Donald R. Houser

Mathematical models used in gear dynamics—A review

With increased demand for high speed machinery, the mathematical modelling of the dynamic analysis of gears has gained importance. Numerous mathematical models have been developed for different purposes in the past three decades. In this paper the mathematical models used in gear dynamics are discussed and a general classification of these models is made. First, the basic characteristics of each class of dynamic models along with the objectives and different parameters considered in modeling are discussed. Then, the early history of the research made on gear dynamics is summarized and a comprehensive survey of the studies involved in mathematical modelling of gears for dynamic analysis is made. Generally, a chronological order is followed in each class studied. The goal is not just to refer to several papers published in this field, but also to give brief information about the models and, sometimes, about the approximations and assumptions made. A considerable number of publications were reviewed and 188 of them are included in the survey.

A ROBUST OPTIMIZATION PROCEDURE WITH VARIATIONS ON DESIGN VARIABLES AND CONSTRAINTS

This paper describes a procedure that incorporates manufacturing and operational variances to achieve designs with robust and optimal performance. The procedure optimizes the expected value of a performance characteristic subject to a set of constraints. It uses concepts from statistical design of experiments to approximate the expected value of a performance characteristic. The procedure incorporates uncertainties in design variables and variations in constraints due to the uncertainty in design variables. This paper discusses the following three methods to incorporate variations in constraints: 1) A method using heuristics that evalutes constraints at the worst combinations of design variables, 2) A method with built-in constraint variation that models constraints using first order Taylor expansion, and 3) A method based on differentiating the K KT optimality conditions. The design of spur and helical gears with minimum transmission error serves as the target application. The key gear design research is...

Dynamic analysis of high speed gears by using loaded static transmission error

Abstract A single degree of freedom non-linear model is used for the dynamic analysis of a gear pair. Two methods are suggested and a computer program is developed for calculating the dynamic mesh and tooth forces, dynamic factors based on stresses, and dynamic transmission error from measured or calculated loaded static transmission errors. The analysis includes the effects of variable mesh stiffness and mesh damping, gear errors (pitch, profile and runout errors), profile modifications and backlash. The accuracy of the method, which includes the time variation of both mesh stiffness and damping is demonstrated with numerical examples. In the second method, which is an approximate one, the time average of the mesh stiffness is used. However, the formulation used in the approximate analysis allows for the inclusion of the excitation effect of the variable mesh stiffness. It is concluded from the comparison of the results of the two methods that the displacement excitation resulting from a variable mesh stiffness is more important than the change in system natural frequency resulting from the mesh stiffness variation. Although the theory presented is general and applicable to spur, helical and spiral bevel gears, the computer program prepared is for only spur gears.

Optimum Profile Modifications for the Minimization of Static Transmission Errors of Spur Gears

A procedure for computing static transmission errors and tooth load sharing was developed for low and high contact ratio internal and external spur gears. A suitable optimization algorithm was used to minimize any combination of the harmonics of gear mesh frequency components of the static transmission error. Different combinations of tip and root relief may be used to achieve optimization. These include varying the starting point of relief and varying the magnitude of relief, and selecting the gear and/or the pinion teeth to be tip and/or root-relieved. Also, there exists an option for using either linear or parabolic relief. In addition to the presentation of optimal profile modifications, the effects of off-design loads, nonoptimum modifications, and random spacing errors are presented.

Dynamic Analysis of Geared Rotors by Finite Elements

Abstract : A finite-element model of a geared rotor system on flexible bearings has been developed. The model includes the rotary inertia of shaft elements, the axial loading on shafts, flexibility and damping of bearings, material damping of shafts and the stiffness and the damping of gear mesh. The coupling between the torsional and transverse vibrations of gears were considered in the model. A constant mesh stiffness was assumed. The analysis procedure can be used for forced vibration analysis of geared rotors by calculating the critical speeds and determining the response of any point on the shaft to mass unbalances, geometric eccentricities of gears and displacement transmission error excitation at the mesh point. The dynamic mesh forces due to these excitation can also be calculated. The model has been applied to several systems for the demonstration of its accuracy and for studying the effect of bearing compliances on system dynamics.

A Procedure Using Manufacturing Variance to Design Gears With Minimum Transmission Error

This paper deals with the design of spur gears that have minimum transmission error and are insensitive to manufacturing variance. We address two stages of design: (1) generation of candidate designs (selection of number of teeth, pressure angle, etc.), and (2) tooth profile modification. The first stage involves a search of discrete combinations of design variables, while the second stage utilizes numerical optimization techniques. The key research issue is finding a candidate design and its profile modification that not only has low transmission error, but is insensitive to variations in the design values caused by the manufacturing process. To achieve this goal, the procedure applies Taguchi’s concept of parameter design. In this paper, we consider a design problem with a set specification: fixed center distance, speed ratio, and transmission torque. We seek to find a limited number of candidate designs by applying conventional design generation techniques and some design heuristics. For each candidate design, the procedure determines the optimum profile modification (linear tip relief) by linking the Load Distribution Program (LDP) for gears with an optimization program package (OPTPAK). From the resulting peak optimum, we further seek the statistical optimum using an algorithm developed in this paper. The statistical optimum shows a nominal increase in the transmission error, but is quite insensitive to typical process error associated with gear manufacturing. The developed algorithm readily applies to other gear designs as well as other types of machine elements. In particular, we foresee our procedure to be particularly effective for helical gears. We hope to further our method by developing a means to add statistical heuristics to the discrete design generation stage.

Detecting Gear Tooth Breakage Using Acoustic Emission: a Feasibility and Sensor Placement Study

Vibration and debris monitoring methods are being increasingly used to detect gear tooth breakage. In this paper an alternate method of detecting gear tooth cracking is investigated. It is based on the phenomenon of acoustic emission (AE). The detectability of growing cracks using AE is established. Before this method can be used to detect crack growth in real systems, the transmissibility of these waves has to be studied. These waves have to propagate across a number of mechanical interfaces as they travel from the source to the sensor. The loss in strength of these waves at various interfaces commonly encountered in mechanical systems is studied in this paper.

Linearization of multibody frictional contact problems

Abstract A Simplex type algorithm is used to impose frictional contact conditions on finite element models of bodies that move close to trajectories that can be determined from kinematic constraints on the bodies. The method is demonstrated by computing the load dependent static transmission error and load sharing of a pair of gears in mesh.

DESIGN OPTIMIZATION FOR ROBUSTNESS USING PERFORMANCE SIMULATION PROGRAMS

Abstract This paper presents a methodology to incorporate manufacturing and operational variances in the design optimization stage to achieve robust and optimal performance. The procedure uses Taguchis orthogonal arrays to approximate the expected value of performance during optimization. This approach reduces the number of function evaluations in problems that use computationally expensive performance simulation programs. The method allows incorporation of variances on many variables simultaneously. This paper uses two illustrative examples: (1) Design of helical gears that have minimum transmission error and at the same time are less sensitive to manufacturing errors, (2) Design of beverage cans where we minimize the effects of errors in tooling on can weight and structural requirements. The optimal robust design shows a considerable decrease in sensitivity to manufacluring and operational variances and, at the same time, has good performance.

A rayleigh-ritz approach to modeling bending and shear deflections of gear teeth

Abstract The Rayleigh-Ritz energy method was used to study the shear effect of an involute gear tooth. The gear tooth was simulated by a tapered plate model subjected to a concentrated load. The plate deflections, including shear deformation, were determined and compared with the theoretical values and experimental data. The comparisons indicate that the deflections of the shear plate model are higher than those computed from the thin plate models which neglects the shear effects. On the other hand, the experimental results are higher than those of the shear plate model due to the base flexibilities of the experimental models. The shear model deflections are also shown to be in excellent agreement with finite element results. The shear plate model could replace the finite element model since it is more computationally efficient and its results are accurate enough for most engineering purposes.