Static And Dynamic Stability Analysis Of A Rotating Taper Beam [624970]

Static And Dynamic Stability Analysis Of A Rotating Taper Beam
Having Elliptical Cross Section Subjected To Pulsating Axial Load
With Thermal Gradient
M Pradhan*, M K Mishra**, P R Dash*
*Faculty, Dept. Of Mechanical Engineering, VSSUT, Burla, India, 768018
**M Tech Student: [anonimizat]. Of Mechanical Engineering, VSSUT, Burla, India, 768018
E-mail: [anonimizat] , [anonimizat] , [anonimizat] ,
Abstract
The static and dynamic stability of a rotating tapered beam having elliptical cross -section
subjected to a pulsating axial load with thermal gradient is investigated under three different
boundary conditions such as, clamped -clamped (C -C), clamped -pinned ( C-P) and pinned -pinned
(P-P). The governing equations of motion have been derived by using Hamilton’s energy
principle. A set of Hill’s equations have been obtained by the application of generalized
Galerkin’s method. The effects of taper parameter, hub ra dius, rotational speed, thermal gradient
and geometric parameter on the static buckling loads and the regions of instability have been
studied and the results are presented graphically.
1. Keywords
Taper parameter, thermal gradient, rotating sandwich beam , rotation parameter .
2. List of Symbols
 A x ,A
: Area of the generic section of the beam
1A
: cross sectional area x=0
0C
: Hub radius
l : length of the beam
0c
: Dimensionless hub radius, (C 0/l)
 b x ,b
: Minor diameter of a generic section of the beam
B
: Diameter at x=0
E
: Young’s modulus
 I x ,I
: Moment of inertia at a generic section
1I
: Moment of inertia x=0
m
: Mass distribution function
0P
: Static axial load
1P
: Dynamic axial load
p
: Dimensionless load
0p
: Dimensionless static axial load
1p
: Dimensionless dynamic axial load
S
: Moment of inertia distribution function
t : time
 w x,t
: Transverse deflection of the beam

q : Geometric parameter
*
: Minor axis taper parameter

: Dimensionless transverse deflection, (=w/ l)

: Dimensionless length, (=x/ l)

: Dimensionless time, (ct)

: Density of the beam material

: uniform angular velocity of the beam about z’ axis
0
: Rotational speed parameter

: Excitation frequency
0
: Dimensionless fundamental natural frequency

: Non-dimensional excitation frequency, (=
 /c)
3. Introduction
The vibration analysis of rotating beams is of great importance in the design of many
engineering examples, such as helicopter rotor blades, turbine blades, turbo engine blades etc.
The effect of thermal gradient is also a vital aspect as the modulus of elasticity for most of the
elastic materials are greatly affected by the temperature. Lo and Renbarger [1] obtain the
differential equation of motion of a cantilever blade mount ed on a rotating disc at a stagger
angle. The natural frequencies a nd mode shapes of a rotating uniform cantilever beam with tip
mass are studied by Bhat [2]. The theoretical expression for the work done due to centrifugal
effects was derived by Carnegie [3]. The effects of rotational speed, disc radius and stagger angle
of the blade on the frequencies of the lateral vibration are studied by Rao [4]. Liu and Yeh [5]
obtained the natural frequencies of a non -uniform rotating beam. Bauer [6] studied the effects of
spin, hub radius and aspect ratio on the vibrational behavior of a rotating beam. The effect of
rotational speed and root flexibilities on static buckling loads and the first order simple resonance
zones are studied by Abbas [7] by using finite element technique. Ishida et al. [8] investigated
the influence of rotat ional speed on the unstable regions of a system consisting of a disc mounted
on an elastic shaft subjected to a pulsating axial load. The explicit stability conditions for a
rotating shaft under parametric excitation are derived by Sri Namachchivaya [9]. R ao [10]
derived the formula for the fundamental flexural frequency of tapered cantilever beam. Taylor
[11] obtained the power series solution of blade natural frequencies for the case of a uniform
beam and a completely taper beam. Abbas and Thomas [12] and Yokoyama [13] studied the
effects of support conditions on the dynamic stability of Timoshenko beams by using finite
element method. The stability properties of a periodically loaded non -linear dynamic system
considering the damping effects are studied by Svensson [14]. Parametric instability of a non –
uniform beam with thermal gradient and elastic end support are studied by Kar and Sujata [15].
The same authors studied the dynamic stability of a rotating beam with various boundary
conditions [16]. The dyna mic stability of a rotating beam with a constrained damping layer are
investigated by Lin and Chen [17]. Rao and Carnegie [18, 19] used the Ritz averaging procedure
to solve the differential equation derived by Carnegie [20] and obtained the non -linear res ponse
of a cantilever blade. The bending frequenc ies of a rotating cantilever beam are determined by
Schilhansl [21]. The results of tor sional vibration of a rectangula r cross -section cantilever beam
are obtained by Vet [22] . Lenci et al. [2 5] obtained the approximate analytical expressions for
the natural frequencies of non -uniform cables and beams by using the asymptotic development

method. The effects of thermal gradient on the frequencies of rotating beams were studied by
Tomar and Jain [ 26].
From the available literature it has been found that no work has been done for the effects
of thermal gradient on the stability analysis of a rotating tapered beam having elliptical cross –
section . The air resistance decreases due to elliptical cross -section, which increases the efficiency
for many engineering applications which can be idealized as rotating beams such as turbine
blades, turbo engine blades and helicopter blades. So the present work deals with the study of
stability analysis of a rotatin g tapered beam having elliptical cross -section with thermal gradien t
under three different boundary conditions. The effect of taper parameter, hub radius, rotational
speed, thermal gradient and geometric parameter on the static buckling loads and the regio ns of
instability have been studied by computational method and the results are presented graphically.
4. Formulation of the problem

Fig 1: System configuration
A rotating tapered beam of length l set off a distance
0C from the axis of rotation and
rotating at a uniform angular velocity
 about a vertical z’ axis and is capable of oscillating in
the x -z plane. A pulsating axial force
01 P( t ) P P cos t   is applied at the end of the beam
along the point of C.G. of the cross -section in the axial direction,
 being the frequency of the
applied load,
t being the time and
0P and
1P being the static and dynamic load amplitudes
respectively.
The following assumptions are made for deriving the equations of motion:
1. The material of the beam is homogeneous & isotropic in nature.
2. The deflections of the beam are sma ll and the transverse deflection w(x, t) is the same for all
points of a cross -section.
3. Extensional deflection of the beam is neglected.
4. The beam obeys Euler -Bernoulli beam theory.
5. A steady one -dimensional temperature gradient is assumed to exist along the central length of
the beam.
6. Extension and rotary inertia effects are negligible.

Applying Hamilton’s principle
 2
10t
P
tT V W  
(1)
The expressions fo r potential energy, kinetic energ y and work done are as follows
 2
22
0011
22llwT A x dx A x w dxt   
(2)
 22 2
2
0 2
0 0 011
22,l l xwwV E x I x dx A x C x dxx x            
(3)
2
01
2l
PwW P t dxx
(4)
Where, w( x ,t ) is transverse deflection of the beam
Solving equation (1), following equation of motion and boundary conditions obtained
  
2
1 0xx xx tt xx
x x xxE x I x w, , A x w, I x w,
N x w, , P t w, 
   
(5)
Where,
2 22
1 0 01
2'N x A x C l C x     

The boundary conditions at
0 x C and
 0 x C l are
  0xx x x E x I x w, , P t w,  

 0xxxlE x I x w, 

0tw,
(6)
In the above expressions
2
2
2
2x xx
t ttwww, ,w,x x
www, ,w,t t 
 

And the dimensionless parameters are



2 0
0 4
2 2
222 2 2 2
2 2 2 2 2
22
22
2
0
1111E x I x C xw, ,c , ct, cl l l A x l
ww,xx
ww,l x x l
wwcl , c ltt
p t lp , p p pEI  







     

       
                          
                     
  
1cos
Non-dimensional equation of motion can be written as
  22
00 0.. '' ''' '' '
g S T m r p q                         
(7)
Now non dimensional boundary conditions are written as

10''' 'S T p
    
  


10''ST
  
 


10''ST
  
 

00,

00',
(8)
In the above expressions

2
1gIr
Al
,
24
2
0
11Al
EI   ,
2
1 2
0
11N x lqEI
 1 A A m
,
 1 I I S ,
 1 bB  
1b
B
,
1 m   ,
3
1 S  
4.1. Approximate solution
Approximate solution to the non -dimensional equations of motion are assumed as

1N
rr
r,f     

(9)
Where
rf is an unknown function of time and
 r being the coordinate function to be so
chosen as to satisfy as many of the bou ndary conditions as possible [ 24]. It is further assumed
that coordinate functions for the various boundary conditions can be approximat ed by the ones
given in Table -1.

Sl.No. End Arrangement Co-ordinate function i=1,2,………r
1 P-P
 sin i   
2 C-P
 1 2 32 2 4 6 2 1i i ii i i           
3 C-C
 1 2 32i i i        
Table 1: Co -ordinate functions
Substitution of the series of solutions in the non -dimensional equations of motion and subsequent
application o f the general Galerkin method leads to the follo wing matrix equations of motion
   01 0..
M f K f p H p cos H f    

(10)
Where,

 2
12..T
Nff and f f ,………., f
d
Various matrix elements are given by
1
0i j ij m d M       

   1
2
0
0'' '' ' '
i j g i j ij S T d q r d K                 

1
0''
i j ij dH       

Where,
12 i, j , ,………….,N
4.2. Static buckling load
Substitution of
10P and
0 f in equation (10) leads to the eigen value problem
1
01K H f fp
. The static buckling loads for the first few modes are obtained as the
reciprocals of the eigenvalues of
1KH using MATLAB.
4.3. Regions of instability
Let [L] be the modal matrix of [M]-1[K]. Then by the introduction of the linear coordinate
transformation, {f}=[L]{v} ,{v} being a new set of generalized coordinates yields ,
2
1 0n v v p cos B v   
(11)
Where
2
n is a spectral matrix corresponding to
1MK
And
11B L M H L
Equation (11) can be written as,
2
1 10mN
n n n nm n mv v p cos b u 
    
, n=1, 2, …, N (12)
Eqn (12) represents a system of N coupled Hill’s equations with complex coefficients.

Here, ω n and b nm are complex quantities, given by

n n,R n,I
n,m nm,R nm,Ij
b b jb  

The boundaries of the region of instability of simple and combination resonances are obtained
using the following conditions by Saito & Otomi [23 ].
4.3.1. Simple resonance
In this case, the regions of instability are given by
When damping is present
22
12
211624,R ,R
,R ,I
,Rp b b 



  
(13)
And, for the undamped case
11
24,R
,R
,Rpb



(14)
For μ=1, 2, …., N

4.3.2. Combination resonance of sum type
This type of resonance occurs when
1 v; , ,…..,N  and the regio ns of Instability are
given by
For the damped case

 2
1
81
22
16v,R ,R
,I v,I
v,R v ,R v,I v ,I ,I v,I
v,R ,R ,I v,Ipb b b b

    
 
 
  
(15)
And, for the un -damped case
1 1
2 2 4v,R v ,R
v,R ,R
v,R ,Rbbp

  
(16)
4.3.3. Combination resonance of difference type
This type of resonance occurs when
1 v; , ,…..,N  and the regions of instability are
given by:
For the damped case

 2
1
81
22
16v,R ,R
,I v,I
v,R v ,R v,I v ,I ,I v,I
v,R ,R ,I v,Ipb b b b

    
 
 
   
(17)
And for the un -damped case
1 1
2 2 4v,R v ,R
v,R ,R
v,R ,Rbb p

  
(18)

Dynamic stability analysis of the elliptically tapered rotating beam with axial pulsating
under various boundary conditions has been ca rried out by using equations (14), (16), ( 18). From
that regions of instability are obtained for various cases.
5. Numerical results and discussion
Numerical results were obtained for various values of the parameters like rotation
parameter, geometric parameter, taper parameter and thermal gradient. The linearly tapered beam
with a n elliptical cross -section is assu med to have a minor diameter varying according to the
relation
 1 bB   .
Where B is the minor diameter of the beam at the end
0 and α* is the minor diameter
taper parameter , where
1b
B .
Consequently, the mass distribution
m and the moment of inertia distribution
S
are given by the relations
1 m  

3
1 S  

The temperature above the reference temperature at any point
 from the end of the
beam is assumed to be
01   . Choosing
0 , the temperature at the end
 = 1 as the
reference temperature, the variation of modulus of elasti city of the beam [15 ] is denoted by

 11 1   1 ,     0 1 EE       ₁

1ET
Where
 is the coefficient of thermal expansion of the beam material,
1  is the thermal
gradient parameter and
 11 T    
Here we are considering,


1 1 1 11
33 3 33 31
2111
11EA E T A
ETA
E AE
A A E

 



Where
1 and
2 are thermal gradient in the top and bottom layer respectively.
Numerical results were obtained for various values of the parameters such as taper
parameter, hub radius, rotational speed, thermal gradie nt and geometric parameter . With
consideration of rectangular cross -section, the equation of motion reduces to that of [ 16] in the
absence of taper parameter and for relevant values of the parameters, results of the present study
were compared with the same and good agreement was observed.
Variations of static buckling loads with different parameters are shown in Fig ures 2 to 6.

Fig.2: Variation of
0critP with
 Fig.3: Variation of
0critP with
0C
As shown in the figure 2, the static buckling load decreases with increase in the value of

. The decrement is non -linear in nature and the rate of decrease is more for the higher modes.
C-C case is most stable and P -P is least among the three boundary conditions.
The figure 3 depicts the effect of
0C on the static buckling loads of the system . The static
buckling load remains almost inde pendent of the parameter
0C . In this case also C -C case is
more stable than the C-P and P -P case.

Fig.4: Variation of
0critP with
 Fig.5: Variation of
0critP with
0
The effect of
 on the static buckling loads is shown in the figure 4. The static buckling
loads decreases with increase in the value of
 for all the considered cases. The variation is quite
linear for the first and second mode. For the third mode the variation is non -linear for P -P and C –
P case.
The variation of static buckling load with increase in the value of
0 is shown in the
figure 5. The stati c buc kling increases up to a point for
00 75. , after that increases linearly
with a higher increasing rate C -C case is more stable than the C -P and P -P case.

Fig.6: Variation of
0critP with
q
The figure 6 addresses the effect of
q on the static buckling loads of the system. For
the first mode the static bu ckling loads increases linearly for all the three boundary conditions.
For the second mode under C -P and C -C case, the static buckling load decreases up to certain
value of
q , then increases linearly for higher values of
q . For P -P case the static
buckling load increases non -linearly with
q . For the third mode the static buckling load
increases with increase in the value of
q for all the boundary conditions.
Effects of different parameters on the regions of instability are shown in Fig ures 7 to 25.
P-P Case

Fig.7: Stability diagrams for
01 ,
01 C,
01. and
05.

Fig.8: Stability diagrams for
03 ,
01 C ,
01. and
05.

Fig.9: Stability diagrams for
05 ,
01 C ,
01. and
05.

C-P Case

Fig.10: Stability diagrams for
01 ,
01 C,
01. and
05.

Fig.11: Stability diagrams for
03 ,
01 C ,
01. and
05.

Fig.12: Stability diagrams for
05 ,
01 C ,
01. and
05.
C-C Case

Fig.13: Stability diagrams for
01 ,
01 C,
01. and
05.

Fig.14: Stability diagrams for
03 ,
01 C ,
01. and
05.

Fig.15: Stability diagrams for
05 ,
01 C ,
01. and
05.
Figures 7 to 1 5 address the effect s of
0 on the in stability regions of the system, for
01
, 3 and 5 with
01 C ,
01. ,
05. . All the instability zones are shifted towards right
and became narrower for all the three boundary conditions. The combination resonance zone
 13
is appeared for
05 , still it improves the stability as its area is very small for the P-
P case. The system stability improves with
0 for C -P and C -C case, as all the instability zones
simple as well as combinatio n zones are shifted towards higher excitation frequencies and at
03
and 5, the zone near
 12 for
01 is disappeared.
P-P Case
Fig.16: Stability diagrams for
02 q. ,
05 ,
05. ,
01.

Fig.17: Stability diagrams for
03 q. ,
05 ,
05. ,
01.

Fig.18: Stability diagrams for
05 q. ,
05 ,
05. ,
01.
C-P Case

Fig.19: Stability diagrams for
02 q. ,
05 ,
05. ,
01.

Fig.20: Stability diagrams for
03 q. ,
05 ,
05. ,
01.

Fig.21: Stability diagrams for
05 q. ,
05 ,
05. ,
01.
C-C Case

Fig.22: Stability diagrams for
02 q. ,
05 ,
05. ,
01.

Fig.23: Stability diagrams for
03 q. ,
05 ,
05. ,
01.

Fig.24: Stability diagrams for
05 q. ,
05 ,
05. ,
01.

The effects of
q on the instability regions are shown in figures 1 6 to 24. With
increase in the value of
q from 0.2 to 0.5 with
05 ,
05. ,
01. , the resonance
zones are shifted towards higher excitation frequencies and reduces in their areas. For P -P case
 13
is appeared for
05 q. but it improves the stability for increasing value of
q as
the zone area is very negligible. For C -P and C -C case combination resonance zones of
 13
and
 23 for all the values of
q . The C -C case is more stable than C -P and
P-P case as the excitation frequencies for C -C case are larger than the C -P and P -P case.
P-P Case

Fig.25: Stability diagrams for
01. ,
05 q. ,
05 ,
05.

Fig.26: Stability diagrams for
02. ,
05 q. ,
05 ,
05.

Fig.27: Stability diagrams for
03. ,
05 q. ,
05 ,
05.
C-P Case

Fig.28: Stability diagrams for
01. ,
05 q. ,
05 ,
05.

Fig.29: Stability diagrams for
02. ,
05 q. ,
05 ,
05.

Fig.30: Stability diagrams for
03. ,
05 q. ,
05 ,
05.
C-C Case

Fig.31: Stability diagrams for
01. ,
05 q. ,
05 ,
05.

Fig.32: Stability diagrams for
02. ,
05 q. ,
05 ,
05.

Fig.33: Stability diagrams for
03. ,
05 q. ,
05 ,
05.
As shown in figures 2 5 to 33, the stability of the system decreases with increase in the
value of
 from 0.1 to 0.3 with
05. ,
05 q. ,
05 , as all the resonance zones are
shifted towards lower excitation frequencies and became wider. For C -P and C -C case
 12
is disappeared for all the values of
 . Under C -C case
 23 is also disappeared
for all values of
 . Hence C -C case is most stable while P -P case is least stable among the three
boundar y conditions.
P-P Case

Fig.34: Stability diagrams for
01 C ,
05 ,
05. ,
01.
C-P Case

Fig.35: Stability diagrams for
01 C ,
05 ,
05. ,
01.
C-C Case

Fig.36: Stability diagrams for
01 C ,
05 ,
05. ,
01.
Figures 3 4 to 36 display the effect of
01 C with
05. ,
05 and
01. . For all
the three considered boundary conditions the instability zones remain unaffected with increase in
the value of
0C from 1 to 3 , so the figures are not shown here. From the figures it is clear that C –
C case is more stable than the other two cases.

Fig.37: Stability diagrams for
02. ,
01 C ,
05 ,
01.

Fig.38: Stability diagrams for
05. ,
01 C ,
05 ,
01.

Fig.39: Stability diagrams for
08. ,
01 C ,
05 ,
01.

Fig.40: Stability diagrams for
02. ,
01 C ,
05 ,
01.

Fig.41: Stability diagrams for
05. ,
01 C ,
05 ,
01.

Fig.42: Stability diagrams for
08. ,
01 C ,
05 ,
01.

Fig.43: Stability diagrams for
02. ,
01 C ,
05 ,
01.

Fig.44: Stability diagrams for
05. ,
01 C ,
05 ,
01.

Fig.45: Stability diagrams for
08. ,
01 C ,
05 ,
01.
From figures 37 to 45, it is seen that with increase in the value of
 from 0.2 to 0.8 with
05
,
05. ,
01 C , the instability regions are shifted towards lower excitation frequencies
and became wider for P -P and C -P case. Although for P -P case one of the combination resonance
zone
 13 disappeared for
08. , still it worsen the stability of the system as the area of
other unstable zones are larger in comparison to
02. and
05. . For C -C condition, th e
combination resonance of
 12 disappeared for all the values of
, where as for
02.
and
08.
 23 disappeared. Hence C -C case is more stable.
6. Conclusion
In this work, a computational analysis of the static and dynamic stability of a rotating
taper beam having elliptical cross -section subjected to pulsating axial load with t hermal gradient
under three different boundary conditions is considered. The programming has been developed
by using MATLAB and comparisons are made with the results of earlier researchers [ 16] to test
the validity of the analysis.

Increase in rotational speed and geometric parameters increase the static buckling load
for all the three modes under the three considered boundary conditions. A higher thermal
gradient and taper parameter have a detrimental effect on the static stability of the syste m. The
static buckling loads are almost independent of the hub radius for all the considered cases.
It is seen that the dynamic stability of the system improves with increase of rotational
speed and geometric parameter. However, increase in thermal gradient and taper parameter
makes the beam more sensitive to periodic forces by shifting the instability regions towards
lower excitation frequencies as well as widening of the instability zones. The stability of the
system remains unaffected with increas e in the value of hub radius. For all the cases the C -C
condition is most stable and P -P case is least among the three boundary conditions. Combination
resonance s of sum type occur for all the three boundary conditions.
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