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  • 1. Adolfsson, J.
    et al.
    Dankowicz, H.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    3D passive walkers: Finding periodic gaits in the presence of discontinuities2001In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 24, no 2, p. 205-229Article in journal (Refereed)
    Abstract [en]

    This paper studies repetitive gaits found in a 3D passive walking mechanism descending an inclined plane. By using direct numerical integration and implementing a semi-analytical scheme for stability analysis and root finding, we follow the corresponding limit cycles under parameter variations. The 3D walking model, which is fully described in the paper, contains both force discontinuities and impact-like instantaneous changes of state variables. As a result, the standard use of the variational equations is suitably modified. The problem of finding initial conditions for the 3D walker is solved by starting in an almost planar configuration, making it possible to use parameters and initial conditions found for planar walkers. The walker is gradually transformed into a 3D walker having dynamics in all spatial directions. We present such a parameter variation showing the stability and the amplitude of the hip sway motion. We also show the dependence of gait cycle measurements, such as stride time, stride length, average velocity, and power consumption, on the plane inclination. The paper concludes with a discussion of some ideas on how to extend the present 3D walker using the tools derived in this paper.

  • 2. Chillingworth, D. R. J.
    et al.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Periodic Orbits Close to Grazing for an Impact Oscillator2013In: Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer, 2013, p. 25-37Conference paper (Refereed)
    Abstract [en]

    We show how the geometric impact surface approach to the dynamics of an impact oscillator provides an immediate visualization of the criteria that determine the existence of an impacting periodic orbit close to grazing. We recover the criteria set out earlier by A. Nordmark and indicate how the geometric setting and singularity geometry may be exploited to yield appropriate criteria in degenerate situations where the Nordmark criteria would not apply.

  • 3. Dankowicz, H.
    et al.
    Adolfsson, J.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Repetitive gait of passive bipedal mechanisms in a three-dimensional environment2001In: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 123, no 1, p. 40-46Article in journal (Refereed)
    Abstract [en]

    The inherent dynamics of bipedal, kneed mechanisms are studied with emphasis on the existence and stability of repetitive gait in a three-dimensional environment, in the absence of external, active control. The investigation is motivated by observations that sustained anthropomorphic locomotion is largely a consequence of geometric and inertial properties of the mechanism. While the modeling excludes active control, the energy dissipated in ground and knee collisions is continuously re-injected by considering gait down slight inclines. The paper describes the dependence of the resulting passive gait in vertically constrained and unconstrained mechanisms on model parameters, such as ground compliance and ground slope. We also show the possibility of achieving statically unstable gait with appropriate parameter choices.

  • 4. Dankowicz, H.
    et al.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    On the origin and bifurcations of stick-slip oscillations2000In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 136, no 04-mar, p. 280-302Article in journal (Refereed)
    Abstract [en]

    A recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis. Particular emphasis is put on the bifurcations associated with the appearance of stick-slip oscillations. In the model it is found that the existence of these oscillations is a result of a periodic orbit straddling a discontinuity in the first derivative of the vector field. A local analysis tool is developed to discuss the stability of such an orbit and its bifurcations due to changes in system parameters. The analysis tool is found to be highly efficient at quantitatively predicting the location and type of bifurcations. It is argued that the method and the general results are applicable to a large class of friction models containing similar discontinuities and thus, hopefully, to actual frictional dynamics. (C)2000 Elsevier Science B.V. All rights reserved.

  • 5. Dankowicz, H.
    et al.
    Piiroinen, P.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Low-velocity impacts of quasiperiodic oscillations2002In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 14, no 2, p. 241-255Article in journal (Refereed)
    Abstract [en]

    A method based on the idea of a discontinuity mapping is derived for predicting the characteristics of system attractors that occur following a grazing intersection of a two-frequency, quasiperiodic oscillation with a two-dimensional impact surface in a three-dimensional state space. Within certain restrictions, the correction to the non-impacting flow afforded by the discontinuity mapping is computable using quantities determined solely by the non-impacting flow and the properties of the impact surface and the associated impact mapping in the immediate vicinity of the initial grazing contact. A model example is discussed to illustrate the quantitative predictive power of the discontinuity-mapping approach even relatively far away in parameter space from the original grazing intersection. Finally, constraints on the applicability of the methodology are described in detail with suggestions for suitable modifications.

  • 6. di Bernardo, M.
    et al.
    Budd, C. J.
    Champneys, A. R.
    Kowalczyk, P.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Tost, G. O.
    Piiroinen, P. T.
    Bifurcations in Nonsmooth Dynamical Systems2008In: SIAM Review, ISSN 0036-1445, E-ISSN 1095-7200, Vol. 50, no 4, p. 629-701Article, review/survey (Refereed)
    Abstract [en]

    A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

  • 7. di Bernardo, M.
    et al.
    Kowalczyk, P.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Bifurcations of dynamical systems with sliding: derivation of normal-form mappings2002In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 170, no 04-mar, p. 175-205Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with the analysis of so-called sliding bifurcations in n-dimensional piecewise-smooth dynamical systems with discontinuous vector field. These novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible. The derivation of appropriate normal-form maps is detailed. It is shown that the leading-order term in the map depends on the particular bifurcation scenario considered. This is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper. A third-order relay system serves as a numerical example.

  • 8. Di Bernardo, M.
    et al.
    Kowalczyk, P.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Sliding bifurcations: A novel mechanism for the sudden onset of chaos in dry friction oscillators2003In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 13, no 10, p. 2935-2948Article in journal (Refereed)
    Abstract [en]

    Recent investigations of nonsmooth dynamical systems have resulted in the study of a class of novel bifurcations termed as sliding bifurcations. These bifurcations are a characteristic feature of so-called Filippov systems, that is, systems of ordinary differential equations (ODEs) with discontinuous right-hand sides. In this paper we show that sliding bifurcations also play an important role in organizing the dynamics of dry friction oscillators, which are a subclass of nonsmooth systems. After introducing the possible codimension-1 sliding bifurcations of limit cycles, we show that these bifurcations organize different types of slip to stick-slip transitions in dry friction oscillators. In particular, we show both numerically and analytically that a sliding bifurcation is an important mechanism causing the sudden jump to chaos previously unexplained in the literature on friction systems. To analyze such bifurcations we make use of a new analytical method based on the study of appropriate normal form maps describing sliding bifurcations. Also, we explain the circumstances under which the theory of so-called border-collision bifurcations can be used in order to explain the onset of complex behavior in stick-slip systems.

  • 9. di Bernardo, M.
    et al.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Olivar, G.
    Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems2008In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 237, no 1, p. 119-136Article in journal (Refereed)
    Abstract [en]

    A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.

  • 10.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Activation dynamics in the optimization of targeted movements2011In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 89, no 11-12, p. 968-976Article in journal (Refereed)
    Abstract [en]

    Human movements, and their underlying muscular recruitment strategies, can be studied in several ways. In order to increase the understanding of human movement planning strategies, the movement problem is here seen as a boundary value problem for a mechanism with prescribed initial and final configurations. The time variations of a set of control actuator forces are supposed to fulfil some optimality criterion while creating this motion. The boundary value problem is discretized by temporal finite element interpolation, where the discrete variables are seen in an optimization context. The present work focusses on the introduction of the activation dynamics of the actuators, introducing a delay in the force production from the stimulation variables. The choice of interpolations of the variables is discussed in the light of the optimization setting. Examples show aspects of the results obtained for different assumptions. It is concluded that the formulation gives a good basis for further improvement of muscular force production models in an optimal movement setting.

  • 11.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Instability of hyper-elastic balloon-shaped space membranes under pressure loads2012In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 237, p. 118-129Article in journal (Refereed)
    Abstract [en]

    This paper discusses the evaluation of quasi-static equilibrium solutions for inflatable space membrane structures. A Mooney-Rivlin hyper-elastic material model, with variable constitutive constants, is considered. A compressible weightless medium is used to introduce within the membrane a one-parametric over-pressure loading compared to an ambient pressure. Analytical instability results are shown for a spherical and derived for a cylindrical case. These are compared to numerical simulations based on a flat linearly interpolated triangular space membrane element. Path-following procedures are used to find generalized equilibrium paths, with different parameterizations. Numerical examples show that the methods developed can give information on the stability of the membranes, but that the medium and means for introducing the internal pressure are of importance for the interpretation of stability.

  • 12.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Instability of thin hyper-elastic space membranes under pressure loads2011In: Proceedings of TCCM-2011, 2011Conference paper (Refereed)
  • 13.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Non-unique response of Mooney-Rivlin model in bi-axial membrane stress2014In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 144, p. 12-22Article in journal (Refereed)
    Abstract [en]

    The commonly used two-parameter Mooney-Rivlin incompressible hyper-elastic material model can show non-intuitive responses under certain conditions. This paper shows that critical states with non-unique responses occur at least at very specific bi-axial stress states. This can happen for cases where the constant related to the second invariant of strain is positive, but not for the case with this constant equal to zero (the Neo-Hookean case). The dependence of the instabilities on the ratio between the two constitutive constants is shown by evaluated fold lines. The instability is shown to be related to the imposed boundary conditions. An analytical treatment of the problem shows that dynamic edge effects correspond to the static instability.

  • 14.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Symmetry aspects in stability investigations for thin membranes2016In: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 58, no 5, p. 747-767Article in journal (Refereed)
    Abstract [en]

    Modelling of structural instability problems is considered for thin square membranes subjected to hydrostatic pressure, with a focus on the effects from symmetry conditions considered or neglected in the model. An analysis is performed through group-theoretical concepts of the symmetry aspects present in a flat membrane with one-sided pressure loading. The response of the membrane is described by its inherent differential eigensolutions, which are shown to be of five different types with respect to symmetry. A discussion is given on how boundary conditions must be introduced in order to catch all types of eigensolutions when modelling only a subdomain of the whole. Lacking symmetry in a FEM model of the whole domain is seen as a perturbation to the problem, and is shown to affect the calculated instability response, hiding or modifying instability modes. Numerical simulations verify and illustrate the analytical results, and further show the convergence with mesh fineness of different aspects of instability results.

  • 15.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Temporal finite element formulation of optimal control in mechanisms2010In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, no 25-28, p. 1783-1792Article in journal (Refereed)
    Abstract [en]

    A temporal finite element discretization of a boundary value problem has several advantages compared to a time-integrating evolution form for optimized target movement simulations. The paper gives some basic aspects on how such a finite element form can be stated, with both displacements and controls discretized and seen as unknowns. Aspects on the resulting formulations are discussed. Important issues are the order, continuity and fineness of the discretizations. When the formulation is seen in an optimization context, minimizing the effort for a prescribed movement, the discretization affects the results obtained in several manners, where some aspects of results are artifacts. The paper discusses these effects from basic principles, but also verifies them in numerical simulations.

  • 16.
    Eriksson, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Patil, Amit
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Zhou, Yang
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Parametric stability investigations for hydro-statically loaded membranes2016In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 174, p. 33-41Article in journal (Refereed)
    Abstract [en]

    This paper discusses instabilities occurring in thin pressurized membranes, important in biological as well as in engineering contexts. The membranes are represented by only their in-plane stress components, for which an incompressible isotropic hyper-elastic behavior can be assumed. A hydro-static pressurization can give instabilities in the form of limit points with respect to a loading parameter, but also bifurcations, and wrinkling. The hyper-elastic material model itself can also, under some circumstances, lead to a bifurcation situation. The instability situations can be included as constraints in a structural optimization. The paper discusses the formulation, the solution methods and some relevant instability situations. Numerical examples considering the pressurization of a flat and a cylindrical pre-stressed membrane illustrate some aspects of instability.

  • 17.
    Eriksson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Drugge, Lars
    KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Vehicle Dynamics.
    An Experimental and Numerical Study of Pantograph Dynamics, with the Application of Dimension Estimation.Manuscript (preprint) (Other academic)
  • 18.
    Essén, Hanno
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    A simple model for the falling cat problem2018In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 39, no 3, article id 035004Article in journal (Refereed)
    Abstract [en]

    We introduce a specific four-particle, four degree-of-freedom model and calculate the rotation that can be achieved by purely internal torques and forces, keeping the total angular momentum zero. We argue that the model qualitatively explains much of the ability of a cat to land on its feet even though released from rest upside down.

  • 19.
    Essén, Hanno
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Static deformation of a heavy spring due to gravity and centrifugal force2010In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 31, no 3, p. 603-609Article in journal (Refereed)
    Abstract [en]

    The static equilibrium deformation of a heavy spring due to its own weight is calculated for two cases: first for a spring hanging in a constant gravitational field, and then for a spring which is at rest in a rotating system where it is stretched by the centrifugal force. Two different models are considered: first a discrete model assuming a finite number of point masses connected by springs of negligible weight, and then the continuum limit of this model. In the second case, the differential equation for the deformation is obtained by demanding that the potential energy is minimized. In this way a simple application of the variational calculus is obtained.

  • 20.
    Essén, Hanno
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Drift velocity of charged particles in magnetic fields and its relation to the direction of the source current2016In: European Physical Journal D: Atomic, Molecular and Optical Physics, ISSN 1434-6060, E-ISSN 1434-6079, Vol. 70, no 10, article id 198Article in journal (Refereed)
    Abstract [en]

    Abstract: Integrable motion of charged particles in magnetic fields produced by stationary currentdistributions is investigated. We treat motion in the magnetic field from an infinite flatcurrent sheet, a Harris current sheath, an infinite rectilinear current, and a dipole inits equatorial plane. We find that positively charged particles as a rule will drift inthe same direction as the current that is the source of the magnetic field in question.The conclusion is that charged particles moving under the influence of currentdistributions tend to enhance the current and that this indicates currentself-amplification. Graphical abstract: [Figure not available: see fulltext.]

  • 21.
    Essén, Hanno
    et al.
    KTH, Superseded Departments, Mechanics.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Hamiltonian of a homogeneous two-component plasma2004In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 69, no 3Article in journal (Refereed)
    Abstract [en]

    The Hamiltonian of one- and two-component plasmas is calculated in the negligible radiation Darwin approximation. Since the Hamiltonian is the phase space energy of the system its form indicates, according to statistical mechanics, the nature of the thermal equilibrium that plasmas strive to attain. The main issue is the length scale of the magnetic interaction energy. In the past a screening length lambda=1/rootr(e)n, with n number density and r(e) classical electron radius, has been derived. We address the question whether the corresponding longer screening range obtained from the classical proton radius is physically relevant and the answer is affirmative. Starting from the Darwin Lagrangian it is nontrivial to find the Darwin Hamiltonian of a macroscopic system. For a homogeneous system we resolve the difficulty by temporarily approximating the particle number density by a smooth constant density. This leads to Yukawa-type screened vector potential. The nontrivial problem of finding the corresponding, divergence free, Coulomb gauge version is solved.

  • 22.
    Essén, Hanno
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Relativistic version of the Feynman-Dyson-Hughes derivation of the Lorentz force law and Maxwell's homogeneous equations2016In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 37, no 5, article id 055201Article in journal (Refereed)
    Abstract [en]

    The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.

  • 23.
    Essén, Hanno
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    Sten, J. C. -E
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Magnetic energy of surface currents on a torus2013In: Progress in Electromagnetics Research B, ISSN 1937-6472, E-ISSN 1937-6472, no 46, p. 357-378Article in journal (Refereed)
    Abstract [en]

    The magnetic energy and inductance of current distribu-tions on the surface of a torus are considered. Specifically, we investigate the influence of the aspect ratio of the torus, and of the pitch angle for helical current densities, on the energy. We show that, for a fixed surface area of the torus, the energy experiences a minimum for a certain pitch angle. New analytical relationships are presented as well as a review of results scattered in the literature. Results for the ideally conducting torus, as well as for thin rings are given.

  • 24. Fiolhais, Miguel
    et al.
    Essén, Hanno
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    Providencia, Constanta
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Magnetic field and current are zero inside ideal conductors2011In: Progress in Electromagnetics Research B, ISSN 1937-6472, E-ISSN 1937-6472, Vol. 27, p. 187-212Article in journal (Refereed)
    Abstract [en]

    We prove a theorem on the magnetic energy minimum in a system of perfect, or ideal, conductors. It is analogous to Thomson's theorem on the equilibrium electric field and charge distribution in a system of conductors. We first prove Thomson's theorem using a variational principle. Our new theorem is then derived by similar methods. We find that magnetic energy is minimized when the current distribution is a surface current density with zero interior magnetic field; perfect conductors are perfectly diamagnetic. The results agree with currents in superconductors being confined near the surface. The theorem implies a generalized force that expels current and magnetic field from the interior of a conductor that loses its resistivity. Examples of solutions that obey the theorem are presented.

  • 25. Fredriksson, M. H.
    et al.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    On normal form calculations in impact oscillators2000In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 456, no 1994, p. 315-329Article in journal (Refereed)
    Abstract [en]

    Normal form calculations are useful for analysing the dynamics close to bifurcations. However, the application to non-smooth systems is a topic for current research. Here we consider a class of impact oscillators, where we allow systems with several degrees of freedom as well as nonlinear equations of motion. Impact is due to the motion of one body, constrained by a motion limiter. The velocities of the system are assumed to change instantaneously at impact. By defining a discontinuity mapping, we show how Poincare mappings can be obtained as an expansion in a local coordinate. This gives the mapping the desired form, thus making it possible to employ standard techniques. All calculations are algorithmic in spirit, hence computer algebra routines can easily be developed.

  • 26.
    Fredriksson, Mats H.
    et al.
    KTH, Superseded Departments, Mechanics.
    Borglund, Dan
    KTH, Superseded Departments, Aeronautical Engineering.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Experiments on the Onset of Impacting Motion Using a Pipe Conveying Fluid1999In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 19, no 3, p. 261-271Article in journal (Refereed)
    Abstract [en]

    The transition from stable periodic nonimpacting motion to impacting motion, due to variations of parameters, is observable in a wide range of vibro-impact systems. Recent theoretical studies suggest a possible scenario for this type of transition. A key element in the proposed scenario is fulfilled if the oscillatory motion involved in the transition is born in a supercritical Hopf bifurcation. If the onset of impacting motion is close to the Hopf bifurcation, the impacting motion is likely to be chaotic. A numerical simulation of a system of articulated pipes conveying fluid is used to illuminate the theory. An experimental setup is presented, where a cantilevered pipe conveying fluid is unilaterally constrained. Results from experiments are found to be in good qualitative agreement with the theory.

  • 27.
    Glendinning, P.
    et al.
    Univ Manchester, Manchester, Lancs, England.
    Kowalczyk, P.
    Manchester Metropolitan Univ, Manchester, Lancs, England.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Attractors near grazing-sliding bifurcations2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 6, p. 1867-1885Article in journal (Refereed)
    Abstract [en]

    In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist.

  • 28. Glendinning, P.
    et al.
    Kowalczyk, P.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Multiple attractors in grazing-sliding bifurcations in Filippov-type flows2016In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 81, no 4, p. 711-722Article in journal (Refereed)
    Abstract [en]

    We describe two examples of three-dimensional Filippov-type flows in which multiple attractors are created by grazing-sliding bifurcations. To the best of our knowledge these are the first examples to show multistability due to a grazing-sliding bifurcation in flows. In both examples, we identify the coefficients of the normal form map describing the bifurcation, and use this to find parameters with the desired behaviour. In the first example this can be done analytically, whilst the second is a dry-friction model and the identification is numerical. This explicit correspondence between the flows and a truncated normal form map reveals an important feature of the sensitivity of the predicted dynamics: the scale of the variation of the bifurcation parameter has to be very carefully chosen. Although no detailed analysis is given, we believe that this may indicate a much greater sensitivity to parameters than experience with smooth flows might suggest. We conjecture that the grazing-sliding bifurcations leading to multistability remained unreported in the literature due to this sensitivity to parameter variations.

  • 29. Kowalczyk, P.
    et al.
    Di Bernardo, M.
    Champneys, A. R.
    Hogan, S. J.
    Homer, M.
    Piiroinen, P. T.
    Kuznetsov, Yu A.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems2006In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 16, no 3, p. 601-629Article in journal (Refereed)
    Abstract [en]

    This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincare map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

  • 30.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Discontinuity mappings for vector fields with higher order continuity2002In: Dynamical systems, ISSN 1468-9367, E-ISSN 1468-9375, Vol. 17, no 4, p. 359-376Article in journal (Refereed)
    Abstract [en]

    This paper concerns the derivation of discontinuity mappings in dynamical systems defined by piecewise smooth vector fields. The cases of trajectories that either cross or are quadratically tangent to a co-dimension 1 surface in state space are treated. The vector field is assumed to have Cn-1 continuity across the surface, and it is shown that the discontinuity mappings are equal to the identity up to order n, and the structure of the mappings and the lowest order term is derived. An example where the mapping can be derived in another way is used to illustrate the theory.

  • 31.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators2001In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 14, no 6, p. 1517-1542Article in journal (Refereed)
    Abstract [en]

    Grazing bifurcations are local bifurcations that can occur in dynamical models of impacting mechanical systems. The motion resulting from a grazing bifurcation can be complex. In this paper we discuss the creation of periodic orbits associated with grazing bifurcations, and we give sufficient conditions for the existence of a such a family of orbits. We also give a numerical example for an impacting system with one degree of freedom.

  • 32.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Dankowicz, H.
    Champneys, A.
    Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law2009In: International Journal of Non-Linear Mechanics, ISSN 0020-7462, E-ISSN 1878-5638, Vol. 44, no 10, p. 1011-1023Article in journal (Refereed)
    Abstract [en]

    This paper concerns the non-smooth dynamics of planar mechanical systems with isolated contact in the presence of Coulomb friction. Following Stronge [impact Mechanics, Cambridge University Press, Cambridge, 2000], a set of closed-form analytic formulae is derived for a rigid-body impact law based on an energetic coefficient of restitution and a resolution of the impact phase into distinct segments of relative slip and stick. Thus, the impact behavior is consistent both with the assumption of Coulomb friction and with the dissipative nature of impacts. The analysis highlights the presence of boundaries between open regions of initial conditions and parameter values corresponding to distinct forms of the impact law and investigates the smoothness properties of the impact law across these boundaries. It is shown how discontinuities in the impact law are associated with discontinuity-induced bifurcations of periodic trajectories, including non-smooth folds and persistence scenarios. Numerical analysis of an example mechanical model is used to illustrate and validate the conclusions.

  • 33.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Dankowicz, Harry
    Champneys, Alan
    Friction-induced reverse chatter in rigid-body mechanisms with impacts2011In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 76, no 1, p. 85-119Article in journal (Refereed)
    Abstract [en]

    The focus of this paper is on the possibility of formulating a consistent and unambiguous forward simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in thepresence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painleve paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-stiffness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward simulation model is afforded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter-a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-stiffness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model.

  • 34.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    Essen, Hanno
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    The comfortable roller coaster-on the shape of tracks with a constant normal force2010In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 31, no 6, p. 1307-1317Article in journal (Refereed)
    Abstract [en]

    A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is, what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.

  • 35.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Essén, Hanno
    KTH, School of Engineering Sciences (SCI), Mechanics.
    The skipping rope curve2007In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 28, no 2, p. 241-247Article in journal (Refereed)
    Abstract [en]

    The equilibrium of a flexible inextensible string, or chain, in the centrifugal force field of a rotating reference frame is investigated. It is assumed that the end points are fixed on the rotation axis. The shape of the curve, the skipping rope curve or troposkien, is given by the Jacobi elliptic function sn.

  • 36.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Kowalczyk, P.
    A codimension-two scenario of sliding solutions in grazing-sliding bifurcations2006In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 19, no 1, p. 1-26Article in journal (Refereed)
    Abstract [en]

    This paper investigates codimension-two bifurcations that involve grazing-sliding and fold scenarios. An analytical unfolding of this novel codimension-two bifurcation is presented. Using the discontinuity mapping techniques it is shown that the fold curve is one-sided and cubically tangent to the grazing curve locally to the codimension-two point. This theory is then applied to explain the dynamics of a dry-friction oscillator where such a codimension-two point has been found. In particular, the presence and the character of essential bifurcation curves that merge at the codimension-two point are confirmed. This allows us to study the dynamics away from the codimension-two point using a piecewise affine approximation of the normal form for grazing-sliding bifurcations and explain the dynamics observed in the friction system.

  • 37.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Piiroinen, P. T.
    Simulation and stability analysis of impacting systems with complete chattering2009In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 58, no 1-2, p. 85-106Article in journal (Refereed)
    Abstract [en]

    This paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chattering-accumulation of impacts-for which local discontinuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations.

  • 38.
    Nordmark, Arne
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Essén, Hanno
    KTH, School of Engineering Sciences (SCI), Mechanics.
    An impacting linear three body system2018In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 39, no 1, article id 015001Article in journal (Refereed)
    Abstract [en]

    We study a system of three identical bodies that can move freely on a horizontal track. Initially one body moves and two are at rest. The moving body impacts with one of the resting bodies which then impacts with the third and so on. The impacts are assumed to be characterised by a coefficient of restitution. We investigate the total number of impacts, the final velocities of the bodies, and the final energy of the system in terms of the initial velocity and the coefficient of restitution. The problem, which originates from mechanics textbooks, can be analysed as a discrete dynamical system with three degrees of freedom. The full solution is more subtle that one might expect.

  • 39.
    Nordmark, Arne
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Essén, Hanno
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Reply to Comment on 'An impacting linear three body system'2018In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 39, no 3, article id 038002Article in journal (Other academic)
  • 40.
    Nordmark, Arne
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Varkonyi, Peter L.
    Univ Technol & Econ, Dept Mech Mat & Struct, Fac Architecture & Engn, H-1111 Budapest, Hungary..
    Champneys, Alan R.
    Univ Bristol, Dept Engn Math, Bristol BS8 1UB, Avon, England..
    Dynamics Beyond Dynamic Jam; Unfolding the Painleve Paradox Singularity2018In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 17, no 2, p. 1267-1309Article in journal (Refereed)
    Abstract [en]

    This paper analyzes in detail the dynamics in a neighborhood of a Genot-Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painleve paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle, trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are nonlocal in momentum space and depend on properties evaluated away from the G-spot. The answer is obtained via an analysis that involves a consistent contact regularization with a stiffness proportional to 1/epsilon(2) for some epsilon. Taking a singular limit as epsilon -> 0, one finds an inner and an outer asymptotic zone in the neighborhood of the G-spot. Matched asymptotic analysis then enables continuation from the G-spot in the limit epsilon -> 0 and also reveals the sensitivity of trajectories to epsilon. The solution involves large-time asymptotics of certain generalized hypergeometric functions, which leads to conditions for the existence of a distinguished smoothest trajectory that remains uniformly bounded in t and epsilon. Such a solution corresponds to a canard that connects stable slipping motion to unstable slipping motion through the G-spot. Perturbations to the distinguished trajectory are then studied asymptotically. Two distinct cases are distinguished according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter beta passes through an integer. Finally, the results are illustrated in a particular physical example, namely the frictional impact oscillator first studied by Leine, Brogliato, and Nijmeijer.

  • 41.
    Patil, Amit
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Free and constrained inflation of a pre-stretched cylindrical membrane2014In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 470, no 2169, p. UNSP 20140282-Article in journal (Refereed)
    Abstract [en]

    This paper presents the free and constrained inflation of a pre-stretched hyperelastic cylindrical membrane and a subsequent constrained deflation. The membrane material is assumed as a homogeneous and isotropic Mooney-Rivlin solid. The constraining soft cylindrical substrate is assumed to be a distributed linear stiffness normal to the undeformed surface. Both frictionless and adhesive contact are modelled during the inflation as an interaction between the dry surfaces of the membrane and the substrate. An adhesive contact is modelled during deflation. The free and constrained inflation yields governing equations and boundary conditions, which are solved by a finite difference method in combination with a fictitious time integration method. Continuity in the principal stretches and stresses at the contact boundary is dependent on the contact conditions and inflation-deflation phase. The pre-stretch has a counterintuitive softening effect on free and constrained inflation. The variation of limit point pressures with pre-stretch and the occurrence of a cusp point is shown. Interesting trends are observed in the stretch and stress distributions after the interaction of the membrane with soft substrate, which underlines the effect of material parameters, pre-stretch and constraining properties.

  • 42.
    Patil, Amit
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Instabilities of wrinkled membranes with pressure loadings2016In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 94, p. 298-315Article in journal (Refereed)
    Abstract [en]

    Wrinkling can affect the functionality of thin membranes subjected to various loadings or boundary conditions. The concept of relaxed strain energy was studied for isotropic, hyperelastic, axisymmetric membranes pressurized by gas or fluid. Non-intuitive instabilities were observed when axisymmetric wrinkled membranes were perturbed with angle dependent displacement fields. A linearized theory showed that static equilibrium states of pressurized membranes, modelled by a relaxed strain energy formulation, are unstable, when the wrinkled surface is subjected to pressure loadings. The theory is extended to the non-axisymmetric membranes and it is shown that these instabilities are local phenomena. Simulations for the pressurized cylindrical membranes with non-uniform thickness and hemispherical membranes support the claims in both theoretical and numerical contexts including finite element simulations.

  • 43.
    Patil, Amit
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Instability investigation on fluid-loaded pre-stretched cylindrical membranes2015In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 471, no 2177, article id 20150016Article in journal (Refereed)
    Abstract [en]

    This paper discusses the evaluation of instabilities on the quasi-static equilibrium path of fluid-loaded pre-stretched cylindrical membranes and the switching to a secondary branch at a bifurcation point. The membrane is represented by only the in-plane stress components, for which an incompressible, isotropic hyperelastic material model is assumed. The free inflation problem yields governing equations and boundary conditions, which are discretized by finite differences and solved by a Newton-Raphson method. An incremental arclength-cubic extrapolation method is used to find generalized equilibrium paths, with different parametrizations. Limit points and bifurcation points are observed on the equilibrium path when fluid level is seen as the controlling parameter. An eigen-mode injection method is employed to switch to a secondary equilibrium branch at the bifurcation point. A limit point with respect to fluid level is observed for a partially filled membrane when the aspect ratio (length/radius) is high, whereas for smaller aspect ratios, the limit point with respect to fluid level is observed at over-filling. Pre-stretch is observed to have a stiffening effect in the pre-limit zone and a softening effect in the post-limit zone.

  • 44.
    Patil, Amit
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Wrinkling of cylindrical membranes with non-uniform thickness2015In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 54, p. 1-10Article in journal (Refereed)
    Abstract [en]

    Thin membranes are prone to wrinkling under various loading, geometric and boundary conditions, affecting their functionality. We consider a hyperelastic cylindrical membrane with non-uniform thickness pressurized by internal gas or fluid. When pre-stretched and inflated, the wrinkles are generated in a certain portion of the membrane depending on the loading medium and boundary conditions. The wrinkling is determined through a criterion based on kinematic conditions obtained from non-negativity of Cauchy principal stresses. The equilibrium solution of a wrinkled membrane is obtained by a specified combination of standard and relaxed strain energy function. The governing equations are discretized by a finite difference approach and a Newton-Raphson method is used to obtain the solution. An interesting relationship between stretch induced softening/stiffening with the wrinkling phenomenon has been discovered. The effects of pre-stretch, inflating medium, thickness variations and boundary conditions on the wrinkling patterns are clearly delineated. (C) 2015 Elsevier Masson SAS. All rights reserved.

  • 45.
    Pettersson, Robert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Free-time optimization of targeted movements based on temporal FE approximation2010In: Proc. CST 2010, 2010Conference paper (Refereed)
  • 46.
    Pettersson, Robert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Free-time optimization of targeted movements based on temporal finite element approximation2010In: Proceedings of the Tenth International Conference on Computational Structures Technology, Civil-Comp Ltd , 2010, Vol. 93Conference paper (Refereed)
    Abstract [en]

    Previous work by the authors has shown that temporal finite element approximations can be used for the representation of targeted optimal control problems, and that a weak equilibrium formulation leads to robust and efficient simulations. A free-time formulation is now introduced to increase the degree of freedom in finding optimal movement. The timescale parameter in relation to the objective function is discussed and verified by numerical examples. For movements with partial contact multiple phases with different mechanical properties are included. The free-time formulation allows these phases to be determined by the optimization. A three-phase two-foot high jump is simulated where the movement optimization finds a prior motion preparing for the subsequent phases with different mechanical properties.

  • 47.
    Pettersson, Robert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Optimisation of multiple phase human movements2013In: Multibody system dynamics, ISSN 1384-5640, E-ISSN 1573-272X, Vol. 30, no 4, p. 461-484Article in journal (Refereed)
    Abstract [en]

    When simulating human movements it is frequently desirable to optimise multiple phase movements where the phases represent, e.g., different contact conditions. The different constraints are usually acting in parts of the movements and their time durations are in most cases unknown. Therefore a multiple phase free-time optimisation method is formulated in this work, with phase times included as variables. Through a temporal finite element approach, a discrete representation is derived and a nonlinear optimisation algorithm solves for the rather high number of variables (similar to 6000) and constraints (similar to 15000) in the presented numerical problem. A four degrees of freedom test problem, representing a standing high jump, is solved in order to test some basic aspects. A more realistic problem shows its performance in its intended applications, biomechanical simulations. This is a sagittal eight degrees of freedom model for a human backward somersault, including preparing movement, flight phase and landing. The numerical performance as well as some application specific results are discussed. The method description is general and applicable to other movements in its presented format.

  • 48.
    Pettersson, Robert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Optimization of multiple phase human movementsIn: Multibody system dynamics, ISSN 1384-5640, E-ISSN 1573-272XArticle in journal (Other academic)
    Abstract [en]

    When simulating human movements it is frequently desirable to optimize multiple phase movements where the phases represent, e.g., different contact conditions. The different constraints are usually acting in parts of the movements and their time durations are in most cases unknown. Therefore a multiple phase free-time optimization method is formulated in this work, with phase times included as variables. Through a temporal finite element approach, a discrete representation is derived and a nonlinear optimization algorithm solves for the rather high number of variables (∼ 6000) and constraints (∼ 15000) in the presented numerical problem. The method is applied to a test problem and a more realistic problem in order to test some basic aspects as well as to see its performance in its intended applications, biomechanical simulations. First a four degrees of freedom test problem, representing a standing high jump, is solved. Then a sagittal eight degrees of freedom model is used with application to a human backward somersault, including preparing movement, flight phase and landing. The numerical performance as well as some application specific results are discussed. The method description is general and applicable to other movements in its presented format.

  • 49.
    Pettersson, Robert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Nordmark, Arne
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Simulation of rowing in an optimization context2014In: Multibody system dynamics, ISSN 1384-5640, E-ISSN 1573-272X, Vol. 32, no 3, p. 337-356Article in journal (Refereed)
    Abstract [en]

    Competitive rowing requires efforts close to the physiological limits, where oxygen consumption is one main aspect. The rowing event also incorporates interactions between the rower, the boat and oars, and water. When the intention is to improve the performance, all these properties make the sport interesting from a scientific point of view, as the many variables influencing the performance form a complex optimization problem. Our aim was to formulate the rowing event as an optimization problem where the movement and forces are completely determined by the optimization, giving at least qualitative indications on good performance. A mechanical model of rigid links was used to represent rower, boat and oars. A multiple phase cyclic movement was simulated where catch slip, driving phase, release slip and recovery were modeled. For this simplified model, we demonstrate the influence of the stated mathematical cost function as well as a parameter study where the optimal performance is related to the planned average boat velocity. The results show qualitatively good resemblance to expected movements for the rowing event. An energy loss model in combination with case specific properties of rower capacities, boat properties, and rigging was required to draw qualitative practical conclusions about the rowing technique.

  • 50. Piiroinen, P. T.
    et al.
    Dankowicz, H. J.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Breaking symmetries and constraints: Transitions from 2D to 3D in passive walkers2003In: Multibody system dynamics, ISSN 1384-5640, E-ISSN 1573-272X, Vol. 10, no 2, p. 147-176Article in journal (Refereed)
    Abstract [en]

    The inherent dynamics of bipedal, passive mechanisms are studied to investigate the relation between motions constrained to two-dimensional (2D) planes and those free to move in a three-dimensional (3D) environment. In particular, we develop numerical and analytical techniques using dynamical-systems methodology to address the persistence and stability changes of periodic, gait-like motions due to the relaxation of configuration constraints and the breaking of problem symmetries. The results indicate the limitations of a 2D analysis to predict the dynamics in the 3D environment. For example, it is shown how the loss of constraints may introduce characteristically non-2D instability mechanisms, and how small symmetry-breaking terms may result in the termination of solution branches.

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