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  • stability  (20)
  • Multibody dynamics  (7)
  • 1
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; finite elementmethod ; QR decomposition ; Cholesky decomposition ; absolute nodal coordinate formulation ; floating frameof reference formulation ; incremental methods ; large deformation ; large rotation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 19 (1999), S. 135-158 
    ISSN: 1573-269X
    Keywords: perturbation methods ; higher-order approximations ; dynamical systems ; codimension ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Higher-order multiple-scale methods for general multiparameter mechanical systems are studied. The role played by the control and imperfection parameters in deriving the perturbative equations is highlighted. The definition of the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. The concept of a reduced dynamical system is then invoked. Different approaches followed in the literature to deal with reconstituted amplitude equations are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Finally, three examples are illustrated to corroborate the findings and to show the quantitative differences between the various approaches.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 19 (1999), S. 173-193 
    ISSN: 1573-269X
    Keywords: fluid conveying pipes ; high-frequency pulsating fluid ; separation of slow and fast motion ; stability ; nonlinear dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 19 (1999), S. 313-332 
    ISSN: 1573-269X
    Keywords: double pendulum system ; double Hopf bifurcation ; stability ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.
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  • 5
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; nonlinear dynamics ; dynamic simulation ; penalty methods ; constraint stabilization ; constraint violation ; constraint convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents stability and convergence results on a novel approach for imposing holonomic constraints for a class of multibody system dynamics. As opposed to some recent techniques that employ a penalty functional to approximate the Lagrange multipliers, the method herein defines a penalized dynamical system using penalty-augmented kinetic and potential energies, as well as a penalty dependent constraint violation dissipation function. In as much as the governing equations are not typically cocreive, the usual convergence criteria for linear variational boundary value problems are not directly applicable. Still numerical simulations by various researchers suggest that the method is convergent and stable. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Likewise, stability and asymptotic stability results for the penalty formulation are derived from well-known stability results available from classical mechanics. Unfortunately, the convergence theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of a typical dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Lyapunov/Invariance Principle analysis. In short, the approach has many advantages as an alternative to other computational techniques: (1) Explicit constraint violation bounds can be derived for a large class of nonlinear multibody dynamics problems (2) Sufficient conditions for the Lyapunov stability, and asymptotic stability, of the penalty formulation are derived for a large class of multibody systems (3) The method can be shown to be relatively insensitive to singular configurations by selecting the penalty parameters to dissipate ‘constraint violation energy’ (4) The Invariance Principle can be employed in the method, in certain cases, to derive the asymptotic behavior of the constraint violation for dissipative multibody systems by identifying ‘constraint violation limit cycles’ Just as importantly, these results for nonlinear systems can be ‘sharpened’ considerably for linear systems: (5) Explicit spectral error estimates can be obtained for substructure synthesis (6) The penalty equations can be shown to be optimal in the sense that the terms represent feedback that minimizes a measure of the constraint violation
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  • 6
    ISSN: 1573-269X
    Keywords: Periodic solutions ; stability ; local bifurcations ; Fourier series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:λ), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.
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  • 7
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; contact-impact ; Hertz models ; flexibility ; structural damping ; crashworthiness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A computer based formulation for the analysis of mechanical systems is investigated as a feasible method to predict the impact response of complex structural systems. A general methodology for the dynamic analysis of rigid-flexible multibody systems using a number of redundant Cartesian coordinates and the method of the Lagrange multipliers is presented. The component mode synthesis is then used to reduce the number of flexible degrees of freedom. In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. This concept is used by associating revolute nonlinear actuators with constitutive relations corresponding to the collapse behavior of the structural components. The contact of the system components is described using a continuous force model based on the Hertz contact law with hysteresis damping. The effect and importance of structural damping schemes in flexible bodies are also addressed here. Finally, the validity of this methodology is assessed by comparing the results of the proposed models with those obtained in different experimental tests where: a beam collides transversally with a rigid block; a torque box impacts a rigid barrier.
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  • 8
    ISSN: 1573-269X
    Keywords: Regularization ; stability ; constrained multibody systems ; dynamics of multibody systems ; non-holonomic ; singularity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the analysis of multibody dynamics, we are often required to deal with singularity problems where the constraint Jacobian matrix may become less than full rank at some instantancous configurations. This creates numerical instability which will affect the performance of the mechanical system. A modification procedure of the constraints when they vanish or become linearly dependent is proposed to regularize the dynamics of the system. A distinction between the asymptotic stability due to the representation of the constraints (at the velocity and acceleration level), and the one due to the singularity is discussed in full in this paper. It is shown that Baumgarte technique could be extended to accommodate the representation of the constraints in the neighborhood of singularity. A two link planar manipulator undergoing large motion and passing through a singular configuration is used to illustrate the proposed stability technique.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 10 (1996), S. 333-357 
    ISSN: 1573-269X
    Keywords: Pipes ; parametric excitation ; nonlinearity ; chaos ; multiple time scales ; harmonic balancing ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Chaotic motions of a simply supported nonlinear pipe conveying fluid with harmonie velocity fluetuations are investigated. The motions are investigated in two flow velocity regimes, one below and above the critical velocity for divergence. Analyses are carried out taking into account single mode and two mode approximations in the neighbourhood of fundamental resonance. The amplitude of the harmonic velocity perturbation is considered as the control parameter. Both period doubling sequence and a sudden transition to chaos of an asymmetric period 2 motion are observed. Above the critical velocity chaos is explained in terms of periodic motion about the equilibrium point shifting to another equilibrium point through a saddle point. Phase plane trajectories, Poincaré maps and time histories are plotted giving the nature of motion. Both single and two mode approximations essentially give the same qualitative behaviour. The stability limits of trivial and nontrivial solutions are obtained by the multiple time scale method and harmonic balance method which are in very good agreement with the numerical results.
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  • 10
    ISSN: 1573-269X
    Keywords: Perturbation methods ; stability ; bifurcation ; codimension two ; periodic and quasi-periodic solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is shown that the logical bases of the static perturbation method, which is currently used in static bifurcation analysis, can also be applied to dynamic bifurcations. A two-time version of the Lindstedt–Poincaré Method and the Multiple Scale Method are employed to analyze a bifurcation problem of codimension two. It is found that the Multiple Scale Method furnishes, in a straightforward way, amplitude modulation equations equal to normal form equations available in literature. With a remarkable computational improvement, the description of the central manifold is avoided. The Lindstedt–Poincaré Method can also be employed if only steady-state solutions have to be determined. An application is illustrated for a mechanical system subjected to aerodynamic excitation.
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