Motion design of cam mechanisms by using non-uniform rational B-spline

Motion Design of Cam Mechanisms

by Using Non-Uniform Rational B-Spline

Bewegungskurven von Kurvengetrieben unter

Verwendung von Non-Uniform Rational B-Spline

Von der Fakultät für Maschinenwesen der Rheinisch-Westfälischen

Technischen Hochschule Aachen zur Erlangung des akademischen Grades

einer Doktorin der Ingenieurwissenschaften

genehmigte Dissertation

vorgelegt von

Thi Thanh Nga Nguyen

Berichter: Univ.-Prof. Dr.-Ing. Dr. h. c. (UPT) Burkhard Corves

Außerplanmäßiger Professor Dr.-Ing. Mathias Hüsing

Tag der mündlichen Prüfung: 21. Juni 2018

Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar.

Acknowledgement iii

Acknowledgement

Foremost, I would like to deeply thank my supervisor Prof. Dr.-Ing. Dr. h. c. Burkhard Corves

for giving me an opportunity to be a PhD student at the Institute of Mechanism Theory, Machine

Dynamics and Robotics (IGMR). I am grateful for his helpful guidance, discussions, comments,

and suggestions on the whole thesis.

Furthermore, I would like to specially thank Prof. Dr.-Ing. Mathias Hüsing as the second

supervisor for giving me important comments, suggestions and encouragements. He also

supported me a chance to be a student job at IGMR.

My big thanks go to Dr.-Ing. Stefan Kurtenbach for his time. He gave me not only many useful

discussions and comments from the start to the end of my thesis but also many valuable

experiences in the research.

I would say to thank Dr.-Ing. Duong Xuan Thang, working at Aachen Institute for Advanced

Study in Computational Engineering Science (AICES), for carefully reading my thesis.

Especially, I would like to thank Prof. Dr-Ing. Rüdiger Schmidt for introducing me to IGMR

to study in the RWTH - Aachen University, Germany.

I would like to thank Mrs. Schneider for helping me administrative procedures at IGMR. Many

thanks go to all my colleagues at IGMR for their warmth and for helping me during the PhD

student’s life.

I would like to thank the Ministry of Education and Training (MOET) of Vietnamese

government for supporting me the living expense during my study time in Germany.

Last but not least, my greatest thanks go to my two litter daughters, Pham Khanh Linh and

Pham Bao Linh, for their love and for understanding me far away from home. I would specially

thank to my mother-in-law and my husband for taking care my daughters during my study time

in Germany.

Aachen, July 2018

Thi Thanh Nga Nguyen

iv Acknowledgement

Abstract v

Abstract

The follower of cam mechanisms may flexibly perform its movement based on the shape of the

cam element and the direct contact with the cam. With this feature, it is convenient to design a

cam mechanism when an output motion is given by working requirements of machines.

The follower motion is characterized by the displacement, velocity, acceleration, and jerk

functions. The acceleration is related to inertial forces of the follower. When an acceleration

curve has abrupt changes, i.e., peak values, this will lead to large inertial forces. Therefore,

contact stresses at the bearing and on the cam surface also change abruptly, which causes noise

and surface wear. Additionally, the peak value of the jerk curve is also important in cam design

since it determines the tendency of vibration in cam-follower systems. Thus, selecting a

mathematic function to describe the motion of the follower is an important step in cam design.

In this thesis, Non-Uniform Rational B-Spline (NURBS) is used to describe motion curves of

the follower. With the properties of NURBS, the motion curves including peak values of the

acceleration and jerk are shown to have advantageous characteristics compared to classical

approaches.

To do this, the displacement, velocity, acceleration, and jerk functions are represented by

NURBS curves. These curves are then determined by solving the system of linear equations

under given boundary conditions of the displacement, velocity, acceleration, and jerk.

Moreover, the main advantage of NURBS compared with other functions is that the NURBS

curve can be controlled by its parameters such as weights and the knot vector. In this thesis, the

computation of the knot vector is presented to evaluate its effect on motion curves.

Furthermore, finding values of the weight factor to reduce peak values of the acceleration and

jerk, the multi-objective functions depended on the weight factor are expressed. For solving

this problem, the simulated annealing algorithm is used to get the optimal value of weights.

Results of this thesis demonstrate that using NURBS for synthesizing motion curves is robust

and effective because this may apply any motion curves of cam-follower systems. In addition,

the kinematics of cam mechanisms is improved by controlling NURBS’s parameters.

vi Abstract

Zusammenfassung vii

Zusammenfassung

Das Eingriffsglied von Kurvengelenken kann seine Bewegung basierend auf der Geometrie der

Kurvenscheibe und dem direkten Kontakt mit der Kurvenscheibe flexibel ausführen. Wegen

dieser Eigenschaft ist es zweckmäßig ein Kurvengelenk zu entwerfen, wenn die

Abtriebsbewegung durch die Betriebsanforderungen einer Maschine gegeben sind.

Die Bewegung des Eingriffsglieds wird charakterisiert durch die Auslenkungs-,

Geschwindigkeits-, Beschleunigungs- und Ruckfunktionen. Die Beschleunigung steht in

Verbindung mit den Trägheitskräften des Eingriffsglieds. Wenn eine Beschleunigungskurve

abrupte Wechsel hat, beispielsweise Spitzenwerte, wird dies zu großen Trägheitskräften führen.

Daher wechseln Kontaktbelastungen im Lager und auf der Kurvenscheibenoberfläche ebenfalls

abrupt, was Geräusche und Oberflächenverschleiß erzeugt. Es kommt hinzu, dass der

Spitzenwert der Ruckkurve auch wichtig bei der Kurvengelenkgestaltung ist, denn er bestimmt

die Tendenz zu Vibrationen des Kurvenscheiben-Eingriffsglied-Systems. Daraus folgt, dass die

Auswahl einer mathematischen Funktion zur Beschreibung der Eingriffsgliedsbewegung ein

wichtiger Schritt der Kurvengelenkgestaltung ist. In dieser Arbeit werden nicht-uniforme

rationale B-Splines (NURBS) zur Beschreibung der Bewegungskurve des Eingriffsglieds

genutzt. Mit Hilfe der Eigenschaften von NURBS wird gezeigt, dass Bewegungskurven, die

Spitzenwerte von Beschleunigung und Ruck beinhalten vorteilhafte Charakteristiken

gegenüber klassischen Ansätzen aufweisen.

Dazu werden die Auslenkungs-, Geschwindigkeits-, Beschleunigungs- und Ruckfunktionen als

NURBS-Kurven dargestellt. Diese Kurven werden im Anschluss bestimmt, indem das lineare

Gleichungssystem unter den gegebenen Randbedigungen von Auslenkung, Geschwindigkeit,

Beschleunigung und Ruck gelöst wird. Ferner ist der Hauptvorteil von NURBS verglichen mit

anderen Funktionen, dass die NURBS-Kurve durch ihre Parameter wie Gewichtungen und den

Knotenvektor kontrolliert werden kann. In dieser Arbeit wird die Berechnung des

Knotenvektors vorgestellt um seine seinen Effekt auf Bewegungskurven zu bewerten.

Außerdem werden, um Werte für den Gewichtungsfaktor zu finden, die Spitzenwerte von

Beschleunigung und Ruck reduzieren, von dem Gewichtungsfaktor abhängige multikriterielle

Funktionen formuliert. Um dieses Problem zu lösen wird der Simulated Annealing￾Algorithmus genutzt, um den optimalen Werte der Gewichtungen zu erhalten.

Ergebnisse dieser Arbeiten zeigen, dass NURBS für das Synthetisieren von Bewegungskurven

robust und effektiv ist, da sie auf jegliche Bewegungskurven von Kurvenscheiben-

viii Zusammenfassung

Eingriffsglied-Systemen angewandt werden können. Zusätzlich wird die Kinematik von

Kurvengetrieben durch die Einstellung der Parameter der NURBS verbessert.

Content List ix

Content List

Acknowledgement....................................................................................................................iii

Abstract .....................................................................................................................................v

Zusammenfassung ..................................................................................................................vii

Equation Signs and Indices...................................................................................................xiii

Abbreviations........................................................................................................................xvii

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

1.1 Motivation................................................................................................................1

1.2 Research objective and scope ..................................................................................4

1.3 Thesis outline...........................................................................................................4

2 State of the Art.....................................................................................................................7

2.1 Cam - follower system.............................................................................................7

2.1.1 Cam and follower classification ..............................................................................7

2.1.2 Displacement program...........................................................................................10

2.2 Follower displacement function ............................................................................11

2.2.1 Polynomial for motion curves ...............................................................................11

2.2.2 Harmonic and cycloidal functions.........................................................................15

2.2.3 Piecewise polynomials...........................................................................................18

2.2.4 Bezier curve ...........................................................................................................20

2.2.5 B-spline..................................................................................................................22

2.2.6 NURBS..................................................................................................................25

2.3 Summary and deficits............................................................................................27

3 General Synthesis of Motion Curves Using NURBS ......................................................29

3.1 Description of NURBS for motion curves.............................................................29

3.2 General synthesis of cam motion using NURBS...................................................35

3.3 Selecting the degree of NURBS used for motion curves ......................................37

3.4 Evaluating weights to motion curves.....................................................................38

3.5 Summary................................................................................................................40

4 Effect of the Knot Vector on Motion Curves ..................................................................43

4.1 Introduction to the knot vector ..............................................................................43

x Content List

4.2 Computation of the knot vector for synthesizing cam motion ..............................46

4.2.1 Parameter calculation.............................................................................................47

4.2.2 Knot vector generation for cam motion.................................................................49

4.2.3 Effect of the power �� on motion curve..................................................................50

4.2.4 Parameter and knot distribution.............................................................................57

4.3 Motion curve evaluation........................................................................................64

4.4 Summary................................................................................................................68

5 Optimizing Weight Factor of Motion Curves by Considering Kinematics..................69

5.1 Optimization problem............................................................................................69

5.2 Multi-objective optimization of motion curves.....................................................70

5.3 Summary................................................................................................................73

6 Simulated Annealing Algorithm for Optimizing Kinematics of Motion Curves.........74

6.1 Methodology of Simulated Annealing...................................................................74

6.1.1 Introduction to simulated annealing ......................................................................74

6.1.2 Physical annealing .................................................................................................75

6.1.3 Simulated annealing algorithm..............................................................................77

6.1.4 Cooling schedule ...................................................................................................79

6.1.5 Generation of neighboring solutions .....................................................................81

6.2 Process of multi-objective optimization by Simulated Annealing for motion curves

...........................................................................................................................82

6.3 Simulated Annealing algorithm for motion curve optimization............................85

6.4 Summary................................................................................................................88

7 Application Examples........................................................................................................90

7.1 Application for a small number of boundary conditions.......................................90

7.1.1 Six boundary conditions........................................................................................90

7.1.2 Eight boundary conditions.....................................................................................92

7.1.3 Nine boundary conditions......................................................................................93

7.1.4 Cam drive engine...................................................................................................94

7.2 Application for a large number of boundary conditions........................................97

7.2.1 Cutting machine.....................................................................................................98

7.2.2 Cam with twenty boundary conditions..................................................................99

7.2.3 Cam mock heart...................................................................................................102

Content List xi

7.2.4 Fourier analysis....................................................................................................107

7.3 Summary..............................................................................................................108

8 Conclusion and Outlook..................................................................................................110

8.1 Conclusion ...........................................................................................................110

8.2 Outlook ................................................................................................................112

List of Figures...................................................................................................................CXIV

List of Tables..................................................................................................................CXVIII

References........................................................................................................................... CXX

Appendix .....................................................................................................................CXXXIV

xii Content List

Equation Signs and Indices xiii

Equation Signs and Indices

Latin Small Letters

y Displacement function [mm or rad]

e Eccentricity [mm]

u Angle of camshaft [rad or degree]

a Lower angle of camshaft [rad or degree]

b Upper angle of camshaft [rad or degree]

n Number of boudary conditions [ - ]

p Degree of functions [ - ]

m Number of knots [ - ]

���� Weights [ - ]

d Number of displacement boundary conditions [ - ]

e Number of velocity boundary conditions [ - ]

f Number of acceleration boundary conditions [ - ]

g Number of jerk boundary conditions [ - ]

���� Parameters [ - ]

s Displacement [mm]

s Velocity [mm/rad]

s Acceleration [mm/rad2

]

s Jerk [mm/rad3

]

t Parameter vector [ - ]

���� Variables [ - ]

x Variable vector [ - ]

hi(x) Equality of constraints [ - ]

gj(x) Inequality of constraints [ - ]

f(x) Opjective function on x [ - ]

f(W) Opjective function on W [ - ]

f1(W) Objective funtion of acceleration [ - ]

f2(W) Objective funtion of jerk [ - ]

upeak_acc Position at maximum acceleration [ - ]

xiv Equation Signs and Indices

upeak_jerk Position at maximum jerk [ - ]

f Difference of cost function [ - ]

kB Boltzmann constant

Latin Capital Letters

O Point [ - ]

A Coefficient of harmonic and cycloidal functions [ - ]

B Coefficient of harmonic and cycloidal functions [ - ]

C Coefficient of harmonic and cycloidal functions [ - ]

D Coefficient of harmonic and cycloidal functions [ - ]

E Coefficient of the cycloidal function [ - ]

V Velocity [mm/rad]

���� Radius of pitch circle [mm]

C(u) Displacement function [length or rad or degree]

C

1

(u) Velocity function [mm/rad]

C

2

(u) Acceleration function [mm/rad2

]

C

3

(u) Jerk function [mm/rad3

]

N(u) B-spline basis function [ - ]

N

1

(u) The first derivative of B-spline basis function [ - ]

N

2

(u) The second derivative of B-spline basis function [ - ]

N

3

(u) The third derivative of B-spline basis function [ - ]

R(u) Rational basis function [ - ]

R

1

(u) The first derivative of Rational basis function [ - ]

R

2

(u) The second derivative of Rational basis function [ - ]

R

3

(u) The third derivative of Rational basis function [ - ]

���� Control points [ - ]

U Knot vector [ - ]

D Vector of input angle [ - ]

����

Input data [ - ]

G Objective function [ - ]

Equation Signs and Indices xv

T0 Initial control parameter [ - ]

T Control parameter (Temperature) [ - ]

Tmin Stopping criterion of control parameter [ - ]

R Matrix with size n x n [ - ]

C Vector of boundary condition [ - ]

P Vector of control point [ - ]

W Weight vector [ - ]

��0

Initial solution of the weight factor [ - ]

���� Current solution of the weight factor [ - ]

Ei Energy at state i [ - ]

Ej Energy at state j [ - ]

E Difference energy between two states [ - ]

P(si) Probability at state i [ - ]

Xi State i [ - ]

Xj State j [ - ]

G(W) Objective function [ - ]

Ni Number of iterations at state i [ - ]

Greek Small Letters

α Coefficient [ - ]

 Coefficient [ - ]

 One cycle of input motion [rad or degree]

 A small fraction [ - ]

 Angle of camshaft [rad or degree]

∆�� Difference of the objective function [ - ]