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  • 1
    Electronic Resource
    Electronic Resource
    Random excitation of nonlinear coupled oscillators (1990)
    Springer
    Nonlinear dynamics 1 (1990), S. 91-116 
    Details
    ISSN: 1573-269X
    Keywords: internal resonance ; random vibrations ; non-Gaussian closure experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents the experimental results of random excitation of a nonlinear two-degree-of-freedom system in the neighborhood of internal resonance. The random signals of the excitation and response coordinates are processed to estimate the mean squares, autocorrelation functions, power spectral densities, and probability density functions. The results are qualitatively compared with those predicted by the Fokker-Planck equation together with a non-Gaussian closure scheme. The effects of system damping ratios, nonlinear coupling parameter, internal detuning ratio, and excitation spectral density level are considered in both studies except the effect of damping ratios is not considered in the experimental investigation. Both studies reveal similar dynamic features such as autoparametric absorber effect and stochastic instability of the coupled system. The experimental results show that the autocorrelation function of the coupled system has the feature of ultra narrow band process and degenerates to a periodic one as the internal detuning departs from the exact internal resonance condition. The measured probability density functions of the response of the main system suggests that the Gaussian representation is sufticient as long as the excitation level is relatively low in the neighborhood of the system internal resonance condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01857587
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  • 2
    Electronic Resource
    Electronic Resource
    Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging (1997)
    Springer
    Nonlinear dynamics 12 (1997), S. 155-197 
    Details
    ISSN: 1573-269X
    Keywords: Nonlinear modeling ; structure-fluid interaction ; parametric excitation ; first- and second-order stochastic averaging ; closure schemes ; noise-induced transition ; on-off-intermittency
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008299615084
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  • 3
    Electronic Resource
    Electronic Resource
    Multiple Internal Resonance in Suspended Cables under Random In-Plane Loading (1997)
    Springer
    Nonlinear dynamics 12 (1997), S. 275-303 
    Details
    ISSN: 1573-269X
    Keywords: Suspended cables ; internal resonances ; intermittency ; random excitation ; closure schemes ; Monte Carlo simulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The random excitation of a suspended cable with simultaneous internal resonances is considered. The internal resonances can take place among the first in-plane and the first two out-of-plane modes. The external loading is represented by a wide-band random process. The response statistics are estimated using the Fokker-Planck-Kolmogorov (FPK) equation, together with Gaussian and non-Gaussian closures. Monte Carlo simulation is also used for numerical verification. The unimodal in-plane motion exists in regions away from the internal resonance condition. The mixed mode interaction is manifested within a limited range of internal detuning parameters, depending on the excitation power spectrum density and damping ratios. The Gaussian closure scheme failed to predict bounded solutions of mixed mode interaction. The non-Gaussian closure results are in good agreement with the Monte Carlo simulation. The on-off intermittency of the autoparametrically excited modes is observed in the Monte Carlo simulation over a small range of excitation levels. The influence of the cable parameters, such as damping ratios, sag-to-span ratio, internal detuning parameters, and excitation level on the autoparametric interaction, is studied. It is found that the internal detuning and excitation level are the two main parameters which affect the autoparametric interaction among the three modes. Due to the system's nonlinearity, the response of the three modes is strongly non-Gaussian and the coupled modes experience irregular modulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008232209273
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  • 4
    Electronic Resource
    Electronic Resource
    Fluid Flow-Induced Nonlinear Vibration of Suspended Cables (1997)
    Springer
    Nonlinear dynamics 14 (1997), S. 377-406 
    Details
    ISSN: 1573-269X
    Keywords: Cable nonlinear dynamics ; co-ordinate transformation ; fluid-structure interaction ; divergence and flutter stability ; two-time-scale asymptotic analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008223909270
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  • 5
    Electronic Resource
    Electronic Resource
    Relationship between multibody dynamics computation and nonlinear vibration theory (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 635-653 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00162235
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  • 6
    Electronic Resource
    Electronic Resource
    Deterministic and Stochastic Response of Nonlinear Coupled Bending-Torsion Modes in a Cantilever Beam (1998)
    Springer
    Nonlinear dynamics 16 (1998), S. 259-292 
    Details
    ISSN: 1573-269X
    Keywords: Beams ; nonlinear bending-torsion dynamics ; parametric excitation ; stochastic stability ; Monte Carlo simulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this study is to understand the main differences between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008096810969
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  • 7
    Electronic Resource
    Electronic Resource
    Parametric Excitation of Nonlinear Elastic Systems Involving Hydrodynamic Sloshing Impact (1999)
    Springer
    Nonlinear dynamics 18 (1999), S. 25-50 
    Details
    ISSN: 1573-269X
    Keywords: liquid sloshing modeling ; impact ; parametric resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The parametric excitation of an elevated water tower experiencing sloshing hydro-dynamic impact is studied using the multiple scales method. The liquid sloshing mass is replaced by a mechanical model in the form of a simple pendulum experiencing impacts with the tank walls. The impact loads are modeled based on a phenomenological representation in the form of a power function with a higher exponent. In this case the system equations of motion include impact nonlinearities (selected to be of fifth power) and cubic structural geometric nonlinearities. When the first mode is parametrically excited the system exhibits hard nonlinear behavior and the impact loading reduced the response amplitude. On the other hand, when the second mode is parametrically excited, the impact loading results in complex response behavior characterized by multiple steady state solutions, where the response switches from soft to hard nonlinear characteristics. Under combined parametric resonance, the system possesses a single steady-state response in the absence and in the presence of impact. However, the system behaves like a soft system in the absence of impact and like a hard system in the presence of impact.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008384709906
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  • 8
    Electronic Resource
    Electronic Resource
    Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 547-571 
    Details
    ISSN: 1573-269X
    Keywords: Nonlinear oscillations ; buckled beams ; internal resonance ; multifurcation ; multiple scales ; numerical simulation ; experimental results
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear response of an initially buckled beam in the neighborhood of 1:1 internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00162232
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  • 9
    Electronic Resource
    Electronic Resource
    The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method (1996)
    Springer
    Nonlinear dynamics 11 (1996), S. 121-141 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00044998
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  • 10
    Electronic Resource
    Electronic Resource
    Random excitation of coupled oscillators with single and simultaneous internal resonances (1996)
    Springer
    Nonlinear dynamics 11 (1996), S. 347-400 
    Details
    ISSN: 1573-269X
    Keywords: Random excitation ; nonlinear inertia ; internal resonance ; Monte Carlo testing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The primary objective of this paper is to examine the random response characteristics of coupled nonlinear oscillators in the presence of single and simultaneous internal resonances. A model of two coupled beams with nonlinear inertia interaction is considered. The primary beam is directly excited by a random support motion, while the coupled beam is indirectly excited through autoparametric coupling and parametric excitation. For a single one-to-two internal resonance, we used Gaussian and non-Gaussian closures, Monte Carlo simulation, and experimental testing to predict and measure response statistics and stochastic bifurcation in the mean square. The mean square stability boundaries of the coupled beam equilibrium position are obtained by a Gaussian closure scheme. The stochastic bifurcation of the coupled beam is predicted theoretically and experimentally. The stochastic bifurcation predicted by non-Gaussian closure is found to take place at a lower excitation level than the one predicted by Gaussian closure and Monte Carlo simulation. It is also found that above a certain excitation level, the solution obtained by non-Gaussian closure reveals numerical instability at much lower excitation levels than those obtained by Gaussian and Monte Carlo approaches. The experimental observations reveal that the coupled beam does not reach a stationary state, as reflected by the time evolution of the mean square response. For the case of simultaneous internal resonances, both Gaussian and non-Gaussian closures fail to predict useful results, and attention is focused on Monte Carlo simulation and experimental testing. The effects of nonlinear coupling parameters, internal detuning ratios, and excitation spectral density level are considered in both investigations. It is found that both studies reveal common nonlinear features such as bifurcations in the mean square responses of the coupled beam and modal interaction in the neighborhood of internal resonances. Furthermore, there is an upper limit for the excitation level above which the system experiences unbounded response in the neighborhood of simultaneous internal resonances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00045332
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