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Lundström, Niklasorcid.org/0000-0003-4040-8142

Open this publication in new window or tab >>Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching### Lundström, Niklas L.P.

### Olofsson, Marcus

### Önskog, Thomas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 475, no 1, p. 13-31Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2019
##### Keywords

variational inequality, existence, viscosity solution, nonlocal operator, partial integro-differential operator, Lévy process, jump diffusion, optimal switching problem, regularity, continuous dependence, well posed
##### National Category

Mathematics Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-153270 (URN)10.1016/j.jmaa.2018.11.003 (DOI)000464490800002 ()
#####

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Available from: 2018-11-14 Created: 2018-11-14 Last updated: 2019-06-12Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integrodiﬀerential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n-dimensional L´evy process. We ﬁrst establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-diﬀerential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as “barrier functions” to prove Ho¨lder continuity in the time variable, and, through Perron’s method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) and of Lundström, Nyström and Olofsson (2014) to hold for more general systems of equations.

Open this publication in new window or tab >>Meeting yield and conservation objectives by harvesting both juveniles and adults### Lundström, Niklas L. P.

### Loeuille, Nicolas

### Meng, Xinzhu

### Bodin, Mats

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); ### Brännström, Åke

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_1_j_idt188_j_idt202",{id:"formSmash:j_idt184:1:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2019 (English)In: American Naturalist, ISSN 0003-0147, E-ISSN 1537-5323, Vol. 193, no 3, p. 373-390Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

University of Chicago Press, 2019
##### Keywords

fisheries management, maximum sustainable yield, pretty good yield, Pareto front, resilience, size structure
##### National Category

Ecology Evolutionary Biology
##### Identifiers

urn:nbn:se:umu:diva-155571 (URN)10.1086/701631 (DOI)000459624900007 ()30794450 (PubMedID)
#####

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Available from: 2019-01-22 Created: 2019-01-22 Last updated: 2019-04-01Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, China.

Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Evolution and Ecology Program, International Institute for Applied Systems Analysis, Laxenburg, Austria.

Sustainable yields that are at least 80% of the maximum sustainable yield are sometimes referred to as "pretty good yields" (PGY). The range of PGY harvesting strategies is generally broad and thus leaves room to account for additional objectives besides high yield. Here, we analyze stage-dependent harvesting strategies that realize PGY with conservation as a second objective. We show that (1) PGY harvesting strategies can give large conservation benefits and (2) equal harvesting rates of juveniles and adults is often a good strategy. These conclusions are based on trade-off curves between yield and four measures of conservation that form in two established population models, one age-structured model and one stage-structured model, when considering different harvesting rates of juveniles and adults. These conclusions hold for a broad range of parameter settings, although our investigation of robustness also reveals that (3) predictions of the age-structured model are more sensitive to variations in parameter values than those of the stage-structured model. Finally, we find that (4) measures of stability that are often quite difficult to assess in the field (e.g., basic reproduction ratio and resilience) are systematically negatively correlated with impacts on biomass and size structure, so that these later quantities can provide integrative signals to detect possible collapses.

Open this publication in new window or tab >>Stochastic and partial differential equations on non-smooth time-dependent domains### Lundström, Niklas

### Önskog, Thomas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 129, no 4, p. 1097-1131Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2019
##### Keywords

Reflected diffusion, Skorohod problem, Oblique reflection, Time-dependent domain, Stochastic differential equations, Non-smooth domain, Viscosity solution, Parabolic partial differential equation, Comparison principle, Existence, Uniqueness
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-152844 (URN)10.1016/j.spa.2018.04.006 (DOI)000462111300001 ()2-s2.0-85047257163 (Scopus ID)
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Available from: 2018-10-27 Created: 2018-10-27 Last updated: 2019-10-11Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden.

In this article, we consider non-smooth time-dependent domains whose boundary is *W*^{1,p} in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.

Open this publication in new window or tab >>Estimates of size of cycle in a predator-prey system### Lundström, Niklas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Söderbacka, Gunnar

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Differential Equations and Dynamical Systems, ISSN 0971-3514, E-ISSN 0974-6870Article in journal (Refereed) Epub ahead of print
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2018
##### Keywords

Locating limit cycle, Locating attractor, Size of limit cycle, Lyapunov function, Lyapunov stability
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-153284 (URN)10.1007/s12591-018-0422-x (DOI)
#####

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Available from: 2018-11-15 Created: 2018-11-15 Last updated: 2019-04-05

We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest.

Open this publication in new window or tab >>How to find simple nonlocal stability and resilience measures### Lundström, Niklas L. P.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 93, no 2, p. 887-908Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2018
##### Keywords

Measure of stability, Measure of resilience, Sensitivity, Initial conditions, Random testing, Bistability, Returntime, Recovery rate, Global stability, Basin stability
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-150760 (URN)10.1007/s11071-018-4234-x (DOI)000435692500047 ()
#####

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Available from: 2018-08-31 Created: 2018-08-31 Last updated: 2018-08-31Bibliographically approved

Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow-Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a high-dimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and methods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.

Open this publication in new window or tab >>Pareto-efficient biological pest control enable high efficacy at small costs### Lundström, Niklas L. P.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Zhang, Hong

### Brännström, Åke

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Ecological Modelling, ISSN 0304-3800, E-ISSN 1872-7026, Vol. 364, p. 89-97Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Amsterdam: Elsevier, 2017
##### Keywords

Pareto frontier, Pest management, Optimization, Dual target, Spodoptera litura
##### National Category

Bioinformatics and Systems Biology
##### Identifiers

urn:nbn:se:umu:diva-140512 (URN)10.1016/j.ecolmodel.2017.08.015 (DOI)000414818900008 ()
#####

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Available from: 2017-10-12 Created: 2017-10-12 Last updated: 2018-06-09Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Financial Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria.

Biological pest control is increasingly used in agriculture as an alternative to traditional chemical pest control. In many cases, this involves a one-off or periodic release of naturally occurring and/or genetically modified enemies such as predators, parasitoids, or pathogens. As the interaction between these enemies and the pest is complex and the production of natural enemies potentially expensive, it is not surprising that both the efficacy and economic viability of biological pest control are debated. Here, we investigate the performance of very simple control strategies. In particular, we show how Pareto-efficient one-off or periodic release strategies, that optimally trade off between efficacy and economic viability, can be devised and used to enable high efficacy at small economic costs. We demonstrate our method on a pest–pathogen–crop model with a tunable immigration rate of pests. By analyzing this model, we demonstrate that simple Pareto-efficient one-off and periodic release strategies are efficacious and simultaneously have profits that are close to the theoretical maximum obtained by strategies optimizing only the profit. When the immigration rate of pests is low to intermediate, one-off control strategies are sufficient, and when the immigration of pests is high, periodic release strategies are preferable. The methods presented here can be extended to more complex scenarios and be used to identify promising biological pest control strategies in many circumstances.

Open this publication in new window or tab >>Constrained growth flips the direction of optimal phenological responses among annual plants### Lindh, Magnus

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Johansson, Jacob

### Bolmgren, Kjell

### Lundström, Niklas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); ### Brännström, Åke

### Jonzén, Niclas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_6_j_idt188_j_idt202",{id:"formSmash:j_idt184:6:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2016 (English)In: New Phytologist, ISSN 0028-646X, E-ISSN 1469-8137, Vol. 209, no 4, p. 1591-1599Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Wiley-Blackwell, 2016
##### Keywords

climate change, constrained growth, flowering, life history, optimal control theory, phenology, productivity, season
##### National Category

Ecology
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-119539 (URN)10.1111/nph.13706 (DOI)000373379800027 ()26548947 (PubMedID)2-s2.0-84948164421 (Scopus ID)
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Available from: 2016-04-22 Created: 2016-04-22 Last updated: 2018-06-07Bibliographically approved

Lund University.

SLU.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Evolution and Ecology Program International Institute for Applied Systems Analysis, Laxenburg, Austria.

Lund University.

Phenological changes among plants due to climate change are well documented, but often hard to interpret. In order to assess the adaptive value of observed changes, we study how annual plants with and without growth constraints should optimize their flowering time when productivity and season length changes. We consider growth constraints that depend on the plant's vegetative mass: self-shading, costs for nonphotosynthetic structural tissue and sibling competition. We derive the optimal flowering time from a dynamic energy allocation model using optimal control theory. We prove that an immediate switch (bang-bang control) from vegetative to reproductive growth is optimal with constrained growth and constant mortality. Increasing mean productivity, while keeping season length constant and growth unconstrained, delayed the optimal flowering time. When growth was constrained and productivity was relatively high, the optimal flowering time advanced instead. When the growth season was extended equally at both ends, the optimal flowering time was advanced under constrained growth and delayed under unconstrained growth. Our results suggests that growth constraints are key factors to consider when interpreting phenological flowering responses. It can help to explain phenological patterns along productivity gradients, and links empirical observations made on calendar scales with life-history theory.

Open this publication in new window or tab >>Phragmén-Lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes### Lundström, Niklas L P

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 44, no 2, p. 313-330Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer Netherlands, 2016
##### Keywords

Global estimates, Growth of p-harmonic functions, Infinity Laplace, Phragmén-Lindelöf, Subharmonic, Quasi-linear
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-110729 (URN)10.1007/s11118-015-9513-2 (DOI)000370368800006 ()
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Available from: 2015-10-26 Created: 2015-10-26 Last updated: 2018-06-07Bibliographically approved

We prove estimates of a *p*-harmonic measure, *p*∈(*n*−*m*,*∞*], for sets in **R** ^{ n } which are close to an *m*-dimensional hyperplane Λ⊂**R** ^{ n }, *m*∈[0,*n*−1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω⊂**R** ^{ n }∖Λ for *p*-subharmonic functions. Moreover, we give local and global growth estimates for *p*-harmonic functions, vanishing on sets in **R** ^{ n }, which are close to an *m*-dimensional hyperplane.

Open this publication in new window or tab >>The boundary Harnack inequality for variable exponent p-Laplacian, Carleson estimates, barrier functions and p(⋅)-harmonic measures### Adamowicz, Tomasz

### Lundström, Niklas L.P.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 195, no 2, p. 623-658Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2016
##### Keywords

Ball condition, Boundary Harnack inequality, Harmonic measure, NTA domain, Nonstandard growth equation, p-harmonic
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-89627 (URN)10.1007/s10231-015-0481-3 (DOI)000373086500019 ()
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##### Note

Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland.

We investigate various boundary decay estimates for p(⋅)-harmonic functions. For domains in R^{n},n≥2satisfying the ball condition (C^{1,1}-domains), we show the boundary Harnack inequality for p(⋅)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(⋅)-harmonic functions in NTA domains in R^{n} and provide lower and upper growth estimates and a doubling property for a p(⋅)-harmonic measure.

arXiv:1405.2678

Available from: 2014-06-07 Created: 2014-06-07 Last updated: 2018-06-07Bibliographically approvedOpen this publication in new window or tab >>Small shape deviations causes complex dynamics in large electric generators### Lundström, Niklas L.P.

### Grafström, Anton

### Aidanpää, Jan-Olov

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2014 (English)In: European Physical Journal: Applied physics, ISSN 1286-0042, E-ISSN 1286-0050, Vol. 66, no 2, p. 20903-Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

EDP Sciences, 2014
##### National Category

Mathematics Physical Sciences
##### Identifiers

urn:nbn:se:umu:diva-91388 (URN)10.1051/epjap/2014130447 (DOI)000338424300012 ()
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Available from: 2014-08-05 Created: 2014-08-04 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Applied Physics and Mechanical Engineering, Luleå University of Technology, SE-97187 Luleå, Sweden .

Department of Forest Resource Management, Swedish University of Agricultural Sciences, SE-90337 Umeå, Sweden .

Department of Applied Physics and Mechanical Engineering, Luleå University of Technology, SE-97187 Luleå, Sweden .

We prove that combinations of small eccentricity, ovality and/or triangularity in the rotor and stator can produce complex whirling motions of an unbalanced rotor in large synchronous generators. It is concluded which structures of shape deviations that are more harmful, in the sense of producing complex whirling motions, than others. For each such structure, we derive simplified equations of motions from which we conclude analytically the relation between shape deviations and mass unbalance that yield non-smooth whirling motions. Finally we discuss validity of our results in the sense of modeling of the unbalanced magnetic pull force.