A. Kahraman

Non-linear dynamics of a spur gear pair

Abstract Non-linear frequency response characteristics of a spur gear pair with backlash are examined in this paper for both external and internal excitations. The internal excitation is of importance from the high frequency noise and vibration control viewpoint and it represents the overall kinematic or static transmission error. Such problems may be significantly different from the rattle problems associated with external, low frequency torque excitation. Two solution methods, namely the digital simulation technique and the method of harmonic balance, have been used to develop the steady state solutions for the internal sinusoidal excitation. Difficulties associated with the determination of the multiple solutions at a given frequency in the digital simulation technique have been resolved, as one must search the entire initial conditions map. Such solutions and the transition frequencies for various impact situations are easily found by the method of harmonic balance. Further, the principle of superposition can be employed to analyze the periodic transmission error excitation and/or combined excitation problems provided that the excitation frequencies are sufficiently apart from each other. Our analytical predictions match satisfactorily with the limited experimental data available in the literature. Using the digital simulation, we have also observed that the chaotic and subharmonic resonances may exist in a gear pair depending upon the mean or design load, mean to alternating force ratio, damping and backlash. Specifically, the mean load determines the conditions for no impacts, single-sided impacts and double-sided impacts. Our results are different from the frequency response characteristics of the conventional, single-degree-of-freedom, clearance type non-linear system. Our formulation should form the basis of further analytical and experimental work in the geared rotor dynamics area.

Load sharing characteristics of planetary transmissions

Abstract A nonlinear time-varying dynamic model of a planetary transmission with an arbitrary number of pinions is developed. The model includes several manufacturing errors and assembly variations, and can accommodate for any pinion spacing arrangement, tooth separations and time-varying gear mesh stiffnesses. A dynamic load sharing factor K DLS , which is defined as the ratio of the actual dynamic load at a given gear mesh to the ideal static load, is predicted using the dynamic model. Key design parameters, assembly variations and manufacturing errors are identified, and their effects on K DLS are quantified.

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

Abstract Frequency response characteristics of a non-linear geared rotor-bearing system with time-varying mesh stiffness k h ( t ) are examined in this paper. First, the single-degree-of-freedom spur gear pair model with backlash is extended to include sinusoidal or periodic mesh stiffness k h ( t ) . Second, a three-degree-of-freedom model with k h ( t ) and clearance non-lineariries associated with gear backlash and rolling element bearings, as excited by the static transmission error e ( t ) under a mean torque load, is developed. The governing equations are solved using digital simulation technique and only the primary resonances are studied. Resonances of the corresponding linear time-varying system associated with parametric and external excitations are identified using the method of multiple scales and digital simulation. Interactions between the mesh stiffness variation and clearance non-linearities have been investigated; a strong interaction between time-varying mesh stiffness k h ( t ) and gear backlash is found, whereas the coupling between k h ( t ) and bearing non-linearities is weak. Finally, our time-varying non-linear formulations yield reasonably good predictions when compared with the benchmark experimental results available in the literature.

Non-linear dynamics of a geared rotor-bearing system with multiple clearances

Abstract Non-linear frequency response characteristics of a geared rotor-bearing system are examined in this paper. A three-degree-of-freedom dynamic model is developed which includes non-linearities associated with radial clearances in the radial rolling element bearings and backlash between a spur gear pair; linear time-invariant gear meshing stiffness is assumed. The corresponding linear system problem is also solved, and predicted natural frequencies and modes match with finite element method results. The bearing non-linear stiffness function is approximated for the sake of convenience by a simple model which is identical to that used for the gear mesh. This approximate bearing model has been verified by comparing steady state frequency spectra. Applicability of both analytical and numerical solution techniques to the multi-degree-of-freedom non-linear problem is investigated. Satisfactory agreement has been found between our theory and available experimental data. Several key issues such as non-linear modal interactions and differences between internal static transmission error excitation and external torque excitation are discussed. Additionally, parametric studies are performed to understand the effect of system parameters such as bearing stiffness to gear mesh stiffness ratio, alternating to mean force ratio and radial bearing preload to mean force ratio on the non-linear dynamic behavior. A criterion used to classify the steady state solutions is presented, and the conditions for chaotic, quasi-periodic and subharmonic steady state solutions are determined. Two typical routes to chaos observed in this geared system are also identified.

Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs

A computational model is proposed for the prediction of friction-related mechanical efficiency losses of parallel-axis gear pairs. The model incorporates a gear load distribution model, a friction model, and a mechanical efficiency formulation to predict the instantaneous mechanical efficiency of a gear pair under typical operating, surface, and lubrication conditions. The friction model uses a new friction coefficient formula obtained by using a validated non-Newtonian thermal elastohydrodynamic lubrication (EHL) model in conjunction with a multiple linear regression analysis. The load and friction coefficient, distribution predictions are used to compute instantaneous torque/ power losses and the mechanical efficiency of a gear pair at any given rotational position. Efficiency measurements from gear pairs having various gear designs and surface treatments are compared to model predictions. Mechanical efficiency predictions are shown to be within 0.1% of the measured values, indicating that the proposed efficiency model is accurate. Results of a parametric study are presented at the end to highlight the influence of key basic gear geometric parameters, tooth modifications, operating conditions, surface finish, and lubricant properties on mechanical efficiency losses.

Effect of Involute Contact Ratio on Spur Gear Dynamics

The influence of involute contact ratio on the torsional vibration behavior of a spur gear pair is investigated experimentally by measuring the dynamic transmission error of several gear pairs using a specially designed gear test rig. Measured forced response curves are presented, and harmonic amplitudes of dynamic transmission error are compared above and below gear mesh resonances for both unmodified and modified gears having various involute contact ratio values. The influence of involute contact ratio on dynamic transmission error is quantified and a set of generalized, experimentally validated design guidelines for the proper selection of involute contact ratio to achieve quite gear systems is presented, A simplified analytical model is also proposed which accurately describes the effects of involute contact ratio on dynamic transmission error.

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.

Effect of Internal Gear Flexibility on the Quasi-Static Behavior of a Planetary Gear Set

Effect of flexibility of an internal gear on the quasi-static behavior of a planetary gear set is investigated. A state-of-the-art finite elements/semi--analytical nonlinear contact mechanics formulation is employed to model a typical automotive automatic transmission planetary unit. The model considers each gear as deformable bodies and meshes them to predict loads, stresses and deformations of the gears. Actual support and spline conditions are included in the model. The rim thickness of the internal gear is varied relative to the tooth height and gear deflections and bending stresses are quantified as a function of rim thickness. Influence of rim thickness on the load sharing amongst the planets is also investigated with and without floating sun gear condition. The results are discussed in detail and guidelines regarding the design of a planetary internal gear are presented.

An Experimental Investigation of Spur Gear Efficiency

In this study, a test methodology was developed for the measurement of spur gear efficiency under high-speed and variable torque conditions. A power-circulating test machine was designed to operate at speeds to 10,000 rpm and transmitted power levels to 700 kW. A precision torque measurement system was implemented, and its accuracy and repeatability in measuring torque loss in the power loop was demonstrated. Tests were conducted on gears with two values of modules and two surface roughness levels, operating in a dry sump jet-lubrication environment with three different gear lubricants. These tests were used to quantify the influence of these parameters on both load-dependent (mechanical), load-independent (spin), and total power loss. Trends in mechanical gear mesh efficiency and total gearbox efficiency were discussed in terms of rotational speed and transmitted torque. Finally, recommendations were made for the design of spur gear pairs, surface roughness, and lubricant selection for improved efficiency.

Dynamic Analysis of a Multi-Shaft Helical Gear Transmission by Finite Elements: Model and Experiment

A dynamic model of a multi-shaft helical gear reduction unit formed by N flexible shafts is proposed in this study. The model consists of a finite element model of shaft structures combined with a three-dimensional discrete model of helical gear pairs. Bearing and housing flexibilities are included in the model as well. Eigenvalue solution and the Modal Summation Technique are used to predict the free and forced vibrations of the system. Results of experimental study on a helical gear-shaft-bearing system are also presented for validation of the model. It is demonstrated that the predictions match well with the experimental data in terms of excited modes and the forced response given in the form of the dynamic transmission error. Forced vibrations of an example system formed by three shafts are also studied to demonstrate the influence of some of the key system parameters.