The Effect of Hertzian, Bending and Shearing Stiffness on Noise and Vibration

2001 ◽  
Author(s):  
Jamil Abdo ◽  
Kambiz Farhang ◽  
Mousa Mohsen
Keyword(s):  
Mathematical Model ◽  
Contact Stiffness ◽  
Hertzian Contact ◽  
Hertz Contact ◽  
Resonant Frequencies ◽  
Tangential Contact ◽  
Apparent Stiffness ◽  
Fundamental Understanding ◽  
The Mathematical Model ◽  
Friction Induced Vibration

Abstract Since the apparent stiffness due to contact of one surface on another relates directly to the localized resonant frequencies, it is believed that accurate account of this property will lead to the fundamental understanding of causes of friction-induced vibration and noise. The mathematical model of contact is utilized to develop formulae for normal and tangential contact stiffness. The inclusion of a study in which the various modes of elastic deflections of an asperity are also considered, as well as their effects. The bending, shear and Hertz contact modes of elastic deflection are assumed to simultaneously occur for an asperity. Investigation of the combined effect of bending, shear and Hertzian contributions to the contact stiffness is indicating that the equivalent contact stiffness is best represented, among the three types of stiffness, by that due to Hertzian contact.

Dynamic Response of Two-Disk Systems With Friction and Misalignment

1998 ◽  
Author(s):  
Ahmed Anabtawi ◽  
Kambiz Farhang
Keyword(s):  
Mathematical Model ◽  
Dynamic Response ◽  
Contact Stiffness ◽  
Stick Slip ◽  
Torsional Response ◽  
Damping Characteristics ◽  
Disk Friction ◽  
Stiffness And Damping ◽  
The Mathematical Model ◽  
Friction Induced Vibration

Abstract Friction induced vibration and noise pose one of the most challenging problems. The complexity of the friction system arises due to the nonlinear nature of friction phenomena and that of contact stiffness and damping. This paper presents a mathematical model for studying the dynamic response of two-disk friction system in the presence of misalignment. The contact stiffness and damping characteristics of the system are represented in the axial as well as the torsional directions. In addition, the axial and torsional responses of the system are coupled by assuming dependency between the torsional response and the normal force between the two disks. Using the mathematical model, various scenarios are examined to study the effect of misalignment. These include cases of symmetric and asymmetric actuation forces as well as forces applied at unequal actuation times. The results suggest that asymmetry in actuation forces has negligible effect on stick-slip behavior of the system.

scholarly journals Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator

2019 ◽  
Vol 98 (3) ◽  
pp. 1795-1806 ◽  
Author(s):  
Sergii Skurativskyi ◽  
Grzegorz Kudra ◽  
Krzysztof Witkowski ◽  
Jan Awrejcewicz
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Physical Model ◽  
Dry Friction ◽  
Stepper Motor ◽  
Nonlinear Phenomenon ◽  
Hertz Contact ◽  
Bifurcation Phenomena ◽  
Statistical Regularities ◽  
The Mathematical Model

Abstract The paper is devoted to the study of harmonically forced impacting oscillator. The physical model for oscillator is a cart on a guide connected to the support with springs and excited by the stepper motor. The support also is provided with limiter of motion. The mathematical model for this system is defined with the second-order piecewise smooth differential equation. Model’s nonlinearity is connected with the incorporation of dry friction and generalized Hertz contact law. Analyzing the classical Poincare sections and inter-impact sequences obtained experimentally and numerically, the bifurcations and statistical properties of periodic, multi-periodic, and chaotic regimes were examined. The development of impact-adding regime as a new nonlinear phenomenon when the forcing frequency varies was observed.

A Mathematical-Numerical Model to Calculate Load Distribution, Contact Stiffness and Transmission Error in Involute Spur Gears

2009 ◽  
Author(s):  
Mehdi Mohammadpour ◽  
Iraj Mirzaee ◽  
Shahram Khalilarya
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Load Distribution ◽  
Contact Stiffness ◽  
Transmission Error ◽  
Spur Gear ◽  
Spur Gears ◽  
Meshing Stiffness ◽  
Definition Of ◽  
The Mathematical Model

This paper firstly presents a mathematical model in order to calculate the load distribution, single contact stiffness and meshing stiffness as well as transmission error. in this way, there is no need to use finite element like methods and also the calculation time is dramatically reduced. Presented method is based on definition of a statically undetermined problem that is formulated using energy method. Some assumptions considered to convert this problem to a statically determined problem and get the mathematical models. Then a numerical method is employed in order to solve the mathematical model using a double iteration flowchart to close the problem. This model is flexible to adapt for any modification in spur gear profile geometry. Finally, this model is verified using previous works that have been utilized finite element and experimental model.

scholarly journals ON APPLICATION OF A SIMPLIFIED TWO-DIMENSIONAL MODEL OF FORCED OSCILLATIONS TO THE POWER ANALYSIS OF FLAT STEELWORKS

2020 ◽  
pp. 109-119
Author(s):  
Холодняк Ю.С. ◽  
Подлєсний С.В. ◽  
Капорович С.В. ◽  
Коротенко Є.Д.
Keyword(s):  
Mathematical Model ◽  
Oscillatory System ◽  
Forced Oscillations ◽  
Simultaneous Action ◽  
Oscillatory Process ◽  
Power Calculation ◽  
Two Dimensional ◽  
Resonant Frequencies ◽  
Computational Tools ◽  
The Mathematical Model

Abstract. An analysis of existing methods of power calculation of steelworks under the influence of forced oscillations is performed. When considering the forced oscillations of flat steelworks, two-dimensional models are used, which are complex for wide practical use. Their implementation requires in-depth mathematical training and complex computational tools. The aim of this work is to develop a simplified two-dimensional mathematical model of forced oscillations of flat steelworks with following use of this model in power calculations. The mathematical model proposed in this paper describes oscillations of a weightless steelwork with a point mass of simultaneous action in vertical and horizontal harmonic disturbing forces acting on them. The model is based on the method of forces, establishes a link between the movements of the steelworks and the forces that act on them. Together with the model the dependences for calculating the resonant frequencies of the oscillatory system are obtained. The performed developments allow to determine the dynamic characteristics of the oscillatory process and to calculate a steelwork strength, stiffness and stability.

Friction-Induced Vibration in a Dynamometer Utilizing a Two-Disk Friction Interaction

1999 ◽  
Author(s):  
Ahmed Anabtawi ◽  
Kambiz Farhang
Keyword(s):  
Mathematical Model ◽  
Coefficient Of Friction ◽  
Measurement Data ◽  
Low Noise ◽  
Polynomial Fit ◽  
The Coefficient Of Friction ◽  
Disk Friction ◽  
Fifth Order ◽  
The Mathematical Model ◽  
Friction Induced Vibration

Abstract A mathematical model is developed for a dynamometer for testing a two-disk friction brake system. The model accounts for the compressive shear properties of the friction disks as well as stiffness and inertial characteristics of the dynamometer. Friction coefficient/velocity relations are established using the measurement data and employed in the mathematical model to perform simulations of the dynamometer’s vibration response and instability. Five different simulations are performed. The first four employ a fifth order polynomial fit from the coefficient friction measured experimentally. One of these four simulations uses data from noisy test while the other three utilize relatively low-noise tests. The fifth and last simulation is carried out using a theoretical coefficient of friction, being assumed to experience certain sinusoidal variation. Such function is used to study the effect of variability in kinetic coefficient of friction of the experimental data. In the first four simulations (i.e. the experimental cases), it is observed that the simulations turn unstable when the friction-velocity slope becomes negative. That phenomenon is alleviated when higher damping is used. In the fifth simulation (i.e. the sinusoidal case), variation in the coefficient of friction is found to increase vibration in the dynamometer system.

DETERMINATION OF PERFORMANCE INDICATORS OF THE TRUCK KrAZ-6510TE

2020 ◽  
pp. 19-26
Author(s):  
Olexandr Pavlenko ◽  
Serhii Dun ◽  
Maksym Skliar
Keyword(s):  
Mathematical Model ◽  
Surface Friction ◽  
Building Structures ◽  
Maximum Torque ◽  
Military Equipment ◽  
Military Vehicles ◽  
Force Magnitude ◽  
Vehicle Mobility ◽  
The Mathematical Model

In any economy there is a need for the bulky goods transportation which cannot be divided into smaller parts. Such cargoes include building structures, elements of industrial equipment, tracked or wheeled construction and agricultural machinery, heavy armored military vehicles. In any case, tractor-semitrailer should provide fast delivery of goods with minimal fuel consumption. In order to guarantee the goods delivery, tractor-semitrailers must be able to overcome the existing roads broken grade and be capable to tow a semi-trailer in off-road conditions. These properties are especially important for military equipment transportation. The important factor that determines a tractor-semitrailer mobility is its gradeability. The purpose of this work is to improve a tractor-semitrailer mobility with tractor units manufactured at PJSC “AutoKrAZ” by increasing the tractor-semitrailer gradeability. The customer requirements for a new tractor are determined by the maximizing the grade to 18°. The analysis of the characteristics of modern tractor-semitrailers for heavy haulage has shown that the highest rate of this grade is 16.7°. The factors determining the limiting gradeability value were analyzed, based on the tractor-semitrailer with a KrAZ-6510TE tractor and a semi-trailer with a full weight of 80 t. It has been developed a mathematical model to investigate the tractor and semi-trailer axles vertical reactions distribution on the tractor-semitrailer friction performances. The mathematical model has allowed to calculate the gradeability value that the tractor-semitrailer can overcome in case of wheels and road surface friction value and the tractive force magnitude from the engine. The mathematical model adequacy was confirmed by comparing the calculations results with the data of factory tests. The analysis showed that on a dry road the KrAZ-6510TE tractor with a 80 t gross weight semitrailer is capable to climb a gradient of 14,35 ° with its coupling mass full use condition. The engine's maximum torque allows the tractor-semitrailer to overcome a gradient of 10.45° It has been determined the ways to improve the design of the KrAZ-6510TE tractor to increase its gradeability. Keywords: tractor, tractor-semitrailer vehicle mobility, tractor-semitrailer vehicle gradeability.

EXPERIMENTAL STUDIES OF CAR PROPERTIES ON A PHYSICAL MODEL

2019 ◽  
pp. 10-16
Author(s):  
Oleksii Timkov ◽  
Dmytro Yashchenko ◽  
Volodymyr Bosenko
Keyword(s):  
Mathematical Model ◽  
Remote Control ◽  
Physical Model ◽  
Model Experiment ◽  
Experimental Studies ◽  
Electrical Energy ◽  
Short Term ◽  
Motor Current ◽  
Memory Card ◽  
The Mathematical Model

The article deals with the development of a physical model of a car equipped with measuring, recording and remote control equipment for experimental study of car properties. A detailed description of the design of the physical model and of the electronic modules used is given, links to application libraries and the code of the first part of the program for remote control of the model are given. Atmega microcontroller on the Arduino Uno platform was used to manage the model and register the parameters. When moving the car on the memory card saved such parameters as speed, voltage on the motor, current on the motor, the angle of the steered wheel, acceleration along three coordinate axes are recorded. Use of more powerful microcontrollers will allow to expand the list of the registered parameters of movement of the car. It is possible to measure the forces acting on the elements of the car and other parameters. In the future, it is planned to develop a mathematical model of motion of the car and check its adequacy in conducting experimental studies on maneuverability on the physical model. In addition, it is possible to conduct studies of stability and consumption of electrical energy. The physical model allows to quickly change geometric dimensions and mass parameters. In the study of highway trains, this approach will allow to investigate the various layout schemes of highway trains in the short term. It is possible to make two-axle road trains and saddle towed trains, three-way hitched trains of different layout. The results obtained will allow us to improve not only the mathematical model, but also the experimental physical model, and move on to further study the properties of hybrid road trains with an active trailer link. This approach allows to reduce material and time costs when researching the properties of cars and road trains. Keywords: car, physical model, experiment, road trains, sensor, remote control, maneuverability, stability.

RESEARCH OF INDICATORS OF A VEHICLE WITH A DIESEL WORKING ON DIESEL AND DIESEL GAS CYCLES USING THE MATHEMATICAL MODEL

2020 ◽  
pp. 14-19
Author(s):  
Serhii Kovbasenko ◽  
Andriy Holyk ◽  
Serhii Hutarevych
Keyword(s):  
Mathematical Model ◽  
Diesel Engine ◽  
Environmental Performance ◽  
Energy Performance ◽  
Dual Fuel ◽  
Fuel System ◽  
Compressed Natural Gas ◽  
Diesel Vehicles ◽  
Diesel Cycle ◽  
The Mathematical Model

The features of an advanced mathematical model of motion of a truck with a diesel engine operating on the diesel and diesel gas cycles are presented in the article. As a result of calculations using the mathematical model, a decrease in total mass emissions as a result of carbon monoxide emissions is observed due to a decrease in emissions of nitrogen oxides and emissions of soot in the diesel gas cycle compared to the diesel cycle. The mathematical model of a motion of a truck on a city driving cycle according to GOST 20306-90 allows to study the fuel-economic, environmental and energy indicators of a diesel and diesel gas vehicle. The results of the calculations on the mathematical model will make it possible to conclude on the feasibility of converting diesel vehicles to using compressed natural gas. Object of the study – the fuel-economic, environmental and energy performance diesel engine that runs on dual fuel system using CNG. Purpose of the study – study of changes in fuel, economic, environmental and energy performance of vehicles with diesel engines operating on diesel and diesel gas cycles, according to urban driving cycle modes. Method of the study – calculations on a mathematical model and comparison of results with road tests. Bench and road tests, results of calculations on the mathematical model of motion of a truck with diesel, working on diesel and diesel gas cycles, show the improvement of environmental performance of diesel vehicles during the converting to compressed natural gas in operation. Improvement of environmental performance is obtained mainly through the reduction of soot emissions and nitrogen oxides emissions from diesel gas cycle operations compared to diesel cycle operations. The results of the article can be used to further develop dual fuel system using CNG. Keywords: diesel engine, diesel gas engine, CNG

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.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

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