Nonlinear Dynamics of an Oilless Reciprocating Compressor

Two modeling methodologies of the dynamics of a motor-compressor system are presented. The first approach considered only the mechanical system subjected to a sinusoidal input force with the pressure term in the equation of motion treated as a nonlinear stiffness term. The second methodology consisted of a mathematical model that couples the electromagnetic and thermodynamic equations to the dynamic equations that describe the motion of the piston. The mathematical model which consisted of a set of four first order simultaneous nonlinear time varying differential equations, was solved by numerical integration routines that use the Adams-Moulton method with an adaptive integration step. The two methodologies are illustrated through an example. Steady-state operation was shown to be reached rapidly after a 0.13s transient. An analysis at various amplitudes and frequencies of the input voltage in the driver-coil of the motor, showed the amplitude dependence of the resonant frequency of the mechanical system, and a heavily damped system when operating at the design amplitude. The most efficient frequency of operation was also determined for a variety of required mass flow rates.

Second Order Averaging Study of an Autoparametric System

The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

Theoretical and Experimental Study of the Nonlinearly Coupled Heave, Pitch, and Roll Motions of a Ship in Longitudinal Waves

The loss of dynamic stability and the resulting large-amplitude roll of a vessel in a head or following sea were studied theoretically and experimentally. A ship model with three degrees of freedom (roll, pitch, heave) was considered. The governing equations for the heave and pitch modes were linearized and their harmonic solutions were coupled with the nonlinear equation governing roll. The resulting equation, which has time-varying coefficients, was used to predict the response in roll. The principal parametric resonance was considered in which the excitation frequency is twice the natural frequency in roll. Force-response curves were obtained. The existence of jump phenomena and multiple stable solutions for the case of subcritical instability was observed in the experiments and found to be in good qualitative agreement with the results predicted by the theory. The experiments also revealed that the large-amplitude roll is dependent on the location of the model in the standing waves.

The Nonstationary Effects in Asymmetric, Angle-Ply Laminated Composite Plates

The system in the title has been subjected to a parametric, nonstationary (NS) linear v(t) = v0 + βt, and cyclic v(t) = v0 ± γ sin βt excitations. The NS linear responses settle on the initial, constant values for extended values of the excitation frequencies, thus they stabilize the stationary (ST) response. This is true for the initial conditions taken on the stationary curve, and for different ply-angles. For the initial conditions (I.C.) beyond the ST plots, the NS responses stay also near the initial conditions, but they have the wavy forms, which increase slightly for the lower values of the forcing frequencies. For the cyclic parametric excitations, the NS responses are cyclic contained within the ranges of the excitation amplitudes ±β and (finite) response amplitudes above the ST initial values, raising up and down within these limits. It appears that they cover the whole area within the above described constraints. The decisive effect of the cyclic NS inputs, i.e., almost instantaneous cyclic responses replacing the ST responses regardless of the I.C. and ST responses, is a bench mark of the cyclic NS. This behavior is distinctly different from the NS cyclic responses of the composite columns.

A Successive Merging and Condensation (SMAC) Method Applied to Transient Analysis of Multi-Shaft Rotor System

A method for performing transient dynamic analysis of multi-shaft rotor system is proposed. The proposed methodology uses the reported Successive Merge and Condensation (SMAC) method [12] and a decoupling technique to decouple the shafts. Multi-shaft rotor systems are treated as systems of many independent single shaft rotor systems with external unknown coupling forces acting at the points of couplings. For each time step, first, the SMAC method is used to get the transient response in terms of the unknown coupling forces. This is followed by the application of the coupling constraints to calculate the coupling forces and, in turn, the response at the end of that time step. The proposed method preserves the efficiency advantages of the SMAC algorithm for single-shaft rotor system. Numerical examples to validate and illustrate the applicability of the method are given. The method is shown to be applicable to linear and non-linear coupling problems.

Splitting of Separatrices in the Rapidly Forced Duffing Oscillator

The splitting of the stable and unstable manifolds of the rapidly forced Duffing oscillator with negative stiffness is investigated. The method used relies on the computation of analytic approximations for the orbits on the perturbed manifolds, and the asymptotic approximation of these orbits by successive integrations by parts. It is shown, that the splitting of the manifolds becomes exponentially small as the perturbation parameter tends to zero, and that the estimate for the splitting distance given by the Melnikov Integral dominates over high order corrections.

Dynamics Response of a Hysteretic Structure Under Random Excitations

The response of a hysteretic structure under horizontal and vertical random excitations is considered. The excitations are modeled by segments of stationary and nonstationary Gaussian white noise and filtered white noise processes. The linearization technique is used and the moments equations of the responses are evaluated. The transition probability density of the response is described and the associated second moment equations are derived. The transient and nonstationary response statistics for a range of values of parameters are obtained. A Monte-Carlo digital simulation study is performed. The results are compared with the theoretical findings and good agreements is observed. Particular attention is given to the amplification effects of the vertical acceleration. It is shown that the effect of the vertical excitation is usually insignificant, unless the load coefficient is quite large.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

The Use of a Simplified Model for the Nonlinear Dynamic Analysis of Fixed Offshore Structures

A simplified model is used in the nonlinear dynamic analysis of fixed offshore platforms. The characteristics of the model are presented and its adequacy for the study is discussed. The action of ocean waves on the model is obtained using typical waves of the Brazilian coast. The nonlinear equation of motion is obtained in its exact form and is expanded up to the cubic term. A comparison between the nonlinear analysis and the linear dynamic analysis is presented. A comparison between experimental results and those obtained with the model is also presented.

Frequency Response Characteristics of a Revolute Impact Pair

Clearances in mechanical joints have deteriorating effects on the dynamic behavior of a machine in increasing noise and vibration and reducing the performance. In order to properly characterize these effects and to develop analytical techniques for machine design, it is necessary to investigate the dynamics associated with basic models of impacting systems. In this paper, we develop a method of harmonic balance to study a revolute impact pair. We focus on the characteristics of nonlinear frequency response of the system for a single frequency excitation. These characteristics include multiply-valued steady state response, multiple jump resonances, and existence and stability of these solutions. The effectiveness of the harmonic balance method combined with the Fast Fourier Transform technique is shown through numerical examples.