# Machine Tool Vibrations And Cutting Dynamics Pdf

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- Machine Tool Vibrations and Cutting Dynamics

## induction mechanism used in milling machines pdf

Unidad Aguascalientes. Circuito Aguascalientes Nte. Luis Pasteur Sur 36, Col. Chatter is a condition of instability that limits productivity of machining processes. This phenomenon was classified as a self-excited vibration problem; therefore, it has been studied under linear and nonlinear approaches. Even though regeneration theory and linear time delay models are the most widely accepted explanation of chatter, nonlinear effects of the process are disregarded.

The nonlinear effects are characterized by the presence of limit cycles, the jump phenomenon, subcritical Hopf and period doubling bifurcations. Experimental results showed that nonlinear behavior can be represented by both structural and regenerative nonlinear terms; even though characterization of these nonlinear terms is under discussion. In this work, a review of the most outstanding models based on both linear and nonlinear approaches is presented.

Additionally, predictive analysis of chatter for typical machining operations is also performed. It can be seen that a linear analysis is enough to obtain a good approach of stability conditions; however, nonlinear analysis is necessary to enhance productivity near unstable machining conditions. Keywords: Chatter, stability analysis, nonlinear dynamics, chaos, bifurcations. Chatter was first identified as a limitation of productivity, by Taylor ; however, theoretical explanations for chatter generation were proposed afterward, such as negative damping, by Arnold ; the theory of regeneration of chip thickness, by Tobias ; structural dynamics, by Tlusty and Merrit ; dry friction and modal coupling, by Wu and Liu a, b.

The regeneration theory is still the most comprehensive explanation for chatter, where the destabilizing term was introduced in the cutting force as a function of the current and the previous cut. Tobias concluded that instability was originated by the auto-excitation of the system from the dynamic variation of chip thickness, because a static chip thickness would not cause instability.

It was Merrit who classified chatter as a kind of self-excited vibrations and presented stability charts in terms of process parameters, such as depth of cut and spindle speed. A first effort on modeling nonlinear behavior of chatter was performed by Hanna and Tobias , who proposed a model with square and cubic terms to represent both structural and regenerative nonlinearities; even though the process of solution was limited, nonlinear behavior as finite amplitude vibrations and the jump phenomenon was found.

More recently, Altintas developed a comprehensive technology for predictive analysis, based on the regenerative theory and supported by modal and frequency measurements. He applied this method for turning, milling and drilling. In addition, Altintas et al. Altintas and Budak , a, b proposed a two-degree of freedom linear model for the cutting tool in milling, where the cutting force was proportional to the chip section and depended on the instantaneous immersion angle of the j-th tooth on the rotating tool, geometry of tool and number of teeth.

The loss of contact between the j-th tooth and the workpiece was modeled by a unit step function; however, this approach is valid for full or half-immersed problems, but not for highly interrupted cutting. Dry friction was identified as a main source of chatter due to its velocity-dependent nature Wu and Liu; a, b.

An extensive research on sources of chatter and theoretical modeling was performed by Wiercigroch and Budak They identified additional causes of instability that confer the chatter phenomenon a highly nonlinear nature. Among these causes were: hardening and softening by deformation, thermal softening, dependence of deformation rate, variable friction, heat generation and conduction, hysteretic behavior for feed, intermittent cutting of tool, and time delay. Likewise, Wiercigroch and Krivtsov found a chaotic behavior due to the effect of dry friction in orthogonal metal cutting.

Tlusty found an experimental power law for the cutting force in terms of the chip thickness, called the three-quarter rule, attributed to the friction between the cutting tool and the workpiece during the cutting process. As a result, period doubling flip bifurcations were found.

Even though the power-law function was reached from experimental data, its work lacks of theoretical validation. Insperger et al. These authors found that variation of spindle speed was highly effective in eliminating chatter at low speeds; whereas improvement was not significant at high speed. Flexibility in the cutting tool has also been taken into account; thus, the effect of compliance between the cutting tool and the work-piece was investigated by Bravo et al. In summary, regenerative chatter is a highly nonlinear vibration problem with chaotic behavior; multiple sources of high nonlinearity are identified and complex models are in good agreement with experimental results.

However, neither a unifying model nor an exact solution has been proposed yet. In this work, predictive analyses for typical machining processes are performed by using both linear and nonlinear approaches.

From the obtained results, nonlinear behavior can be attributed to structural nonlinearities. Hence, future work to model nonlinearities from both the cutting force and the structural stiffness is under preparation, such that a better understanding of the phenomenon will be obtained. Modeling of Self-Excited Vibrations. Regeneration theory and linear stability analysis. The regeneration theory was proposed by Tobias , and it establishes that the onset of chatter is due to an excitation of one of the mode shapes of the system, such that vibration causes a wavy surface on the workpiece and chip thickness varies at the subsequent cutting pass.

Thus, current vibrations depend of both dynamics of the system and vibrations from the previous period of revolution. In a condition of severe chatter, vibrations can grow exponentially that the cutting tool can loose contact with the workpiece due to large amplitude vibrations.

The first approach to represent the dynamics of the process of metal cutting was also proposed by Tobias through a single degree of freedom model, as shown in Figure 1 for a turning process. The cutting process originates a motion in the cutting tool, defined by x t , such that the equation of motion for the cutting tool is defined as follows:.

According to the theory of regeneration, the dynamic chip thickness is given by the following expression:. If the Merchant model is used for the cutting force Altintas, , Equation 1 becomes:. A stability analysis of Equation 3 can be seen in Altintas Equation 3 is carried to the frequency domain by the Laplace transform with initial conditions set to zero. Afterward, from an expression of the chip thickness in frequency domain, the characteristic equation of the system is obtained by considering a pure imaginary root as condition of critical stability.

Finally, the system will perform a chatter-free condition whenever the depth of cut is less than the critical depth of cut, which is given by:. From Equations 4 and 5 it can be seen that the minimal depth of cut is obtained when G is minimum. Then, G keeps decaying and reaches a minimum, after that, it starts growing slow and asymptotically to the horizontal axis.

Thus, chatter frequency can be calculated as the frequency ratio, r, ranges from 1 to 1. In addition, the spindle speed associated to the critical depth of cut is calculated as follows:.

Even though any structure has an infinite number of modes of vibration, chatter analysis is usually performed only around the first mode of vibration, because it is the most flexible. Then, stability lobes from the second or third mode would appear above the corresponding to them of the first mode. Sometimes, any intersection between the upper part of the first mode lobes and the lower part of the second mode lobes occur such that first mode lobes look incomplete.

For the purpose of this analysis, only the first mode of vibration will be taken into consideration. On the other hand, the model of Budak and Altintas a, b for milling was more refined since the cutting force was time and directional dependant, as well as loss of contact between the j-th tooth and the workpiece was modeled by a unit step function, such that it took a value of 1 during contact time and zero otherwise.

Thus, equations of motion, which represents the cutting phenomenon from Figure 2 , are expressed as follows:. Kt and Kr represent the cutting coefficient in thrust and radial direction, respectively. Periodic terms of A t are calculated as a Fourier series:. Summing the cutting forces contributed by all teeth, matrix A t can be represented by time-dependant directional dynamic milling force coefficients:. The number of harmonics, r, of the tooth passing frequency to be considered is reflected in the accuracy of A t , which depends on both the immersion conditions and the number of engaged teeth.

Hence, Equation 10 becomes:. Linear stability analysis of Equation 8 was performed in order to obtain the characteristic equation in frequency domain, such that the critical depth of cut for a chatter-free machining was given by:. Additionally, the spindle speed can be calculated by Equation 6 ; however, the phase shift of the system for milling is obtained as follows:. This model is useful to predict stability for a milling process where the cutting tool is totally immersed or half-immersed into the material.

However, when the width of cut is very small, the milling force is intermittent such that the average coefficients from the matrix A t are limited for the analysis. Low immersion and intermittency of cutting tool is analyzed in the next Section. Nonlinearities from the structure and the cutting force. The cutting force F is approximated by a Taylor series respect to h, such that stability conditions can be represented by the following equations:.

Even though, a theoretical validation of coefficients of the Taylor series was not presented. The advantage of this method is that only the delayed terms are discretized while the actual domain terms are unchanged, in contrast to full discretization techniques; thus, a finite dimensional discrete map approximation of the delayed differential equation is constructed. Consequently, the stability of the system depended on the nature of the eigenvalues of the Floquet transition matrix.

Modeling of a 1 degree of freedom milling process is described below:. The discretization scheme proposed by Insperger et al. A series of integers mi, which can be considered as an approximation parameter of the length of the time delay, is introduced:.

If ti is denoted by i, Equation 19 can be approximated in the i-th interval as:. The solution of Equation 19 for the initial condition. The discrete map of Equation 26 can be expressed as :.

In this Section, stability analyses for turning, as well as full and partial immersion milling are performed by applying the predictive models described in the last Section. Both linear and nonlinear models are tailored to the machining process through the dynamic parameters of the system, which can be obtained from the impact testing, as mentioned above.

The cutting coefficient can be obtained according to the material of the workpiece and geometry in terms of the immersion of the cutting tool. A linear stability analysis of the turning process is shown in Figure 3a , the stability chart is constructed with Equations 4 and 6 , where the chatter frequency is swept around the natural frequency of the system. On the other hand, limits of stability from the nonlinear analysis are constructed with Equations 15 and 16 and shown in Figure 3b.

A general agreement between stable and unstable zones is obtained, even though some discrepancies are found. An unconditionally stable depth of cut of 0. On the other hand, nonlinear lobes look displaced respect to the linear lobes because shift phase between the cutting tool and the workpiece is taken into consideration in the linear analysis.

If this shift phase is disregarded, both nonlinear and linear lobes are totally in phase. The critical depth of cut for full immersion is 0. In fact, the stability borderline is also raised at the peaks of lobes for half-immersion process. Even though the discretized map approximation of Equation 27 was derived for the delayed single degree of freedom Equation 17 , these results can be used to show the effect of low immersion milling.

Good agreement with linear stability, in the arrangement of lobes along the spindle speed axis is found; however, stability is dropped when nonlinearities are taken into account; the critical and the maximum depth of cut at every lobe are smaller than in the linear analysis, whether for full or half-immersion process.

On the other hand, stability is enhanced as the immersion rate is 0. However, an interesting result is obtained when the immersion rate is 0. From the preceding results, it can be summarized that linear stability analysis shows good accuracy for continuous turning, as well as full or half immersion milling. Models based on the power law-function showed nonlinear behavior as measured experimentally, even though theoretical validation of the nonlinear terms was not presented.

## Machine Tool Vibrations and Cutting Dynamics ||

Machine Tool Vibrations and Cutting Dynamics covers the fundamentals of cutting dynamics from the perspective of discontinuous systems theory. It shows the reader how to use coupling, interaction, and different cutting states to mitigate machining instability and enable better machine tool design. In addition, this volume also:. Machine Tool Vibrations and Cutting Dynamics is an ideal volume for researchers and engineers working in mechanical engineering and engineering systems. Skip to main content Skip to table of contents. Advertisement Hide. This service is more advanced with JavaScript available.

Gegg, C. It shows the reader how to use coupling, interaction, and different cutting states to mitigate machining instability and enable better machine tool design. Among the topics discussed are- underlying dynamics of cutting and interruptions in cutting motions- the operation of the machine-tool systems over a broad range of operating conditions with minimal vibration and the need for high precision, high yield micro- and nano-machining. Buy Premium To. High Lonesome New and Selected Stories Peace I Virations taking a philosophy class in my university and my professor is teaching us o. Lori Evert book.

Cutting Tools for Turning and Milling Operations. A lathe is a machine tool used for turning or facing opera- tions. In these.. The clamping method used to firmly keep the insert in the toolholder. The mechanisms of noise generation depend on the particularly noisy pressing and shearing, lathes, milling machines and grinders, as well as textile In many countries, noise-induced hearing loss is one of the examples of the most common machines used in the work environment.

## Machine Tool Vibrations and Cutting Dynamics

Brandon C. GeggDynacon Inc. Albert C. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.

*Abakumov, Y. Vidmanov, and V.*

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