Effects of the Gear Tooth Modification on the Nonlinear Dynamics of Gear Transmission System

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

An original application of Genetic Algorithms (GAs) is developed in order to optimize spur gear pairs toward vibration and noise reduction. The approach takes into account the most important parameters of micro-geometric modifications, namely tip and root relief, therefore the parameter space is eight dimensional. The objective function of the GA depends on the static transmission error (STE) that is related to teeth flexibility. STE is estimated by means of a nonlinear finite element approach: either the amplitude of the STE fluctuation or its harmonic content are considered as objective functions. The effectiveness of the approach is checked on an actual test case: GAs are able to find the optimum after a reasonable number of steps; such optimum is obtained on static basis and gives a strong vibration reduction. The reliability test proves that GAs lead to robust optima.

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.

<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>

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, 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.

In this study, the interactions between backlash, bearing clearance and bearing flexibility are studied in geared flexible rotors. For this purpose, the non-linear mathematical model developed in a recent previous study for geared rotors on flexible bearings is extended further to handle clearance nonlinearity in bearings. The model consists of elastic shafts on elastic bearings with clearance, and coupled by a non-linear gear mesh interface. Shafts are modeled by using finite elements. The mathematical model includes the axial loading on shafts, rotary inertia, material damping of shafts, backlash nonlinearity of gear mesh and flexible bearings with clearances nonlinearity. The model considers also the effects of gear errors and profile modifications. The excitation effect of time varying mesh stiffness is indirectly included into the analysis through a periodic displacement representing loaded static transmission error (STE). STE input is modeled by the highest n harmonic terms of the Fourier series representation of the measured or calculated data. The model developed is believed to be one of the most advanced ones: It includes several linear and non-linear effects and capable of handling various different gear-rotor-bearing configurations. Through several case studies, the combined effects of backlash, bearing clearance, and bearing flexibility on dynamic mesh force and bearing loads are studied.

The static transmission error is one of the key incentives of gear dynamics and addressed by many scholars. However, the traditional calculation method of static transmission errors of spur gears does not take into account the influence of thermo-elastic coupling caused by the gear temperature field, and it limits the accuracy of the dynamic characteristic analysis. Thus, in this study, the calculation method of static transmission errors associated with thermo-elastic coupling is proposed. Furthermore, the differences between static transmission errors associated with thermo-elastic coupling contacts and traditional method of gear is discussed. The study is helpful to improve the accuracy of dynamic analysis of gear transmission system.

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.

It is difficult to obtain a characteristic, periodical signal of gears, that is, the transmission error, as many irregular signals due to lubrication, frictional losses and so on are relatively larger and involved. This report tries to obtain the transmission error by processing method with average, using a new developed machine which can directly and rapidly show an error spectrum and phases of the static transmission error.(1) The processing method with average is very useful to obtain the transmission error of gears during rotation.(2) The limit of rotational speed to measure the static transmission error of the above mentioned machine is about 300 rpm, and agrees with the theoretical limit. Therefore the limit can be estimated theoretically in general case.(3) The dynamic motion of gears, which have a known static transmission error, obtained by the experiment agrees with the theoretical motion. Therefore the measured values by the above mentioned machine are very useful to estimate any motion of gears.

The principal source of vibratory excitation of gear systems is the static transmission error which measures the deviation from uniform relative rates of (slow) rotation of pairs of meshing gears. The static transmission error can be decomposed into tooth bending, profile modification, and machining error components. The contributions of each of these components to the frequency spectrum of the static transmission error, and the effects of the mesh geometry parameters of helical gears on the spectrum, will be discussed.

Formulation of the equations of motion of a generic gear system in the frequency domain is shown to require the Fourier-series coefficients of the components of vibration excitation; these components are the static transmission errors of the individual pairs of meshing gears in the system. A general expression for the static transmission error is derived and decomposed into components attributable to elastic tooth deformations and to deviations of tooth faces from perfect involute surfaces with uniform lead and spacing. The component due to tooth-face deviations is further decomposed into appropriately defined mean and random components. The harmonic components of the static transmission error that occur at integral multiples of the tooth-meshing frequency are shown to be caused by tooth deformations and mean deviations of the tooth faces from perfect involute surfaces. Harmonic components that occur at the remaining multiples of gear-rotation frequencies are shown to be caused by the random components of the tooth-face deviations. Expressions for the Fourier-series coefficients of all components of the static transmission error are derived in terms of two-dimensional Fourier transforms of local tooth-pair stiffnesses and stiffness-weighted weighted deviations of tooth faces from perfect involute surfaces. Results are valid for arbitrary, specified tooth-face contact regions and include spur gears as the special case of helical gears with zero helix angle.

A loaded tooth contact analysis model was developed to assess transmission error by incorporating profile errors in a helical gear pair. Profile errors for all teeth of both the pinion and wheel were measured, and the mean profile error was computed and integrated into the model. To validate the model, static transmission error (STE) was measured experimentally. The experimental results, particularly the peak-to-peak static transmission error (PPSTE) in relation to torque and tooth-meshing harmonics, demonstrated consistency with the simulation outcomes. Subsequently, the model was employed to investigate the impact of surface waviness (SW) on the STE, focusing on SW amplitude, spatial frequency, and initial phase. It was observed that both the amplitude and spatial frequency of the SW influenced the PPSTE and the harmonic characteristics of the transmission error, while the initial phase of the waviness had no significant effect.

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.