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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 18 (1999), S. 185-202 
    ISSN: 1573-269X
    Keywords: Identification ; nonlinear vibration ; vibration of continuous systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In previous papers, the authors proposed a new experimental identification technique applicable to elastic structures. The proposed technique is based on the principle of harmonic balance and can be classified as a frequency domain technique. The technique requires the excitation force to be periodic. This is, in some cases, a restriction. So another technique free from this restriction is of use. In this paper, as a first step for developing such techniques, a technique applicable to beams is proposed. The proposed technique can be classified as a time domain technique, two variations of which are proposed. The first method is based on the usual least-squares method. The second is based on solving a minimization problem with constraints. The latter usually yields better results. But in this method, an iteration procedure is used which requires initial values for the parameters. To obtain the initial values, the first method can be used. So both methods are useful. Finally, the applicability of the proposed technique is confirmed by numerical simulation as well as experiments.
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  • 2
    ISSN: 1573-269X
    Keywords: Backward Kolmogorov equation ; Duffing oscillator ; finite elements ; first passage problem ; Fokker-Planck equation ; linear oscillator ; nonlinear vibration ; probability distribution ; random vibration ; stochastic processes ; Van der Pol oscillator
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.
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  • 3
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; nonlinear vibration ; internal resonance ; energy balance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents the ground-work of implementing the multibody dynamics codes to analyzing nonlinear coupled oscillators. The recent developments of the multibody dynamics have resulted in several computer codes that can handle large systems of differential and algebraic equations (DAE). However, these codes cannot be used in their current format without appropriate modifications. According to multibody dynamics theory, the differential equations of motion are linear in the acceleration, and the constraints are appended into the equations of motion through Lagrange's multipliers. This formulation should be able to predict the nonlinear phenomena established by the nonlinear vibration theory. This can be achieved only if the constraint algebraic equations are modified to include all the system kinematic nonlinearities. This modification is accomplished by considering secondary nonlinear displacements which are ignored in all current codes. The resulting set of DAE are solved by the Gear stiff integrator. The study also introduced the concept of constrained flexibility and uses an instantaneous energy checking function to improve integration accuracy in the numerical scheme. The general energy balance is a single scalar equation containing all the energy component contributions. The DAE solution is then compared with the solution predicted by the nonlinear vibration theory. It also establishes new foundation for the use of multibody dynamics codes in nonlinear vibration problems. It is found that the simulation CPU time is much longer than the simulation of the original equations of the system.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 19 (1999), S. 237-260 
    ISSN: 1573-269X
    Keywords: energy method ; periodic solution ; nonlinear vibration ; nonautonomous system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.
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  • 5
    ISSN: 1573-269X
    Keywords: Vibration of continuous rotor ; nonlinear vibration ; subharmonic resonance ; combination resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.
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  • 6
    ISSN: 1573-269X
    Keywords: d'Alembert principle ; reduced multibody method ; constrained flexibility ; nonlinear vibration ; Galerkin's method ; checking function ; differential and algebraic equations (DAE) ; bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 20 (1999), S. 85-98 
    ISSN: 1573-269X
    Keywords: fuzzy chip ; fuzzy controller ; fuzzy logic ; nonlinear vibration ; vibration control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The fuzzy logic ability to manage nonlinear vibration control problems has one major drawback: the slow reaction time offered by a software-implemented controller. This justifies the adoption of integrated fuzzy chip controllers. Nevertheless, designing a fuzzy controller for structural control purposes requires some fine tuning work. This can be conveniently carried out over an equivalent electronic circuit which emulates a civil structure, rather than by an expensive laboratory environment for testing frame specimens or by using computer simulations which would not allow a real-time study. Conceiving and implementing an electronic circuit equivalent to a single-degree-of-freedom system is the topic of this paper. A case study is finally presented. It discusses the major features of the adopted 'fuzzy chip controller'. A brief overview of the results of the laboratory tests is also given. They were performed in order to properly tune the fuzzy project with civil structure applications.
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  • 8
    ISSN: 1573-269X
    Keywords: Aeroelasticity ; nonlinear vibration ; panel flutter ; bifurcation ; attractor ; chaos ; attraction basin
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.
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  • 9
    ISSN: 1573-269X
    Keywords: Dynamics ; finite element method ; triangular elements ; multibody dynamics ; component mode synthesis ; nonlinear vibration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.
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  • 10
    ISSN: 1573-269X
    Keywords: Composite plates ; nonlinear vibration ; finite element method ; multibody dynamics ; stress analysis ; and numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The development of a shear-deformable laminated plate element, based on the Mindlin plate theory, for use in large reference displacement analysis is presented. The element is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotrophic material property sets. Coordinate mapping is utilized so that non-rectangular elements may be modeled. The Gauss quadrature method of numerical integration is utilized to evaluate volume integrals. A comparative study is done on the use of full Gauss quadrature, reduced Gauss quadrature, mixed Gauss quadrature, and closed form integration techniques for the element. Dynamic analysis is performed on the RSSR (Revolute-Spherical-Spherical-Revolute) mechanism, with the coupler modeled as a flexible plate. The results indicate the differences in the dynamic response of the transverse shear deformable eight-noded element as compared to a four-noded plate element. Dynamically induced stresses are examined, with the results indicating that the primary deformation mode of the eight-noded Mindlin plate model being bending.
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