An approximate analytical solution of the gear meshing problem based on MSM using DynPy Python library

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system.

This research aims to advance the understanding and application of dynamic models for gears within agricultural machinery drive trains by developing analytical solutions. Despite the significant advancements in vibration analysis, there is a notable scarcity of comprehensive research that addresses the analytical modeling of gear dynamics, particularly using advanced mathematical techniques such as the multiple scale method (MSM). A new approach to modeling gear meshing is introduced, where the Fourier expansion of a rectangular signal is utilised to simulate the time-varying mesh stiffness (TVMS). Such an approach allows the use of the MSM as an efficient tool to obtain solutions for parametrically induced vibrations. A dynamic model of a simple helical gear system is introduced in the form of Hill’s equation, and a sensitivity analysis is conducted for the main parameters of the system based on the solution obtained with the MSM. The results show the high credibility of the provided method when compared with a well-known state-of-the-art model and that the model reacts to parameter variations as expected. Additionally, an analysis of the energy harvesting possibilities is presented, which shows that, for the default parameter values, the harvester should be tuned to 5694.31 Hz to generate the maximum energy. It is concluded that the proposed model and MSM approach can serve as suitable tools for gear analysis, and the future paths for research are defined.

Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system

Effects of gear center distance variation on time varying mesh stiffness of a spur gear pair

The investigation of planetary gear dynamics including dynamic modeling and dynamic response analysis is a crucial approach in vibration reduction of industrial power transmission systems. In this paper, the nonlinear, time-varying dynamic model of a spur planetary gear system under consideration of the translational and rotational motions is investigated. The three subsystems for sun-planet, ring-planet and planet-carrier are analyzed subsequently, and the dynamic equations of the system are obtained. Moreover, different planet phasing and spacing configurations can be described by means of this model. In addition, the dynamic response is investigated by the multiple-scale method. First, the analytical solutions of the primary, superharmonic and subharmonic resonances are obtained. Then the frequency amplitude curves of different resonance modes are compared and the influence of some parameters on the vibration amplitude is studied. Meanwhile, the accuracy of the analytical solutions is evaluated by the numerical integration simulations. The results show that the frequency-amplitude curves of the primary and superharmonic resonance are similar in shape, the three solutions coexist, and the types of unstable solutions and stable solutions are identical. Furthermore, the softening nonlinearity of the subharmonic amplitude-frequency curve is more obvious under the three-dimensional model. This research is an important development of three-dimensional dynamic modeling and vibration prediction of planetary gears, and also improves the efficiency and accuracy of dynamic response calculation.

To obtain exact analytical solutions of differential equations of gear system dynamics due to the difficulty of solving complicated differential equations. Only the approximate analytical solutions can be determined. The method of multiple scales is one of the most powerful, popular perturbation methods. The dynamic model which describes the torsional vibration behaviors of gear system has been introduced accurately in this paper. The differential equation of gear system nonlinear dynamics exhibiting combined nonlinearity influence such as time-varying stiffness, tooth backlash and dynamic transmission error (DTE) has been proposed. The theory of multiple scales method has been presented in solving nonlinear differential equations of gear systems and the frequency response equation has been obtained. The fact that the approximate analytical solution by using the method of multiple scales is in good agreement with the exact solutions by numerically integrating differential equations has proved that the method of multiple scales is one of the most frequently used methods in solving differential equations, especially for large and complicated differential equations.

Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. At resonance, the resulting vibration causes tooth separation leading to nonlinear effects such as jump phenomena and subharmonic resonance. This work examines the nonlinear dynamics of planetary gears by numerical and analytical methods over the meaningful mesh frequency ranges. Concise, closed-form approximations for the dynamic response are obtained by perturbation analysis. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. Correlation between the amplitude of response and external torque demonstrates that tooth separation occurs even under large torque. The harmonic balance method with arclength continuation confirms the perturbation solutions. The accuracy of the analytical and harmonic balance solutions is evaluated by parallel finite element and numerical integration simulations.

A gear mesh dynamic model of spur gears is presented in this paper. Initially, the influence of variation in center distance due to geometric eccentricity, assembly error and bending deformation of shaft on gear mesh parameters such as pressure angle, backlash and time-varying mesh stiffness were determined. Then, the gear mesh dynamic model was formulated by replacing the constant backlash and time-varying mesh stiffness under quasi-static conditions of the previous model with continuously calculated time-varying backlash and time-varying mesh stiffness under dynamic conditions. A six-degree-of-freedom lateral-torsional coupled nonlinear dynamic model was established with the gear mesh dynamic model. A gear pair was analyzed to demonstrate and qualify the impact of shaft stiffness and operating conditions on the differences between the dynamic responses predicted by the nonlinear dynamic model established in this study and the previous nonlinear dynamic model. The results showed clear differences between the dynamic responses predicted by different models, especially for the gear rotor bearing system with high compliant shafts. A strong coupling effect between dynamic response and gear mesh parameters was found. The comparison between the gear mesh forces within a wide range of torque and speed also revealed differences, especially in the vicinity of the primary resonance speed and 1/3 of this speed. With increasing torque, the differences increased and the range of speeds corresponding to same difference also increased. The differences reached to a minimum near half of the resonance speed under different input torques.

Nonlinear dynamic response of a spur gear pair based on the modeling of periodic mesh stiffness and static transmission error

Time-varying mesh stiffness (TVMS) is a vital internal excitation source for the spiral bevel gear (SBG) transmission system. Spalling defect often causes decrease in gear mesh stiffness and changes the dynamic characteristics of the gear system, which further increases noise and vibration. This paper aims to calculate the TVMS and establish dynamic model of SBG with spalling defect. In this study, a novel analytical model based on slice method is proposed to calculate the TVMS of SBG considering spalling defect. Subsequently, the influence of spalling defect on the TVMS is studied through a numerical simulation, and the proposed analytical model is verified by a finite element model. Besides, an 8-DOF dynamic model is established for SBG transmission system. Incorporating the spalling defect into TVMS, the dynamic responses of spalled SBG are analysed. The numerical results indicate that spalling defect would cause periodic impact in time domain. Finally, an experiment is designed to verify the proposed dynamic model. The experimental results show that the spalling defect makes the response characterized by periodic impact with the rotating frequency of spalled pinion. Conflict of Interest Statement The authors declare no conflicts of interest.

Due to the complex working environment, gear systems always suffer from multiple excitations in actual engineering. This paper concerns the frequency response characteristics of a nonlinear time-varying spur gear system subjected to multi-frequency excitation. Firstly, a single degree-of-freedom gear pair model is established with consideration of the gear backlash, time-varying mesh stiffness and multiple harmonic excitations. Then, using the multiple time scales method, a comprehensive theoretical study is conducted to analyze various resonant cases including primary, parametric and combination resonances. Besides, parametric studies are accomplished to reveal the effects of the multi-frequency excitation on gear dynamics and to provide some useful references for reducing the vibration level. With the help of the fifth-order Runge–Kutta method, the numerical results are obtained to verify the validity of the analytical solutions and to emphasize the significances of the multi-frequency excitation. In addition, a comparison is performed between the numerical results and the published experimental results to validate the proposed gear model. Results show that the presence of the multi-frequency excitation will introduce the interaction between different harmonic excitations, which significantly affects the nonlinear vibration characteristics of a spur gear system. The proposed gear model with multi-frequency excitation could be more reliable and universal than that with single-frequency excitation. In addition, the results of parametric study could provide some suggestions to designers and researchers attempting to obtain desirable dynamic behaviors of a gear system subjected to multi-frequency excitation.

Coupled nonlinear dynamics of acoustic modes in a quasi 1-D duct with inhomogeneous mean properties and mean flow

The main purpose of this dissertation is to study coupled thermoelastic behaviors in disks subjected to thermal shock loads based on the generalized and classic theories of coupled thermoelasticity. To this end, this research has been carried out in two stages. In the first stage, thermoelasticity problems in an axisymmetric rotating disk with constant thickness made of a homogeneous isotropic material are analytically solved and closed-form formulations are presented for temperature and displacement fields. Since, the analytical solution is not always feasible, the finite element (FE) method can be employed for more sophisticated coupled thermoelasticity problems. Accordingly, in the second stage of the research, a novel refined 1D finite element approach with 3D-like accuracies are developed for theories of coupled thermoelasticity. Then, the developed FE models are applied for a 3D solution of the dynamic generalized coupled thermoelasticity problem in disks. Use of the reduced models with low computational costs may be of interest in a laborious time history analysis of the dynamic problems. The obtained analytical and numerical solutions are in good agreement with the results available in the literature. It is further shown that the proposed analytical and FE methods are quite efficient with very high rate of convergence.

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The solution is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As presented in previous papers, deriving such permeability weighting functions appears as a complicated calculation, available only for special cases. This paper presents a simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc), the associated permeability weighting function can be immediately derived. Among other examples, we calculate and discuss the well pressure solution for a horizontal well producing from a heterogeneous reservoir. Introduction The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media. Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems, involving some simple symmetry: layered reservoir, single linear discontinuities, radially composite systems etc. Rosa and Horner computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al. proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods. This paper is related to these first-order approximate solutions of the well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous works in the domain. After that, we present our method to calculate the weighting permeability functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well. The Permeability Weighting Function The perturbation method is a well known technique to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms. In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The permeability k0 may vary in the space, i.e k0 (xD) What is important is that the background problem has a known exact analytical solution, PDO (XD, tD). The full problem has the same boundary conditions of the background problem but the permeability k(XD) differs from ko(XD) in arbitrary regions of the space. Rigorously speaking, k(XD) / k0 (XD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts, say up to 10, between these permeabilities, depending on the specific problem. P. 457