Motion design of cam mechanisms by using non uniform rational b spline

These curves are then determined by solving the system of linear equationsunder given boundary conditions of the displacement, velocity, acceleration, and jerk.Moreover, the main advanta

Motion Design of Cam Mechanismsby Using Non-Uniform Rational B-SplineBewegungskurven von Kurvengetrieben unterVerwendung von Non-Uniform Rational B-Spline

Von der Fakultät für Maschinenwesen der Rheinisch-WestfälischenTechnischen Hochschule Aachen zur Erlangung des akademischen Grades

einer Doktorin der Ingenieurwissenschaften

genehmigte Dissertation

vorgelegt vonThi 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 Corvesfor 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 secondsupervisor for giving me important comments, suggestions and encouragements He alsosupported 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 manyuseful discussions and comments from the start to the end of my thesis but also manyvaluable experiences in the research

I would say to thank Dr.-Ing Duong Xuan Thang, working at Aachen Institute for AdvancedStudy 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 IGMRto 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 thePhD student’s life

I would like to thank the Ministry of Education and Training (MOET) of Vietnamesegovernment 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 andPham Bao Linh, for their love and for understanding me far away from home I wouldspecially thank to my mother-in-law and my husband for taking care my daughters during mystudy time in Germany

Aachen, July 2018Thi Thanh Nga Nguyen

iv Acknowledgement

Abstract v

Abstract

The follower of cam mechanisms may flexibly perform its movement based on the shape ofthe cam element and the direct contact with the cam With this feature, it is convenient todesign a cam mechanism when an output motion is given by working requirements ofmachines

The follower motion is characterized by the displacement, velocity, acceleration, and jerkfunctions The acceleration is related to inertial forces of the follower When an accelerationcurve 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 causesnoise and surface wear Additionally, the peak value of the jerk curve is also important in camdesign since it determines the tendency of vibration in cam-follower systems Thus, selectinga mathematic function to describe the motion of the follower is an important step in camdesign In this thesis, Non-Uniform Rational B-Spline (NURBS) is used to describe motioncurves of the follower With the properties of NURBS, the motion curves including peakvalues of the acceleration and jerk are shown to have advantageous characteristics comparedto classical approaches

To do this, the displacement, velocity, acceleration, and jerk functions are represented byNURBS curves These curves are then determined by solving the system of linear equationsunder given boundary conditions of the displacement, velocity, acceleration, and jerk.Moreover, the main advantage of NURBS compared with other functions is that the NURBScurve 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 andjerk, the multi-objective functions depended on the weight factor are expressed For solvingthis 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 robustand 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

Die Bewegung des Eingriffsglieds wird charakterisiert durch die Auslenkungs-,Geschwindigkeits-, Beschleunigungs- und Ruckfunktionen Die Beschleunigung steht inVerbindung mit den Trägheitskräften des Eingriffsglieds Wenn eine Beschleunigungskurveabrupte Wechsel hat, beispielsweise Spitzenwerte, wird dies zu großen Trägheitskräftenführen Daher wechseln Kontaktbelastungen im Lager und auf der Kurvenscheibenoberflächeebenfalls abrupt, was Geräusche und Oberflächenverschleiß erzeugt Es kommt hinzu, dassder Spitzenwert der Ruckkurve auch wichtig bei der Kurvengelenkgestaltung ist, denn erbestimmt die Tendenz zu Vibrationen des Kurvenscheiben-Eingriffsglied-Systems Darausfolgt, dass die Auswahl einer mathematischen Funktion zur Beschreibung derEingriffsgliedsbewegung ein wichtiger Schritt der Kurvengelenkgestaltung ist In dieserArbeit werden nicht-uniforme rationale B-Splines (NURBS) zur Beschreibung derBewegungskurve des Eingriffsglieds genutzt Mit Hilfe der Eigenschaften von NURBS wirdgezeigt, dass Bewegungskurven, die Spitzenwerte von Beschleunigung und Ruck beinhaltenvorteilhafte Charakteristiken gegenüber klassischen Ansätzen aufweisen.

Dazu werden die Auslenkungs-, Geschwindigkeits-, Beschleunigungs- und Ruckfunktionenals NURBS-Kurven dargestellt Diese Kurven werden im Anschluss bestimmt, indem daslineare Gleichungssystem unter den gegebenen Randbedigungen von Auslenkung,Geschwindigkeit, Beschleunigung und Ruck gelöst wird Ferner ist der Hauptvorteil vonNURBS verglichen mit anderen Funktionen, dass die NURBS-Kurve durch ihre Parameterwie Gewichtungen und den Knotenvektor kontrolliert werden kann In dieser Arbeit wird dieBerechnung des Knotenvektors vorgestellt um seine seinen Effekt auf Bewegungskurven zubewerten

Außerdem werden, um Werte für den Gewichtungsfaktor zu finden, die Spitzenwerte vonBeschleunigung und Ruck reduzieren, von dem Gewichtungsfaktor abhängige multikriterielleFunktionen 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 Bewegungskurvenrobust und effektiv ist, da sie auf jegliche Bewegungskurven von Kurvenscheiben-

Zusammenfassung 8

Eingriffsglied-Systemen angewandt werden können Zusätzlich wird die Kinematik vonKurvengetrieben durch die Einstellung der Parameter der NURBS verbessert

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

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 𝜶 on motion curve 50

4.2.4 Parameter and knot distribution 57

4.3 Motion curve evaluation 64

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 11

7.2.4 Fourier analysis 1077.3 Summary 108

8 Conclusion and Outlook .110

8.1 Conclusion 1108.2 Outlook 112

List of Figures CXIVList of Tables CXVIIIReferences CXXAppendix CXXXIV

xii Content List

Equation Signs and Indices 13

Equation Signs and Indices

Latin Small Letters

yeuabn

pm

�

�

Displacement functionEccentricity

Angle of camshaftLower angle of camshaft Upper angle of camshaft Number of boudary conditionsDegree of functions

Number of knotsWeights

[mm or rad][mm][rad or degree][rad or degree][rad or degree]

[ - ][ - ][ - ][ - ]

��

s

ParametersDisplacement

[ - ][mm]

[ - ][ - ]

Equation Signs and Indices 14

Latin Capital Letters

OABCDEV

��𝑝

PointCoefficient of harmonic and cycloidal functionsCoefficient of harmonic and cycloidal functionsCoefficient of harmonic and cycloidal functionsCoefficient of harmonic and cycloidal functionsCoefficient of the cycloidal function

VelocityRadius of pitch circle

[ - ][ - ][ - ][ - ][ - ][ - ][mm/rad][mm]

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 function[ - ]

R1(u) The first derivative of Rational basis function[ - ]

R2(u) The second derivative of Rational basis function[ - ]

R3(u) The third derivative of Rational basis function[ - ]��

U

Control pointsKnot vector

[ - ][ - ]

��

G

Input dataObjective function

[ - ][ - ]

Equation Signs and Indices 15

�0��

Ei

Initial solution of the weight factorCurrent solution of the weight factorEnergy at state i

[ - ][ - ][ - ]

Greek Small Letters

Equation Signs and Indices 16

Abbreviations xvii

Abbreviations

IGMR Institut für Getriebetechnik, Maschinendynamik und Robotik (In

English: Institute of Mechanism Theory, Machine Dynamics andRobotics)

NURBS Non-Uniform Rational B-SplineSA Simulated Annealing

deg degree

xviii Abbreviations

1.1 Motivation

Cam-follower systems play an important role in the operation of many machine classes,especially those of the automatic type, such as cutting machines, screw machines, textilemachines, internal combustion engines, and control systems In any class of machinery inwhich the automatic control and accurate timing are paramount, a cam mechanism is anessential part of machines In the design of cam systems, the versatility and flexibility, whichallow a wide variety of cam shapes and forms, are among their more attractive features Cammechanisms therefore are applied for many machines

In general, designing cam-follower systems has several steps The detail of these steps forcam design can be found in [VDI17] They include:

(a) Setting up the structure of cam mechanisms(b) Structure of the motion program

(c) Selection of a mathematic function for the displacement program(d) Determination of main dimensions

(e) Calculation of cam profiles(f) Construction of the cam mechanism(g) Making technical drawing of all components(h) Manufacture and assembly of cam-follower systemIn these design steps of cam mechanisms, a displacement program of the follower isestablished based on working requirements and application situations of cam-followersystems For instance, Fig 1-1 shows an application of cam mechanisms The cam is designedby given the

Fig 1-1: Example for application of cam mechanisms [VDI17]

On the other hand, the rise or return motion of the displacement program can satisfy boundaryconditions at the start and the end points as shown in Fig 1-2a This motion is well-known andcommon in cam design The other one is shown in Fig 1-2b; boundary conditions are not onlyon the start and end points but also on the points between the start and end points Theseconditions can be found in the cutting machine and the glass bottle handing device [VDI14]

Fig 1-2: Boundary conditions of the normalized rise motion

Besides, designing the non-traditional motion of the follower is in the interval [0, ��] asshown in Fig 1-3 The motion curve satisfies arbitrary boundary conditions of thedisplacement,

velocity, acceleration, and jerk on the whole cycle if the value 𝜑 is equal to 2𝜋 or a part ofcycle

if 𝜑 is smaller than 2𝜋 of cam-follower systems

1 Introduction 3

Fig 1-3: Arbitrary boundary conditions of the displacement function

With these steps of cam design as discussed above, the selection of a mathematic function stepis one of the important tasks because the function would affect to the shape of motion curvesand the peak value of these curves Several well-known functions can be used for synthesizingthe displacement program, such as polynomial, harmonic, cycloidal functions [Che82; Nor99;UPS03] However, these functions have been calculated for specific cases Moreover, usingthese functions is limited by the number of boundary conditions A large number of boundaryconditions as shown Fig 1-3 can be satisfied with a polynomial function, but the displacementcurve can display oscillations Due to this reason, the acceleration and jerk curves may exhibitlarge values on these curves

On the other hand, piecewise polynomial functions can be used to synthesize motion curves[KAK02; MA04] In this case, the continuity of the acceleration and jerk curves depends on aset of continuous conditions in each piecewise polynomial The Bezier [CZJ13] and B-spline[SGS16; SRJ09; XZ09] functions have been also described for motion curves These functionscan satisfy any boundary conditions but they are not convenient to adjust motion curves ofcam mechanisms In recent years, Non-Uniform Rational B-Spline (NURBS) is applied forthe follower motion [GG12; Sat14; XSL12] These researches used NURBS for theapproximation of basic functions such as harmonic and cycloidal functions In theseresearches, computing motion curves is not flexible since control points of NURBS are takenon the displacement curve from a basic function Due to this, motion curves are approximatedwhen the displacement curve of the follower is given Moreover, boundary conditions ofmotion curves have not been mentioned

4 1 Introduction

In this thesis, NURBS is generally synthesized for any cases of displacement programsregarding boundary conditions of the displacement, velocity, acceleration, and jerk Forexamples, these cases are shown in Fig 1-2 and Fig 1-3 Furthermore, using NURBS formotion curves can be controlled by parameters, e.g., the knot vector and weights This impliesthat, peak values of motion curves can be reduced by controlling these parameters Due to thisreason, the design of cams using NURBS can reduce inertial forces and vibrations

1.2 Research objective and scope

The main objective of this work is to apply the NURBS function for synthesizing motioncurves, such that the curves satisfy arbitrary boundary conditions of the displacement,velocity, acceleration, and jerk Furthermore, motion curves that are designed by NURBSwill be optimized with regards to peak values of the acceleration and jerk

From the objective, the main tasks are performed as follows Generally synthesize motion curves satisfying arbitrary boundary

conditions of displacement, velocity, acceleration, and jerk. Evaluate the parameters of NURBS on motion curves Compute the optimal parameters for NURBS to reduce peak values of the

acceleration and jerk

Motion curves are established in chapter 3 From this, chapter 4 evaluates the knot vector,which is also one of the important parameters of NURBS In this chapter, several methods,which are

4 1 Introduction

suitable to compute the knot vector of NURBS for motion curves, are presented Then, theeffect of the knot vector on motion curves is described; with this, the satisfactory method willbe chosen to calculate the knot vector for motions curves

Motion curves using NURBS are controlled by two parameters that are weights and the knotvector After choosing the method for computing the knot vector in chapter 4 and motioncurves as shown in chapter 3, the multi-objective functions depended on variables of weightsare built in chapter 5 These functions are created by minimizing peak values of theacceleration and jerk

Chapter 6 presents the method for solving the optimization problem in chapter 5 Thesimulated annealing algorithm is used for this thesis The background and the procedure ofthis method are presented in this chapter

With the optimal value of weights that is computed from solving the multi-objective functions(chapters 5 and 6) and with the method that is used to calculate the knot vector (chapter 4), therobustness of this proposed computation is illustrated by various examples of the applicationfor arbitrary boundary conditions, which is presented in chapter 7 Finally, this thesis issummarized and concluded in chapter 8

6 1 Introduction

Introduction (Chap 1)State of the art (Chap 2)General synthesis of motion curves

- Methods for computing the knotvector

- Distribution of parameters and knots- Evaluation of the knot vector onmotion curves

Optimizing weight factor (Chap 5)

- Exhibition of multi-objective functionsconsidering the acceleration and jerk

Simulated Annealing Algorithm (Chap 6)

- Presentation of the algorithm to computeoptimal weights: methodology and

6 1 Introduction

2 State of the Art

The aim of this chapter is to present an introduction into cam mechanisms and to provide apart of the work, which has been done in the past of this area The literature review focuses oncam- follower systems and follower displacement functions Finally, summaries and deficitsconclude this chapter

2.1 Cam - follower system

This section introduces the background of cam-follower systems, and motion programs of thefollower are discussed

2.1.1 Cam and follower classification

A cam mechanism usually consists of two moving components, e.g the cam and the follower,which are mounted on a fixed frame The possible applications of cam mechanisms are almostunlimited, and their shapes have great variety The shape of a cam depends on the transferfunction of the follower

In general, the classification of cam-follower systems is based on shapes of cams andfollowers The basic shapes of a cam are shown in Fig 2-1 [UPS03] They are a plate cam ora disk cam (Fig 2-1a), a wedge cam (Fig 2-1b), a cylindrical cam or barrel cam (Fig 2-1c),and a face cam (Fig 2-1d) In these types of cams, plate cams are commonly used inmachines For this cam, with a given continuous input motion of the cam element, thefollower moves forward and backward in a straight line

8 2 State of the Art

Fig 2-1: Type of cams: (a) plate cam, (b) wedge cam, (c) cylindrical cam, and (d) face cam

[UPS03]

The classification of followers is depicted in Fig 2-2 The basic shapes of the followercomprise four different types, such as a knife-edge follower (Fig 2-2a), a flat-face follower(Fig 2-2b), a roller follower (Fig 2-2c), and a spherical face (Fig 2-2d) [UPS03]

For the other way of classification based on the motion of the follower, they arereciprocating (translating) followers (Fig 2-1 a, b, d and Fig 2-2 a, b) and oscillating(rotating) followers as shown in Fig 2-1 c and Fig 2-2 c, d [UPS03] On the other hand,reciprocating followers are cataloged by the centerline of the follower as the offset cam (Fig.2-2 a) and the radial cam (Fig

2-2 b)

8 2 State of the Art

Fig 2-2: Type of followers: (a) offset reciprocating knife-edge follower, (b) reciprocating face follower, (c) oscillating roller follower, and (d) oscillating spherical face follower [UPS03]

flat-Additionally, the spherical and spatial cam mechanisms are shown in Fig 2-3 [CN06] Theyare both oscillating roller followers The intersection at point � of three axes of the

cam,follower, and roller is used in the spherical cam (Fig 2-3 a) The spatial cam with skew jointaxis is shown in Fig 2-3 b

Fig 2-3: (a) Spherical cam mechanism, (b) Spatial cam mechanism [CN06]

10 2 State of the Art

During the motion of cam-follower systems in machines, the contact between the cam andfollower is kept in a whole motion time In order to make this, Fig 2-1c shows that the contactis enforced by the groove in the cam surface In other manners, the shape of the follower canmaintain the contact as the constant-breadth cam (Fig 2-4a) and the conjugate cam (Fig 2-4b)[UPS03]

Fig 2-4: (a) Constant-breadth cam with a reciprocating flat-face follower, (b) Conjugate cams

with an oscillating roller follower [UPS03]

2.1.2 Displacement program

A displacement program is created by motion of the follower that is accomplished by a shapeof cams during the whole cycle of input motion For example, a type of the displacementprogram is shown in Fig 2-5 with the rise, the dwell, the return and the dwell motion On adisplacement program, the follower moves away from the cam center, which is called a rise.Otherwise, a return is a motion of the follower that leads to the follower towards the camcenter Lift is the maximum value in a rise motion for a case of cam mechanisms withreciprocating followers A dwell is defined as no motion of the follower in a period of theentire cycle while motion of cams is continuously maintained [UPS03] An important featureof cam mechanisms is to design exact dwells In order to design linkage mechanisms with oneor two dwells, it requires at least a six bar linkage, and dwell motion is only achieved as anapproximate result for using linkage mechanisms [Nor99]

10 2 State of the Art

Fig 2-5: Displacement program [UPS03]

Apart from the displacement program as shown in Fig 2-5, the displacement programs such asrise-return, rise-return-dwell, and an arbitrary motion program with an unlimited number ofsegments can be used in cam mechanisms They depend on working requirements andapplication situations of the underlying motion task However, in these displacementprograms, each of these cases has a set of constraints on the behavior of the cam functionsbetween segments, which control the rise, the return, and the dwell In general, boundaryconditions are used to match these functions of segments These boundary conditions are notonly these displacement functions but also their derivatives

2.2 Follower displacement function

Mathematic functions, describing a motion of the follower, are presented in this section.Selecting a mathematical function for a motion of the follower is an important task in camdesign because this can affect not only kinematics and dynamics but also cam profiles Thesefunctions are presented in the following sections below

2.2.1 Polynomial for motion curves

Traditionally, several mathematical basic functions have been applied to cam motion curves,e.g polynomial, harmonic, and cycloidal functions The properties of these functions are well-known in [Che82; Nor02; Nor99; UPS03] The application of these functions for the camworkbench can be found in [CNM16] Polynomial is widely used for motion curves in camdesign

A general polynomial function can be expressed by

12 2 State of the Art

Many researches used polynomial curves as shown in Eq (2.1) to design cam mechanisms asdisplacement functions Motion curves using the polynomials with degree � = 3, � = 5,and

� = 7 are depicted in [Che82; KS13; Nor02; Nor99; UPS03] They are normalized andshown

in Fig 2-6 For the polynomial function with degree � = 3, four boundary conditionsof

displacement and velocity at the start and end points are used The acceleration curve hasdiscontinuities at the start and end points Thus, the jerk values are infinitive at these points.Motion curves with degree � = 5 require six boundary conditions of displacement,velocity,

and acceleration at the start and end points Likewise, eight boundary conditions ofdisplacement, velocity, acceleration, and jerk at the start and end points are used for thepolynomial with degree � =

7

Fig 2-6: SVAJ diagram of polynomial function with degrees 𝒑 = 𝟑, 𝟓, 𝟕.

12 2 State of the Art

Using polynomial functions for synthesizing motion curves was presented in [YL95] Thepolynomial function has been also used for motion curves of cam mechanisms with oscillatingflat-face followers as shown in [YL96] In this work, the polynomial displacement wasdetermined by minimizing the size of this cam A method of varying the speed of the cam wasproposed by [YTH96] In this method, a 3-4-5 polynomial displacement function has beenused

12 2 State of the Art

to reduce the peak values of the follower output motion Subsequently, using the polynomialfor displacement function to optimize the cam size with the translating roller follower wasinvestigated by Q Yu and H P Lee [YL98] By using Monte Carlo simulation to create thecam design tool, several basic functions including polynomials were used for motion curves tocompute the optimum cam design as shown in [CS98] Parabolic, 3-4 and 3-4-5 polynomialshave been also used in this calculation The polynomial displacement functions have been alsoused in [CL13; FM98; JTR12; NSJ14; SL13; YTZ14] J Yang et al [YTZ14] used thepolynomial for the displacement function to analyze the reciprocating roller follower of thecam system This research presented a design of cam mechanisms, which is applied fortransient and heavy loads in the operating system of a large hydraulic press A cammechanism applied for the control machine was designed by some aspects of kinematics of amotion control as shown in [NSJ14] The method for analyzing cam mechanisms with bothoscillating and translating flat-face followers is that the condition for the cam profile is convexin the whole segments [CL13] Applying polynomial functions for motion curves with theflat-face and roller followers to determine cam profile was presented in [VDI14]

Besides the polynomial function used for the displacement function, it can also be used withcurve fitting of motion curves Sadek and Daadbin [SD90] used polynomial curve fitting forthe acceleration diagram In this research, the polynomial was used for fitting with the givendata of the acceleration The obtained acceleration curve eliminated the narrow pulses andcusps in the acceleration diagram compared to the original harmonic motion curve.Synthesizing cam profiles for arbitrarily prescribed acceleration is investigated by Fan YuChen [Che73] This work presented the finite integration method to synthesize the motioncurve using polynomial functions An analytical method for synthesizing cam mechanismswas presented in [SL13] In this work, the displacement function using the polynomial can beobtained by a given diagram of acceleration

For high-speed cam-follower systems, reducing vibration is important in the dynamic area.Many researches used polynomial functions for cam displacement and output motion (see Fig.2-7) The work by Kanzaki and Itao [KI72] and Ulf Andresen and William Singhose [AS04]used the polynomial function for the follower motion In this research, the displacementfunction was determined upon consideration of the boundary conditions and the characteristicsto reduce the residual vibration The polynomial output motion has been used to obtain theoptimum design of cam-follower system as shown in [Ber82] M Chew and C H Chuang[CC95] presented the polynomial function for the acceleration curve The objective of thisresearch is to reduce the residual vibration in high speed cam system The residual vibration atthe end of the rise motion of a dwell-rise-dwell cam motion was minimized with the boundaryconditions on the displacement function A method to design high-speed cam systems for

14 2 State of the Art

vibration reduction by using command smoothing technique was proposed by Zan Liang andJie Huang [LH14] To determine the command smoother, a 3-4-5 polynomial was used formotion curves Motion curves using the polynomial for cam dynamic systems with one andtwo degrees of freedom models were investigated in [MT76a] In this research, the dynamicsof the modeled cam system has been used to validate a number of fundamental design rulesconcerning the motion distortion due to off-speed operation This cam-follower system alsoreduced the peak value of acceleration

Fig 2-7: Dynamic cam systems [KI72]

To develop a repetitive control system for high speed cam follower system with a singledegree of freedom, Woosuk Chang [Woo96] used the third degree polynomial function for thedesired output motion As also a single degree of freedom, Pridgen and Singhose [PS12]proposed an alternative method called input shaping to reduce the residual vibration Theycompared the reduction of vibration between 3-4-5 and 4-5-6-7 polynomials, which are usedfor motion curves Using polynomial function for the dynamic synthesis and analysis of theone degree of freedom model was presented in [MT76b]

14 2 State of the Art

As presented above, the main advantage of a polynomial function is that boundary conditionsof displacement, velocity, acceleration, and jerk values are unlimited for calculating motioncurves of cam mechanisms However, with a large number of boundary conditions,displacement functions must use a high order of polynomials Thus, motion curves can exhibitunusable oscillations; therefore, peak values of the acceleration and jerk can occur For thatreason, this will increase the vibration and force in cam-follower systems

2.2.2 Harmonic and cycloidal functions

Harmonic and cycloidal functions are trigonometric curves Harmonic function can beexpressed as

Using these functions, boundary conditions are limited; for example, the harmonic functiononly satisfies four boundary conditions, and the cycloidal function satisfies five boundaryconditions (see Eq (2.2) and Eq (2.3)), respectively In addition, the disadvantage ofharmonic function is that the acceleration curve is discontinuous at the start and end points.Thus, the jerk will be infinite at these points (see the curves with black line in Fig 2-8)

16 2 State of the Art

Fig 2-8: Harmonic and cycloidal displacement functions

Many researches used trigonometric functions for motion curves as shown in [DG04; JTR12;LHC08; MET11; WR75; ZCJ09] E.E Zayas et al [ZCJ09] presented the analysis andsynthesis of motion curves for the constant-breadth cam mechanisms as shown in Fig 2-4 a.In this work, the cosine function has been used to compute the displacement curve.Subsequently, the cosine function used for rise and return in the rise-dwell-return motionprogram of translating flat-face follower is depicted in [MET11] Using harmonic function formotion curves, Anirvan DasGupta and Amitabha Ghosh [DG04] presented the analysis of thejamming problem for the translating roller follower The harmonic displacement function wasconsidered to determine the basic dimensions of the cam The development of a computeraided design and manufacturing system for spatial cams was investigated by Yuan L Lai, JuiP Hung, and Jian H Chen [LHC08] An example of the cylindrical cam was designed byusing harmonic function for the follower motion

16 2 State of the Art

The cycloidal function has been used for the follower displacement curve to calculate the camcontour [Che69] This function was also used in [TMB13] This work presented the optimumdesign of a cam mechanism with translating flat-face follower considering the cam size, theinput torque, and the contact stress Another work by Zhonghe Ye, Michael R Smith [YS02]exposed the synthesis and analysis of constant-breadth cam mechanisms with both oscillatingand translating flat-face followers Using the cycloidal function for the rise-return and thedwell-rise-dwell-return motions was also presented in [GM10; KLH04; SBK15] The designof cylindrical cams with translating follower was proposed in [TW93] In this work, thecycloidal function has been used for a displacement program with rise-return-dwell toapproach the cam profile based on the theory of envelopes

Using the basic function for cam mechanisms with the translating roller follower was depictedin [SPB09] The aim of this work is to present the software, which is applied to design cams.By using Pro/E software, Sun Jianping and Tang Zhaoping [ST11] proposed the displacementdiagram with harmonic and cycloidal functions to create cam profiles The design of camswith flat-face and roller followers was presented in [CS98] In this work, several basicfunctions, e.g polynomial, harmonic and cycloidal functions have been used for motioncurves to calculate the cam-follower systems In high-speed cam, harmonic and cycloidalfunctions have been also used for motion curves as shown in [SMA10; TSS16] Theseresearches handed out the dynamic analysis of cam-follower systems

Another direction is combination of different functions, such as polynomial and trigonometricfunctions in order to minimize the peak value of acceleration Many researches presented thedisplacement function that uses the combination of these functions as shown in [Che82;KLH04; LLW14; Nor02; Nor99; SMH16; UPS03] Forrest W Flocker [For12] proposed thesynthesis of the cam motion to adjust the value of the acceleration In this work, theacceleration function was associated by trigonometric and polynomial functions H D Desaiand V K Patel [DP10] used the combination of several functions, i.e., the harmonic,cycloidal, polynomial for the motion program This work presented the kinematic anddynamic force analysis for the cam with the reciprocating roller follower

It is obviously seen that harmonic and cycloidal functions are very simple functions.However, these functions are limited by the number of boundary conditions Thus, for a largenumber of boundary conditions, these functions do not satisfy all boundary conditions

18 2 State of the Art

2.2.3 Piecewise polynomials

Piecewise polynomials are normally used to describe a curve, which is constructed by manypieces of polynomials For given n data points {(�� , �� ), � = 1, 2, … , �} with (��+1 >�� ) and a set of (� − 1) polynomial functions, �1, �2, … , ��−1, a piecewise function �(�) can be

expressed as [MB85]

�(�) = �1 (�), �1 ≤

� ≤ �2�2 (�), �2 ≤

� ≤ �3⋮ ⋮�� (�), �� ≤ �

≤ ��+1⋮ ⋮{��−1 (�), �� ≤ � � 1 − ≤

2-10

Fig 2-9: Normalized cam displacement using cubic piecewise polynomials [MB85]

In recent years, using piecewise polynomials for smoothing curves was presented in [NK07a;NK07b] In these researches, a synthesis method for cam design has been proposed by asmoothing technique for sets of given constraints in the displacement, velocity, andacceleration of the follower Hansong Xiao and Jean W Zu [XZ09] used six-order piecewisepolynomials for synthesizing the displacement curve for cam driven engine Coefficients ofthe polynomial functions were determined by optimizing the output torque of the engine Six-order piecewise polynomials have been also used to synthesize the motion curves with givendisplacement, velocity and acceleration boundary conditions as found in [FCL13]

20 2 State of the Art

Fig 2-10: Motion curves using piecewise polynomials [MA04]

20 2 State of the Art

where

� ! � �−���,� (�) =

�! (� − �)! �(1 − �)

Kwun-shown in Fig 2-11 Control points on the displacement curve �(��) are also Kwun-shown ascircle

points in this figure

22 2 State of the Art

Fig 2-11: Motion curves using Bezier for constant-breadth cam mechanisms [CZJ13]

As shown in Eq (2.5) and Eq (2.6), the degree of Bezier curve used for cam synthesisdepends on the number of boundary conditions With a large number of boundary conditions,high order Bezier must be used for motion curves Due to this reason, Bezier curve has beenrarely used for motion programs when the number of boundary conditions becomes large

2.2.5 B-spline

B-spline is one of the most popular and successful methods for modeling curves and surfacesdue to its smoothness properties Especially, for fitting curves or surfaces with a given set ofpoints, B-spline can be used for approximation A B-spline curve with p-degree can beexpressed as [HK98]

�

�(�) = ∑ ��,𝑝 (�)�� , � ∈ [�, �],

(2.7)

�=0