On the dynamics of rings rotating with variable spin speed

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.     

Dynamics of Slider-Crank Mechanisms with Flexible Supports and Non-Ideal Forcing

Transient and steady state dynamic response of a class of slider-crank mechanisms is investigated. Specifically, the class of mechanisms examined involves rigid members but compliant supporting bearings. Moreover, the mechanisms are subjected to non-ideal forcing. Namely, both the driving and the resisting loads are expressed as a function of the angular coordinate describing the crank rotation. First, an appropriate set of equations of motion is derived by applying Lagrange's equations. These equations are strongly nonlinear due to the large rigid body rotation of the crank and the connecting rod, as well as due to the nonlinearities associated with the bearing action and the form of the driving and the resisting loads. Consequently, the dynamics of the resulting dynamical system is examined by solving the equations of motion numerically. More specifically, transient response is captured by direct integration, while determination of complete branches of steady state response is achieved by applying appropriate numerical methodologies. Initially, mechanisms whose crankshaft is supported by bearings with rolling elements and linear stiffness characteristics are examined. Then, numerical results are presented for rolling element bearings with nonlinear stiffness characteristics. Finally, the study is focused on mechanisms supported by hydrodynamic bearings. In all cases, the attention is focused on investigating the influence of the system parameters on its dynamics. Moreover, models with constant crank angular velocity are first analysed, since they provide valuable insight into some aspects of the system dynamics. Eventually, the emphasis is shifted to the general case of non-ideal forcing, originating from the dependence of the driving and the resisting moments on the crankshaft motion.     

On periodic steady state response and stability of Filippov-type mechanical models

In the first part of this study, the basic steps of a methodology are presented, leading to a long time response of a class of periodically excited mechanical models with contact and dry friction. In particular, the models examined belong to the special class of Filippov-type dynamical systems, which possess continuous displacements and velocities, but exhibit discontinuities in their accelerations. The direct determination of periodic steady state response of this class of models is achieved by combining suitable numerical integration of the equations of motion with an appropriate technique yielding the corresponding monodromy matrix. This matrix, which arises from a linearization of the motion around a located periodic solution, involves saltations (jumps) and is also useful in predicting its stability properties. The analytical part is complemented by a suitable continuation procedure, enabling evaluation of complete branches of periodic motions. In the second part of the study, the effectiveness of the methodology developed is confirmed by presenting representative sets of numerical results obtained for selected examples. The first two of them are single degree of freedom oscillators. Besides investigating some interesting aspects of regular periodic response, some cases involving rich dynamics of the class of the system examined are also studied in a systematic way. The last example is a more involved and challenging model, related to the function of an engine valve and characterized by large numerical stiffness.     

Dynamics of Oscillators with Strongly Nonlinear Asymmetric Damping

Dynamics of a class of strongly nonlinear single degree of freedom oscillators is investigated. Their common characteristic is that they possess piecewise linear damping properties, which can be expressed in a general asymmetric form. More specifically, the damping coefficient and a constant parameter appearing in the equation of motion are functions of the velocity direction. This class of oscillators is quite general and includes other important categories of mechanical systems as special cases, like systems with Coulomb friction. First, an analysis is presented for locating directly exact periodic responses of these oscillators to harmonic excitation. Due to the presence of dry friction, these responses may involve intervals where the oscillator is stuck temporarily. Then, an appropriate stability analysis is also presented together with some quite general bifurcation results. In the second part of the work, this analysis is applied to several example systems with piecewise linear damping, in order to reveal the most important aspects of their dynamics. Initially, systems with symmetric characteristics are examined, for which the periodic response is found to be symmetric or asymmetric. Then, dynamical systems with asymmetric damping characteristics are also examined. In all cases, emphasis is placed on investigating the low forcing frequency ranges, where interesting dynamics is noticed. The analytical predictions are complemented with results obtained by proper integration of the equation of motion, which among other responses reveal the existence of quasiperiodic, chaotic and unbounded motions.     

Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances

An analysis is presented for a class of two degree of freedom weakly nonlinear oscillators, with symmetric restoring force. Conditions of one-to-three internal resonance and subharmonic external resonance of the lower vibration mode are assumed to be satisfied simultaneously. As a consequence, the second vibration mode may also be under the action of external primary resonance. Initially, a set of slow-flow equations is derived, governing the amplitudes and phases of approximate long time response of these oscillators, by applying an asymptotic analytical method. Determination of several possible types of steady-state motions is then reduced to solution of sets of algebraic equations. For all these solution types, appropriate stability analysis is also performed. In the second part of the study, this analysis is applied to an example mechanical system. First, a systematic search is performed, revealing effects of system parameters on the existence and stability properties of periodic motions. Frequency-response diagrams are presented and attention is focused on understanding the evolution and interaction of the various solution branches as the external forcing and nonlinearity parameters are varied. Finally, numerical integration of the equations of motion demonstrates that the system exhibits quasiperiodic or chaotic response for some parameter combinations.     

Free Vibration in a Class of Self-Excited Oscillators with 1:3 Internal Resonance

Free vibration of a two degree of freedom weakly nonlinear oscillator is investigated. The type of nonlinearity considered is symmetric, it involves displacement as well as velocity terms and gives rise to self-excited oscillations in many engineering applications. After presenting the equations of motion in a general form, a perturbation methodology is applied for the case of 1:3 internal resonance. This yields a set of four slow-flow nonlinear equations, governing the amplitudes and phases of approximate motions of the system. It is then shown that these equations possess three distinct types of solutions, corresponding to trivial, single-mode and mixed-mode response of the system. The stability analysis of all these solutions is also performed. Next, numerical results are presented by applying this analysis to a specific practical example. Response diagrams are obtained for various combinations of the system parameters, in an effort to provide a complete picture of the dynamics and understand the transition from conditions of 1:3 internal resonance to non-resonant response. Emphasis is placed on identifying the effect of the linear damping, the frequency detuning and the stiffness nonlinearity parameters. Finally, the predictions of the approximate analysis are confirmed and extended further by direct integration of the averaged equations. This reveals the existence of other regular and irregular motions and illustrates the transition from phase-locked to drift response, which takes place through a Hopf bifurcation and a homoclinic explosion of the averaged equations.     

Self-Excited Oscillators with Asymmetric Nonlinearities and One-to-Two Internal Resonance

An analysis is presented on the dynamics of asymmetric self-excited oscillators with one-to-two internal resonance. The essential behavior of these oscillators is described by a two degree of freedom system, with equations of motion involving quadratic nonlinearities. In addition, the oscillators are under the action of constant external loads. When the nonlinearities are weak, the application of an appropriate perturbation approach leads to a set of slow-flow equations, governing the amplitudes and phases of approximate motions of the system. These equations are shown to possess two different solution types, generically, corresponding to static or periodic steady-state responses of the class of oscillators examined. After complementing the analytical part of the work with a method of determining the stability properties of these responses, numerical results are presented for an example mechanical system. Firstly, a series of characteristic response diagrams is obtained, illustrating the effect of the technical parameters on the steady-state response. Then results determined by the application of direct numerical integration techniques are presented. These results demonstrate the existence of other types of self-excited responses, including periodically-modulated, chaotic, and unbounded motions.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.