After the stable one of the curves (4) had been concluded, the expressions of the power characteristics can be established as the functions as the stimulating dimensionless frequen-cies. It was pointed out, that if the relations of
2 2
2 3 2 2 2
0, 1, SH 1, 1S
R F R
L r F L
were fulfilled at
one time and the system is stimulated at its natural frequency, then the device would oper-ate among optimal power conditions. The analytical results have been checked by numeri-cal investigations. The following diagram displays the extremums of some of the power characteristics (Ebe the total electric energy consumption of the system, ER the electric energy consumption of the ohmic resistance, Eh the electric energy consumption of the Coulomb-damping, which is considered to be equivalent to the finishing process), which refer to the optimum power conditions (Figure 3). is the dimensionless frequency.
Dynamical Investigation of a Superfinishing Device 121
Figure 3. The extremums of some power characteristics at resonance frequency The following table contains the numerical values of each power characteristic by which the power conditions of the nonlinear models of the prototype and the new devices are compared to each other, whilst the manufacturing parameters are assumed to be the same for both models (Table 2). It can be seen clearly, that adding an appropriate flexible ele-ment to the prototype device, the optimum manufacturing parameters can be achieved by less pulling force
F0 and non-significant wattles power
Pm , whilst the electric energy
Ebe and the current consumption
i0 is significantly reduced comparing to those of the prototype device.Table 2 The power conditions of the prototype and the properly tuned devices
based on nonlinear models
Power characteristics Prototype device (k0) New device (k0, tuned to its natural frequency:1)
sini 0,63 0,083
cosi 0,77 0,996
A [mm] 1 1
i0[A] 23,19 3,74
F0[N] 728,16 76,40
Pm[VAr] 605,04 4,62
Ph[W] 733,93 54,95
P[VA] 951,17 55,15
R 0,99 0,86
Rh 98,98 6,57
ER [J] 24,21 1,59
Ebe [J] 24,46 1,83
These optimum power conditions make it possible for us to construct such a linear motor driven device, whose overall dimensions and weigh is much more favourable than those of
the prototype device, and thus can be adopted on a certain ultraprecise hard-turning ma-chine without significant reconstructions and producing thermal and vibrational distur-bances which would have some effects on the work-room of the base-machine.
8. SUMMARY
This article is a brief summary of the results of some dynamic and power calculations emerging in the development process of a superfinishing prototype-device. The subject of the investigation is a 1-phase electromechanical superfinishing device operated by electric-ity and performing an oscillating movement of short-stroke. Linear and non-linear models are assigned to the device, which enables us to explore the power characteristics of the de-vice and to establish the optimum superfinishing frequencies. The positive effect of the ap-propriate flexible element was demonstrated, i.e. considering constant manufacturing pa-rameters, and exciting the non-linear system at its natural frequency, it becomes possible to construct an improved device of a significantly smaller size and of increased power con-sumption efficiency than the prototype.
Acknowledgement
“This research was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support by the European Union, co-financed by the European Social Fund.”
References
[1] Harris–Crede: Shock and vibration handbook. McGraw-Hill Co. Inc., 1956.
[2] A. Heck: Bevezetés a Maple használatába. Juhász Gyula Felsőoktatási Kiadó, Zenon Kft., Szeged, 1999.
[3] J. D. W.ordan–P. Smith: Nonlinear ordinary differential equations. Oxford University Press, 1999.
[4] J. L. Meriam–L. G. Kraige– B. D. Harper: Dynamics – Solving dynamics problems in Maple.
John Wiley & Sons, Inc., 2007.
[5] A. H. Nayfeh–D.T. Mook: Nonlinear oscillations. John Wiley & Sons, Inc. 1995.
[6] J. M. T Thompson–H. B. Stewart: Nonlinear dynamics and chaos. John Wiley & Sons, 2001.
[7] Gy. Patkó: Dinamikai eredmények és alkalmazások a gépészetben. A Miskolci Egyetem Ha-bilitációs Füzetei, Miskolc, 1998.
[8] Gy. Patkó: Közelítő módszer nemlineáris rezgések vizsgálatára. Kandidátusi értekezés, Miskolc, 1984.
Design of Machines and Structures, Vol. 2, No. 2 (2012), pp. 123–135.
ANALYTICAL MODEL TO DETERMINE MESHING STIFFNESS OF SPUR GEARS
RENÁTA SZŰCS–LÁSZLÓ KAMONDI
Department of Machine and Product Design, University of Miskolc H-3515 Miskolc-Egyetemváros
szucs.renata@citromail.hu; machkl@uni-miskolc.hu
Abstract. During dynamic analysis of gears correct determination of the meshing stiffness bears great importance. In order to determine meshing stiffness first of all the contact points have to be located. In this work a general method is presented for determining the contact points i.e. the contact curve, and a special method for involute profiles. An analytical method for deflection analysis of spur gears is formulated. During deflection analysis elas-tic deformation of the teeth is taken into account and as a consequence of the model and the calculation method failure in the base pitch and the profile can be treated.
Keywords: gear mesh, meshing stiffness, elastic deflection, non-linear dynamics 1. INTRODUCTION
Due to the increasing demands analysis of reasons of noise sources and dynamic effects is important, it is also indicated by the numbers of works published even nowadays in this area. On one hand installation inaccuracies and on the other hand elastic deformation of the teeth and base pitch and profile errors are the most common reasons of the undesired dy-namic effects.
Endeavour to increase volume performance and service life of drive gears results in in-stallation of gears of smaller accuracy grade. In case of these accuracy grades scale of elas-tic deformations and teeth errors can be the same. In order to decrease the disturbing effect of elastic deformation trimming of the tooth tip is a common procedure, but from the point of analysis of stress conditions, knowing deflections of the tooth at the whole contact region is essential.
Before wide-spreading of computers only analytical methods were available for deflec-tion analysis. In that time authors [Karas, Weber] modelled the tooth as a cantilever. Be-tween these models the main difference was the description of the tooth profile i.e. some models only approximate the involute curve, while others deal with the real curve. Of course these models vary on approximation of other conditions too. Nowadays papers deal-ing with tooth deflection of external spur gears can be divided into two main groups. Stud-ies belonged to the first groups [2–5] model the tooth as a cantilever beam even nowadays, but owing to development of calculation and modelling methods approximations are more accurate and on the other hand in most cases authors supplement their results with FEA analysis. Studies of the second group [6–9] use FEM analysis for determine tooth deflection of gears. Regarding the considered and neglected stresses (e.g. pressing, bending, shearing) causing deflection studies also differ from each other.
By the help of Finite Element Method analysis of gears is realizable in 3Ds, but it bears more importance in case of helical gears. From this point of view authors study tooth de-flection by analytical methods in which the tooth is modelled by a cantilever, in this work
this method is presented in case of involute profiles. In this paper the model established by the authors neglects installation inaccuracies and friction, only takes into account the effects of elastic deformation. In later studies our aim is to verify results of this paper by FEM.