| 1. |
- Adolfsson, J., et al.
(författare)
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3D passive walkers : Finding periodic gaits in the presence of discontinuities
- 2001
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Ingår i: Nonlinear dynamics. - 0924-090X .- 1573-269X. ; 24:2, s. 205-229
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Chillingworth, D. R. J., et al.
(författare)
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Periodic Orbits Close to Grazing for an Impact Oscillator
- 2013
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Ingår i: Recent Trends in Dynamical Systems. - Basel : Springer. - 9783034804509 ; , s. 25-37
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 3. |
- Dankowicz, H., et al.
(författare)
-
Low-velocity impacts of quasiperiodic oscillations
- 2002
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Ingår i: Chaos, Solitons & Fractals. - 0960-0779 .- 1873-2887. ; 14:2, s. 241-255
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Dankowicz, H., et al.
(författare)
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On the origin and bifurcations of stick-slip oscillations
- 2000
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Ingår i: Physica D. - 0167-2789 .- 1872-8022. ; 136:04-mar, s. 280-302
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Dankowicz, H., et al.
(författare)
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Repetitive gait of passive bipedal mechanisms in a three-dimensional environment
- 2001
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Ingår i: Journal of Biomechanical Engineering. - : ASME International. - 0148-0731 .- 1528-8951. ; 123:1, s. 40-46
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 6. |
- di Bernardo, M., et al.
(författare)
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Bifurcations in Nonsmooth Dynamical Systems
- 2008
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Ingår i: SIAM Review. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1445 .- 1095-7200. ; 50:4, s. 629-701
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Forskningsöversikt (refereegranskat)abstract
- 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.
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| 7. |
- di Bernardo, M., et al.
(författare)
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Bifurcations of dynamical systems with sliding : derivation of normal-form mappings
- 2002
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Ingår i: Physica D. - 0167-2789 .- 1872-8022. ; 170:04-mar, s. 175-205
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- di Bernardo, M., et al.
(författare)
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Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems
- 2008
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Ingår i: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 237:1, s. 119-136
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Di Bernardo, M., et al.
(författare)
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Sliding bifurcations : A novel mechanism for the sudden onset of chaos in dry friction oscillators
- 2003
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 13:10, s. 2935-2948
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
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| 11. |
- Essén, Hanno, et al.
(författare)
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Drift velocity of charged particles in magnetic fields and its relation to the direction of the source current
- 2016
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Ingår i: European Physical Journal D. - : Springer. - 1434-6060 .- 1434-6079. ; 70:10
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Tidskriftsartikel (refereegranskat)abstract
- 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.]
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| 12. |
- Essén, Hanno, et al.
(författare)
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Hamiltonian of a homogeneous two-component plasma
- 2004
-
Ingår i: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics. - 1063-651X .- 1095-3787. ; 69:3
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 13. |
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| 14. |
- Fredriksson, Mats H., et al.
(författare)
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Experiments on the Onset of Impacting Motion Using a Pipe Conveying Fluid
- 1999
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Ingår i: Nonlinear dynamics. - 0924-090X .- 1573-269X. ; 19:3, s. 261-271
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 15. |
- Fredriksson, M. H., et al.
(författare)
-
On normal form calculations in impact oscillators
- 2000
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Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : The Royal Society. - 1364-5021 .- 1471-2946. ; 456:1994, s. 315-329
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 16. |
- Glendinning, P., et al.
(författare)
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Attractors near grazing-sliding bifurcations
- 2012
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Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 25:6, s. 1867-1885
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 17. |
- Glendinning, P., et al.
(författare)
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Multiple attractors in grazing-sliding bifurcations in Filippov-type flows
- 2016
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Ingår i: IMA Journal of Applied Mathematics. - : Oxford University Press (OUP). - 0272-4960 .- 1464-3634. ; 81:4, s. 711-722
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 18. |
- Kowalczyk, P., et al.
(författare)
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Two-parameter discontinuity-induced bifurcations of limit cycles : Classification and open problems
- 2006
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 16:3, s. 601-629
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 19. |
- Nordmark, Arne B., et al.
(författare)
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A codimension-two scenario of sliding solutions in grazing-sliding bifurcations
- 2006
-
Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 19:1, s. 1-26
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 20. |
- Nordmark, Arne B., et al.
(författare)
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Discontinuity-induced bifurcations in systems with impacts and friction : Discontinuities in the impact law
- 2009
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Ingår i: International Journal of Non-Linear Mechanics. - : Elsevier BV. - 0020-7462 .- 1878-5638. ; 44:10, s. 1011-1023
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 21. |
- Nordmark, Arne B.
(författare)
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Discontinuity mappings for vector fields with higher order continuity
- 2002
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Ingår i: Dynamical systems. - : Informa UK Limited. - 1468-9367 .- 1468-9375. ; 17:4, s. 359-376
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 22. |
- Nordmark, Arne B.
(författare)
-
Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators
- 2001
-
Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 14:6, s. 1517-1542
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 23. |
- Nordmark, Arne B., et al.
(författare)
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Friction-induced reverse chatter in rigid-body mechanisms with impacts
- 2011
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Ingår i: IMA Journal of Applied Mathematics. - : Oxford University Press (OUP). - 0272-4960 .- 1464-3634. ; 76:1, s. 85-119
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 24. |
- Nordmark, Arne B., et al.
(författare)
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Simulation and stability analysis of impacting systems with complete chattering
- 2009
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Ingår i: Nonlinear dynamics. - : Springer Science and Business Media LLC. - 0924-090X .- 1573-269X. ; 58:1-2, s. 85-106
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 25. |
- Nordmark, Arne B., et al.
(författare)
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The comfortable roller coaster-on the shape of tracks with a constant normal force
- 2010
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Ingår i: European journal of physics. - : IOP Publishing. - 0143-0807 .- 1361-6404. ; 31:6, s. 1307-1317
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 26. |
- Nordmark, Arne B., et al.
(författare)
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The skipping rope curve
- 2007
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Ingår i: European journal of physics. - : IOP Publishing. - 0143-0807 .- 1361-6404. ; 28:2, s. 241-247
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 27. |
- Piiroinen, P. T., et al.
(författare)
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Breaking symmetries and constraints : Transitions from 2D to 3D in passive walkers
- 2003
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Ingår i: Multibody system dynamics. - 1384-5640 .- 1573-272X. ; 10:2, s. 147-176
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 28. |
- Piiroinen, P. T., et al.
(författare)
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On a normal-form analysis for a class of passive bipedal walkers
- 2001
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Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 11:9, s. 2411-2425
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Tidskriftsartikel (refereegranskat)abstract
- This paper implements a center-manifold technique to arrive at a normal-form for the natural dynamics of a passive, bipedal rigid-body mechanism in the vicinity of infinite foot width and near-symmetric body geometry. In particular, numerical schemes are developed for finding approximate forms of the relevant invariant manifolds and the near-singular dynamics on these manifolds. The normal-form approximations are found to be highly accurate for relatively large foot widths with a range of validity extending to widths on the order of the mechanisms' height.
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