**3. Numerical tests and experimental results**

## **3.1. DSM results**

When the spindle was modeled using simply supported boundary conditions at the bearing locations, the fundamental natural frequency of the system was found to be just below 1400 Hz, i.e., higher than the nominal value provided by the manufacturer. The boundary condi‐ tions were then updated and the simple supports (bearings) were replaced by spring ele‐ ments. The new spring-supported model was then updated/calibrated to achieve the spindle's nominal fundamental natural frequency by varying spring stiffness values, *Ks*, all assumed to be identical. It was observed that as the spring stiffness value increase the natural frequency of the system increases. The natural frequency then levels out and reaches an asymptote as the springs start behaving more like simple supports at high values of spring stiffness. It was also found that at spring stiffness value of *Ks* =2.1×10<sup>8</sup> N/m the system achieves the natural frequency reported by the spindle manufacturer. This value of spring stiffness will be used for any further analysis of the system. These results are shown in Figure 7.

Using the above results the natural frequency of the spindle was also found for multiple ro‐ tational speeds (Figure 8). It was observed that, as expected, as the spindle rotation speed increases the natural frequency of the system decreases. It was also found that the spindle critical spindle speed is 2.3×10<sup>6</sup> RPM which is well above the operating rotational speed of the spindle, i.e., 3 *.*5×10<sup>4</sup> RPM.

**Figure 7.** System Natural Frequency vs. Bearing Equivalent Spring Constant (in log scale).

**Figure 8.** Spindle Natural Frequency vs. Spindle RPM.

It is assumed that the entire system is made from the same material and the properties of tooling steel were used for all section. It was also assumed that the system is simply sup‐ ported at the locations of the bearings. The simply supported boundary conditions were then modified and replaced by linear spring elements (Figure 6); the spring stiffness values were varied in an attempt to achieve a fundamental frequency equivalent to the spindle sys‐

tem's natural frequency reported by the manufacturer.

80 Advances in Vibration Engineering and Structural Dynamics

**Figure 6.** Spindle model, with bearings modeled as linear spring elements(modified BC).

When the spindle was modeled using simply supported boundary conditions at the bearing locations, the fundamental natural frequency of the system was found to be just below 1400 Hz, i.e., higher than the nominal value provided by the manufacturer. The boundary condi‐ tions were then updated and the simple supports (bearings) were replaced by spring ele‐ ments. The new spring-supported model was then updated/calibrated to achieve the spindle's nominal fundamental natural frequency by varying spring stiffness values, *Ks*, all assumed to be identical. It was observed that as the spring stiffness value increase the natural frequency of the system increases. The natural frequency then levels out and reaches an asymptote as the springs start behaving more like simple supports at high values of spring stiffness. It was

frequency reported by the spindle manufacturer. This value of spring stiffness will be used

Using the above results the natural frequency of the spindle was also found for multiple ro‐ tational speeds (Figure 8). It was observed that, as expected, as the spindle rotation speed increases the natural frequency of the system decreases. It was also found that the spindle

for any further analysis of the system. These results are shown in Figure 7.

N/m the system achieves the natural

**3. Numerical tests and experimental results**

also found that at spring stiffness value of *Ks* =2.1×10<sup>8</sup>

**3.1. DSM results**

## **3.2. Preliminary Experimental results**

The experimentally evaluated Frequency Response Function (FRF) data were collected for a machine over the period of twelve months. A 1-inch diameter blank tool with a 2-inch pro‐ trusion was used. A typical shrink fit tool holder was also used (See Figure 9). This type of holder was selected for its rigid contact surface with the tool. Therefore, any play in the whole system was going to be attributed to the spindle. The tested machine was used to pro‐ duce typical machined parts and was not restricted to one type of cut. This was done to ob‐ serve the spindle decay over time while operating under normal production conditions. The tool was placed in the spindle and the spindle was returned to its neutral position as shown in Figure 9. Acceleration transducers were placed in both the *X* and *Y* direction. The tool was struck with an impulse hammer in both the *X* and *Y* directions and corresponding bending natural frequencies were evaluated over the time. Figure 10 shows the bending nat‐ ural frequencies of the non-spinning spindle vs. machine hours. As can be seen, system nat‐ ural frequencies in both *X* and *Y* directions reduce with spindle's life, which can be attributed to bearings decay. Further reseatrch is underway to analyze more spindles and to model the system decay by establishing a relationship between bearings stiffness, *Ks*, and machine hours. This, in turn, can be used to predict the optimum machining parameters as a function of spindle age.

**4. Conclusion**

themachine Chatter.

**Author details**

Seyed M. Hashemi\*

**References**

**Acknowledgements**

The effects of spindle system's vibrational behavior on the stability lobes, and as a result on the Chatter behavior of machine tools have been established. It has been observed that the service life changes the vibrational behavior of spindles, i.e., reduced natural frequency over the time. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation and exact within the limits of the Euler-Bernoulli beam bending theory, was developed. The beam exhibits coupled Bending-Bending (B-B) vibra‐ tion and, as expected, its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model using two different models; rigid, simplysupported,frictionless pins and flexible linear spring elements. The linear spring element stiffness, *Ks*, was then calibrated so that the fundamental frequency of the system matched the nominal data provided by the manufacturer. This step is vital to the next phase of the authors' ongoing research, where the bearing wear would be modeled in terms of spindle's service time/age, to investigate the consequent effects on the stability lobes and, in turn, on

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

http://dx.doi.org/10.5772/51174

83

The authors wish to acknowledge the support provided by Ryerson University and Natural

[1] Adetoro, O. B. , Wen, P. H., Sim, W. M., & Vepa, R. (2009, 1-3 July). Stability Lobes Prediction in Thin Wall Machining. Paper presented at World Congress on Engineer‐

[2] Altintas, Y., & Budak, E. (1995). Analytical Prediction of Stability Lobes in Milling.

[3] Arnold, R. N. (1946). Discussion on the Mechanism of Tool Vibration in the Cutting

Science and Engineering Research Council of Canada (NSERC).

and Omar Gaber

\*Address all correspondence to: smhashem@ryerson.ca

Department of Aerospace Eng., Ryerson University, Toronto (ON), Canada

ing, WCE 2009, London, U.K. 520-525, 978-9-88170-125-1.

*CIRP Annals- Manufacturing Technology*, 44(1), 357-362, 0007-8506.

of Steel. *Proceedings of the Institution of Mechanical Engineers*, 154, 429-432.

**Figure 9.** Blank Tool (left) and Blank Tool in Spindle (middle and right).

**Figure 10.** Natural Frequency vs. Machine Hours.
