Estimation of Natural Frequencies and Mode Shapes of a Shaft Supported by more than Three Bearings

The present paper emphasizes on the estimation of natural frequencies and mode shapes of a shaft supported by more than three bearings . In advent of this, a counter shaft of already developed experimental setup has been considered. The natural frequencies and mode shapes of counter shaft are deteremined analytically by adopting Holzer's method and the results obtained are then compared with the results of software based approach . By adopting the same method , Stresses in the shaft and amplitudes of rotors are also estimated. Based on the obtained data the generallised mathematical models have been formulated for the prediction of stresses in shaft and amplitudes of rotors. Significant independent variables which influence the phenomenon i.e development of stresses and amplitudes of rotor have been accounted in terms of a group of pie terms. These group of pie terms are accomplished with the help of dimensional analysis technique and they are used to formulate the mathematical models. In this paper the qualitative and quantitative analyses of the established mathematical models are also carried out.

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Keywords: countershaft, natural frequency, mode shapes, shear stress, mathematical model

Journal of Mechanical Design and Vibration, 2014 2 (1), pp 11-24.
DOI: 10.12691/jmdv-2-1-2

Received January 30, 2014; Revised May 19, 2014; Accepted May 20, 2014

Cite this article:

Meshram, Atul B., Girish D. Mehta, and Jayant P. Modak. "Estimation of Natural Frequencies and Mode Shapes of a Shaft Supported by more than Three Bearings." Journal of Mechanical Design and Vibration 2.1 (2014): 11-24.

Meshram, A. B. , Mehta, G. D. , & Modak, J. P. (2014). Estimation of Natural Frequencies and Mode Shapes of a Shaft Supported by more than Three Bearings. Journal of Mechanical Design and Vibration, 2(1), 11-24.

Meshram, Atul B., Girish D. Mehta, and Jayant P. Modak. "Estimation of Natural Frequencies and Mode Shapes of a Shaft Supported by more than Three Bearings." Journal of Mechanical Design and Vibration 2, no. 1 (2014): 11-24.

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1. Introduction

Rotor dynamics have mesmerized the intellect since medieval times. In the past era rotors were most often used at slower speed for the applications like moving a bullock cart and running of oil extraction units. However, as the time passed, the interest of human in the rotors aroused and different attempts were made to understand the dynamics of rotors in the past 300 years ^[10]. With the advent of modern technology, it has become much easier and less time consuming to analyze the behavior of a rotor system with the help of advanced software’s. The natural frequency is an inherent property of a rotating shaft and is a major entity in determining shaft’s stability in dynamic conditions. In industrial applications, it becomes inevitable to transfer the power of a motor to the end user through various mechanical components. These mechanical components could be pulleys, gears, chains, couplings etc. The shafts may be a driving shaft, a counter shaft or a driven shaft. These shafts handle the time varying loads which ultimately act on the machine components. The bearing elements help the shaft to bear such loads. Generally it is seen that, the shafts are placed on two or three bearings but in rare cases it is supported by more than three bearings. Such shafts are referred as a counter shaft (intermediate shaft) which bears the excessive load and becomes a node member. Hence it is required to investigate its behavior under different loading condition. Also, it is essential to find its natural frequencies and mode shapes. In this regards, a countershaft of already developed experimental set-up is selected and by using analytical and software based approaches the results are obtained.

2. Description of already Developed Experimental Setup

The detailed construction of the experimental setup ^[7], which is shown in Figure 1 is given below, A 0.5 HP motor is used to feed the power to various machine components. In this experimental setup a V-Belt drive is used for the first stage speed reduction. For this stage the velocity ratio is taken as 2.4. A smaller pulley P1 is keyed on the motor shaft S1, whereas, a bigger pulley P2 is placed on Shaft S2. This shaft S2 is placed on two antifriction bearings B1 and B2. The second stage speed reduction is with velocity ratio as 2 and it is accomplished with gear pair PI2GE2. Pinion PI2 is placed on the same shaft, where the bigger pulley P2 is placed. The GE2 is placed on shaft S4. The third stage speed reduction is accomplished by the gear pairs PI1GE1 and PI3GE3. The velocity ratio of this reduction is 2. Pinion PI1 and PI3 are placed on shafts S3 and S5 respectively, and Gear GE1 and GE3 are placed on shaft S6 and S7. A unique assembly is known as counter shaft which is formed by connecting the three shafts S3, S4 and S5 with the help of two rigid couplings CP1 and CP2. The six different antifriction bearings B3, B4, B5, B6, B7 and B8 withstand the loads and forces acting on the countershaft. The load drums BRD1, BRD2 are placed on shafts S6 and S7, where already GE1 and GE3 are placed. These shafts are supported by antifriction bearings B9, B10, B11 and B12. The antifriction bearings B1 to B12 all are fixed with the help of the pedestal to the frame. A frame is constructed with steel angles of size 30 mm X 30 mm X 5 mm and 50 mm x 50 mm x 5 mm. These steel angles are joined by welding process. This complete assembly of experimental set-up is then fixed to the foundation by using four foundation bolts.

Figure 1. Pictorial and schematic view of experimental setup

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