Literature Review -150115
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Machining processes involve removal of material from workpieces whatever whichmechanism material removal is used. Then machining as a manufacturing processis evaluated by two main aspects; productivity and product quality. Usually theproductivity of machining processes is measured by material removal rate (MRR),and product quality is measured by surface roughness.
To achieve these main claims of machining, there are many constrains. Oneof the major constrains is mechanical vibrations that affects the whole MFTW(Machine-Fixture-Tool-Workpiece) system.
2 Machine tool vibrations
Vibrations in metal cutting can be classified into three main categeories; freevibrations, forced vibrations and self-excited vibrations  .The three types ofvibrations can be very obviously differentiated with respect to the equation ofmotion represented as follows
mx+ cx+ kx = F
In case of having zero external forces (F = 0) provided a damped structure(c > 0), then free vibrations occur and diminish in short time due to the dampingdissipation of energy. In case of having an external force (F 6= 0) while theoverall damping is positive (c > 0) forced vibrations occur, which have an ampli-tude according to the force amplitude and frequency equal to the force frequency.Finally, in case of having negative damping (c < 0) an exponential increase in thevibration amplitude occurs which may lead to high damaging results and this iscalled self-excited vibrations.
Self-excited vibrations in machining known as chatter is the most damagingwhile being the least controllable. Chatter in machining is caused by the inter-action between the MFTW system and the cutting process, this interaction givesnegative damping to the vibrating MFTW system and therefore causes high am-plitudes and alot of negative effects.
Effects of chatter phenomenon on the cutting process have its impacts on theworkpiece shown by having poor surface quality and inaccurate dimensions. Alsoits effects on the tool can increase till its total damage, this can be extended alsoto any element in the machine tool . Chatter and its effect can be avoided byhaving small depth of cut which therefore decreases the material removal rate and
thus decreasing productivity. It can be concluded then, that chatter is an obviouslimitation to the machining processes productivity and thats why it is studiedextensively through a whole century.
Although the other machining processes such as milling, turning and drillinghave been studied relatively broader and deeper, there were only a few attemptsto model the cutting forces and stability in boring .
The enlargement of holes is achieved via boring operations. The hole diameteris either enlarged with a single insert attached to a long boring bar, or with aboring head which has a diameter equal to the diameter of the hole to be enlarged.Long boring bars statically and dynamically deform under the cutting forces dur-ing boring operations. Excessive static deflections may violate the dimensionaltolerance of the hole, and vibrations may lead to poor surface, short tool life andchipping of the tool.
The problem of vibration becomes more significant when a flexible tool isused,as in the case of internal turning operations. . Boring bars have gener-ally high length to diameter ratio in order to generate internal surfaces.Thatswhy boring process is a very specific case for machine tool chatter that need amore comprehensive work.
Due to the insufficient rigidity of boring bars, chatter is more likely to occurin boring than in any other machining operation, which results in a poor surfacequality, shorter tool life and limited production rate. Extensive investigations havebeen carried out to avoid chatter vibrations. Several types of vibration dampershave been suggested by previous investigators. However, due to the complexity,high expenses, and size limitations of such dampers they have found only lim-ited practical applications. More satisfactory results can still be attained by anadequate selection of cutting conditions as was proved by previous investigationscarried out into chatter in turning. 
Research in machine tool vibrations has involved two main paths. The firstone involved the investigation of the chatter behaviour itself and the parametersaffecting it. The second one involved researching different methods for suppressing,avoiding or eliminitating chatter.
This chapter summarizes the literature in the context of a) vibrations thatexist in machine tools and its effects on the metal cutting process, b) the chatterbehaviour and the theories regarding its explanation and modeling, and c) thesuppression methods in the case of boring bars.
3 Modeling of machine tool chatter
The first observation of chatter was in 1907 by F.Taylor stating chatter behaviourand its negative effects, he also tried to explain these vibrations by variable periodic
shearing forces as the metal was removed. The first thorough investigation ofchatter return back to 1945,when Arnold studied the chatter vibrations in thedirection of cutting speed. Due to the negative slope of variation between cuttingforces and the relative cutting speed, the vibrating tool may undergo dynamicinstability which leads to chatter vibrations. 
Tobias discussed in the 1960s the chatter behaviour and its modeling. Heexplained the initiation of the machine tool self excited vibrations to be due toany disturbance to the MFTW dynamic system such as material hard spots. Hereturned the cause of the dynamic instability to the regeneratiion effect.
The regenerative effect occurs due to the chip thickness variation when thecutting edge of the tool traverses a surface on the workpiece that experienced aprevious cut. When overlapping occurs between the previous undulations on theworkpiece and the current cut undulations, regeneration effects takes place whichmay increase the amplitudes exponentially with respect to time. 
Tobias and Fishwick modeled the chatter behaviour by putting the equation ofmotion of the whole MFTW system. They introduced variable acting forces thatare function of vibration velocity and displacement, so in case of having negativedamping (negative coefficient of velocity) present ,dynamic instability occurs. 
The stabilty measure set in Tobias and Fishwick research is considered theeffective amplification factor Qe which represents the damping effect on the struc-ture, therefore minimum Qe at a specific rotational speed means more tendency tochatter and vice versa . The effective amplification factor for any single degreeof freedom system can be presented as
where is the damping ratioThis means that the presented system needs a higher damping or a lower Qe to
achieve stability, therefore the stability of the system can be measured by plottingthe effective amplification factor against the rotational speed. The plotted stabil-ity chart would have successive lobes that have some local minima at certainrotational speed called the stability lobes.
Several authors researched the stability borderline of chatter thoroughly in the60s, Tlusty  considered a single degree of freedom system for the MFTW systemsubjected to cutting dynamics, his analysis is done -unlike Tobias- in the complexdomain. Through this analysis, he could present a mathematical model for thechatter stability borderline by taking the widthof cut as a measure of stability.The presented model is as simple as follows
where k1 is the cutting stiffness that represents the relation between the chipthickness and the cutting force, while G() is the real part of the MFTW systemtransfer function.
Tlusty and Polacek also presented another theory for chatter which is themode-coupling, this means that the system has a higher tendency for chatter incase of have more than one degree of freedom .
Several models representing regenerative chatter are put to identify the chatterborderline of stability. Tobias represented some method for determining the sta-bility border line using analytical methods  and graphical methods . AlsoMerrit  modeled the chatter phenomenon using the feedback control theory.He modeled the interdependance of cutting dynamics and structural dynamics asa closed loop that has an input of a given uncut chip thickness and an ouptut ofthe actual chip thickness.
Nigm  criticized Merritts graphical method to determine the stability bor-derline. He pointed to the limitation of that method in not accounting for the metalcutting dynamics, also he pointed to the complexity of using it due to the needof using a specially prepared chart before plotting the transfer function. Instead,he proposed both a graphical method and a corresponding analytical method thataccounts for the dynamics of the metal cutting process. This method uses also thefeedback control theory but having much more simpler approach graphically, anda very simple analytical solution for the stability borderline equation.
Shi and Tobias  further investigated the possible causes of chatter. Thispaper represents the finite amplitude instability theory, which puts the non-linear forces (if present) as an initiation cause for the chatter behaviour providedthat the width of cut is within a specific range. Non-linear forces existence isexplained by large hammer force or interrupted cutting. This work concludes thatbelow a certain limit of the width of cut no chatter occurs even if non-linear forcesare subjected to the system, while above a certain limit of the width of cut chatteroccurs even if no non-linear forces existed. Finally, it concludes that within theupper and lower limites of the width of cut