Interactions Between Backlash and Bearing Clearance Nonlinearity in Geared Flexible Rotors

2007 ◽  
Author(s):  
N. Ersan Gu¨rkan ◽  
H. Nevzat O¨zgu¨ven
Keyword(s):  
Mathematical Model ◽  
Series Representation ◽  
Calculated Data ◽  
Axial Loading ◽  
Transmission Error ◽  
Gear Mesh ◽  
Bearing Clearance ◽  
Flexible Rotors ◽  
Non Linear ◽  
Clearance Nonlinearity

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.

Effect of cracks and pitting defects on gear meshing

2012 ◽  
Vol 226 (11) ◽  
pp. 2805-2815 ◽  
Author(s):  
Alfonso Fernandez del Rincon ◽  
Fernando Viadero ◽  
Miguel Iglesias ◽  
Ana de-Juan ◽  
Pablo Garcia ◽  
...  
Keyword(s):  
Finite Element ◽  
Computational Models ◽  
Inequality Constraints ◽  
Variable Stiffness ◽  
Element Model ◽  
Equation System ◽  
Transmission Error ◽  
Computational Effort ◽  
Gear Mesh ◽  
Non Linear

The development of vibration-based condition monitoring techniques, especially those focused on prognosis, requires the development of better computational models that enable the simulation of the vibratory behaviour of mechanical systems. Gear transmission vibrations are governed by the so-called gear mesh frequency and its harmonics, due to the variable stiffness of the meshing process. The fundamental frequency will be modulated by the appearance of defects which modify the meshing features. This study introduces an advanced model to assess the consequences of defects such as cracks and pitting on the meshing stiffness and other related parameters such as load transmission error or load sharing ratio. Meshing forces are computed by imposing the compatibility and complementarity conditions, leading to a non-linear equation system with inequality constraints. The calculation of deformations is subdivided into a global and a local type. The former is approached by a finite element model and the latter via a non-linear Herztian-based formulation. This procedure enables a reduced computational effort, in contrast to conventional finite element models with contact elements. The formulation used to include these defects is described in detail and their consequences are assessed by a quasi-static analysis of a transmission example.

scholarly journals Dynamic Analysis of Geared Rotors by Finite Elements

1992 ◽  
Vol 114 (3) ◽  
pp. 507-514 ◽  
Author(s):  
A. Kahraman ◽  
H. Nevzat Ozguven ◽  
D. R. Houser ◽  
J. J. Zakrajsek
Keyword(s):  
Finite Element ◽  
Vibration Analysis ◽  
Axial Loading ◽  
Element Model ◽  
Transmission Error ◽  
Analysis Procedure ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Transverse Vibrations ◽  
Forced Vibration Analysis

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 shafts to mass unbalances, geometric eccentricities of gears, and displacement transmission error excitation at the mesh point. The dynamic mesh forces due to these excitations 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.

Transverse Vibration of Geared-Rotor Integrated With Active Magnetic Bearings in Identification of Multiple Faults

2021 ◽  
Vol 143 (9) ◽  
Author(s):  
Gargi Majumder ◽  
Rajiv Tiwari
Keyword(s):  
Mathematical Model ◽  
Transverse Vibration ◽  
Transmission Error ◽  
Magnetic Bearings ◽  
Mesh Deformation ◽  
Identification Algorithm ◽  
Active Magnetic Bearings ◽  
Full Spectrum ◽  
Gear Mesh ◽  
Fault Parameters

Abstract This paper presents a novel concept of the modeling, active control of transverse vibration responses, and identification of fault parameters in a geared-rotor system integrated with active magnetic bearings (AMBs). The sources of error in gears while in the operation are the gear mesh deformation, transmission error, and runout, resulting in dynamic forces, excessive vibration, and noise. To avoid any undesirable effect on the gear-pair and other supporting structures, it is essential to investigate these forced vibrations in time and frequency domain. Hence, an approach to monitor and control the transverse vibration of mating gears is presented with the help of AMBs. The AMBs are capable of suppressing the vibration of the system (transients as well as steady-state) by controlled electromagnetic forces considering the rotor vibrational displacement with a closed-loop feedback system. A mathematical model has been developed with geared rotor faults, like the mesh deformation, gear run-out, and asymmetric transmission error. The transmission error has been modeled as the sum of mean and varying components of error in two orthogonal transverse directions. Based on the mathematical model, an identification algorithm has been developed. Considering full spectrum analysis of the rotor vibration and AMB current information, estimation of system parameters, i.e., the equivalent mesh stiffness, mesh damping, gear runouts, the mean and varying transmission error magnitude and phase angles, and the current and displacement constants of AMBs has been performed. Gaussian noise in responses and modeling errors in mathematical models have been added to test the robustness of the proposed algorithm to comply with the experimental settings.

Imprecision of Laboratory Determinations and Diagnostic Accuracy: Theoretical Considerations

1974 ◽  
Vol 13 (03) ◽  
pp. 151-158 ◽  
Author(s):  
D. A. B. Lindbebo ◽  
Fr. R. Watson
Keyword(s):  
Mathematical Model ◽  
Diagnostic Accuracy ◽  
Diagnostic Process ◽  
Clinical Laboratories ◽  
Non Linear ◽  
Theoretical Considerations ◽  
Made In

Recent studies suggest the determinations of clinical laboratories must be made more precise than at present. This paper presents a means of examining benefits of improvement in precision. To do this we use a mathematical model of the effect upon the diagnostic process of imprecision in measurements and the influence upon these two of Importance of Diagnosis and Prevalence of Disease. The interaction of these effects is grossly non-linear. There is therefore no proper intuitive answer to questions involving these matters. The effects can always, however, be calculated.Including a great many assumptions the modeling suggests that improvements in precision of any determination ought probably to be made in hospital rather than screening laboratories, unless Importance of Diagnosis is extremely high.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

scholarly journals MODELING AND PRODUCTIVITY PREDICTION OF THE COMPANIES-INTERNAL CROWDSOURCING-BASED IDEATION

2021 ◽  
Vol 1 ◽  
pp. 2147-2156
Author(s):  
Pavel Livotov
Keyword(s):  
Mathematical Model ◽  
Finite Number ◽  
Research Question ◽  
Performance Function ◽  
Non Linear ◽  
Series Of Experiments ◽  
Productivity Prediction ◽  
A Company

AbstractThe internal crowdsourcing-based ideation within a company can be defined as an involvement of its staff, specialists, managers, and other employees, to propose solution ideas for a pre-defined problem. This paper addresses a question, how many participants of the company-internal ideation process are required to nearly reach the ideation limit for the problems with a finite number of workable solutions. To answer the research question, the author proposes a set of metrics and a non-linear ideation performance function with a positive decreasing slope and ideation limit for the closed-ended problems. Three series of experiments helped to explore relationships between the metric attributes and resulted in a mathematical model which allows companies to predict the productivity metrics of their crowdsourcing ideation activities such as quantity of different ideas and ideation limit as a function of the number of contributors, their average personal creativity and ideation efficiency of a contributors’ group.

Dynamic Analysis of Geared Rotor-Bearing Systems by the Transfer Matrix Method

1995 ◽  
Author(s):  
Siu-Tong Choi ◽  
Sheng-Yang Mau
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Analysis ◽  
Transmission Error ◽  
Mode Shapes ◽  
Rotor Systems ◽  
Gear Mesh ◽  
Mass Unbalance ◽  
Rotor Bearing

Abstract In this paper, an analytical study of the dynamic characteristics of geared rotor-bearing systems by the transfer matrix method is presented. Rotating shafts are modeled as Timoshenko beam with shear deformation and gyroscopic effects taken into account. The gear mesh is modeled as a pair of rigid disks connected by a spring-damper set and a transmission-error exciter. The transfer matrix of a gear mesh is developed. The coupling motions of the lateral and torsional vibration are studied. In free vibration analysis of geared rotor systems, natural frequencies and corresponding mode shapes, and the whirl frequencies under different spin speeds are determined. Effects of bearing stiffness, isotropic and orthotropic bearings, pressure angle of the gear mesh are studied. In steady-state vibration analysis, responses due to the excitation of mass unbalance and the transmission error are studied. Parametric characteristics of geared rotor systems are discussed.

Dynamic Analysis of a Multi-Mesh Helical Gear Train

1992 ◽  
Author(s):  
Ahmet Kahraman
Keyword(s):  
Loading Condition ◽  
Helix Angle ◽  
Transmission Error ◽  
Helical Gear ◽  
Forced Response ◽  
Gear Train ◽  
Loading Conditions ◽  
Gear Mesh ◽  
Natural Modes ◽  
Static Transmission Error

Abstract In this paper, the dynamic behavior of a multi-mesh helical gear train is studied. The gear train consists of three helical gears, with one of the gears in mesh with the other two. An 18-degree-of-freedom dynamic model which includes transverse, torsional, axial and rotational (rocking) motions of the flexibly mounted gears is developed. Two different loading conditions are identified. For case I, the system is driven by the gear in the middle, and for case II, the system is driven by one of the gears at either end of the gear train. Gear mesh phases under each loading condition are determined. The natural modes are predicted, and effects of the helix angle and the loading condition on the natural modes are explained. The forced response, which includes dynamic mesh and bearing forces, due to the static transmission error excitation is found. Effects of loading conditions and asymmetric positioning on the response are also explored. The results suggest that the dynamic forces are lower if the number of teeth of the gear in the middle is (i) an odd number for case I type loading, and (ii) an even number for case II type loading.

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