Frequency response analysis of the gear box in a lathe ... · Frequency response analysis of the...

International Journal of Scientific and Research Publications, Volume 4, Issue 11, November 2014 1 ISSN 2250-3153

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Frequency response analysis of the gear box in a lathe

machine using transfer functions

V Suma Priya 1a

, B V S Raghu Vamsi2b*

, E Kavitha 1c

, K. Srividya 1d

1a

– PG Student, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada 2b* – Assistant Professor, Dept. of Mechanical Engineering, Gudlavalleru Engineering College, Gudlavalleru

1c – Assistant Professor, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada 1d – Assistant Professor, Dept. of Mechanical Engineering, PVP Siddhartha Institute of Technology, Vijayawada

Abstract- In this research work, frequency response of the gear

box in the medium duty lathe machine is studied using transfer

functions. The effect of the torque acting at a particular rotor on

the amplitude of vibration of the other rotors is studied. Initially,

equation of motion is developed for the multi-rotor system in the

gearbox and later, Laplace transforms are applied to find the

transfer functions. Torque acting at various rotors is also

calculated. The obtained characteristic equation and the transfer

functions are solved for poles and zeros (frequencies to attenuate

the inputs at every rotor) by writing programming in MATLAB.

Plots of the frequency response curves are plotted by writing the

programming in MATLAB and the final conclusions are drawn.

Index Terms- Frequency response curves, Laplace transforms,

MATLAB, Poles, and Zeros

Symbols and notations

Den – denominator of the transfer function

G – shear modulus of the shaft

J – mass moment of inertia

Jeq – equivalent mass moment of inertia for the mating gears

Kt – torsional stiffness of the shaft

l –length of the shaft

N - speed of the shaft

- torque acting at the rotor

Z(s) – Laplace transform

Z11 (num) – numerator of the transfer function (similar notation

for the other functions)

– Angular deflection -

–angular acceleration

I. INTRODUCTION

he study of Torsional vibrations in a multi-rotor system

plays an important role while designing the power

transmission systems in the machines and internal combustion

engines. The torque acting at a particular rotor will make it to

vibrate with large amplitudes and the effect of the same torque

on the vibration characteristics of the other rotors in the

transmission system is also significant. When the forcing

frequency becomes equal to the natural frequency of any of the

rotors, a state of resonance will occur. Apart from the torque

acting on the rotors, the forces developed in the mating of two

gears and forces during the operation of the corresponding

machine also amplifies the magnitude of vibration to a much

higher value. Hence, there is a need to study the effect of these

torques and additional forces on the vibration characteristics of

the entire power transmission system. In this research work, the

effect of torques acting at the different rotors on the other rotors

is studied.

Guo Rui et. al [1] developed a 10-DOF lumped parameter

model for the machine tool spindle system with geared

transmission, for the purpose of analyzing the torsional

vibrations caused by the gravitational torque arisen in a spindle

system when machining a heavy work piece. By using the

elementary method and Runge-kutte method in MATLAB, the

eigen values problem was solved and the pure torsional vibration

responses were obtained and examined. Wu Hao et. al [2]

established a numerical model of the bending stiffness of the

tapered roller bearing through mechanics and deformation

analysis. On the base of the model, a new TMM (transfer matrix

method) for bearing-rotor system was established; the new TMM

considers the influences of the bearing structure on the vibration

of the rotor system. The method is validated by analyzing the

same problem (modal analysis of air-blower) using FEM. B.B.

Maharathi et. al [3] presented a general formulation for the

problem of the steady-state unbalance response of a dual rotor

system with a flexible intershaft bearing using an ‘extended’

transfer matrix method, where the transfer matrix assumes a

dimension of (33x33) and the formulation is validated through a

computer program. M. Aleyaasin et. al [4] considered the

distributed-lumped model for the analysis of the flexural

vibrations of a rotor-bearing system and derived a general

formula for the determinant of the tri-diagonal partitioned matrix

description of the system. The obtained results are compared to

those acquired from the transfer matrix method.

HU Qinghua et. al [5] developed a five degrees of freedom

(5-DOF) model for aeroengine spindle dual-rotor system

dynamic analysis. The proposed model mathematically

formulates the nonlinear displacements, elastic deflections and

contact forces of bearings with consideration of 5-DOF and

coupling of dual rotors. The nonlinear equations of motions of

dual rotors with 5-DOF are solved using Runge-Kutta-Fehlberg

algorithm. Vishwajeet kushwaha and Prof. N.kavi [6] used finite

element methods to find the natural frequencies for different

possible cases of multi-rotor and gear-branched systems. The

various mode shapes for several cases are also shown to illustrate

the state of the system at natural frequencies. The results

obtained have been compared with Holzer’s method and transfer

matrix method to establish the effectiveness of finite element

method for such systems.

T

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International Journal of Scientific and Research Publications, Volume 4, Issue 11, November 2014 2

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In section-I, literature review is presented. In section-II,

arrangement of gears in the gear box with the calculation of

stiffness of the shaft and mass moment of inertia of the rotors is

discusses. In section-III, mathematical modeling of the rotor

system is presented. In section-IV, the procedure of the transfer

function analysis, involving the method to derive transfer

functions is presented in detail. In section-V, the results are

discussed in detail with the conclusion in section-VII.

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II. GEAR BOX ARRANGEMENT IN THE LATHE MACHINE

In this work, the gear box in a medium duty lathe machine is considered for the analysis. The frequency response of every

rotor to the torque acting at the self as well as other rotors is studied, to understand the vibration characteristics of a gear box

much better. The estimated details of the components in the gear box i.e. the stiffness of the shafts and moment of inertia of the

rotors (gears) are presented in the table 1. The arrangement of the various rotors in the gear box adopted in this study is presented

in the fig. 1. Stiffness of the shaft is calculated using the torsion equation:

tG GJ

J l R l (1)

III. MATHEMATICAL MODELING OF MULTI ROTOR SYSTEM

The spring-mass representation of the gear box is presented in fig. 2. Wherever two gears are mating, the equivalent mass

moment of inertia is calculated and it is considered to be as an equivalent rotor. The equation of motion can be derived by

Newton’s law as: ..

tJ K (2)

The equation of motion derived for the entire gear box is presented in as eq. (3) These equations are written in matrix form for

simplicity:

..

tJ K (4)

Where

J inertia matrix tk stiffness matrix

Torque matrix

Angular Acceleration matrix:

.. .. .. .. .. .. .. .. .. .. .. .. .. ..

1 2 3 4 5 6 7 8 9 10 11 12 13

T

Angular displacement matrix:

1 2 3 4 5 6 7 8 9 10 11 12 13

T


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..

11 12 1 2 1

..

22 12 2 1 23 2 3 2

..

33 23 3 2 34 3 4 3

..4 4

44 34 4 3 45 4 5 47 4 7 48 4 8 4

6 6

..

55 45 5 4 5

..

66 67 6 7 6

..

77 67 7

eq

eq

eq

eq

eq

eq

eq

J k

J k k

J k k

Z ZJ k k k k

Z Z

J k

J k

J k

4

6 47 7 4 7

6

..4

88 48 8 4 89 8 9 8

6

..10 10

99 89 9 8 910 9 10 912 9 12 913 9 13 9

12 12

..

1010 910 10 9 10

..

1111 1112 11 12 11

..

12

eq

eq

eq

eq

eq

Zk

Z

ZJ k k

Z

Z ZJ k k k k

Z Z

J k

J k

J

1012 1112 12 11 912 12 9 12

12

..10

1313 913 13 9 13

12

eq

Zk k

Z

ZJ k

Z

Torque matrix:

1 2 3 4 5 6 7 8 9 10 11 12 13

T

Inertia matrix:

1

2

3

4

5

6

7

8

9

10

11

12

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

eq

eq

eq

eq

eq

eq

eq

eq

eq

eq

eq

eq

J

J

J

J

J

J

JJ

J

J

J

J

J

130 0 0 0 0 0 0 0 0 eqJ

Stiffness matrix:

(3)


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12 12

12 12 23 23

23 23 34 34

34 45

34 45 47 48447 48

6

45 45

67 67

447 67 47 67

6

448 48 89

6

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0t

K K

K K K K

K K K K

K K

K K K KZK K

Z

K K

K K

ZK K K K

Z

K ZK K K

Z

89

89 910

89 910 912 91310912 913

12

910 910

1112 1112

10912 1112 1112 912

12

10913 913

12

0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

K

K K

K K K KZK K

Z

K K

K K

ZK K K K

Z

ZK K

Z

IV. TRANSFER FUNCTION ANALYSIS – PROCEDURE

These are the functions, which helps to study the effect of torque Ti acting at the rotor Rj. The procedure to find the transfer

functions is given in detail below:

Initially derive the expression for equation of motion for the each rotor.

Apply Laplace transform to them and convert them into s-form.

Find the transfer functions and characteristic equation for the each rotor, corresponding to each torque.

Write the program to plot the frequency response curves in MATLAB.

Step-1 presented in the above section i.e. eq. (3).

Step-2: Laplace transforms:

Applying Laplace transforms to the eq. (4), we get

2Js K z s s (5)


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Where,

2

1 12 12

2

12 2 12 23 23

2

23 3 23 34 34

2

4 34 45

34 45 47 484

47 48

6

2

45 5 45

2

6 67 67

2

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0

eq

eq

eq

eq

eq

eq

J S K K

K J S K K K

K J S K K K

J S K K

K K K KZK K

Z

K J S K

J S K K

JS K

24

47 67 7 47 67

6

24

48 8 48 89 89

6

2

9 89 910

89 910 912 91310

912 913

12

2

910 10 910

2

11 1112 1112

912

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

eq

eq

eq

eq

eq

ZK K J S K K

Z

ZK J S K K K

Z

J S K K

K K K KZK K

Z

K J S K

J S K K

K

210

1112 12 1112 912

12

210

913 13 913

12

0 0

0 0 0 0 0 0 0 0 0 0 0

eq

eq

ZK J S K K

Z

ZK J S K

Z

Z-matrix is given as 1 2 3 4 5 6 7 8 9 10 11 12 13

T

z s z s z s z s z s z s z s z s z s z s z s z s z s z s

Torque matrix is given as

22.38 22.38 22.38 22.38 22.38 70.6 70.6 70.6 70.6 70.6 52.42 52.42 52.42

T

ss s s s s s s s s s s s s


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Step-3: Deriving for transfer functions.

From

2JS K matrix, we will get transfer functions by applying below procedure and transfer functions

are arranged in a matrix as Transfer function matrix shown below.

Z11 represents the effect of torque τ1 acting at rotor 1 on the rotor 1 itself. Similarly, Z119 represents the

effect of torque τ9 acting at rotor 9 on the rotor 11.

11 12 13 14 15 16 17 18 19 110 111 112 113

21 22 23 24 25 26 27 28 29 210 211 212 213

31 32 33 34 35 36 37 38 39 310 311 312 313

41 42 43 44 45 46 47 48 49 410 411 412 413

51 52 53 54 55 56 57 58 59

z z z z z z z z z z z z z




z z z z z z z z z z510 511 512 513

61 62 63 64 65 66 67 68 69 610 611 612 613

71 72 73 74 75 76 77 78 79 710 711 712 713

81 82 83 84 85 86 87 88 89 810 811 812 813

91 92 93 94 95 96 97 98 99 910 911 912 913

101 102 103

z z z





z z z z104 105 106 107 108 109 1010 1011 1012 1013

111 112 113 114 115 116 117 118 119 1110 1111 1112 1113

121 122 123 124 125 126 127 128 129 1210 1211 1212 1213

131 132 133 134 135 136 137 138 139 1310 131

z z z z z z z z z



z z z z z z z z z z z 1 1312 1313z z

Let us consider

11

det 12 12

det 13 13Z

det 12 12= By elimination of 1

st row & 1

st column of

2JS K matrix.

12

det 12 12

det 13 13Z

det 12 12= By elimination of 1

st row & 2

nd column of

2JS K matrix.

Similar procedure is followed for the remaining transfer functions.

det 12 12= denominator = den

det 13 13= numerator = num

Sample calculations for transfer functions:

1 11

11

1

z z numZ

den

1

22.38

s

11

11

22.38z numZ

den s

45 24 32 22 21 20 10 18 16 12 14

11 11

22 12 32 10 42 8 51 6 60 4 68 2 74

22.38 2.05 1.01 4.57 8.13 68.76 2.88

6.19 6.73 3.26 4.610 2.24 2.32 7.43

Z num Z num e s e s e s e s s e s

e s e s e s e s e s e s e


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49 27 36 25 24 23 13 21 19 8 17 19 15

29 13 38 11 48 9 56 7 64 5 71 3 76

5.02 2.48 1.12 1.910 0.02 7.06 1.53

1.66 8.02 1.17 5.63 6.23 3.74 1.49

Den e s e s e s e s s e s e s

e s e s e s e s e s e s e s

Step-4: Plotting Frequency response curves

Once the transfer functions are derived, then the programming is written to plot the frequency domain

behavior of each transfer function, using MATLAB. Frequency domain behavior means identifying the

magnitude and phase characteristics of each transfer function, showing how they change as the frequency of the

forcing function is varied over a frequency range. Each transfer function is evaluated in the frequency domain

by evaluating it ass j

, where w – is the frequency of the forcing function, radians /sec. For plotting the

frequency response, different methods are available.

In this work, transfer functions are given as input, in the form of ‘num’ and ‘den’ for the each and ‘bode’

command with no left hand arguments is used by choosing the frequency range and the graphs plotting

magnitude and phase generating automatically. On x-axis, range of frequencies is taken and on the y-axis,

magnitude of vibration and phase angles are plotted. The obtained response curves are tabulated below with the

titles as their transfer functions.

V. ANALYSIS OF FREQUENCY RESPONSE CURVES

From the results it has been observed that

The peak points on the response curves indicate the poles and the valley points on the curves indicates

the zeros. The poles are the roots of the characteristic equation. They show the frequencies where the

system will amplify the inputs. All the transfer functions will have same characteristic equation i.e.

denominator (‘den’). The poles depend only on the distribution of mass and stiffness throughout the

system under analysis. But not on where the toques are applied or where the displacements are

measured. The zeros are the roots of the numerator of a given transfer function. Zeros show the

frequencies where the system will attenuate inputs. They are different for different transfer functions.

Some transfer functions may have no zeros.

In the low frequency range (< 1000 rad / sec), the response of all transfer functions ij i j

Z

,i=j=1,

2, 3 ….13, the response curve profiles are similar and presents higher gain. This is because of all the

rotors acted upon by the torque directly and the rigid body motion is falling off at the 1 rate. In that

operating range, all the rotors are running out of phase with the applied torque.

In the high frequency range (> 1000 rad / sec), the response of all transfer functions ij i j

Z

, i=j=1,

2, 3 ….13, the response curve profiles are similar and presents lesser gain. This is because of all the

rotors acted upon by the torque directly and the rigid body motion is observed to be falling off at the

31 rate. In that operating range, all the rotors are running out of phase with the applied torque.

When the effect of the torques acting on the rotors

1jR

, on the rotor R1, is considered, all the

response curves are looking alike for the transfer functions, 16 113z z. At the higher

frequencies, the rigid body motion is falling off at the 11 17

1 1

rate with different gains.

In that operating range, the rotor R1 is running in phase with some torques and out of phase

with remaining torques. At the lower frequencies, the effect of all the torques (except 1 ) is

similar and the rigid motion is falling off at the 1 rate with same gains.


,2

jR j

, on the rotor R2, is considered,

all the response curves are looking alike for the transfer functions, 26 211z z and for the

transfer functions, 212 213z z

. At the higher frequencies, the rigid body motion is falling


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off at the 9 15

1 1

rate and at the

5 91 1

rate with lesser but different gains. In

that operating range, the rotor R2 is running in phase with some torques and out of phase with

remaining torques. At the lower frequencies, the effect of all the torques (except 2 ) is similar

and the rigid motion is falling off at the 1 rate with same gains.


,3

jR j


all the response curves are looking alike for the transfer functions 31 35z z and for

36 37,z zand for 38 313z z

. At the higher frequencies, the rigid body motion is falling off at

the 5 9

1 1

rate and with the

11 131 1

rate with lesser but different gains. In that

operating range, the rotor R3 is running in phase with some torques and out of phase with

remaining torques. These many variations are because of mating of two gears, which develops

forces affecting the vibrating characteristics. This increases the amplitude of vibration also. At

the lower frequencies, the effect of all the torques (except 3 ) is similar and the rigid motion is

falling off at the 1 rate with same gains.


,4

jR j


all the response curves are looking alike for the transfer functions 41 42,z z and for

43 44,z zand for 45 48z z

, and for 49 413z z .The magnitude of vibration is high for the

transfer functions 43 44,z z , which is because of the mating of gears resulting in the

increasing of forces and amplitudes. At the higher frequencies, the rigid body motion is falling

off at the 5 9

1 1

rate with lesser but different gains. In that operating range, the rotor

R4 is running in phase with some torques and out of phase with remaining torques. At the

lower frequencies, the effect of all the torques (except3 4,

) is similar and the rigid motion

is falling off at the 1 rate with same gains.


,5

jR j


all the response curves are looking alike for the transfer functions 51 53z z and for 54 55,z z

and for 56 513z z . At the higher frequencies, the rigid body motion is falling off at the

9 131 1

rate with lesser but different gains. In that operating range, the rotor R5 is

running in phase with some torques and out of phase with remaining torques. At lower

frequencies, the effect of all the torques (except 5 ) is similar and the rigid motion is falling

off at the 1 rate with same gains.


,6

jR j



and for 68 613z z . At the higher frequencies, the rigid body motion is falling off at the

7 151 1

rate with lesser but different gains. In that operating range, the rotor R6 is


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running in phase with some torques and out of phase with remaining torques. At lower

frequencies, the effect of all the torques (except 6 ) is similar and the rigid motion is falling

off at the 1 rate with same gains.


,7

jR j



and for 76 77,z z, and for 78 713z z

. At the higher frequencies, the rigid body motion is

falling off at the 5 11

1 1

rate with lesser but different gains. In that operating range, the

rotor R7 is running in phase with some torques and out of phase with remaining torques. At

lower frequencies, the effect of all the torques (except 7 ) is similar and the rigid motion is

falling off at the 1 rate with same gains.


,8

jR j



84 87z z and for 89 813z z


off at the 5 11

1 1

rate with lesser but different gains. In that operating range, the rotor

R8 is running in phase with all the torques. At lower frequencies, the effect of all the torques

(except 8 ) is similar and the rigid motion is falling off at the 1 rate with same gains. Due

to the generation of forces developed during mating, at rotor R9, it has a significant effect in

amplifying the amplitudes at rotors R8, R9, R12, and R13.


,9

jR j



97 98z z and for 99 913z z


off at the 5 13

1 1

different rates and with lesser but different gains. In that operating

range, the rotor R9 is running in phase with all the torques. At lower frequencies, the effect of

all the torques (except 9 ) is similar and the rigid motion is falling off at the 1 rate with

same gains. Due to the generation of forces developed during mating, at rotor R9, it has a

significant effect in amplifying the amplitudes at rotors R8, R9, R12, and R13.


,10

jR j



108 109z z and for 1011 1013z z

. At the higher frequencies, the rigid body motion is

falling off at the 7 13

1 1

different rates and with lesser but different gains. In that

operating range, the rotor R10 is running in phase with all the torques. At lower frequencies,

the effect of all the torques (except 10) is similar and the rigid motion is falling off at the

1 rate with same gains.


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,11

jR j


all the response curves are looking alike for the transfer functions 111 117z z and

for 118 1113z z. At the higher frequencies, the rigid body motion is falling off at the

11 171 1

rate and at the

5 91 1

rate with different rates respectively and with

lesser but different gains. In that operating range, the rotor R11 is running in phase with some

of the torques and out of phase with the remaining torques. At lower frequencies, the effect of

all the torques (except 11) is similar and the rigid motion is falling off at the

1 rate with

same gains.


,12

jR j



for 129 1211z z and for 1213z

. At the higher frequencies, the rigid body motion is falling off

at the 7 11

1 1

and at the

5 71 1

different rates respectively and with lesser but

different gains. In that operating range, the rotor R12 is running in phase with the torques

1 8 and out of phase with the remaining torques

9 13 . At lower frequencies, the

effect of all the torques (except 12) is similar and the rigid motion is falling off at the

1

rate with same gains.


,13

jR j



for 138 1312z z and for 1313z

. At the higher frequencies, the rigid body motion is falling off

at the 9 15

1 1

and at the

5 71 1

different rates respectively and with lesser but

different gains. In that operating range, the rotor R13 is running in out of phase with the

torques 1 7

and in phase with the remaining torques 8 12

. At lower frequencies,

the effect of all the torques (except 13) is similar and the rigid motion is falling off at the

1 rate with same gains.

VI. CONCLUSION

This study reveals that there is a need to study the effect of torques acting at a particular rotor on another

rotor and the affect is significant. At lower forcing frequencies, all the rotors are exhibiting same behavior

irrespective of the magnitude of torques acting at different rotors and the rate of fall of rigid body motion is

1 . At higher forcing frequencies, rotors are exhibiting different behaviors with the fall in rigid body motion

at the 5 15

1 1

rate. The gain in amplitude is also varying from rotor to rotor and the effect of forces

generated in the mating of gears is also a reason for that.

REFERENCES

[1] Guo Rui, Jang Sung-Hyun, Choi Young-Hyu, Torsional vibration analysis of lathe spindle system with unbalanced workpiece, J. Cent. South Univ. Technol. (2011) 18: 171−176, Springerlink


ISSN 2250-3153

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[2] Wu Hao, Zhou Qiong1, Zhang Zhiming and An Qi, Vibration analysis on the rolling element bearing-rotor system of an air blower, Journal of Mechanical Science and Technology 26 (3) (2012) 653~659, Springerlink

[3] B.B. Maharathi, P.R. Dash, A.K. Behera, Dynamic Behaviour Analysis of a Dual-Rotor System Using the Transfer Matrix Method, International Journal of Acoustics and Vibration, Vol. 9, No. 3, 2004 115

[4] M. Aleyaasin, M. Ebrahimi, R. Whalley, Vibration analysis of distributed-lumped rotor systems, Comput. Methods Appl. Mech. Engrg. 189 (2000) 545±558, Elsevier

[5] Hu Qinghuaa, Deng Sierb, Teng Hongfeia, A 5-DOF Model for Aeroengine Spindle Dual-rotor System Analysis, Chinese Journal of Aeronautics 24 (2011) 224-234, Elsevier

[6] Vishwajeet Kushwaha, Prof. N.Kavi, Professor, Analysis of torsional vibration characteristics for multi-rotor and gear-branched systems using finite element method, A Thesis, National Institute of Technology, Rourkela, 2011-2012

[7] Dr. Rajiv Tiwari, A short term course on Theory and Practice of Rotor Dynamics, Department of Mechanical Engineering, Indian Institute of Technology Guwahati, (15-19 Dec 2008)

Table. 1

S.No Shaft Stiffness of the shaft K, in N-m/rad

S.No Moment of inertia, J in kg-m²

1 S12 79521.56 1 J₀ = 5.593 10-3

2 S23 2395628.02 2 J₁ = 2.571 10-4

3 S34 106028.75 3 J₂ = 1.553 10-4

4 S45 212057.50 4 J₃ = 1.607 10-5

5 S67 4021238.59 5 J₄ = 1.973 10-5

6 S47 5089380.09 6 J₅ = 1.598 10-5

7 S48 103059947 7 J₆ = 4.817 10-5

8 S89 90881.78 8 J₇ = 1.850 10-4

9 S910 1272345.02 9 J₈ = 7.707 10-4

10 S1112 10553338.57 10 J₉ = 2.397 10-3

11 S912 636172.51 11 J₁₀ = 9.633 10-6

12 S913 4690372.69 12 J₁₁ = 7.253 10-4

13 J₁₂ = 4.415 10-4

14 J₁₃ = 4.415 10-4

15 J₁₄ = 0.0111

16 J₁₅ = 9.633 10-5


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Fig 1: Layout of gear box in all geared lathe

Fig 2. Spring-mass representation of the gear box


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Frequency response curves for various rotors corresponding to different transfer functions

Z11 Z12 Z13

Z14 Z15 Z16


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Z17 Z18 Z19

Z110 Z111 Z112


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Z113 Z21 Z22

Z23 Z24 Z25


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Z26 Z27 Z28

Z29 Z210 Z211


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Z212 Z213 Z31

Z32 Z33 Z34


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Z35 Z36 Z37

Z38 Z39 Z310


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Z311 Z312 Z313

Z41 Z42 Z43


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Z44 Z45 Z46

Z47 Z48 Z49


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Z410 Z411 Z412

Z413 Z51 Z52


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Z53 Z54 Z55

Z56 Z57 Z58


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Z59 Z510 Z511

Z512 Z513 Z61


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Z62 Z63 Z64

Z65 Z66 Z67


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Z68 Z69 Z610

Z611 Z612 Z613


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Z71 Z72 Z73

Z74 Z75 Z76


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Z77 Z78 Z79

Z710 Z711 Z712


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Z713 Z81 Z82

Z83 Z84 Z85


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Z86 Z87 Z88

Z89 Z810 Z811


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Z812 Z813 Z91

Z92 Z93 Z94


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Z95 Z96 Z97

Z98 Z99 Z910


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Z911 Z912 Z913

Z101 Z102 Z103


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Z104 Z105 Z106

Z107 Z108 Z109


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Z1010 Z1011 Z1012

Z1013 Z111 Z112


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Z113 Z114 Z115

Z116 Z117 Z118`


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Z119 Z1110 Z1111

Z1112 Z1113 Z121


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Z122 Z123 Z124

Z125 Z126 Z127


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Z128 Z129 Z1210

Z1211 Z1212 Z1213


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Z131 Z132 Z133

Z134 Z135 Z136


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Z137 Z138 Z139

Z1310 Z1311 Z1312


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Z1313

AUTHORS

First Author – V Suma Priya, PG Student, PVP Siddhartha

Institute of Technology, Vijayawada, [email protected]

Second Author – B V S Raghu Vamsi, M.tech, Assistant

Professor, Gudlavalleru Engineering College, Gudlavalleru

[email protected]

Third Author – E Kavitha, M.Tech, Assistant Professor, PVP

Siddhartha Institute of Technology, Vijayawada,

[email protected]

Fourth Author – K Srividya, M.Tech, Assistant Professor, PVP

Siddhartha Institute of Technology, Vijayawada,

[email protected]

Correspondence Author – B V S Raghu Vamsi,

[email protected], [email protected],

Contact number: +919492508145

Frequency response analysis of the gear box in a lathe ... · Frequency response analysis of the...

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