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The use of straight beam finite elements for analysis of vibrations of curved beams
Journal of Sound mid Vibration (1973) 26 (1), 155-158
LETTERS TO THE EDITOR THE USE OF STRAIGHT BEAM FINITE ELEMENTS FOR ANALYSIS OF VIBRATIONS OF CURVED BEAMS Tile use of straight beam finite elements for the analysis of the vibration of curved beams is discussed. For two sample problems, the element is shown to give results which converge satisfactorily, less rapidly than with the best curved elements but more rapidly than with other curved elements. It is shown that, for a twisted curved beam-like structure (the end winding of an alternator), the straight beam element gives superior results than a solid 20 node isoparametric element. 1. INTRODUCTION
Recently a number of curved beam finite elements have been developed [1, 2]. Although some of these elements give a very good rate of convergence, it is difficult to use them for the analysis of certain structures since, in general, the curvature of the beam has to be calculated for each element. In this note, it is shown that results of an acceptable accuracy can be obtained by using a straight beam element and applying a transformation to each element matrix. For a particular curved twisted beam (the end winding of an alternator), it is shown that this technique is also appreciably more efficient than using an isoparametric " b r i c k " element. 2. DESCRIPTIONOF ELEMENT The beam element used in the present analysis was used previously for beams parallel to the coordinate axes by Wilson and Brebbia [3]. It has six degrees of freedom at each of two nodes, allowing cubic variations for the two transverse displacements and linear variations for the torsional and longitudinal displacements. The mass and stiffness matrices, M t and K/, are given in a local coordinate system (4, r/, 0 chosen so that the beam lies along the f-axis and the/I- and ~-axes are the axes about which the second principal moments of area are defined. The element matrices in the global system, M and K, are expressed as M = TTMzT, K = TTK~T,
(1)
where T is a transformation matrix containing the direction cosines of the local axes [4].
Figure I. Specification of orientation of beam element. 155
156
LETTERSTO TIIE EDITOR
The direction cosines of the ~-axis can be obtained directly from the coordinates of the nodes P1 and P2. However, if the two principal second moments of area are not equal, additional information is required to fix the position of the ~1- and (-axes in respect of rotation about the length of the beam. Instead of specifying the coordinates of the nodes PI and P2, the coordinates of four points, PI', P I " , P2' and P2", are given (Figure 1). Pt and P2 are assumed to lie midway between PI' and P I " and P2' and P2", respectively. The normal PIN~ to the plane PI'PIP2 defines the direction of the second principal axis of the beam at PI and similarly the normal P2N2 to the plane PIP2P2 ' gives the second principal axis at P2. The (-axis is taken as the bisector of PIN 1 and a line through P~ parallel to P2N2. The 1/-axis is a line through P normal to the ~-( plane in the sense required to ensure that (3, tl, 0 forms a right-handed system. 3. RESULTSAND DISCUSSION The natural frequencies for the in-plane vibration of a ring were found by idealizing a quarter of the ring as a series of straight segments and using the technique described above. The results (Table l) are compared with the exact solution given by Timoshenko [5]. These results show a convergence which is slower than that of the curved element described by Davis, Henshell and Warburton I l l or the best two described by Sabir and Ashwell [2] but faster than their other two elements. TABLE I
1n-plane vibration of a rh~g Number of elements Number of d.o.f.
3 8
6 17
12 35
Mode Shape
Percentage error
3.49 4.23 19.2 2.3
0.85 0.91 1.94 0"57
0:21 0.21 0-28 0.14
2nd circum. 4th circum. 8th circum. 1st extensional
TABLE 2
Out-of-plane vibration of 180 ° arc Number of elements Number of d.o.f.
Percentage error
curved element [1] straight element
2 3
4 9
6 15
1.48
0-40
0.32 3.79
12 33
24 69
1.07
0.38
Table 2 shows the percentage errors in the fundamental frequency for out-of-plane vibration of a 180 ° arc clamped at both ends, compared with an approximate solution given by Brown [6]. It can be seen that the straight element converges satisfactorily but much less rapidly than the curved element [1]. The straight beam element was also used to analyse the vibrational behaviour of a complex twisted curved beam (an alternator end winding), which had been analysed previously by using solid parabolic isoparametric elements with 20 nodes and three degrees
157
LETTERS TO TIlE EDITOR
o f freedom per node I-4]. Figure 2 shows the finest meshes used for the analyses with each of the two elements. For the solid element, node condensation was used to keep the number of degrees of freedom in the eigenvalue routine at an acceptable level. The nodes used as "masters" are shown in the figure. The results are shown in Table 3. TABLE 3
Natural freqttencies of an alternator end whlding
Mesh A No. of elements No. of d.o.f. Natural frequencies (Hz)
Beam element Mesh B Mesh C
Brick element Mesh A Mesh B
15
21
32
86 5"58 5"99 23"43 31"15 34"33
122 5"53 5"91 22"87 30"09 33" 13
188 5"47 5"85 22"58 29"56 32"84
696t 6"15 6"43 24"40 31.65 35"78
1"7
4"2
6"1
Relative cost
1
19
30 1092t 5-76 6"13 23 "74 30-35 34"22 9"0
t After node condensation 96 d.o.f, remain.
(o)
(b)
Figure 2. Meshes used for end winding: (a) beam element, mesh C; (b) brick element, mesh B. O, Master node. It can be seen f r o m the table that the beam element converges more rapidly than the solid element. It might have been expected that the ability of the solid element to represent the curved path of the conductor and its twist would enable it to produce superior results. However, the transverse displacements vary quadratically along the length of this element whereas they vary cubically in the beam element. This higher order displacement function apparently is more important for accuracy of the frequencies than the superior geometrical representation of the twisted curved beam by the solid element.
158
LETTERSTO THE EDITOR 4. CONCLUSIONS
The orientation o f a two-noded straight beam finite element can be conveniently described by specifying the coordinates of an additional point at e a c h end o f the element. By this means a straight beam element can be used to give a satisfactory solution o f the vibrational behaviour of curved beams. The frequencies converge more slowly than with the best curved beam elements, but more rapidly than with others in the literature. For a complex twisted curved beam such as an alternator end winding, the straight beam element gives results superior to those obtained with curved-sided solid elements. ACKNOWLEDGMENT This work was carried out at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board. CentralElectricityResearch Laboratories, Kelvbt Avenue, Leatherhead, Surrey, England.
D . L . TIIOMAS AND R. R. WILSON
(Received 5 October 1972) REFERENCES I. R. DAVIS,R. D. HENSHELLand G. B. WARBURTON1971 British Acoustical Society paper 71/1 I, presented at Symposhtm on Finite Element Techniqttes in Structural Vibration, Sottthampton, March 1971. New beam elements. 2. A. B. SABra and D. G. ASnWELL 1971 JottrnalofSototdand Vibration 18, 555-563. A comparison of curved beam elements when used in vibration problems. 3. R. R. WILSONand C. A. BREBBIA1971 Journal of Sound attd Vibration 18, 405--416. Dynamic behaviour of steel foundations for turbo-alternators. 4. O. C. ZIENKIEWICZ1971 The Finite Element Methodin Engineerhtg Science. London: McGrawHill, second edition. 5. S. P. TIMOSHENKO1964 Vibration Problems hi Enghteering. New York: Van Nostrand, third edition, pp. 427-430. 6. F. H. BROWN 1934 Journal of the Franklin Institute 218, 41--48. Lateral vibration of ring-shaped frames.
LETTERS TO THE EDITOR THE USE OF STRAIGHT BEAM FINITE ELEMENTS FOR ANALYSIS OF VIBRATIONS OF CURVED BEAMS Tile use of straight beam finite elements for the analysis of the vibration of curved beams is discussed. For two sample problems, the element is shown to give results which converge satisfactorily, less rapidly than with the best curved elements but more rapidly than with other curved elements. It is shown that, for a twisted curved beam-like structure (the end winding of an alternator), the straight beam element gives superior results than a solid 20 node isoparametric element. 1. INTRODUCTION
Recently a number of curved beam finite elements have been developed [1, 2]. Although some of these elements give a very good rate of convergence, it is difficult to use them for the analysis of certain structures since, in general, the curvature of the beam has to be calculated for each element. In this note, it is shown that results of an acceptable accuracy can be obtained by using a straight beam element and applying a transformation to each element matrix. For a particular curved twisted beam (the end winding of an alternator), it is shown that this technique is also appreciably more efficient than using an isoparametric " b r i c k " element. 2. DESCRIPTIONOF ELEMENT The beam element used in the present analysis was used previously for beams parallel to the coordinate axes by Wilson and Brebbia [3]. It has six degrees of freedom at each of two nodes, allowing cubic variations for the two transverse displacements and linear variations for the torsional and longitudinal displacements. The mass and stiffness matrices, M t and K/, are given in a local coordinate system (4, r/, 0 chosen so that the beam lies along the f-axis and the/I- and ~-axes are the axes about which the second principal moments of area are defined. The element matrices in the global system, M and K, are expressed as M = TTMzT, K = TTK~T,
(1)
where T is a transformation matrix containing the direction cosines of the local axes [4].
Figure I. Specification of orientation of beam element. 155
156
LETTERSTO TIIE EDITOR
The direction cosines of the ~-axis can be obtained directly from the coordinates of the nodes P1 and P2. However, if the two principal second moments of area are not equal, additional information is required to fix the position of the ~1- and (-axes in respect of rotation about the length of the beam. Instead of specifying the coordinates of the nodes PI and P2, the coordinates of four points, PI', P I " , P2' and P2", are given (Figure 1). Pt and P2 are assumed to lie midway between PI' and P I " and P2' and P2", respectively. The normal PIN~ to the plane PI'PIP2 defines the direction of the second principal axis of the beam at PI and similarly the normal P2N2 to the plane PIP2P2 ' gives the second principal axis at P2. The (-axis is taken as the bisector of PIN 1 and a line through P~ parallel to P2N2. The 1/-axis is a line through P normal to the ~-( plane in the sense required to ensure that (3, tl, 0 forms a right-handed system. 3. RESULTSAND DISCUSSION The natural frequencies for the in-plane vibration of a ring were found by idealizing a quarter of the ring as a series of straight segments and using the technique described above. The results (Table l) are compared with the exact solution given by Timoshenko [5]. These results show a convergence which is slower than that of the curved element described by Davis, Henshell and Warburton I l l or the best two described by Sabir and Ashwell [2] but faster than their other two elements. TABLE I
1n-plane vibration of a rh~g Number of elements Number of d.o.f.
3 8
6 17
12 35
Mode Shape
Percentage error
3.49 4.23 19.2 2.3
0.85 0.91 1.94 0"57
0:21 0.21 0-28 0.14
2nd circum. 4th circum. 8th circum. 1st extensional
TABLE 2
Out-of-plane vibration of 180 ° arc Number of elements Number of d.o.f.
Percentage error
curved element [1] straight element
2 3
4 9
6 15
1.48
0-40
0.32 3.79
12 33
24 69
1.07
0.38
Table 2 shows the percentage errors in the fundamental frequency for out-of-plane vibration of a 180 ° arc clamped at both ends, compared with an approximate solution given by Brown [6]. It can be seen that the straight element converges satisfactorily but much less rapidly than the curved element [1]. The straight beam element was also used to analyse the vibrational behaviour of a complex twisted curved beam (an alternator end winding), which had been analysed previously by using solid parabolic isoparametric elements with 20 nodes and three degrees
157
LETTERS TO TIlE EDITOR
o f freedom per node I-4]. Figure 2 shows the finest meshes used for the analyses with each of the two elements. For the solid element, node condensation was used to keep the number of degrees of freedom in the eigenvalue routine at an acceptable level. The nodes used as "masters" are shown in the figure. The results are shown in Table 3. TABLE 3
Natural freqttencies of an alternator end whlding
Mesh A No. of elements No. of d.o.f. Natural frequencies (Hz)
Beam element Mesh B Mesh C
Brick element Mesh A Mesh B
15
21
32
86 5"58 5"99 23"43 31"15 34"33
122 5"53 5"91 22"87 30"09 33" 13
188 5"47 5"85 22"58 29"56 32"84
696t 6"15 6"43 24"40 31.65 35"78
1"7
4"2
6"1
Relative cost
1
19
30 1092t 5-76 6"13 23 "74 30-35 34"22 9"0
t After node condensation 96 d.o.f, remain.
(o)
(b)
Figure 2. Meshes used for end winding: (a) beam element, mesh C; (b) brick element, mesh B. O, Master node. It can be seen f r o m the table that the beam element converges more rapidly than the solid element. It might have been expected that the ability of the solid element to represent the curved path of the conductor and its twist would enable it to produce superior results. However, the transverse displacements vary quadratically along the length of this element whereas they vary cubically in the beam element. This higher order displacement function apparently is more important for accuracy of the frequencies than the superior geometrical representation of the twisted curved beam by the solid element.
158
LETTERSTO THE EDITOR 4. CONCLUSIONS
The orientation o f a two-noded straight beam finite element can be conveniently described by specifying the coordinates of an additional point at e a c h end o f the element. By this means a straight beam element can be used to give a satisfactory solution o f the vibrational behaviour of curved beams. The frequencies converge more slowly than with the best curved beam elements, but more rapidly than with others in the literature. For a complex twisted curved beam such as an alternator end winding, the straight beam element gives results superior to those obtained with curved-sided solid elements. ACKNOWLEDGMENT This work was carried out at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board. CentralElectricityResearch Laboratories, Kelvbt Avenue, Leatherhead, Surrey, England.
D . L . TIIOMAS AND R. R. WILSON
(Received 5 October 1972) REFERENCES I. R. DAVIS,R. D. HENSHELLand G. B. WARBURTON1971 British Acoustical Society paper 71/1 I, presented at Symposhtm on Finite Element Techniqttes in Structural Vibration, Sottthampton, March 1971. New beam elements. 2. A. B. SABra and D. G. ASnWELL 1971 JottrnalofSototdand Vibration 18, 555-563. A comparison of curved beam elements when used in vibration problems. 3. R. R. WILSONand C. A. BREBBIA1971 Journal of Sound attd Vibration 18, 405--416. Dynamic behaviour of steel foundations for turbo-alternators. 4. O. C. ZIENKIEWICZ1971 The Finite Element Methodin Engineerhtg Science. London: McGrawHill, second edition. 5. S. P. TIMOSHENKO1964 Vibration Problems hi Enghteering. New York: Van Nostrand, third edition, pp. 427-430. 6. F. H. BROWN 1934 Journal of the Franklin Institute 218, 41--48. Lateral vibration of ring-shaped frames.