NON-SMOOTH DYNAMICS USING DIFFERENTIAL-ALGEBRAIC EQUATIONS PERSPECTIVE: MODELING AND NUMERICAL SOLUTIONS. A Thesis PRIYANKA GOTIKA

Transcription

1 NON-SMOOTH DYNAMICS USING DIFFERENTIAL-ALGEBRAIC EQUATIONS PERSPECTIVE: MODELING AND NUMERICAL SOLUTIONS A Thesis by PRIYANKA GOTIKA Submitte to the Office of Grauate Stuies of Texas A&M University in partial fulfillment of the requirements for the egree of MASTER OF SCIENCE December 211 Major Subject: Mechanical Engineering

2 NON-SMOOTH DYNAMICS USING DIFFERENTIAL-ALGEBRAIC EQUATIONS PERSPECTIVE: MODELING AND NUMERICAL SOLUTIONS A Thesis by PRIYANKA GOTIKA Submitte to the Office of Grauate Stuies of Texas A&M University in partial fulfillment of the requirements for the egree of MASTER OF SCIENCE Approve by: Chair of Committee, Committee Members, Hea of Department, Kalyana B. Nakshatrala Junuthula N. Rey Chii-Der Suh Jeral Caton December 211 Major Subject: Mechanical Engineering

3 iii ABSTRACT Non-smooth Dynamics Using Differential-algebraic Equations perspective: Moeling an Numerical Solutions. (December 211) Priyanka Gotika, B.E.; M.S., Birla Institute of Technology & Science-Pilani Chair of Avisory Committee: Dr. Kalyana B. Nakshatrala This thesis aresse non-smooth ynamics of lumpe parameter systems, an was restricte to Filippov-type systems. The main objective of this thesis was twofol. Firstly, moeling aspects of Filippov-type non-smooth ynamical systems were aresse with an emphasis on the constitutive assumptions an mathematical structure behin these moels. Seconly, robust algorithms were presente to obtain numerical solutions for various Filippov-type lumpe parameter systems. Governing equations were written using two ifferent mathematical approaches. The first approach was base on ifferential inclusions an the secon approach was base on ifferential-algebraic equations. The ifferential inclusions approach is more amenable to mathematical analysis using existing mathematical tools. On the other han, the approach base on ifferential-algebraic equations gives more insight into the constitutive assumptions of a chosen moel an easier to obtain numerical solutions. Bingham-type moels in which the force cannot be expresse in terms of kinematic variables but the kinematic variables can be expresse in terms of force were consiere. Further, Coulomb friction was consiere in which neither the force can be expresse in terms of kinematic variables nor the kinematic variables in terms of force. However, one can write implicit constitutive equations in terms of kinematic quantities an force. A numerical framework was set up to stuy such systems an

4 iv the algorithm was evise. Towars the en, representative ynamical systems of practical significance were consiere. The evise algorithm was implemente on these systems an the results were obtaine. The results show that the setting offere by ifferential-algebraic equations is appropriate for stuying ynamics of lumpe parameter systems uner implicit constitutive moels.

5 To my mom, a an sister v

6 vi ACKNOWLEDGMENTS I wish to express my eepest gratitue to my avisor, Dr. Kalyana B. Nakshatrala, who has been supporting an encouraging me towars research in the fiel of computational mechanics right from the beginning of my Master s stuy. I am inebte a ton for the innumerable iscussions after class an thought-provoking research meetings that mae me take the big step of committing myself to this thesis. Without his constant support, this thesis woul not have taken its final form. I thank Dr. J. N. Rey for setting high stanars towars research that helpe this thesis to come out efficiently. Also, I thank Dr. Steve Suh for his valuable avice an for serving as a committee member for this thesis. I woul like to thank all my professors who have set examples an lai a strong founation that forge my thought process which was crucial while working on this thesis. Finally, I acknowlege my parents, sister an friens, for their priceless moral support an love, without which this thesis woul not have been realise.

7 vii TABLE OF CONTENTS CHAPTER Page I INTRODUCTION AND MOTIVATION A. Main contributions B. Organization of the thesis II VARIOUS EXAMPLES: SMOOTH AND NON-SMOOTH DYNAMICAL SYSTEMS A. Classical simple harmonic oscillator B. Spring-mass-ash-pot systems C. Coulomb friction moel D. System with unilateral constraint III NOTATION AND DEFINITIONS A. Convex analysis preliminaries B. Orinary ifferential equations (ODEs) C. Differential inclusions (DIs) D. Differential-algebraic equations (DAEs) IV MATHEMATICAL MODELING A. Problem efinition an governing equations B. Lumpe parameter system with a Bingham-type ashpot of non-linear monotonic characteristics C. Lumpe parameter system with a Bingham-type ashpot of non-linear non-monotonic characteristics D. Lumpe parameter system with Coulomb friction E. Lumpe parameter system with a Bingham-type ashpot of linear characteristics an a constraint V NUMERICAL ALGORITHMS A. Algorithm for the system with Bingham-type ash-pot of non-linear monotonic characteristics B. Algorithm for the system with Bingham-type ash-pot of non-linear non-monotonic characteristics C. Algorithm for the system with Coulomb friction

8 viii CHAPTER Page D. Algorithm for the system with Bingham-type ash-pot of linear characteristics an a constraint VI REPRESENTATIVE NUMERICAL EXAMPLES A. Numerical results for the system with Bingham-type ash-pot of non-linear monotonic characteristics B. Numerical results for the system with Bingham-type ash-pot of non-linear non-monotonic characteristics C. Numerical results for the system with Bingham-type ash-pot representing Coulomb friction D. Numerical results for system with Bingham-type ashpot of linear characteristics an a constraint VII CONCLUSIONS REFERENCES APPENDIX A VITA

9 ix LIST OF FIGURES FIGURE Page 1 Constitutive relation for Bingham-type flui. Force ue to ashpot cannot be written as a function of kinematic variables (that is, velocity of the mass). On the other han, as shown in the figure, velocity can be written as a function of the force in the ash-pot Spring-mass-ash-pot system that is represente as a lumpe parameter system. m is the mass of the boy subject to external force f(t) an x enotes the isplacement of the system A simple harmonic oscillator: (a) Pictorial escription (b) A typical isplacement versus response uner free vibration (i.e., the external force is zero) A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by visco-elastic moel A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Binghamtype flui of linear characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Binghamtype flui of non-linear monotonic characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function.... 1

10 x FIGURE Page 7 A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Binghamtype flui of non-linear non-monotonic characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function A pictorial representation of constitutive relation between velocity an frictional force for a system with friction governe by Coulomb friction moel A spring-mass-ash-pot system representing a lumpe parameter system with unilateral constraint place at a istance of L units from mass m restricting the isplacement x of the system. Here, f(t) is the external force acting on the system A lumpe parameter moel of an electronic circuit where i enotes current, R enotes resistance an v enotes voltage source Spring-mass-ash-pot system representing a lumpe parameter system subjecte to an external force f(t). Displacement of the system is given by x(t) A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Binghamtype flui of non-linear monotonic characteristics, a cubic function in this case. In Figure (a), the notation is lower-case for f to represent a stanar function i.e., velocity can be written as a stanar function of force ue to ash-pot. In Figure (b), the notation of force ue to ash-pot is written as F in upper case to represent a multi-value function i.e., force ue to ash-pot can be written in terms of velocity only by using multi-value function.. 26

11 xi FIGURE Page 13 A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Binghamtype flui of non-linear non-monotonic characteristics, a polynomial function in this case. In Figure (a), the notation is lower-case for f to represent a stanar function i.e., velocity can be written as a stanar function of force ue to ash-pot. In Figure (b), the notation of force ue to ash-pot is written as F in upper case to represent a multi-value function i.e., force ue to ash-pot can be written in terms of velocity only by using multi-value function A spring-mass-system involving friction governe by Coulomb friction moel in this case. Here f(t) enotes the external force x enotes the isplacement of the system A pictorial representation of constitutive relation between velocity an frictional force for a system with friction governe by Coulomb friction moel. It shoul be note that neither kinematic quantities can be expresse as a function of ynamic quantities nor ynamic quantities as a function of kinematic quantities A spring-mass-ash-pot system with a massless support place at a istance of L units such that there is a constraint on the isplacement of the system. Here f(t) enotes the external force, x 1 enotes the isplacement of mass m an x 2 enotes the isplacement of massless support measure from the reference point as shown in the figure Constitutive relation between velocity an force ue to ash-pot of a Bingham-type ash-pot with linear characteristics. In Figure (a), the notation is lower-case for f to represent a stanar function i.e., velocity can be written as a stanar function of force ue to ash-pot. In figure (b), the notation of force ue to ash-pot is written as F in upper case to represent a multivalue function i.e., force ue to ash-pot can be written in terms of velocity only by using multi-value function Pictorial representation of the intersection of the straight line with constitutive relation of the ash-pot at a particular instant of.. 42

12 xii FIGURE Page 19 Dynamics of a spring-mass-ash-pot system with ash-pot of nonlinear monotonic characteristics. The initial value problem is a non-zero external forcing conition with zero initial isplacement an initial force ue to ash-pot Dynamics of a spring-mass-ash-pot system with ash-pot of nonlinear monotonic characteristics. The initial value problem is a non-zero initial isplacement conition with zero external force f(t) an initial force ue to ash-pot A plot velocity versus force ue to ash-pot. The arrowe lines show the path traverse when solve by using the approach in which jumps in velocity profile are avoie Dynamics (by Approach 1) of a spring-mass-ash-pot system with ash-pot of non-linear non-monotonic characteristics. The initial value problem is a non-zero external forcing conition with zero initial isplacement an force ue to ash-pot Dynamics (by Approach 1) of a spring-mass-ash-pot system with ash-pot of non-linear monotonic characteristics. The initial value problem is a non-zero initial isplacement conition with zero external force f(t) an initial force ue to ash-pot A plot for velocity versus force ue to ash-pot. The arrowe lines show the path traverse when solve by using the approach in which rate of issipation is maximize Dynamics (by Approach 2) of a spring-mass-ash-pot system with ash-pot of non-linear non-monotonic characteristics. The initial value problem is a non-zero external forcing conition with zero initial isplacement an force ue to ash-pot Dynamics (by Approach 2) of a spring-mass-ash-pot system with ash-pot of non-linear monotonic characteristics. The initial value problem is a non-zero initial isplacement conition with zero external force f(t) an initial force ue to ash-pot

13 xiii FIGURE Page 27 Dynamics of a spring-mass system involving Coulomb friction. The initial value is a non-zero external forcing conition with zero initial isplacement an initial velocity, initial force ue to ash-pot f crit / Dynamics of a spring-mass system involving Coulomb friction. The initial value problem is a non-zero initial isplacement conition with zero external force an initial velocity being, initial force ue to ash-pot as f crit / Dynamics of system spring-mass-ash-pot system with a massless support connecte to a rigi wall by a linear spring of spring constant k support = 1 place at a istance of L units from mass m Dynamics of system spring-mass-ash-pot system with a massless support connecte to a rigi wall by a linear spring of spring constant k support = 1 place at a istance of L units from mass m Dynamics of system spring-mass-ash-pot system with a massless support connecte to a rigi wall by a linear spring of spring constant k support = 1 place at a istance of L units from mass m Dynamics of system spring-mass-ash-pot system with a massless support connecte to a rigi wall by a linear spring of spring constant k support = 1 place at a istance of L units from mass m. 65

14 1 CHAPTER I INTRODUCTION AND MOTIVATION Mechanical systems can be broaly ivie into two classes: smooth an nonsmooth. This classification is base on the system ynamics observe with the ai of their corresponing mathematical moels. Stuy of smooth systems ha been extensively carrie out in the past, mostly moele using orinary ifferential equations (ODEs). One classical example is the simple harmonic oscillator which is a continuous moel governe by a secon-orer orinary ifferential equation. There are several existence an uniqueness results for the solution of ODEs. For example, Cauchy-Peano theorem [1], Carathéoory theorem [1] an, many others. Non-smooth systems are significantly more ifficult to stuy in terms of moeling, obtaining analytical solutions, an mathematical analysis [2, 3]. More mathematical aspects of non-smooth systems can be seen from References [4] an [5]. Non-smooth systems can be further ivie into three basic classifications: non-smooth continuous systems or piece-wise continuous systems, iscontinuous systems of Filippov-type, an impacting systems [6]. Non-smooth continuous systems are stuie using the theory of orinary ifferential equations uner some assumptions. Till ate, Filippov systems are stuie using ifferential inclusion formalism, an impacting systems using the theory of complementarity systems. Though large volumes of literature exist for all the three kins of systems, stuy of non-smooth systems is still an active area of research. A review paper on bifurcations in non-smooth systems gives the avances of research in this irection [7]. Most physical systems have one or more non-smooth characteristics involve in their ynamics. Some of the popular examples The journal moel is IEEE Transactions on Automatic Control.

15 2 involving non-smooth ynamics are friction, granular mechanics simulations, multiboy ynamics applications an robotics. Among these, iscontinuous systems of Filippov-type form the main subject of this thesis. Filippov-type systems are informally efine as systems whose mathematical moels consist of a set of first-orer ifferential equations with iscontinuous right-han sie [6]. For example, systems that involve non-smooth characteristics like ry friction have a iscontinuous function in their mathematical moels. The mathematical moeling of friction [8, 9, 1] ha been revolving as an amusing subject among many physicists an mathematicians for ecaes now. Most of the existing literature on moels of ry friction are base on the Amontons-Coulomb moel. However, it shoul be note that the moel cannot be use for general applications, an has many limitations. Application of this moel to ynamical systems with ry friction gives rise to ifferential equations with a iscontinuity in the right-han sie. This is one of the most important motivations behin the stuy of iscontinuous systems of Filippov-type. This motivation for the stuy of Filippov-type systems rawn from friction moels is from mechanics stan point. But Filippov-type ifferential equations arise in various fiels like electrical systems, control systems or for that matter any system which has an on/off switching mechanism [11, 12]. Filippov an others have mae enormous contributions towars the evelopment of theories to stuy such systems [13, 14, 15]. In particular, Filippov formulate the so-calle Filippov s convex metho using which he extene the concept of iscontinuous ifferential equations to ifferential inclusions. He also formulate the most celebrate Filippov s theorem escribing the conitions for the existence of solution for such ifferential inclusions. This theorem is iscusse in etail in forth-coming chapters. In mathematical literature, Filippov systems are typically stuie using ifferential inclusions approach [16, 17]. Though this theory is rich in its own sense,

16 3 misnomers like multi-value functions ha to be introuce to support the arguments [13]. In this thesis, a ifferent kin of a framework is evelope to unerstan the non-smooth behaviour of Filippov-type systems. This approach lies in moeling the systems to form a set of ifferential-algebraic equations (DAEs) or in some cases ifferential-algebraic inequalities (DAIs). In a crue sense with respect to ynamical systems, DAEs can be escribe as a set containing ifferential an algebraic equations. These governing equations arise from the system ynamics like linear momentum balance an the constitutive relations of energy storage/issipating mechanisms [18, 19]. It is important to mention that frequently, these set of DAEs can be simplifie into a system of orinary ifferential equations (ODEs) [2] making their analysis easier. Occasionally they are neee to be solve as a system of DAEs which is somewhat complex yet straight forwar compare to the methos use to solve ifferential inclusions. For example, if we consier a simple ynamical system with Bingham-type ash-pot, the constitutive equation of ash-pot cannot be written in a way in which force ue to ash-pot is expresse as a stanar function of velocity. This can be seen from Figure 1. Writing the constitutive equations using multi-value functions is the only way to o that. On the other han, the same constitutive relation can be written as a stanar function if velocity is expresse in terms of force ue to ash-pot [21]. Definitions for multi-values function an stanar function are iscusse in Chapter III. This way of expressing kinematic quantities in terms of ynamic quantities gives rise to a set of ifferential-algebraic equations. Such system of equations are generally referre to as semi-explicit DAEs. In some systems like those involving Coulomb friction, only implicit constitutive relations are possible [22]. In this thesis, a spring-mass system involving Coulomb friction is consiere an this accounts for a broaer class of ifferential-algebraic equations calle implicit DAEs.

17 4 v f crit 1 γ f crit f Fig. 1.: Constitutive relation for Bingham-type flui. Force ue to ash-pot cannot be written as a function of kinematic variables (that is, velocity of the mass). On the other han, as shown in the figure, velocity can be written as a function of the force in the ash-pot. At this point it shoul be clear that this way of expressing kinematic an ynamic quantities shoul not be confuse with the causality relationship. This ifference is clearly explaine in Reference [23]. For example, in many situations in Physics, the relation F = ma is confuse as a cause an effect relationship but it is not. Similarly, for the present work even if ynamic quantity is expresse in terms of kinematic quantity or the other way roun, either way it nee not be a cause-effect relationship. All these concepts introuce here are iscusse in etail in forth-coming chapters. On the whole, it can be summarize that ifferential inclusions (DIs) are an extension to the concept of iscontinuous ifferential equations with more etails from Reference [13], an orinary ifferential equations(odes) form a special class of ifferential-algebraic equations (DAEs). Details on this can be seen from Reference [24].

18 5 spring ash-pot m x f(t) Fig. 2.: Spring-mass-ash-pot system that is represente as a lumpe parameter system. m is the mass of the boy subject to external force f(t) an x enotes the isplacement of the system. From the present work without introucing misnomers like multi-value functions as in ifferential inclusions it can be shown that the Filippov systems can be moele using ifferential-algebraic equations [21]. At this point it can be claime that DAEs are mathematically more elegant an numerically more robust. The purpose of the rest of the thesis is to rive home this point. To keep things simple, we consiere a lumpe parameter system of a spring-mass-ash-pot system as in Figure 2 for analysis. Moeling the spring-mass-ash-pot system as a lumpe parameter system makes the stuy simple. This is iscusse in Reference [25]. A. Main contributions The main contribution of this stuy is to provie robust numerical techniques to solve the set of ifferential-algebraic equations obtaine from governing equations of lumpe parameter systems consiere. In orer to accommoate for ifferent situations, three cases are consiere whose characteristics are ifferent an representative. The cases consiere are: (1) Systems with non-linear constitutive relations governe by semi-explicit DAEs.

19 6 (2) Systems with Coulomb friction governe by implicit DAEs. (3) Systems with unilateral constraints governe by ifferential-algebraic inequalities. A generalize numerical framework is evelope to accommoate all the cases. B. Organization of the thesis The remainer of this thesis is organize as follows. Chapter II iscusses various Filippov systems that can be treate using ifferential-algebraic equations approach. Chapter III is about notation an efinitions that are neee for mathematical moeling. In Chapter IV, each of the systems iscusse in Chapter II are mathematically moele using the ifferential inclusions an the ifferential-algebraic equations approaches. Numerical algorithms that are evelope to treat these problems are iscusse in Chapter V. In Chapter VI, representative examples are consiere an the numerical algorithms are verifie for correctness. Conclusions are rawn in Chapter VII.

20 7 CHAPTER II VARIOUS EXAMPLES: SMOOTH AND NON-SMOOTH DYNAMICAL SYSTEMS This chapter presents various ynamical systems, which will be further stuie in subsequent chapters. These ynamical systems can be mathematically moele using the framework provie by ifferential-algebraic equations, which is the main focus of this thesis. An alternate way to moel these systems (which is common in the mathematics literature) is using ifferential inclusions. As mentione earlier it is not always possible to express ynamic quantities like force in terms of kinematic quantities such as isplacement an velocity. Below few examples shall be iscusse of how such situations arise in practicality. These examples woul serve for a better unerstaning of the concepts of ifferential-algebraic equations an ifferential inclusions in all. A. Classical simple harmonic oscillator Firstly, starting off with a smooth system, a classic example is a simple harmonic oscillator. An unampe spring hanging on its own weight as in Figure 3 can be consiere as one of the examples of simple harmonic oscillator. The response will be smooth in both kinematic an ynamic quantities (if the forcing function is smooth). That is, the linear spring is incapable of proucing non-smooth ynamics. B. Spring-mass-ash-pot systems For all the below cases consier a system as in Figure 2. The Filippov-type iscontinuity arises in these cases ue to the ynamics of flui in the ash-pot consiere.

21 8 x k t m (a) (b) Fig. 3.: A simple harmonic oscillator: (a) Pictorial escription (b) A typical isplacement versus response uner free vibration (i.e., the external force is zero). 1. For the first case consier a visco-elastic ash-pot. The constitutive relation between velocity of the system an force ue to ash-pot can be written such that velocity v is written as a function of force ue to ash-pot f an viceversa. For such systems the governing equations reuce to a set of ODEs. A graphical representation of the constitutive relation can be seen in Figure 4. This system also comes uner the category of smooth systems. 2. Now consier a Bingham-type ash-pot with linear characteristics. In this moel it is not possible to write force ue to ash-pot f in terms of velocity v but the other way roun is possible. This can be seen from Figure A Bingham-type ash-pot with non-linear but monotonic characteristics is consiere. Even in this case, the force ue to ash-pot F cannot be written as a function of velocity v but v can be written as a function of f. The only ifference is that since it is a non-linear moel, obtaining analytical solutions is not straight-forwar. For example, a Bingham-type ash-pot moel governe by a monotonic function can be seen in Figure 6.

22 9 v f c 1 f 1 c v (a) v versus f (b) f versus v Fig. 4.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by visco-elastic moel. 1 f crit γ v f crit f f crit F γ 1 f crit v (a) v versus f (b) F versus v Fig. 5.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Bingham-type flui of linear characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function.

23 1 v F f crit f crit f f crit f crit v (a) v versus f (b) F versus v Fig. 6.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Bingham-type flui of non-linear monotonic characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function. 4. Another special case among non-linear moels coul be a Bingham-type ashpot with non-monotonic characteristics. This moel coul be challenging in the sense that it being non-monotonic it may give rise to bifurcations or limit points. Hence, the choice of the appropriate numerical metho makes it more interesting case. A representative example of such a non-linear non-monotonic moel is shown in Figure 7. The same problem arises when we try to write force in terms of kinematic variables as in the previous case. C. Coulomb friction moel The next an most important example of Filippov-type system is a ynamical system involving Coulomb friction. This type of friction arises ue to the movement between soli surfaces. It is not even possible to write force ue to friction in terms of

24 11 v F f crit f crit f f crit f crit v (a) v versus f (b) F versus v Fig. 7.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Bingham-type flui of non-linear nonmonotonic characteristics. In Figure (a), the notation is lowercase for f to represent a stanar function whereas in Figure (b), the notation of force ue to ashpot is written as F in upper case to represent a multi-value function. kinematic quantities. The constitutive relations obtaine are only implicit relations. Many of the mechanical systems in inustry involve friction, Coulomb friction moel is a popular one. Thus, this stuy is important from both theoretical an practical points of view. Figure 8 gives a pictorial representation between velocity of the system an frictional force, an it is clear that writing explicit expressions between velocity an force is not possible. Remark C.1 The non-linearity in Bingham-type moels is ue to flui friction. But the non-linearity in Coulomb moel is ue to ry friction between soli surfaces. D. System with unilateral constraint The stuy of system ynamics when subjecte to unilateral constraints is much more interesting. The ynamics of such systems with constraints gives rise to a sys-

25 12 v F f crit f crit f crit f v f crit (a) v versus f (b) f versus v Fig. 8.: A pictorial representation of constitutive relation between velocity an frictional force for a system with friction governe by Coulomb friction moel. tem of ifferential-algebraic inequalities. For example, consier a spring-mass system with Bingham-type ash-pot of linear characteristics as in Figure 5, an a unilateral constraint place at a certain istance such that there is a constraint on the isplacement as shown in Figure 9. This forms a special case an oes not come uner Filippov-type system since at the of impact the iscontinuity in the system of equations is ifferent. A contact-resolution algorithm is use at impact to fin the of impact an ynamics is suitably moifie uring impact an release. Thus it can be seen from this chapter that in various practical situations where the constitutive relations cannot be expresse such that ynamic quantities are written as stanar functions or explicit expressions of kinematic quantities we may resort to writing the constitutive relations in a way giving rise to DAEs/DAIs. With this motivation we can procee to the next chapter which gives in etail all the notations, efinitions an existence theorems that woul be helpful in unerstaning the point presente in this thesis coherently.

26 13 spring ash pot m x f(t) L Fig. 9.: A spring-mass-ash-pot system representing a lumpe parameter system with unilateral constraint place at a istance of L units from mass m restricting the isplacement x of the system. Here, f(t) is the external force acting on the system.

27 14 CHAPTER III NOTATION AND DEFINITIONS In this chapter, relevant notations an efinitions shall be introuce, which will be use in the remainer of this stuy. A. Convex analysis preliminaries Definition A.1 (Function) Let X R an Y R be two sets. A real-value function f : X Y associates to each element x X a unique value in Y, which is enote by f(x). The sets X an Y are respectively calle the omain an co-omain. The graph of the function f is efine as graph[f] := {(x, f(x)) x X} (3.1) Definition A.2 (Continuous function) Let f be a function such that f : R R an let c be an element of the omain. The function f is sai to be continuous at the point c if for every ɛ > there exists a δ > such that x c < δ implies f(x) f(c) < ɛ, x R. This is also calle the epsilon-elta efinition for a continuous function. Definition A.3 (Lipschitz continuity) A function f : R R is sai to be Lipschitz continuous if there exists a real constant K such that for all x 1 an x 2 in R, the conition f(x 2 ) f(x 1 ) K x 2 x 1 satisfies. The constant K is also calle the Lipschitz constant. Definition A.4 (Convex set) A set C R is convex set if for each x C an y C also (1 q)x + qy C for arbitrary with q 1

28 15 Definition A.5 (Open set) A subset U of R is calle an open set if, given any point x in U there exists a real number ɛ > such that given any point y in R whose Eucliean istance from x is smaller than ɛ, y also belongs to U. Equivalently, a subset U of R is open if every point in U has a neighborhoo in R containe in U. This efinition will be useful to efine the close set. Definition A.6 (Close set) A close set is a set whose compliment is an open set Definition A.7 (Set-value function/multi-value function) A set-value function/multivalue function F : R R is a map from R to the subsets of R, that is for every x R, we associate a (potentially empty) set F (x), x R. Another efinition that can be use is, a set-value function/multi-value function F (x), x R is a function almost everywhere except at a finite number of isolate points where F (x) forms a subset of R. A set-value function can therefore contain vertical segments on its graph. To istinguish between a function an a set-value function, lower case shall be use for a function (e.g., f(x)), an upper case for a set-value function (e.g., F (x)). By efinition, a function is single-value, which exclues vertical lines, loops an surfaces on its graph. Hence, herein the usage of multi-value functions is not prescribe to. This efinition of set-value function or multi-value function loses its valiity in the sense of the efinition of a function. Definition A.8 (Measurable multi-value function) A multi-value function F : S R is measurable if for every open(close) C R, x S : F (x) C is Lebesgue measurable. Definition A.9 (Upper semi-continuous function) A set-value function F (x) is upper semi-continuous in x if for y x implies

29 16 sup a F (y) inf b F (x) a b (3.2) Having unerstoo the efinition of set-value/multi-value function, let us give the signum notation here. For the present thesis, two ifferent signum notations were use. Matlab an Wolfram Mathworl uses the following notation, where every mapping is one-one satisfying the efinition of a function. 1 x < sign[x] = x = 1 x > (3.3) This notation is commonly use to moel ifferential inclusions, which involves set-value functions Sign[x] { 1} x < [ 1, 1] x = (3.4) {1} x > B. Orinary ifferential equations (ODEs) With the above efinitions an notations, we are almost equippe with everything to unerstan ifferential inclusions an ifferential-algebraic equations. But both the concepts share a ifferent relationship with ODEs. Differential inclusions are an extension to the concept of orinary iscontinuous ifferential equations where as the ODEs form a special class of DAEs. So, this is the right place to efine ODEs an their properties in general. Before jumping into the efinitions of orinary ifferential equations an their existence theorems, let us introuce some notation, which is the same as in Reference

30 17 [1] as follows: I : an open interval on the real line < t < (or) two real numbers a an b exist such that a < t < b C k (I) : the set of all complex-value functions having k-continuous erivatives on I f C k (I) : If k f t k exists an continuous on I D : An open connecte set (Domain), in the real (t, x) plane C k (D) : the set of all complex-value functions having k-continuous partialerivatives on D f C k (D) : If all the k th -orer partial erivatives exist an are continuous on D k f t p x q, where (p + q = k) C(I) an C(D) : the continuous functions C (I) an C (D) on I an D Let f C(D) be a real-value function continuous in the omain D, then the efinition of the orinary ifferential equation can be evelope as follows. Definition B.1 [1] To fin a ifferentiable function φ efine on a real t interval, I such that 1. (t, φ(t)) D; (t I) 2. φ (t) = f(t, φ(t)); (t I, = ) t This problem is calle an orinary ifferential equation of first orer an is enote by x = f(t, x) ( = t ) (3.5)

31 18 If at least one such interval I an function φ exists, then φ is terme as solution of the orinary ifferential equation (3.5). It shoul be note that there coul be many or no solution existing for (3.5) too. There are several existence an uniqueness theorems for the solution of orinary ifferential equations evelope by numerous mathematicians. In this report, two most important an relevant ones of them are quote. For the following theorems, the notations as escribe above will be use. Theorem B.2 (Cauchy-Peano existence theorem) If f C on the rectangle R, then there exists a solution φ C 1 of the orinary ifferential equation (3.5) on t τ α for which φ(τ) = ξ. where, R is a rectangle efine as R : t τ α x ξ b (a, b > ) (3.6) Theorem B.3 (Carathéoory theorem) Let f be efine on R, an suppose it is measurable in t for each fixe x, continuous in x for each fixe t. If there exists a Lebesgue-integrable function m on the interval t τ a such that f(t, x) m(t) ((t, x) R) (3.7) then there exists a solution φ of (3.5) in the extene sense on some interval t τ β, (β > ), satisfying φ(τ) = ξ. Mathematical proofs to the above two theorems can be foun in Reference [1]. C. Differential inclusions (DIs) The stuy of ifferential inclusions an its application starte almost in the mithirties of 2th century. It has gaine a mathematical form with the evelopment of control theory. There were several mathematicians who worke on eveloping existence an uniqueness conitions for the solution of ifferential inclusions. Filippov

32 19 along with Weżewski evelope the most celebrate Filippov-Ważewski relaxation theorem uner some weak assumptions [26]. Filippov [13] extene the concept of iscontinuous ifferential equations to ifferential inclusions an evelope existence an uniqueness of the solutions. He consiere a ifferential inclusion of the form ẋ F(x, t), x(t ) = x (3.8) where F(x, t) is a set-value function as efine before. The following theorem states the conition for existence an uniqueness of the solution for such a ifferential inclusion. Theorem C.1 (Filippov s theorem [13]) Assuming that F(t, x) is an upper semicontinuous function of x, measurable in t, an that F(t, x) is a close, convex set for all t an x, existence of the solutions for the initial value problem (3.8) for a sufficiently small interval [t, t + ɛ], ɛ > is then followe. If F oes not blow-up then global existence can also be shown. An the uniqueness is prove with an aitional conition that F(t, x) satisfies a one-sie Lipschitz conition which states as (x 1 x 2 ) T (F(t, x 1 ) F(t, x 2 )) C x 1 x 2 2 (3.9) for some C, x 1 an x 2. As mentione earlier, it can be seen that solutions to ifferential inclusions in the sense of Filippov exist when moele using multi-value functions. Having sai that, in further chapters when specific lumpe parameter systems are consiere, moeling using ifferential inclusions becomes clear.

33 2 D. Differential-algebraic equations (DAEs) The system of equations of the form K(t, x, ẋ) = (3.1) are calle ifferential-algebraic equations(daes). If K x is non-singular then the above equation can be written as an explicit orinary ifferential equation(ode) [18]. Thus a generalization can be mae here that (ODEs) are a special class of DAEs. An explicit ODE is of the form, x t f(t, x) =. This is a special case of a more general system of the form (3.1). The equations of the form (3.1) are calle implicit ODEs an popularly DAEs. Definition D.1 The system of equations of the form ẋ 1 = f(t, x 1, x 2 ) x 1 (t) R p, x 2 (t) R q (3.11a) = g(t, x 1, x 2 ) g R q (3.11b) are calle semi-explicit DAEs. The DAEs given by (3.11a) an (3.11b) can be thought of as a limiting case of the singularly perturbe ODE. ẋ 1 = f(t, x 1, x 2 ) ɛẋ 2 = g(t, x 1, x 2 ) (3.12a) (3.12b) where g x 2 is assume to be non-singular. The limiting case where ɛ = of (3.12a) an (3.12a) is calle the reuction problem as in (3.11a) an (3.11b). Definition D.2 (Differential inex of a DAE) Differential inex of a DAE is a non-negative number that enotes the complexity level of a ifferential-algebraic equa-

34 21 tion to be numerically solve an analyze. It can be taken as a measure of how close/far from an orinary ifferential equation i.e., a smaller inex inicates it is quite simpler compare to a higher-inex DAE. For example, if the algebraic equation (3.11b) is ifferentiate once with respect to t resulting in a set of implicit ODEs. Then the ifferential inex of DAE is sai to be one. Another notation for inex of a DAE is perturbation inex an etails on this can be seen from Reference [19]. If the algebraic constraint (3.11b) is an inequality then the governing equations turn out to be a system of ifferential-algebraic inequalities (DAIs). A moel problem that gives rise to DAIs is consiere to give more insight. Research on similar lines can be foun in Reference [27]. Equippe with these efinitions an notations we can now go ahea an efine the problem an the governing equations for various cases consiere in Chapter II using ifferential-algebraic equations perspective.

35 22 CHAPTER IV MATHEMATICAL MODELING In this thesis, only lumpe parameter systems are consiere for analysis to keep the stuy simple. A lumpe parameter system is a way to approximate the istribute system behavior uner certain assumptions. This greatly simplifies the stuy of the system uner consieration [25]. It is mainly useful in the fiels of electronics, mechanics an control theory. Iealization of energy storage/issipation is one of the general applications of lumpe parameter systems. A goo example of a lumpe parameter moel is the representation of an electrical network, represente by a circuit iagram as shown in Figure 1. The parameter resistance is lumpe an represente by iealize resistor instea of consiering the Maxwell equations of actual system. Another goo example is a thermal system where average temperature of the system is consiere as a lumpe parameter. A. Problem efinition an governing equations The lumpe parameter system consiere for all problems in this thesis is a springmass-ash-pot system with a linear spring an varying characteristics of ash-pot as i v R Fig. 1.: A lumpe parameter moel of an electronic circuit where i enotes current, R enotes resistance an v enotes voltage source.

36 23 iscusse in Chapter I. For the rest of the thesis, let x enote the isplacement an v enote the velocity of the system which are consiere to be the kinematic variables of the system. Let f s be the force ue to spring an f be the force ue to ash-pot on the mass m. With these notations, let us first see how governing equations look like in ifferential inclusions approach in the most general form for the same system with linear spring an ash-pot with Bingham-type flui. The momentum balance equation as a ifferential inclusion is written as v(t) 1 m [F (t) F s(t) F (t)] (4.1) The constitutive relations for the spring an ash-pot are: α(x, F s ) = β(v, F ) = (4.2a) (4.2b) For the various examples consiere in this thesis, the constitutive relations (4.2a) an (4.2b) are written as inclusions. Inclusions are witten using the signum notation Sign[ ] as in Chapter III. Equations (4.1), (4.2a) an (4.2b) form the governing equations in ifferential inclusions approach. Depening on the characteristics of the ash-pot we wish to consier for analysis β(v, F ) is varie an the system ynamics is stuie. With that insight into the way in which the governing equations in ifferential inclusion approach woul look like, let us now see the general form of governing equations of the same system in DAE approach. Momentum balance gives rise to a ifferential equation an written as v(t) = 1 m [f(t) f s(t) f (t)] (4.3)

37 24 The constitutive relations for the energy storage an issipation mechanisms are written as α(x, f s ) = β(v, f ) = (4.4a) (4.4b) where α(x, f s ) = is x(t) = fs k for linear spring. The relation (4.4b) woul turn out to be either explicit or implicit algebraic equations which keeps varying specific to the problem consiere from linear to non-linear monotonic an non-linear non-monotonic characteristics. Equations (4.3), (4.4a) an (4.4b) form the set of governing equations in DAE approach. Also, it is to be note that upper-case notation is use for moeling using ifferential inclusion approach an lower-case for ifferential-algebraic equations approach for all problems consiere. In orer to unerstan the robustness of the numerical framework evelope, we consier ifferent moels for analysis to accommoate non-linear constitutive relations which give rise to semi-explicit ifferential-algebraic equations, lumpe parameter system with Coulomb friction which gives rise to implicit ifferential-algebraic equations, an system with constraint which gives rise to ifferential-algebraic inequalities. In the following sections, each problem is ealt separately. B. Lumpe parameter system with a Bingham-type ash-pot of non-linear monotonic characteristics A lumpe parameter system as shown in Figure 11 is consiere. The spring consiere is a linear spring an the ash-pot is consiere such that velocity of the system is a function of force ue to ash-pot. For the present case, the function is consiere to be non-linear monotonic in nature. In Reference [21], a similar problem

38 25 spring ash-pot m x f(t) Fig. 11.: Spring-mass-ash-pot system representing a lumpe parameter system subjecte to an external force f(t). Displacement of the system is given by x(t). was solve for a ash-pot with Bingham-type flui where the function was consiere to be linear in nature. The governing equations of the system in ifferential inclusions approach will be similar to the equations (4.1), (4.2a) an (4.2b) with the constitutive relations written as follows: F s (t) = kx(t) (4.5a) F (t) = Sign[v(t)]f crit + ηβ(v(t)) (4.5b) f crit is the critical force for Bingham-type fluis an β(v) will be consiere as a nonlinear monotonic function which is cubic in nature for this case as shown in Figure 12(a). From this figure, it is evient that F is expresse in terms of v only as an inclusion an not as a stanar function. The governing equations for this system in ifferential-algebraic equations approach is as follows. Momentum balance is a ifferential equation an written as v(t) = 1 m [f(t) f s(t) f (t)] (4.6)

39 26 F v f crit f crit v f crit f crit f (a) v vs. f (b) F vs. v Fig. 12.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Bingham-type flui of non-linear monotonic characteristics, a cubic function in this case. In Figure (a), the notation is lower-case for f to represent a stanar function i.e., velocity can be written as a stanar function of force ue to ash-pot. In Figure (b), the notation of force ue to ash-pot is written as F in upper case to represent a multi-value function i.e., force ue to ash-pot can be written in terms of velocity only by using multi-value function.

40 27 an x(t) = f s k v(t) = f f crit γ(f ) 2 (f sign[f ]f crit ) f > f crit (4.7a) (4.7b) For the purpose of serving as an example of monotonic nature, the Bingham-type ash-pot is characterize by a function that is cubic in nature as in Figure 12(b). Equations (4.6), (4.7a) an (4.7b) form the governing equations in ifferential-algebraic equations approach. C. Lumpe parameter system with a Bingham-type ash-pot of non-linear nonmonotonic characteristics The secon case consiere is a similar lumpe parameter system but the characteristics of the ash-pot are moifie such that a non-linear non-monotonic function characterizes the ash-pot. This problem is consiere for analysis because the iea is to evelop a generalize framework to treat a wier class of Filippov systems. Some of the numerical techniques that are use to solve the non-linear monotonic functions fail for non-monotonic cases at limit points or bifurcation points. This leas to the challenge of eveloping such a generalize framework. Hence, the stuy of such a non-smooth system with a ash-pot of non-linear non-monotonic characteristics bears practical importance. Let us consier the same lumpe parameter system with ash-pot characteristics as shown in Figures 13(a) an 13(b).

41 28 F v f crit f crit v f crit f crit f (a) v vs. f (b) F vs. v Fig. 13.: A pictorial representation of constitutive relation between velocity an force ue to ash-pot for a ash-pot governe by Bingham-type flui of non-linear nonmonotonic characteristics, a polynomial function in this case. In Figure (a), the notation is lower-case for f to represent a stanar function i.e., velocity can be written as a stanar function of force ue to ash-pot. In Figure (b), the notation of force ue to ash-pot is written as F in upper case to represent a multi-value function i.e., force ue to ash-pot can be written in terms of velocity only by using multi-value function. To write the governing equations in ifferential inclusions approach, it is the same as in previous case with (4.1), (4.5a) an (4.5b) but with β(v) as a non-linear non-monotonic function which is shown in Figure 13(a). The governing equations of this system in ifferential-algebraic equations approach, is written as follows. The momentum balance equation is the same ifferential equation v(t) = 1 m [f(t) f s(t) f (t)] (4.8)

42 29 The constitutive relations are as follows: x = f s k v = f f crit (1 +.1(f 2 4)(f 2 9))(f sign[f ]f crit ) f > f crit (4.9a) (4.9b) Equations (4.8), (4.9a), an (4.9b) represent the governing equations of the system in the ifferential-algebraic equations approach. The relation (4.9b) is only for the purpose of representating a non-monotonic function. D. Lumpe parameter system with Coulomb friction Moels with Coulomb friction are of great interest among Filippov systems as iscusse in Chapter I. For the present case, the system consiere is shown in Figure 14 involving Coulomb friction. Mechanical systems with Coulomb friction is a classical example of Filippov systems. One important thing to be note here is that force f is ue to ry friction unlike flui friction as in previous cases. Hence, the governing equations nees moification which woul affect the system ynamics. The system with such friction is pictorially represente in Figure 14. Coulomb moel is graphically represente by the Figures 15(a) an 15(b). From Figures 15(a) an 15(b) it is seen that either way it is not possible to express one quantity in terms of the other but one can write implicit relations. Keeping the notations for isplacement, velocity an the forces in the system the same the governing equations for the system in ifferential inclusions approach are written as v(t) 1 m [f(t) F s(t) F (t)] (4.1)