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Random response of periodic structures by a finite element technique
Journal of Sound and Vibration(1975) 43(l), l-8
RANDOM
RESPONSE
OF PERIODIC
STRUCTURES
BY A
FINITE ELEMENT TECHNIQUE? RUTH M.
ORRISAND M.
PENT
Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 SNH, England (Received 16 January 1975 ) A finite element method is presented for analysing the response of periodic structures to convected random pressure fields. It is shown that the problem reduces to one of finding the response of a single periodic section to a harmonic pressure wave. In this case the inertia, stiffness and damping matrices become functions of the phase difference between the pressures at corresponding points in adjacent sections. The method is applied to a skin-rib type structure,
1. INTRODUCTION
The standard method of analysing the random response of structures to distributed loadings is the modal method. Generally, it is assumed that the modal frequencies are widely separated. Thus, the correlations between different modes are negligible and the complexity of the modal analysis therefore can be reduced. For many practical structures this assumption is not valid. Examples of such configurations are skin-stringer and skin-rib arrays encountered in aircraft structures. A method of analysing the response of these structures by a finite difference approach is described by Olson [l]. A finite element analysis is presented by Lindberg and Olson [2]. The limitation of the method described is that it does not give an adequate representation of the pressure field on each element. A consistent finite element representation of the pressure field has been formulated by Olson 131. Frequently, a complex structure consists of a number of identical, repetitive units, linked in identical ways. The analysis of such a structure may be simplified if its periodic nature is utilized. The propagation of free flexural waves in periodic structures has been studied by Mead [4]. He shows that when a system is vibrating in one of its free waves, there is a constant ratio between the amplitude of the motion in one bay and that at the corresponding point in the adjacent bay. The ratio of the amplitudes is equal to e*, where p is the complex propagation constant. The possibility of applying approximate methods to the study of wave propagation in periodic structures has been discussed by Mead [5] and Abrahamson [6]. Orris and Petyt [7] have used the finite element displacement method to obtain the propagation constants of periodic structures. The forced response of a finite, periodic structure to a random pressure field is equal to the sum of the forced response of an infinite structure and the free response of the finite structure. This latter contribution accounts for the reflection of waves at the boundaries. The forced vibration of infinite, periodic structures has been studied by Sen Gupta [S], Sen Gupta and Mead [9, lo], and Mead and Pujara [l I]. They find that if the structure contains more than five bays, and provided damping is present, then its response, averaged over the bays, is close to the response of an infinite structure. For a five bay structure, the maximum responses did not exceed the infinite structure response by more than 50 %. This conclusion has been t Presented at the Eighth International Congress on Acoustics, London, July
23-31, 1974.
2
RUTH M. ORRIS AND M. PETYT
validated by experimental evidence 1121. Thus, the simpler procedure for analysing the response of an infinite structure may be used to estimate the response of a finite structure. This paper describes a finite element method for analysing the response of infinite, periodic structures to convected random pressure fields. By way of introduction, the response of a finite structure to such a pressure field is first described. A wider range of applications than presented here is contained in reference [ 131. A random pressure field can be regarded as a continuous assembly of harmonic pressure waves of all different frequencies and wave-numbers. Hence at each frequency w, there are harmonic waves travelling at all velocities a = o/k, where k is the wave-number. The spectral information therefore can be described by a two-dimensional wave-number/frequency spectrum [l 11.A random pressure field which is convected along at a uniform velocity a, can be analysed into an assembly of harmonic waves of all different frequencies. In this case each frequency w is associated with a unique wave-number k, = w/a,,. The spectral information therefore can be described by a one-dimensional frequency spectrum. These concepts will be utilized in the following sections. 2. RANDOM RESPONSE OF A FINITE STRUCTURE In the finite element displacement method of analysis, the structure is divided into a number of discrete elements. The response at the point r0 = (xe,y,) within the ith element can be expressed in the form w&09r) = lNcJlm,{~~~ (1) where [N(rJJ is a row matrix of element shape functions. The matrix [T] expresses the relation between the element nodal degrees of freedom and the degrees of freedom of the complete structure denoted by {q}. The stress at r0 is given by a&, r) =
14~o)h[~ld~>9
(2)
where p(ro)J is obtained by differentiating [A&)] to obtain the strains, and then using the stress-strain relationships. If the structure is subjected to a convected random pressure field, then the mean square stress is given by (see reference [ 111) +m +m (&ro, r)> = 1 s I J’,h, w,k) I2 %@A k) dk do, (3) --m -m where Y,,(r,,,w, k) is the complex stress response at r0 due to a harmonic pressure wave of unit amplitude, frequency w and wave-number k. The function S,(w, k) is the wave-number/ frequency spectrum of the pressure field. In the case of a random acaustic plane pressure field equation (3) reduces to <6(x0, r)> = r I Y& 0, k(w)) I2 &W dw, (4) --m where S,(w) is the power spectral density of the pressure field. In order to determine Y,(r,, w, k), the response {q} of the complete structure is obtained and then substituted into equation (2). Substituting equation (1) into the energy expressions for a single structural element and then summing over all the elements gives expressions of the form T=
3WTW1k%
u = f-{q>‘[~l{cr~9 w = WT~~~~
(5)
for the kinetic and potential energies and the work done by the applied pressures. In equations (5), [M] and [K] are the inertia and stiffness matrices of the complete structure and {F} is a
3
RANDOM RESPONSE OF PERIODIC STRUCTURES
column matrix of equivalent nodal forces. This latter matrix is of specific interest in the present context and therefore will be dealt with in detail. In the case of distributed pressures p(r, t), it is given by {F] = y [TIT s lN(r)l%(r, t) dA, (6) i-1 Al where NE represents the number of elements exposed to the pressure field. A harmonic pressure wave of unit amplitude, frequency w and wave-number k travelling in the direction of the x-axis is given by p(r, t) = exp i (wt - kx). (7) Substituting equation (7) into equation (6) gives {F} = {F’(k)} e’“‘,
(8)
where {F(k)} = y
[TIT 1 [N(r)jTexp (-ikx)dA.
1-l
(9)
4
The equations of motion of the complete structure are obtained by substituting equations (5) into Lagrange’s equations to give
Mb71 + [N4) + PM1 = V? = PW; eta’,
(10)
where [D] is a damping matrix. The solution of this equation is {q] = b@4lVW e’“‘, where [or(io)] is a matrix of receptances, and is given by
(11)
[a(io)] = p - wz M + iwD]-‘.
(12)
Substituting equation (11) into equation (2) gives Y&o, w, k) =
1~~~~~1~~~1~~~~~~~1~~~~~~~‘“‘.
03)
3. RANDOM RESPONSE OF AN INFINITE PERIODIC STRUCTURE Consider the one-dimensional periodic structure shown schematically in Figure 1. The calculation of the response of such a structure to a random pressure field, which is being convected along the chain, can be simplified if the periodic nature of the structure is taken into account. The mean square stress is once again given by either equation (3) or (4) and therefore the problem is reduced to finding the response of the structure to a harmonic pressure wave. Left-hand bcundory I
Rqht-hand boundary I
L
I
Perlodtc s&on
Figure 1. Schematic idealisation of a one-dimensional
periodic system.
Representing one of the periodic sections by a finite element model leads to the following equation of motion : [ir - co2M + iwD]{g} = {F} . (14) In this equation the motion has been assumed to be harmonic and the common factor eim’ has been omitted.
4
RUTH M. ORRIS AND M. PETYT
The column matrix of nodal degrees of freedom will contain freedoms corresponding to nodes on the left-hand boundary QL,the right-hand boundary qR and all other nodes Qr. Thus
(15)
Corresponding to each nodal freedom there will be a generalized force. The column matrix of generalized forces therefore may be partitioned in a similar manner to give FL {F} = F,
.
(16)
FR
H Now this matrix consists of forces due to the external pressure field and also boundary forces due to the motion of the adjacent sections. It therefore may be written in the form
(17)
where the superscripts e and b denote external and boundary loading, respectively. If the structure is,subjected to a harmonic pressure wave of frequency w and wave-number k, then forces and displacements at corresponding points in adjacent sections are equal in magnitude, but differ in phase by E = kl, where 1is the periodic distance. In this case Fi = -e-i” FbL, Ft;= e-*apaL, qR = eeiEijL. (18)
_ I1
Thus
9L
{4>= Qr e-‘C
(19)
-
QL
and
.
(20)
K L 0 6:K,,1
The matrices K, M and D also may be partitioned in a manner corresponding to the three sets of nodes. For example
[Kl =
KIR
.
(21)
KRI KRR 10 It is not usual to have K LR,KRL# 0 when using finite elements, but they could be included if necessary. Substituting equations (19) to (21) into equation (14) and eliminating Pi gives an equation of the form
[g(-is) - cc2M(-is) + i.sD(-is)]
II[ 1 QL
_ = qr.
2Ft
Fg ’
RANDOM RESPONSE OF PERIODIC STRUCTURES
where [I((-is)] =
Ku + KRR
5
1
KL, + e” KRI
K,, + e-” KIR KII
(23)
and [M(-is)], [D(-is)] are defined in a similar manner. The definitions of K(-is) and M(-is) coincide with the definitions of K(p) and M(p) as derived for free wave propagation in reference [7]. The solution of equation (22) is (24) where [a(o,
-is)] = [K(-is) - w2 M(-is) + ioD(- is)]-‘.
4. APPLICATION
TO A SKIN-RIB
(25)
STRUCTURE
As an example of an infinite, periodic structure excited by convected random loading, the skin-rib structure studied by Sen Gupta and Mead [8,93, which is shown in Figure 2, has been considered. They present curvature spectra for a structure excited by random plane waves which are convected at a velocity given by CV = 2.0, where CV = (ph/D)“‘(1/2) U. Here, p, h, 1 and D are, respectively, the density, thickness, rib pitch and flexural rigidity of the identical upper and lower skins, and U is the convection velocity of the loading. The power spectrum of the excitation is assumed to be
(26)
S,(o) = s,(Q) = 1.0 for 0 < Q ,< 6.5, = 0 elsewhere, where R = (ph/D)‘l’(l/2)” w.
Soar Smp
i-supported
edges
Figure 2. Plate model of a tail plane structure.
The curvature response per unit pressure also is made non-dimensional by multiplying by the factor (D/12). The dimensions of the structure are rib pitch:rib depth = 2.2: 1.0, rib pitch:rib length = 13*2:36*0, rib thickness : skin thickness = O-1: 1.0. The periodic section chosen to represent the structure is shown in Figure 3. The loading is convected along the upper skin surface in the x-direction, which corresponds to the infinite direction. In the y-direction the structure is symmetrically loaded. Thus it is only necessary to consider one half the periodic section, since the response will be symmetrical about an axis mid-way between the spars. At the spar edges the structure is assumed to be simply supported. Twelve quadrilateral plate bending elements, as derived in reference [ 141,were used to represent one half the periodic section. The element arrangement is shown in Figure 3.
6
RUTH M. ORRIS AND M. PETYT
Figure 3. Finite element idealisation of a periodic section of a tail plane.
In performing the analysis, the damping was assumed to be hysteretic. Therefore, in equation (14), COD was replaced by qK, where q is the flexural loss factor. The value of this factor was taken to be 0.01. The quadrilateral element of reference [14] was obtained by combining the four triangles defined by inserting the two diagonals. The evaluation of the generalized forces, as given by equation (9), was simplified by assuming that within each triangle the exponential term could be approximated by a linear variation as follows [3] : exp (-ikx) = exp - gV5 = elL, + elLl + e,L,, (’ 1
(27)
where the Lt are the area co-ordinates of the triangle. The coefficients el are determined by evaluating (27) at the three vertices of the triangle. The generalized forces for the four triangles are combined and the terms corresponding to the freedoms at the internal node are eliminated by an analogous procedure to that outlined for the stiffness matrix in reference [ 141.It should be noted that for accurate determination of the generalized forces, the size of the quadrilateral is limited by the value of the parameters defining the excitation: namely, Q/W = al/U.
dL_-__Jo 2.5
2.6
2.7
2.6
Frequency pammeter, n
Figure 4. Response of upper skin centre. -,
Finite element; -----, exact.
The power spectral density of the curvature response at the upper skin centre, obtained from the finite element analysis is compared with the exact solution in Figure 4, for the range 2.5 G D G 3.0. The finite element results correspond to the average value of the curvatures obtained from each triangle meeting at a node. The finite element results for the same point over a wider frequency range (0 G B < 6.5) are shown in Figure 5.
RANDOM RESPONSE OF PERIODIC STRUCTURES
8 10-l a” 0 3
I
I.0
2.0 Frequency
3.0
4.0
5.0
6.0 6.5
parameter, 0.
Figure 5. Response of upper skin centre.
All the finite element results in Figures 4 and 5 are larger than the exact solutions. One of the reasons is that the pressure distribution between the spars has been assumed to be uniform, whereas in references [8,9] it is a half sine-wave. The peaks in the finite element results also occur at slightly higher frequencies than the peaks in the exact solution. In Figure 4 the finite element results peak at Sz= 2.735, 2.86 and 2.89 compared with 51= 2.697, 2.855 and 2.885 for the exact solution. The second peak occurs because the convection velocity matches the phase velocity of the primary wave with the skins vibrating in-phase with one another. The first and third peaks correspond to the convection velocity coinciding with the phase velocity of primary waves with the skins vibrating in counter-phase. In the first of these, rib motion predominates. The fourth peak, which is shown in Figure 5, corresponds to the convection velocity coinciding with the phase velocity of a secondary wave with skins in counterphase. At zero frequency the finite element results differ from the exact value by nearly 50 %. This error is due to the coarse mesh used for the analysis. This deduction was confirmed by performing a static analysis of a simply supported plate subject to a uniform load. This gave an error of 20 % in curvature, which indicates an error of nearly 50 % in the square of the curvature. With increase in frequency the error in curvature is less. It is felt that the idealization used here is adequate for predicting displacement response. In order to predict curvature or stress response a finer idealization should be used. However, the results presented do indicate the feasibility of the method.
5. CONCLUSIONS
A finite element method has been presented for analysing the response of periodic structures to convected random pressure fields. The advantages of the method are as follows : (1) only one periodic section need be modelled by a finite element mesh, thus reducing the number of degrees of freedom ; (2) the basic periodic section may be any complex shape; (3) no knowledge of the frequencies and modes of the structure is required; (4) the method makes use of existing finite element programs. The full potential of the method has yet to be exploited; in particular experimental verification is required. However, the feasibility of the method has been demonstrated.
8
RUTH M. ORRIS AND M. PETYT
REFERENCES 1. M. D. OLSON 1967 National Research Council of Canada, Aero. Report LR-479. A numerical approach to random response problems. 2. G. M. LINDBERGand M. D. OLSON 1967 National Research Council of Canada, Aero. Report LR-492. Vibration modes and random response of a multi-bay panel system using finite elements. 3. M. D. OLSON 1972 International Journal of Computers and Structures 2, 163-180. A consistent finite element method for random response problems. 4. D. J. MEAD 1970 Journal of Soundand Vibration 11,181-197.Free wave propagation in periodically supported, infinite beams. 5. D. J. MEAD 1973 Journal of Sound and Vibration 21,235-260. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. 6. A. L. ABRAHAMSON1973 Journal of Sound and Vibration 28,247-258. Flexural wave mechanicsan analytical approach to the vibration of periodic structures forced by convected pressure fields. 7. RUTH M. ORRISand M. PETYT 1974 Journal of Sound and Vibration 33,223-236. A finite element study of harmonic wave propagation in periodic structures. 8. G. SEN GUPTA 1970 Ph.D. Thesis, University of Southampton. Dynamics of periodically stiffened structures using a wave approach. 9. G. SEN GUPTA and D. J. MEAD 1970 Proceedings of Symposium on Structural Dynamics, Loughborough University of Technology. Wave group theory applied to the analysis of the forced vibrations of rib-skin structures. 10. D. J. MEAD and G. SEN GUPTA 1970 Proceedings of Conference on Current Developments in Sonic Fatigue, University of Southampton. Wave group theory applied to the response of finite structures. Il. D. J. MEAD and K. K. PUJARA1971 Journal of Sound and Vibration 14,525-541. Space-harmonic analysis of periodically supported beams : response to convected random loading. 12. J. M. O’KEEFE 1971 M.Sc. Thesis, University of Southampton. A study of the forced response of a highly damped periodic structure. 13. R. M. ORRIS 1974 Ph.D. Thesis, University of Southampton. A finite element study of the vibration of skin-rib structures. 14. R. M. ORRIS and M. PETYT 1973 Journal of Sound and Vibration 27, 325-344. A finite element study of the vibration of trapezoidal plates.
RANDOM
RESPONSE
OF PERIODIC
STRUCTURES
BY A
FINITE ELEMENT TECHNIQUE? RUTH M.
ORRISAND M.
PENT
Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 SNH, England (Received 16 January 1975 ) A finite element method is presented for analysing the response of periodic structures to convected random pressure fields. It is shown that the problem reduces to one of finding the response of a single periodic section to a harmonic pressure wave. In this case the inertia, stiffness and damping matrices become functions of the phase difference between the pressures at corresponding points in adjacent sections. The method is applied to a skin-rib type structure,
1. INTRODUCTION
The standard method of analysing the random response of structures to distributed loadings is the modal method. Generally, it is assumed that the modal frequencies are widely separated. Thus, the correlations between different modes are negligible and the complexity of the modal analysis therefore can be reduced. For many practical structures this assumption is not valid. Examples of such configurations are skin-stringer and skin-rib arrays encountered in aircraft structures. A method of analysing the response of these structures by a finite difference approach is described by Olson [l]. A finite element analysis is presented by Lindberg and Olson [2]. The limitation of the method described is that it does not give an adequate representation of the pressure field on each element. A consistent finite element representation of the pressure field has been formulated by Olson 131. Frequently, a complex structure consists of a number of identical, repetitive units, linked in identical ways. The analysis of such a structure may be simplified if its periodic nature is utilized. The propagation of free flexural waves in periodic structures has been studied by Mead [4]. He shows that when a system is vibrating in one of its free waves, there is a constant ratio between the amplitude of the motion in one bay and that at the corresponding point in the adjacent bay. The ratio of the amplitudes is equal to e*, where p is the complex propagation constant. The possibility of applying approximate methods to the study of wave propagation in periodic structures has been discussed by Mead [5] and Abrahamson [6]. Orris and Petyt [7] have used the finite element displacement method to obtain the propagation constants of periodic structures. The forced response of a finite, periodic structure to a random pressure field is equal to the sum of the forced response of an infinite structure and the free response of the finite structure. This latter contribution accounts for the reflection of waves at the boundaries. The forced vibration of infinite, periodic structures has been studied by Sen Gupta [S], Sen Gupta and Mead [9, lo], and Mead and Pujara [l I]. They find that if the structure contains more than five bays, and provided damping is present, then its response, averaged over the bays, is close to the response of an infinite structure. For a five bay structure, the maximum responses did not exceed the infinite structure response by more than 50 %. This conclusion has been t Presented at the Eighth International Congress on Acoustics, London, July
23-31, 1974.
2
RUTH M. ORRIS AND M. PETYT
validated by experimental evidence 1121. Thus, the simpler procedure for analysing the response of an infinite structure may be used to estimate the response of a finite structure. This paper describes a finite element method for analysing the response of infinite, periodic structures to convected random pressure fields. By way of introduction, the response of a finite structure to such a pressure field is first described. A wider range of applications than presented here is contained in reference [ 131. A random pressure field can be regarded as a continuous assembly of harmonic pressure waves of all different frequencies and wave-numbers. Hence at each frequency w, there are harmonic waves travelling at all velocities a = o/k, where k is the wave-number. The spectral information therefore can be described by a two-dimensional wave-number/frequency spectrum [l 11.A random pressure field which is convected along at a uniform velocity a, can be analysed into an assembly of harmonic waves of all different frequencies. In this case each frequency w is associated with a unique wave-number k, = w/a,,. The spectral information therefore can be described by a one-dimensional frequency spectrum. These concepts will be utilized in the following sections. 2. RANDOM RESPONSE OF A FINITE STRUCTURE In the finite element displacement method of analysis, the structure is divided into a number of discrete elements. The response at the point r0 = (xe,y,) within the ith element can be expressed in the form w&09r) = lNcJlm,{~~~ (1) where [N(rJJ is a row matrix of element shape functions. The matrix [T] expresses the relation between the element nodal degrees of freedom and the degrees of freedom of the complete structure denoted by {q}. The stress at r0 is given by a&, r) =
14~o)h[~ld~>9
(2)
where p(ro)J is obtained by differentiating [A&)] to obtain the strains, and then using the stress-strain relationships. If the structure is subjected to a convected random pressure field, then the mean square stress is given by (see reference [ 111) +m +m (&ro, r)> = 1 s I J’,h, w,k) I2 %@A k) dk do, (3) --m -m where Y,,(r,,,w, k) is the complex stress response at r0 due to a harmonic pressure wave of unit amplitude, frequency w and wave-number k. The function S,(w, k) is the wave-number/ frequency spectrum of the pressure field. In the case of a random acaustic plane pressure field equation (3) reduces to <6(x0, r)> = r I Y& 0, k(w)) I2 &W dw, (4) --m where S,(w) is the power spectral density of the pressure field. In order to determine Y,(r,, w, k), the response {q} of the complete structure is obtained and then substituted into equation (2). Substituting equation (1) into the energy expressions for a single structural element and then summing over all the elements gives expressions of the form T=
3WTW1k%
u = f-{q>‘[~l{cr~9 w = WT~~~~
(5)
for the kinetic and potential energies and the work done by the applied pressures. In equations (5), [M] and [K] are the inertia and stiffness matrices of the complete structure and {F} is a
3
RANDOM RESPONSE OF PERIODIC STRUCTURES
column matrix of equivalent nodal forces. This latter matrix is of specific interest in the present context and therefore will be dealt with in detail. In the case of distributed pressures p(r, t), it is given by {F] = y [TIT s lN(r)l%(r, t) dA, (6) i-1 Al where NE represents the number of elements exposed to the pressure field. A harmonic pressure wave of unit amplitude, frequency w and wave-number k travelling in the direction of the x-axis is given by p(r, t) = exp i (wt - kx). (7) Substituting equation (7) into equation (6) gives {F} = {F’(k)} e’“‘,
(8)
where {F(k)} = y
[TIT 1 [N(r)jTexp (-ikx)dA.
1-l
(9)
4
The equations of motion of the complete structure are obtained by substituting equations (5) into Lagrange’s equations to give
Mb71 + [N4) + PM1 = V? = PW; eta’,
(10)
where [D] is a damping matrix. The solution of this equation is {q] = b@4lVW e’“‘, where [or(io)] is a matrix of receptances, and is given by
(11)
[a(io)] = p - wz M + iwD]-‘.
(12)
Substituting equation (11) into equation (2) gives Y&o, w, k) =
1~~~~~1~~~1~~~~~~~1~~~~~~~‘“‘.
03)
3. RANDOM RESPONSE OF AN INFINITE PERIODIC STRUCTURE Consider the one-dimensional periodic structure shown schematically in Figure 1. The calculation of the response of such a structure to a random pressure field, which is being convected along the chain, can be simplified if the periodic nature of the structure is taken into account. The mean square stress is once again given by either equation (3) or (4) and therefore the problem is reduced to finding the response of the structure to a harmonic pressure wave. Left-hand bcundory I
Rqht-hand boundary I
L
I
Perlodtc s&on
Figure 1. Schematic idealisation of a one-dimensional
periodic system.
Representing one of the periodic sections by a finite element model leads to the following equation of motion : [ir - co2M + iwD]{g} = {F} . (14) In this equation the motion has been assumed to be harmonic and the common factor eim’ has been omitted.
4
RUTH M. ORRIS AND M. PETYT
The column matrix of nodal degrees of freedom will contain freedoms corresponding to nodes on the left-hand boundary QL,the right-hand boundary qR and all other nodes Qr. Thus
(15)
Corresponding to each nodal freedom there will be a generalized force. The column matrix of generalized forces therefore may be partitioned in a similar manner to give FL {F} = F,
.
(16)
FR
H Now this matrix consists of forces due to the external pressure field and also boundary forces due to the motion of the adjacent sections. It therefore may be written in the form
(17)
where the superscripts e and b denote external and boundary loading, respectively. If the structure is,subjected to a harmonic pressure wave of frequency w and wave-number k, then forces and displacements at corresponding points in adjacent sections are equal in magnitude, but differ in phase by E = kl, where 1is the periodic distance. In this case Fi = -e-i” FbL, Ft;= e-*apaL, qR = eeiEijL. (18)
_ I1
Thus
9L
{4>= Qr e-‘C
(19)
-
QL
and
.
(20)
K L 0 6:K,,1
The matrices K, M and D also may be partitioned in a manner corresponding to the three sets of nodes. For example
[Kl =
KIR
.
(21)
KRI KRR 10 It is not usual to have K LR,KRL# 0 when using finite elements, but they could be included if necessary. Substituting equations (19) to (21) into equation (14) and eliminating Pi gives an equation of the form
[g(-is) - cc2M(-is) + i.sD(-is)]
II[ 1 QL
_ = qr.
2Ft
Fg ’
RANDOM RESPONSE OF PERIODIC STRUCTURES
where [I((-is)] =
Ku + KRR
5
1
KL, + e” KRI
K,, + e-” KIR KII
(23)
and [M(-is)], [D(-is)] are defined in a similar manner. The definitions of K(-is) and M(-is) coincide with the definitions of K(p) and M(p) as derived for free wave propagation in reference [7]. The solution of equation (22) is (24) where [a(o,
-is)] = [K(-is) - w2 M(-is) + ioD(- is)]-‘.
4. APPLICATION
TO A SKIN-RIB
(25)
STRUCTURE
As an example of an infinite, periodic structure excited by convected random loading, the skin-rib structure studied by Sen Gupta and Mead [8,93, which is shown in Figure 2, has been considered. They present curvature spectra for a structure excited by random plane waves which are convected at a velocity given by CV = 2.0, where CV = (ph/D)“‘(1/2) U. Here, p, h, 1 and D are, respectively, the density, thickness, rib pitch and flexural rigidity of the identical upper and lower skins, and U is the convection velocity of the loading. The power spectrum of the excitation is assumed to be
(26)
S,(o) = s,(Q) = 1.0 for 0 < Q ,< 6.5, = 0 elsewhere, where R = (ph/D)‘l’(l/2)” w.
Soar Smp
i-supported
edges
Figure 2. Plate model of a tail plane structure.
The curvature response per unit pressure also is made non-dimensional by multiplying by the factor (D/12). The dimensions of the structure are rib pitch:rib depth = 2.2: 1.0, rib pitch:rib length = 13*2:36*0, rib thickness : skin thickness = O-1: 1.0. The periodic section chosen to represent the structure is shown in Figure 3. The loading is convected along the upper skin surface in the x-direction, which corresponds to the infinite direction. In the y-direction the structure is symmetrically loaded. Thus it is only necessary to consider one half the periodic section, since the response will be symmetrical about an axis mid-way between the spars. At the spar edges the structure is assumed to be simply supported. Twelve quadrilateral plate bending elements, as derived in reference [ 141,were used to represent one half the periodic section. The element arrangement is shown in Figure 3.
6
RUTH M. ORRIS AND M. PETYT
Figure 3. Finite element idealisation of a periodic section of a tail plane.
In performing the analysis, the damping was assumed to be hysteretic. Therefore, in equation (14), COD was replaced by qK, where q is the flexural loss factor. The value of this factor was taken to be 0.01. The quadrilateral element of reference [14] was obtained by combining the four triangles defined by inserting the two diagonals. The evaluation of the generalized forces, as given by equation (9), was simplified by assuming that within each triangle the exponential term could be approximated by a linear variation as follows [3] : exp (-ikx) = exp - gV5 = elL, + elLl + e,L,, (’ 1
(27)
where the Lt are the area co-ordinates of the triangle. The coefficients el are determined by evaluating (27) at the three vertices of the triangle. The generalized forces for the four triangles are combined and the terms corresponding to the freedoms at the internal node are eliminated by an analogous procedure to that outlined for the stiffness matrix in reference [ 141.It should be noted that for accurate determination of the generalized forces, the size of the quadrilateral is limited by the value of the parameters defining the excitation: namely, Q/W = al/U.
dL_-__Jo 2.5
2.6
2.7
2.6
Frequency pammeter, n
Figure 4. Response of upper skin centre. -,
Finite element; -----, exact.
The power spectral density of the curvature response at the upper skin centre, obtained from the finite element analysis is compared with the exact solution in Figure 4, for the range 2.5 G D G 3.0. The finite element results correspond to the average value of the curvatures obtained from each triangle meeting at a node. The finite element results for the same point over a wider frequency range (0 G B < 6.5) are shown in Figure 5.
RANDOM RESPONSE OF PERIODIC STRUCTURES
8 10-l a” 0 3
I
I.0
2.0 Frequency
3.0
4.0
5.0
6.0 6.5
parameter, 0.
Figure 5. Response of upper skin centre.
All the finite element results in Figures 4 and 5 are larger than the exact solutions. One of the reasons is that the pressure distribution between the spars has been assumed to be uniform, whereas in references [8,9] it is a half sine-wave. The peaks in the finite element results also occur at slightly higher frequencies than the peaks in the exact solution. In Figure 4 the finite element results peak at Sz= 2.735, 2.86 and 2.89 compared with 51= 2.697, 2.855 and 2.885 for the exact solution. The second peak occurs because the convection velocity matches the phase velocity of the primary wave with the skins vibrating in-phase with one another. The first and third peaks correspond to the convection velocity coinciding with the phase velocity of primary waves with the skins vibrating in counter-phase. In the first of these, rib motion predominates. The fourth peak, which is shown in Figure 5, corresponds to the convection velocity coinciding with the phase velocity of a secondary wave with skins in counterphase. At zero frequency the finite element results differ from the exact value by nearly 50 %. This error is due to the coarse mesh used for the analysis. This deduction was confirmed by performing a static analysis of a simply supported plate subject to a uniform load. This gave an error of 20 % in curvature, which indicates an error of nearly 50 % in the square of the curvature. With increase in frequency the error in curvature is less. It is felt that the idealization used here is adequate for predicting displacement response. In order to predict curvature or stress response a finer idealization should be used. However, the results presented do indicate the feasibility of the method.
5. CONCLUSIONS
A finite element method has been presented for analysing the response of periodic structures to convected random pressure fields. The advantages of the method are as follows : (1) only one periodic section need be modelled by a finite element mesh, thus reducing the number of degrees of freedom ; (2) the basic periodic section may be any complex shape; (3) no knowledge of the frequencies and modes of the structure is required; (4) the method makes use of existing finite element programs. The full potential of the method has yet to be exploited; in particular experimental verification is required. However, the feasibility of the method has been demonstrated.
8
RUTH M. ORRIS AND M. PETYT
REFERENCES 1. M. D. OLSON 1967 National Research Council of Canada, Aero. Report LR-479. A numerical approach to random response problems. 2. G. M. LINDBERGand M. D. OLSON 1967 National Research Council of Canada, Aero. Report LR-492. Vibration modes and random response of a multi-bay panel system using finite elements. 3. M. D. OLSON 1972 International Journal of Computers and Structures 2, 163-180. A consistent finite element method for random response problems. 4. D. J. MEAD 1970 Journal of Soundand Vibration 11,181-197.Free wave propagation in periodically supported, infinite beams. 5. D. J. MEAD 1973 Journal of Sound and Vibration 21,235-260. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. 6. A. L. ABRAHAMSON1973 Journal of Sound and Vibration 28,247-258. Flexural wave mechanicsan analytical approach to the vibration of periodic structures forced by convected pressure fields. 7. RUTH M. ORRISand M. PETYT 1974 Journal of Sound and Vibration 33,223-236. A finite element study of harmonic wave propagation in periodic structures. 8. G. SEN GUPTA 1970 Ph.D. Thesis, University of Southampton. Dynamics of periodically stiffened structures using a wave approach. 9. G. SEN GUPTA and D. J. MEAD 1970 Proceedings of Symposium on Structural Dynamics, Loughborough University of Technology. Wave group theory applied to the analysis of the forced vibrations of rib-skin structures. 10. D. J. MEAD and G. SEN GUPTA 1970 Proceedings of Conference on Current Developments in Sonic Fatigue, University of Southampton. Wave group theory applied to the response of finite structures. Il. D. J. MEAD and K. K. PUJARA1971 Journal of Sound and Vibration 14,525-541. Space-harmonic analysis of periodically supported beams : response to convected random loading. 12. J. M. O’KEEFE 1971 M.Sc. Thesis, University of Southampton. A study of the forced response of a highly damped periodic structure. 13. R. M. ORRIS 1974 Ph.D. Thesis, University of Southampton. A finite element study of the vibration of skin-rib structures. 14. R. M. ORRIS and M. PETYT 1973 Journal of Sound and Vibration 27, 325-344. A finite element study of the vibration of trapezoidal plates.