Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train

In this paper, nonlinear dynamic model of spur gear pairs with fractional-order damping under the condition of time-varying stiffness, backlash and static transmission error is established. The general formula of fractional-order damping term is derived by using the incremental harmonic balance method (IHBM), and the approximate analytical solution of the system is obtained by use of the iterative formula. The correctness of the results is verified by comparing with the numerical solutions in the existing literature. The effects of mesh stiffness, internal excitation amplitude and fractional order on the dynamic behavior of the system are analyzed. The results show that changing the fractional order can effectively control the resonance position and amplitude in the meshing process. Both the mesh stiffness and internal excitation can control the collision state and the stability.

<div class="section abstract"><div class="htmlview paragraph">Motivated by accurate representations in gear dynamics models, this work analyzes the force-deflection relationship between spur gear pairs when the gear teeth have tooth root cracks. A finite element/contact mechanics approach is used to accurately capture the elastic deformations of the gear mesh incorporating kinematic gear motion; elastic deflections of the teeth, root, and blank; and elastic contact between the mating gear teeth. Tooth root crack damage of fixed sizes are analyzed, and the resulting static transmission error and mesh stiffness are calculated. These FE/CM model outputs are relatively insensitive to important gear crack geometry, including the initial crack location, the path it follows, and its final location. Crack-induced changes in static transmission error and mesh stiffness are driven by the remaining amount of the tooth that is healthy. Calculations of average-slope and local-slope mesh stiffness are included because both are used in gear dynamic models.</div></div>

In order to accurately describe the complex transmission systems and clarify the effects of internal and external excitation on the vibration characteristics for large complex transmission systems, a dynamics model considering gear meshing error and time-varying meshing stiffness is established at two-stage planetary gears systems. The coupling model is established by matrix set method for two-stage planetary gears and multistage spur gears transmission systems. The random load excitation verified by experiments is the external excitation in the system, and the excitation caused by the time-varying meshing stiffness and meshing error is the internal excitation. The research on the response of the vibration systems shows that the internal excitation caused by gear meshing stiffness and gear meshing error has important influence on the dynamic load behavior of spur gears transmission in complex systems. For the planetary system with heavy external load, the dynamic behavior mainly depends on the external load, and the influence of internal excitation on the dynamic meshing force is less than that of the external excitation for the planetary system. The external excitation mainly affects the low-frequency region of the dynamic meshing force for the system, and the internal excitation affects the high-frequency region of the dynamic force for the system. The frequency influence interval of the internal excitation is much larger than that of the external excitation.

This paper investigates the dynamics of a spur gear system with time-varying and square nonlinearities, by both analytical method and numerical simulation. First, the equations of motion of a 2 degree-of-freedom system are established and the harmonic balance method is used to analyze the stability and the steady-state response of the system under the main resonant condition. Then the perturbation method is used to analyze the parameter instability under the main, subharmonic and nonresonance conditions. Finally, the interactions between the main and subharmonic resonant amplitudes and the nonlinearity and the contact ratio are analyzed. The results reveal that the system response contains various frequency components, such as meshing frequency and its higher harmonic terms due to the nonlinearity and the time-varying stiffness. The existence of the time-varying meshing stiffness can also result in the subharmonic resonance, and even chaos through period-doubling bifurcations as the input torque increases.

The effects of tooth modification on the nonlinear dynamic behaviors are studied in this paper. Firstly, the static transmission error under load combined with misalignment error and modification are deduced. These effects can be introduced directly in the meshing stiffness and static transmission error models. Then the effect of two different type of tooth modification combined with misalignment error on the dynamic responses are investigated by using numerical simulation method. The numerical results show that the misalignment error has a significant effect on the static transmission error. The tooth crowning modification is generally preferred for absorbing the misalignment error by comparing with the tip and root relief. The tip and root relief can not resolve the vibration problem induced by misalignment error but the crowning modification can reduce the vibration significantly.

Nonlinear dynamic modeling and analysis of spur gears considering dynamic contact state under misalignment errors

Abstract. A noncircular face gear (NFG) conjugated with a pinion is a new type of face gear which can transmit variable velocity ratio and in which two time-varying excitations exist, namely the meshing stiffness excitation and instantaneous center excitation. Considering the tooth backlash, static transmission error and multifrequency parametric excitation, a nonlinear dynamic model of the NFG pair is presented. Based on the harmonic balance method and discrete Fourier transformation, a semi-analytic approach for the nonlinear dynamic model is given to analyze the dynamic behaviors of the NFG. Results demonstrate that, with increase in the eccentric ratio, input velocity and error amplitude, the NFG will undergo a non-rattle, unilateral rattle and bilateral rattle state in succession, and a jump phenomenon will appear in the dynamic responses when the rattle state of the gears is transformed from unilateral rattle to bilateral rattle.

Analytical modeling of spur gear corner contact effects

Spur gears for many applications make use of profile shapes that are modifications of the mathematically defined involute to avoid premature tooth contact and unfavourable transmission error. It is well known that specification of the involute modification is important as it relates to the resulting performance, vibration and gear noise. In this research, a hybrid analysis method using finite elements and contact mechanics was used to investigate the effects of linear, parabolic and exponential tooth profile modifications on performance of an involute spur gear pair. For each type of modification, 10 different magnitudes of tip relief were used to study its influence on static transmission error (STE), tooth loads, root stresses and contact stresses. Peak-to-peak transmission error, root mean square transmission error and total area per cycle transmission error were used to characterise STE. Multiple output responses were calculated including the STE characteristics, tooth loads, tooth root stresses, contact stresses, gear moments and bearing forces to compare and contrast the type and magnitude of the tooth modification. Results show that linear modification produced low root stresses; parabolic modification produced low STE, contact stress, gear moment variations, and bearing force variations; exponential modification produced nearly the same STE characteristics over a wide range of tip modification magnitudes.

The measurement of the meshing stiffness in gear pairs is a technological problem. Many studies have been conducted, but a few results are available. A tailored test bench was designed and realized to measure the Static Transmission Error in two mating gears to address this issue. The bench is capable of testing several kinds of gears, e.g., spur, helical, conical, and internal, and it measures the transmission error concerning the applied torque. The Static Transmission Error is due to the variable stiffness of the gear teeth during a mesh cycle. In this paper, a dynamical method for measuring gear mesh stiffness is presented. The tooth stiffness is estimated from the torsional modal behavior of the rotating parts of the test bench. The dynamics of the system are acquired using accelerometers and very precise encoders to measure the angular accelerations and displacements of rotating parts. The torsional mode shapes are identified; those that show a vibrational behavior of the gears that do not follow the transmission ratio’s sign of the mating kinematic condition are selected because they depend on the flexibility of the teeth. In such a way, the engagement stiffness is estimated from the natural frequencies of the selected mode-shapes and the known inertia of gears and shafts. The experimentally identified results are also compared with numerical values computed with a commercial software for mutual validation.

In this paper, the harmonic balance method has been extended to investigate the nonlinear dynamics of a compound planetary gear sets. The lumped parameter dynamic model, incorporating parametric gear mesh stiffness fluctuations, transmission errors and gear backlash, is established. The responses of the system, mesh stiffness, transmission error, and gear backlash have all been represented by harmonic functions, which have been utilized to derive the algebraic equations via Galerkin process. The nonlinear dynamic characteristics of the gear sets are researched, and the effect of nonlinearities on the frequency response characteristic has been investigated by changing the value of the non-dimensional backlash, mesh stiffness, and amplitude of error excitations. The multiple value, jump and discontinuities phenomena have been revealed in the presented frequency response curves. The formulation derived here can also be extended to incorporate incremental harmonic balance method and parameter continuation scheme for further nonlinear dynamics investigation.

This study analyzes the nonlinear static and dynamic response in spur gear pairs with tooth root crack damage. A finite element/contact mechanics (FE/CM) model is used that accurately captures the elastic deformations on the gear teeth due to kinematic motion, tooth and rim deformations, vibration, and localized increases in compliance due to a tooth root crack. The damage is modeled by releasing the connectivity of the finite element mesh at select nodes near a tooth crack. The sensitivity of the calculated static transmission errors and tooth mesh stiffnesses is determined for varying crack initial locations, final locations, and the path from the initial to final location. Gear tooth mesh stiffness is calculated for a wide range of tooth root crack lengths, including large cracks that extend through nearly all of the tooth. Mesh stiffnesses are meaningfully reduced due to tooth root crack damage. The dynamic response is calculated for cracks of varying length. Larger cracks result in increased peak dynamic transmission errors. For small tooth root cracks the spectrum of dynamic transmission error contains components near the natural frequency of the gear pair. The spectrum of dynamic transmission error has broadband frequency response for large tooth root cracks that extend further than one-half of the tooth’s thickness.

The results of modeling nonlinear parametric vibrations in aircraft tooth gears are presented in the paper. Resonance vibration with loss of tooth contact is analyzed. A lumped-parameter dynamic model of cylindrical gears is investigated using the results of gear stiffness modeling obtained from a finite element model for simulating elastic-damping connection for each step during one phase of mesh. Parametric vibrations of a system with base excitation due to variable mesh stiffness and non-linear effects determined by modeling the conditions of loss of tooth contact are studied. It is shown that the maximal level of dynamic loads does not exceed a limit value determined by the presence of discontinuous vibrations in the system. A number of parametric studies are presented to demonstrate the influence of major gear contact ratio on the resultant dynamic behavior. The results obtained by the numerical simulation of steady-state responses of two different pairs of spur gears are in satisfactory agreement with the experimental results.

A new effective mesh stiffness calculation method with accurate contact deformation model for spur and helical gear pairs

In the present paper, two different methodologies for simulate the dynamic behavior of gears are described and compared. The nonlinear torsion vibrations of geared systems are studied at resonance frequency with the contact loss of the meshing teeth. In the proposed model a time-varying mesh stiffness of tooth pairs and a viscous damping proportional to the meshing stiffness are considered. Gear errors of each meshing tooth pair are also included. Dynamic motion of the spur gear pair over wide range of excitation frequency using the finite element method was simulated. Results of bending stress distribution during overall operation speeds were obtained by FEA. It may serve as a tool for aiding the gear fault diagnosis.