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Machine
Dynamics Problems
2005, Vol. 29, No 1
Contents
1.
Jarosław Dyk
Tooth Friction in
Models of Gears
2.
Jan Freundlich, Jerzy Jachimowicz
Investigation into
the Influence of Structural Parameters of a Machine Frame on its Static and
Dynamic Properties
3.
Jan Freundlich, Piotr Żach, Jerzy Osiński
Investigation of
Dynamic Properties of Thick Wall Composite Shafts with FEM
4.
Krzysztof
Gołoś, Bogdan Wasiluk, Toshihiko Hoshide
The Crack Propagation Model for Elastic-Plastic Materials
5.
Michał Hać
Influence of Static
and Dynamic Deflection on Shaft Pins on Strength Parameters of Toothed Gear
6.
Jerzy
Osiński
Analysis of Conditions of the Existence of Combined Parametric Resonances in
Damped Systems by Partial Decoupling of Governing Equations
7.
Piotr M. Przybyłowicz
Resonant Behaviour
of Geared System with Active Journal Bearings
8.
Zbigniew
Starczewski
Vibration of the Toothed Gears Shaft Supported on the Slide Bearings
9.
Piotr Żach
Reverse Running
Simulation of Power Transmission System
1. Tooth Friction in Models of Gears
Jarosław Dyk
Warsaw University of Technology
jaroslaw.dyk@ipbm.simr.pw.edu.pl
Abstract
The aim of this report is research on dynamics in gears on the basis of models with regard to tooth friction, considered as Coulomb’s dry friction, as well as an evaluation of its influence on dynamic behaviour of gears. The models of isolated and non-isolated gear transmissions are under our consideration. The non-isolated one includes additionally the input and output parameters of shafts. The mesh description includes tooth stiffness varying on the line of action and the function of the mesh parameters that depends on the mesh phase, clearances, pitch and profile deviations as well as the mesh displacements. Generally, the form of this function is consistent with the modified Müller’s model (see Müller, 1996). Assuming frictional resistance in the mesh, on the basis of the carried out researches some detailed conclusions considering the applied levels of the equivalent damping and damping distribution of dissipative forces in the gear models are formulated. The influence of friction on the maximal values of dynamic forces appears in the low range of rotational speeds and also in the resonant zones. Introduction of the friction forces into the models gives us more accurate mesh description, and thus additional explanation for the sources of dynamic loads in gears.
2. The Influence of Structural Parameters of a Machine Frame on its Static and Dynamic Properties
Jan
Freundlich, Jerzy Jachimowicz
Warsaw University of Technology
jfr@simr.pw.edu.pl
Abstract
An investigation of the influence of selected structural parameters of a machine frame on its static and dynamic properties has been presented. Comparative calculations of frames in a static area were accomplished in order to perform comparison with experimental results presented in work (Iżykowski et al., 1994). Dynamic calculations of hypothetical machine frames were performed in order to examine effects of selected constructional parameters on their dynamic properties. Their influence on natural frequencies and vibration modes was investigated. The calculations were performed for various thickness of a frame flange, various initial forces in pre-stressed bolts, using or not stiffening ribs. Next, several models of a frame attachment to foundation were tested to find less sophisticated FE model of the tested system for further dynamic calculations. In the last stage, an influence of application of a special layer made of a damping material to frame walls on natural frequencies and modal damping factors of a machine frame was investigated.
3. Investigation of Dynamic Properties of Thick Wall Composite Shafts with FEM
Jan Freundlich, Piotr Żach,
Jerzy Osiński
Warsaw University of Technology
jfr@simr.pw.edu.pl, pzach@simr.pw.edu.pl
Abstract
The investigation of dynamics thick walled composite shafts has been presented. The wall of examined shafts comprised three layers and the middle layer was made of high damping material. The influence of a lamination angle q of outer and inner layers onto dynamic properties of composite shafts was performed. Calculations of natural frequencies and modes, resonant curves and modal loss factors were accomplished. The obtained results have shown existence of extreme in relationship between maximum resonant amplitude, a modal loss factor and a lamination angle.
4. The Crack Propagation Model for Elastic-Plastic Materials
Krzysztof Gołoś1), Bogdan Wasiluk, Toshihiko Hoshide2)
1) Warsaw University of
Technology
2) Kyoto
University, Japan
kgo@simr.pw.edu.pl
Abstract
The problem of inclined crack in plate subjected to biaxial stresses is investigated in this study. A model for crack growth process under mixed mode loading conditions is proposed and analysed. The model is based on analysis of the energy dissipation processes situated ahead of the crack tip. The plastic zone shape as well as the plastic zone size effect is taken into account. The W-criterion is applied for prediction of the crack growth angles. A new formula for estimation of the crack growth trajectory is also proposed. The crack propagation process is numerically simulated and the crack propagation angles as well the crack growth paths are determined for LYJ2-CZ aluminium alloy.
5. Influence of Static and Dynamic Deflection of Shaft Pins on Strength Parametrers of Toothed Gear
Michał Hać
Warsaw University of Technology
Abstract
In the paper the evaluation of strength of
tooth gear due to shaft deflection is under consideration. There is taken into
account the machining tolerance and its influence on the deflection of the
shaft. The finite element method is used for determination of the shaft
deflection and the geometrical non-linear model is considered. The load
distribution factor used in calculation of strength of toothed gear for the
given example is evaluated. The dynamic equations for rotating shafts are
presented and the vibration due to changes of shaft stiffness with the angle of
rotation is analysed. The results of analysis show that in toothed gears much larger significance has
the preciseness of machining of shaft pins than taking into account non-linear
phenomena.
6. Analysis of Conditions of the Existence of Combined Parametric Resonances in Damped Systems by Partial Decoupling of Governing Equations
Jerzy Osiński
Warsaw University of Technology
jerzy.osinski@ipbm.simr.pw.edu.pl
Abstract
In systems subject to parametric excitation, combined resonances that appear in the case of the excitation frequency coinciding with a linear combination of eigenfrequencies are of great importance. Such resonances however are eliminated by damping in the system. This problem has been poorly studied so far. A method of the assessment of the effect of damping on combined resonances by making use of transformation of the equations of motion into quasi-normal variables is presented. The thus obtained equations are decoupled with respect to damping. The equations have been solved numerically – the results confirmed that the combined resonances will occur if the newly determined governing equations written in quasi-normal coordinates remain coupled by terms corresponding to the parametric excitation.
7. Resonant Behaviour of Geared System with Active Journal Bearings
Piotr M.
Przybyłowicz
Warsaw University of Technology
piotr.przybylowicz@ipbm.simr.pw.edu.pl
Abstract
The paper is concerned with the problem of active control of resonant vibration of a geared transmission system with a rigid shaft supported on journal bearings. Toothed gears generate vibration due to variable meshing stiffness and other dynamic phenomena affecting the course and amplitude of contact forces between teeth. Moreover, geared systems exhibit resonance behaviour characterized by considerable variations of the meshing force that reaches more than a double of the static level. Such strongly oscillating forces cause vibrations of the entire gear assembly and deteriorate its operational features. In the paper, it is assumed that the gear shaft is supported on journal bearings instead of roll ones as in typical designs. Additionally, the bearings are active elements adjustable to current working conditions. The control system is based on the application of piezoelectric elements which are used to change the thickness of the oil gap with growing angular velocity of the rotor. The size of the oil gap greatly changes the dynamic properties of the bearings, including those related to damping capability. It is shown that the assumption of the simplest proportional control strategy, i.e. proportional to the rotation speed, yields an apparent drop in the amplitude of resonant vibration.
8. Vibration of the Toothed Gears Shaft Supported on the Slide Bearings
Zbigniew
Starczewski
Warsaw University of Technology
zst@simr.pw.edu.pl
Abstract
In the paper, vibrations of a toothed gear supported on journal bearings are investigated. Two cases are considered. In the first case, the pre-resonant range is studied, which corresponds to constant loading of the gear system. In the second one, the meshing force has exactly the resonant character. Appropriate relationships and diagrams are presented.
9. Reverse Running Simulation of Power Transmission System
Piotr Żach
Warsaw University of Technology
Abstract
In this paper we took into consideration dynamic problems that occur in system of electric motor – gear transmission – machine that is propel during its running with reversal of rotation. The rotation is described with system of equations that contain mechanical and electrical parameters. It was elaborated two dynamic models of the system:
– general – described total movement with transmission of flexibility teeth,
– detailed – focused on gear transmission problems.
Verification of the system was performed on test stand in laboratory of Institute of Machine Design Fundamentals – Warsaw University of Technology.