Development of finite elements for analysis of biomechanical structures using flexible multibody formulations

The absolute nodal coordinate formulation was originally developed for the analysis of structures undergoing large rotations and deformations. This dissertation proposes several enhancements to the absolute nodal coordinate formulation based finite beam and plate elements. The main scientific contribution of this thesis relies on the development of elements based on the absolute nodal coordinate formulation that do not suffer from commonly known numerical locking phenomena. These elements can be used in the future in a number of practical applications, for example, analysis of biomechanical soft tissues. This study presents several higher-order Euler–Bernoulli beam elements, a simple method to alleviate Poisson’s and transverse shear locking in gradient deficient plate elements, and a nearly locking free gradient deficient plate element. The absolute nodal coordinate formulation based gradient deficient plate elements developed in this dissertation describe most of the common numerical locking phenomena encountered in the formulation of a continuum mechanics based description of elastic energy. Thus, with these fairly straightforwardly formulated elements that are comprised only of the position and transverse direction gradient degrees of freedom, the pathologies and remedies for the numerical locking phenomena are presented in a clear and understandable manner. The analysis of the Euler–Bernoulli beam elements developed in this study show that the choice of higher gradient degrees of freedom as nodal degrees of freedom leads to a smoother strain field. This improves the rate of convergence.

Optimization of the procedure for removing iron deposits on ceramic filter media

In this thesis, cleaning of ceramic filter media was studied. Mechanisms of fouling and dissolution of iron compounds, as well as methods for cleaning ceramic membranes fouled by iron deposits were studied in the literature part. Cleaning agents and different methods were closer examined in the experimental part of the thesis. Pyrite is found in the geologic strata. It is oxidized to form ferrous ions Fe(II) and ferric ions Fe(III). Fe(III) is further oxidized in the hydrolysis to form ferric hydroxide. Hematite and goethite, for instance, are naturally occurring iron oxidesand hydroxides. In contact with filter media, they can cause severe fouling, which common cleaning techniques competent enough to remove. Mechanisms for the dissolution of iron oxides include the ligand-promoted pathway and the proton-promoted pathway. The dissolution can also be reductive or non-reductive. The most efficient mechanism is the ligand-promoted reductive mechanism that comprises two stages: the induction period and the autocatalytic dissolution.Reducing agents(such as hydroquinone and hydroxylamine hydrochloride), chelating agents (such as EDTA) and organic acids are used for the removal of iron compounds. Oxalic acid is the most effective known cleaning agent for iron deposits. Since formulations are often more effective than organic acids, reducing agents or chelating agents alone, the citrate¿bicarbonate¿dithionite system among others is well studied in the literature. The cleaning is also enhanced with ultrasound and backpulsing.In the experimental part, oxalic acid and nitric acid were studied alone andin combinations. Also citric acid and ascorbic acid among other chemicals were tested. Soaking experiments, experiments with ultrasound and experiments for alternative methods to apply the cleaning solution on the filter samples were carried out. Permeability and ISO Brightness measurements were performed to examine the influence of the cleaning methods on the samples. Inductively coupled plasma optical emission spectroscopy (ICP-OES) analysis of the solutions was carried out to determine the dissolved metals.

Studies on integrating kinematic design method with mechanical systems simulation techniques

Theultimate goal of any research in the mechanism/kinematic/design area may be called predictive design, ie the optimisation of mechanism proportions in the design stage without requiring extensive life and wear testing. This is an ambitious goal and can be realised through development and refinement of numerical (computational) technology in order to facilitate the design analysis and optimisation of complex mechanisms, mechanical components and systems. As a part of the systematic design methodology this thesis concentrates on kinematic synthesis (kinematic design and analysis) methods in the mechanism synthesis process. The main task of kinematic design is to find all possible solutions in the form of structural parameters to accomplish the desired requirements of motion. Main formulations of kinematic design can be broadly divided to exact synthesis and approximate synthesis formulations. The exact synthesis formulation is based in solving n linear or nonlinear equations in n variables and the solutions for the problem areget by adopting closed form classical or modern algebraic solution methods or using numerical solution methods based on the polynomial continuation or homotopy. The approximate synthesis formulations is based on minimising the approximation error by direct optimisation The main drawbacks of exact synthesis formulationare: (ia) limitations of number of design specifications and (iia) failure in handling design constraints- especially inequality constraints. The main drawbacks of approximate synthesis formulations are: (ib) it is difficult to choose a proper initial linkage and (iib) it is hard to find more than one solution. Recentformulations in solving the approximate synthesis problem adopts polynomial continuation providing several solutions, but it can not handle inequality const-raints. Based on the practical design needs the mixed exact-approximate position synthesis with two exact and an unlimited number of approximate positions has also been developed. The solutions space is presented as a ground pivot map but thepole between the exact positions cannot be selected as a ground pivot. In this thesis the exact synthesis problem of planar mechanism is solved by generating all possible solutions for the optimisation process ¿ including solutions in positive dimensional solution sets - within inequality constraints of structural parameters. Through the literature research it is first shown that the algebraic and numerical solution methods ¿ used in the research area of computational kinematics ¿ are capable of solving non-parametric algebraic systems of n equations inn variables and cannot handle the singularities associated with positive-dimensional solution sets. In this thesis the problem of positive-dimensional solutionsets is solved adopting the main principles from mathematical research area of algebraic geometry in solving parametric ( in the mathematical sense that all parameter values are considered ¿ including the degenerate cases ¿ for which the system is solvable ) algebraic systems of n equations and at least n+1 variables.Adopting the developed solution method in solving the dyadic equations in direct polynomial form in two- to three-precision-points it has been algebraically proved and numerically demonstrated that the map of the ground pivots is ambiguousand that the singularities associated with positive-dimensional solution sets can be solved. The positive-dimensional solution sets associated with the poles might contain physically meaningful solutions in the form of optimal defectfree mechanisms. Traditionally the mechanism optimisation of hydraulically driven boommechanisms is done at early state of the design process. This will result in optimal component design rather than optimal system level design. Modern mechanismoptimisation at system level demands integration of kinematic design methods with mechanical system simulation techniques. In this thesis a new kinematic design method for hydraulically driven boom mechanism is developed and integrated in mechanical system simulation techniques. The developed kinematic design method is based on the combinations of two-precision-point formulation and on optimisation ( with mathematical programming techniques or adopting optimisation methods based on probability and statistics ) of substructures using calculated criteria from the system level response of multidegree-of-freedom mechanisms. Eg. by adopting the mixed exact-approximate position synthesis in direct optimisation (using mathematical programming techniques) with two exact positions and an unlimitednumber of approximate positions the drawbacks of (ia)-(iib) has been cancelled.The design principles of the developed method are based on the design-tree -approach of the mechanical systems and the design method ¿ in principle ¿ is capable of capturing the interrelationship between kinematic and dynamic synthesis simultaneously when the developed kinematic design method is integrated with the mechanical system simulation techniques.

Mass transfer and particleinteractions in granular systems and suspension flows

The purpose of this study was to investigate some important features of granular flows and suspension flows by computational simulation methods. Granular materials have been considered as an independent state ofmatter because of their complex behaviors. They sometimes behave like a solid, sometimes like a fluid, and sometimes can contain both phases in equilibrium. The computer simulation of dense shear granular flows of monodisperse, spherical particles shows that the collisional model of contacts yields the coexistence of solid and fluid phases while the frictional model represents a uniform flow of fluid phase. However, a comparison between the stress signals from the simulations and experiments revealed that the collisional model would result a proper match with the experimental evidences. Although the effect of gravity is found to beimportant in sedimentation of solid part, the stick-slip behavior associated with the collisional model looks more similar to that of experiments. The mathematical formulations based on the kinetic theory have been derived for the moderatesolid volume fractions with the assumption of the homogeneity of flow. In orderto make some simulations which can provide such an ideal flow, the simulation of unbounded granular shear flows was performed. Therefore, the homogeneous flow properties could be achieved in the moderate solid volume fractions. A new algorithm, namely the nonequilibrium approach was introduced to show the features of self-diffusion in the granular flows. Using this algorithm a one way flow can beextracted from the entire flow, which not only provides a straightforward calculation of self-diffusion coefficient but also can qualitatively determine the deviation of self-diffusion from the linear law at some regions nearby the wall inbounded flows. Anyhow, the average lateral self-diffusion coefficient, which was calculated by the aforementioned method, showed a desirable agreement with thepredictions of kinetic theory formulation. In the continuation of computer simulation of shear granular flows, some numerical and theoretical investigations were carried out on mass transfer and particle interactions in particulate flows. In this context, the boundary element method and its combination with the spectral method using the special capabilities of wavelets have been introduced as theefficient numerical methods to solve the governing equations of mass transfer in particulate flows. A theoretical formulation of fluid dispersivity in suspension flows revealed that the fluid dispersivity depends upon the fluid properties and particle parameters as well as the fluid-particle and particle-particle interactions.

Drug loading of mesoporous silicon particles

Porous silicon (PSi) is a promising material to be utilized in drug delivery formulations. The release rate of the drug compound can be controlled by changing the pore properties and surface chemistry of PSi. The loading of a poorly soluble drug into mesoporous silicon particles enhances its dissolution in the body. The drug loading is based on adsorption. The attainable maximum loaded amount depends on the properties of the drug compound and the PSi material, and on the process conditions. The loading solvent also essentially affects the adsorption process. The loading of indomethacin into PSi particles with varying surface modification was studied. Solvent mixtures were applied in the loading, and the loaded samples were analyzed with thermal analysis methods. The best degree of loading was obtained using a mixture of dichloromethane and methanol. The drug loads varied from 7.7 w-% to 26.8 w-%. A disturbing factor in the loading experiments was the tendency of indomethacin to form solvates with the solvents applied. In addition, the physical form and stability of indomethacin loaded in PSi and silica particles were studied using Raman spectroscopy. In the case of silica, the presence of crystalline drug as well as the polymorph form can be detected, but the method proved to be not applicable for PSi particles.

Feature selection using Fuzzy Entropy measures with Yu's Similarity measure

In this study, feature selection in classification based problems is highlighted. The role of feature selection methods is to select important features by discarding redundant and irrelevant features in the data set, we investigated this case by using fuzzy entropy measures. We developed fuzzy entropy based feature selection method using Yu's similarity and test this using similarity classifier. As the similarity classifier we used Yu's similarity, we tested our similarity on the real world data set which is dermatological data set. By performing feature selection based on fuzzy entropy measures before classification on our data set the empirical results were very promising, the highest classification accuracy of 98.83% was achieved when testing our similarity measure to the data set. The achieved results were then compared with some other results previously obtained using different similarity classifiers, the obtained results show better accuracy than the one achieved before. The used methods helped to reduce the dimensionality of the used data set, to speed up the computation time of a learning algorithm and therefore have simplified the classification task

Gelatin as an additive in bio-based barrier films

The main objective of the present study was to verify the approach on starch-gelatin blending for the paperboard coating formulations with enhanced barrier and mechanical properties. Based on that, another objective was to find out, how the approach will function with wood-based polysaccharides (CMC, EHEC and HPC) by analyzing their barrier properties and convertibility. The last objective was to find out, if pigments can be used in the composition of polysaccharide-protein blends without causing any negative effect on stated properties. The whole process chain of the barrier coating development was studied in the research. The methodology applied included pilot-scale coating and converting trials for the evaluation of mechanical properties of obtained coatings, namely their exposure to cracking with the loss of barrier properties. The results obtained indicated that the combination of starch with gelatin, in fact, improves the grease barrier properties and flexibility of starch-based coatings, thereby confirming the offered approach. The similar results were obtained for CMC, exhibited elevated barrier properties and surface coverage, proving that the approach also functions with wood-based polysaccharides. The introduction of equal amounts of talc gave various effects at different gelatin dosages on barrier properties of wood-based polysaccharides. Mainly, the elevation of grease barrier properties was observed. The convertibility of talc-filled coatings was not sufficient.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Vibrations in Rotating Machinery Arising from Minor Imperfections in Component Geometries

Vibrations in machines can cause noise, decrease the performance, or even damage the machine. Vibrations appear if there is a source of vibration that excites the system. In the worst case scenario, the excitation frequency coincides with the natural frequency of the machine causing resonance. Rotating machines are a machine type, where the excitation arises from the machine itself. The excitation originates from the mass imbalance in the rotating shaft, which always exists in machines that are manufactured using conventional methods. The excitation has a frequency that is dependent on the rotational speed of the machine. The rotating machines in industrial use are usually designed to rotate at a constant rotational speed, the case where the resonances can be easily avoided. However, the machines that have a varying operational speed are more problematic due to a wider range of frequencies that have to be avoided. Vibrations, which frequencies equal to rotational speed frequency of the machine are widely studied and considered in the typical machine design process. This study concentrates on vibrations, which arise from the excitations having frequencies that are multiples of the rotational speed frequency. These vibrations take place when there are two or more excitation components in a revolution of a rotating shaft. The dissertation introduces four studies where three kinds of machines are experiencing vibrations caused by different excitations. The first studied case is a directly driven permanent magnet generator used in a wind power plant. The electromagnetic properties of the generator cause harmonic excitations in the system. The dynamic responses of the generator are studied using the multibody dynamics formulation. In another study, the finite element method is used to study the vibrations of a magnetic gear due to excitations, which frequencies equal to the rotational speed frequency. The objective is to study the effects of manufacturing and assembling inaccuracies. Particularly, the eccentricity of the rotating part with respect to non-rotating part is studied since the eccentric operation causes a force component in the direction of the shortest air gap. The third machine type is a tube roll of a paper machine, which is studied while the tube roll is supported using two different structures. These cases are studied using different formulations. In the first case, the tube roll is supported by spherical roller bearings, which have some wavinesses on the rolling surfaces. Wavinesses cause excitations to the tube roll, which starts to resonate at the frequency that is a half of the first natural frequency. The frequency is in the range where the machine normally operates. The tube roll is modeled using the finite element method and the bearings are modeled as nonlinear forces between the tube roll and the pedestals. In the second case studied, the tube roll is supported by freely rotating discs, which wavinesses are also measured. The above described phenomenon is captured as well in this case, but the simulation methodology is based on the flexible multibody dynamics formulation. The simulation models that are used in both of the last two cases studied are verified by measuring the actual devices and comparing the simulated and measured results. The results show good agreement.

New families of highly efficient, halogen-free flame retardants for polypropylene (PP)

The driving forces for current research of flame retardants are increased fire safety in combination with flame retardant formulations that fulfill the criteria of sustainable production and products. In recent years, important questions about the environmental safety of antimony, and in particular, brominated flame retardants have been raised. As a consequence of this, the current doctoral thesis work describes efforts to develop new halogen-free flame retardants that are based on various radical generators and phosphorous compounds. The investigation was first focused on compounds that are capable of generating alkyl radicals in order to study their role on flame retardancy of polypropylene. The family of azoalkanes was selected as the cleanest and most convenient source of free alkyl radicals. Therefore, a number of symmetrical and unsymmetrical azoalkanes of the general formula R-N=N-R’ were prepared. The experimental results show that in the series of different sized azocycloalkanes the flame retardant efficacy decreased in the following order: R = R´= cyclohexyl > cyclopentyl > cyclobutyl > cyclooctanyl > cyclododecanyl. However, in the series of aliphatic azoalkanes compounds, the efficacy decreased as followed: R = R´= n-alkyl > tert-butyl > tert-octyl. The most striking difference in flame retardant efficacy was observed in thick polypropylene plaques of 1 mm, e.g. azocyclohexane (AZO) had a much better flame retardant performance than did the commercial reference FR (Flamestab® NOR116) in thick PP sections. In addition, some of the prepared azoalkane flame retardants e.g. 4’4- bis(cyclohexylazocyclohexyl) methane (BISAZO) exhibited non-burning dripping behavior. Extrusion coating experiments of flame retarded low density polyethylene (LDPE) onto a standard machine finished Kraft paper were carried out in order to investigate the potential of azoalkanes in multilayer facings. The results show that azocyclohexane (AZO) and 4’4-bis (cyclohexylazocyclohexyl) methane (BISAZO) can significantly improve the flame retardant properties of low density polyethylene coated paper already at 0.5 wt.% loadings, provided that the maximum extrusion temperature of 260 oC is not exceeded and coating weight is kept low at 13 g/m2. In addition, various triazene-based flame retardants (RN1=N2-N3R’R’’) were prepared. For example, polypropylene samples containing a very low concentration of only 0.5 wt.% of bis- 4’4’-(3’3’-dimethyltriazene) diphenyl ether and other triazenes passed the DIN 4102-1 test with B2 classification. It is noteworthy that no burning dripping could be detected and the average burning times were very short with exceptionally low weight losses. Therefore, triazene compounds constitute a new and interesting family of radical generators for flame retarding of polymeric materials. The high flame retardant potential of triazenes can be attributed to their ability to generate various types of radicals during their thermal decomposition. According to thermogravimetric analysis/Fourier transform infrared spectroscopy/MS analysis, triazene units are homolytically cleaved into various aminyl, resonance-stabilized aryl radicals, and different CH fragments with simultaneous evolution of elemental nitrogen. Furthermore, the potential of thirteen aliphatic, aromatic, thiuram and heterocyclic substituted organic disulfide derivatives of the general formula R-S-S-R’ as a new group of halogen-free flame retardants for polypropylene films have been investigated. According to the DIN 4102- 1 standard ignitibility test, for the first time it has been demonstrated that many of the disulfides alone can effectively provide flame retardancy and self-extinguishing properties to polypropylene films at already very low concentrations of 0.5 wt.%. For the disulfide family, the highest FR activity was recorded for 5’5’-dithiobis (2-nitrobenzoic acid). Very low values for burning length (53 mm) and burning time (10 s) reflect significantly increased fire retardant performance of this disulfide compared to other compounds in this series as well as to Flamestab® NOR116. Finally, two new, phosphorus-based flame retardants were synthesized: P’P-diphenyl phosphinic hydrazide (PAH) and melamine phenyl phosphonate (MPhP). The DIN 4102-1 test and the more stringent UL94 vertical burning test (UL94 V) were used to assess the formulations ability to extinguish a flame once ignited. A very strong synergistic effect with azoalkanes was found, i.e. in combination with these radical generators even UL94 V0 rate could be obtained.

Vaahtoamisen estäminen vesiohenteisissa sisäseinämaaleissa

Maalit ovat hyvin komplekseja seoksia, jotka sisältävät erilaisia pinta-aktiivisia aineita. Erilaisia pinta-aktiivisia apuaineita käytetään varsinkin vesiohenteisissa maaleissa veden korkean pintajännityksen takia. Nämä maalin komponentit saat-tavat kuitenkin aiheuttaa maaliin vaahtoamisongelmia. Maalien vaahtoaminen aiheuttaa esimerkiksi erilaisia pintavikoja maalikalvossa ja epäotpimaalista tuo-tantosäiliöiden ja pakkausten täyttöä. Joissain tapauksissa myös maalin funktio-naalisuus saattaa häiriintyä. Maalien vaahtoamista pyritään vähentämään pääasiassa vaahdonestäjiksi kutsu-tuilla apuaineilla. Tärkeimmät vesipohjaisissa maaleissa käytettävät vaahdonestäjät ovat mineraaliöljyjä sekä silikonipohjaisia vaahdonestäjiä. Tässä työssä tutkitaan kolmen eri vesiohenteisen sisäseinämaalin vaahtoamista ja mahdollisia keinoja vähentää sitä. Laboratoriossa kokeillaan nykyisten vaahdonestäjien tilalle mahdollisia korvaavia apuaineita. Lisäksi kullekin maalille luodaan koesuunni-telma, jonka mukaan toteutetaan tehtaalla sarja koeajoja, joilla pyritään löytämään sopiva vaahdonestäjän annostus sekä lisäystapa kullekin maalille, siten että maalien vaahtoaminen vähenee. Laboratoriokokeiden perusteella korvaavaa vaahdonestäjää nykyisten tilalle ei löydetty. Valmistusprosessin yhteydessä sen sijaan havaittiin, että vaahdonestäjän lisäystapaa muuttamalla voisi olla mahdollista vähentää maalien vaahtoamista merkittävästi lisäämättä käytettävää vaahdonestäjän annostusta.

Sekaelementit absoluuttisten solmukoordinaattien menetelmässä

Tässä lisensiaatintyössä käsitellään sekaelementtien sovellusmahdollisuuksia absoluuttisten solmukoordinaattien menetelmässä. Absoluuttisten solmukoordinaattien menetelmä on uudentyyppinen lähestymistapa elementtimenetelmän elementtien koordinaattien määrittämiseksi ja sen yhtenä tavoitteena on tehostaa suuria siirtymiä tai kiertymiä sisältävien elementtien laskentatehokkuutta. Tässä työssä absoluuttisten solmukoordinaattien menetelmä esitellään pääpiirteittäin sekä annetaan esimerkkejä muutamista tyypillisimmistä elementeistä lausuttuna edellä mainittujen koordinaattien perusteella. Sekaelementeiksi kutsutaan elementtityyppejä, missä tuntemattomien muuttujien joukkoja on aina enemmän kuin yksi. Sekaelementit erottavat redusoitumattomista elementeistä siirtymäkentän sisältyminen muuttujaryhmään ja hybridielementeistä muuttujien identtiset ulottuvuudet. Sekaelementtejä käytetään esimerkiksi kokoonpuristumattomien materiaalien rakenneanalyyseissä, alentamaan elementiltä vaadittavia jatkuvuusehtoja tai mallintamaan ilmiöitä, missä fysikaaliset ominaisuudet ovat jostain syystä voimakkaasti toisistaan riippuvaisia. Tämän lisensiaatintyön kirjoittamiseksi on tehty tutkimusta sekaelementtien mahdollisuuksista toimia absoluuttisten solmukoordinaattien menetelmässä. Tutkimuksen tuloksena on saatu aikaan kaksi tässä työssä esiteltävää, varsin rajatun toimintakyvyn omaavaa sekaelementtityyppiä, joiden siirtymäkentät on määritelty globaalien koordinaattien suhteen sisältäen myös orientaatiotermit. Tutkimusaihe vaatii kuitenkin vielä paljon lisätyötä, ennen kuin sekaelementtityyppejä voidaan kauttaaltaan soveltaa absoluuttisten solmukoordinaattien menetelmällä toteutetuissa rakenneanalyyseissä.

Studies of rotor dynamics using a multibody simulation approach

The non-idealities in a rotor-bearing system may cause undesirable subcritical superharmonic resonances that occur when the rotating speed of the rotor is a fraction of the natural frequency of the system. These resonances arise partly from the non-idealities of the bearings. This study introduces a novel simulation approach that can be used to study the superharmonic vibrations of rotor-bearing systems. The superharmonic vibrations of complex rotor-bearing systems can be studied in an accurate manner by combining a detailed rotor and bearing model in a multibody simulation approach. The research looks at the theoretical background of multibody formulations that can be used in the dynamic analysis of flexible rotors. The multibody formulations currently in use are suitable for linear deformation analysis only. However, nonlinear formulation may arise in high-speed rotor dynamics applications due to the cenrrifugal stiffening effect. For this reason, finite element formulations that can describe nonlinear deformation are also introduced in this work. The description of the elastic forces in the absolute nodal coordinate formulation is studied and improved. A ball bearing model that includes localized and distributed defects is developed in this study. This bearing model could be used in rotor dynamics or multibody code as an interface elements between the rotor and the supporting structure. The model includes descriptions of the nonlinear Hertzian contact deformation and the elastohydrodynamic fluid film. The simulation approaches and models developed here are applied in the analysis of two example rotor-bearing systems. The first example is an electric motor supported by two ball bearings and the second is a roller test rig that consists of the tube roll of a paper machine supported by a hard-bearing-type balanceing machine. The simulation results are compared to the results available in literature as well as to those obtained by measuring the existing structure. In both practical examples, the comparison shows that the simulation model is capable of predicting the realistic responses of a rotor system. The simulation approaches developed in this work can be used in the analysis of the superharmonic vibrations of general rotor-bearing systems.