10.IJAEST-Vol-No-5-Issue-No-1-STUDY-OF-INTERACTIONS-BETWEEN-THE-LATERAL-AND-TORSIONAL-ROTOR-MOTIONS- by ISERP ISERP

K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

K.V.S.SESHENDRA KUMAR*

Assistant Professor, Dept. of Industrial Production Engineering, GITAM Institute of Technology, Gitam University, Visakhapatnam-533045 Andhra Pradesh India venkat_seshendra@yahoo.co.in Ph: 9295757588

STUDY OF INTERACTIONS BETWEEN THE LATERAL AND TORSIONAL ROTOR MOTIONS IN A GEAR-PINION SYSTEM USING COMPLEX VARIABLE APPROACH

B.S.K.SUNDARASIVA RAO

Professor, Dept. of Mechanical Engineering, Andhra University College of Engineering, Visakhapatnam India hodmechau@gmail.com

ISSN: 2230-7818

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

1.INTRODUCTION

Analysis of the gear system as rotors is a very challenging problem because the system is a composite rotor; therefore possesses very complicated dynamic behavior. For example, the whirling motion and critical speed of the system has to be found by considering all the rotors in the system as well as interactions between them, which makes the solution process as well as proper interpretation of the solution extremely complicated. Complex variable approach, which was proposed for the analysis of a single rotor system, is very powerful for this purpose. The approach is expanded to the analysis of combined rotor system to apply it to the gear system analysis.

Early geared rotor dynamic models concentrated on the effects of mass imbalance and eccentricity of the gear on the shaft, virtually neglecting the actual dynamics of gear mesh. Hamad and Seireg[1] studied the whirling of geared rotor systems supported on hydrodynamic bearings. Torsional vibrations were not considered in this model and the shaft of the gear was assumed to be rigid. Iida et.al., [2] considered the same problem by assuming one of the shafts to be rigid and neglecting the compliance of the gear mesh and obtained a three degree of freedom model that determined the first three vibration modes and the forced vibration response due to unbalance and the geometric eccentricity of one of the gears. They also showed that their theoretical results confirmed experimental measurements. Later Iida.et.al., [3] applied their model to a larger system consisting of three shafts coupled by two gear meshes. Hagiwara, Iida and Kikuchi[4] developed a simple model that included the transverse flexibilities of the shafts by using discrete stiffness values that took the damping and compliances of the journal bearings into account and that assumed the mesh stiffness to be constant. With their model they studied the forced response of geared shafts due to unbalance and runout errors. The backlash detection and its influence in geared systems

The complex variable description of planar motion incorporates directivity as inherent information which is therefore very convenient in vibration analysis of rotors. This paper proposes to use the directional information explicitly when the equation of motion of a rotor is formulated in complex variables. It is shown that the free vibration solution to the equation of motion formulated as such can be defined as the directional natural mode because it describes not only the shape and frequency but also the direction of the free vibration response. The directional frequency response functions (dFRFs) that have been used recently are obtained as the solution to the forced vibration solution to the equation of motion.

ABSTRACT

The system under consideration consists of two gears in mesh, a driving motor and the load. The bearing stiffness and shaft flexibility are taken into account in two directions. The system responses are obtained for two cases: when only the torsional motions are considered without rotor effect, and when both the torsional and lateral motions caused by rotor effect are considered. The differences in the responses of the two models show the effect of neglecting rotor effects in gear dynamics simulation. It is shown that the lateral vibrations have considerable effect when the natural frequencies of the lateral vibration and torsional vibration are close to each other, which is well expected. By studying the responses of the system with strong lateral-torsional coupling, the nature of the coupling effect is discussed.

KEYWORDS

ISSN: 2230-7818

[6]

Fregolent determined modal parameters of spur gear system using Harmonic Balance Method. Study of designing compact spur gears taking into consideration of tooth stress and dynamic response [7]

was done by PH Lin et al . Dynamic behavior of Spur gears to varying mesh stiffness and tooth error [8]

has been studied by J Kuan and A Lin . A mathematical model for gear geometry error and mounting error has been developed by P Velex and M [9]

[10]

Maatar . S Theodassiades and S Nastiavas studied non-linear behavior of gear system with backlash and varying stiffness. They also studied nonlinear influence of bearings characteristics on gear pair system[11]. Modal Analysis of compliant multibody gear systems has been analyzed by H Vinayak [12]

and R Singh . The influences of non-uniform gear speed, and timevariant meshing stiffness on dynamic behavior using Finite Element Modeling was investigated by Y. Wang et al

Rotor dynamics, lateral-torsional coupling, bearing stiffness, shaft flexibility, gear dynamics, complex variable approach.

[5]

has been studied by N Sarkar et al . M Ambili and A

[13]

Effect of shaft flexibility and non-linear

vibration in gear pair was studied by G Litak and M [14]

Friswell . Method to calculate dynamic gear tooth force and bearing forces and its effects were studied

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

[16]

[15]

. Y Cheng and T

Lim derived exact gear geometry from manufacturing parameters and used it to study dynamic behavior of hypoid gears. Rotating shafts tend to bow out and whirl at certain [17]

speeds called critical speed or whirling speed . Various factors that can cause the shaft to whirl are mass unbalance of rotating system, gyroscopic forces, unsymmetrical stiffness etc. Study of whirling of rotor systems can be found in numerous literatures [21]

[18,19,20]

G. Genta has conducted studies on dynamics of rotor and discussed related assumptions. Modal analysis of undamped rotor system with gyroscopic effect has been studied by many researchers including Wang & Kirkhope

[22]

[24]

Dutt & Bakra

[23]

most important advantage of the complex variable description that it carries directionality as built-in information. Two fundamental concepts in modal analysis, the natural mode and the frequency response function, are obtained from the proposed procedure. The natural modes of general anisotropic rotors obtained from the procedure can be considered to be directional natural modes which define the frequency, mode shape and direction of the motion in a single complex variable expression.

, Genta and

Tonoli etc. Gyroscopic effect is what couples the forward and backward modes of rotors as discussed [25]

The main goal of this study is to develop a finite element model for the dynamic analysis of geared rotor systems and to study the effect of bearing flexibility, which is usually neglected in simple gear dynamics models, on the dynamics of the system. In many gear dynamics analyses, the effect of the lateral vibration has been ignored in modeling the system with the underlying assumption that the effect of the lateral deflection in a typical gear system is negligible compared to that of the torsional deflection. This assumption may not be valid in some configurations such as overhung type gears rotating at a high speed, in which therefore the whirling resonance speed occur within the operating frequency range. To understand the effect of possible interactions between the lateral and torsional responses, a simple gear system is considered in this study whose parameters are chosen so that its lateral and torsional natural frequencies are close to each other.

by Kessler and Kim . Complex mode description was first proposed by C.W. Lee and his colleagues, in which they represented two-dimensional motion of rotors by using complex variables. The real and imaginary parts of a complex variable can be used to represent the coordinates of a point in planar motion. This description has been used to describe twodimensional motion by many researchers including Crandall[26], Dimentberg [27],, KraÂ¨mer [28], Childs [29], Ehrich [30], Laws [31], Muszynska [32], Kessler and Kim [33], and Lee [34-41],Ehrich [30], used the complex variable representation to describe general elliptical motion observed in rotors. Laws [31], applied the representation and related interpretations to the diagnostics of rotating machinery. Muszynska [32], used the concept to perform modal testing using nonsynchronous rotating perturbation. Lee and his colleagues started the extensive use of complex variables in the forced and natural response analysis of rotors [34-41]. They developed the concept, coined the terminology of directional frequency response functions ~dFRFs! [34,35],solved various free and forced vibration problems of rotors using complex variables [38,39,40], and developed signal processing techniques and testing theories [35,36] ,necessary for complex modal analysis. In this work, a new procedure for complex variable based rotor analysis is proposed. Compared to previous works, the directional relationships are used from the first step in the proposed procedure, when the equation of motion is formulated. This foundation results in a unique form of equations of motion and enhanced physical insight. As will be shown, the procedure makes maximum utility of the

2.FINITE ELEMENT MODEL OF GEAR PAIR SYSTEM

by L Vedmar and A Anderson

ISSN: 2230-7818

Pinion

Motor

Gear

Load

Fig.1 Mathematical model of the geared rotor system

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077



Pinion Kx Θm

Θp







Θg Kt

Gear

-----------(2.2)

Lagrange equation states that;

d  T  dt  q i

Load

Kx Ky



 T V    Fi   qi qi

------------(2.3)

Ky Motor

Θl



K r   y  r   y 2  p g g g  m p p  1 2 2 2 2  V   K y y p  yg  K x x p  xg   2  K (   ) 2  K (   ) 2  t g l  t p m 

the equation of motion is obtained by an eight by eight matrix equation as follows: Fig.2 FEA model of the gear pair system

The system consists of two gears (pinion, P and gear, G) in mesh and driven by a motor M and driving a load L. The shafts on which gears are mounted have two translational degrees of freedom each, one in horizontal direction (Z direction) and other in vertical direction (Y direction). The gears, motor and load have torsional degree of freedom. Therefore, there are four degrees of freedom (x , y , x , and y ) in the p

lateral direction and four degrees of system (θ , m

θ , θ , and θ ) in the torsional direction in the p

m p 0  0  0 0   0 0   0

0 mg 0 0 0 0 0 0

0 0 mp 0 0 0 0 0

0 0 0 mg 0 0 0 0

0 0 0 0 Ip 0 0 0

Equation of motion in real description:



system. The shaft stiffness (K for shaft 1 and K 1

for shaft 2) is assumed to be constant in all directions making the system isoparametric.

The equations of motion can be derived by using the Lagrange method.

K y  Km   Km   0  0  rp K m    rg K m  0  0 

0 0  Km K y  Km 0 0 0 Kx 0 0 0 Kx

0 0 0 0 0 Ig 0 0

0 0 0 0 0 0 Jm 0

0   y p  0   yg  0   xp    0   xg   0  p    0  g  0  m    J l   l 

 rg K m

0 0

rg K m 0

rp K m  rp K m

 rp K m

2 rp K m  K t

rg K m 0

 rp rg K m

 Kt 0

 rp rg K m  K t 2 rg K m  K t 0 0 Kt 0  Kt

   0  0  0    Kt  0  K t  0 0

The kinetic energy of the system T is;

2  2 2 2  2 1 I p p  I g  g  J m m  J l  l  m p y p   T   2 m y 2  m x 2  m x 2  p p g g  g g  ----------(2.1)

The strain energy of the system V is;

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 U p  2 Cost yp    y  U N 2  2 Cos ( Nt   )   g  g 2  xp   U p  Sin t     2 2  x g  =  U g N  Sin ( Nt   )    p   Tp        T  g g     Tm   m     l   Tl

p1  p1 f e jt  p1b e  jt p2  p2 f e jt  p2b e  jt  y1   p1 (t )  y   p (t )  2  2     T    z1   p1 (t )   z 2   p 2 (t ) -----------(2.6) where, T is the transformation matrix;

----------(2.4) The external forces considered in the above equation are: a. Torques on motor (T ), load (T ), pinion m

T

b. Unbalance masses U and U acting on pinion p

1 I 2  jI

----------(2.7) P is the complex variable with P representing the

and gear respectively. ψ defines the phase angle between the two unbalance masses in the pinion and gear. These unbalance forces act in the lateral directions, however can cause torsional responses because the equations are coupled in two directions.  Equation of motion in complex variable description:

of size 2x2. P1 (t )and P2 (t ) stands for the complex conjugates of P (t) and P (t). 1

A 







 

 j t j t    j k t   j t  p (t )   (Y e k  Y k e )  Y e k  k  Y ke k 0  k  k 



p(t )   p fk e jk t  pbk e  jk t



---------(2.5)

k 0

where ω is 2kπ/T, T is the fundamental period of k

motion and Y is the k component of the Fourier series.

To transform equation (2.4) to complex variable form following to the procedure proposed by Kessler and [13]

Kim , the displacement vector is transformed as follows. the displacement vector is transformed as follows.

ISSN: 2230-7818

It has been modified for a partial complex transformation as:

 1  I 22 I 22    T  2  jI 22 jI 22   Null 44 

Expanding y(t) and z(t) as Fourier series and collecting +ω and -ω terms, above equation can be represented as:



backward motion component. I is the identity matrix

p(t )  y(t )  jz (t )

 

forward motion component and P representing the

Any planar motions can be represented as complex variable of function of time by matching the real and imaginary parts of the complex variable to y and z coordinates. For example, any point P(t) in complex plane can be represented by vector:



I jI 

 Null 44   I 44 

-----------(2.8) The equation of motion after transformation is:

m p 0  0  0 0   0  0   0

0 mg 0 0 0 0 0 0

0 0 mp 0 0 0 0 0

0 0 0 mg 0 0 0 0

0 0 0 0 Ip 0 0 0

0 0 0 0 0 Ig 0 0

0 0 0 0 0 0 Jm 0

  0  P 1    0   P 2     0 P1      0  P 2      0 p   0  g  0  m    J l   l 

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 Km

rp K m

 rg K m

K y  Km

 rp K m

rg K m 0

 rp K m

2 rp K m  K t

rg K m 0

 rp rg K m

 Kt

 rp rg K m  K t 2 rg K m  K t 0 0 Kt 0  Kt

 P1     0  P  2 0      P1 0     P 0   2=    0  p      K t   g  0    m K t    l 

Mass Moment of Inertia for load, J = 0.00575 kg-m l

Bearing stiffness in x-direction, K = 2.75 x 10 N-m x

Bearing stiffness in y-direction, K = 2.75 x 10 N-m y

Mesh stiffness, K = 2 x 10 N-m m

Torsional stiffness, K = 115 N-m/rad t

Radius of pinion, r = 0.0445 m p

Radius of gear, r = 0.047 m g

 U p  2 e j t   2 2 jNt  U g N  e   U  2 e  jt   p 2 2  jNt  U g N  e    Tp     Tg   Tm     Tl

Mass unbalance on pinion, U = 0.0003 kg-m p

Mass unbalance on gear, U = 0.00028 kg-m g

The gear ratio is given as 0.9468 from the ratio of radius of the pinion and gear. Also notice that K is

taken the same for the motor side shaft and the load side shaft.

3. FREE VIBRATION RESPONSE OF THE GEAR SYSTEM :

-----------(2.9)

Torsional vibration analysis :



Once the system equation is set up this form, positive frequency solutions indicate whirling and rotational motion in the forward direction (counter-clockwise direction) and negative frequency solutions indicate motions in the backward direction (clockwise direction). 

Mass Moment of Inertia for motor, J = .0115 kg-m

System parameters :

K y  K m   Km   0  0  rp K m    rg K m  0  0 

Mass Moment of Inertia for gear, I = 0.001016 kg-m

Because the main purpose of this study is to understand the interaction between the lateral and torsional vibrations of gear systems in theoretical terms, while system parameters were selected so that the lateral and torsional mode frequencies exist in a close range to induce relatively strong coupling effect, they were also selected as reasonably practical by studying various past works selected are:

[25,26,27]

First, we consider torsional vibration of the system without considering the lateral vibration effect. The bearing stiffness can be set to an infinite value in both x and y directions to remove the lateral vibration effect in equation (2.4). Now the system becomes a four degree of freedom system. Equations of motion described in matrix form are:

I p 0 0  0

2   p  r p K m  K t r r K 0   g  p g m     0   m    K t J l  l    0 0

0  r p rg K m  K t 2 0  Kt rg K m  K t 0

 Kt

  p  T p     T   g    g    m  Tm     l   Tl 

---(3.1)

The parameters

For free vibration analysis, letting the external force vector to be zero and assume harmonic responses;

Mass of pinion, m = 1.84 kg p

Mass of gear, m = 0.92 kg g

Mass Moment of Inertia for pinion, I = 0.001821 kg-m

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 p   p       g   g  it     e  m   m    l    l 

external force to be zero and assuming harmonic responses:

 y p   Yp  y  Y   g  g  xp  X p       x g   X g  it     e  p   p   g   g       m   m        l  l

---------------(3.2)

the free vibration equation becomes: 0

2   p  r p K m  K t r r K 0   g  p g m     0  m    K t J l   l    0 0

0  r p rg K m  K t 2 0  Kt rg K m  K t 0

 Kt

  p  T p     T   g    g    m  Tm     l   Tl 

-----(3.3)

 Km

K y  Km   mg

rp K m

 rg K m

 rp K m

rg K m

Kx   mp

0  Kt

Natural frequencies of the system are obtained by solving the eigenvalues of the determinant of equation (3.3). The modal frequencies obtained as: ±0, ±122.32, ±309.7, ±25727.92 (rad/s).

K y  K m   2m p  K m   0  0   rp K m   rg K m  0   0

-------------(3.4)

I p 0 0  0

The natural modes are obtained by solving equation (3.3) after substituting the above natural frequencies back to the equation. Taking only the positive frequencies, the natural modes are obtained as follows, which are shown in Table 3.1.

As expected, the first mode associated with the zero frequency is a rigid body mode. The mode shape of this mode indicates the gear ratio 1.e., 0.9468. From the mode shape, it is seen that the second mode is dominated by the motions of the motor and load relative to the pinion and gear, respectively. The third mode is dominated by the motions of the gear and pinion masses relative to each other. The highest mode is isolated vibration participated only by the pinion and gear associated by the spring that represents the tooth stiffness. 

Coupled analysis:

lateral-torsional

Kx   mp 0

 rp K m

2 2 rp K m  K t   I p

rg K m

 rp rg K m

 rp rg K m 2 2 rg K m  K t   I g

 Kt

2 Kt   J m

 Kt

-----(3.5)

Solving for the eigenvalues and eigenvectors of the determinant in equation (3.5) with the physical parameters, the natural frequencies are obtained as: ±0, ±122.32, ±309.70, ±23446.05, ±38659.61, ±40474.36, ±54672.94, ±57304.6 (rad/s) Substituting these frequencies back to equation (4.1), the mode shapes are obtained. The frequencies and modes are summarized in Table 3.2.

Vibration

Free vibration analysis is conducted using the equation of motion described in real variables, which provides the same solutions. The solutions obtained from the real analysis can be transformed to the complex variable description using equation (2.8), which provides the directional information. From the equation of motion in equation(2.4), letting the

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 Y  p  0   Yg  0 X p    X g  0     0 0  p    Kt  g     m 0 2   l  Kt   Jl  0

K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

Table 3.1: Mode shapes after normalization for Θ Frequency (rad/s)

0 1

122.32 1

309.7 1

25727.92 1

0.9468

0.9476

0.9467

-1.9012

-2.0153

-0.1164

0.9468

3.7617

-0.2494

p g m l

Table 3.2: Mode shapes after normalization for θ

0 0

122.32 0

309.70 0

Frequency (rad/s) 23446.05 38659.61 -0.01 0

40474.36 0.2508

54672.94 0

57304.6 0.0403

0.0533

-0.4899

0.01

0.9468

0.9477

0.9467

-1.9012

-2.0154

-0.1164

0.9468

3.7619

-0.2494

The system equation is an 8 x 8 equation; therefore 8 modes are obtained. There are 4 lateral modes and 4 torsional modes. As before, the first frequency of vibration is zero, which corresponds to the rigid body torsional mode. The second and third modes are torsional modes without any participation by lateral motion modes. Notice the mode component ratio 0.9468 is the gear ratio. The second mode is predominantly the relative motion of the motor and load, and the third mode is predominantly the relative motion between the pinion-gear set and the motor-load set. The relative motion between the pinion and gear is not involved in this motion as the pinion-gear set moves without inducing any deformation in the gear teeth mesh. The fifth and seventh modes are un-coupled lateral modes in x-direction, which represent the single degree of freedom motions of the gear and pinion respectively. Modes 4, 6 and 8 are the modes with coupling effect between the lateral and torsional

modes. The 4 mode shows the weakest coupling. It is interesting to see only one torsional motion ratio (1: 1.9012) is involved in all these coupled modes.

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As can be seen from the table, the lateral modes are active only at very high frequencies. For all lower frequencies, their mode participations are nearly zero, which suggest they can be safely ignored. The first three lower frequency modes are purely torsional modes. The higher frequency modes are bendingtorsion combined modes with two in-between frequency modes being purely transverse. For these purely transverse modes, the displacement occurs in the direction perpendicular to the mesh line. Along the mesh line (Y direction) where coupling is present, no displacement occurs. For the system under study, if only torsional analysis had been carried out, the lateral motions would have been ignored; therefore only first three modes would have been picked up by the analysis. As the frequency increases, the lateral modes become more and more significant. If we compare the above modes to torsional only system, it can be seen that first three modes are almost same in both the systems, even though third modal frequency is different because of the action of the lateral spring provides extra stiffness in this torsional mode. These three motions are not of main

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

Now we consider that the motor torque T varies m

The response characteristics calculated in this way is from the system model considering only the torsional degree-of-freedoms, which will be later compared with the response calculated from the combined lateral-torsional system model. Therefore the force vector is given as:

---------(4.1)

0 0   F   e jt 1 0

The response spectrum is calculated for the frequency range between 0 rad/s and 26,000 rad/s in Figure (4.1). The first two peaks correspond to the responses that involve mainly the relative motions between the motor and pinion, and between the gear and the load. The peak at around 25,000 rad/s corresponds to vibration due to tooth deformation, which is of main concern in gear dynamics. This vibration is translated into gear force induced vibration and accompanying noise.

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Magnitude of force

-2

-4

10 10 Log10 Frequency (rad/s)

Fig 3. Mesh force in 4 DOF system in frequency range of 0 to 26000 rad/s

5. SUMMARY AND CONCLUSION :

harmonically with a 1N-m in equation (3.1) as the only active component in the force vector. The force acting on the gear mesh, which can be calculated as K m (rp p  rg g ) , is considered as the response.

4. FORCED VIBRATION RESPONSE OF THE GEAR SYSTEM :

Mesh force in torsional 4DOF system

Log

concern in gear dynamics because they involve with rigid body relative motions of the gear-pinion set. The coupled modes, mode 4, 6 and 8, correspond to the torsional mode at 25,727 rad/s in torsion only mode (see Table 3.1). Two lateral degrees of freedom, motions of the gear and pinion in y axis, are coupled with the torsional modes; therefore three modes are observed. If only torsional degrees of freedom would have been considered in the analysis, only one torsional mode at 25,727 rad/s would have been observed instead of these three modes.

In the present work, the dynamic behavior of geared rotor system for combined torsional-lateral vibrations has been carried out. In the analysis, transverse and torsional vibrations of the shafts and the transverse vibrations of the bearings have been considered. The gear-pinion system is modeled as a combined rotor system in the lateral directions and a torsional system driven by a motor and driving a load inertia in the rotationary direction. The equations of motion for eight degree of freedom gear system are derived using Lagrange’s equation and converted to complex variable form using the method developed by Kessler and Kim. The complex variable notation helps in understanding the whirling characteristics of rotor systems in a more simple way. Initially the system model that considered only the torsional motion has been analyzed, and then the results were later compared with the responses obtained by solving for the combined torsional-lateral system. The combined system has been analyzed in two different forms: one in the real co-ordinates description to compare it with the torsion only system, and other in the complex co-ordinates description to understand the directional information of the whirling motion of the system. Different types of inputs have been used to analyze the combined system like external torque on motor, unbalance on pinion only and unbalance on gear only. It has been demonstrated that the lateral motions interact significantly with torsional degrees of freedom if the resonance frequencies are similar. For the system under study, this interaction occurs around 26,000 rad/s where two close critical speeds exist

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

REFERENCES:

[1] HAMAD. B.M. AND SEIREG.A. Simulation of whirl interaction in pinion-gear systems supported on oil film bearings. Journal of Engineering Power ,1980 ,vol.102 pp 508-510. [2] IIDA et.al., Coupled Torsional-flexural vibration of a shaft in a geared system of rotors, Japanese society of mechanical Engineers, 1980, vol.23 pp 2111-211. [3] IIDA et.al., Coupled torsional-flexural vibration of a shaft in a geared system,International conference on vibration in rotating machinery 3 rd Mechanical Engineering publications, London, 1984. [4] HAGIWARA.N. IDA.M. KIKUCHI.K. Forced vibration of a pinion gear system supported on journal bearings, Proceedings of International symposium of gearing and power transmissions JSME Tokyo, 1981, pp 85-90. [5] N. SARKAR, RE ELLIS AND TN MOORE, Backlash Detection in Geared Mechanism: Modeling, Simulation, and Experimentation, Mechanical Systems and Signal Processing, 1997, vol. 11 issue 3 pp.391-408. [6] M. AMABILI AND A. FREGOLENT, A Method to Identify Modal Parameters and Gear Errors by Vibrations of a Spur Gear Pair, Journal of Sound and Vibration, 1998 vol. 214 issue 2 pp.339-357. [7] PH LIN, HH LIN, FB OSWALD AND DP TOWNSEND, Using Dynamics Analysis for Compact Gear Design, Design Engineering Technical Conference, 1998, pp.1-8. [8] JH KUAN AND AD LIN, Theoretical Aspects of Torque Responses in Spur Gearing due to Mesh Stiffness Variation, Mechanical Systems and Signal Processing, 2003, vol. 17 issue 2 pp.255-271.

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instead of one as found by torsional only system. Comparison of the resulting mesh forces in two systems when the motor torque varies with amplitude of 1 N-m is applied on the motor for the torsionlateral motion combined system and the torsion only system. The analysis certainly shows that the lateral whirling motions of the rotors (that model the gear and pinion) and torsional motions of the gears interact with each other. The degree of interaction depends on the proximity of the natural frequencies of the lateral and torsional motions. Neglecting the coupled effect in design can have serious effects on the estimation of the performance of the system if their frequencies are close to one another.

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K.V.S.SESHENDRA KUMAR*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 072 - 077

[37] JOH, Y. D., AND LEE, C. W., ‘Excitation Methods and Modal Parameter Identification in Complex Modal Testing for Rotating Machinery,’’ The Inter- national Journal of Analytical and Experimental Modal Analysis, 1993, pp.179 – 203. [38] LEE, C. W., ‘‘Rotor Dynamics and Control in Complex Modal Space,’’ Keynote Speech of 1st International Conference on Motion and Vibration Control, 1992. [39] LEE, C. W., and Kim, J. S., ‘‘Modal Testing and Suboptimal Vibration Control of Flexible Rotor Bearing System by Using a Magnetic Bearing,’’ ASME J. Dyn. Syst., Meas., Control, 1992, June, pp. 244 – 252. [40] JOH, C. Y., AND LEE, C. W., ‘‘Use of dFRFs for Diagnosis of Asymmetric/ Anisotropic Properties in Rotor-Bearing system,’’ ASME Journal of Vibration &.Acoustics., January 1996. [41] KESSLER, C., AND KIM, J., ‘‘Application of Triaxial Force Sensor to Im- pact Testing of Spinning Rotor Systems,’’ Proceedings of the 17th International Modal Analysis Conference, 1999, pp. 1699 – 1705.

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