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  • 1
    ISSN: 1573-269X
    Keywords: Internal resonances ; bifurcations ; quasiperiodic motions ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a collection of experimental results on the influence of modal interactions (i.e., internal or autoparametric resonances) on the nonlinear response of flexible metallic and composite structures subjected to a range of resonant excitations. The experimental results are provided in the form of frequency spectra, Poincaré sections, pseudo-phase planes, dimension calculations, and response curves. Experimental observations of transitions from periodic to chaotically modulated motions are also presented. We also discuss relevant analytical results. The current study is also relevant to other internally resonant structural systems.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-269X
    Keywords: Flexible robotic manipulator ; advanced composites ; mathematical modeling ; finite element method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A dynamic model for a two degree-of-freedom planar robot arm is derived in this study. The links of the arm, connected to prismatic and revolute joints, are considered to be flexible. They are assumed to be fabricated from either aluminum or laminated composite materials. The model is derived based on the Timoshenko beam theory in order to account for the rotary inertia and shear deformation. These effects are significant in modeling flexible links connected to prismatic joints. The deflections of the links are approximated by using a shear-deformable beam finite element. Hamilton's principle is implemented to derive the equations describing the combined rigid and flexible motions of the arm. The resulting equations are coupled and highly nonlinear. In view of the large number of equations involved and their geometric nonlinearity (topological and quadratic), the solution of the equations of motion is obtained numerically by using a stiff integrator. The digital simulation studies examine the interaction between the flexible and the rigid body motions of the robot arm, investigate the improvement in the accuracy of the model by considering the flexibility of all rather than some of the links of the arm, assess the significance of the rotary inertia and shear deformation, and illustrate the advantages of using advanced composites in the structural design of robotic manipulators.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 235-262 
    ISSN: 1573-269X
    Keywords: Burgers' equation ; stabilizability ; optimal controls ; approximations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A stabilization problem for Burgers' equation is considered. Using linearization, various controllers are constructed which minimize certain weighted energy functionals. These controllers produce the desired degree of stability for the closed-loop nonlinear system. A numerical scheme for computing the feedback gain functional is developed and several numerical experiments are performed to illustrate the theoretical results.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 317-317 
    ISSN: 1573-269X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 353-365 
    ISSN: 1573-269X
    Keywords: Chaotic motions ; diagnostics ; specepstrum
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A technique based on the power cepstrum has been developed to analyze and characterize data of various nonlinear and chaotic motions. For repeatability and ready availability, nonlinear response data of a Duffing oscillator and van der Pol oscillator, generated numerically by the fourth order Runge-Kutta algorithm, were used in the investigation. Results obtained by the proposed technique using spectrum, cepstrum, and specepstrum which is defined as the spectrum of the logarithmic cepstrum, indicate that it is superior to methods previously available in the literature.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1573-269X
    Keywords: Railway dynamics ; limit-cycles ; bifurcations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 187-198 
    ISSN: 1573-269X
    Keywords: Nonlinear ; beam ; vibrations ; modes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The investigation reported herein analyzes the vibration of a uniform beam with hinged ends which are restrained. The beam is subjected to a linearly-varying distributed load which has a maximum intensity w 0 at the center and is released from rest when the load is suddenly removed. The motion is found to be inherently nonlinear, even for small vibrations, and there is dynamic mode-coupling. The mode frequencies are functionally related to initial conditions, particularly the amplitudes of all modes.
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  • 8
    ISSN: 1573-269X
    Keywords: Nonlinear rotor ; chaotic whirling ; subsynchronous motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new HB (Harmonic Balance)/AFT (Alternating Frequency Time) method is further developed to obtain synchronous and subsynchronous whirling response of nonlinear MDOF rotor systems. Using the HBM, the nonlinear differential equations of a rotor system can be transformed to algebraic equations with unknown harmonic coefficients. A technique is applied to reduce the algebraic equations to only those of the nonlinear coordinates. Stability analysis of the periodic solutions is performed via perturbation of the solutions. To further reduce the computational time for the stability analysis, the reduced system parameters (mass, damping, and stiffness) are calculated in terms of the already known harmonic coefficients. For illustration, a simple MDOF rotor system with a piecewise-linear bearing clearance is used to demonstrate the accuracy of the calculated steady-state solutions and their bifurcation boundaries. Employing ideas from modern dynamics theory, the example MDOF nonlinear rotor system is shown to exhibit subsynchronous, quasi-periodic and chaotic whirling motions.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 291-304 
    ISSN: 1573-269X
    Keywords: Chaos ; Fokker-Planck-Equation ; probability density ; global description of motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An observation of single trajectories exhibiting chaotic motion turns out to be disadvantageous because even smallest variations of the initial conditions grow exponentially in time and result in an unpredictable long-time behaviour. The paper gives a different approach based on a probability distribution of the state space variables which is invariant on the area of attraction and results in a global description of chaotic motion.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 335-351 
    ISSN: 1573-269X
    Keywords: Non-linear oscillation ; critical values ; singularities ; impacting
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Elementary applications of singularity theory to dynamical systems are outlined. The main idea is to consider the geometry of the wrinkled surface generated by the time development of a one variable solution as a function of initial conditions or system parameters. Applications of singularity theory to vibro-impacting systems are discussed in some detail and further applications to dynamics are briefly reviewed.
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