Longitudinal-Torsional and Two Plane Transverse Vibrations ...

© 2016 IAU, Arak Branch. All rights reserved.

Journal of Solid Mechanics Vol. 8, No. 2 (2016) pp. 418-434

Longitudinal-Torsional and Two Plane Transverse Vibrations of a Composite Timoshenko Rotor

M. Irani Rahagi *, A. Mohebbi , H. Afshari

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

Received 22 March 2016; accepted 18 May 2016

ABSTRACT

In this paper, two kinds of vibrations are considered for a composite Timoshenko rotor:

longitudinal-torsional vibration and two plane transverse one. The kinetic and potential

energies and virtual work due to the gyroscopic effects are calculated and the set of six

governing equations and boundary conditions are derived using Hamilton principle.

Differential quadrature method (DQM) is used as a strong numerical method and

natural frequencies and mode shapes are derived. Effects of the rotating speed and the

lamination angle on the natural frequencies are studied for various boundary conditions;

meanwhile, critical speeds of the rotor are determined. Two kinds of critical speeds are

considered for the rotor: the resonance speed, which happens as rotor rotates near one

of the natural frequencies, and the instability speed, which occurs as value of the first

natural frequency decreases to zero and rotor becomes instable.

© 2016 IAU, Arak Branch.All rights reserved.

Keywords : Longitudinal-torsional vibration; Transverse vibration; Composite rotor;

DQM.

1 INTRODUCTION

INCE composite materials have the potential of the innovative and cost effective manufacturing technology,

they have attracted a lot of attention for thick or thin structures. A number of different elements of composite

structures such as aircraft wings, helicopter rotor blades, robots arms, bridges and structural elements in civil

engineering constructions can be idealized as thin- or thick-walled beams, which can be studied by considering

simpler governing equations. Zu et al. [1] have investigated the free vibration behavior of a spinning metallic beam,

where the classical theory of differential equations which couple flexural motions in both principal planes, but

ignores torsional deformation is used, with general boundary conditions. In another study, Zu et al. [2] considered

natural frequencies and normal moods for externally damped spinning Timoshenko beam with general boundary

conditions. Reis et al. [3] published analytical investigations on thin-walled layered composite cylindrical tubes,

concluding that bending-stretching coupling and shear-normal coupling effects will alter the frequency values. A

research was conducted by Gupta and Singh [4] on the effects of shear-normal coupling on rotor natural frequencies

and modal damping. Amongst all the investigators, it seems that Bert [5] is the first one who presented a simple

method for critical speed analysis of composite shafts by taking coupled bending-torsion composite beam theory

into consideration. Before long, Kim and Bert [6] used a more accurate shell theory for composite shaft and made a

direct comparison with the beam theory. In another attempt, Banerjee and Su [7] developed the dynamic stiffness

matrix for spinning composite beam by including bending-torsion coupling effects and then analyzed free vibration

characteristics. Chang et al. [8] published the vibration behaviors of the rotating composite shafts. In the model, the

______ *Corresponding author. Tel.: +98 31 55912420; Fax: +98 31 55559930.

E-mail address: [email protected] (M. Irani Rahagi).

S

419 M. Irani Rahagi et al.

© 2016 IAU, Arak Branch

transverse shear deformation, rotary inertia, and gyroscopic effects, as well as the coupling effects due to the

lamination of composite layers have been incorporated.

In this paper, longitudinal-torsional and two plane transverse vibrations of a composite Timoshenko rotor are

investigated. Shear deformation, rotating inertia and gyroscopic effect are considered. Differential quadrature

method is employed and natural frequencies and mode shapes are derived numerically. Effects of the rotating speed

and the lamination angle on the natural frequencies are studied for various boundary conditions; meanwhile, critical

speeds of the rotor are determined.

2 THE GOVERNING EQUATION AND BOUNDARY CONDITION

As depicted in Fig. 1, a composite rotor rotating at a constant angular velocity is considered. The following

displacement field is assumed by choosing the coordinate axis x to coincide with the shaft axis [9]:

0

0

0

, , , , , ,

, , , , ,

, , , , ,

y zU x y z t U x t z x t y x t

V x y z t V x t z x t

W x y z t W x t y x t

(1)

where U, V and W are displacements of any point on the cross-section of the shaft in the x, y and z directions,

respectively; U0, V0 and W0 are value of the U, V and W on the shaft’s axis, while x and y

are rotation angles of

the cross-section, about the y and z axes, respectively and is the angular rotation of the cross-section due to the

torsion deformation of the shaft.

Fig.1

Composite rotor.

The velocity of each point on the shaft can be stated as:

v r r (2)

In which

i (3)

Substituting Eqs. (1) and (3) for Eq. (2), the velocity of each point on the shaft can be derived as:

0 0 0

0 0

y zU V W

v z y i z W y j y V z kt t t t t t t

(4)

Kinetic energy of the shaft can be stated as:

0

1.

2

L

kA

E v v dA dx

(5)

where is density of the shaft. Using Eqs. (4) and (5), kinetic energy can be stated as:

Longitudinal-Torsional and Ttwo Plane Transverse Vibrations…. 420


2 2 2

2 2 20 0 0 0 0

0 0 0 0

0

2 22

2 2

0 0

12

2

1 1

2 2

L

k m

L Ly z

p d

U V W W VE I W V V W dx

t t t t t

I dx I dxt t t

(6)

In which mass, transverse and polar mass moment of inertias of the rotor are defined respectively as:

2 2 2 2 2m d p dA A A A

I dA I y dA z dA I y z dA I (7)

For a composite rotor with k layers, these parameters can be calculated using following relations:

2 2 4 4 4 4

1 1 1

1 1 1

24 2

k k k

m n n n d n n n p n n n d

n n n

I R R I R R I R R I

(8)

where 1, n nR and

nR are density, inner and outer radius of the thn layer, respectively.

Applying the relation between stress and strain in the composite materials and strain-displacement relations in

the cylindrical coordinate, the potential energy of the rotor can be derived as [9]:

22 2 2

0

11 11 66

0 0 0

0 0 0

16

0

2 2

20 0

55 66

1 1 1

2 2 2

12

2

1

2

L L Ly z

u

Ly yz z

z y

y

UE A dx B dx kB dx

x x x x

U V WkA dx

x x x x x x x x

V Wk A A

x x

2 0 0

0

2 2

L

z y z

W Vdx

x x

(9)

In which the following axial, polar and transverse flexural rigidities are defined:

2 2 3 3

11 11 1 16 16 1

1 1

2 2 4 4

55 55 1 11 11 1

1 1

2 2 4 4

66 66 1 66 66 1

1 1

2

3

2 4

2 2

k k

n n n n n n

n n

k k

n n n n n n

n n

k k

n n n n n n

n n

A C R R A C R R

A C R R B C R R

A C R R B C R R

(10)

and iinC are the effective elastic constants of the thn layer which are introduced in Appendix A.

Also, the virtual work due to the gyroscopic moments can be considered as [10]:

0

Ly z

p z yW I dxt t

(11)

Now, using Hamilton principle as:

2

1

0

t

k u

t

E W E dt

(12)

The set of governing equations and boundary conditions can be derived as:



2 22

0 0

0 11 162 2 2

2 2 220

16 662 2 2

2 2 2

20 0 0

0 16 55 66 02 2 2

22

0

0 16 55 662 2

: 0

: 0

1: 2 0

2

1:

2

m

p

y z

m

yz

U UU A kA I

x x t

UkA kB I

x x t

V W VV kA k A A I V

x tx x t

WW kA k A A

xx x

2

2 0 0

0 2

2 22

0 0

16 55 66 16 112 2 2

2 2 2

0 0

16 55 66 16 112 2 2

2 0

1: 0

2

1: 0

2

m

y yz z

y y p d

y yz z

z z p d

V WI W

t t

V WkA k A A kA B I I

x x tx x t

W VkA k A A kA B I I

x x tx x t

(13a)

0

11 16 0

0

0

66 16

0

0

16 55 66 0

0

0

16 55 66 0

0

0

11 16

0

0

10

2

10

2

1

2

x L

x

x L

x

x L

y

z

x

x L

z

y

x

y

z

UA kA U

x x

Uk B A

x x

Vk A A A V

x x

Wk A A A W

x x

VB kA

x x

0

0

11 16

0

0

10

2

x L

y

x

x L

z

y z

x

WB kA

x x

(13b)

Applying method of separation of variables as:

0 0 0 0 0 0, , ,

, , ,

t t t

t t t

y y z z

U x t Lu x e V x t Lv x e W x t Lw x e

x t x e x t x e x t x e

(14)

where is natural frequency of vibration and also using following dimensionless parameters:

2 2 2 2

2 2 16 55

16 55

11 11 11 11

66 6611

66 11 662 2 2 2

11 11 11

2

m m

pd

d p d

m m

A AI L I Lx

L A A A L A

IA B IB

A A L A L I L I L

(15)

In which L is the length of the rotor, dimensionless governing equation and boundary conditions can be rewritten

as:



2 220

16 02 2

2 22 20

16 662 2

2 2

2 216 0

55 66 0 0 02 2

22

2 216 0

55 66 0 0 02 2

2

16 0

55 662

0

2 2 0

2 02

2 02

2

d d

y z

yz

d u dk u

d d

d u dk k

d d

dk d vdk v w v

dd d

dk d wdk w v w

dd d

k d vk

d

2

20

16 11 2

2 2

216 0 0

55 66 16 112 2

2 0

2 02

yz

y d z d y

y z

z d y d z

ddw dk

d d d

dk d w dv dk k

d dd d

(16a)

0

16 0

0

66 16

16 0

55 66 0

16 0

55 66 0

16 0

11

16 0

11

0 0

0 0

0 02

0 02

0 02

0 02

y

z

z

y

y

z y

z

y z

duk or u

d

dudor

d d

d dvor v

d d

dwdor w

d d

d k dvor

d d

k dwdor

d d

(16b)

As shown in Eq. (16), the coupled longitudinal-torsional vibrations and coupled two plane transverse vibration

occur separately.

3 DIFFERENTIAL QUADRATURE METHOD (DQM)

The differential quadrature method is based on the idea that all derivatives of a function can be easily approximated

by means of weighted linear sum of the function values at "N" pre-selected grid of points as:

1i

r Nr

ij jrjx x

d fA f

dx

(17)

where ( )rA is the weighting coefficient associated with the thr order derivative given by [11]



(1)

( ) (1) (r 1)

1,

1

1

1

, 1, 2,3,..., ;

1, 2,3,...,

2 1r

N

i m

mm i j

N

j mij

mm j

N

i m

mm i

ij ij ij

x x

i j N i j

x xA

x x i j N

A A A r N

(18)

In this paper, for simplifying, following notations are considered:

1 2

.A A B A (19)

Distribution of the grid points is an important aspect in convergence of the solution. A well-accepted set of the

grid points is the Gauss–Lobatto–Chebyshev points given for interval [0,1] as:

111 cos 1,2,3,..., .

2 1i

ix i N

N

(20)

4 DQM SOLUTION

4.1 Longitudinal-torsional vibration

Using Eq. (17), the set of governing equation for longitudinal-torsional vibration can be written as:

2

lt ltK p M p (21)

where

16 0

2

16 66

0

2 0 2lt lt

d d

B k B I uK M p

k B k B I

(22)

and boundary conditions can be modeled as:

0ltT p (23)

where matrix 1tT is presented in Appendix B for various boundary conditions.

4.2 Transverse vibration

Using Eq. (17), the set of governing equation for transverse vibration can be written as:

2 0t t tK C M q (24)

where



2

55 66 16 55 66

2

55 66 55 66 16

16 55 66 55 66 11 16

55 66 16 16 55 66 11

0 0.5

0 0.5

0.5

0.5

0 2 0 0

2 0 0 0

0 0 0 2

0 0 2 0

t

t

d

d

k B I k B k A

k B I k A k BK

k B k A k I B k A

k A k B k A k I B

I

IC

I

I

0

0

0 0 0

0 0 0

0 0 0

0 0 0

t

yd

d z

vI

wIM q

I

I

(25)

and boundary conditions can be modeled as:

0tT q (26)

where matrix tT is presented in Appendix B for various boundary conditions.

Let us divide grid points of solution into two groups; first and final points as boundary points (b) and others as

domain ones (d). The equations of motion should be written only for the domain points [12]; thus Eqs. (21) and (24)

should be rewritten as:

2

lt ltK p M p (27a)

2 0t t tK C M q (27b)

In which, bar signs show the corresponded truncated non-square matrices. Eqs. (27a) and (27b) may be

partitioned in order to separate the boundary and domain components as [12]:

2

lt lt lt ltd b d bd b d bK p K p M p M p (28a)

2 0t t t t t tb d b d b db d b db dK q K q C q C q M q M q

(28b)

and in a similar manner, Eqs. (23) and (26) can be written as:

0lt ltb db dT p T p (29a)

0t tb db dT q T q (29b)

Using Eqs. (28) and (29), following eigen values equations can be derived:

2

lt ltd dk p m p (30a)

2 0t t td d dk q c q m q (30b)

In which

1 1

1 1 1

lt lt lt lt lt lt lt lt lt ltb d b dd b d b

t t t t t t t t t t t t t t tb d b d b dd b d bd b

k K K T T m M M T T

k K K T T c C C T T m M M T T

(31)

Unlike Eq. (30a), Eq. (30b) is not a standard eigen value equation and can be converted to a standard one as:



0 0

0

d d

t t td d

q qI I

q qk c m

(32)

5 EXACT SOLUTION

In order to be able to validate the proposed numerical solution, an exact solution for a specific case is presented. For

a rotor with immovable supports, an exact solution can be found for longitudinal-torsional vibration. According to

Eq. (16b), for longitudinal-torsional vibration following spatial functions can be considered for a rotor with

immovable supports:

0 sin sinm nu A m B n (33)

Substituting Eq. (33) into the Eq. (16a), following relation can be found for mn :

2 2 2 2 2

16

2 2 2 2 2 2

16 66

02

mn

d mn

m n k

m k k n

(34)

6 NUMERICAL RESULTS AND DISCUSSION

In all of the numerical examples, a rotor made of Boron-Epoxy with the following properties is considered [13]:

1 2 3 12 13 23

3

12 13 23

146.85 11.03 6.21 3.86

0.28 0.5 1578

E Gpa E E Gpa G G Gpa G Gpa

Kg m

Table 1

First six longitudinal-torsional and transverse frequencies of a clamped-clamped rotor.

Longitudinal-torsional

DQM 1.927018 3.141813 3.86278 5.796597 6.283627 7.729931

Exact 1.927377

(m=0,n=1)

3.141813

(m=1,n=1)

3.863499

(m=0,n=2)

5.797675

(m=0,n=3)

6.283627

(m=2,n=2)

7.731365

(m=0,n=4)

Transverse (Forward modes)

DQM 0.541546 1.181134 2.065827 3.133866 4.337743 5.645814

Transverse (Backward modes)

DQM 0.243661 0.888069 1.778292 2.851663 4.060101 5.371739

Also dimension of the rotor are considered as [9]:

00.1321 12.69 2.47plyh mm D cm L m

and shear correction factor is considered as 0.503sk [9]. Consider a rotor made of 10 layers with lamination

angle 45 . Table 1. shows the values of the first six longitudinal-torsional and transverse frequencies of a

clamped-clamped rotor rotating with the angular velocity 0.15 . Number of grid points is considered as 15N .

Also, exact results obtained by Eq. (34) are presented in this table. As shown, results with high accuracy can be

obtained; this table also confirms an evident conclusion that transverse frequencies are smaller than longitudinal-

torsional ones. It indicates that rotor is more flexible in transverse deflection in comparison with longitudinal and

torsional displacements. Also corresponding longitudinal-torsional modes are depicted in Fig. 2. There is an

agreement between the modes obtained by DQM and corresponding values of m and n presented in Table 1., which

confirms the accuracy and versatility of the proposed solution.



0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

U

0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

U

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

U

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

U

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

U

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

U

Fig.2 Corresponding longitudinal-torsional modes.

In what follows, a rotor made of 10 layers as 5

/ is considered. Fig. 3 shows variation of the first three

longitudinal-torsional frequencies of a clamped-clamped rotor versus rotating speed (Campbell diagram) for various

values of the lamination angle and similar diagrams are depicted for clamped-free one in Fig. 4. As depicted in these

figures, all frequencies decrease as value of the rotating speed increases. For both longitudinal-torsional vibration

and two plane transverse one, two kind of critical speeds are considered for the rotor; the resonance speed (cr ),

which happens as rotor rotates near one of the natural frequencies, and the instability speed (in ),which occurs as

value of the first natural frequency decreases to zero and rotor becomes instable. Both resonance rotating speed and

instability one are detectable in these figures.

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

n

=0

n=1

n=2n=3

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

2

2.5

3

n

=30

n=1

n=2n=3



0 0.5 1 1.5 20

1

2

3

4

5

n

=60

n=1

n=2n=3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5

1

1.5

2

2.5

3

n

=90

n=1

n=2n=3

Fig.3 Campbell diagram for the first three longitudinal-torsional frequencies of clamped-clamped rotor.

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

n

=0

n=1

n=2n=3

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

n

=30

n=1

n=2n=3

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

n

=60

n=1

n=2n=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.5

1

1.5

2

n

=90

n=1

n=2n=3

Fig.4 Campbell diagram for the first three longitudinal-torsional frequencies of clamped-free rotor.

Figs. 5-8 show the variation of the first four backward and forward transverse frequencies of a clamped-clamped,

clamped-simple, simple-simple and clamped-free rotors, versus rotating speed for various values of the lamination

angle.As shown in these figures, for a non-rotating rotor, the value of the forward frequencies (ascending lines) and

backward ones (descending lines)are the same; but because of gyroscopic effects, as the value of the velocity of spin

increases, forward frequencies increase and backward ones decrease. These figures also show the line of

synchronous whirling; intersection of this line with the Campbell diagram determines the critical speeds, which

should be avoided.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.1

0.2

0.3

0.4

0.5

0.6

1

=0

=20

=40=60

=80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

2

=0 =20 =40 =60 =80



0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1

1.2

1.4

1.6

1.8

2

2.2

3

=0 =20 =40 =60 =80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.5

2

2.5

3

3.5

4

=0 =20 =40 =60 =80

Fig.5 Campbell diagram for the first four transverse frequencies of clamped-clamped rotor.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

1

=0

=20

=40=60

=80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

2

=0 =20 =40 =60 =80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.8

1

1.2

1.4

1.6

1.8

2

3

=0 =20 =40 =60 =80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

4

=0 =20 =40 =60 =80

Fig.6 Campbell diagram for the first four transverse frequencies of clamped-simple rotor.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1

=0

=20

=40=60

=80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.4

0.5

0.6

0.7

0.8

0.9

2

=0 =20 =40 =60 =80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

3

=0 =20 =40 =60 =80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

4

=0 =20 =40 =60 =80

Fig.7 Campbell diagram for the first four transverse frequencies of simple-simple rotor.



0 0.01 0.02 0.03 0.04 0.050

0.02

0.04

0.06

0.08

0.1

0.12

1

=0

=20

=40=60

=80

0 0.01 0.02 0.03 0.04 0.05

0.25

0.3

0.35

0.4

0.45

2

=0

=20

=40=60

=80

0 0.01 0.02 0.03 0.04 0.050.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

3

=0 =20 =40 =60 =80

0 0.01 0.02 0.03 0.04 0.05

1.2

1.4

1.6

1.8

2

4

=0 =20 =40 =60 =80

Fig.8 Campbell diagram for the first four transverse frequencies of clamped-free rotor.

Also for 40 and 0.05 , the corresponding first two forward and backward modes are depicted in Figs. 9

to 12 for clamped-clamped, clamped-simple, simple-simple and clamped-free rotors, respectively.

0

0.5

1

-1

0

10

0.1

0.2

0.3

0.4

Forward mode

0

0.5

1

0

0.1

0.2

0.3

0.4-0.1

-0.05

0

Backward mode

0

0.5

1

-1

0

1-1

-0.5

0

0.5

1

x 10-3

Forward mode

0

0.5

1

-1

0

1

x 10-3

-0.2

-0.1

0

0.1

0.2

Backward mode

Fig.9 Corresponding modes for the first two transverse modes of clamped-clamped rotor with 45 and 0.05 .



0

0.5

1

-1

0

10

0.05

0.1

0.15

0.2

0.25

Forward mode

0

0.5

1

-0.4

-0.2

0-0.025

-0.02

-0.015

-0.01

-0.005

0

Backward mode

0

0.5

1

-1

0

1-0.01

-0.005

0

0.005

0.01

Forward mode

0

0.5

1

-0.01

0

0.01-0.2

-0.1

0

0.1

0.2

Backward mode

Fig.10 Corresponding modes for the first two transverse modes of clamped-simple rotor with 45 and 0.05 .

0

0.5

1

-1

0

1-0.2

-0.15

-0.1

-0.05

0

Forward mode

0

0.5

1

0

0.1

0.2-0.2

-0.15

-0.1

-0.05

0

Backward mode

0

0.5

1

-1

0

1-1

-0.5

0

0.5

1

x 10-4

Forward mode

0

0.5

1

-1

0

1

2

x 10-4

-0.2

-0.1

0

0.1

0.2

Backward mode

Fig.11 Corresponding modes for the first two transverse modes of simple-simple rotor with 45 and 0.05 .



0

0.5

1

-1

0

1-0.8

-0.6

-0.4

-0.2

0

Forward mode

0

0.2

0.4

0.6

0.8

1

-0.8-0.6

-0.4-0.2

0-0.06

-0.04

-0.02

0

Backward mode

0

0.5

1

-1

0

1-0.1

-0.05

0

0.05

0.1

0.15

Forward mode

0

0.5

1

-2

0

2

x 10-3

-0.2

-0.1

0

0.1

0.2

0.3

Backward mode

Fig.12 Corresponding modes for the first two transverse modes of clamped-free rotor with 45 and 0.05 .

Figs. 13 and 14 show the variation of the longitudinal-torsional and transverse critical speeds versus lamination

angle. As these figures show, for both kind of vibrations, both resonance and instability speeds increase as the value

of the lamination angle grows from 0 to near 60 and then decrease as value of the lamination angle grows from near

60 to 90 . It should be noted that in order to be able to see variation of the critical frequencies for all boundary

conditions, all of them are divided to the corresponding values of 0 ; in other words / ( 0)cr cr cr and

/ ( 0)in in in .

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

cc

cc

cf

cf

Resonance speed

Instability speed

Fig.13

Variation of the longitudinal-torsional critical speeds

versus lamination angle.

0 10 20 30 40 50 60 70 80 901

1.05

1.1

1.15

1.2

1.25

1.3

cr

cc

csss

cf

0 10 20 30 40 50 60 70 80 90

1

1.05

1.1

1.15

1.2

1.25

1.3

in

cc

csss

cf

Fig.14 Variation of the transverse critical speeds versus lamination angle.



7 CONCLUSIONS

Using Hamilton principle, the set of governing equations and boundary conditions of longitudinal-torsional and two

plane transverse vibration analyses of a composite Timoshenko rotor were derived and solved numerically by DQM.

Comparison between obtained numerical results and exact solution which was proposed for a special case,

confirmed the accuracy of the proposed solution. Numerical results showed that for longitudinal-torsional vibration,

all frequencies decrease as the value of the rotating speed increases. Moreover, for transverse one, for a non-rotating

rotor, the value of the forward and backward frequencies are the same. Owing to gyroscopic effects, however, as the

value of the velocity of spin increases, forward frequencies increase and backward frequencies decrease. Numerical

results showed that for both longitudinal-torsional and transverse vibration, all natural frequencies and therefore

both resonance and instability critical rotating speeds increase as value of the lamination angle grows from 0 to near

60 and then decrease as value of the lamination angle grows from near 60 to 90 .

APPENDIX A

As said in Eq. (10), ijnC are the effective elastic constants of the thn layer which connect components of stress and

strain as:

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 0

0 0

0 0

0 0 0 0

0 0 0 0

0 0

xx xx

rr rr

r r

xr xr

x x

C C C C

C C C C

C C C C

C C

C C

C C C C

(A.1)

These constants are related to the elastic constants of principal axes ( ijC ) and lamination angle as [14]:

4 2 2 4

11 11 12 66 22

3 3

16 11 12 66 66 12 22

2 2

55 55 44

2 2 4 4

66 11 22 12 66 66

cos 2 2 sin cos sin

2 sin cos 2 sin cos

cos sin

2 2 sin cos sin cos

C C C C C

C C C C C C C

C C C

C C C C C C

(A.2)

Also elastic constants of principal axes can be calculated using following relations [14]:

2

23 12 32 13 13 31

11 12 22 44 23 55 13 66 12

2 3 1 3 1 3

12 21 23 32 31 13 21 32 13

1 2 3

1 1

1 2

C C C C G C G C GE E E E E E

E E E

(A.3)

APPENDIX B

For longitudinal-torsional vibration, the matrix of boundary conditions appeared in Eq. (23) can be defined for

clamped (C), simple (S) or free (F) conditions as:



1

1 1

1

16

16

66

,

1, 1, ,

1, 1, , 2, 1, , 2

2, 1, , 2 3, 1, ,

3, 1, , 3, 1, , 2

4, 1, , 2 4, 1, ,

0 4,

j

j j N

Njj N

Nj N j Nlt lt

NjN j N

N j N

CC CS or SS CF

i j N

i j N i j N N

i j N N A i j N

i j N k A i j N NT T

i j N N A i j N

else A i j

1, ,2

0

N N

else

(B.1)

and for transverse vibration, the matrix of boundary conditions appeared in Eq. (26) can be defined as:

1

1

1

1

1 2

1 3

2

3

1, 1, ,

1, 1, ,

2, 1, , 2

3, 2 1, ,3

4, 3 1, , 4

5, 1, ,

6, 1, , 2

7, 2 1, ,3

8, 3 1, , 4

0

j

j

j

j N

j N

j N

Njt t

N j N

N j N

N j N

CC CS

i j N

i j N

i j N N

i j N N

i j N N

i j NT T

i j N N

i j N N

i j N N

else

1 2

1 3

16

11 2

16 3

16

16

2, 1, , 2

3, 2 1, ,3

4, 3 1, , 4

5, 1, ,

6, 1, , 2

0.5 7, 1, ,

7, 2 1, ,3

0.5 7, 3 1, , 4

0.5 8, 1, , 2

0.5

N

j N

j N

Nj

N j N

Nj

N j N

N j N

N j N

i j N N

i j N N

i j N N

i j N

i j N N

k A i j N

A i j N N

k i j N N

k A i j N N

k

2

11 3

1

1

16 1

11 1 2

16 1 3

16 1

8, 2 1, ,3

8, 3 1, , 4

0

1, 1, ,

1, 1, , 2

0.5 2, 1, ,

2, 2 1, ,3

0.5 3, 3 1, , 4

0.5

N j N

N j N

j

j N

j

j N

j N

j

t

i j N N

A i j N N

else

SS CF

i j N

i j N N

k A i j N

A i j N N

k i j N N

k A

T

16 1 2

11 1 3

16

11 2

16 3

16

3, 1, , 2

0.5 4, 2 1, ,3

4, 3 1, , 4

5, 1, ,

6, 1, , 2

0.5 7, 1, ,

7, 2 1, ,3

0.5 7, 3 1, , 4

0.5 8, 1, ,

N

j N

j N

Nj

N j N

Nj

N j N

N j N

N j N

i j N N

k i j N N

A i j N N

i j N

i j N N

k A i j N

A i j N N

k i j N N

k A i j N

1

1

1 2

1 3

55 66

16

16 2

11 3

1, 1, ,

2, 1, , 2

3, 2 1, ,3

4, 3 1, , 4

5, 1, ,

0.5

2

0.5 8, 2 1, ,3

8, 3 1, , 4

0

j

j N

j N

j N

Nj

t

N j N

N j N

i j N

i j N N

i j N N

i j N N

A i j N

A

T

N

k i j N N

A i j N N

else

2

55 66 3

55 66

55 66 2

16 3

16

11 2

16 3

5, 2 1, ,3

5, 3 1, , 4

6, 1, , 2

6, 2 1, ,3

0.5 6, 3 1, , 4

0.5 7, 1, ,

7, 2 1, ,3

0.5 7, 3 1,

N j N

N j N

Nj

N j N

N j N

Nj

N j N

N j N

i j N N

i j N N

A i j N N

i j N N

A i j N N

k A i j N

A i j N N

k i j N

16

16 2

11 3

, 4

0.5 8, 1, , 2

0.5 8, 2 1, ,3

8, 3 1, , 4

0

N j N

N j N

N j N

N

k A i j N N

k i j N N

A i j N N

else

(B.2)



REFERENCES

[1] Zu J., Hans R.P., 1992, Natural frequencies and normal modes of a spinning beam with general boundary conditions,

Journal of Applied Mechanics, Transactions ASME 59:197-204.

[2] Zu J., Melanson J., 1998, Natural frequencies and normal modes of for externally damped spinning Timoshenko beams

with general boundary conditions, Journal of Applied Mechanics, Transactions ASME 65:770-772.

[3] Dos Reis H. L. M., Goldman R. B., Verstrate P. H., 1987, Thin-walled laminated composite cylindrical tubes—part III:

critical speed analysis, Journal of Composites Technology and Research 9(2):58-62.

[4] Gupta K., Singh S. E., 1996, Dynamics of composite rotors, Proceedings of the Indo-US Symposium on Emerging

Trends in Vibration and Noise Engineering, New Delhi, India.

[5] Bert C.W., 1992, The effects of bending twisting coupling on the critical speed of drive shafts, Composite Materials,

6th Japan/US Conference, Orlando Lancaster.

[6] Kim C.D., Bert C.W., 1993, Critical speed analysis of laminated composite, hollow drive shafts, Composites

Engineering 3:633-643.

[7] Banerjee J.R., Su H., 2006, Dynamic stiffness formulation and free vibration analysis of a spinning composite beam,

Computers and Structures 84: 1208-1214.

[8] Chang M.Y., Chen J.K., Chang C.Y., 2004, A simple spinning laminated composite shaft model, International Journal

of Solids and Structures 41(4): 637-662.

[9] Boukhalfa A., Hadjoui A., 2010, Free vibration analysis of an embarked rotating composite shaft using the hp-version

of the FEM, Latin American Journal of Solids and Structures 7: 105-141.

[10] Choi S.H., Pierre C., Ulsoy A.G., 1992, Consistent modeling of rotating Timoshenko shafts subject to axial loads,

Journal of Vibration and Acoustics 114: 249-259.

[11] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, Applied Mechanics

Reviews 49: 1-28.

[12] Du H, Lim M.K., Lin N.R., 1994, Application of generalized differential quadrature method to structural problems,

International Journal for Numerical Methods in Engineering 37:1881-1896.

[13] Sun J., Ruzicka M., 2006, A calculation method of hollow circular composite beam under general loadings, Bulletin of

Applied Mechanics 3(12): 105-114.

[14] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca

Raton, FL.

http://www.scimagojr.com/journalsearch.php?q=20949&tip=sid

http://www.scimagojr.com/journalsearch.php?q=20949&tip=sid

Longitudinal-Torsional and Two Plane Transverse Vibrations ...

Documents

Transcript of Longitudinal-Torsional and Two Plane Transverse Vibrations ...