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NASA's Ingenuity Mars Helicopter has recently performed the first powered controlled flight on another planet. However, in spite of success of this mission, rotary wing solution might not be the optimal technology for flying in thin Martian atmosphere. The low density of the atmosphere (only 1% of the Earth's) forces the aircraft to fly at low Re numbers, which significantly deteriorates the performance of a rotary wing. On the other hand, insects fly very efficiently at the same Reynolds numbers, with great aerial capabilities, which makes the insect-type aircraft a promising concept for Mars exploration. To this end, the novel optimization algorithm for development of insect-type aerial vehicle, capable of flight in Martian atmosphere, is proposed. The optimization algorithm combines Discrete Mechanics and Optimal Control (DMOC) approach with quasi- steady aerodynamical model. Results of the numerical experiment indicate that the developed optimization algorithm can be successfully used for computationally efficient optimization of design and dynamics of an insect-type flapping aerial vehicle for Mars exploration purposes — where higher fidelity fluid-structure coupled procedures fail to deliver because of computational non-efficiency. The next natural step, in the development process of the presented algorithm and research of flapping flight on Mars, is to expand the existing aerodynamic model with forward flight features.

Insect-like flapping-wing vehicle (FWV) concepts have great potential that implies excellent aerial capabilities, like hovering, robustness, great maneuverability, and very efficient flight in low Reynolds number regimes. However, complex multiphysics of flapping-wing systems hinders sufficient progress in the FWV development. The absence of a reliable and computationally efficient optimization tool is a significant part of that problem. To address this, the novel flapping wing optimization algorithm is developed. The algorithm combines Discrete Mechanics and Optimal Control method and quasisteady aerodynamical model. The algorithm is then applied to minimize the energy consumption of the insect- like flappingwing aerial vehicle, more precisely, to find the most efficient flapping pattern for hovering. Four types of numerical optimizations were conducted, based on how the flapping frequency is treated and which flapping pattern is used as an initial guess, and all four converged to almost identical solution and achieved better energy efficiency. The results of the numerical experiments prove that this approach to flapping wing vehicle dynamics optimization is efficient, robust and well suited for design optimization numerical algorithms. To this end, presented approach can be utilized in the design campaigns of flying vehicles in completely new environments, as well as for more conventional engineering applications.

In this work we study dynamics of flapping-wing micro unmanned aerial vehicle (MUAV) of the insect type. We present a novel control algorithm based on the reduced mechanical model and Discrete Mechanics and Optimal Control (DMOC) approach. In recent years, the control of such systems has drawn considerable attention of both researchers and engineers. However, there are still many challenges in the process: for example, synthesis of wing mechanism - that allows for complex kinematics with high flapping frequencies - is cumbersome due to involved aerodynamics of the insect type flapping wings (small Re number, unsteady flow, formidable vortex dynamics on the both edges etc.). Also, due to the coupling between fluid and high-frequency mechanism, the optimal control of such aerial vehicle represents a challenging task.

In order to study dynamics of flapping wing moving at low Reynolds number in ambient fluid, we adopt geometric modelling approach of fully coupled wing-fluid system, incorporating Boundary Integral Method for calculating added masses, and Lie group rigid body integrator. Our aim is to explore numerical advantages of such an approach in comparison to the standard procedures that comprise volume discretization of a fluid domain.

By using principles of geometric mechanics - under assumption of its Hamiltonian structure - it is possible to treat solid-fluid system in a monolithic way without separating its fluid and solid part, as it is standard procedure in the contemporary computational practice. This strategy involves symplectic and Lie-Poisson reduction of the airfoil-fluid Hamiltonian system and allows for significantly improved numerical efficiency, since full discretization of the fluid domain becomes non-obligatory. However, in order to take into account viscous effects of fluid flow, numerical imposing of Kutta condition on the airfoil trailing edge is mandatory. To this end, a new hybrid approach - combining analytical and numerical modeling techniques as well as model reduction procedures - for dynamics simulation of airfoil motion in fluid flow with vortices will be described.

Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential- equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential- algebraic equations (DAEs) that requires numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion- normalisation constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations. This can be done due to the isomorphism of the Lie algebras of the SO(3) group and the unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem (by using incremental rotation vector as integration variable at Lie algebra level), the proposed algorithm may also exhibit numerical advantages. Depending on the numerical case at hand - as it is in the presented flight example with 'slower' dynamics and steady rotation component - the method allows for utilization of longer integration steps and overall better numerical accuracy. This is because numerical integration of three non-linear ODEs in terms of incremental rotation vector within the proposed novel algorithm follows a non-linear kinematical rotation update more accurately than integration of a four (linear!) ODEs (as it appears in the standard algorithm), resulting in more accurate simulation results. The effect of better numerical accuracy will be more accented if overall flight vehicle dynamics allows for longer integration steps and aircraft motion pattern involves steady rotational component within its 3D motion.

In order to study dynamics of a rigid body moving at low Reynolds number in ambient fluid, we take geometric modelling approach of fully coupled body-fluid system, incorporating Boundary Integral Method of Galerkin type for calculating added masses, and Lie group rigid body integrator. Our aim is to explore numerical advantages of such an approach in comparison to the ’standard’ procedures that comprise volume discretization of a fluid domain.

In order to study dynamics of multibody system (MBS) moving at low Reynolds number in ambient fluid, we take geometric modelling approach of fully coupled MBS-fluid system, incorporating Boundary Integral Method of Galerkin type for calculating added masses, and Lie group MBS integrator. Our aim is to explore numerical advantages of such an approach in comparison to the ’standard’ procedures that comprise volume discretisation of a fluid domain.

U fokusu istraživanja je prilagodba i primjena novoprojektiranih geometrijskih integracijskih algoritama na numeričke zadaće dinamike letjelica te je ocijenjena njihova pogodnost i kvaliteta za primjenu u okviru numeričkih simulacijskih procedura za izračunavanje dinamičkih odziva zračnih i svemirskih letjelica. Za razliku od standardnih numeričkih metoda integracije običnih diferencijalnih jednadžbi koje operiraju u vektorskim prostorima, ciljani geometrijski algoritmi projektirani su na mnogostrukostima i Lievim grupama. Polazne geometrijske metode uobličene su na način da algoritamski zadovoljavaju određena kinematička i dinamička ograničenja koja proistječu iz diferencijalno-geometrijske strukture sustava. Nakon pojedinačnog opisa izvoda matematičkih modela odabranih algoritama i njihove prilagodbe za ciljane zadaće integracije dinamike letjelica, prezentirani su primjeri primjene geometrijskih integracijskih metoda te njihova usporedba s klasičnim procedurama vektorskih prostora. The research focuses on the adaptation and application of a newly designed geometric integration algorithms on numerical tasks of flight vehicle dynamics and evaluation of their benefits and quality of adaptation in the framework of numerical simulation procedures for calculating the dynamic response of aircrafts and spacecrafts. With regard to the standard numerical methods for integrating ordinary differential equations that operate on vector spaces, targeted geometric algorithms are designed on manifolds and Lie groups. Starting geometric methods are shaped in the form that algorithmically satisfies kinematic and dynamic constraints that come from the differential geometric structure of system. Upon completion of description of the individual derivation of mathematical models of the selected algorithms and their adaptation to the target tasks of integration of flight vehicle dynamics, the examples of applying geometric integration methods are presented and compared with classical procedures of linear vector spaces. This thesis is organized in seven chapters, as follows: Chapter 1: Introduction. This chapter presents the relevance of the research. It gives an overview of the relevant literature related to the topic of the thesis and explains the scope of the research. The first chapter is divided into five sections which includes: motivation, overview of the previous research, research goals and hypotheses, methodology of the research, expected scientific contributions and structure of the thesis. Chapter 2: Manifolds. In this chapter manifolds are introduced and explained. Firstly manifolds are explained and then vector fields, vector spaces, curves and vector fields commutation are shortly described. The mathematics and terminology described here enables a better understanding of the following sections. Chapter 3: Rotations and the SO(3) group. In this chapter rotations and their mathematical description in terms of groups and manifolds is presented. Until now no formal mathematical mention of the rotations was made and the dynamics was mostly focused on particles and body translations. After the detailed presentation of rotations of the following chapter the description of the individual derivation of mathematical models of the selected algorithms is presented. Chapter 4: Lie-group integration method for constrained multibody systems in state space. Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is described, adapted and applied to the helicopter forward dynamics in this chapter. Mathematical model of MBS dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on the system state space modelled as a Lie-group. Since formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local rotational coordinates are necessary and kinematical differential equations (that are prone to singularities in the case of 3-parameters-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe-Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. Dynamic simulation procedures of aircraft 3D motion need robust and efficient integration methods in order to allow for reliable (and possibly real-time) simulation missions. To this end, derivation of such integration schemes in coordinate-free Lie-group setting could be a promising start, since Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and additional algebraic constraints as well as kinematical differential equations. These integration characteristics should be especially beneficial for the flight vehicle simulation missions since a realization of the aircraft complex 3D maneuvers often requires numerical forward dynamics that includes complete 3D rotation domain. In such cases, the utilization of the 'standard' vector-space-based modeling procedures (with the local rotation parameters) leads toward kinematical singularities and re-parameterization of the rotation domain, which requires further computational burden. Along this line, a numerical integration scheme in Lie-group settings for the helicopter forward dynamics is presented and discussed in this chapter. The presented formulation provides a compact integration platform for smooth flight vehicle dynamics numerical integration, independently of the character of the 3D rotations involved and without introducing additional 'artificial' algebraic constraints. For the initial case study a simple maneuver of a helicopter is selected with generic transport helicopter modeled as a single 6DOF rigid body problem. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for a MBS with kinematical constraints of general type and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (3-parameters-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations). Chapter 5: Singularity-free time integration of rotational quaternions using non redundant ordinary differential equations. In this chapter a novel time stepping scheme for solving rotational kinematics, formulated as an ODE in terms of unit quaternions, is described, adapted and applied to the numerical forward dynamics of a fixed-wing aircraft. This scheme inherently respects the unit-length condition without including it explicitly as a further equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAE). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form an ordinary differential-equations (ODE) on Lie-algebra so(3) of the rotation group SO(3). This rotation vector then defines an incremental rotation (and thus the associated unit-quaternion), and the rotation update is determined by projection of the incremental vector on the quaternion group via pertinent exponential mapping. Since the kinematic ODE on so(3) can be solved by using any standard (possibly higher-order) ODE integration scheme, the result of the procedure is a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions that allows for application of general ODE integration schemes, and thus avoids integration of DAE equations. This solves a long-standing problem of necessity to deal with DAE’s during integration of rotational kinematics, which has been a major drawback of using quaternions. As a numerical example, a 3D motion of a general aviation airplane, modeled as a flat-earth 6DOF single rigid body problem, is presented. Chapter 6: An angular momentum and energy conserving Lie-group integration schemes for rigid body rotational dynamics originating from Störmer-Verlet algorithm. This chapter describes two novel 2nd order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. Firstly described algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Störmer-Verlet integration algorithm for solving ordinary differential equations, which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a non-linear regime, such as non-linear SO(3) rotational group. Then, is described an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two 'main' motion integrals of a rotational rigid body, i.e. spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the thesis, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to which the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body systems. The system under consideration is a satellite modeled as a single rigid body problem. Chapter 7: Conclusion. This chapter summarises the main contributions of the thesis and gives several recommendations for future research.

Dynamic simulation procedures of flight vehicle (fixed-wing, rotorcraft, UAV, satellite) 3D manoeuvres need robust and efficient integration methods in order to allow for reliable, and possibly real-time, simulation missions. Since flight vehicle 3D manoeuvres necessary include complete 3D rotation domain, such procedures also require an efficiency way of dealing with large 3D rotation. Usually, in this context, the simulation procedures built around the standard numerical ordinarydifferential-equations (ODE) based on three-parameters rotation variables (such as Euler angels) have their limitations, as they impose discontinuities or even singularities in the flight vehicle attitude integral curves. Most commonly, the quaternion representation is widely used in flight simulation to overcome the mentioned deficiency [1]. However, if quaternions are used for a parameterisation of the rotation manifold, the standard model leads to integration of differential-algebraic equations (DAE) that requires additional stabilisation of the algebraic constraint due to quaternions normalisation equation. Recently, a method of integration of rotational quaternions based on non-vectorial geometric Lie-group integration that leads to a minimal-form ODE integration (avoiding thus DAE integration) has been introduced in [2]. By adopting such an approach, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form an ODE on Lie-algebra so(3) of the rotation group SO(3), after which the integration incremental update on the configuration quaternion group Sp(1) is determined by the exponential map. Consequently, only a system of three independent ODEs is integrated and hence no stabilization of the unit-length constraint is necessary.

U radu je predstavljena integracijska procedura za numeričku integraciju dinamike rotacije krutog tijela koja koristi kvaternione (engl. quaternions) za parametrizaciju mnogostrukosti velikih rotacija. Na taj se način izbjegava upotreba lokalnih koordinata rotacije (npr. Eulerovih kuteva i sl.) koja nužno dovodi do kinematičkih singulariteta. Za razliku od standardnog algoritma koji se zasniva na redundantnoj formulaciji diferencijalno-algebarskih (DAE) jednadžbi na jediničnim kvaternionima (i pri tome nameće nužno rješavanje dodatne jednadžbe kinematičkog ograničenja modula jediničnog kvaterniona), predstavljena metoda koristi jednadžbe lokalnog rotacijskog vektora u minimalnoj formi koje operiraju na Lievoj algebri rotacijske grupe SO(3). Nakon što se izračuna lokalni rotacijski vektor za trenutni korak integracije korištenjem standardnog ODE rješavača (eventualno metode višeg reda), stanje rotacije se izravno rekonstruira na Sp(1) grupi kvaterniona preko odgovarajuće eksponencijalne mape. Na taj se način tijekom integracije prirodno slijedi konfiguracijski prostor rotacijske mnogostrukosti, modeliran pomoću grupe jediničnih kvaterniona Sp(1). Opisanim postupkom predlaže se izvorna metoda integracije rotacijskog tenzora pomoću jediničnih kvaterniona kojom se rješava standardni problem neredundantnog matematičkog modela koji je nužno iziskivao integraciju DAE jednadžbi (kao i numeričku stabilizaciju dodatnog kinematičkog ograničenja), dok u novo-opisanoj formi predloženi algoritam vodi jednostavnijem i numerički efikasnijem rješavanju minimalnog skupa ODE jednadžbi u vektorskom prostoru Lieve algebre.

In many engineering applications, such as satellite dynamics or various case-studies of the specific locomotion patterns in mechatronics and biomechanics, motion integrals of the system need to be conserved during numerical integration in order to reflect global physical properties of the analysed motion. Derivation of such integration schemes in Lie-group settings should be especially efficient since Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and artificial algebraic constraints as well as kinematical differential equations (differential equations that relate body rotation parameters’ derivatives and angular velocity). In this paper, different mathematical models and angular momentum conserving integration schemes, based on Lie-group DAE index 1 formulation of the system dynamics, will be investigated and tested through the several case-studies, such as satellite dynamics with the internal kinematical chain.

Dynamic simulation procedures of the aircraft 3D motion need robust and efficient integration methods in order to allow for reliable (and possibly real-time) simulation missions. Derivation of such integration schemes in coordinate-free Liegroup settings is especially efficient since Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and artificial algebraic constraints as well as kinematical differential equations. These integration features of the coordinate-free formulations should be especially beneficial for the flight vehicle simulation missions since a realization of the aircraft complex 3D maneuvers often requires numerical forward dynamics that includes complete 3D rotation domain. In such cases, the utilization of the ‘standard’ vector-space-based modeling procedures (with the local rotation parameters) leads toward kinematical singularities and re-parameterization of the rotation domain, which requires further computational burden. Along this line, a numerical integration scheme in Lie-group settings for the helicopter forward dynamics as well as inverse dynamics control problem is presented and discussed in the paper.

Störmer-Verlet integracijski algoritam, izvorno definiran kao integracijska shema za obične diferencijalne jednadžbe u linearnom vektorskom prostoru (s učestalom primjenom u okviru zadaća molekularne dinamike), ima više pogodnih značajki stabilne numeričke integracije: shema je eksplicitna, drugog reda točnosti, omogućava očuvanje integrala gibanja te je simplektična za Hamiltonove sustave. S polazištem u izvornom Störmer-Verlet algoritmu linearnog prostora, u ovom se radu predlaže konstrukcija Störmer-Verlet algoritma na Lievoj rotacijskoj grupi s ciljem njegove upotrebe za numeričku integraciju dinamike mehaničkih sustava. Opisan je izvod matematičkog modela Störmer-Verlet algoritma na Lievoj grupi konfiguracijskog prostora krutog tijela te je prezentirano više numeričkih primjera koji ukazuju na izvrsne numeričke značajke predloženog algoritma.

Design of the numerical integration methods that operate on manifolds and Lie-groups, instead of linear vector spaces like 'standard' ODE and DAE integrators, offer some attractive features such as numerical robustness and avoidance of the kinematical singularities as well as numerical efficiency of the code. The possibility of using global rotational coordinates for kinematic definition of the system certainly appeals for numerous applications. The additional motivation for taking this approach can be based on the fact that computational geometric formulation provides integration schemes that can preserve global characteristics of motion (such as conservation of the first integrals) in 'more naturally' manner than 'classical' methods that operate in the vector spaces.

A coordinate free Lie-group integration method for the constrained multibody systems (MBS) is proposed in the paper. Mathematical model of MBS dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on the system state-space modeled as a Lie-group. Since formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local coordinates are necessary and kinematical differential equations are completely avoided. A basis of the integration procedure is the Munthe- Kaas algorithm for ODE integration on the Lie- groups, which is re-formulated and expanded to be applicable for the integration of constrained MBS in DAE-index-1 form. In order to eliminate numerical constraint violation for the generalized positions and velocities during the integration procedure, two constraint stabilization algorithms are introduced: a projection method based on the constrained least square minimization algorithm via system global coordinates and the one-step non-iterative constraint manifold projection method by using local exponential coordinates. Two numerical examples, heavy top dynamics and satellite with mounted 5 DOF manipulator, are presented and documented. The proposed Lie-group DAE-index-1 integration method is easy-to-use for the MBS with the kinematical constraints of a general type and it is especially suitable for dynamics of the mechanical systems with a large 3D rotation motion domain.

A Lie-group integration method for constrained multibody systems is proposed in the paper and applied for numerical simulation of a satellite dynamics. Mathematical model of multibody system dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on Lie-group introduced as system ‘state-space formulation’. The basis of the method is Munthe-Kaas algorithm for ODE on Liegroups, which is re-formulated and expanded to be applicable for the integration of constrained multibody dynamics in DAE index 1 form. The constraint violation stabilization algorithm at the generalized position and velocity level is introduced by using two different algorithms: a first one that operates directly on the ‘state-space’ manifold and, a second one, that uses Cartesian rotation vectors as local coordinates for the generalized positions. A numerical example of ‘dual-spin’ satellite that demonstrates the proposed integration procedure is described and discussed at the end of the paper.

Analiza kinematike samohodnog kotača pogonjenog ekscentričnom masom provedena u ovom dokumentu obuhvaća sintezu i analizu kinematičkog modela sustava s ciljem određivanja funkcija promjena pogonskih pomaka aktuatora.

Analiza dinamike samohodnog kotača pogonjenog ekscentričnom masom provedena u ovom dokumentu obuhvaća sintezu i analizu dinamičkog modela te modeliranje jednadžbe algoritma vođenja.

A Lie-group integration method for constrained multibody systems is proposed in the paper. Mathematical model of multibody system dynamics is shaped as DAE system of equations of index 1, while dynamics is evolving on Lie-group introduced as system ‘state space formulation’. Integration algorithm operates directly with angular velocities and rotational matrices and no local (generalized) coordinates are introduced. The basis of the method is Munthe-Kaas algorithm for ODE on Lie-groups, which is re-formulated and expanded to be applicable for the integration of constrained multibody dynamics, where constraint violation stabilization is one of the important issues that must be successfully solved.

Provedena je statička analiza sustava koja se temelji na MSC Adams kinematičkoj analizi.

The paper surveys geometric mathematical framework for computational modeling of multibody system dynamics. Starting with the configuration space of rigid body motion and analysis of it’s Lie group structure, the elements of respective Lie algebra are addressed and basic relations pertinent to geometrical formulations of multibody system dynamics are surveyed. Dynamical model of multibody system on manifold introduced, along with the outline of geometric characteristics of holonomic and non-holonomic kinematical constraints.

U radu je izvršeno dimenzioniranje dimnjaka na koji je priključeno jedno plinskog trošilo snage 24 kW. Korišten je postupak proračuna dimnjaka prema europskoj normi EN 13384, te je načinjen računalni program u programskom jeziku Python. Rezultati proračuna su uspoređeni s rezultatima komercijalnog programa KESA ALADIN za proračun dimnjaka, koji se temelji na istoj normi, a ima ugrađeno automatsko pridruživanje podataka (standardnih promjera, izbora koeficijenata i sl.), te baze podataka koje specificiraju proizvođači opreme. Slaganje rezultata vlastitog proračuna s rezultatima komercijalnog programa je vrlo dobro. U drugom dijelu rada su analizirane prijelazne pojave strujanja dimnih plinova prilikom uključivanja plinskog trošila u ljetnim mjesecima. Poznato je da se za vrijeme vrućih ljetnih dana događaju trovanja ugljičnim monoksidom, koja su uzrokovana neispravnošću plinskog trošila (nepotpunim izgaranjem) i povratom dimnih plinova u prostoriju. Ako dimnjak prolazi zidovima klimatiziranih prostorija, zrak u dimnjaku je hladniji od okolnog te se u dimnjaku ustaljuje natražno strujanje zraka, iz okoline u prostoriju. Nakon uključivanja plinskog trošila dimni plinovi prvo trebaju zaustaviti to natražno strujanje zraka, te okrenuti smjer strujanja u dimnjaku. Za to vrijeme dimni plinovi će ulaziti u prostoriju umjesto u dimnjak. Analizom utjecajnih parametara je zaključeno da će na vrijeme potrebno da se prekine strujanje dimnih plinova u prostoriju utjecati duljina horizontalnog dijela vezne cijevi (spoj trošila na dimnjak). Simulirano je uključivanje plinskog trošila za tri različite duljine vezne cijevi: 0,7, 1,5 i 2,5 m, te je dobiveno da vrijeme uspostave normalnog rada dimnjaka kod cijevi duljine 0,7 m iznosi oko 15 s, kod cijevi duljine 1,5 m oko 135 s, dok kod cijevi duljine 2,5 m, u 95-toj sekundi trend pokazuje da vjerojatno neće niti doći do uspostave normalnog rada dimnjaka. In this work the sizing of the chimney that is connected to a gas appliance power 24 kW has been preformed. The used calculation method was in according to the European standard EN 13384. An own computer program was developed in the programming language Phyton. The calculation results of the developed program were compared with the results of commercial program KESA ALADIN which is based on the same standards, and has a builtin automatic data assignation (standard pipe diameters, selection of needed coefficients, etc.) and data bases which are specified by the equipment manufacturers. The agreement of the obtained results with the results of commercial programs is very good. In the second part of the transients of the gas flow caused by the gas appliance turn on while the chimney is cold (the summer time). It is known that during hot summer days the accidents due to carbon monoxide poisoning occurs. These are consequences of improperly maintenance of the gas appliances (because of the incomplete combustion) and the returning of the gas fumes back into the room. If the chimney passes through the walls of airconditioned rooms, the air in the chimney is cooler than the surrounding one, and the chimney air flow is backwards from outside into the room. After turning on the gas appliance, the gas fumes should first stop the chimney backward airflow and after that turn the flow direction from the room to outside. During that time the gas fumes will enter the room instead of the chimney. From the analysis of influence parameters it has been concluded that the time needed to establish right flow direction depends on the length of the horizontal part of the tie tube (the connection of the appliance into the chimney). Transient gas flows were simulated for three lengths of tie tube. For the tube length of 0,7 m, the normal chimney flow was achieved in 15 s, for the tube length of 1,5 m in 135 s, and obtained trends in the simulation with tube length of 2,5 m shows that the normal chimney flow will probably not be achieved.

U radu je opisan standardni proračun dimnjaka prema norme EN13384. Opisani proračun je primijenjen na jedan konkretni slučaj dimnjaka za trošilo nazivne snage 33 kW. Proračun je također provjeren primjenom dijagrama za izbor dimnjaka koji daje proizvođač. Dodatno je provedena analiza toplinskih pojava u dimnjaku za vrijeme njegova početka rada, te itermitentnog rada trošila. Pokazano je da se za slučaj kontinuiranog rada plinskog trošila uvjeti kondenzacije pojavljuju, prema očekivanjima samo u početku, dok se za analizirani intermitentni rad pojavljuju uvjeti pojave kondenzacije, ako je omjer vremena rada trošila i vremena isključenog trošila manji od nekog kritičnog omjera. Za okolišnu temperaturu od 5°C, kritični omjer je oko 2, 5, kod okolišne temperature od 0°C, kritični omjer je oko 4, dok kod okolišne temperature -15°C taj omjer ide na približno 10.

Cilj ovog rada jest odrediti utjecaj elemenata dinamičke sprege na energetsku učinkovitost mehanizma mahokrila. Uz pomoć programskog paketa MSC ADAMS propisuje se funkcije gibanja krila s obzirom na kriterij postizanja traženog kinematičkog profila uzorka gibanja mahokrila. U ADAMS se ubacuje geometrija modela rađenog u SolidWorks programu na koju se zatim propisuju odgovarajuća kinematska struktura primjenom dinamike više tijela (engl. Multybody Dynamics). Dodavanjem elemenata dinamičke sprege nastoji se poboljšati energetsku učinkovitost mehanizma. Na temelju usporedbe rezultata dobivenih za dvoje različitih (s obzirom na raspon krila i frekvenciju mahanj) modela mahokrila zaključuje se kako se dodavanjem linearne, jednako kao i nelinearne opruge značajno poboljšava učinkovitost mehanizma. The goal of this thesis is to analyse the influence of dynamic coupling elements on energy efficiency of the flapping wing mechanism. The flapping wing motion, which follows the kinematic profile of a flapping wing, can be simulated using ADAMS (on two different models based on their wingspan and flapping frequency of the wings). The 3D model, which was made in SolidWorks, is set with its kinematical constraints in ADAMS using multybody dynamics. By adding dynamic coupling elements we are trying to improve the energy efficiency of the flapping wing mechanism. Based on the given results, we can determine, by using linear as well as nonlinear springs, a significant improvement in the efficiency of the mechanism.

Sustavi s velikim brojem krutih tijela predstavljaju vrlo složen zadatak i predmet su proučavanja brojnih znanstvenika diljem svijeta. Njihova analiza je veoma zahtjevna zbog velikog broja stupnjeva slobode gibanja pa se numeričke simulacije nameću kao jedino rješenje za njihovo proučavanje. Međutim, čak i takve simulacije su veoma zahtjevne za provedbu, prvenstveno sa stajališta računalnog vremena i složenosti proračuna. U ovom radu je provedena numerička simulacija jednog takvog sustava na primjeru kretanja rovera namijenjenog istraživanju Marsa. Rad je podijeljen na četiri cjeline. U prvoj cjelini se opisuje i analizira konstrukcija sustava suspenzije promatranog rovera. Nakon toga su opisane jednadžbe koje upravljaju simulacijom, a posebna pozornost je dana modelu kontaktne dinamike temeljenom na diferencijalnim varijacijskim nejednadžbama. U trećoj cjelini je detaljno opisan postupak pripreme numeričke simulacije u programskom paketu Project Chrono s primjerima koda korištenog za izradu simulacije. Na kraju su grafički prikazani rezultati dobiveni simulacijom te su uspoređeni s rezultatima ekvivalentnog eksperimenta.

U ovom se radu razmatraju kinematička svojstva mehanizama mahokrilnog tipa i njihova mogućnost da svojim gibanjem opisuju gibanje krila insektnog tipa. Usporedbom s traženom kinematikom mahanja krila vinske mušice odabiru se dva mehanizma na temelju čijih se kinematičkih shema izrađuje CAD model mehanizama. U programskom paketu ADAMS modelira se kinematička struktura željenog kinematičkog profila vinske mušice kao i modeli dva razmatrana mehanizma te se primjenom dinamike više tijela pronalaze i uspoređuju profili gibanja krila kao i potrebna snaga za pogon mehanizma pri različitim frekvencijama mahanja. Dodavanjem elemenata dinamičke sprege utvrđuje se mogućnost poboljšanja energetske učinkovitosti razmatranih mehanizama. Prilagodbom CAD modela aditivnim se tehnologijama proizvode razmatrani modeli mehanizama i ocjenjuje se njihova funkcionalnost. This thesis presents an overview of the kinematical properties of flapping-wing mechanisms and their ability to reproduce the movement of the wings of a real insect. Considering the desired kinematic profile of a fruit fly, two mechanisms are selected. According to the kinematic diagrams of the two mechanisms, the CAD models are created. Using the ADAMS software, the kinematic structure of the desired kinematic model of a fruit fly is modeled as well as the kinematic structure of the two selected mechanisms. Using the Multibody Dynamics the flapping patterns are determined and compared as well as the required power at different flapping frequencies. Improving the efficiency of the selected mechanisms is evaluated by adding coupling elements. The CAD models of the mechanisms are adapted for the additive manufacturing of physical models whose functionality is evaluated.

Tema ovog rada je implementacija integracijske procedure kinematičke rekonstrukcije orijentacije kvadrirotorske letjelice te provjera simulacijskog modela u stvarnom vremenu na računalu Raspberry Pi. U prvom poglavlju su između formalizama za opis rotacija odabrane i opisane matrice rotacije kao elementi SO(3) grupe te jedinični kvaternioni iz SU (2) grupe nakon čega su izvedene relacije temeljene na Eulerovim parametrima. Poglavlje završava izvodom jednadžbe kinematičke rekonstrukcije orijentacije. U drugom poglavlju dan je pregled nove metode vremenske integracije jediničnih kvaterniona koja inkrementalnu integraciju vrši na tangentnom prostoru rotacijske mnogostrukosti SO(3) uz izravnu rekonstrukciju orijentacije na mnogostrukosti jediničnih kvaterniona SU (2). Poglavlje završava opisom i ko- mentarom detalja implementacije pojedinih relevantnih funkcija implementirane integracijske procedure. U trećem poglavlju opisane su i uspoređene programske paradigme u kontekstu numeričkih simulacija i znanstvenog računanja te je dan kratak opis hardverske infrastrukture na kojoj su simulacije realizirane. U zadnjem poglavlju, nakon validacije modela i rezultata, dan je niz usporedbi procesorskih vremena za ’off-line’ i ’on-line’ simulacije s različitim koracima integracije. The topic of this paper is the implementation of the integration procedure for kinematic reconstruction of attitude for a quadrotor aerial vehicle and validation of a real-time simulation model on a Raspberry Pi computer. In the first chapter, between the formalisms for the description of rotations, the rotation matrices were selected and described as elements of the SO(3) group and unit quaternions from the SU (2) group, after which relations based on Euler parameters were derived. The chapter ends with a derivation of the equation of kinematic reconstruction of attitude. The second chapter provides an brief overview of a new integration method of unit quaternions that performs incremental integration on the tangential space of the rotational manifold SO(3) with a direct reconstruction of the orientation on the manifold of unit quaternions SU (2). The chapter ends with a description and commentary on the details of the implementation of certain relevant functions. The third chapter describes and compares programming paradigms in the context of numerical simulations and scientific computing, and gives a brief description of the hardware infrastructure on which the simulations were implemented. In the last chapter, after model and result validation, a series of comparisons of processor times for ‘off-line’ and ‘on-line’ simulations with different integration steps are given.