Modeling of a rotor-bearing system with a tapered part and the correspondent dynamic response.

Visan, Dana ; Gheorghiu, Horia ; Parausanu, Ioan 等

1. INTRODUCTION

The paper presents the dynamic modeling of a tapered rotor-bearing system. This study came from a practical necessity of design and dynamic analysis of rotor machines and from the need of knowledge of their dynamic response.

2. THEORETICAL CONSIDERATIONS

The calculus model for the dynamic study of a rotor-bearing system is made using the finite element method. This technique has a great possibility of modeling for all the elements, effects and interactions between the components of rotor machines.

A MATLAB code called DRACULA, for rotor dynamics calculations, was developed in the Department of Strength of Materials from University Politehnica of Bucharest. Bernoulli-Euler and Timoshenko shaft elements were implemented in this code, while [C.sup.1] conic elements are used for modeling the tapered parts.

3. CALCULUS EXAMPLE

A consequence of shaft tapering is presented in the proposed example. In this case, "tapering" means an increase of the inclination angle of the external surface of the shaft, keeping constant the volume of material.

This tapering is very useful for drilling and milling cutting tools to increase the frequency of bending vibration and thus the stiffness of the tool compared to the ones of constant section shafts having the same volume and manufactured from the same material (Kim et al., 1999a, Kim et al., 1999b)

A hollow tapered shaft is considered in this example, having a volume of 59 [cm.sup.3] (Fig. 1). The characteristics of the material of the shaft are: Young's modulus E = 207 GPa and mass density p = 7700 kg/[m.sup.3]. The rotative speed of the rotor is 400 rad/s, this means 63.66 Hz.

The following notations are used in this paper:

[b.sub.1] = the external great radius of the truncated cone;

[b.sub.2] = the external small radius of the truncated cone;

L = length of the tapered shaft;

t = wall thickness of the tapered shaft.

[FIGURE 1 OMITTED]

The inclination ratio TR or "tapering" of the shaft is defined as:

[T.sub.R] = ([b.sub.1] - [b.sub.2]/L (1)

For different values of the inclination ratio (TR = 0, TR = 1, TR = 2 and TR = 3) and using models with different number of elements (2 and 4 elements respectively), the first eigen pulsation was calculated using the DRACULA code. The variation of this parameter with respect to the inclination ratio was plotted.

In order to keep the 59 [m.sup.3] constant volume of the shaft, the following parameters were kept constant:

t = 5.4 mm ; L = 240 mm ; [b.sub.1] + [b.sub.2] = 20 mm.

The external radii of the tapered shaft were considered as variable.

The values of the first eigen pulsation of the shaft for different inclination ratios and different tapering are presented in Fig. 2.

From this figure, one can conclude that the greater the inclination ratio for the same volume of material, the greater the first eigen pulsation. Therefore, the stiffness of the shaft is increased. In these conditions, a cutting tool can work at higher rotative speeds in safety conditions.

The variation of the eigen pulsation ratio having i = 0, 1, 2 and 3 with the inclination ratio is shown in Fig. 3. One can observe the increase of the pi /p0 ratio with the tapering and therefore the increase of the stiffness of the rotor and the tool.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

4. CONCLUSIONS

The tapered part of the shaft was modeled with Timoshenko conical elements, this variant being satisfactory from the point of view of the accuracy of the results.

The modeling procedure of the tapered part is simple and easy.

It is not recommended to model the tapered part with only one conical element, because important errors can appear.

5. REFERENCES

Hohn, A. (2006). Die mechanische Auslegung von Dampfturbogruppen, BBC Aktiengesellschaft Brown, Boveri & Cie, Druckschrift Nr. CH-T 110273 D (in German)

Kim, W., Argento, A. & Scott, R.A. (1999a). Free Vibration of a Rotating Tapered Composite Timoshenko Shaft, Journal of Sound and Vibrations, 226(1) pp 125-147

Kim, W., Argento, A. & Scott, R.A. (1999b). Forced Vibration and Dynamic Stability of a Rotating Tapered Composite Timoshenko Shaft, 1999 ASME Design Engineering Technical Conferences, September 12-15, 1999, Las Vegas, Nevada, USA

Rades, M. (1996). Dinamica sistemelor rotor-lagare (Dynamics of the rotor-bearings systems), vol. I, University Politehnica of Bucharest, Department of Strength of Materials, Bucharest, Romania (in Romanian)

Rades M., Dynamics of Machinery, Part II (1995). University Politehnica of Bucharest, Department of Engineering Sciences, Division of Mechanical Engineering, Bucharest, Romania (in Romanian)

Visan, D.C. (2008). Modelarea dinamica a sistemelor rotorlagare (Dynamic modeling of the rotor-bearings systems), Printech Publishing House, Bucharest, Romania (in Romanian)