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• 11
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 243-264
ISSN: 1573-269X
Keywords: Aeroelastic flutter ; influence of maneuvering ; chaotic vibrations ; routes to chaos ; dissipative dynamical systems ; computational methods
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.
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• 12
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 293-311
ISSN: 1573-269X
Keywords: Oil whirl ; bifurcation ; unbalance ; instability
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The nonlinear behavior of an unbalanced rotor supported in a fluid film bearing is analyzed. A simplified two dimensional model is adopted which uses the long-bearing approximation with a π-film to account for cavitation. This model has been thoroughly studied by Myers [1] in the balanced case, where it is shown that the whirl instability is the result of a Hopf bifurcation. The implications of imbalance are studied in this paper. This leads to the study of a periodically perturbed Hopf bifurcation. It is shown that the dynamics in this situation can, especially under certain nonlinear resonance conditions, have an extremely complicated dependence on the system parameters and the rotor speed. Complete bifurcation diagrams are presented for a particular rotor model which demonstrate this dependence.
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• 13
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 313-339
ISSN: 1573-269X
Keywords: Power systems ; chaos ; bifurcations ; loss of synchronism
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We investigate some of the instabilities in a single-machine quasi-infinite busbar system. The system's behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency Ω. When Ω≈ω0, where ω0 is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop an approximate first-order closed-form expression for the period-one responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted period-one solutions and conditions for its instability are in good agreement with the digital-computer results.
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• 14
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 265-291
ISSN: 1573-269X
Keywords: Cables ; autonomous systems ; primary resonance ; first-order approximations ; three-dimensional vibrations with quadratic nonlinearity
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations. In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity. Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.
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• 15
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 359-377
ISSN: 1573-269X
Keywords: Vibrations ; parametric excitation ; Kane's equations ; Floquet theory ; axial motions ; cantilever beam
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Studied in this work are the formulation of equations of motion and the response to parametric excitation of a uniform cantilever beam moving longitudinally over a single bilateral support. The equations of motion are generated by using assumed modes to discretize the beam, by regarding the support as a kinematic constraint, and by employing an alternate form of Kane's method that is particularly well suited to systems subject to constraints. Instability information is developed using the results of perturbation analysis for harmonic longitudinal motions of small amplitude and with Floquet theory for general periodic motions of any amplitude. Results demonstrate that definitive instability information can be obtained for a beam moving longitudinally over supports based on the frequencies of free transverse vibration of a beam that is longitudinally fixed.
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• 16
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 341-358
ISSN: 1573-269X
Keywords: Rotating shaft ; critical speed ; nonstationary response ; summed-and-differential harmonic oscillation
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration λ, the initial angular position of the unbalance ψn and the initial rotating speed ω on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on λ but also on ψn and ω: (2) when the initial angular position ψn changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of λ must exceed a certain critical value.
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• 17
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 379-400
ISSN: 1573-269X
Keywords: Space-platform ; system-modes ; dynamics
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The paper studies libration/vibration interaction dynamics of the space station Freedom during its evolutionary phases. To that end, a relatively general nonlinear formulation, applicable to a system of interconnected plate and beam-type structural members forming a tree topology is developed. System modes, obtained through a finite element analysis, are employed in the discretization process and the response study is purposely confined to the orbital plane to emphasize interactions between librational dynamics and flexibility. Finally, an approximate closed-form nonlinear solution of the problem is attempted using the variation of parameters method. Its validity is assessed through comparison with the numerically obtained results over a range of system parameters and initial conditions. Results provide information pertaining to the levels of librational and vibrational response and the associated acceleration tield, which may prove helpful in appropriately locating experiments and monitoring instruments. It may also aid in the planning of the control system. The analytical approach provides surprisingly close correlation with the more elaborate numerical procedure, thus promising better physical appreciation of the complex interactions as well as a considerable saving in the computational cost.
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• 18
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 401-420
ISSN: 1573-269X
Keywords: Chaos ; perturbation methods ; elliptic functions ; differential equations
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We investigate the system $$\ddot x - x\cos \varepsilon 1 + x^3 = 0$$ in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the $$x - \dot x$$ phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 〉 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
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• 19
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 433-447
ISSN: 1573-269X
Keywords: Multibody systems ; clastic bodies ; geometric nonlinearities ; centrifugal stiffening
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Systems of rigid and flexible bodies undergoing large rigid body motions but small elastic deformations are investigated. In order to get the correct equations of motion one has to consider geometric nonlinearities even in the elastic coordinates. Different possibilities of independently choosing these coordinates are presented. The flexible bodies are discretized using a Ritz-Galerkin approximation. This discretization leads to ordinary differential equations for the description of the clastic vibrations of the flexible bodies as well as for the description of the rigid body motions. The modelling of deformable bodies is one aspect of our investigations. Another aim of our research is the consideration of clastic bodies in existing multibody programs, in particular in the program AUTOLEV. This is a symbol manipulating program for the formulation of the equations of motion running on PCs. Two examples are considered showing that it is possible to treat clastic bodies in this and in similar programs with the concept of modelling presented above.
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• 20
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 421-432
ISSN: 1573-269X
Keywords: Rotating cable structures ; weight-excited oscillations ; snap-through phenomena ; experiment and theory
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The combined effect of gravity and a centrifugal field on the dynamics of a heavy cable with negligible bending stiffness is treated. The inextensible length of the cable is preassigned in such a way that the cross-sectional shape of the cable structure is circular if the contiguration rotates about its central axis with a constant speed and no external forces act. Under the additional influence of gravity, complicated nonlinear weight-excited vibrations occur. To understand the variety of vibrational phenomena. The dynamic system is studied by some experiments first. In order to explain the experimental results theoretically, the governing nonlinear boundary value problem is derived next. Subsequently, an appropriate variational formulation for application of a Rayleigh Ritz procedure is suggested. The resulting nonlinear ordinary differential equations approximately deseribe a part of the observed vibrational behaviour. Both the experiments and the calculations demonstrate that, for high speeds, nonlinear weight-excited vibrations about the circular reference configuration occur. On the other hand, for low velocitites, periodie and even chaos-like snap-through phenomena appear.
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