Showing papers in "Nonlinear Dynamics in 1992"
TL;DR: It is shown by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbations methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems.
Abstract: Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions
138 citations
TL;DR: In this article, the near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model, which captures the interaction of a symmetric in-plane mode and an out-ofplane mode with near commensurable natural frequencies in a 2:1 ratio.
Abstract: The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.
127 citations
TL;DR: In this paper, the authors investigated self-sustained oscillations due to dry friction in engineering systems and showed that chaotic motions are also possible, depending on the system parameters.
Abstract: Self-sustained oscillations due to dry friction often occur in engineering systems, cf. Magnus [1]. One example, the phenomenon of machine tool chattering which deteriorates the production quality and increases the tool wear is — at least partly — caused by friction forces with a decreasing characteristic. Another phenomenon is the curving noise of tram wheels, induced by nonlinear slip forces, which annoys passengers and city dwellers. Recent investigations of oscillations induced by dry friction show that beside the well-known limit cycle behaviour, chaotic motions are also possible, depending on the system parameters.
92 citations
TL;DR: In this article, a percussion drilling machine is examined as an example for mechanical systems with unilateral contacts, and the constrained motion of the system and simultaneously the constraint forces are taken into account by algebraic relations.
Abstract: In the following a percussion drilling machine is examined as an example for mechanical systems with unilateral contacts. It is characteristic for such systems that the number of degrees of freedom changes during motion. To avoid a description of each possible system state using different sets of minimal coordinates, the constrained motion is taken into account by algebraic relations. This method has the advantage that the motion of the system and simultaneously the constraint forces are available, which is necessary to obtain conditions for a change in the state of the system. Furthermore different combinations of constraints can be easily taken into consideration in this way.
62 citations
TL;DR: In this article, the quadrature method was applied to flexural vibration analysis of a geometrically nonlinear beam and the numerical results by OM agree with the results by the finite element method.
Abstract: The quadrature method (OM) has been used in structural analysis only in recent years. In this study, OM is applied to flexural vibration analysis of a geometrically nonlinear beam. The numerical results by OM agree with the results by the finite element method. It is believed that this is the first attempt to solve a nonlinear dynamic problem by the quadrature method.
60 citations
TL;DR: In this paper, the equations of motion for a flexible body are derived based on the principles of continuum mechanics and the finite element method, and the use of a lumped mass formulation and the description of the nodal accelerations relative to a nonmoving reference frame lead to a simple form of these equations.
Abstract: The problem of formulating and numerically solving the equations of motion for a multibody system undergoing large motion and clasto-plastic deformations is considered here. Based on the principles of continuum mechanics and the finite element method, the equations of motion for a flexible body are derived. It is shown that the use of a lumped mass formulation and the description of the nodal accelerations relative to a nonmoving reference frame lead to a simple form of these equations. In order to reduce the number of coordinates that describe a deformable body, a Guyan condensation technique is used. The equations of motion of the complete multibody system are then formulated in terms of joint coordinates between the rigid bodies. The kinematic constraints that involve flexible bodies are introduced in the equations of motion through the use of Lagrange multipliers.
52 citations
TL;DR: In this article, the authors studied the common features in regular and strange behavior of three classic dissipative softening type driven oscillators: (a) twinwell potential system, (b) single-well potential unsymmetric system and (c) singlewell potential symmetric system.
Abstract: The paper is devoted to the study of common features in regular and strange behavior of the three classic dissipative softening type driven oscillators: (a) twin-well potential system, (b) single-well potential unsymmetric system and (c) single-well potential symmetric system.
45 citations
TL;DR: In this paper, a new physical model for a shock absorber is presented which provides a more realistic representation of the stiffness characteristics than previous simple models, and validated on experimental data.
Abstract: A new physical model for a shock absorber is presented which provides a more realistic representation of the stiffness characteristics than previous simple models. The new model is validated on experimental data.
42 citations
TL;DR: In this paper, the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈ 2ω1 to a harmonic excitation of the third mode was investigated.
Abstract: An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈2ω1 to a harmonic excitation of the third mode, where the ω
m
are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.
37 citations
TL;DR: In this paper, a piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior.
Abstract: Steady-state solutions of a piecewise-linear oscillator under multi-forcing frequencies are obtained using the fixed point algorithm (FPA). Stability analysis is also performed using the same technique. For the periodic solutions of a piecewise-linear oscillator with single forcing frequency, the harmonic balance method (HBM) is also used along with the FPA. Although both FPA and HBM generate accurate solutions, it is observed that the HBM failed to converge to solutions in the superharmonic range of the forcing frequency. The fixed point algorithm was also applied to the oscillator under multifrequency excitation. The algorithm proved to be very effective in obtaining torus solutions and in locating corresponding bifurcation thresholds. A piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior. A second order Poincare mapping technique reveals the hidden fractal-like nature of the resulting chaotic response. A parametric study is performed for the response of the ALP.
32 citations
TL;DR: In this article, a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations is presented, which fully accounts for geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case.
Abstract: Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.
TL;DR: In this article, the amplitude and phase of the bifurcating cyclic solutions of a representative non-conservative four-dimensional autonomous system with quadratic nonlinearities were studied.
Abstract: We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.
TL;DR: In this article, the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system was investigated, where the system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes.
Abstract: In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.
TL;DR: In this article, the fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis.
Abstract: The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a ‘1−1’ internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses ‘nonlinear normal modes.’ For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.
TL;DR: In this paper, the influence of initial geometric imperfections on the dynamic behavior of simply supported rectangular plates subjected to the action of periodic in-plane forces is analyzed by the first-order generalized asymptotic method.
Abstract: The present work deals with the influence of initial geometric imperfections on the dynamic behavior of simply supported rectangular plates subjected to the action of periodic in-plane forces The nonlinear large-deflection plate theory used in this analysis corresponds to the dynamic analog of von Karman's theory The temporal response is analyzed by the first-order generalized asymptotic method The solution for the temporal equations of motion takes into account the possibility of existence of simultaneous forced and parametric vibrations The results indicate that the presence of initial imperfections may significantly raise the resonance frequencies, cause the plate to exhibit a soft spring behavior and improve slightly the stability of the plate by reducing the area of its instability zones Furthermore, the presence of initial imperfections induces forced vibrations which interact with parametric vibrations in order to generate a competitive hesitation phenomenon in the transition zone
TL;DR: In this article, the dynamic behavior of a harmonically excited, preloaded mechanical oscillator with dead-zone nonlinearity is described quantitatively and the governing strongly nonlinear differential equation is solved numerically.
Abstract: The dynamic behavior of a harmonically excited, preloaded mechanical oscillator with dead-zone nonlinearity is described quantiatively. The governing strongly nonlinear differential equation is solved numerically. Damping coefficient-force ratio maps for two different values of the excitation frequency have been formed and the boundaries of the regions of different motion types are determined. The results have been compared with the results of the forced Duffing's equation available in the literature in order to identify the differences between cubic and dead-zone nonlinearities. Period-doubling bifurcations, which take place with a change of any of the system parameters, have been found to be the most common route to chaos. Such bifurcations follow the scaling rule of Feigenbaum.
TL;DR: In this article, a technique for robust identification of nonlinear dynamic systems is developed and illustrated using both simulations and analog experiments, which is based on the minimum model error optimal estimation approach.
Abstract: A technique for robust identification of nonlinear dynamic systems is developed and illustrated using both simulations and analog experiments. The technique is based on the Minimum Model Error optimal estimation approach. A detailed literature review is included in which fundamental differences between the current approach and previous work is described. The most significant feature of the current work is the ability to identify nonlinear dynamic systems without prior assumptions regarding the form of the nonlinearities, in constrast to existing nonlinear identification approaches which usually require detailed assumptions of the nonlinearities. The example illustrations indicate that the method is robust with respect to prior ignorance of the model, and with respect to measurement noise, measurement frequency, and measurement record length.
TL;DR: In this article, a general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented, which fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case.
Abstract: A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.
TL;DR: In this paper, the stability and control characteristics of two twin-lift helicopter configurations are analyzed in order to address the issue of configuration selection from a handling qualities viewpoint, their open-and closed-loop characteristics are compared.
Abstract: The stability and control characteristics of two twin-lift helicopter configurations are analyzed in this paper. In order to address the issue of configuration selection from a handling qualities viewpoint, their open-and closed-loop characteristics are compared. The two twin-lift configurations considered are the twin-lift with spreader bar and twin-lift without spreader bar. The nonlinear models describing the dynamics of these two configurations in the lateral/vertical plane are derived. The open-loop characteristics of the two systems are compared by linearizing the nonlinear models about a symmetric hovering equilibrium condition. The closed-loop characteristics of the two systems are compared using nonlinear controllers based on feedback linearization schemes. The performance of the resulting closed-loop systems in carrying out a typical twin-lift mission is evaluated through nonlinear simulation. Also, the effects of helicopter performance degradation and measurement errors on the overall system performance are discussed.
TL;DR: In this article, the authors consider the dynamics of "roller-coaster" type experimental models used as analog devices for nonlinear oscillators, and show how to choose the shape of the track in order to achieve a desired oscillator equation, in terms of the length coordinate or its projection onto the horizontal.
Abstract: We consider the dynamics of ‘roller-coaster’ type experimental models used as analog devices for nonlinear oscillators. It is shown how to chose the shape of the track in order to achieve a desired oscillator equation, in terms of the are length coordinate or its projection onto the horizontal. Explicit calculations are carried out for the linear oscillator, the so-called ‘escape equation’, the two-well Duffing oscillator, and the pendulum.
TL;DR: In this article, the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity was investigated for ground tests of space-station structural components.
Abstract: A critical problem in designing large structures for space applications, such as space stations and parabolic antennas, is the limitation of testing these structures and their substructures on earth. These structures will exhibit very high flexibilities due to the small loads expected to be encountered in orbit. It has been reported in the literature that the gravitational sag effect under dead weight is of extreme importance during ground tests of space-station structural components [1–4]. An investigation of a horizontal, pinned-pinned beam with complete axial restraint and undergoing large-amplitude oscillations about the statically deflected position is presented here. This paper presents a solution for the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity.
TL;DR: In this article, a new mechanical model for simulating both ship oscillations and the induced twisting of the hull in the case of longitudinal seas is presented, which may result from non-linearly coupled heave-pitch-roll motions.
Abstract: A new mechanical model for simulating both the ship oscillations and the induced twisting of the hull in the case of longitudinal seas is presented. Particular attention is given to the onset of parametric rolling, which may result from non-linearly coupled heave-pitch-roll motions. It is shown that in these sea conditions the phenomenon of twisting is likely to occur under a mechanism similar to that of parametric rolling.
TL;DR: In this paper, an extension of the Cumulant-Neglect closure scheme is utilized for the random vibration analysis of a single degree of freedom system with a general pinching hysteresis restoring force.
Abstract: In this paper, an extension of the Cumulant-Neglect closure scheme is utilized for the random vibration analysis of a single degree of freedom system with a general pinching hysteresis restoring force. The hysteresis element used in the system model can simulate commonly observed forms of stiffness, strength and pinching degradations. The second order statistics of the system response to a stationary Gaussian white noise input are derived using an Ito-based approximation technique. The validity of these response statistics are then verified by comparing them to Monte Carlo simulation results. The numerical studies performed for different combinations of degradation parameters and excitation levels show that the response estimates obtained by this solution method are in good agreement with Monte Carlo simulation. These studies also indicate the applicability of this technique for response analysis of complicated forms of non-linearities.
TL;DR: In this paper, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically, and the results obtained using the three and six node triangular element are compared with the higher order six-node triangular element.
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.
TL;DR: In this paper, the scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales.
Abstract: The scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales. The scaling law for coupled Duffing oscillators is that the coupling intensity should be proportional to the total energy of the system.