- Email: [email protected]
A finite element free vibration analysis of a thermally stressed spinning plate
Pergamon
0045-7949000245-6
A FINITE ELEMENT THERMALLY
Compurers & Srrucrures Vol. 59, No. 2. pp. 377-385, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/96 Sl5.00 + 0.00
FREE VIBRATION ANALYSIS STRESSED SPINNING PLATE
OF A
G. Venkateswara Rao, G. Sinha, N. Mukherjee and M. Mukhopadhyayt Indian Institute of Technology, Kharagpur-721302, West Bengal, India (Received
24 August 1994)
Abstract-A Lagrangian quadratic finite element formulation is used to analyse the free vibration problem in spinning isotropic plates exposed to a thermal environment. The plate is oriented arbitrarily with respect to the spinning hub. The inplane component of the distributed centrifugal force produces inplane tensile stresses that add to the bending strain energy. The normal component is added to the inertia force. A finite element steady-state heat conduction analysis of the plate yields the two-dimensional temperature field (convection terms due to spinning inertia are accounted for). In the thermo-structural interface, the inplane thermal forces, and subsequently the inplane thermal stresses, are evaluated. These stresses further affect the bending strain energy. The formulation includes the effects of transverse shear deformation and rotary inertia of the plate. In the numerical examples considered, the effects of the speed of rotation, the hub radius and the high temperature gradients on the non-dimensional frequency parameter, are clearly brought out.
time independent component of {d} thermal conductivity of plate in x and y directions temperature conductive stiffness matrix coefficient of thermal expansion of the material thermal force vector thermal stresses displacement due to temperature geometric stiffness matrix due to thermal stresses rotating speed of the blade natural frequency of the blade at zero speed speed ratio = n/w0 radius of the disc radius ratio (r/L) frequency parameter
NOTATION global coordinate system local coordinate system shape function at node i length and breadth of the plate thickness of the plate Young’s modulus of elasticity shear modulus of elasticity Poisson’s ratio Jacobian matrix Jacobian determinant strain membrane strain bending strain stress of any point inside element normal force per unit length shear force per unit length bending moment per unit length twisting moment per unit length shear force per unit length rigidity matrix shear correction factor strain energy due to bending element stiffness matrix element mass matrix shape function matrix assembled stiffness matrix assembled mass matrix displacement vector centrifugal force components in x, y, z directions setting angle of the blade lateral displacement of the blade additional stiffness due to in-plane loading stress due to in-plane loading geometric stiffness matrix normal component of the centrifugal force density of the material time dependent component of {d} t To whom all correspondence
should be addressed.
INTRODUCTION Spinning plates form key structural units of several prime movers and their ancillaries. With the current emphasis on fuel efficiency and high specific strength,
the analysis of spinning plates is receiving enormous design attention for turbomachinery blades in steam and aircraft gas turbine engines, in centrifugal fans and blowers, in centrifugal compressors, etc. Most of these components are exposed to a severe heat environment. This necessitates a deeper probe into the elastic deformations of the plate due to thermal disturbances, apart from the centrifugal forces. Vibration analysis of spinning plates has interested a host of researchers during the past two decades. These investigations have been systematically reviewed by Leissa [l]. In a majority of cases, a turbomachinary blade is modelled as a beam. Leissa [2] presents a comparison of the beam and the shell 377
G. Venkateswara Rao et al.
378
model. Amongst the beam-model investigations Rao [3] and Banerjee and Rao [4] established the fundamental effects of pretwist, camber, taper and rotational velocity of the free vibration characteristics of turbomachinery blades. Dokainish and Rawtani [5] used the finite element technique to evaluate natural frequencies of rotating cantilever plates. Henry and Lalanne [6] and Nagamatsu and Michimura [7] analysed the same problem to reveal the variation of the vibration characteristics with the setting angle. In a series of investigations [8-lo] Leissa et al. used the shallow shell theory and the Ritz method to study vibration of blades with variable thickness, curvatures or pre-twists. In other finite element investigations, Sreenivasamurthy and Ramamurthy [I 11 included the effect of rotational stiffness and tip mass; Gupta [12] included the effect of the coriolis stiffness and Chattopadhyay and Bhattacharyya [13] analysed spinning orthotropic plates with variable thickness, using tapered finite elements. Although an exhaustive array of investigations have been carried out on the vibration analysis of rotating plates, scanty information is available on the effect of temperature gradients on the natural frequencies of rotating plates. Jones [14] studied the effect of high temperature damping on turbomachinery components. However, an exclusive analysis of the thermal effects on spinning plates requires a parallel heat transfer analysis of the plate to extract the temperature distributions. Similar problems have so far been attempted on the basis of assumed temperature distribution in the interface, and no attempt has been made to-date to peform a more refined study of the thermal aspects coupled with the investigation of the vibration characteristics. Furthermore, as the plate spins, rotational convection reinforces the conduction term. Such investigations have not been reported. In the present free-vibration analysis, a nine-noded Lagrangian, isoparametric, thin-shell element is used to discretize the spinning plate. Both the structural and the heat conduction analysis of the plate have been carried out using the same Lagrangian finite element. Convection effects due to spinning inertia are considered in the heat conduction analysis. The effect of transverse shear deformation and rotary inertia of the plate are also taken into account. The plate is considered to have an arbitratry orientation with the spinning hub. Effects of the rotating speed, hub radius and temperature gradient on the natural frequencies of the plate are presented. Comparisons have been made with the published results wherever possible. FINITE ELEMENT
FREE VIBRATION
(2) the thermo-mechanical properties of the plate material are independent of the temperature; (3) the bending deformation follows Mindlin’s hypothesis; (4) amplitudes of vibration are small; (5) the cantilever plate is fixed to the rotating hub; (6) the thickness temperature gradient of the plate is negligible. Free vibration equation The free vibration equation for the undamped non-rotating plate can be written as
where [K,] is the elastic stiffness matrix, [M] denotes the mass matrix and 1 = l/p*, p is the natural frequency of the system. Linear stiflness matrix A nine noded Lagrangian quadratic isoparametric shell element as depicted in Fig. 1 is employed for the analysis of the problem. The element has five degrees of freedom per node, viz., u, v, w, 0, and 0,. The following relationships for the coordinates and field vectors at any point inside the element are valid:
{;}
[zi]{;:} t2)
=i
WX
I
(a) Element
coordinates
7 t 4(-1,l) 0
7(0,1)
3
(1,l)
0
ANALYSIS
Assumptions The following assumptions have been made: (1) the material of the plate is isotropic and obey’s Hooke’s law;
1 (-1,-l)
(b)
<(0,-l)
Isoparametric
1(1,-l) coordinates
Fig. 1. Nine-noded isoparametric shell element.
Free vibration analysis of a thermally stressed spinning plate and IJI is the determinant
379
of the matrix
(11)
where [It] _. is an identity matrix of sixe 5 x 5. The shape functions are explicitly given by N,=$(1+5~,)(1+~,)
fori=5,7
Ni=~(l-~2)(1+~
fori=6,8
-t*)(l
-s*)
The element mass matrix is given by
foriS
Ni=f(l-{2)(1+~~i)~~i
Ni=(l
~~~~ matrix
Ml=
’ ’ ~NITblPIIJl d5 dtl, ss-I -I
where
for i=9.
(4)
The elastic stiffness matrix is given by
[PI= kl
(12)
= ’ ’ PITPIPWI dt dtl, ss-I -1
0
0
0
0
0
pt
0
0
0
0
0
pt
0
0
1 0
0
0
0
0
0
0
N,x
0
0
0
0
0
0
0
0
0
Ni.y
Ni,x
0
- Ni,,
0
- Ni,y
0
N,
0
0
- Nia
- N,,
Ni
0
Ni,y
(7)
(8)
(13)
ooop”o12
(5)
where
pt
0
12 Pt3 i
0
(6)
Finite element thermal stress analysis If heat conduction takes place in a plate moving with velocities U, and U,,, the differential equation of steady state heat conductions, as described by Carlslaw and Jaegar [15], includes the convective effect due to the motion as
(9)
Et3 EDb]= 1q1 -g) lv
0
0
0
vl
0
0
0
0
0
o
o
00 X
o 00
o
l-v 2 5(1-v) 6t2 0
(10)
o
~5(1-v) 6t2
Using a typical Galerkin weighted residual method, minimization of the residual yields the guiding finite element equation in the form
G. Venkateswara Rao et al.
380 where the matrices
can be expressed
as
I [f&l = t
Kc1 = t
I
[BJT[k][&]IJ 1d<
S.ls_I
I
1
ss-I
-I
[NflTIU] [B,]IJ1dc
For a typical rotating plate as shown in Fig. 2, the Cartesian velocity components can be expressed as u, = -!A(x -a/2) and U,=Q(R +y) I
2 W,il{T,I
Solution of eqn ture field of the temperature field face, the thermal
(16)
(15) gives the steady-state temperaplate. With the knowledge of the data in the thermo-structural interinplane forces can be formed as
where Tav denotes the average element temperature. The coefficient of thermal expansion vector is given by
{C(}T=
9
,=I
dq = rotational convectivity matrix
(17)
Temperature boundary conditions are specified over surface S 1, while external flux q is specified over surface S2 denotes the strength of any internal heat generation source, while C is the heat capacity of the plate material. The temperature field can be related to the nodal temperatures through the expression
{TJ =
dq = conductivity matrix
(18)
with [N,i] denoting the temperature shape functions discussed explicitly in eqns (3)-(5). The temperature slope-nodal temperature matrix can be expressed as
(19)
{a,
tL,
0
0
0
0
0
0
0
0).
The static structural equation is next solved, yield the thermal displacements as
(21) to
(22) Subsequently, the thermal using the relation {Q,} =
stresses
can be found
PIPI{&,) = {N:, N:>N:,.).
(23)
Centrifugal and thermal eflects on free vibration Centrifugal force effects have been treated in the same way as given in Ref. [5]. A typical cantilever plate rotating at speed !LI is shown in Fig. 2. The centrifugal force per unit volume at any point in the global axes is given by F, = pR’(x + r); &=pR’
ycos’0
F2=p02
;
-:sin20
>
( -$sin20
.
+w sin20
>
(24)
F,, Fp are the inplane force components in the midplane of the plate and for small amplitudes Fig. 2. Rotating cantilever plate.
F, = pn2(x + r);
F, = pn’y cos’ 0.
(25)
Free vibration analysis of a thermally stressed spinning plate
The additional strain energy stored in the plate due to the inplane centrifugal stresses produced by these forces, is given by
If {JJ’} denotes the constant component of {F,,} which is independent of displacements, eqn (31) gives l-1 Pl
{hJ={P’)+Pt
J. J. -I
where {crC}== {NY, NP, NT,} denotes the centrifugal stresses. This additional strain energy leads to an increase in the bending stiffness by an amount
381
-I
x [N,IT(R2w sin2 0 - ti) d
(33)
(Fb} = {p’} + a2 sin2 f3[M]{di} - [M]{di}.
(34)
After global assembly of the stiffness and mass matrices of the rotating plate element, the guiding equation of free vibration assumes the form ([k] - (w2 + R2 sin2 0)[M]){d,}
This matrix is often referred to as the centrifugal stiffness or geometric stiffness matrix, due to centrifugal action. [G] is defined as
NUMERICAL EXAMPLES Examples solved by the proposed approach are given below: Example
F=F,--F2,=pR2(-y
sinBcosO+w
The nodal force vector equivalent tributed force takes the form
to this
1. Rotating
1[WIT dv, Jr
where
[NJ = [
0
1
N,
0
0
N,
0 .
0
0
N,
cantilever plate: free
(30)
TT
dis-
1
P
B
0
0
Mode no.
1
(31) 2
1
45
for a
Frequency parameter A Present method Ref. 151
5
5.094 9.884 23.016 27.782 32.847
5.126 9.844 22.994 28.172 32.445
1 2 3 4 5
10.790 12.810 29.212 34.044 41.495
IO.756 13.893 29.184 34.681 41.283
2 3 4
(32)
vibration
A cantilever plate rotating at a fixed angular velocity has been analysed by using a finite element mesh 4 x 4. Non-dimensional frequency parameters for the first five frequencies for variation of si, V and 0 are given in Table 1, which also incorporates the results of Dokaimish and Rawtani [5]. Graphical comparison of the results are given in Figs 3 and 4, where excellent agreement between them has been obtained.
n
{fir) =
finite element
Table 1. Frequency parameter 1 ( =oL2m rotating square cantilever plate
sin2B)+pC.
(35)
Since non-linear strain terms occur in the geometric stiffness matrix, an iterative procedure is adopted, as discussed earlier [ 111, to evaluate the centrifugal stresses with an increase in the speed of rotation. The thermal geometric stiffness matrix, on the contrary, is evaluated once and for all. A computer program in FORTRAN 77 has been developed based on the above procedure to solve different problems.
Identically, the inplane thermal stresses also affect the bending strain energy, and the additional bending stiffness due to heat loads is given by
It is to be noted at this stage that the centrifugal stiffness invariably has a positive contribution to the overall stiffness, as centrifugal forces would necessarily induce tensile stresses in the plate. On the contrary, thermal stresses can be compressive in certain regions of the plate and can consequently deplete the overall bending stiffness, producing a weaker structure. From D’Alembert’s principle, the total force acting in the z direction on the plate is given by
= {O}.
FEM MESH: 4 x 4 for both E = IO.92 x lo6 kgcm-2, v = 0.3, t = 0.01 cm, p = 100 kg cm-j, L = 1cm.
G. Venkateswara Rao et al.
382
2.1
-
Present
1.9 -
1.a Y 1.7
t t
a
2
1 Speed
ratio
Tg=57JK
i
4
3
0
Y la
(WlWo)
Fig. 3. Variation of natural frequency with speed of rotation for 0 = 0”.
Fig. 5. Comparison of the present heat conduction solution (FEM) with the analytical solution for a moving platetemperature gradient at X = a/2.
Example
following the procedure Jaegar [15] as
2. Rotating
cantilever plate: heat conduction
Figure 5 depicts a square plate with Dirichlet boundary conditions imposed at all edges moving in the y direction with a velocity UY= Ll,pC/k, = 0.39. The analytical solution to the problem can be derived
described
cc2tg(/-cosnn)
T(x, y) = C n=l
nx (emz,emr3
by Carslaw
and
(emzu- e”lY)sin y (34)
where
--0-
Present
--*--
Ref
(5 1
Llb
m2 = ( iTJy- d-)/2.
~1
(37)
The close agreement of the analytica solution and finite element solution validates the present heat transfer formulation. Example 3. Rotating vironment
00 0
2
1 Speed
ratio
3
1
(W/W,)
Fig. 4. Variation of natural frequency with speed of rotation for 19= 45”.
cantilever plate in thermal en-
A cantilever plate rotating clockwise at a = 1 (0 denotes the ratio of the angular speed to the lowest natural frequency of the static plate) is exposed to a tip flux (q = 40 w cmm2). The support edge is maintained at a constant temperature of 343 K. The temperaure field over the plate is presented in Fig. 6 while the stress contours are illustrated in Fig. 8. The non-dimensional stress resultants are expressed as RX = RX/(EccTt) and nY = &,/(EaTt). The effects of the thermal environment on the natural frequencies of the rotating plate are explicitly presented in Table 2. As the radius increases the effect of the convective term in the direction of U, (in the direction of
Free vibration analysis of a thermally stressed spinning plate
383
Fig. 6. Temperature contours for the plate at R = 18 cm. rotation) predominates, leaving the plate at an almost uniform temperature, except for the very hot trailing edge. Consequently, as the trailing edge gets hotter due to an increase in R, the thermal stresses grow to reduce the natural frequencies.
SUMMARY
AND CONCLUSIONS
A nine-node1 Lagrangian shell element is employed to model a spinning plate. The finite element formu-
lation is based on Mindlin’s hypothesis and considers the effects of transverse shear deformation and rotary inertia. The heat conduction equation is solved employing the same element. The thermal forces and stresses are subsequently evaluated. The additional geometric stiffness matrices due to centrifugal and thermal inplane stresses are considered. The normal component of the distributed centrifugal force is added to the inertial force. A simultaneous iteration routine [16] is used to solve the relevant eigenproblem.
6
0 0
3
IA 6
9
-0.16 I 12
I
!
15
18
21
Fig. 7. Normal stress (RX) contours at R = 18 cm.
384
G. Venkateswara
Rao et al.
1
n
L P
0
2.25
4.50
6.75
9.00
11.25
13.50
15.75
18.00
20.25
Fig. 8. Normal stress (nY) contours at R = 18 cm. The results of this work show an increase frequency parameter with the speed of rotation. an
increase
magnify,
Table
in the resulting
setting
angle,
in a structural
the
inertia
damping
in the With forces of
the
2. Natural frequencies w (rads-‘) for a rotating square cantilever plate in a thermal environment
Mode
OJ
0
no.
0.75
1 2 3 4 5
29.078 70.368 177.586 222.273 255.656
28.512 68.943 175.689 222.411 250.543
- 1.903 -2.021 - 1.068 - 1.707 -2.000
1.00
1 2 3 4 5
29.085 70.371 177.592 226.274 255.661
27.923 65.219 172.891 216.123 237.125
- 3.995 -7.321 -2.647 -4.486 - 7.250
1.125
1 2 3 4 5
29.086 70.374 177.597 226.275 255.664
26.983 62.196 169.237 210.529 229.862
- 7.230 -11.621 -4.707 -6.959 - 10.092
1.25
1 2 3 4 5
29.089 70.378 177.601 226.277 255.667
26.283 60.012 166.982 206.122 223.812
- 9.646 - 14.729 - 5.979 - 8.907 - 12.460
heat
with heat
X
system, and the frequency parameter lowers. The frequencies also increase with radius ratio. However, steep thermal gradients produce a high density of compressive zones within the rotating plate. Consequently the natural frequencies lower. The first torsional mode is affected most crucially by temperature gradients.
%
R/a
without
22.50
change
E = 210 x lo6 kg cm-*, Y = 0.3, f = 1.0 cm, a = 24 cm, p = 7.85 g cm-‘, e = 900, a= 1.0, k =29WmK-‘, a = 20 x lO-6 K-‘.
REFERENCES 1. A. W. Leissa, Vibrational aspects of rotating turbomachinery blades. Appl. Mech. Rev. 34, 629635 (1981). 2. A. W. Leissa and M. S. Ewing, Comparison of beam and shell theories for the vibrations of thin turbomachinery blades. J. Engng Power Trans. ASME 105, 383-392 (1983). 3. J. S. Rao, Turbomachinery blade vibration. Shock Vibr. Dig. 12, 19-26 (1977). 4. S. Banerjee and J S. Rao, Coupled bending-torsion vibrations of rotating blades. ASME Paper no. 76 GT-43 (1976). 5. M. A. Dokainish and S. Rawtani, Vibration analysis of rotating cantilever plates. Inr. J. numer. Meth. Engng 3, 233-248 (1971). 6. R. Henry and M. Lalanne, Vibration analysis of rotating compressor blades. J. Engng Ind. Tram ASME 96, 1028-1035 (1974). 7. A. Nagamastsu and S. Michimura, Vibration of blades of rotating impeller. Theor. appl. Mech. 92, 327-338 (1978). 8. A. W. Leissa, J. K. Lee and A. J. Wang, Vibrations of cantilevered shallow cylindrical shells of rectangular planform. J. Sound Vibr. 78, 31 l-328 (1981). 9. A. W. Leissa, J. K. Lee and A. J. Wang, Rotating blade vibration analysis using shells. J. Engng Power Trans. ASME 104, 296-302 (1982).
Free vibration analysis of a thermally stressed spinning plate 10. A. W. Leissa, J. K. Lee and A. J. Wang, Vibration of blades with variable thickness and curvature by shell theory. J. Engng Gas Turb. Power Trans. ASME 106, 1l-16 (1984). 11. S. Sreenivasamurthy and V. Ramamurthy, A parametric study of vibration of rotating, twisted and tappered low aspect ratio cantilever plates. J. Sound Vibr. 70, 598-601 (1980). 12. K. K. Gupta, A special structural element for the analysis of spinning plates. Int. J. numer. Meth. Engng 7, 223-234.
CAS 59/2--M
385
13. T. K. Chattopadhyay and S. Bhattacharyya, Free flexma1 vibration analysis of spinning orthotropic and isotropic plates. Proc. Ind. Nat1 Sci. Acad. 56, 427441 (1990).
14. D. I. G. Jones, High temperature damping of dynamic systems. Shock Vibr. Dig. 11, 13-18 (1979). 15. J. S. Carslaw and C. Jaegar, Conduction of Heat in Solids. Clarendon, Oxford (1959). 16. R. B. Corr and A. Jennings, A simultaneous iteration algorithm for symmetric eigenvalue problem. Inr. J. numer. Meth. Engng 10, 647-663. (1976).
0045-7949000245-6
A FINITE ELEMENT THERMALLY
Compurers & Srrucrures Vol. 59, No. 2. pp. 377-385, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/96 Sl5.00 + 0.00
FREE VIBRATION ANALYSIS STRESSED SPINNING PLATE
OF A
G. Venkateswara Rao, G. Sinha, N. Mukherjee and M. Mukhopadhyayt Indian Institute of Technology, Kharagpur-721302, West Bengal, India (Received
24 August 1994)
Abstract-A Lagrangian quadratic finite element formulation is used to analyse the free vibration problem in spinning isotropic plates exposed to a thermal environment. The plate is oriented arbitrarily with respect to the spinning hub. The inplane component of the distributed centrifugal force produces inplane tensile stresses that add to the bending strain energy. The normal component is added to the inertia force. A finite element steady-state heat conduction analysis of the plate yields the two-dimensional temperature field (convection terms due to spinning inertia are accounted for). In the thermo-structural interface, the inplane thermal forces, and subsequently the inplane thermal stresses, are evaluated. These stresses further affect the bending strain energy. The formulation includes the effects of transverse shear deformation and rotary inertia of the plate. In the numerical examples considered, the effects of the speed of rotation, the hub radius and the high temperature gradients on the non-dimensional frequency parameter, are clearly brought out.
time independent component of {d} thermal conductivity of plate in x and y directions temperature conductive stiffness matrix coefficient of thermal expansion of the material thermal force vector thermal stresses displacement due to temperature geometric stiffness matrix due to thermal stresses rotating speed of the blade natural frequency of the blade at zero speed speed ratio = n/w0 radius of the disc radius ratio (r/L) frequency parameter
NOTATION global coordinate system local coordinate system shape function at node i length and breadth of the plate thickness of the plate Young’s modulus of elasticity shear modulus of elasticity Poisson’s ratio Jacobian matrix Jacobian determinant strain membrane strain bending strain stress of any point inside element normal force per unit length shear force per unit length bending moment per unit length twisting moment per unit length shear force per unit length rigidity matrix shear correction factor strain energy due to bending element stiffness matrix element mass matrix shape function matrix assembled stiffness matrix assembled mass matrix displacement vector centrifugal force components in x, y, z directions setting angle of the blade lateral displacement of the blade additional stiffness due to in-plane loading stress due to in-plane loading geometric stiffness matrix normal component of the centrifugal force density of the material time dependent component of {d} t To whom all correspondence
should be addressed.
INTRODUCTION Spinning plates form key structural units of several prime movers and their ancillaries. With the current emphasis on fuel efficiency and high specific strength,
the analysis of spinning plates is receiving enormous design attention for turbomachinery blades in steam and aircraft gas turbine engines, in centrifugal fans and blowers, in centrifugal compressors, etc. Most of these components are exposed to a severe heat environment. This necessitates a deeper probe into the elastic deformations of the plate due to thermal disturbances, apart from the centrifugal forces. Vibration analysis of spinning plates has interested a host of researchers during the past two decades. These investigations have been systematically reviewed by Leissa [l]. In a majority of cases, a turbomachinary blade is modelled as a beam. Leissa [2] presents a comparison of the beam and the shell 377
G. Venkateswara Rao et al.
378
model. Amongst the beam-model investigations Rao [3] and Banerjee and Rao [4] established the fundamental effects of pretwist, camber, taper and rotational velocity of the free vibration characteristics of turbomachinery blades. Dokainish and Rawtani [5] used the finite element technique to evaluate natural frequencies of rotating cantilever plates. Henry and Lalanne [6] and Nagamatsu and Michimura [7] analysed the same problem to reveal the variation of the vibration characteristics with the setting angle. In a series of investigations [8-lo] Leissa et al. used the shallow shell theory and the Ritz method to study vibration of blades with variable thickness, curvatures or pre-twists. In other finite element investigations, Sreenivasamurthy and Ramamurthy [I 11 included the effect of rotational stiffness and tip mass; Gupta [12] included the effect of the coriolis stiffness and Chattopadhyay and Bhattacharyya [13] analysed spinning orthotropic plates with variable thickness, using tapered finite elements. Although an exhaustive array of investigations have been carried out on the vibration analysis of rotating plates, scanty information is available on the effect of temperature gradients on the natural frequencies of rotating plates. Jones [14] studied the effect of high temperature damping on turbomachinery components. However, an exclusive analysis of the thermal effects on spinning plates requires a parallel heat transfer analysis of the plate to extract the temperature distributions. Similar problems have so far been attempted on the basis of assumed temperature distribution in the interface, and no attempt has been made to-date to peform a more refined study of the thermal aspects coupled with the investigation of the vibration characteristics. Furthermore, as the plate spins, rotational convection reinforces the conduction term. Such investigations have not been reported. In the present free-vibration analysis, a nine-noded Lagrangian, isoparametric, thin-shell element is used to discretize the spinning plate. Both the structural and the heat conduction analysis of the plate have been carried out using the same Lagrangian finite element. Convection effects due to spinning inertia are considered in the heat conduction analysis. The effect of transverse shear deformation and rotary inertia of the plate are also taken into account. The plate is considered to have an arbitratry orientation with the spinning hub. Effects of the rotating speed, hub radius and temperature gradient on the natural frequencies of the plate are presented. Comparisons have been made with the published results wherever possible. FINITE ELEMENT
FREE VIBRATION
(2) the thermo-mechanical properties of the plate material are independent of the temperature; (3) the bending deformation follows Mindlin’s hypothesis; (4) amplitudes of vibration are small; (5) the cantilever plate is fixed to the rotating hub; (6) the thickness temperature gradient of the plate is negligible. Free vibration equation The free vibration equation for the undamped non-rotating plate can be written as
where [K,] is the elastic stiffness matrix, [M] denotes the mass matrix and 1 = l/p*, p is the natural frequency of the system. Linear stiflness matrix A nine noded Lagrangian quadratic isoparametric shell element as depicted in Fig. 1 is employed for the analysis of the problem. The element has five degrees of freedom per node, viz., u, v, w, 0, and 0,. The following relationships for the coordinates and field vectors at any point inside the element are valid:
{;}
[zi]{;:} t2)
=i
WX
I
(a) Element
coordinates
7 t 4(-1,l) 0
7(0,1)
3
(1,l)
0
ANALYSIS
Assumptions The following assumptions have been made: (1) the material of the plate is isotropic and obey’s Hooke’s law;
1 (-1,-l)
(b)
<(0,-l)
Isoparametric
1(1,-l) coordinates
Fig. 1. Nine-noded isoparametric shell element.
Free vibration analysis of a thermally stressed spinning plate and IJI is the determinant
379
of the matrix
(11)
where [It] _. is an identity matrix of sixe 5 x 5. The shape functions are explicitly given by N,=$(1+5~,)(1+~,)
fori=5,7
Ni=~(l-~2)(1+~
fori=6,8
-t*)(l
-s*)
The element mass matrix is given by
foriS
Ni=f(l-{2)(1+~~i)~~i
Ni=(l
~~~~ matrix
Ml=
’ ’ ~NITblPIIJl d5 dtl, ss-I -I
where
for i=9.
(4)
The elastic stiffness matrix is given by
[PI= kl
(12)
= ’ ’ PITPIPWI dt dtl, ss-I -1
0
0
0
0
0
pt
0
0
0
0
0
pt
0
0
1 0
0
0
0
0
0
0
N,x
0
0
0
0
0
0
0
0
0
Ni.y
Ni,x
0
- Ni,,
0
- Ni,y
0
N,
0
0
- Nia
- N,,
Ni
0
Ni,y
(7)
(8)
(13)
ooop”o12
(5)
where
pt
0
12 Pt3 i
0
(6)
Finite element thermal stress analysis If heat conduction takes place in a plate moving with velocities U, and U,,, the differential equation of steady state heat conductions, as described by Carlslaw and Jaegar [15], includes the convective effect due to the motion as
(9)
Et3 EDb]= 1q1 -g) lv
0
0
0
vl
0
0
0
0
0
o
o
00 X
o 00
o
l-v 2 5(1-v) 6t2 0
(10)
o
~5(1-v) 6t2
Using a typical Galerkin weighted residual method, minimization of the residual yields the guiding finite element equation in the form
G. Venkateswara Rao et al.
380 where the matrices
can be expressed
as
I [f&l = t
Kc1 = t
I
[BJT[k][&]IJ 1d<
S.ls_I
I
1
ss-I
-I
[NflTIU] [B,]IJ1dc
For a typical rotating plate as shown in Fig. 2, the Cartesian velocity components can be expressed as u, = -!A(x -a/2) and U,=Q(R +y) I
2 W,il{T,I
Solution of eqn ture field of the temperature field face, the thermal
(16)
(15) gives the steady-state temperaplate. With the knowledge of the data in the thermo-structural interinplane forces can be formed as
where Tav denotes the average element temperature. The coefficient of thermal expansion vector is given by
{C(}T=
9
,=I
dq = rotational convectivity matrix
(17)
Temperature boundary conditions are specified over surface S 1, while external flux q is specified over surface S2 denotes the strength of any internal heat generation source, while C is the heat capacity of the plate material. The temperature field can be related to the nodal temperatures through the expression
{TJ =
dq = conductivity matrix
(18)
with [N,i] denoting the temperature shape functions discussed explicitly in eqns (3)-(5). The temperature slope-nodal temperature matrix can be expressed as
(19)
{a,
tL,
0
0
0
0
0
0
0
0).
The static structural equation is next solved, yield the thermal displacements as
(21) to
(22) Subsequently, the thermal using the relation {Q,} =
stresses
can be found
PIPI{&,) = {N:, N:>N:,.).
(23)
Centrifugal and thermal eflects on free vibration Centrifugal force effects have been treated in the same way as given in Ref. [5]. A typical cantilever plate rotating at speed !LI is shown in Fig. 2. The centrifugal force per unit volume at any point in the global axes is given by F, = pR’(x + r); &=pR’
ycos’0
F2=p02
;
-:sin20
>
( -$sin20
.
+w sin20
>
(24)
F,, Fp are the inplane force components in the midplane of the plate and for small amplitudes Fig. 2. Rotating cantilever plate.
F, = pn2(x + r);
F, = pn’y cos’ 0.
(25)
Free vibration analysis of a thermally stressed spinning plate
The additional strain energy stored in the plate due to the inplane centrifugal stresses produced by these forces, is given by
If {JJ’} denotes the constant component of {F,,} which is independent of displacements, eqn (31) gives l-1 Pl
{hJ={P’)+Pt
J. J. -I
where {crC}== {NY, NP, NT,} denotes the centrifugal stresses. This additional strain energy leads to an increase in the bending stiffness by an amount
381
-I
x [N,IT(R2w sin2 0 - ti) d
(33)
(Fb} = {p’} + a2 sin2 f3[M]{di} - [M]{di}.
(34)
After global assembly of the stiffness and mass matrices of the rotating plate element, the guiding equation of free vibration assumes the form ([k] - (w2 + R2 sin2 0)[M]){d,}
This matrix is often referred to as the centrifugal stiffness or geometric stiffness matrix, due to centrifugal action. [G] is defined as
NUMERICAL EXAMPLES Examples solved by the proposed approach are given below: Example
F=F,--F2,=pR2(-y
sinBcosO+w
The nodal force vector equivalent tributed force takes the form
to this
1. Rotating
1[WIT dv, Jr
where
[NJ = [
0
1
N,
0
0
N,
0 .
0
0
N,
cantilever plate: free
(30)
TT
dis-
1
P
B
0
0
Mode no.
1
(31) 2
1
45
for a
Frequency parameter A Present method Ref. 151
5
5.094 9.884 23.016 27.782 32.847
5.126 9.844 22.994 28.172 32.445
1 2 3 4 5
10.790 12.810 29.212 34.044 41.495
IO.756 13.893 29.184 34.681 41.283
2 3 4
(32)
vibration
A cantilever plate rotating at a fixed angular velocity has been analysed by using a finite element mesh 4 x 4. Non-dimensional frequency parameters for the first five frequencies for variation of si, V and 0 are given in Table 1, which also incorporates the results of Dokaimish and Rawtani [5]. Graphical comparison of the results are given in Figs 3 and 4, where excellent agreement between them has been obtained.
n
{fir) =
finite element
Table 1. Frequency parameter 1 ( =oL2m rotating square cantilever plate
sin2B)+pC.
(35)
Since non-linear strain terms occur in the geometric stiffness matrix, an iterative procedure is adopted, as discussed earlier [ 111, to evaluate the centrifugal stresses with an increase in the speed of rotation. The thermal geometric stiffness matrix, on the contrary, is evaluated once and for all. A computer program in FORTRAN 77 has been developed based on the above procedure to solve different problems.
Identically, the inplane thermal stresses also affect the bending strain energy, and the additional bending stiffness due to heat loads is given by
It is to be noted at this stage that the centrifugal stiffness invariably has a positive contribution to the overall stiffness, as centrifugal forces would necessarily induce tensile stresses in the plate. On the contrary, thermal stresses can be compressive in certain regions of the plate and can consequently deplete the overall bending stiffness, producing a weaker structure. From D’Alembert’s principle, the total force acting in the z direction on the plate is given by
= {O}.
FEM MESH: 4 x 4 for both E = IO.92 x lo6 kgcm-2, v = 0.3, t = 0.01 cm, p = 100 kg cm-j, L = 1cm.
G. Venkateswara Rao et al.
382
2.1
-
Present
1.9 -
1.a Y 1.7
t t
a
2
1 Speed
ratio
Tg=57JK
i
4
3
0
Y la
(WlWo)
Fig. 3. Variation of natural frequency with speed of rotation for 0 = 0”.
Fig. 5. Comparison of the present heat conduction solution (FEM) with the analytical solution for a moving platetemperature gradient at X = a/2.
Example
following the procedure Jaegar [15] as
2. Rotating
cantilever plate: heat conduction
Figure 5 depicts a square plate with Dirichlet boundary conditions imposed at all edges moving in the y direction with a velocity UY= Ll,pC/k, = 0.39. The analytical solution to the problem can be derived
described
cc2tg(/-cosnn)
T(x, y) = C n=l
nx (emz,emr3
by Carslaw
and
(emzu- e”lY)sin y (34)
where
--0-
Present
--*--
Ref
(5 1
Llb
m2 = ( iTJy- d-)/2.
~1
(37)
The close agreement of the analytica solution and finite element solution validates the present heat transfer formulation. Example 3. Rotating vironment
00 0
2
1 Speed
ratio
3
1
(W/W,)
Fig. 4. Variation of natural frequency with speed of rotation for 19= 45”.
cantilever plate in thermal en-
A cantilever plate rotating clockwise at a = 1 (0 denotes the ratio of the angular speed to the lowest natural frequency of the static plate) is exposed to a tip flux (q = 40 w cmm2). The support edge is maintained at a constant temperature of 343 K. The temperaure field over the plate is presented in Fig. 6 while the stress contours are illustrated in Fig. 8. The non-dimensional stress resultants are expressed as RX = RX/(EccTt) and nY = &,/(EaTt). The effects of the thermal environment on the natural frequencies of the rotating plate are explicitly presented in Table 2. As the radius increases the effect of the convective term in the direction of U, (in the direction of
Free vibration analysis of a thermally stressed spinning plate
383
Fig. 6. Temperature contours for the plate at R = 18 cm. rotation) predominates, leaving the plate at an almost uniform temperature, except for the very hot trailing edge. Consequently, as the trailing edge gets hotter due to an increase in R, the thermal stresses grow to reduce the natural frequencies.
SUMMARY
AND CONCLUSIONS
A nine-node1 Lagrangian shell element is employed to model a spinning plate. The finite element formu-
lation is based on Mindlin’s hypothesis and considers the effects of transverse shear deformation and rotary inertia. The heat conduction equation is solved employing the same element. The thermal forces and stresses are subsequently evaluated. The additional geometric stiffness matrices due to centrifugal and thermal inplane stresses are considered. The normal component of the distributed centrifugal force is added to the inertial force. A simultaneous iteration routine [16] is used to solve the relevant eigenproblem.
6
0 0
3
IA 6
9
-0.16 I 12
I
!
15
18
21
Fig. 7. Normal stress (RX) contours at R = 18 cm.
384
G. Venkateswara
Rao et al.
1
n
L P
0
2.25
4.50
6.75
9.00
11.25
13.50
15.75
18.00
20.25
Fig. 8. Normal stress (nY) contours at R = 18 cm. The results of this work show an increase frequency parameter with the speed of rotation. an
increase
magnify,
Table
in the resulting
setting
angle,
in a structural
the
inertia
damping
in the With forces of
the
2. Natural frequencies w (rads-‘) for a rotating square cantilever plate in a thermal environment
Mode
OJ
0
no.
0.75
1 2 3 4 5
29.078 70.368 177.586 222.273 255.656
28.512 68.943 175.689 222.411 250.543
- 1.903 -2.021 - 1.068 - 1.707 -2.000
1.00
1 2 3 4 5
29.085 70.371 177.592 226.274 255.661
27.923 65.219 172.891 216.123 237.125
- 3.995 -7.321 -2.647 -4.486 - 7.250
1.125
1 2 3 4 5
29.086 70.374 177.597 226.275 255.664
26.983 62.196 169.237 210.529 229.862
- 7.230 -11.621 -4.707 -6.959 - 10.092
1.25
1 2 3 4 5
29.089 70.378 177.601 226.277 255.667
26.283 60.012 166.982 206.122 223.812
- 9.646 - 14.729 - 5.979 - 8.907 - 12.460
heat
with heat
X
system, and the frequency parameter lowers. The frequencies also increase with radius ratio. However, steep thermal gradients produce a high density of compressive zones within the rotating plate. Consequently the natural frequencies lower. The first torsional mode is affected most crucially by temperature gradients.
%
R/a
without
22.50
change
E = 210 x lo6 kg cm-*, Y = 0.3, f = 1.0 cm, a = 24 cm, p = 7.85 g cm-‘, e = 900, a= 1.0, k =29WmK-‘, a = 20 x lO-6 K-‘.
REFERENCES 1. A. W. Leissa, Vibrational aspects of rotating turbomachinery blades. Appl. Mech. Rev. 34, 629635 (1981). 2. A. W. Leissa and M. S. Ewing, Comparison of beam and shell theories for the vibrations of thin turbomachinery blades. J. Engng Power Trans. ASME 105, 383-392 (1983). 3. J. S. Rao, Turbomachinery blade vibration. Shock Vibr. Dig. 12, 19-26 (1977). 4. S. Banerjee and J S. Rao, Coupled bending-torsion vibrations of rotating blades. ASME Paper no. 76 GT-43 (1976). 5. M. A. Dokainish and S. Rawtani, Vibration analysis of rotating cantilever plates. Inr. J. numer. Meth. Engng 3, 233-248 (1971). 6. R. Henry and M. Lalanne, Vibration analysis of rotating compressor blades. J. Engng Ind. Tram ASME 96, 1028-1035 (1974). 7. A. Nagamastsu and S. Michimura, Vibration of blades of rotating impeller. Theor. appl. Mech. 92, 327-338 (1978). 8. A. W. Leissa, J. K. Lee and A. J. Wang, Vibrations of cantilevered shallow cylindrical shells of rectangular planform. J. Sound Vibr. 78, 31 l-328 (1981). 9. A. W. Leissa, J. K. Lee and A. J. Wang, Rotating blade vibration analysis using shells. J. Engng Power Trans. ASME 104, 296-302 (1982).
Free vibration analysis of a thermally stressed spinning plate 10. A. W. Leissa, J. K. Lee and A. J. Wang, Vibration of blades with variable thickness and curvature by shell theory. J. Engng Gas Turb. Power Trans. ASME 106, 1l-16 (1984). 11. S. Sreenivasamurthy and V. Ramamurthy, A parametric study of vibration of rotating, twisted and tappered low aspect ratio cantilever plates. J. Sound Vibr. 70, 598-601 (1980). 12. K. K. Gupta, A special structural element for the analysis of spinning plates. Int. J. numer. Meth. Engng 7, 223-234.
CAS 59/2--M
385
13. T. K. Chattopadhyay and S. Bhattacharyya, Free flexma1 vibration analysis of spinning orthotropic and isotropic plates. Proc. Ind. Nat1 Sci. Acad. 56, 427441 (1990).
14. D. I. G. Jones, High temperature damping of dynamic systems. Shock Vibr. Dig. 11, 13-18 (1979). 15. J. S. Carslaw and C. Jaegar, Conduction of Heat in Solids. Clarendon, Oxford (1959). 16. R. B. Corr and A. Jennings, A simultaneous iteration algorithm for symmetric eigenvalue problem. Inr. J. numer. Meth. Engng 10, 647-663. (1976).