Vibration analysis of polar orthotropic annular discs

Journal of Sound and Vibration (1979) 63 l), 97-l 05

VIBRATION

ANALYSIS OF POLAR ORTHOTROPIC ANNULAR DISCS-l F. GINESU, B. PICASSO AND P. PRIOLO Universitd di Cagliari, 09100 Cagliari, Italy

(Received 18 October 1978, and in revisedjbrm 8 February 1979)

With the aim of designing structural elements of filament wound composite materials, a first analysis of the vibrating modes of an annular orthotropic disc was performed. To check the reliability of both the analytical and experimental approach, a uniform steel disc was previously tested. Natural frequencies were computed by means of a finite element program, while the experimental analysis was based on real time and time average holographic interferometry. For orthotropic discs, whose material properties are generally difficult to determine, the holographic analysis of vibrations can be used as a complementary instrument for testing the material and detecting defects.

1. INTRODUCTION

The use of filament wound materials for medium-high energy density flywheels is, at the present time, a widespread practice. Moreover, the usefulness of a basic approach to polar orthotropic materials can derive from their increasing use in designing turbomachinery components. The authors found that transverse flexural vibrations can constitute a major problem in designing flywheels. With the above considerations in mind, and in order to make a first theoretical-experimental study of the problem, a plane orthotropic nonrotating annular disc of constant thickness was analyzed. In preceding papers the Kirchhoff plate theory has been applied to take into account thickness variations, the effect of rotary inertia and transverse shear deformations [l], material orthotropy [2], and eventual rotation [3]. The Rayleigh-Ritz method has also been used to determine natural frequencies of polar orthotropic discs [4]. The method enables modes with nodal circles and diameters to be determined, provided that suitable admissible functions are chosen. Kirkhope and Wilson [.5] have recently used the finite element method to compute natural frequencies of isotropic thin rotating discs, introducing an additional stiffness matrix related to the stress state induced by rotation. This approach has been adopted by the authors, by extending its applicability to polar orthotropic materials. The experimental analysis on small scale models was mainly performed by means of real time holographic interferometry with mismatch fringes. The presence of initial fringes, generally considered a drawback in real time holography can, however, be particularly useful for the vibration

analysis;

when the disc is excited

at the frequency

of a resonant

mode, initial fringes suddenly disappear at the points where the amplitude is not zero, remaining visible along nodal lines only. Two models were tested: the first, a plane isotropic steel disc, was used to check the experimental set up; the second was a filament wound glass fibre-polyester resin composite plane disc. t Presented

at the Sixth Symposium

on Mechanisms

and Gears, Miskolc,

Hungary,

5-7 September

1978.

97 0022460X/79/130097+09

%02.00/O

0

1979 Academic

Press Inc. (London)

Limited

98

F. GINESU,

B. PICASSO

AND P. PRIOLO

2. FINITE ELEMENT

APPROACH

The finite element of Figure 1 has two nodes, with 2 degrees of freedom/node [S]. If {d} represents the column vector where 4 degrees of freedom of the element are listed,

the following relation between internal normal displacement w(r, 4) and {d} is assumed : w(r, 5) = CN,N2 N3 NJ ~0sm5IdI,

(1)

I

Figure

1. Two node finite element [5]

where N, N, N, N, are Hermitian polynomials of the third degree and m is the number of nodal diameters. If one now introduces the TOW vector [r] = [l r r* r3], relation (1) can be written as w(r, 0 = CrlC&l cm m5Id 1. Assuming the strain-displacement

16W

h2W

Er=-S’ one can write

relations [63

“5=----’ r6r

162w r2 St2

&r5 =2

1 d2w ___f$)

( r 6rS[

1

1s) = Cd [&I Wt. Now, introducing stresses (that are bending and shearing moments) M,, M, and M,.,, one

can write for a polar orthotropic elastic material 1W = [DFlI&Jt, with [DF] the elastic bending matrix. The stiffness and mass matrices can now be obtained:

L&l =

C&ITCrFIT CW CrFlC&l Wol) = CJU’Cbl CRJt, svol Cm1=

t The

expression

for the unknown

CNI’CNId(vol) = C&lTCmdlC~dlt~ sWI matrix of this relation

is given in the Appendix.

99

POLAR ORTHOTROPIC ANNULAR DISCS

Matrix [kF] differs from that obtained in reference [S] because it takes into account the orthotropic material properties. If the disc rotates, with radial and circumferential stresses arising from the centrifugal force field, a correcting stiffness matrix can be easily introduced [S]. To compute natural frequency values an iterative method [7] was adopted. The complete program for computing natural bending frequencies and stresses for rotating discs of non-uniform thickness was named EACO; several checks of the program were carried out for various disc configurations (see Table 1). TABLE 1 Finite element program (EACO) checks for various disc configurations

Case

Steel disct w=o C-F

Reference

Prog. EACO

c51 PI

r n=O

m=O

, n=l

497 493 490

3244

638 644

3396

3200

c41 Steel disc? w = 400 C-F

EACO

Orthotropic disc$ w=o F-C

EACO

Orthotropic disc1 w=o F-SS

EACO

Orthotropic discf w=o F-F

EACO

c51

c41

c41

c41

Natural frequency, 52(rad/s) m=l m=2 I \ n=O II= 1 n=O n=l

507 514 500 507

3308

714 720

3479

3262

561 559 553 561 919 907

502 495

949 939

191 190

687 682

354 348

147 145

3500 3449 3723

w = rotational speed (rad/s); f2 = natural frequency (rad/s) m = no. of nodal diameters: n = no. of nodal circles C-F = clamped at inner edge; free at outer edge F-C = free at inner edge; clamped at outer edge FPSS = free at inner edge; simply supported at outer edge F-F = free at inner edge; free at outer edge t Disc with circular central hole; inner radius = 101.6 mm, outer radius = 203.2 mm, thickness = 1.016 mm 1 Disc of orthotropic material having EJE, = 0.5,with the same dimensioning.

3. EXPERIMENTAL ANALYSIS The experimental apparatus for frequency measurements and mode recording was the classical set up of holographic interferometry [9] (see Figure 2). The model was excited by a piezoelectric shaker driven by an audio oscillator over the frequency range of interest. Two basic techniques were used: namely, real time holography with mismatch fringes, and time average holography, which does not differ, in principle, from the double exposure technique for measuring static displacements. Figure 5(a) shows the fringe pattern of a resonant mode of the orthotropic disc recorded by using the second method. No doubt exists that the mode is axisymmetric, but the number of nodal circles is not easily determined. This ambiguity can be overcome by using the real time technique. Figure 4(b) shows the fringe pattern obtained with this technique for the same mode of Figure 5(a),

100

F. GINESU, B. PICASSO AND P. PRIOLO MR

MR

Figure 2. Experimental apparatus. PS, Piezoelectric shaker; M, model; BE, beam expander; BS, beam splitter; MR, mirror; HP, holographic plate; L, laser.

which clearly appears to be the fundamental mode. A single real time hologram allows the whole frequency range to be analyzed, for observing mode shapes and measuring corresponding frequencies. Figures 3(a) to 3(d) show some illustrative photos taken during real time analysis for the steel plate. Generally, nodal patterns were clearly recognizable up to frequencies at which resonant modes were not excited at sufficient amplitude by the shaker. Nodal patterns are sometimes slightly irregular probably due to unsymmetrical boundary conditions. In Table 2 the natural frequencies measured are compared with expected values obtained from the finite element programme and from a purely analytical method [8]. Experimental values are slightly lower than those predicted as was to be expected, although the differences are never considerable even for the higher frequencies. It should be mentioned that the finite element results are closer to experimental values than those shown in reference [S]. Figures 4(a) to 4(e) refer to the analysis of the orthotropic disc, the dimensions of which

Figure 3. Steel disc of Table 2. Real time holography. n indicates the number of nodal circles and m the number of nodal diameters. (a) No excitation; (b) n = 0, m = 0: (c) n = 0,m = 4; (d) n = 1, m = 2.

101

POLAR ORTHOTROPIC ANNULAR DISCS TABLE 2

Natural frequei ncies Q (rad/s) of‘a steel disc having inner radius 4.8 mm, outer radius 61.9 mm, thickness 3.2 mm, clamped at inner edge No. of nodal circles 0 0 0 0 1 1 1 1 2 2

No. of nodal diameters 0 1 2 4 0 1 2 4 0 1

Program

EACO

5297 4161 1147 28 151 31180 34 305 46 994 95 157 91239 96 084

cw

E xperimental value

5290 4104 1343 33044 35 636 48 379

Figure 4. Ortha Itropic disc of Table 3. Real time holography. (a) No excitation; (b) n = 0, n‘1 m = 1; (d) n = 0, m = 2; (e) n = 1, M = 0.

5212 6719 28 260 28 762 30 080 46 158 93 572 86 036 82 268

=

0; (c) n = 0,

102

F. GINESU, B. PICASSO AND P. PRIOLO

are given in Table 3. The material used for the model was a polyester resin-glass fibre composite obtained by filament winding. Difficulties of two kinds are encountered when this type of disc is considered: (a) the elastic properties of the material cannot be determined, without ambiguity, by using the normal tensile/compression test performed on straight specimens; (b) the probability that the working process ensures no defects and perfect homogeneity of the material over the whole structure is very small. In order to reduce the uncertainties mentioned in (a), the elastic properties were determined by means of a static flexural test performed on the same disc. The strain on both surfaces was measured by means of electric strain gauges, with the disc simply supported at the outer boundary and loaded along the inner circumference. With reference to (b), it should be pointed out that careful examination of the photos, showing mode patterns obtained by both real time and time average holography, can be of aid in revealing inhomogeneities and structural defects such as circumferential delaminations. TABLE 3

Natural frequencies R (rad/s) of an annular orthotropic disc simply supported at inner edge having inner radius 19.5 mm, outer radius 85.9 mm, thickness 5.5 mm, and elastic properties E, = 42 GN/m’, E, = 10 GN/m’, v,.<= 0.24

No. of nodal circles

No. of nodal diameters

0 0

0

0

1 1 1 2 2 2 3

1 2 0 1 2 0 1 2 0

Program EACO 3494 2521 4606 16349 17996 25 052 45464 47 974 56 891 90 732

Experimental value 3654 28162323 4019 14632 49 298 86 664

The photos of Figure 4 show various fringe patterns recorded during the real time analysis. The high value of the damping factor of the material results in a larger width of nodal lines producing some difficulty in mode identification. This is particularly evident for Figure 4(b). The high damping was a limiting factor also for the number of modes recognizable. A more powerful exciter would probably have enabled analyses at higher frequencies. Table 3 gives the comparison between experimental and analytical frequency values. The experimental values are not shown where the corresponding nodal pattern was not observed. The photos of Figures 5(a) to 5(d), recorded by time average holography, confirm what has already been said about the difftculty in mode identification. Particularly for the mode identified by the real time technique as having n = 2, m = 2, the symmetry and continuity of nodal lines are completely lost. However, the comparisons summarized in Table 3 show that resonant frequencies calculated by the finite element method are reasonably close to experimental values at least for the cases where interpretation is sufficiently unambiguous. The two values given for the &l mode truly

POLAR

Figure 5 Orthotropic (d) n = 2, m = 2.

ORTHOTROPIC

disc. Time average

holography.

ANNULAR

(a) n=O,

103

DISCS

m=O;

(b) n=O,

m=l;

(c) n=

1, m= 0:

correspond to two resonant frequencies with a nodal diameter. This could probably be justified by different bending stiffness in the various radial planes caused by inhomogeneity of the material.

4. CONCLUSIONS The results of the vibration analysis on the models tested seem to confirm the validity of the numerical method used which is an extension of that in reference [S]. For the case of a steel disc, tested to check the experimental apparatus, the experimental approach, based substantially on the use of real time holography with mismatch fringes, revealed itself satisfactorily simple and accurate for the determination of the first resonant frequencies as well as for the identification of the modes. Application of the same technique, and of the usual time average holography, to the flexural vibration analysis of a filament wound composite disc resulted in more difficulties due, for the most part, to the high damping effect of the material. However, at least for the determinable values of resonant frequency, the experimental results agree reasonably well with the expected values. Some inhomogeneity of the material was evidenced by distorted or unexpected nodal patterns, thus confirming the potential usefulness of holography in the field of material characterization and non-destructive testing.

F. GINESU, B. PICASSO AND P. PRIOLO

104

ACKNOWLEDGMENTS This paper represents the results of research carried out at the “Istituto and at the “Centro di Calcolo dell’Universit8 di Cagliari”, under 77.00100.07, sponsored by the C.N.R. (Centro Nazionale delle Ricerche).

di Meccanica” contract No.

REFERENCES 1. R. D. MINDLIN 195 1 JournalofApplied Mechanics 18,3 l-38. Influence of rotatory inertia and shear on flexural motion of isotropic elastic plate. 2. S. R. SONI and C. L. AMBA RAO 1975 Journal of Sound and Vibration 42, 57-63. On radially symmetric vibrations of orthotropic non-uniform discs including shear deformation. 3. R. Y. SOUTHWELL 1922 Proceedings of the Royal Society, London 101, 133-153. On the free transverse vibration of a uniform circular disc clamped at its center and on the effect of rotation. 4. G. K. RAMAIAH and K. VIJAYAKUMAR 1973 Journal of Sound and Vibration 26, 5 17-53 1. Natural frequencies of polar orthotropic annular plates. 5. J. KIRKHOPE and G. J. WILSON 1976 Journal of Sound and Vibration 44, 461-474. Vibration and stress analysis of thin rotating discs using annular finite elements. 6. S. P. TIMOSHENKO and S. WOINOWSKY-KRIEGER 1959 Theory of Plates and Shells. New York: McGraw-Hill, second edition. 7. I. S. RAJO, G. VENKATESWARA RAO and T. V. G. K. MURTY 1974 Computers and Structures 4, 549-558. Eigenvalues and eigenvectors of large order bounded matrices. 8. S. M. VOGEL and D. W. SKINNER 1965 Journal of .4pplied Mechanics 32, 926-931. Natural frequencies of transversely vibrating uniform annular plates. 9. R. K. ERF (editor) 1974 Holographic Nondestructive Testing. New York: Academic Press.

APPENDIX: Matrix

[Bd]

EXPRESSIONS

FOR MATRICES

:

2

1

L3

L2

-_

L3

--

1 L2

[rf] :

Matrix

0 9

0 F(m2-1)

-2a

- 6ar

a(m2-2)

ar(m2 - 3) c( =

2mB _

2

r2

0

- 2rnp

- 4mpr

cos m<,

/I = sin mc.

POLAR

ORTHOTROPIC

ANNULAR

105

DISCS

Matrix [DF] :

4 -vr&

w

w

4 V*< -V&)

0

E, 12(1 -V,rVgJ

SYMM.

0 G rT 12

Matrix [kdF] (symmetric): (1,l): [D&m4

+ DF3;,4m2] P-3,

(1,2): [DF&-mqe,, (1,3): [-DF,,2mz+DF,,(m4-2m2)-DF,,4mqP_,, (1,4): [DF2,6m2

+ DF2,(m4-3m2)

- DF,,8m2]Po,

(2,2): [DF2,(m2 - l)“] I--,, (2,3): [-DF2,2(m2-1)+DF2,(m4-3m2+2)]P,,, (2,4): [-DF2,6(m2-1)+DF2,(m4-4m2+3)]P,, (3,3): [DF,,4-

DF,2(m2-2)+DF22(m2-2)2+DF,,4m2]P,,

(3,4): [DF,, 12- DF,,(8m2-18)+

DF2,(m4-5m2+6)+

DF,,8m2]

P2,

(4,4):[DF,,36-DF,212(m2-3)+DF22(m2-3)2+DF,,16m2]P,, Pi = Cn

R2 s RI

h3 (r) ri dr,

Matrix [md]

C=2

ifm=O,

C=l

ifmB1;

h(r) = thickness.

: Ql

Q2

Q3

Q4

Q3

Q4

Q5

Qs

Qs

SYMM.

Q7 Qi = Crcp

R2h (r) ri dr,

C=2

ifm=O,

s Rl

p

=

mass density.

C=l

ifm>l;