modeling and process analysis of free and forced ......the machine tool vibrations are classified as...

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International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 5, September - October 2018, pp. 37–55, Article ID: IJARET_09_05_005

Available online at http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=9&IType=5

ISSN Print: 0976-6480 and ISSN Online: 0976-6499

© IAEME Publication

MODELING AND PROCESS ANALYSIS OF

FREE AND FORCED VIBRATIONS IN MILLING

VVSH Prasad

Associate Professor, Department of Mechanical Engineering,

Institute of Aeronautical Engineering, Hyderabad

Dr. V. Kamala

Retd. Deputy General Manager

Bharath Heavy Electrical Limited- Corporate R&D Division, Hyderabad

ABSTRACT

Milling is an important machining operation for production of prismatic

components using multipoint cutting tools both in vertical and horizontal spindle

orientations at low speeds to very high speeds. These cutting tool edges progressively

penetrate into work piece producing harmonic forces. The cutting operation is

modeled as forced vibration system where as idle running of machine as free

vibration. Unbalanced forces (belt tensions) in Main spindle drive and gear forces

lead to set up natural vibrations in idle running of the machine tool. In this paper, free

and forced vibrations study is conducted experimentally in complex milling operations

to characterize process parameters. The machine tool-work piece system behaves like

a damped spring mass system. Amplitude and velocity measurement are done using

vibrometer during machining. Further Matlab code is developed to generate process

parameter response plots in 2D and 3D envelop for the amplitude of vibrations in time

domain for universal milling machine to identify sudden changes during metal

removal at entry and exit of cutting tool on to the work piece.

Key words: free and forced vibrations, milling, Matlab.

Cite this Article: VVSH Prasad and Dr. V. Kamala, Modeling and Process Analysis

of Free and Forced Vibrations in Milling. International Journal of Advanced Research

in Engineering and Technology, 9(5), 2018, pp 37–55.

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=9&IType=5

1. INTRODUCTION

The main consideration of vibration analysis is to find the natural frequency of vibration,

which is one of the characteristic frequencies of vibration of a body when it is under free

vibration. This can be further extended in the case of machine tools as forced vibration

analysis when there is cutting motion. If the excitation frequency is very near to the natural

frequency, the amplitude of vibration will be excessively large which readily leads to failure

of machine members due to resonance. In general to limit the maximum stress within the

proportionality limit, the displacements must be small. Similarly, if the displacements are not

Modeling and Process Analysis of Free and Forced Vibrations in Milling


small, the equations involved become non-linear and solutions become complex. When a

machine tool structure is subjected to a periodic force, it will vibrate at the forcing frequency.

There will be resonance when angular frequency of the external harmonic force equals to

natural angular frequency of the structure. The machine tool & work piece system behaves

like a damped spring mass system. In order to minimize the amplitude of vibration of the

system, the damping should be large and natural frequency of the system should be

significantly less than the frequency of the disturbing force. Forced vibrations are caused by

period forces within the system.

The machine tool is subjected to cyclic forced vibrations during face milling or slab

milling [1]. The frequency of forced vibration will be equal to the product of the tool angular

frequency and number of teeth on the cutter. The spindle torque variation during a slab

milling operation will dependent upon the number of teeth. The frequency of torque variations

increases and the peak torque decreases with the increase of number of teeth. The machine

tool should be designed so that the natural frequencies of the various parts of the structure do

not approach the forcing frequencies. The harmonics of the periodic forces should also be

avoided to safeguard against resonance. In a universal milling machine the important modes

of vibration are those which cause relative displacement of tool with respect to work piece.

This will deteriorate the surface finish of work piece.

2. MACHINE TOOL VIBRATIONS

The work piece, cutting tool and machine form a structure system with complicated dynamic

characteristics. The cutting tool applies force on the work piece during machining. The cutting

tool receives its cutting forces from the machine. Under certain operating conditions, the

structural system may experience vibrations. The presence of vibration results in poor surface

finish, cutting tool edge damage and irritating noise. It is therefore very important to study the

cause and control all types of free and forced vibrations due to interaction between the cutting

process and the machine tool structure.

Machine tools are complex structures consisting of beds, columns, gears, shafts, slides,

cutting tools, work pieces, etc. These are distributed mass system with infinite number of

mass points and therefore infinite degrees of freedom. The cutting forces can be resolved into

steady or constant component and time dependent dynamic components. The steady

component of cutting forces along with dead loads can cause static deflections in the elastic

work piece- tool- machine system. These deflections disturb orientation and motion of tool

relative to work piece. The point of application of cutting load changes with time and hence

the deflection. The machine tool must have static stiffness to resist the constant loads of the

combined system

The machine tool vibrations are classified as

1. Free vibrations or transient vibrations or random vibrations.

2. Forced vibrations.

3. Self-excited vibrations or machine tool chatter.

2.1. Forced Vibrations

These are caused by periodic force within the system. The sources of such forces are:

(i) Unbalanced rotating masses

(ii) Intermittent engagement of multi point cutters like milling

When a machine tool structure is subjected to a periodic force, it will vibrate at the forcing

frequency. There will be resonance when angular frequency of the external harmonic force

equals the natural angular frequency of the structure. The machine tool work piece system

VVSH Prasad and Dr. V. Kamala


behaves like a damped mass spring system. In order to minimize the amplitude of vibration of

the system, the damping should be large and natural frequency of machine should be

significantly less than the frequency of the disturbing force.

The natural frequency of the universal milling machine Model -221 (praga-gambian)

determined by using FEM technique .The major components (base, column, saddle, table ) are

assembled in series with individual stiffness using CST elements under plain strain condition

[2].

Combined stiffness K Total = 5.6 x 107 N/m, Mass of the machine= 1640 kg

Natural frequency: Angular velocity ωn = (k/m)1/2

= 184.78 rad/sec

Frequency = 29.41 Hz

(iii) The system frequency should be significantly less than the frequency of the disturbing

force.

(iv) Consider a single degree of freedom system with hysteresis damping subjected to a

harmonic force F(t)= .

(v) From free vibrations under hysteresis damping ,the damping force c=

(vi) +

+ Kx = . ( 1 )

(vii)

=

Figure 1

(viii) Mathematical modeling of forced vibrations with hysteresis damping shown in

Figure:1

(ix) The steady state solution can be written as xp(t) = X sin (wt – Ø)

(x) Substituting in equation (1),

The amplitude of vibration

√

(2)

3. EXPERIMENTATION WITH VIBROMETER –MODEL MV310:

The machine is operated at different speeds with spindle rotation to measure amplitude of free

vibration X (t) using eqn. (2) (Here F0 is the unbalanced force component found out by belt

tension).

The machine is operated under machining/milling operation.



√

(Here the cutting force fx also adds into F0 component).This amplitude

value gives the amount of displacement of machine during forced vibration under machining.

3.1. Free Vibrations

3.1.1. Vertical Milling

Considering the successive amplitudes at two successive speeds of the machine,

The logarithmic ratio of the system at successive amplitudes is:

(

)

√

√ = √

√ = 606102.3

it is under damped system hence free vibration response is given by

x (t) = √

√ √ } (3)

Are the initial velocity and initial displacement given to the machine (milling

machine) at the start of free vibration. It is observed that for milling machine the damping

factor can be considered as

.

Values obtained in vibrometer and theoretical are as shown in (Table 1)

Using eqn. (3), the displacement values obtained and vibrometer observations are as

follows:

Table 1 Amplitude of vibration theoretical and vibrometer measurements

S. No. Speed

‘N’ (rpm) Vibrometer Theoretical

Displacement

‘x0’(m)

velocity

dx/dt(mm/sec)

displacement

x(t) m

1 250 4x10-7

0.01 2.14x10-7

2 400 6x10-7

0.1 3.253x10-7

3 630 7x10-7

0.14 3.808x10-7

4 1000 20x10-7

0.16 10.75x10-7

5 1600 18x10-7

0.22 9.72x10-7

3.1.2. Horizontal Milling

Free horizontal vibrations as shown in Table 2 recorded using vibrometer



Table 2 Amplitude of vibration recorded using vibrometer

S. No. Speed ‘N’

(rpm)

Displacement

‘x0’(m)-vertical

Displacement

‘x0’(m)-

horizontal

1 250 4x10-7

6x10-7

2 400 6x10-7

8x10-7

3 630 7x10-7

9x10-7

4 1000 20x10-7

6x10-7

3.2. Forced Vibrations


The displacement values obtained by using vibrometer are as follows at different speeds and

different depth of cuts (Table 3). Setup is shown in figure 2.

Figure 2 Experimental setup on universal milling machine using vibrometer – MV310

Table 3 Amplitude of vibration with different process parameters

Speed

in rpm

Depth of cut is 0.5

mm

Depth of cut is 0.75

mm

Depth of cut is 1 mm

Displacement(X) m Displacement(X) m Displacement(X) m

250 12x10-7

18x10-7

10x10-7

400 6x10-7

7x10-7

20x10-7

630 6x10-7

11x10-7

0.9x10-7

1000 6x10-7

20x10-7

6x10-7

1600 20x10-7

30x10-7

9.5x10-7

Theoretical cutting force calculations: [7]

Power at the spindle (Pw) = UKhKrQ KW

Material removal rate,

⁄

U=Unit power Kw/cc/min ( from tables),Kh =correction factor for flank wear( from

tables), Kf= correction factor for rake angle(from tables),b = width of cut (mm), U=unit power

in KW/cm3/min ,. t = depth of cut (mm),Sm = feed rate mm/min



= 9.45

Pw,power at spindle = 31 x 10 -3

x 9.45 x 1.12 x 1 = 0.32

PZ=

= 1.484 x10-6

m

√

=

√

= 1.2x10-6

m

Model calculation:

√

C √

= 0.12 x√ = 36366.13

√

X = 1.2 x 10-6

x

=1.35 x 10-6

m

Summary of amplitude of forced vibration in vertical milling is shown in Table 4.

Table 4 Summary of displacement of vibration in vertical milling

Theoretical

(x) m

Vibrometer

(x) m Speed in rpm

Dia of cutter

(mm)

Vc

(m/min) cutting

15.1x10-7

12x10-7

250 30 23.56

9.7x10-7

6x10-7

400 30 37.69

6.675x10-7

6x10-7

630 30 59.37

5.45x10-7

6x10-7

1000 12 37.69

10.26x10-7

20x10-7

1600 12 60.31


Critical machining operations are carried out for three speeds as set up involves cutter with

long arbor using yoke attachment support at the other end as shown in the figure 3 and values

are shown in Table 5.

Figure 3 Experimental setup on universal milling machine for slab milling



Table 5 Amplitude of vibration theoretical and vibrometer measurements

Theoretical

(x) m

Vibrometer

(x) m Speed in rpm

Dia of cutter

(mm)

Vc

(m/min) cutting

3.2x10-7

4.04x10-7

160 95 47.75

2x10-7

2.59x10-7

250 95 74.16

1.4x10-7

1.4x10-7

400 95 119.38

4. MAT LAB PROGRAM FOR VIBRATION ANALYSIS

There are four test conditions:

1. Free vibrating condition when the spindle is vertical.

2. Forced vibrating condition when the spindle is vertical.

3. Free vibrating condition under horizontal milling.

4. Forced vibrating condition in horizontal milling.

As the system is under damped, the governing equation (iii) for free vibrating condition

both in horizontal and vertical milling operation is

x (t) = √

√ √ }

The mat lab programme for an under damped system is as follows:

% gives mass of the system

m=1640;

% gives stiffness of the system

k=56000000;

wn=sqrt(k/m);

(Initial displacement) = 4e-7

(Initial velocity) = 1e-4

Time span = 2:50 where dt= 0.06

The governing equation for forced vibrating condition is:

+

+ Kx = (which includes cutting force Pz added to unbalanced free

vibrating force)

5. RESULTS AND DISCUSSIONS

5.1. Comparative Study of Theoretical and Experimental Measurements

Figure 4 Comparison of theoretical and vibrometer displacements



The graphical representation pattern for free vibration is shown in fig 4, is following the

similar theoretical trend with structural damping as shown in fig 5 and fig 6. The variation for

vibrometer to theoretical values is attributed due to unbalanced belt tension forces, belt

slippage, gear clearances, rubbing inertial forces and worn out components.

Figure 5 Magnification factor vs frequency ratio

Figure 6 Phase angle vs frequency ratio

At 1000 rpm the amplitude of vibration has reached the maximum value and at subsequent

speed it is reduced, the peak value is 20x10-7

in vertical milling and it is 10x10

-7 in horizontal

milling. The phase angle verses frequency ratio is shown in fig 7 for theoretical observation.

Figure 7 Comparison of vertical and horizontal free vibrations



5.2. Validation of Theoretical and Experimental Results using MATLAB

5.2.1. Free Vibrations - Vertical Milling

Amplitude of vibration plots shown in figures from fig 8 to fig 13 are in agreement with table

2 data. It is observed that as the speed increases the plots are sharper close to simple harmonic

motion with time interval of 0.2 seconds. As the speed is increasing the amplitude of vibration

also increases.

Figure 8 Amplitude at 250 rpm



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-8

-6

-4

-2

0

2

4

6

8x 10

-7

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-8

-6

-4

-2

0

2

4

6

8x 10

-7

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-6

time

dis

plac

emen

t

displacement - time





Figure 13 Free vibration plots for four different speeds in vertical spindle

Vertical free vibrations plots as shown in fig 11: Yellow – at 250 rpm, Red – at 400 rpm,

Magenta – at 630 rpm Green at 1000 rpm

5.2.2. Free Vibrations - Horizontal Milling

Figures from fig 14 to fig 18 are in agreement with table 2 which indicates that in horizontal

milling the amplitude is on higher side even at low speeds. This is because of complex milling

setup as shown in fig 3 and need attention while selecting the process parameters i.e., cutting

speed, feed and depth of cut for acceptable level of machining accuracy such as T-slot and

slab milling etc.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-6

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-6

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1x 10

-5

time

dis

pla

cem

ent

displacement - time







0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1

1.5x 10

-6

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1

1.5x 10

-6

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-6

time

dis

plac

emen

t

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1

1.5x 10

-6

time

dis

plac

emen

t

displacement - time



Figure 18 Horizontal free vibrations plots

The displacement in free vibrations has gradually increased in vertical milling setup

operation from lower rpm to higher rpm where as in horizontal milling setup the free

vibrations are very high even at lower rpm’s (overhang setup)

The max displacement in vertical milling is 1*10-6m and in horizontal milling 1.5*10-6 m

which is greater amplitude in horizontal milling,(observation at 630 speed)

The critical speed phenomena adds a lot in free vibrating conditions in horizontal milling

and more over the secondary critical speed adds to it which makes higher amplitudes. The

sharp curves of displacement-time are approximated to be harmonic (SHM).

5.2.3. Forced Vibrations - Vertical Milling

The MATLAB plots for forced vibration with single degree of freedom is as follows:



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1

1.5x 10

-6

time

disp

lacem

ent

displacement - time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-6

t

x(t)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6

t

x(t)

forced vibraions



Figure 21 Amplitude at 630



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6

t

x(t)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6

t

x(t)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6

t

x(t)

forced vibraions



Figure 24 Plots for forced vibrations

Plots for forced vibrations: yellow-at 250 rpm, red –at 400 rpm, magenta –at 630 rpm

green – at 1000 rpm, blue – at 1600 rpm. All the above 2D plots are superimposed in the same

plane with different cutting forces as shown fig 24.

An interesting observation is noted that during the initial contact of tool with work piece

in the time period of 0 – 0.1 sec indicates the machine metal cutting system is subjected to

random vibrations. This is because cutting force is vectorially adding to the unbalanced free

vibrating force.

5.2.4. Forced Vibrations - Horizontal Milling

The MATLAB plots for forced vibration with single degree of freedom is as follows:


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-6

t

x(t

)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

-7

t

x(t)

forced vibraions





Figure 28 Super imposed 2D plots for forced vibrations

Super imposed 2D plots for forced vibrations: blue 160 rpm, red- 250 rpm, magenta -400 rpm.

In horizontal forced milling at the point of engagement of the cutter with work piece

which induces impulsive forces and causes random vibrations during time period of 0 – 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

4

5x 10

-7

t

x(t)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-7

t

x(t)

forced vibraions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-7

t

x(t)

forced vibraions



sec. And because of the complexity in the setup, machining operation is being carried out at 3

speeds only. Results show that the amplitude of vibration is high at the lower speeds in

comparison with vertical milling.

5.3. Beat Phenomenon Observation in Vertical and Horizontal Milling using

MATLAB

The phenomenon of beat (interference) occurs when two harmonic motions are added which

results in maximum amplitude and after a gap of certain time minimum amplitude.


Superimposed plot of free and forced vibrations in vertical milling is shown in fig 29 at 1600

rpm.

Figure 29 Phenomenon of beat in vertical milling at 1600 rpm

The curve in black color indicates free vibration whereas curve in blue indicates forced

vibration. Beats phenomenon has occurred after time peroid of 0.1 sec.


Superimposed plot of free and forced vibrations in horizontal milling is shown in fig 30 at 160

rpm.

Figure 30 Phenomenon of beat in horizontal milling at 160 rpm



The curve in black color indicates free vibration whereas curve in blue indicates forced

vibration. Beats phenomenon has occurred after time peroid of 0.1 sec.

5.4. Interpretation of Process Parameters for Milling using MATLAB 3D Plots

The amplitudes obtained (table 3) for forced vibrations from various process parameters

applied in milling were used to draw the MATLAB 3D plots and for visualization.

Figure 31 3D plot for forced vibration in horizontal milling

Figure 32 3D plot for forced vibration in vertical milling

From the graphs the process parameters can be interpreted at 0.75 mm depth of cut the

amplitude of vibration is on higher side compared to 0.5 mm and 1 mm depth of cut. If

vibration (referred to as chattering) occurs in the milling machine during the cutting process,

the speed should be reduced and the feed increased. Vibrations will appear more in up milling

when compared to down milling as there will be opposing force to the motion of tool.

Tendency of chatters and vibrations are more in up milling. For special purpose applications

using ball nose end mill for profiling aerofoil machining the amplitude of vibration in case of

continuous milling producing ruled surfaces, this experimental setup is useful for finding the

chatter phenomenon. The objective of profile milling is to produce surfaces less than 0.8 µ-m

surface roughness in wing machining.



6. CONCLUSIONS

The phenomena of vibration which measured as a factor of displacement, using

vibrometer and theoretically is in agreement which was observed from figs 4, 5, 6 & 7.

It is observed that with the change in cutting speed, the “force component” changes due to

which the amplitude of vibration changes (fig 24, 28 & table 3).

Results show that in vertical milling the cutting speed varies from 23.56 to 60.31 m/min

the amplitude of vibration increases from 18x10-7 to30 x10-7 m respectively at a depth of

cut 0.75 mm, where as in horizontal milling the cutting speed varies from 47.75 to 119.38

m/min the amplitude of vibration increases from 1.04x10-7 to 4.04 x10-7m respectively at

a depth of cut 0.75 mm as shown in tables 3&5.

There is a phenomena of critical speeds as the speed increases which makes free

vibrations more predominant than forced vibrations, which has increased gradually from

lower rpm to higher rpm which can be clearly observed from 1000rpm to 1600 rpm ie

6x10-7 to 20x10-7 in vertical milling of forced vibration from table 4..

The amplitudes of free and forced vibrations are in phase but there is some phase lag

between them. Once the speed increases the phase difference decreases and the resultant

of force increases which is observed in fig 29 and 30.This phenomena is called beat in

which amplitude records maximum and resulting chatter vibrations.

The effect of beat phenomena can be reduced by minimizing the variation of belt

tensions in the motor drive of free vibrations (estimated tight and slack tensions are

T1=128.8N,T2=36.71N respectively) .

Since viscous damping is neglected and only structural damping is considered, the

“vibrations never reaches to infinity and the value never approaches to zero” because

structural damping causes continuous vibrations of small amplitude even at free vibration

conditions. This is mainly due to the unbalanced forces (like belt tensions etc) in system

and suggested periodic condition monitoring of machine tool.

In horizontal milling, in forced vibrations at the point of engagement of cutter with the

work piece generating “impulse forces” resulting high amplitude of vibration at the

starting of cutting operation as shown in figs 25, 26, 27 & 28 in initial time period of 0.1

sec.

In the middle of cutting “chatter phenomena” is predominant which makes the peaks very

sharp where larger displacement is observed at a point and then eventually the motion

becomes “simple harmonic” as observed in figs 24&28. Thus the vibration of metal

cutting is a complex phenomena and in milling the complexity increases due to the

method of cutting

In vertical milling sharp peaks have been observed than in horizontal milling, which

shows that in vertical milling cutting action produces more vibration and in horizontal

milling spindle running that is at free vibrations there is high displacement observed in

figs 24 &18

In general free vibrations can be even considered in cutting operation dynamic phenomena

by which free running of the spindle is considered as free vibration and cutting operation

of the spindle is considered as forced vibration.

ACKNOWLEDGMENTS

The authors would like to thank the Principal of Institute of Aeronautical Engineering for his

encouragement, support and granting permission to publish this work. The authors also

express their sincere thanks to the authorities of Osmania University, Hyderabad for their

support and permission to present this work.



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[7] Machine tool design handbook by CMTI (TATA MC GRAW HILL)-2015 ISBN:978-0-07-

451564-8

[8] Mechanical vibrations by S.S.Rao Pearson Fourth Edition ISBN:978-81-7758-874-3

[9] APPLICATION OF MATLAB IN HIGH SPEED DRILLING RESEARCHESY. Karavaev, P.

ekomtsev, A. Panyukov.Izhevsk state technical university, Russia

[10] Monitoring and Analysis of Vibration Signal in Machine Tool Structures Puneet Bansal, I.

S. RajayVedaraj International Journal of Engineering Development and Research Volume 2,

Issue 2 | ISSN: 2321-9939 2014

[11] An experimental investigation into the impact of vibration on the surface roughness and its

defects of Al\l6061-t6 n. zeelanbasha, v.senthil, b. sharon Sylvester ISSN 0543-5846

METABK 57(1-2) 121-124 (2018)UDC – UDK 69.715:621.7:621.385.833:621.795=111.

modeling and process analysis of free and forced ......the machine tool vibrations are classified as...

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