TORSION VIBRATION AND PARAMETRIC INSTABILITY ANALYSIS OF A SPUR GEAR SYSTEM WITH TIME-VARYING AND SQUARE NONLINEARITIES

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.

This paper is concerned with the identification problem of two degree of freedom robot arm’s joints’ time-varying stiffness under time-varying torque loads. The industrial robot arms can be simplified into series hinged linkage mechanism. The unrestraint dynamic equation of two degrees of freedom robot arm can be obtained by using analytical mechanics method. The model which was established in Pro/E had been imported into ADAMS for simulation. And the simulation consisted of free vibration and forced vibration. Then by choosing limited memory least square method and using the post-processing measured variables, time-varying stiffness could have been identified. Finally, the calculative stiffness was compared to the “real” stiffness which was simulated in ADAMS. The whole process shows that the robot arm’s dynamic model and the method of identification are both effective. DOI: http://dx.doi.org/10.11591/telkomnika.v10i8.1644 Full Text: PDF

The nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.

Weight reduction is a recurring concern in the design of modern mechanical systems. This search may lead engineers to resort to using gears with holes in order to meet their requirements. This paper presents a methodology to carry out nonlinear dynamic analyses of a gear transmission with holes in the gear blanks subjected to a multiharmonic internal excitation. This work investigates the influence of these holes on the vibration levels, occurrence of contact loss and possible bifurcations. The numerical model features two flexible shafts coupled by a spur gear pair with holes. The gear model consists in two lumped masses and inertias and includes the gear backlash as well as the internal excitation sources that are the time-varying stiffness and the static transmission error (STE). The resulting mechanical system is solved in the frequency domain by the Harmonic Balance Method (HBM) coupled with an arc-length continuation algorithm and its stability is evaluated with Hill’s method. Results show that adding holes not only impacts the STE but also the mesh stiffness. The interactions of these two quantities has a substantial influence on the bifurcation structure along the main solution branch and leads to a decrease of the span of vibro-impact regions. As the applied static torque is increased, it is found that using holed gear blanks can effectively prevent contact loss and lead to a linear response.

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

In this paper, a dynamical system, which consists of two linear mechanical oscillators, coupled with a nonlinear damping device is considered. First, the dynamic equations are derived, then, an analytical method such as harmonic balance method, is applied to obtain the response to a harmonic base excitation. The response of the system depends on the excitation characteristics. A parametric study is carried out based on different base excitation amplitudes, frequencies, and different nonlinear damping values and the response of the system is fully described. For validation, time domain simulations are carried out to obtain the nonlinear response of the coupled system.

Subharmonic resonance of a symmetric ball bearing–rotor system

APPROXIMATE METHOD FOR THE DETECTION OF CHAOTIC MOTION IN A TYPE OF DUFFING'S OSCILLATOR

One has been interested in periodic motions for along time. For instance, in 1788, Lagrange [1] developed the method of averaging for periodic motions of three-body problem. To continue Langrange’s work, in 1890, Poincare [2] developed the perturbation theory for periodic motions of celestial bodies. In 1920, van der Pol [3] applied the method of averaging for the periodic solutions of oscillation systems in circuits. In 1928, Fatou [4] gave the proof of the asymptotic validity through the solution existence theorems of differential equations. In 1935, Krylov and Bogoliubov [5] further developed the method of averaging for nonlinear oscillations in nonlinear vibration systems. Since then, researchers extensively used the perturbation method to investigate periodic solutions in nonlinear dynamical systems. In 1964, Hayashi [6] discussed perturbation methods, averaging method, and principle of harmonic balance. The stability of approximate periodic solutions in nonlinear dynamical systems is determined by the improved Mathieu equation. In 1973, Nayfeh [7] systematically presented the perturbation theory and multi-scale methods. In 1976, Holmes and Rand [8] used the method of averaging to discuss the stability and bifurcation of the Duffing oscillator through the catastrophe theory. The catastrophe theory was used to determine the singular characteristics of frequency-amplitude curves. In 1979, Nayfeh and Mook [9] used the multiple-scale perturbation methods for approximate solutions of periodic motions. The frequency-amplitude responses were discussed. In 1987, Rand and Armbruster [10] presented the perturbation method and bifurcation theory to determine the stability of periodic solutions. To determine frequency-amplitude characteristics of periodic motions in nonlinear dynamical systems, the harmonic balance method is often used as one of the most popular techniques. In 1997, Luo and Han [11] analytically studied the stability and bifurcations of periodic solutions of Duffing oscillators through the first-order harmonic balance method. To obtain accurate results of periodic solutions in nonlinear vibration, many harmonic terms are included in the harmonic balance method. In 2008, Peng et al. [12] presented the approximate period-1 solution for the Duffing oscillator by the HB3 method compared with the fourth-order Runge–Kutta method. Such results still cannot give a satisfactory solution.

Harmonic balancing method has more powerful superiority than other perturbation methods in the respect of solving steady-state response of strong nonlinear system. The amplitude-frequency equation of dry-friction vibration system considering cubic nonlinear displacement under harmonic excitation was derived by harmonic balancing method. Because the effect of even nonlinear term on vibration response of system was not evident, even nonlinear term of elastic restoring force was neglected. The first term coefficient and third term coefficient obtained by static experimental data for constitutive relationship of metallic rubber were substituted into frequency response equation. Amplitude-frequency characteristics curves of vibration response for cubic nonlinear displacement dry-friction system of metallic rubber were obtained. Natural frequencies obtained by computation were compared with natural frequencies by experiment, and the two Natural frequencies consisted with each other. It showed that the method that vibration response for cubic nonlinear displacement dry-friction system of metallic rubber was computed is reasonable.

The nonlinear equation for the rotational vibration of a pair of spur gears has a restriction that the analytical solution of the equation cannot be obtained. In this paper, the linear equation of vibration is derived theoretically and its physical model, i.e. the linear model of vibration is presented. The analytical solution of the linear equation, which is derived by analytical method, agrees well with the numerically calculated result by the nonlinear equation. By analyzing the analytical solution of the linear equation in detail, we clarified the relation between the waveforms of the vibration and the profile error of gear pairs, and also found that the effect of the contact ratio to the vibration is large and complex. The equivalent error, accounting for effects of the static load, the time-varying stiffness and the profile error of gear pairs, is proposed in this paper. It can be considered as promising for evaluating the profile error, because the vibration of gear pairs is excited mainly by the equivalent error. Finally, for confirming the above results, the vibration of two tested gear pairs has been measured by an experimental set-up for this purpose.

This study is aimed at investigating the effects of intermediate support stiffness on nonlinear vibration response of transmission system. A coupled bending–torsional–lateral dynamic model with 14 degrees of freedom of transmission system is proposed comprehensively considering rubber dynamic stiffness, time-varying stiffness, static transmission error, and backlash. Meanwhile, a model of intermediate support with rubber nonlinear properties is developed to improve the accuracy of analysis. On the basis of the nonlinear differential equations derived by the lumped-parameter method, the dynamic responses of the key parts in the transmission system are obtained using the Runge–Kutta method and the effects of intermediate support stiffness on the dynamic characteristics are analyzed. Numerical results show that complicated interaction occurs and intermediate support stiffness has a great influence on the vibration of the coupled system. With the increase in the intermediate support stiffness, the vibration displacements in the three directions of the gear are progressively reduced and the nature frequencies are changed. The influence of the radial stiffness is greater than that of the axial stiffness. An experiment is performed subsequently to validate the effect of intermediate support stiffness on system, and the experimental results are consistent with numerical ones.

Piecewise-type nonlinearities, such as clutch dampers in a torsional system, induce complex nonlinear dynamic behaviors that resemble super- and sub-harmonic responses. This study focuses on investigating the sub-harmonic responses induced by piecewise-type nonlinearities in the middle of various dynamic behaviors in a torsional vibratory system. To examine the dynamic characteristics in a sub-harmonic regime, the harmonic balance method (HBM) was implemented. Its results were compared with the numerical simulation (NS). To reveal the sub-harmonic responses, the input conditions of the HBM were modified with a small number of input values. In addition, bifurcation diagrams were numerically determined and projected onto stable and unstable solutions of the HBM to examine the effective dynamic behaviors within the unstable regimes. The results of the HBM with the modified input conditions reveal the sub-harmonic effects well, and the comparisons of bifurcation diagrams under unstable conditions lead to an understanding of the complex dynamic behaviors. Overall, this study suggests the first analytical technique to determine the sub-harmonic responses with the HBM, and second investigates the complex dynamic behaviors in a practical vibratory system by considering the bifurcations in the unstable regimes.

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.

Analysis of periodically excited non-linear systems by a parametric continuation technique