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 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 viii Zusammenfassung Zusammenfassung 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 AnnealingAlgorithmus 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 Zusammenfassung Eingriffsglied-Systemen angewandt werden können Zusätzlich wird die Kinematik von Kurvengetrieben durch die Einstellung der Parameter der NURBS verbessert xContent List Content List Content List Acknowledgement iii Abstract v Zusammenfassung vii Equation Signs and Indices xiii Abbreviations xvii Introduction 1.1 Motivation 1.2 Research objective and scope 1.3 Thesis outline State of the Art 2.1 Cam - follower system 2.1.1 Cam and follower classification 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 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 Effect of the Knot Vector on Motion Curves 43 4.1 Introduction to the knot vector 43 xContent List Content List 10 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 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 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 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 xContent List Content List 11 7.2.4 Fourier analysis 107 7.3 Summary 108 Conclusion and Outlook 110 8.1 Conclusion 110 8.2 Outlook 112 List of Figures CXIV List of Tables CXVIII References CXX 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N(u) B- spline basis function [-] N1(u) The first derivative of B- spline basis function [-] N2(u) The second derivative of B- spline basis function [-] N3(u) The third derivative of B- spline basis... 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