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  • 1
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; internal resonance ; parametric vibrations ; quadratic nonlinearities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.
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  • 2
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; hysteretic oscillators ; periodic and non-periodic orbits
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincaré map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; control structure interaction ; multibody dynamics ; finite element method ; Lagrangian dynamics ; inverse dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 605-633 
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; amplitude equations ; orthogonal pendulum ; bifurcation analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 7 (1995), S. 217-229 
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; chaos ; elastic-plastic material ; Euler arch
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the dynamics of a truss structure, the Euler arch. The bars are made of elastic-plastic material, and the structure can exhibit large displacements. The aim of this paper is to give evidence of the possible chaotic behavior of this structure, even in the presence of a hardening plastic branch. The tools used are the diagrams of bifurcation, the measure of the dimension of the attractor, the Kolmogorov entropy, and the maximum Lyapunov exponent. This study emphasizes the sensitivity to the initial conditions by means of generalized basins of attraction.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 11 (1996), S. 329-346 
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; geometric elastic nonlinearities ; inertia nonlinearities ; rotary inertia ; shear deformation ; robotics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a method for the dynamic analysis of geometrically nonlinear elastic robot manipulators is presented. Robot arm elasticity is introduced using a finite element method which allows for the gross arm rotations. A shape function which accounts for the combined effects of rotary inertia and shear deformation is employed to describe the arm deformation relative to a selected component reference. Geometric elastic nonlinearities are introduced into the formulation by retaining the quadratic terms in the strain-displacement relationships. This has lead to a new stiffness matrix that depends on the rotary inertia and shear deformation and which has to be iteratively updated during the dynamic simulation. Mechanical joints are introduced into the formulation using a set of nonlinear algebraic constraint equations. A set of independent coordinates is identified over each subinterval and is employed to define the system state equations. In order to exemplify the analysis, a two-armed robot manipulator is solved. In this example, the effect of introducing geometric elastic nonlinearities and inertia nonlinearities on the robot arm kinematics, deformations, joint reaction forces and end-effector trajectory are investigated.
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  • 7
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; stochastic excitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper deals with the analysis of stochastic mechanical systems with one degree of freedom and proposes a simple procedure to obtain a representation of the dynamical response. In particular, approximate solution of the FPK equation is obtained for a system subjected to a stochastic force term. The resolving procedure is implemented with reference to a polynomial expansion of the restoring force function. Numerical tests are performed with reference to Duffing and van der Pol oscillators, showing good agreement with simulated response.
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  • 8
    ISSN: 1573-269X
    Keywords: Nonlinear dynamics ; control structure interaction ; multibody dynamics ; finite element method ; Lagrangian dynamics ; and inverse dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The inverse dynamics problem for articulated structural systems such as robotic manipulators is the problem of the determination of the joint actuator forces and motor torques such that the system components follow specified motion trajectories. In many of the previous investigations, the open loop control law was established using an inverse dynamics procedure in which the centrifugal and Coriolis inertia forces are linearized such that these forces in the flexible model are the same as those in the rigid body model. In some other investigations, the effect of the nonlinear centrifugal and Coriolis forces is neglected in the analysis and control system design of articulated structural systems. It is the objective of this investigation to study the effect of the linearization of the centrifugal and Coriolis forces on the nonlinear dynamics of constrained flexible mechanical systems. The virtual work of the inertia forces is used to define the complete nonlinear centrifugal and Coriolis force model. This nonlinear model that depends on the rate of the finite rotation and the elastic deformation of the deformable bodies is used to obtain the solution of the inverse dynamics problem, thus defining the joint torques that produce the desired motion trajectories. The effect of the linearization of the mass matrix as well as the centrifugal and Coriolis forces on the obtained feedforward control law is examined numerically. The results presented in this investigation are obtained using a slider crank mechanism with a flexible connecting rod.
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