H. Nevzat Özgüven

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.

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.

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.

Structural modifications using frequency response functions

Abstract A general method using frequency response functional (FRFs) is developed for reanalysing a structure subjected to structural modification. In this method, theoretically calculated or experimentally measured FRFs can be used in determining the FRFs of the modified structure. Structural modifications may be in the form of additional structural components which may be expressed in terms of additional mass, stiffness and damping matrices. The formulation is given for two possible cases where there are and are not additional degrees of freedom due to structural modification. The application of the method is demonstrated with numerical examples in which theoretically calculated receptances of the original system are used.

Iterative receptance method for determining harmonic response of structures with symmetrical non-linearities

Abstract Although there are several techniques available for the harmonic vibration analysis of non-linear systems, the application of these methods to multi-degree of freedom systems is rather limited. In this study a new method (the Iterative Receptance Method—IRM) is developed for the harmonic vibration analysis of non-linear multi-degree of freedom systems with non-linear stiffness and damping forces. Different kinds of non-linearities are covered by expressing non-linear spring and damping forces as polynomials in displacements and velocities, respectively. Quasilinear receptance matrix of a non-linear system for a given level of external forcing is determined by using the receptance matrix of the linear part of the system and a matrix representing the non-linearity in the system. An iterative algorithm is employed in the numerical solution. The iterative receptance method is also modified for an efficient dynamic analysis of structures with local non-linearities. The application of the method is demonstrated by numerical examples, and the results are found to be quite encouraging for the application of the method to practical structures with large degrees of freedom.

Analytical Prediction of Part Dynamics for Machining Stability Analysis

An analytical procedure is developed to predict workpiece dynamics in a complete machining cycle in order to obtain frequency response functions (FRF) which are needed in chatter stability analyses. For this purpose, a structural modification method which is an efficient tool for updating FRFs is used. The removed mass by machining is considered as a structural modification in order to determine the FRFs at different stages of the process. The method is implemented in a computer code and demonstrated on different geometries. The predictions are compared and verified by FEA. Predicted FRFs are used in chatter stability analyses, and the effect of part dynamics on stability is studied. Different cutting strategies are compared for increased chatter free material removal rates considering part dynamics.

Vibration measurements predict the mechanical properties of human tibia

BACKGROUND Vibration analysis is a promising technique in diagnosing metabolic bone diseases such as osteoporosis and monitoring fracture healing. The aim of this study is to observe the structural dynamic property changes of the tibia extracted from the vibration analysis data. METHODS In this study, bone mineral density and vibration measurements were made both in in vivo and in vitro conditions. The relationship between structural dynamic properties, obtained and bone mineral densities measured were investigated. Also, the effect of soft tissues on measured structural dynamic properties was analyzed. FINDINGS Natural frequency of the tibia decreased with decreasing bone mineral density that presented a weak correlation with the bone mineral density values measured by dual energy X-ray densitometer of the femur. In the case of in vitro experiments, it was observed that the effect of muscles on measurement results is higher than that of the effect of the skin and the fibula which makes the modal identification procedure difficult. However, having very large percentage changes in the loss factors when mineral content and collagen are reduced is an encouraging result to believe that damping measurements may yield a promising technique in diagnosing progressing osteoporosis and monitoring fracture healing period. INTERPRETATION The utilization of natural frequency alone as a diagnosing tool does not seem to be a sufficient method although there is a correlation between this parameter and bone mineral density. However, in vitro experiments showed that the identification of the loss factor is a promising technique in diagnosing progressing osteoporosis.

Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion

Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated structures such as satellites require even the smallest nonlinear behavior to be considered for better performance. Therefore, it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, in most of the cases, test data will be incomplete. These handicaps make most of the methods given in the literature difficult to apply to engineering structures. The aim of this study is to improve a previously developed method considering these practical limitations. The approach proposed can be used for detection, localization, characterization and parametric identification of nonlinear elements by using incomplete FRF data. In order to reduce the effort and avoid the limitations in using footprint graphs for identification of nonlinearity, describing function inversion is used. Thus, it is made possible to identify the restoring force of more than one type of nonlinearity which may co-exist at the same location. The verification of the method is demonstrated with case studies.

Optimization of an Energy Harvester Coupled to a Vibrating Membrane

Resonant energy harvesters are generally designed for the operating frequency of the host structure. However, when the flexibility and mass parameters of the host structure are comparable to those of the energy harvester, mounting an energy harvester on the host structure may change the vibration characteristics of the host structure considerably. In this study, modeling and optimization of an energy harvester coupled to a vibrating membrane are presented to show the effect of dynamic coupling on maximum voltage output of the energy harvester. First, the optimum design parameters of an energy harvester are calculated to obtain maximum voltage output from piezoelectric material for a given excitation at a specified frequency by considering the dynamics of the energy harvester only. Then, by using the finite element (FE) models of the membrane and the energy harvester, coupled analysis is made and design parameters are optimized to obtain maximum voltage output from piezoelectric material. Finally, the voltage output results from two optimization approaches are compared and it is observed that maximum voltage obtained by considering the coupling effects is several times higher than the voltage output obtained from the energy harvester designed traditionally. The study shows that disregarding the host structure that energy harvesters are mounted in optimization stage may prevent us to find the optimum design parameters.

Obtaining Linear FRFs for Model Updating in Structures with Multiple Nonlinearities Including Friction

Most of the model updating methods used in structural dynamics are for linear systems. However, in real life applications structures may have nonlinearity. In order to apply model updating techniques to a nonlinear structure, linear FRFs of the structure have to be obtained. The linear dynamic behavior of a nonlinear structure can be obtained experimentally by using low forcing level excitations, if friction type of nonlinearity does not exist in the structure. However when the structure has multiple nonlinearities including friction type of nonlinearity, nonlinear forces due to friction will be more pronounced at low forcing level excitations. Then it will not be possible to measure linear FRFs by using low level forcing. In this study a method is proposed to obtain linear FRFs of a nonlinear structure having multiple nonlinearities including friction type of nonlinearity by using experimental measurements made at low and high forcing levels. The motivation is to obtain FRFs of the linear part of the system that can be used in model updating of a nonlinear system. The method suggested can also be used as a nonlinear identification method for nonlinear systems. The proposed method is validated with different case studies using SDOF and lumped MDOF systems and simulated experimental data. The effect of the excitation frequency, at which experiments are carried out, on the accuracy of the proposed method, is also demonstrated.