Nonlinear vibration of the spiral bevel gear under periodic torque considering multiple elastic deformation evaluations due to different bearing supports

This paper investigates two parameters effect on vibrational responses of the spiral bevel gear. Changing the gear system overall stiffness (GSOS) considering elastic deformation and periodic torques are the two parameters which are represented as the main goals of this study. In order to investigate the effects of shaft stiffness and elastic deformation, two different cases with different support locations are considered. The first case is presented by locating the support close to the gear, and in the latter one, the distance between gear and support is increased. Besides, to study the effect of torque, two main types are considered: constant and periodic excitation torque. To illustrate the dynamic behavior, the governing differential equations are solved numerically according to the Runge-Kutta method. The equations are nonlinear due to backlash and time-varying coefficients as the results of GSOS variation. Vibrational phenomena are illustrated by means of bifurcation diagrams, RMS, and Poincare maps. Particular vibrational behaviors such as "chaos" and "period-doubling" phenomena are illustrated with details. By investigating the effect of shaft stiffness, results show that when the support is far away from gear, the vibration response increased by 67.5%. Moreover, while the input torque is constant, the support movement does not cause undesirable responses such as chaotic or period-doubling responses. The periodic torque causes undesirable responses such as chaos and bifurcation and period-doubling responses

Application of linear and nonlinear vibration absorbers for the nonlinear beam under moving load

Recently, a large amount of studies have been related to nonlinear systems with multi-degrees of freedom as well as continuous systems. The purpose of this paper is to optimize passive vibration absorbers in linear and nonlinear states for an Euler-Bernoulli beam with a nonlinear vibratory behavior under concentrated moving load. The goal parameter in the optimization is maximum deflection of the beam. The large deformation for beam modeling is considered, i.e. the relation between strains and deflections is nonlinear. The force magnitude and beam length are two effective factors for the beam deflection. Vibration absorber with linear damping and linear or nonlinear stiffness is also considered in this manuscript. The results show that, for normal forces and short beams, linear and nonlinear models have similar behaviors, while surveying nonlinear behavior is necessary by increasing the force and length of the beam, i.e. large deflections. Moreover, the difference between linear and nonlinear beam models for regular force magnitudes and beam lengths is negligible. For higher loads and longer beams, beam model nonlinearity can be important. Results demonstrate that,in the presented numerical values (train bridge application) for cubic nonlinear vibration absorber, there are two optimal locations for vibration absorber installation: one inclined from the middle of the beam to the direction of moving loads and the second which is more interestingly inclined from the middle of the beam to moving loads in the opposite direction. Moreover, depending on the model's numerical parameters, for short beams, linear vibration absorber is more effective, while for long beams, cubic nonlinear beam behaves better than the linear one

Nonlinear vibration of crowned gear pairs considering the effect of Hertzian contact stiffness

This study aims to analyze the influence of lead crowning modification of teeth on the vibration behavior of a spur gear pair. Two dynamic rotational models including an uncrowned and crowned gear are examined. Hertzian mesh stiffness is computed using tooth contact analysis in quasi-static state along a complete mesh cycle of teeth mesh. The dynamic orbits of the system are observed using some useful attractors which expand our understanding about the influence of crown modification on the vibration behavior of the gear pair. Nonlinear impact damper consists of non-integer compliance exponents identify energy dissipation of the system beneath the surface layer. By augmenting tooth crown modification, the surface penetration increases and consequently normal pressure of the contact area becomes noticeable. Finally, the results show modification prevents gear pair to experience period doubling bifurcation as the numerical results proved. Using this new method in dynamic analysis of contact, broaden the new horizon in analyzing of the surface of bodies in contact

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Nonlinear vibration of the bevel gear with teeth profile modification

The prediction of gear vibration and noise has always been a major concern in gear design. Noise and vibration are inevitable problems that are involved in transmission systems; they have intensified when some nonlinear phenomena such as jump phenomenon, tooth separation and period-doubling bifurcation appear in the system. Tip and/or root modifications are well-known solutions that improve dynamic performance of gears. The present work investigates the complex, nonlinear dynamic behavior of three bevel gear models: (1) model with pure involute profile, (2) model with statically optimized tooth profile, and (3) model with dynamically optimized tooth profile. Tooth profile modification is employed in models by means of genetic algorithm in order to extract the best amount and length of modifications. The dynamic responses obtained from dynamic analyzer were compared qualitatively and quantitatively. By augmenting tooth profile modification, the average value of the dynamic responses is decreased intensely for both statically and dynamically optimized gear pairs. Dynamic load factor is calculated and compared with the involute tooth profile model and the two optimized gear sets. Employing teeth optimization leads to elimination of period- (Formula presented.) in both optimized simulations

Spiral Bevel Gears Nonlinear Vibration Having Radial and Axial Misalignments Effects

In gear transmissions, vibration causes noise and malfunction. In actual applications, misalignments contribute to intensifying the destructive effect of vibrations. In this paper, the nonlinear dynamics of a spiral bevel gear pair, with small helix angle, considering different misalignments, are deeply investigated. Axial misalignment, radial misalignment, and the combination of these two types are considered in this study. The governing equation is numerically solved through an implicit Runge–Kutta scheme. Since the main goal of this study is the analysis of the dynamic scenario, the mesh stiffness of the gear pair is obtained from the literature. The dynamical system is nonlinear and time-varying; it is analyzed through time responses, phase portraits, Poincaré maps, and bifurcation diagrams. Results show that, among the considered three cases with different types of misalignments, the spiral bevel gear with axial misalignment is the worst destructive case; aperiodic, subharmonic, and multiperiod responses are observable for this case. It is interesting that the chaotic responses for the case, having both types of misalignments, are less likely for the case with axial misalignment, only

Nonlinear vibration of the spiral bevel gear with a novel tooth surface modification method

The issue of gear noise is fairly common in power transmission systems. This noise largely stems from the gear pairs vibration triggered by transmission error excitation. This is mainly caused by tooth profile errors, misalignment and tooth deflections. This research endeavors to examine nonlinear spiral bevel gears vibration with the innovative method of tooth surface modification. To design spiral bevel gears with the higher-order transmission error (HTE), the non- linear vibration of a novel method is investigated. The meshing quality of the HTE spiral bevel gears, as the results demonstrate, sounds more suitable than of the meshing quality gears. Their design was made by means of the parabolic transmission error (PTE). The maximum time response root mean square of the HTE method decreases by 44% concerning the PTE method. The peak-to-peak of the transmission error is decreased by 35% via HTE overall frequency range. However, HTE method is not able to decrease the vibration level on all frequency ratios