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Vibration modes of a disc of wood laminate
WC-722.5184s3.M)+ .w 0 1984PergamonPress Ltd.
ht. 1. EngngSci. Vol. 2.2.No. I. pp. 77-86.1984 Printedin Great Britain.
VIBRATION MODES OF A DISC OF WOOD LAMINATE D. S. DUGDALE Department of Mechanical Engineering, University of Sheffield, England (Communicated by B. A. Bilby) Abstract-Orthotrophic elastic constants were obtained from tensile specimens cut at various directions relative to the grain direction. A strain energy function was derived for the flexure of a centrally clamped disc with a free rim. Test functions specifying deflection were used, with parameters having values that gave stationary frequencies according to the Rayleigh principle. Calculated frequencies and mode shapes for the five lowest principal modes showed good agreement with experimental measurements, and illustrated the distortion of the mode shapes caused by anisotropy. I. INTRODUCTION NATURALFREQUENCYmeasurements may be carried out on steel discs to within a margin of error less than 1 Hz, on account of low damping and sharp resonances. During tests on steel discs that were prepared so as to be free from internal stress and so as to have a high degree of geometrical accuracy, it has been observed that frequencies in the cos 0 and cos 28 modes often varied with angular position of the nodal lines by amounts up to 2%, while no such variation existed for the higher modes [ 1,2]. This suggested the presence of an orthotropic type of anisotropy set up during the rolling of the steel plate. In attempting to formulate this anisotropy in precise terms, the difficulty was encountered that the anisotropy of steel plates was not sufficiently pronounced to provide a stringent test of any theoretical analysis. For the present tests, wood laminate was chosen. Each layer had a Young’s modulus varying directionally by a factor of about forty. Hence this material was suitable for use in exploring the distortion of principal mode shapes caused by anisotropy. Veneer for making laminate is produced by peeling logs in a lathe. This gives a layer of wood having principal axes parallel to the longitudinal and tangential axes of the log, and such sheets may be modelled as orthotropic material. As a preliminary, the idealization of a thin plate should be briefly discussed. For a plate lying in the x-y plane, the basic assumption of thin plate theory is that the through-thickness stress cZ is zero. Now consider a plate of thickness t with an imposed deflection varying as cos (2rx/h), where A is the wavelength. When expressions are set up for displacements in the x and z directions it is found that shear strain e,, contains an extra factor t/h relative to longitudinal strain e,. Hence it is implicit in thin plate theory that provided the wavelength is large compared with plate thickness, the contributions of shear stresses uI* and cryZto the total strain energy may be neglected. It may be argued that if the glue is imagined to be removed from the joint in a laminate, then flexure will cause slip to occur on the surface separating the laminae, so that strain will no longer vary linearly through the whole thickness of the composite plate. Some shear stress must therefore be sustained by the glued joint, but nevertheless, this shear stress is of a lower order of magnitude than in-plane stresses uX,oy and uXy.
2. STRAIN ENERGY FUNCTION
FOR AN ORTHOTROPIC
DISC
A homogeneous plate has principal axes denoted (1) and (2), the W-axis being taken in the direction of the highest Young’s modulus, i.e. in the direction of the grain of the wood. Using the assumption that the through-thickness stress o, is zero in a thin plate, the in-plane strain components may be expressed in terms of stresses by ,,=ELa,
E, En
e*=!2_2L e,22!2~ E, E,; 2G
D. S. DUGDALE
78
Four elastic constants are needed [31, as the reciprocal theorem for an elastic body requires that the same constant E,, should serve in the first two of these equations. Now suppose that the given stress components happen to be referred to some other axes (x, y) inclined at angle 0 to axes (1, 2), as in Fig. 1. These stress components may be resolved into components referred to the principal axes (1, 2), since the process of transforming stress or strain is unaffected by anisotropy of the material. When the strain components referred to axes (1, 2) have been found from eqn (1) these may be transformed back to original axes (x, y). When this procedure is followed through, it is found that strain components can be written in the form
l-acos28-pcos48
e,
-v+pc0s4e (Y sin
e,
=:_1 E
28 + 2p sin 48
- v+p c0s4e i+oc0s2e-pc0s4e a: sin 28 - 2p sin 48
exs
fly sin 28 + p sin 48 $ZIsin 28 - p sin 48 itvt2pc0s4e
Here, combinations of the constants in (1) have been expressed as new constants to facilitate the analysis of circular disc problems. Constants (Yand p express the degree of anisotropy. For an isotropic material, these constants vanish, and then Young’s modulus E and Poisson’s ratio v take on their usual meanings, but for an anisotropic material, E and v are understood to have the special meanings defined by eqn (2). This matrix can now be inverted to give stress in terms of strain,
L,tMcos2etNcos4e L,-Ncos48 - M sin 28 - 2N sin 48
ex
Lz- N cos4B L,-Mcos2etNcos4e
-Msin28+2Nsin48
eY
- iM sin 28 -N sin 48 - fM sin 28 + N sin 48 L3 - 2N
cos 48
exy
(3)
19
Vibration modes of a disc of wood laminate
Fig. I. Co-ordinates for disc.
where D=(l+ v+2/3)[(1- v)(lt v-2/3)-(~*1 L,= 1tv-$*2--2p~ L2= v(1t v)+~a*-2~* LJZ l- &;J M=CY(ltvt2p) N=j&t(l-V)/3 The strain energy function is now required, for an element of a disc of area SA and thickness 6z, i.e. 6U = i(e,u, t e,u, t 2e,,u,,) 6ASz.
(4)
In this function, stresses may be replaced by strains from (3). In a plate, strains eii are given in terms of curvature components cij by the relation eij = - Zcii,where z is measured normal to the central plane of the plate. At this point, integration of (4) for a homogeneous plate may be carried out with respect to z between limits +-it, where t is the plate thickness. For a disc, co-ordinates (p, 19)are used, where p is the radius ratio r/r1 (Fig. 1). Curvature components may be written in a non-dimensional form W c, = -2 a, rl
c,=Tb, rl
W
Crtl=--Tc, rl
where W is an amplitude of displacement, specified later, and a, b and c are functions also specified later. If the (x, y) axes are now identified with the (p, 13)axes, the strain energy function (4) assumes the form W2t3E ’ = 24r,‘D
(a2t b2)(L, t N cos 40) t (a’- bZ)M cos 28
1
A feature of this expression is the cross-products of a and b with c, which may be associated with the well-known observation that when a strip inclined to the principal axes is bent by means of terminal couples, bending is accompanied by twist. We now consider a vibration mode having an axis of symmetry coinciding with the (I)-axis of the orthotropic disc. Displacement, at its maximum value with respect to time, is expressed
D. S. DUGDALE
80 by
w=
w2
&R,(p) cos ne.
Each of the functions R,(p) is arranged to have a value R,(l) = 1. Of the numerical multipliers h,, one may be taken as unity, the amplitude of rim displacement of this term being W. More specific expressions for curvature functions are now introduced, a = z AA cos no, b = 2 X,b, cos nt?, c = c h,c, sin no. When these expressions are inserted into (5), products of the trigonometrical terms may be integrated between 19= 0 and 27~.Finally, the curvature coefficients
may be calculated as tabulations for selected intervals of p, in a manner to be described later. Products of the tabulated values can then be integrated numerically to reduce strain energy (5) to an expression involving multipiers An. When the mode considered has a displacement skew-symmetrical about 0 = 0, that is, when it is expressed by a sine series, eqn (5) remains valid, but the new products of trigonometrical terms need to be integrated again, and the negative sign of c, as shown above becomes positive.
3. TESTS ON THE MATERIAL
The three-ply wood laminate consisted of two outer layers having grain in the (1) direction, with a central layer having its grain in the (2) direction, i.e. at right angles to the grain of the outer layers, as shown in Fig. 1. AlI specimens were taken from a single sheet of size 2m square, The average overall thickness t of the laminate was 3.70 mm and the average thickness of each outside layer was 1.19 mm. The three layers were of the same kind of wood, which was Finnish birch, and were bonded with phenolic resin. Specimens were stored in a room maintained at a relative humidity of 40%. The measured moisture content. was 8%, as expected for this humidity[4]. The measured density was m = 670 kg/m3. Tensile specimens having a parallel portion 40 mm wide and 200 mm long were cut at various orientations relative to the grain direction. Two layers were then stripped off with a knife and the adhesive was polished away so that the remaining single layer could be tested. Fracture strength limited the stress range that could be applied to transverse-grain specimens to about 1 N/mm*, and load was applied by suspended weights, but for strips with longitudinal grain, stresses up to lON/mm* were applied in a testing machine. An opposed pair of Huggenburger double-lever extensometers of 25 mm gauge length was moved to various positions on the strip to obtain three or more sets of readings, and satisfactoriliy linear stress-strain curves were obtained. The elastic modulus in the direction of the grain, for both outer and inner layers of the laminate, was found to be 17,400N/mm*, while the value in the direction perpendicular to the grain was 410 N/mm’. Tensile tests were carried out on a single layer having its grain direction at various angles 6 relative to the specimen axis, and stress-strain curves were obtained with the extensometers set to measure the strain both in the direction (x) along the specimen axis and in the direction (y) transverse to the specimen axis. These results are presented in Fig. 2. It can be seen that Poisson’s ratio was negative except at orientations near 6 = 0 and 0 = 90”, which is as expected for a material of large modulus ratio. For interpreting the results in Fig. 2, the following equations from matrix (2) are required,
Vibration modes of a disc of wood laminate
81
Fig.2. Experimental valuesof strain-stress gradientsversus specimen orientation, with fitted curves (Stress units N/mm’).
Curves of this kind were fitted to the experimental results to obtain elastic constants E = 870 N/mm*, v = - 0.070, cx= 1.040, /3 = - 0.090. With these values, the constants required for eqns (3) and (5) become D = 0.0796, N = 0.1741, L, = 0.643, L2= 0.188, L3= 0.454. When a laminate lying in the x-y plane is deformed, there will be continuity of all strain components through the thickness, on account of the glued joints, but as the strong axis of the, outer layers is at right angles to the strong axis of the inner layer, there will be discontinuity of stresses a,, cr, and uXYacross the glued joints. However, by the method used here, strain energy is obtained from strains without reference to stresses, so the stress distribution need not be examined. As the principal axes of the three-ply laminate are taken to be those of the outer layers, the integration of the strain energy function through the outer layers can proceed directly, but further attention must be given to the central layer. As seen from equation (2), properties of the layers are defined by terms OLcos 20 and /3 cos 48. The central layer has properties the same as the outer layers, provided that angle 0 is replaced by angle (0 + 90”). Allowance can be made for this by leaving constant /3 unchanged, and by reversing the sign of (Y. Referring to the groupings L,M and N in eqn (3), this means that the same constants L and N may be used for integrating through the whole thickness of the laminate. However, the constant M, which is linearly proportionate to (Y,should be reversed in sign over the part of the integration extending over the inner layer. From dimensions already given, it can be seen that the half-thickness of the inner layer occupied a fraction E = 0.357 of the half-thickness of the whole laminate. Term M is assigned a negative value for its contribution between limits 0 to E, and a positive value for the remaining part of the thickness range from E to 1. It follows that in order to obtain a valid average value through the whole thickness, the value of M for the outside layer should be modified by a factor (1 - 2~~)= 0.909. Hence the value of M as specified in (3) becomes M = 0.709. Some further explanation may be given here about the method adopted for determining the mechanical properties. The elastic modulus of the composite laminate may be measured experimentally as a function of orientation angle, and tests of this kind were carried out. However, due to averaging of the properties of the inner layer and outer layers, the modulus in the longitudinal direction of the outer layer was found to exceed that in the transverse direction by a factor of only about 1.5. So although it should be possible, in principle, to deduce properties of the individual layers from results of tensile tests on the entire laminate, the values so obtained are likely to be subject to a large degree of uncertainty. For this reason, the laminae were separated and tested singly. It might also be argued that mechanical properties ES Vol. 22, No.
I-F
82
D. S. DUGDALE
can be obtained from static flexure tests. When such tests were carried out, it was found that a
good value of the higher modulus was calculated when the strip was bent in its strong direction, as the weak inner layer contributed very little to stiffness. However, on flexing the strip in its weak direction, the large contribution to stiffness made by the strong inner layer led to a large uncertainty in the calculated value of the lower modulus. Hence the constants needed for the present analysis could not be satisfactorily obtained from flexure tests. 4. GUIDE-LINES
FOR FREQUENCY
CALCULATIONS
For applying the Rayleigh method to a continuous disc, it is necessary to select some deflected shape in which slope varies continuously with radius, while the slope and deflection are arranged to be zero at the radius of the rigid central clamps. In doing this, no notice need be taken of any conditions of equilibrium of shear forces or moments. Therefore, any radial variation of radial curvature may be assigned, provided only that when this curvature is integrated twice between clamping radius p = 0.3 and rim radius p = 1, a deflection equal to unity is obtained. As a first approximation, a parabolic variation of curvature was used for all Fourier terms, R”(p) = 16.66 (1 - p)*, as suggested by known solutions for an isotropic disc. Numerical integration with p-increments of 0.05 was then carried out to find slope R’ and deflection R, and from these tabulations, further tabulations of curvature coefficients a,, b, and c, were prepared for each Fourier term. The existence of two axes of symmetry in a uniform orthotropic disc requires that all principal modes of vibration should have one of four available types of symmetry. When the mode shape is symmetrical about both (1) and (2) axes, it is represented by a series of cosines of exclusively even multiples of 8. Symmetry about (1) and skew symmetry about (2) requires a series of cosines of odd multiples of 8. Symmetry about (2) and skew symmetry about (1) requires a series of sines of odd multiples of 8. Finally, skew symmetry about both axes requires a series of sines of even multiples of 8. Therefore, the first four principal modes have respectively a dominant term which is the lowest term in each of these series, these terms being the axisymmetrical term and the terms having the factor cos 6, sin 8 or sin 20. As each of these principal modes has the lowest frequency of all modes of its generic kind, the frequency of each will be a minimum with respect to all variations that may be applied to its Fourier series. In an isotropic disc, each principal mode consists of a single Fourier term, and as any angular disposition of reference axes may be chosen, the cos 0 and sin 0 modes are the same. However, in an orthotropic disc, each principal mode requires the whole of its appropriate series for its full description. Suppose that some term in the Fourier series other than the lowest is chosen to be the dominant term, so that a particular principal mode is identified. Further, suppose that an additional term is selected, which may be higher or lower in the series than the dominant term, the multiplier of this additional term being A. A quadratic equation may be solved to find the values of A which give stationary values of frequency, one a minimum and the other a maximum. The natural frequency in the mode identified by the selected dominant term was then found to be a minimum with respect to multipliers of terms higher in the series, and a maximum with respect to multipliers of terms lower in the series. Each principal mode is orthogonal to every other principal mode. Thus, if w(p, 0) and w’(p, 19)are the deflections in any two principal modes of vibration, and the plate is of uniform thickness,
then
I
ww’dA = 0,
the integral being taken over the surface area A of the plate. This is a consequence of symmetry about its leading diagonal of the stiffness matrix for a system in which mass is discretized at a series of points (see e.g., Karman and Biot[S]). This symmetry follows from the reciprocal theorem when applied to any two such points, and this in turn is valid for any linear elastic body, whether it is isotropic or not.
Vibration modes of a disc of wood laminate
83
Referring to the four series mentioned above, two modes each belonging to different series are bound to be orthogonal regardless of the radial curvature functions chosen. For the four lowest modes, the lowest term in each series was selected, and this was accompanied by the next term in its series. After the strain energy function (5) had been integrated with respect to 8, products and cross-products of curvature coefficients were also integrated with respect to p and were then inserted to obtain strain energy. Kinetic energy was obtained from integrations of I?:. This led to an expression for frequency j,
The physical factor on the right was found, from the values already given, to have a value 11,100SC’. The numerical coefficient n was obtained, as described above, as a function of multipliers A,, an example of which is given later. 5. VIBRATION
TEST
RESULTS
The discs of wood laminate used had an outside diameter of 400mm and were clamped between steel discs of thickness 25 mm and diameter 120 mm, that is, a diameter equal to 0.3 times the disc diameter. Three such discs were tested and results were averaged. Owing to the high damping of the laminate, it was found that two vibrators applied at opposite points on the rim of the disc gave a more symmetrical vibration than was obtained by using only one, and this arrangement also made it easier to obtain pure modes when two modes had similar frequencies. A single oscillator supplied the vibrators, with a counter in circuit. The rim of the disc was marked off in lo” intervals relative to the (l)-axis. With the greatest range of rim motion equal to about 1 mm, the range of movement of each point on the rim was measured during steady vibration using a microscope having a graduated scale, while the disc was illuminated by a stroboscope. Theoretical rim profiles at each of the first four resonances are shown in Fig. 3. The measured values, scaled to coincide with the peak of the theoretical curve are also shown, Physically, it can be appreciated that the modes shown in Figs. 3(a) and 3(b) became distorted in such a way as to reduce flexure along the stiff (l)-axis. Thus, Fig. 3(a) shows the nominally axi-symmetric mode to have a shape approximating to a cylindrical surface. While the type of distortion in each of these four modes was correctly predicted by calculation, the measurements show a distortion somewhat more accentuated than that predicted.
2.0
x-X'
x’
El 2e
l-O.90 cos
J
,'
I.0
./
.'
(al
0
*
Sin & 0.20Sin
36
I.0
*_+
sin 28-0.0.9Sin 48
X X X X IC)
E X/X 9-“’ 0 (11
45
80
90 (21
Fig. 3. Measured rim deflections with calculated curves for first four vibration modes.
D. S. DUGDALE
84
In Table 1, the first column shows the frequencies obtained from the Rayleigh met’hod by using only one term of each Fourier series. The second column shows frequencies obtained using the two-term expressions given in Fig. 3. Each of the first four of these is reduced, as expected, by using the more accurate mode shape. The third column of Table 1 gives the experimental values of natural frequencies. Calculated values differ from these by not more than 7%. It may be questioned whether the elastic modulus obtained from tensile loading over a period of a rhinute or so can be applied to a vibrating plate. Some experiments were carried out with a cantilever beam of wood laminate with a steel weight attached to its end. By calculating, with small corrections, the frequency expected from the static deflection produced by placing the weight in position, and comparing this with the measured natural frequency, it was concluded that the dynamic modulus was not appreciably different from the quasi-static modulus.
6. IMPROVED
MODE SHAPE
Assigning a parabolic variation of radial curvature for all Fourier terms is clearly a crude method. Other functions were tried and gave marginal reductions in calculated frequencies, but a random or intuitive selection of functions is unsatisfactory. As a more logical approach, radial curvature was represented by a step function. Radial intervals were determined in such a way that equal increments of curvature in any of three radial intervals produced equal increments in rim deflection. For a clamping radius ratio of 0.3, the radial intervals satisfying this condition are shown in Fig. 4. With these intervals, the height of any step may be varied in
TableI.Natural frequencies (Hz) Nominal mode
Calculated (single term)
Axi-sym.
Calculated (two terms)
Experimental
93
50
50
cos 8
116
78
74
sin Q
58
47
47
sin 28
00
77
72
cos 20
137 l
143'
138
Three terms
RADIUS
*Or
RATIO
p
12.00 10.24
: z 2 -10 Fig. 4. Calculated
AXI-SYM
step functions
cos
28
I--10.00 Cos 48
of radial curvature for three terms in the series describing the nominal cos 28 mode.
85
Vibration modes of a disc of wood laminate O-37 +cOS
29 + 0.106
COS 40
(11
Fig. 5. Measured rim deflections with calculated curve, and nodal line for vibration in nominal cos 20 mode.
any way, provided that the three step values when summed, remain equal to 12.24, so as to give unit deflection at the rim. This method was applied to the cos 28 mode, having deflection w = W[hoR,,(p) + R,(p) cos 20 + A,R&J) cos 461. Variations in the curvature steps of each of the three functions RI were applied so that the step values converged upon the values which gave a minimum value of frequency. The final stage in this process gave the functions shown in Fig. 4. At this stage, the frequency coefficient required for eqn (7) was obtained as rj = 0.0265
9.63 + 7.9A. + 25.2A: - 10.7& + 104.2Az,* - 27.7AoAd 0.151+ 0.457A02+ 0.0714
The stationary value of this function, i.e. a maximum with respect to A0and a minimum with respect to A4 is reached when A,,= 0.37 and A4= 0.106, giving frequency coefficient ?I = 1.85. The calculated natural frequency shown in Table 1 exceeds the experimental value by about 4 per cent. Fig. 5 shows a fairly satisfactory comparison of theoretical and experimental findings for rim deflection and for the disposition of the nodal lines. Orthogonality of this cos 28 mode, as calculated, with the axisymmetric mode as shown in Fig. 3(a) was checked by means of (6) and appeared to be satisfactory. Possibly because modifications to the basic cos 20 term due to the two additional Fourier terms are of secondary importance, there seems to be no very simple physical interpretation of the curvature step functions for the additional terms as shown in Fig. 4. The curvature function for the axisymmetric term appears to be determined by a requirement for an appreciable contribution to kinetic energy, together with a favourable integral of curvature product aoc2 which reduces the numerical coefficient of A0in (8). The curvature function for the cos 48 term converges to the rather unusual form shown in Fig. 4 because this leads to favourable integrals of curvature products such as aqc2 and aoc4 which increase the negative numerical coefficients of A4 and Aoh in (8). Inclusion of this third Fourier term in the calculation reduced the frequency by about 4%. CONCLUSIONS
It was found from tensile tests on wood laminate that this material could be modelled as an orthotropic material with four elastic constants. For such a material, a strain energy function was set up and was integrated over the surface of a disc, for which suitable deflections were assumed in the form of truncated Fourier series. By taking stationary values of frequency in accordance with the Rayleigh principle, the values found for multipliers of the Fourier terms led to fairly trustworthy predictions of mode shapes and natural frequencies for the five lowest
86
D. S. DUGDALE
modes of vibfation. However, in order to obtain even an approximately correct frequency, Table 1 shows that it was necessary to take account of the distortion of the mode shape induced by anisotropy. REFERENCES Int. J. Engng Sci. 17, 745 (1979). Proc, 6th Wood Machining Seminar, University California Forest Products Lab., Richmond, Calif., p. 194 (1979). Theory of Elasticity of an Anisotropic Elastic Body, p. 20. Holden-Day Inc., San Francisco [31S. G. LEKHNITSKII, (1963). [41H. E. DESCH and J. M. DINWOODIE, Timber, its Structural Properties and Utilization, 6th Edn, p. 165, Macmillan Press, LONDON (1981). PI T. VON KARMAN and M. A. BIOT, Mathematical Methods in Engineering, 3rd Edn, p. 172. McGraw-Hill, New York (1940).
Ul D. S. DUGDALE, PI D. S. DUGDALE,
(Receiued
I December 1982)
ht. 1. EngngSci. Vol. 2.2.No. I. pp. 77-86.1984 Printedin Great Britain.
VIBRATION MODES OF A DISC OF WOOD LAMINATE D. S. DUGDALE Department of Mechanical Engineering, University of Sheffield, England (Communicated by B. A. Bilby) Abstract-Orthotrophic elastic constants were obtained from tensile specimens cut at various directions relative to the grain direction. A strain energy function was derived for the flexure of a centrally clamped disc with a free rim. Test functions specifying deflection were used, with parameters having values that gave stationary frequencies according to the Rayleigh principle. Calculated frequencies and mode shapes for the five lowest principal modes showed good agreement with experimental measurements, and illustrated the distortion of the mode shapes caused by anisotropy. I. INTRODUCTION NATURALFREQUENCYmeasurements may be carried out on steel discs to within a margin of error less than 1 Hz, on account of low damping and sharp resonances. During tests on steel discs that were prepared so as to be free from internal stress and so as to have a high degree of geometrical accuracy, it has been observed that frequencies in the cos 0 and cos 28 modes often varied with angular position of the nodal lines by amounts up to 2%, while no such variation existed for the higher modes [ 1,2]. This suggested the presence of an orthotropic type of anisotropy set up during the rolling of the steel plate. In attempting to formulate this anisotropy in precise terms, the difficulty was encountered that the anisotropy of steel plates was not sufficiently pronounced to provide a stringent test of any theoretical analysis. For the present tests, wood laminate was chosen. Each layer had a Young’s modulus varying directionally by a factor of about forty. Hence this material was suitable for use in exploring the distortion of principal mode shapes caused by anisotropy. Veneer for making laminate is produced by peeling logs in a lathe. This gives a layer of wood having principal axes parallel to the longitudinal and tangential axes of the log, and such sheets may be modelled as orthotropic material. As a preliminary, the idealization of a thin plate should be briefly discussed. For a plate lying in the x-y plane, the basic assumption of thin plate theory is that the through-thickness stress cZ is zero. Now consider a plate of thickness t with an imposed deflection varying as cos (2rx/h), where A is the wavelength. When expressions are set up for displacements in the x and z directions it is found that shear strain e,, contains an extra factor t/h relative to longitudinal strain e,. Hence it is implicit in thin plate theory that provided the wavelength is large compared with plate thickness, the contributions of shear stresses uI* and cryZto the total strain energy may be neglected. It may be argued that if the glue is imagined to be removed from the joint in a laminate, then flexure will cause slip to occur on the surface separating the laminae, so that strain will no longer vary linearly through the whole thickness of the composite plate. Some shear stress must therefore be sustained by the glued joint, but nevertheless, this shear stress is of a lower order of magnitude than in-plane stresses uX,oy and uXy.
2. STRAIN ENERGY FUNCTION
FOR AN ORTHOTROPIC
DISC
A homogeneous plate has principal axes denoted (1) and (2), the W-axis being taken in the direction of the highest Young’s modulus, i.e. in the direction of the grain of the wood. Using the assumption that the through-thickness stress o, is zero in a thin plate, the in-plane strain components may be expressed in terms of stresses by ,,=ELa,
E, En
e*=!2_2L e,22!2~ E, E,; 2G
D. S. DUGDALE
78
Four elastic constants are needed [31, as the reciprocal theorem for an elastic body requires that the same constant E,, should serve in the first two of these equations. Now suppose that the given stress components happen to be referred to some other axes (x, y) inclined at angle 0 to axes (1, 2), as in Fig. 1. These stress components may be resolved into components referred to the principal axes (1, 2), since the process of transforming stress or strain is unaffected by anisotropy of the material. When the strain components referred to axes (1, 2) have been found from eqn (1) these may be transformed back to original axes (x, y). When this procedure is followed through, it is found that strain components can be written in the form
l-acos28-pcos48
e,
-v+pc0s4e (Y sin
e,
=:_1 E
28 + 2p sin 48
- v+p c0s4e i+oc0s2e-pc0s4e a: sin 28 - 2p sin 48
exs
fly sin 28 + p sin 48 $ZIsin 28 - p sin 48 itvt2pc0s4e
Here, combinations of the constants in (1) have been expressed as new constants to facilitate the analysis of circular disc problems. Constants (Yand p express the degree of anisotropy. For an isotropic material, these constants vanish, and then Young’s modulus E and Poisson’s ratio v take on their usual meanings, but for an anisotropic material, E and v are understood to have the special meanings defined by eqn (2). This matrix can now be inverted to give stress in terms of strain,
L,tMcos2etNcos4e L,-Ncos48 - M sin 28 - 2N sin 48
ex
Lz- N cos4B L,-Mcos2etNcos4e
-Msin28+2Nsin48
eY
- iM sin 28 -N sin 48 - fM sin 28 + N sin 48 L3 - 2N
cos 48
exy
(3)
19
Vibration modes of a disc of wood laminate
Fig. I. Co-ordinates for disc.
where D=(l+ v+2/3)[(1- v)(lt v-2/3)-(~*1 L,= 1tv-$*2--2p~ L2= v(1t v)+~a*-2~* LJZ l- &;J M=CY(ltvt2p) N=j&t(l-V)/3 The strain energy function is now required, for an element of a disc of area SA and thickness 6z, i.e. 6U = i(e,u, t e,u, t 2e,,u,,) 6ASz.
(4)
In this function, stresses may be replaced by strains from (3). In a plate, strains eii are given in terms of curvature components cij by the relation eij = - Zcii,where z is measured normal to the central plane of the plate. At this point, integration of (4) for a homogeneous plate may be carried out with respect to z between limits +-it, where t is the plate thickness. For a disc, co-ordinates (p, 19)are used, where p is the radius ratio r/r1 (Fig. 1). Curvature components may be written in a non-dimensional form W c, = -2 a, rl
c,=Tb, rl
W
Crtl=--Tc, rl
where W is an amplitude of displacement, specified later, and a, b and c are functions also specified later. If the (x, y) axes are now identified with the (p, 13)axes, the strain energy function (4) assumes the form W2t3E ’ = 24r,‘D
(a2t b2)(L, t N cos 40) t (a’- bZ)M cos 28
1
A feature of this expression is the cross-products of a and b with c, which may be associated with the well-known observation that when a strip inclined to the principal axes is bent by means of terminal couples, bending is accompanied by twist. We now consider a vibration mode having an axis of symmetry coinciding with the (I)-axis of the orthotropic disc. Displacement, at its maximum value with respect to time, is expressed
D. S. DUGDALE
80 by
w=
w2
&R,(p) cos ne.
Each of the functions R,(p) is arranged to have a value R,(l) = 1. Of the numerical multipliers h,, one may be taken as unity, the amplitude of rim displacement of this term being W. More specific expressions for curvature functions are now introduced, a = z AA cos no, b = 2 X,b, cos nt?, c = c h,c, sin no. When these expressions are inserted into (5), products of the trigonometrical terms may be integrated between 19= 0 and 27~.Finally, the curvature coefficients
may be calculated as tabulations for selected intervals of p, in a manner to be described later. Products of the tabulated values can then be integrated numerically to reduce strain energy (5) to an expression involving multipiers An. When the mode considered has a displacement skew-symmetrical about 0 = 0, that is, when it is expressed by a sine series, eqn (5) remains valid, but the new products of trigonometrical terms need to be integrated again, and the negative sign of c, as shown above becomes positive.
3. TESTS ON THE MATERIAL
The three-ply wood laminate consisted of two outer layers having grain in the (1) direction, with a central layer having its grain in the (2) direction, i.e. at right angles to the grain of the outer layers, as shown in Fig. 1. AlI specimens were taken from a single sheet of size 2m square, The average overall thickness t of the laminate was 3.70 mm and the average thickness of each outside layer was 1.19 mm. The three layers were of the same kind of wood, which was Finnish birch, and were bonded with phenolic resin. Specimens were stored in a room maintained at a relative humidity of 40%. The measured moisture content. was 8%, as expected for this humidity[4]. The measured density was m = 670 kg/m3. Tensile specimens having a parallel portion 40 mm wide and 200 mm long were cut at various orientations relative to the grain direction. Two layers were then stripped off with a knife and the adhesive was polished away so that the remaining single layer could be tested. Fracture strength limited the stress range that could be applied to transverse-grain specimens to about 1 N/mm*, and load was applied by suspended weights, but for strips with longitudinal grain, stresses up to lON/mm* were applied in a testing machine. An opposed pair of Huggenburger double-lever extensometers of 25 mm gauge length was moved to various positions on the strip to obtain three or more sets of readings, and satisfactoriliy linear stress-strain curves were obtained. The elastic modulus in the direction of the grain, for both outer and inner layers of the laminate, was found to be 17,400N/mm*, while the value in the direction perpendicular to the grain was 410 N/mm’. Tensile tests were carried out on a single layer having its grain direction at various angles 6 relative to the specimen axis, and stress-strain curves were obtained with the extensometers set to measure the strain both in the direction (x) along the specimen axis and in the direction (y) transverse to the specimen axis. These results are presented in Fig. 2. It can be seen that Poisson’s ratio was negative except at orientations near 6 = 0 and 0 = 90”, which is as expected for a material of large modulus ratio. For interpreting the results in Fig. 2, the following equations from matrix (2) are required,
Vibration modes of a disc of wood laminate
81
Fig.2. Experimental valuesof strain-stress gradientsversus specimen orientation, with fitted curves (Stress units N/mm’).
Curves of this kind were fitted to the experimental results to obtain elastic constants E = 870 N/mm*, v = - 0.070, cx= 1.040, /3 = - 0.090. With these values, the constants required for eqns (3) and (5) become D = 0.0796, N = 0.1741, L, = 0.643, L2= 0.188, L3= 0.454. When a laminate lying in the x-y plane is deformed, there will be continuity of all strain components through the thickness, on account of the glued joints, but as the strong axis of the, outer layers is at right angles to the strong axis of the inner layer, there will be discontinuity of stresses a,, cr, and uXYacross the glued joints. However, by the method used here, strain energy is obtained from strains without reference to stresses, so the stress distribution need not be examined. As the principal axes of the three-ply laminate are taken to be those of the outer layers, the integration of the strain energy function through the outer layers can proceed directly, but further attention must be given to the central layer. As seen from equation (2), properties of the layers are defined by terms OLcos 20 and /3 cos 48. The central layer has properties the same as the outer layers, provided that angle 0 is replaced by angle (0 + 90”). Allowance can be made for this by leaving constant /3 unchanged, and by reversing the sign of (Y. Referring to the groupings L,M and N in eqn (3), this means that the same constants L and N may be used for integrating through the whole thickness of the laminate. However, the constant M, which is linearly proportionate to (Y,should be reversed in sign over the part of the integration extending over the inner layer. From dimensions already given, it can be seen that the half-thickness of the inner layer occupied a fraction E = 0.357 of the half-thickness of the whole laminate. Term M is assigned a negative value for its contribution between limits 0 to E, and a positive value for the remaining part of the thickness range from E to 1. It follows that in order to obtain a valid average value through the whole thickness, the value of M for the outside layer should be modified by a factor (1 - 2~~)= 0.909. Hence the value of M as specified in (3) becomes M = 0.709. Some further explanation may be given here about the method adopted for determining the mechanical properties. The elastic modulus of the composite laminate may be measured experimentally as a function of orientation angle, and tests of this kind were carried out. However, due to averaging of the properties of the inner layer and outer layers, the modulus in the longitudinal direction of the outer layer was found to exceed that in the transverse direction by a factor of only about 1.5. So although it should be possible, in principle, to deduce properties of the individual layers from results of tensile tests on the entire laminate, the values so obtained are likely to be subject to a large degree of uncertainty. For this reason, the laminae were separated and tested singly. It might also be argued that mechanical properties ES Vol. 22, No.
I-F
82
D. S. DUGDALE
can be obtained from static flexure tests. When such tests were carried out, it was found that a
good value of the higher modulus was calculated when the strip was bent in its strong direction, as the weak inner layer contributed very little to stiffness. However, on flexing the strip in its weak direction, the large contribution to stiffness made by the strong inner layer led to a large uncertainty in the calculated value of the lower modulus. Hence the constants needed for the present analysis could not be satisfactorily obtained from flexure tests. 4. GUIDE-LINES
FOR FREQUENCY
CALCULATIONS
For applying the Rayleigh method to a continuous disc, it is necessary to select some deflected shape in which slope varies continuously with radius, while the slope and deflection are arranged to be zero at the radius of the rigid central clamps. In doing this, no notice need be taken of any conditions of equilibrium of shear forces or moments. Therefore, any radial variation of radial curvature may be assigned, provided only that when this curvature is integrated twice between clamping radius p = 0.3 and rim radius p = 1, a deflection equal to unity is obtained. As a first approximation, a parabolic variation of curvature was used for all Fourier terms, R”(p) = 16.66 (1 - p)*, as suggested by known solutions for an isotropic disc. Numerical integration with p-increments of 0.05 was then carried out to find slope R’ and deflection R, and from these tabulations, further tabulations of curvature coefficients a,, b, and c, were prepared for each Fourier term. The existence of two axes of symmetry in a uniform orthotropic disc requires that all principal modes of vibration should have one of four available types of symmetry. When the mode shape is symmetrical about both (1) and (2) axes, it is represented by a series of cosines of exclusively even multiples of 8. Symmetry about (1) and skew symmetry about (2) requires a series of cosines of odd multiples of 8. Symmetry about (2) and skew symmetry about (1) requires a series of sines of odd multiples of 8. Finally, skew symmetry about both axes requires a series of sines of even multiples of 8. Therefore, the first four principal modes have respectively a dominant term which is the lowest term in each of these series, these terms being the axisymmetrical term and the terms having the factor cos 6, sin 8 or sin 20. As each of these principal modes has the lowest frequency of all modes of its generic kind, the frequency of each will be a minimum with respect to all variations that may be applied to its Fourier series. In an isotropic disc, each principal mode consists of a single Fourier term, and as any angular disposition of reference axes may be chosen, the cos 0 and sin 0 modes are the same. However, in an orthotropic disc, each principal mode requires the whole of its appropriate series for its full description. Suppose that some term in the Fourier series other than the lowest is chosen to be the dominant term, so that a particular principal mode is identified. Further, suppose that an additional term is selected, which may be higher or lower in the series than the dominant term, the multiplier of this additional term being A. A quadratic equation may be solved to find the values of A which give stationary values of frequency, one a minimum and the other a maximum. The natural frequency in the mode identified by the selected dominant term was then found to be a minimum with respect to multipliers of terms higher in the series, and a maximum with respect to multipliers of terms lower in the series. Each principal mode is orthogonal to every other principal mode. Thus, if w(p, 0) and w’(p, 19)are the deflections in any two principal modes of vibration, and the plate is of uniform thickness,
then
I
ww’dA = 0,
the integral being taken over the surface area A of the plate. This is a consequence of symmetry about its leading diagonal of the stiffness matrix for a system in which mass is discretized at a series of points (see e.g., Karman and Biot[S]). This symmetry follows from the reciprocal theorem when applied to any two such points, and this in turn is valid for any linear elastic body, whether it is isotropic or not.
Vibration modes of a disc of wood laminate
83
Referring to the four series mentioned above, two modes each belonging to different series are bound to be orthogonal regardless of the radial curvature functions chosen. For the four lowest modes, the lowest term in each series was selected, and this was accompanied by the next term in its series. After the strain energy function (5) had been integrated with respect to 8, products and cross-products of curvature coefficients were also integrated with respect to p and were then inserted to obtain strain energy. Kinetic energy was obtained from integrations of I?:. This led to an expression for frequency j,
The physical factor on the right was found, from the values already given, to have a value 11,100SC’. The numerical coefficient n was obtained, as described above, as a function of multipliers A,, an example of which is given later. 5. VIBRATION
TEST
RESULTS
The discs of wood laminate used had an outside diameter of 400mm and were clamped between steel discs of thickness 25 mm and diameter 120 mm, that is, a diameter equal to 0.3 times the disc diameter. Three such discs were tested and results were averaged. Owing to the high damping of the laminate, it was found that two vibrators applied at opposite points on the rim of the disc gave a more symmetrical vibration than was obtained by using only one, and this arrangement also made it easier to obtain pure modes when two modes had similar frequencies. A single oscillator supplied the vibrators, with a counter in circuit. The rim of the disc was marked off in lo” intervals relative to the (l)-axis. With the greatest range of rim motion equal to about 1 mm, the range of movement of each point on the rim was measured during steady vibration using a microscope having a graduated scale, while the disc was illuminated by a stroboscope. Theoretical rim profiles at each of the first four resonances are shown in Fig. 3. The measured values, scaled to coincide with the peak of the theoretical curve are also shown, Physically, it can be appreciated that the modes shown in Figs. 3(a) and 3(b) became distorted in such a way as to reduce flexure along the stiff (l)-axis. Thus, Fig. 3(a) shows the nominally axi-symmetric mode to have a shape approximating to a cylindrical surface. While the type of distortion in each of these four modes was correctly predicted by calculation, the measurements show a distortion somewhat more accentuated than that predicted.
2.0
x-X'
x’
El 2e
l-O.90 cos
J
,'
I.0
./
.'
(al
0
*
Sin & 0.20Sin
36
I.0
*_+
sin 28-0.0.9Sin 48
X X X X IC)
E X/X 9-“’ 0 (11
45
80
90 (21
Fig. 3. Measured rim deflections with calculated curves for first four vibration modes.
D. S. DUGDALE
84
In Table 1, the first column shows the frequencies obtained from the Rayleigh met’hod by using only one term of each Fourier series. The second column shows frequencies obtained using the two-term expressions given in Fig. 3. Each of the first four of these is reduced, as expected, by using the more accurate mode shape. The third column of Table 1 gives the experimental values of natural frequencies. Calculated values differ from these by not more than 7%. It may be questioned whether the elastic modulus obtained from tensile loading over a period of a rhinute or so can be applied to a vibrating plate. Some experiments were carried out with a cantilever beam of wood laminate with a steel weight attached to its end. By calculating, with small corrections, the frequency expected from the static deflection produced by placing the weight in position, and comparing this with the measured natural frequency, it was concluded that the dynamic modulus was not appreciably different from the quasi-static modulus.
6. IMPROVED
MODE SHAPE
Assigning a parabolic variation of radial curvature for all Fourier terms is clearly a crude method. Other functions were tried and gave marginal reductions in calculated frequencies, but a random or intuitive selection of functions is unsatisfactory. As a more logical approach, radial curvature was represented by a step function. Radial intervals were determined in such a way that equal increments of curvature in any of three radial intervals produced equal increments in rim deflection. For a clamping radius ratio of 0.3, the radial intervals satisfying this condition are shown in Fig. 4. With these intervals, the height of any step may be varied in
TableI.Natural frequencies (Hz) Nominal mode
Calculated (single term)
Axi-sym.
Calculated (two terms)
Experimental
93
50
50
cos 8
116
78
74
sin Q
58
47
47
sin 28
00
77
72
cos 20
137 l
143'
138
Three terms
RADIUS
*Or
RATIO
p
12.00 10.24
: z 2 -10 Fig. 4. Calculated
AXI-SYM
step functions
cos
28
I--10.00 Cos 48
of radial curvature for three terms in the series describing the nominal cos 28 mode.
85
Vibration modes of a disc of wood laminate O-37 +cOS
29 + 0.106
COS 40
(11
Fig. 5. Measured rim deflections with calculated curve, and nodal line for vibration in nominal cos 20 mode.
any way, provided that the three step values when summed, remain equal to 12.24, so as to give unit deflection at the rim. This method was applied to the cos 28 mode, having deflection w = W[hoR,,(p) + R,(p) cos 20 + A,R&J) cos 461. Variations in the curvature steps of each of the three functions RI were applied so that the step values converged upon the values which gave a minimum value of frequency. The final stage in this process gave the functions shown in Fig. 4. At this stage, the frequency coefficient required for eqn (7) was obtained as rj = 0.0265
9.63 + 7.9A. + 25.2A: - 10.7& + 104.2Az,* - 27.7AoAd 0.151+ 0.457A02+ 0.0714
The stationary value of this function, i.e. a maximum with respect to A0and a minimum with respect to A4 is reached when A,,= 0.37 and A4= 0.106, giving frequency coefficient ?I = 1.85. The calculated natural frequency shown in Table 1 exceeds the experimental value by about 4 per cent. Fig. 5 shows a fairly satisfactory comparison of theoretical and experimental findings for rim deflection and for the disposition of the nodal lines. Orthogonality of this cos 28 mode, as calculated, with the axisymmetric mode as shown in Fig. 3(a) was checked by means of (6) and appeared to be satisfactory. Possibly because modifications to the basic cos 20 term due to the two additional Fourier terms are of secondary importance, there seems to be no very simple physical interpretation of the curvature step functions for the additional terms as shown in Fig. 4. The curvature function for the axisymmetric term appears to be determined by a requirement for an appreciable contribution to kinetic energy, together with a favourable integral of curvature product aoc2 which reduces the numerical coefficient of A0in (8). The curvature function for the cos 48 term converges to the rather unusual form shown in Fig. 4 because this leads to favourable integrals of curvature products such as aqc2 and aoc4 which increase the negative numerical coefficients of A4 and Aoh in (8). Inclusion of this third Fourier term in the calculation reduced the frequency by about 4%. CONCLUSIONS
It was found from tensile tests on wood laminate that this material could be modelled as an orthotropic material with four elastic constants. For such a material, a strain energy function was set up and was integrated over the surface of a disc, for which suitable deflections were assumed in the form of truncated Fourier series. By taking stationary values of frequency in accordance with the Rayleigh principle, the values found for multipliers of the Fourier terms led to fairly trustworthy predictions of mode shapes and natural frequencies for the five lowest
86
D. S. DUGDALE
modes of vibfation. However, in order to obtain even an approximately correct frequency, Table 1 shows that it was necessary to take account of the distortion of the mode shape induced by anisotropy. REFERENCES Int. J. Engng Sci. 17, 745 (1979). Proc, 6th Wood Machining Seminar, University California Forest Products Lab., Richmond, Calif., p. 194 (1979). Theory of Elasticity of an Anisotropic Elastic Body, p. 20. Holden-Day Inc., San Francisco [31S. G. LEKHNITSKII, (1963). [41H. E. DESCH and J. M. DINWOODIE, Timber, its Structural Properties and Utilization, 6th Edn, p. 165, Macmillan Press, LONDON (1981). PI T. VON KARMAN and M. A. BIOT, Mathematical Methods in Engineering, 3rd Edn, p. 172. McGraw-Hill, New York (1940).
Ul D. S. DUGDALE, PI D. S. DUGDALE,
(Receiued
I December 1982)