difference estimation fortransient zone of backlash phenomenon by including varying reduction ratio. As discussed insection 6.2.1, it is crucial to include reduction ratio variation to relate input-output positionsand estimate input and output position differences accurately. According to evaluation in

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section 6.2.1 compliance of the structure seems to be the most essential element to be modeledin this respect.

A drive system model with backlash that closely approximates the system response, whichis the outcome of this thesis work, is built to be the first step in order to solve the technicalproblem of backlash compensation for servo systems with standard gear pairs.

Next step would be to use the backlash model as suggested in various studies in literature[7, 30, 33, 34]. Basic approach is to directly implement the model into the controller and usepredicted torque in feedforward path to correct the control input. Implemented controllercan even be extended to have self-learning characteristics [34] and has its system parametersupdated dynamically. However, integrating system model into a real time controller requirescalculation times to be improved. Although there are some precautions explained in the scopeof thesis work, they are implemented to increase simulation speed but it is not aimed to makethe model run in real time. Model optimization issue should be studied and model should bemodified carefully since faster models will result in less accurate estimation of the response,unavoidably.

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REFERENCES

[1] US Patent, US 6,247,377 B1, 19.06.2001.

[2] US Patent, US 2010/0050799 A1, 04.03.2010.

[3] US Patent, US 7,086,302 B2, 08.08.2006.

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APPENDIX A

EXPERIMENTAL SETUP

Experimental setup originally has a preloaded mechanism shown in Fig.A.3. Driver assemblyis able to rotate around the hinge point pointed in Fig.A.2. Cylinder highlighted in Fig.A.1contains number of conical springs. These conical springs give required preload for gear meshand preloads the driving pinion against the ring gear, to sustain minimum backlash during op-eration. Mentioned preloading piece has been modified to provide constant backlash. Conicalsprings have been removed from the cylinder. And the cylinder is restrained in a desired po-sition by fixing from both end along the threaded shaft it is carried by. By doing so, centerdistance between gears, thus backlash has been fixed.

Clearance issue at the pin (which can be distinguished with its black handle and yellow releasebutton in Fig.A.1 ) has also been solve by changing it with a replica having tighter tolerances.By this way, unwanted structural clearance has been eliminated.

Figure A.1: Spring preloaded anti-backlash mechanism

Distances marked in Fig.A.3 namely r1 and r2 are known quantities so it is possible to esti-mate center distance by making use of a reference measurement taken from preloading piece.Formulas for mentioned estimations are as follows,

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Figure A.2: Spring preloaded anti-backlash mechanism hinge position

∆C = dr2

r1cos(ϕ) (A.1)

B = 2∆Ctan(φ) (A.2)

BLA = Bcos(φ) (A.3)

where

∆C change in the center distance in mm

B linear clearance for gear mesh in mm

BLA linear backlash along line of action (on pitch circle)

φ gear tooth pressure angle in degrees

r1 distance between hinge pivot and spring preloading axis in mm

r2 distance between hinge pivot and pinion center of rotation in mm

ϕ angle between ring gear & pinion mesh tangent line at contact point and line crossingbracket hinge point and center of rotation of pinion in degrees

d set distance along the axis of preloading piece from zero backlash in mm

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Figure A.3: Spring preloaded anti-backlash mechanism 3D wireframe top view

Motor shaft has its own position sensor with 0.0055◦ accuracy. This accuracy enables themeasurement of a minimum travel of 0.4µm on the pitch circle of the motor pinion as follows,

npm2

θminmot = 0.4µm (A.4)

where,

np number of teeth on the motor pinion

m module of the motor pinion gear

θminmot resolution for motor side position sensor reduced to output pinion of driver assembly inrad

An absolute encoder is being used to measure the angular position of the ring gear. There isanti-backlash pinion attached to encoder which prevents backlash between pinion of sensorand ring gear. Minimum linear travel that can be detected by encoder is evaluated as 0,05µm.

Experimental setup without any additional weight plates on it has inertia of about 40kgm2. Itcan be increased by 10kgm2 additions approximately up to 110kgm2.

For data logging purpose, an external computer is used and data is sampled at 1kHz.

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APPENDIX B

MEASUREMENTS FOR IDENTIFICATION

TableB.1: Inertia values of driving system calculated for different excitations

Motor Speed 1 Excitation Frequency Torque Amplitude Inertia(deg/s) (Hz) (Nm) kgm2

7.98 1.0 6.4 7.2

14.17 1.5 16.2 7.0

20.89 1.0 17.5 7.8

23.37 0.6 10.6 6.9

28.06 0.6 12.8 6.8

28.87 1.0 22.2 6.9

29.24 1.0 22.8 7.11 Motor speed is reduced to load speed

TableB.2: Inertia values of total system calculated for different excitations for Configuration1

Load Speed Excitation Frequency Torque Amplitude Inertia(deg/s) (Hz) (Nm) kgm2

4.93 1.0 20.1 35.8

4.95 1.0 22.2 41.9

5.91 1.0 21.4 33.8

7.08 1.0 29.2 38.5

7.22 1.0 33.4 43.2

7.44 1.0 30.4 38.2

8.97 1.0 38.4 40.0

9.85 1.0 41.8 39.7

10.63 1.0 47.9 42.1

11.79 1.0 46.5 36.8

15.58 1.0 60.2 36.1

15.60 1.0 66.9 40.0

20.03 1.0 82.0 38.2

31.11 1.0 127.0 38.1

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TableB.3: Total friction measured for Configuration 1

Mean Standart Deviation

Rotational Speed (deg/s) Torque Applied (p.u) Rotational Speed Torque Applied (p.u)

-90.03 -105.4 0.29 18.1

-80.01 -112.9 0.25 17.9

-50.00 -97.7 0.26 17.1

-40.00 -94.3 0.26 17.8

-39.99 -95.3 0.24 17.7

-29.99 -90.6 0.22 20.2

-19.99 -85.9 0.24 22.4

-19.99 -84.4 0.24 22.2

-10.00 -80.5 0.18 10.1

-9.99 -78.8 0.17 10.1

10.00 80.6 0.17 11.1

10.01 80.1 0.18 11.4

19.99 87.2 0.21 22.4

19.99 85.7 0.20 22.3

29.99 89.3 0.23 20.6

39.99 92.7 0.26 18.3

40.00 95.2 0.28 18.4

50.00 95.7 0.28 16.8

80.02 102.9 0.26 17.7

90.05 107.0 0.28 19.2

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