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Higher order tapered beam finite elements for vibration analysis
Journal
ofSoundand
Vibration (1979) 63( I), 33-50
HIGHER ORDER TAPERED BEAM FINITE ELEMENTS FOR VIBRATION ANALYSIS C. W. S. To Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH. England (Received 17 July 1978)
In this paper, explicit expressions for mass and stiffness matrices of two higher order tapered beam elements for vibration analysis are presented. One possesses three degrees of freedom per node and the other four degrees of freedom per node. The four degrees of freedom of the latter element are the displacement, slope, curvature and gradient of curvature. Thus, this element adequately represents all the physical situations involved in any combination of displacement, rotation, bending moment and shearing force. The explicit element mass and stiffness matrices eliminate the loss ofcomputer time and round-off errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. Comparisons with existing results in the literature concerning tapered cantilever beam structures with or without an end mass and its rotary inertia are made. The higher order tapered beam elements presented here are superior to the lower order one in that they offer more realistic representations of the curvature and loading history of the beam element. Furthermore, in general the eigenvalues obtained by employing the higher order elements converge more rapidly to the exact solution than those obtained by using the lower order one.
1. INTRODUCTION
The vibration analysis and design of ship mast antenna structures, and similar structures to withstand shock impact upon the ship or wind induced vibrations, can be investigated by using one dimensional beam elements. In this paper, the element properties (namely, the mass and stiffness matrices) of a linearly tapered beam finite element in which a seventh degree polynomial displacement function is used, representing four degrees of freedom per node (the transverse displacement w, the slope dw/ax, the curvature a2w/ax2, and the gradient of curvature @w/8x3) are derived and the explicit expressions for the mass and stiffness matrices are given. These element properties are subsequently used for the vibration analysis of mast antenna structures individually treated as a tapered cantilever beam with a mass, incorporating rotary inertia, attached to its free end. The development of this higher order tapered beam element is motivated by two main factors. First, it has been shown in the cases of axial vibration of a bar [l] and both transverse impact problems [2] and vibration analysis [3] of uniform beams that improved accuracy can be obtained more efficiently with an increase of the number of degrees of freedom in the element rather than with an increase of the number of elements of the same or fewer degrees of freedom. Furthermore, in general, finite elements based on low degree polynomial displacement functions incorporate only crude curvature distributions and usually yield discontinuous bending moments among elements. In particular, the linearly tapered beam finite element, in which a cubic displacement function developed by Lind33 0022-460X/79/050033
+ 18 $02.00/O
0 1979 Academic
Press Inc. (London)
Limited
34
C. W. S. TO
berg [4] is used, has two inherent disadvantages: namely, the curvature, $2w/8x2 varies linearly along the length of the element, which is not realistic, and the rate of loading a4w/8x4 is zero, which does not resemble the true loading history of the element. The same element but with a quintic displacement function as first considered by Handa [5], and later by Thomas and Dokumaci [6], leads to contradictory results. According to the results obtained by Handa one would draw the conclusion, contrary to that drawn by Thomas and Dokumaci, that a more correct representation of the displacement distribution within elements can make up for the loss of accuracy resulting from coarser subdivision. Because of this contradiction, an examination of the tapered beam finite element with a quintic displacement function is in order. Before this examination and the derivation of the properties of the linearly tapered beam element with a septimal (polynomial of seventh degree) displacement function the element geometrical properties will be considered first in the following section.
2. GEOMETRICAL PROPERTIES OF CROSS-SECTIONAL AREA One class of non-uniform beam element is the linearly tapered element as shown diagrammatically in Figure 1. A linear, homogeneous, isotropic material is assumed, so that the cross-sectional area and second moment of area are given by (a list of notation is given in the Appendix) A(x) = c,&)d(x),
I(x) = QWd3(x),
(la, b)
where c1 and c2 are constants that depend on the shape of the beam cross-section. For an
ni
YLL
Node
3
/-I
(bl
T d
a
Figure 1. The linearly tapered beam element. (a) Beam element with positive edge forces in which 0 = aw/?x. 4 = Pw/dx2 and JI = d3w/ax3; (b) tapered beam element; (c) beam cross-sections at section s - s in (b).
TAPERED
35
BEAM FINITE ELEMENTS
elliptic-type closed curve cross-section, they are given by [7] Cl
=
r(l//fi+
l)l-(l/pL, + l)lr(li&
(>a)
+ l//J? + l),
c2 = (1112)1-(11~~+ 1)r(3/p(, + l)/r(lk + 3/i+ + I), (2b) where r( ) is the gamma function and pi and pz are real positive numbers which need not be integers. With pi = pz = 1,the cross-section is a triangle (in this case the factor l/12 in c2 should be replaced by 1/9) whereas with p, = pL2= 2 it is an ellipse. As p, and I”? each approaches infinity it is a rectangle. The cross-sectional dimensions, b(x) and d(x), vary linearly along the length of the elements so that h(x) = b,_ j [I -I- (a - 1)x/L],
d(x) = d,- Jl
+ (/I - 1)x/L],
(3a, b)
where a = bJb,_ 1, and @ = di/di_ 1 are the taper ratios for the beam element. Substituting equation (3) into equation (1) results in A(x) = Ai_ ,(l + ;‘, i’ + y,:2), I(x) = Ii- ,(l + A, + i&i’+
(4a)
&t3 + &<4),
(4b)
where 5 = x/L, 6, =(a-
y1 = (a - i) + (B -
1) +3(/?-l),
y2 = (a - 1MB -
I),
6,=3(a-
6, = 3(a - I)(/? - 1)’ +(/I - 1)3, A,_ 1 and Ii_ 1 are respectively the cross-sectional ated with node i - 1.
l)(p-
1) +3(8-
6, = (a -
I)(B -
I),
1)‘. 1)3:
area and second moment of area associ-
It should be noted that in applying equation (4) to hollow beams, of square or circular cross-section, for example, either the ratio b/d must be small or the ratio b/d must be constant because in equation (1) for a square hollow cross-section c, = 4 and c2 = (2/3)[1 + (b/d)*], and for a circular hollow cross-section c, = TCand c? = (7rr/8)[I + (b/d)2].
3. PROPERTIES OF THE TAPERED BEAM ELEMENT WITH A QUINTIC DISPLACEMENT FUNCTION The quintic displacement function employed by Handa [5], and later by Thomas and Dokumaci [6], is w = ,i,
cjx- ‘,
or, in matrix form,
w = Cf#ml~
(5)
where [4] = [l x x2 x3 x4 x”] and (i} = [i, iz c3 [, cs ;,I’, in which the superscript 7 designates “the transpose of”. Introducing the co-ordinates successively into equation (5) leads to the nodal displacements (41 = where
[Cl{ij,
(6)
36
C. W. S. TO
1
0
0
0
0
0
0
1
0
0
0
0 (7)
0
1
2L
3L2
4L3
5L4
0
0
2
6L
12L2
2oL- I
By applying the so-called Frobenius-Schurs of [C] is
[cl-’
relation [S], it can be shown that the inverse
1
0
0
0
0
0
0
1
0
0
0
0
0
0
112
0
0
0
= - lo/L3
- 6/L2
15/L4 - 6/L’
- 3/2L
8/L3
3/2L2
- 3/L4
- 1/2L3
1O/L3
- 4/L2
- 15JL4
7JL3
6/LS
1/2L
(8)
- l/L2 1/2L 3
- 3/L4
By applying equation (8) in conjunction with [$J] as given in equation (5), the shape functions are found to be (9)
[N] = [NII N12N13N14Nl,Nld where N,, = 1 - lot3 + 155” - 65’,
N,, =
Nl3 = +((L)2(1 - 35 + 342 - <3),
N,, = 4L(7c3 - 4c2 - 3t4),
N,, = 10c3 - 1554 + 6cs,
N,, = $
On application of the definitions and going through a lengthy algebraic manipulation, the mass and stiffness matrices of the element are found to be those given in equations (10) and (11). When a = p = 1 the results from equations (10) and (11) tally with those obtained by using Hermitian polynomials as given in reference [9]. Note that there are typographical errors in reference [S] concerning matrices [Cl, [Cl- ‘, [m] and [k]. Table 1 gives these errors.
TABLE 1
Typographical
errors in CC], [Cl-
Matrix
Row
[Cl
6 6 6 5 5 6
ccl-’
icorrection
‘, [m] and [k] in reference [5] Column 5 6 3 3 4 2
for the common factor in this matrix is also necessary.
[k]
=
Ezl-l
13860L3
(1584 + 15846, :;;;;TJ)$ 11226, (198 t 996 -+776, +66& + 60 6,)L4
(42768 + 213846, +150486, + 118806, +9720 6,)L*
(-1584 + 2646, +3306, + 3606,)p
(118800 + 712806, +514806, + 396006, +313206,)L
-(5940 + 19806, +6606, - 3906&Z?
-(5940 + 39606, +26406, + 19806, + 1590 6,)LZ
(1188 + 2976, +1546,+ 996, +726,)L"
f277206, + 198006, +15 480 6,)L
-(118800+475206,
(156 + 65y, +3%,)L4
+79200& + 594006, +468006,)
1188006,
(4356 + 23766, +13206, + 8586, +6306,)L3
(5940 + 39606, +26406, + 19806, +15906&Z
-(237600+
(76032 + 261366, +126726,+ 79206, f57606,)L'
Symmetric
(1794 + 702~~
(21632 + 7800~~ +3264y,)L* +310y2)L3
(7306 + 25747, +1062y,)Lz
(97032 + 300567, + I1 52Oy,)L
(118800 +47520$ +277206, f 198006, +15 4806,)L
(237600 + 1188006, +792006, + 594006, +468006,)
(564720 + 127920,, t42800 y2)
-Cl352 + 702~~
(4706 + 25747,
(5940 + 19806, +6606, - 3906.JZ.2
-(118800 + 712806, +514806, + 39600~5, + 3 I 320 6,)L
(237600 + 1188006, +792006, + 594OOd, + 468OOb.J
-(1794 + 1092y, +700;~z)z.3
(21632 + 13832y, +9296 yz)L2
-(4356 + 19806, +9246, + 3306, - 306JL3
(76032 + 498968, f364326, + 277206, + 21 6006,)L'
Symmetric
(7306 + 47327, +322O;l,)Z.z
-(97032 + 669767, + 4844OyJL
(564720+4368007, +35168oy,)
(1188 + 89lh, +l4862 + 6606, +6006,)L4
(156 + 9ly, +56y,)L4
+35y,)L4
+ 392 yJL3
(130 + 65y,
+34Oy,)L3
(1352 + 65Oy,
(4706 + 2132:, tl070y-JZ.Z
+1512y$
+1422Oy,)L
-Cl3832 + 6916y, +3756y,)L'
-(47112 + 222567, +11 620 yJL
(47112 + 248567,
(156000 +78000~, +43000y2)
38
c. 4. PROPERTIES
W.S.TO
OF THE TAPERED BEAM ELEMENT DISPLACEMENT FUNCTION
WITH A SEPTIMAL
The septimal displacement function considered here is w = j$, or, in matrix form,
ijxj-',
(12)
w = r+lK>~
where [$J] = [l x x2 x3 x4 x5 x6 x71 and {i} = [i, i2 13 i, As in section 3, the nodal displacements are expressed as
is
16 c-7 kc,]‘.
(4) = [Wf
(13)
where
(4) = Cwi- ‘i- 14i1
1
1 tii-
‘i
wi
I(/ilT,
4i
a3w/ax3,
J/ =
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
6
0
0
0
0
[Cl = 1
L
L2
L3
L4
LS
L6
L7
0
1
2L
3L2
4L3
5L4
6L5
7L6
0
0
2
6L
12L2
20L3
30L4
42Ls
0
0
0
6
24L
60L2
120L3
21OL4
(14)
The inverse of [C] is found to be 1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 1 6 -35/L4 -20/L3 -5/L' -2/3L 0
[cl-l
=
84/L’
0
45/L4
1 T
1O/L3
35/L4 - 15/L3
r/L= - 84/L’
- IO/L6-36/L' - 15/2L4 - 2/3L3 20/L’
IO/L6 2/L5
1/6L4
39/L4 - 7/L3
IO/L6- 34/L5 - 2O/L7
5/2L2 - 1/6L
(15)
l/2L2
13/2L4 - l/2L3
lo/L6-.2/L5
ll6LQ
The shape functions for this beam element are [N] = [N,, N,,N,,
N,,N,,
N,, NI: NH],
(16)
where N,, =
N,, = 1 - 35t4 + 844’ - 70c6 + 205’,
N,, = $(i’L)2(1 - IO<= + 20c3 - 15r4 +4:5) N,, = 35t4 - 845’ + 70t6 - 20t7,
N,, = $(
N,, =
Nr, = $(5L)2(5<2 - 14i3 + 13c4 - 45’),
N,, = 35L)3(352 - i - 3i3 + i4).
TAPERED BEAM FINITE ELEMENTS
39
As the element mass and stiffness matrices are symmetric only half of these matrices are given here. The diagonal and super-diagonal elements of the element mass matrix are as follows : + 1314432Oy,],
=
n[178557120
ml2
=
qL[33 268 320 + 10 208 160 'r', + 3 805 920 y2],
ml3
=
‘1L2[3 353 760 + 1 168 920 y, + 468 648 yZ]r
“11
+ 402696007,
ml4 = qL3[156264 + 583447, m15 = q[41983200
+ 24468y2],
+ 20991 6OOy, + 11 360 t6oy,],
qL[ 13 990 320 + 6 695 280 ;‘! + 3 470 760 y2],
m16
=
-
ml:
=
qL2[l
m18
=
-nL3[106284
897 200 + 878 220 ;‘, + 440 784 y2], + 47940~~ + 23472~~1,
mz2 = qL2[8078400 + 286416Oy,
+ 1 161 216~~1,
mz3 = qL3[899 640 + 345 168 y, + 147420~~1, mz4 = r,rL4[44064 + 17 748 y1 + 7848 y2], qL[ 13 990 320 + 7 295 040 y1 + 40 70 520 y2],
m25
=
m26
=
“2:
=
qL3[608 940 + 294 984 y1 + 153072 y,],
m28
=
-nL4[33660
-nL2[4565520
+ 228276Oy, + 1221 912y,],
+ 1591211, + 8064~~1,
m33 = qL4[105264 +4284Oy,
+ 1908Oy,],
m34 = qL5[5304 + 2244y, + 1029y,] m35
=
qL2[l
897200 + 101898Oy, + 581 544~~1,
%6 = -nL3[608940 m37
m 38
=
=
+ 313956y,
nL4[80172 + 40086y, -qL5[4386
+ 172044y,],
+ 21 312;‘?],
+ 2142y, + 1113yJ,
m44 = qL6[272 + 119 y1 + 56 y2], m45 = qL3[106284 + 58 344~~ + 33 876~~1, -nL4[33 660 + 17 748 y1 + 9900 y2],
rn46
=
m4:
=
qL5[4386 + 2244 y1 + 1315 y2],
m48
=
-qL6[238
+ 119 y, + 63 y,],
m55 = t(l78557120 =
m5:
=
m58
=
-nL3[156264
=
qL2[8078400
)7166
+ 111
16224Oy,],
qL[33 268 320 + 23 060 160 ;‘, + 16 657 920 yz],
m56
-
+ 1382875207,
qL2[3 353 760 + 2 184 840 ;‘1 + 1484 568 y2], + 979207, + 64044y,], + 521424Oy, + 3511 296~~1,
C. W. S. TO
rn6: m68
=
-qL3[899640
+ 5544727,
=
~I,~[44064 + 263167,
+ 356724yJ,
+ 16416y,],
= qL4[105 264 + 62424;1, + 38664~~1,
%
rnT8 = -@[5304 11288
=
+ 3060;‘, + 1845yJ,
uL6[272 + 153 ;‘, + 9Oy,], q = pA,_ ,I,/441 080640.
The diagonal and sub-diagonal elements of the element stiffness matrix are k 11 = H[27518400
+ 137592006,
+ 84672006,
+ 58212006,
+ 42336006,-j,
k,, = HL[l3759200
+ 58968006,
+ 3250800&
+ 21168006,
+ 15120006,],
k,,
= HL*[8424000
+ 32292006,
+ 15552006,
+9072OOfi, + 6048006,-J,
k 31
=
HL2[l
k 32
=
HL3[886860 + 4212006,
k,,
= HL4[234000
k,,
= HL3[49140
+ 327606,
+ 214206,
+ 15 1206, + 113406,],
k,,
= HL4[37440
+ 210606,
f 118806,
+ 73806, + 50406,-j,
k,,
= HL5[10920
+ 39006, + 19866, + 11886, + 7926,],
k,,
= HZ?[624 + 2346, + 1208, + 726, + 486,],
k,,
= -H[27518400
k,,
= -HL[13759200
k,,
= -HL*[l
k,,
= -HL3[49
k,,
= H[27518400
k,,
310400 + 7043406,
+ 702006,
+ 4208406? + 2835006, + 2197806,
+ 344406,
+ 1314006, + 202506,
+ 2066406,], + 880206,],
+ 133926,],
+ 137592006,
+ 84672006,
+ 58212006,
+ 58968006,
+ 32508006,
+ 2 1168006, + 15120006,],
310400 + 7043406, 140 + 327606,
+ 420840&
+ 214206,
+ 2835006,
+ 4233600&j,
+ 2066406,],
+ 15 1206, + 113406,],
+ 137592006,
+ 84672006,
+ 5821200 + 42336006,],
= HL[l3759200
+ 78624006,
+ 52164006,
+ 37044006,
+ 27216006,],
k 62 = HL*[5335200
+ 26676006,
+ 16956006,
+ 12096006,
+ 9072006,],
k,,
= HL3[423 540 + 283 1406, + 2010606,
k 64 = HL4[ll
+ 152 1006, + 1186206,-j,
700 + 117006, + 95406, + 77406, + 63006,],
k,,
= -HL[13759200
k,,
= HL*[8424000
k,, = -HL2[1310400 k:, = -HL3[423540
+ 78624006,
+ 52164006,
+ 5 1948006,
+ 35208006,
+ 6060606,
+ 3225606,
+ 1404006, + 583206,
+ 37044006, + 24948006,
+ 1764006,
+ 252006,
k 73 =
HL4[-2340
k 74
ZfL5[1950 + 7806, + 4716, + 3486, + 2946,],
=
k7, = HL2[1310400
+ 27216006,], + 18144006,],
+ 907206,],
+ 75606,-j,
- 11706, + 6606, + 15756, + 21966,],
+ 6060606,
+ 3225606,
+ 1764006, t- 907206,-j,
41
TAPERED BEAM FINITE ELEMENTS
k7, = -HL3[886860+ k,: = HL4[234000+
4656606, + 2642406, + 1512006, +831606,-j, 1638006, + 128040d2 t1064706, +922326,-j,
k,, = HL3[49140 + 163806, + 50406, - 252OS,], k,, = HL4[11 700 - 21606, - 25206, - 25206,], k,, = -HL5[1950 + 11706, + 8616, + 675b, + SSSS,], k,, = -Z-fL6[234-I+1176, + 756, + 546, + 426,-j, k,, = HL3[-49140-
163806, - 50406, + 25206,],
k,, = HL4[37440 + 163806, + 72006, + 25206,], ks7 = -HL5[10920+
70206, + 51066, + 39906, + 3276b,],
k,, = HL6[624 + 3906, + 2766, + 2106, + 1686,], H = Eli_ #081080L”. For uniform beams, that is when a = /I = 1, the above results reference [9], as obtained by using Hermitian polynomials.
5. VIBRATION
CHARACTERISTICS OF LINEARLY ELEMENTS
On assuming simple harmonic motion with radian of motion for the assembled structure can be written ([K]
- m2 CM])
frequency as
agree with those given in
TAPERED
BEAM
w the governing
equation
(Q,)= {O),
TABLE 2
Comparison of eigenvalues, (Al)* = wl*J’~ of three linearly tapered beam elements (cantilever) (a) a, = fi, = 0, pyramid First mode 8.7192
Second
mode
Third mode
21.146
38.453
Fourth
mode
Fifth mode
-_____ I
2
tB3 TB5 TB7
8.73517 8.74506 8.75096
25.18142 23.18407 22.83025
68.85968 55.37000
78.22758 169.30409
78.22758 178.53337
TB3 TB5
21.71782 21.25307 21.27920
41.92455 40.39639 40.25180
9486600 72.27633 70.13591
103.73917 128.51998 I1 7.87558
TB7
8.72388 8.72021 8.72089
4
TB3 TB5 TB7
8.71961 8.71927 8,71936
21.20345 21.14939 21.15169
39 10222 38.54472 38.57220
62.76983 61.57421 6 1.60905
98.77 139 92.29457 91.80001
IO
TB3 TB5 TB7
8.7 1927 8.71927 8.7 1927
21.14733 21.14571 21.14571
38.47887 38.45479 38.45556
6085846 6069236 60.69902
88.60268 87.92289 87.95364
(b) a, = fi, = 02 First mode Second mode
Third mode
Fourth
mode
71.242
Fifth mode 112.828
6.1964
18.3855
39.834
TB3 TB5 TB7
6.19986 6.20819
18.95919 19.03870
56.85606 48.69267
63.82743 147.25952
63.82743 154 19323
2
TB3 TB5 TB7
6.20186 6.19648 6.19667
18.88848 18.38837 1840247
41.11316 39.96101 4009945
92.82321 73.73720 73.37800
105.03897 131.03760 123.68543
4
TB3 TB5 TB7
6.19676 6.19641 6.19640
18.42262 18.38641 18.38556
4041611 39.84572 39.83513
75.30089 71.30942 71.25365
117.85937 113+)4873 112.89501
TB3 TB5 TB7
6.19640 6.19639 6.19639
18.38641 18.38547 18.38547
39.84648 3983367 39.83359
7 I.32205 7 1.24245 71.24177
113.16316 112.83180 I 12.82753
1
10
(c) a, = 8, = O-4 First mode
Second mode
Third mode
Fourth
mode
85.344
Fifth mode 138.0352
5.009032
19.06488
45.73836
TB3 TB5 TB7
501026 50J9760
19.11975 1916211
53-98838 4917535
6@98298
6098298
138.67416
148.06058
TB3 TB5 TB7
5.01154 5.009145 5009033
19.52378 1907756 19.06526
5001809 45.90603 45.74914
112.90697 86.13375 85.54191
13025277 146.41576 149.8967 I 138.38380 138.03534 138.37866 138.03620 138.03534
--
1
2
4
TB3 TB5 TB7
5.00925 5+IO9034 5.009033
1908500 19*06509 19.06483
46.06530 4574411 45.73838
88.90501 85-41801 85.34387
10
TB3 TB5 TB7
5w904 5.009033 5009033
1906569 1906483 19.06483
45.751 11 45.73838 45.73838
85.42747 85.34387 85.34378
140.99705
(d) a, = /3, = 1.O, uniform beam First mode 3.516015
1
TB3 TB5 TB7
Second
mode
22.03449
_~_____~_______~-________ 3.53270 34.8066 1 3.516019 22.22655 3.516019 22.03545
Third mode 61.69717
Fourth
mode
120.90192
Fifth mode 199.85953 __
_._.
6476539 61.76748
7421237 136.28122
7421237 151.70892
2
TB3 TB5 TB7
3.51772 3.516019 3.516019
22.22151 22.03474 2203449
75.15707 61.87149 61.69745
218.13809 123.27721 12093808
239.24153 206.46548 200.02710
4
TB3 TB5 TB7
3.51613 3.516019 3.516019
22.06019 2203449 22-03449
62.17492 61.69753 61.69719
122.65805 120.90648 12090220
228313769 200.13470 199*85971
10
TB3
3.516019
22.03525
T35
3.516019
2203448
61.71290 61.69718
121.01749 120-90220
200.36356 199.85969
TB7
3.516019
22.03448
61.69718
12090220
199.85969
43
TAPERED BEAM FINITE ELEMENTS (e) a, =
First mode
1 and 8, = 0, wedge Second
mode
Third
mode Fourth mode
Fifth mode
5.3150
15.2012
30.0198
TB3 TB5 TB7
5.3 1872 5.33 166 5.33991
17.30040 16.59495 16.57530
52.12928 43.70635
58.5744 1 137.69459
58.57441 146.05962
2
TB3 TB5 TB7
5.3 1585 5.31629 5.31735
15.44434 15.32421 15.37628
32.70286 3 1.92000 32.02802
78.57037 6099690 59.47300
84.80926 I 13.58248 104.34379
4
TB3 TB5 TBI
5.31515 5.31518 5.3 1526
15.22990 15.21513 1522119
30.37696 30.17491 30245 19
5 1.48493 51+0310 51.20251
85.8075 1 79.84908 79.69246
10
TB3 TB5 TB7
5.31510 5.3 I 502 5.31502
15.20778 15.20736 15.20761
30.03408 30.024 17 30.02785
49.88682 49.80433 49.83131
75.01 171 74.67446 74.78104
1
49.161
where ([K] - o2 [M]) is the dynamic stiffness matrix of the structure in question and {Q,} the column matrix of amplitude of displacement. The solution to the above algebraic equation is known as the eigenvalue solution [lo]. The computed results presented here are confined to vibration characteristics of clampedfree tapered beam structures. Consistent with the notation employed in Figure 1, the boundary conditions are as follows: at
x = 0,
w= 0
and
at
x = I,
a2~l/ax2 = 0
awlax = 0: and
~/dx(El(d2tllld.u2)) = 0.
It is to be noted that for the class of tapered beams considered, I is a function of x and therefore the last boundary condition amounts to EI(d3w/dx3) + E(al/dx)(Pw/Bx?
= 0.
Table 2 provides a comparison of the squared eigenvalues, US = of/*,!pA,/EI,, for five different cases. The values obtained by using the finite elements due to Lindberg, Handa, and the author are referred to as TB3, TB5 and TB7, respectively. The analytical solution is termed “exact”. In the case of a pyramid the exact solution is deduced from results in reference [4]. Exact solutions for the two cases a, = j?, = 02 and CI,= fi, = 04 are deduced from results presented in reference [I I], where a, = b,,/b, and j?, = d,,/d,, in which b, and d,,, and b, and d,, are the values of the cross-sectional dimensions at the tip and base respectively. The exact solution for a uniform beam is derived from reference [ 121 whereas that for a wedge is deduced from reference [6]. The mode shapes for the three special cases-namely, a pyramid, wedge and uniform beam-are presented in Table 3. The exact solutions for the first two cases are drawn from reference [4] while those of a uniform beam are obtained by using the computer program included in reference [ 121. Results in Table 2 lead to the following conclusions. 1. In general, a more correct representation of the displacement function within tapered beam elements can make up for the loss of accuracy resulting from coarser subdivision. In other words, the convergence of higher order tapered beam finite elements is more
0 0.0062 @0259 00614 01149 0.1893 0.2877 04138 0.5715 07656 i-0000
0 OGO62 @0259 O-0614 01149 @1892 0.2876 @4136 0.5713 0.7653 1GooO 0 00168 O-0639 Ck1365 0.2299 0.3395 04611 0.5909 0.1255 08624 1Woo
(cl a, = B, = 1, uniform beam 0.0 0 0 0.1 00168 0.0168 02 0.0639 0.0639 0.3 01365 @1365 D4 0.2299 0.2299 o-3395 05 Q3395 @6 0.4611 O-461 1 07 0.5909 0.5909 0.8 0.7255 07255 0.9 08624 0.8624 10000 I .o 1Woo 0.8168 0.0639 01365 0.2299 a3395 04611 0.5909 Cl.7255 08624 1NtOO
o-0501
00964 01635 02557 0.3787 G-5388 0.7435 l*ooOO
o-0206
- 0.0926 - 0.3011 - 05261 - 06834 - 07137 - 05895 -03171 00700 0.5238 1NWo
0
0 - 00926 - 0.3011 -0.5261 -0-6835 -0.7137 - 05895 -03171 0.0700 @5238 1SQOO
0
- 0.0078 - 0.0302 - oa640 - 01023 -0.1330 -0.1360 - 00803 PO801 0.4112 1@000
0
1moo
-- 0.0926 - 0.3011 - @5261 -0.6835 - 07137 - 0.5895 -03171 0.0700 0.5238
0
- 00037 -00147 -@0315 - 0.0498 -00612 - 00500 0.0097 0.1589 0.4604 1OOOo
- OGO78 - 0.0301 - 0.0637 -0.1018 - 0.1323 -0.!352 - 0.0198 0.0796 0.4090 1-0000
0
TB5
0 - 00078 - 00300 - @0636 -01017 -@I322 -0.1352 - @0798 00796 0.4087 1GOOil
TB3 _--^_----__-__--
w of second mode h
0 - 00037 - (fO146 -00313 - O-0495 - 00608 - 0.0496 0.0097 01580 0.4576 1MIO
I Exact
0 0.0037 00146 0.0313 CM495 QO607 bO496 0.0096 0.1578 04571 lN)OO
-
2048
TB7
0 00048 O-0206 00501 O-0964 0.1634 o-2556 03785 05386 0.7432 1WMO
TB5
0 0 00 @1 OGO48 QOO48 00206 00206 0‘2 00500 0.3 00500 00963 00963 04 01632 01632 @5 02553 0,6 02553 03781 @7 @3781 05379 &8 05379 0.7423 07423 0.9 1WOO 1*OoOO 1.0 (b) a, = 1 and & = 0, wedge 0 0.0 0 0.1 o&62 cGO62 02 00259 00259 @3 0.0613 00613 o-4 O-1148 01148 0.5 O-1891 01891 66 0.2874 0.2874 0.7 0.4133 04133 0.8 05709 0.5709 @9 0.7648 07648 1.0 l%IOtKI IWO
(a) a, = 8, = 0, pyramid (cone)
TB3
w of first mode
19ooo
- 00926 -@3011 - W261 - 0.6835 -0.7137 - @5895 - 0.3171 00700 05238
0
-t30078 -DO303 - @0642 --@lo26 -0.1333 -0.1363 - 00805 00803 0.4122 1@000
0
0 -00037 -00148 -0.0316 - QOJOO -0.0613 -@0501 MO97 01593 04615 1+KKlO
TB7
\
0.2281 0.6045 0.7562 0.5259 c-0197 - 0.4738 - 0.6574 - 0.3949 0.2285 1GOOO
0
0.0086 00304 QO552 00681 0*0515 -00063 -00915 -0.1282 00878 1WOtl
0
0 00030 O-0107 vo191 0.0216 00100 -@0196 -0-0541 - 00340 O-1991 l.OOOo
Exact
0
0.228 I 06045 07562 0.5259 00197 - 0.4738 - 06574 - 0.3949 0.2285 1GOOO
0
00088 0.0311 00566 OG698 @0527 - 00065 -00937 -0.1313 oWOO 1QGilo
0
0 00031 00109 @0196 00221 @0103 -002OI -0.0554 - 0.0348 0.2040 1MOO
TBS
o-228 I 0.6045 @7562 0.5259 0.0197 - 0.4738 - 06574 - 0.3949 0.2285 1QOOo
@0087 00306 0.0556 0%X85 00517 - 00064 - 0.0920 -@1286 0.0890 1%@00
0
00030 00108 00193 @0218 @OlOl -00198 --@OS44 -00339 0.201 I 1WOO
0
TB3
w of third mode
Comparison of mode shapes of cantilever beams (deflection w)
TABLE 3
0228 1 0.6045 0.7562 O-5259 0.0197 - Q4738 - 0.6574 - 0.3949 02285 l%IOOO
0
00089 0.0314 0.057 1 0.0704 0.0531 - 0.0066 - 0.0946 - a1324 0.0909 1@000
0
00031 0~0110 00197 0.0223 O-0104 - 0.0203 - 00558 -&0351 02055 1WOO
0
TB?
45
TAPERED BEAM FINITE ELEMENTS
rapid. This is particularly so for higher mode eigenvalues. This point is further confirmed by the results obtained for a uniform beam. In particular, results involving the approximation of a wedge by two elements show that increase in accuracy due to more correct representation of the displacement function within elements cannot make up for the loss of accuracy resulting from coarser subdivision. This is precisely the case considered and the conclusion reached by Thomas and Dokumaci [6]. In the cases of a pyramid and wedge anomalous results occur in TB7. The anomality indicates that the convergence of TB7 is less rapid than that of TB5. One possible explanation for this phenomenon is that it is due to the zero value of the crosssectional area (and also of the second moment of area) at the free end of the structure. In Table 3 two points are worth noting. First, for the pyramid and wedge the values of the mode shapes obtained by using elements TB3, TB5 and TB7 show excellent agreement with the exact values though the results by TB3 give the best values. Second, for a uniform beam results indicate that TB3, TB5 and TB7 all converge equally rapidly. Nevertheless. when a comparison to the exact solution obtained by the computer program given in reference [ 121 was made for the beam approximated by four elements, it was observed that the values of the mode shapes obtained by TB7 are superior, particularly for higher modes. 6. VIBRATION
OF TAPERED CANTILEVER BEAM INCORPORATING END MASS AND ITS ROTARY INERTIA
Having examined the three linearly tapered beam elements, one can now consider, the effects of taper ratio, end mass, and rotary inertia of the end mass on the eigenvalues and mode shapes of a double-tapered cantilever beam structure as predicted by using elements TB5 and TB7. To enable this investigation to be made, two discrete mass element matrices, one compatible with TB5 and the other with TB7, are required. By considering the kinetic energy, it can easily be shown that the discrete mass matrix associated with element TB5 is
[m] = IO0
I
,
md
0
0
0 0
0 Jd
0 0
I 1
111111,
,_ _ _.-
-.-
--------________
I
,
I
I
,-
A./
- /.
-,-.-
I,,
--_-______-_---
----
IO-
3
0.01
I
I
t,,,,,,
0.1
I
,
11,111
I.0
Figure 2. Eigenvalues u! of a double tapered cantilever beam with a free end. a, = 8,. -, -, third mode.
second mode: -
First mode:
-
-.
Mass ratio,+
3. Eigenvalues u? or a double second ~-, first mode:---,
Figure K/I
=
tapered cantilever beam with an end mass and its mode: ~.-. -. third mode.
rotary
inertia.
zr = B,
/ _._.&&::: 1 _.-.i
0;
p,-08 5
‘---_
.._
_
----‘_._
-.-
t
-
----._,
-.-
_.
-.--ilIII____,
Mass
Figure K/I
=
ratq
+
ui of a double tapered cantilever beam with an end mass and its rotary first mode; ---, second mode: -. .-. third mode.
4. Eigenvalues
02: e,
.
inertia.
2, = b,
TAPERED
Figure 5. Eigenvalues uf of a double tapered
K/I = 0.4: - --, first mode:
cantilever beam with an end mass and its rotary --, second mode: -.-.-. third mode.
and that associated with TB7 is
[ml=
47
BEAM FINITE ELEMENTS
I 1 mn 0 0 Jd
0 0
0 0
0
0
0
0’
0
0
00
inertia.
5~~= /I,:
where md and Jd are the discrete mass and its attached rotary inertia, respectively. Also, Jd = m$, where K is the radius of gyration of the attached mass. The computed results are presented in Figures 2 through 5. In all these computations, the beam structures were approximated by ten elements. It may be appropriate to point out that there is no exact solution available in the literature concerning a tapered cantilever beam with an end mass and rotary inertia of the end mass. TWO features appear in Figures 3 through 5. They can be best studied with Figure 4. First, the curve of the third mode with /?V= O-2 intersects those of the second mode with j?, = O-8 and 1.O at about 4 = 0.15 and 0.3, respectively. The other interesting feature in
48
C. W. S. TO
Figure 4 is to be found at about 4 = 0.35 and 0.9 for the first mode plots. At about C$= 0.35 all three cases with taper ratios /I, = 0.5, 08 and 1.0 have the same eigenvalue. Also, at about C$= 0.9 the two cases with /I, = 0.8 and 1.0 have a common eigenvalue. This feature is different from the first one in that the common eigenvalue occurs in the first mode. It implies that in the range of 0.5 d B, < 1.0 with the given mass and length ratios the lowest eigenvalue of the beam structure is independent of taper ratio.
7. CONCLUSIONS
To recapitulate, the investigation presented in this paper encompasses two integral parts: namely, (1) the examination and development of higher order tapered beam finite elements, and (2) the application of the higher order tapered beam elements to the transverse vibration of tapered cantilever beam structures with end mass and rotary inertia of the end mass representing a class of tapered mast antenna structures. The conclusions drawn from the first part of the investigation are as follows. (1) In general, the eigenvalues obtained by using higher order tapered beam elements converge more rapidly to the exact solution, but, for a pyramid and wedge, in particular, the adverse trend indicates that a more correct representation of the displacement function within elements cannot make up for the loss of accuracy resulting from coarser subdivision. This leads to a further conclusion that the contradictory general conclusions drawn by Handa and by Thomas and Dokumaci are both correct only with respect to the cases considered by them. Neither one of these two contradictory conclusions drawn by them should be regarded as a general conclusion for tapered beam elements. (2) The values of mode shapes obtained are in excellent agreement with the exact solution. (3) Additional information concerning curvature, a2w/ax2, and its gradient, a”w/ax’, is available. Thus, application of higher order tapered beam elements, such as the septimal order element presented here, permits problems of stress analysis of beam structures, including tapered ones, to be dealt with, circumventing the difficulties caused by discontinuous bending moments between elements when using a lower order tapered beam element. (4) The explicit element mass and stiffness matrices eliminate the loss of computer time and round-off errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. (5) Now that explicit expressions of mass and stiffness matrices of higher order tapered beam elements are available comparison between results obtained by using these expressions and those obtained by employing numerical integration and matrix operations for the element matrices can be made. (6) Greater accuracy can be achieved by using higher order uniform beam elements, which are special cases of the corresponding tapered beam elements. This finding is consistent with that of Fried [3]. In respect to the second part of the investigation, two points are to be noted. First, graphs such as those presented in Figures 2 through 5 are most useful in the design of similar structures. Second, the vibration characteristics of composite beam structures, with or without lumped masses and their rotary inertias, can be investigated by using the higher order tapered beam elements presented here.
TAPERED BEAMFINITE ELEMENTS
ACKNOWLEDGMENTS
The author gratefully acknowledges support by the Admiralty Surface Weapons Establishment, Ministry of Defence. He also wishes to express his gratitude to his supervisor, Professor B. L. Clarkson, for guidance; and to Dr M. Petyt for advice during the course of this investigation.
REFERENCES 1. T. Y. YANG and C. T. SUN 1973 Journal of Sound and Vibration 27,297-311. Finite elements for the vibration of framed shear walls. 2. C. T. SUN and S. N. HUANG 1975 Computers and Structures 5, 297-303. Transverse impact problems by higher order beam finite element. 3. I. FRIED 1971 American Institute of Aeronautics and Astronautics Journal 9, 2071-2073. Discretization and computational errors in high-order finite elements. 4. G. M. LINDBERG1963 Aeronautical Quarterfy 14,387-395. Vibration of non-uniform beams. 5. K. N. HANDA 1970 Ph.D. Thesis, University ofSouthampton. The response of tall structures to atmospheric turbulence. 6. J. THOMASand E. DOKUMACI 1973 Aeronautical Quarterly 24,39-46. Improved finite elements for vibration analysis of tapered beams. 7. W. J. WORLEYand F. D. BREUER1957 Product Engineering 28, 141-144. Areas, centroids and inertias for a family of elliptic-type closed curves. 8. E. BODEWIG1959 Matrix Calculus. Amsterdam: North-Holland Publishing Company, second edition. 9. E. C. PESTKL1965 Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, U.S.A., October 1965. Dynamic stiffness matrix formulation by means of hermitian polynomials. 10. 0. C. ZIENKIEWICZ1971 The Finite Element Method in Engineering Science. London: McGrawHill, second edition. 11. H. H. MABIEand C. B. ROGERS1974 Journal of the Acoustical Society of America 55, 986991. Transverse vibrations of double-tapered cantilever beams with end support and with end mass. 12. C. W. S. TO 1976 Institute of Sound and Vibration Research Contract Report No. 7717. A second mathematical model of ship mast antenna structures subjected to transient base disturbances.
APPENDIX:
NOTATION
In addition to the notation defined in what follows, symbols have been defined throughout text whenever it was felt necessary for clarity. Matrices, vectors, Roman upper and lower cases cross-sectional area A(x), A
[Bl b(x), b d(x), d E I(x), I
Jd
1;’ L
CM1
differential of matrix of element shape functions, [N] breadth of cross-section depth of cross-section Young’s modulus of elasticity second moment of area of cross-section node number rotary inertia of discrete mass, md integer assembled stiffness matrix of the structure element stiffness matrix length of element length of the structure assembled mass matrix of the structure
the
C. W. S. TO
50 element mass matrix discrete mass matrix of element shape functions nodal displacement vector surface area Greek lower case a breadth
; BS K P
4 w
taper ratio of the beam element breadth taper ratio of the structure depth taper ratio of the beam element depth taper ratio of the structure radius of gyration of the discrete mass, m,, density of material mass ratio: that is, the ratio of discrete mass md to mass of the structure radian frequency
ofSoundand
Vibration (1979) 63( I), 33-50
HIGHER ORDER TAPERED BEAM FINITE ELEMENTS FOR VIBRATION ANALYSIS C. W. S. To Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH. England (Received 17 July 1978)
In this paper, explicit expressions for mass and stiffness matrices of two higher order tapered beam elements for vibration analysis are presented. One possesses three degrees of freedom per node and the other four degrees of freedom per node. The four degrees of freedom of the latter element are the displacement, slope, curvature and gradient of curvature. Thus, this element adequately represents all the physical situations involved in any combination of displacement, rotation, bending moment and shearing force. The explicit element mass and stiffness matrices eliminate the loss ofcomputer time and round-off errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. Comparisons with existing results in the literature concerning tapered cantilever beam structures with or without an end mass and its rotary inertia are made. The higher order tapered beam elements presented here are superior to the lower order one in that they offer more realistic representations of the curvature and loading history of the beam element. Furthermore, in general the eigenvalues obtained by employing the higher order elements converge more rapidly to the exact solution than those obtained by using the lower order one.
1. INTRODUCTION
The vibration analysis and design of ship mast antenna structures, and similar structures to withstand shock impact upon the ship or wind induced vibrations, can be investigated by using one dimensional beam elements. In this paper, the element properties (namely, the mass and stiffness matrices) of a linearly tapered beam finite element in which a seventh degree polynomial displacement function is used, representing four degrees of freedom per node (the transverse displacement w, the slope dw/ax, the curvature a2w/ax2, and the gradient of curvature @w/8x3) are derived and the explicit expressions for the mass and stiffness matrices are given. These element properties are subsequently used for the vibration analysis of mast antenna structures individually treated as a tapered cantilever beam with a mass, incorporating rotary inertia, attached to its free end. The development of this higher order tapered beam element is motivated by two main factors. First, it has been shown in the cases of axial vibration of a bar [l] and both transverse impact problems [2] and vibration analysis [3] of uniform beams that improved accuracy can be obtained more efficiently with an increase of the number of degrees of freedom in the element rather than with an increase of the number of elements of the same or fewer degrees of freedom. Furthermore, in general, finite elements based on low degree polynomial displacement functions incorporate only crude curvature distributions and usually yield discontinuous bending moments among elements. In particular, the linearly tapered beam finite element, in which a cubic displacement function developed by Lind33 0022-460X/79/050033
+ 18 $02.00/O
0 1979 Academic
Press Inc. (London)
Limited
34
C. W. S. TO
berg [4] is used, has two inherent disadvantages: namely, the curvature, $2w/8x2 varies linearly along the length of the element, which is not realistic, and the rate of loading a4w/8x4 is zero, which does not resemble the true loading history of the element. The same element but with a quintic displacement function as first considered by Handa [5], and later by Thomas and Dokumaci [6], leads to contradictory results. According to the results obtained by Handa one would draw the conclusion, contrary to that drawn by Thomas and Dokumaci, that a more correct representation of the displacement distribution within elements can make up for the loss of accuracy resulting from coarser subdivision. Because of this contradiction, an examination of the tapered beam finite element with a quintic displacement function is in order. Before this examination and the derivation of the properties of the linearly tapered beam element with a septimal (polynomial of seventh degree) displacement function the element geometrical properties will be considered first in the following section.
2. GEOMETRICAL PROPERTIES OF CROSS-SECTIONAL AREA One class of non-uniform beam element is the linearly tapered element as shown diagrammatically in Figure 1. A linear, homogeneous, isotropic material is assumed, so that the cross-sectional area and second moment of area are given by (a list of notation is given in the Appendix) A(x) = c,&)d(x),
I(x) = QWd3(x),
(la, b)
where c1 and c2 are constants that depend on the shape of the beam cross-section. For an
ni
YLL
Node
3
/-I
(bl
T d
a
Figure 1. The linearly tapered beam element. (a) Beam element with positive edge forces in which 0 = aw/?x. 4 = Pw/dx2 and JI = d3w/ax3; (b) tapered beam element; (c) beam cross-sections at section s - s in (b).
TAPERED
35
BEAM FINITE ELEMENTS
elliptic-type closed curve cross-section, they are given by [7] Cl
=
r(l//fi+
l)l-(l/pL, + l)lr(li&
(>a)
+ l//J? + l),
c2 = (1112)1-(11~~+ 1)r(3/p(, + l)/r(lk + 3/i+ + I), (2b) where r( ) is the gamma function and pi and pz are real positive numbers which need not be integers. With pi = pz = 1,the cross-section is a triangle (in this case the factor l/12 in c2 should be replaced by 1/9) whereas with p, = pL2= 2 it is an ellipse. As p, and I”? each approaches infinity it is a rectangle. The cross-sectional dimensions, b(x) and d(x), vary linearly along the length of the elements so that h(x) = b,_ j [I -I- (a - 1)x/L],
d(x) = d,- Jl
+ (/I - 1)x/L],
(3a, b)
where a = bJb,_ 1, and @ = di/di_ 1 are the taper ratios for the beam element. Substituting equation (3) into equation (1) results in A(x) = Ai_ ,(l + ;‘, i’ + y,:2), I(x) = Ii- ,(l + A, + i&i’+
(4a)
&t3 + &<4),
(4b)
where 5 = x/L, 6, =(a-
y1 = (a - i) + (B -
1) +3(/?-l),
y2 = (a - 1MB -
I),
6,=3(a-
6, = 3(a - I)(/? - 1)’ +(/I - 1)3, A,_ 1 and Ii_ 1 are respectively the cross-sectional ated with node i - 1.
l)(p-
1) +3(8-
6, = (a -
I)(B -
I),
1)‘. 1)3:
area and second moment of area associ-
It should be noted that in applying equation (4) to hollow beams, of square or circular cross-section, for example, either the ratio b/d must be small or the ratio b/d must be constant because in equation (1) for a square hollow cross-section c, = 4 and c2 = (2/3)[1 + (b/d)*], and for a circular hollow cross-section c, = TCand c? = (7rr/8)[I + (b/d)2].
3. PROPERTIES OF THE TAPERED BEAM ELEMENT WITH A QUINTIC DISPLACEMENT FUNCTION The quintic displacement function employed by Handa [5], and later by Thomas and Dokumaci [6], is w = ,i,
cjx- ‘,
or, in matrix form,
w = Cf#ml~
(5)
where [4] = [l x x2 x3 x4 x”] and (i} = [i, iz c3 [, cs ;,I’, in which the superscript 7 designates “the transpose of”. Introducing the co-ordinates successively into equation (5) leads to the nodal displacements (41 = where
[Cl{ij,
(6)
36
C. W. S. TO
1
0
0
0
0
0
0
1
0
0
0
0 (7)
0
1
2L
3L2
4L3
5L4
0
0
2
6L
12L2
2oL- I
By applying the so-called Frobenius-Schurs of [C] is
[cl-’
relation [S], it can be shown that the inverse
1
0
0
0
0
0
0
1
0
0
0
0
0
0
112
0
0
0
= - lo/L3
- 6/L2
15/L4 - 6/L’
- 3/2L
8/L3
3/2L2
- 3/L4
- 1/2L3
1O/L3
- 4/L2
- 15JL4
7JL3
6/LS
1/2L
(8)
- l/L2 1/2L 3
- 3/L4
By applying equation (8) in conjunction with [$J] as given in equation (5), the shape functions are found to be (9)
[N] = [NII N12N13N14Nl,Nld where N,, = 1 - lot3 + 155” - 65’,
N,, =
Nl3 = +((L)2(1 - 35 + 342 - <3),
N,, = 4L(7c3 - 4c2 - 3t4),
N,, = 10c3 - 1554 + 6cs,
N,, = $
On application of the definitions and going through a lengthy algebraic manipulation, the mass and stiffness matrices of the element are found to be those given in equations (10) and (11). When a = p = 1 the results from equations (10) and (11) tally with those obtained by using Hermitian polynomials as given in reference [9]. Note that there are typographical errors in reference [S] concerning matrices [Cl, [Cl- ‘, [m] and [k]. Table 1 gives these errors.
TABLE 1
Typographical
errors in CC], [Cl-
Matrix
Row
[Cl
6 6 6 5 5 6
ccl-’
icorrection
‘, [m] and [k] in reference [5] Column 5 6 3 3 4 2
for the common factor in this matrix is also necessary.
[k]
=
Ezl-l
13860L3
(1584 + 15846, :;;;;TJ)$ 11226, (198 t 996 -+776, +66& + 60 6,)L4
(42768 + 213846, +150486, + 118806, +9720 6,)L*
(-1584 + 2646, +3306, + 3606,)p
(118800 + 712806, +514806, + 396006, +313206,)L
-(5940 + 19806, +6606, - 3906&Z?
-(5940 + 39606, +26406, + 19806, + 1590 6,)LZ
(1188 + 2976, +1546,+ 996, +726,)L"
f277206, + 198006, +15 480 6,)L
-(118800+475206,
(156 + 65y, +3%,)L4
+79200& + 594006, +468006,)
1188006,
(4356 + 23766, +13206, + 8586, +6306,)L3
(5940 + 39606, +26406, + 19806, +15906&Z
-(237600+
(76032 + 261366, +126726,+ 79206, f57606,)L'
Symmetric
(1794 + 702~~
(21632 + 7800~~ +3264y,)L* +310y2)L3
(7306 + 25747, +1062y,)Lz
(97032 + 300567, + I1 52Oy,)L
(118800 +47520$ +277206, f 198006, +15 4806,)L
(237600 + 1188006, +792006, + 594006, +468006,)
(564720 + 127920,, t42800 y2)
-Cl352 + 702~~
(4706 + 25747,
(5940 + 19806, +6606, - 3906.JZ.2
-(118800 + 712806, +514806, + 39600~5, + 3 I 320 6,)L
(237600 + 1188006, +792006, + 594OOd, + 468OOb.J
-(1794 + 1092y, +700;~z)z.3
(21632 + 13832y, +9296 yz)L2
-(4356 + 19806, +9246, + 3306, - 306JL3
(76032 + 498968, f364326, + 277206, + 21 6006,)L'
Symmetric
(7306 + 47327, +322O;l,)Z.z
-(97032 + 669767, + 4844OyJL
(564720+4368007, +35168oy,)
(1188 + 89lh, +l4862 + 6606, +6006,)L4
(156 + 9ly, +56y,)L4
+35y,)L4
+ 392 yJL3
(130 + 65y,
+34Oy,)L3
(1352 + 65Oy,
(4706 + 2132:, tl070y-JZ.Z
+1512y$
+1422Oy,)L
-Cl3832 + 6916y, +3756y,)L'
-(47112 + 222567, +11 620 yJL
(47112 + 248567,
(156000 +78000~, +43000y2)
38
c. 4. PROPERTIES
W.S.TO
OF THE TAPERED BEAM ELEMENT DISPLACEMENT FUNCTION
WITH A SEPTIMAL
The septimal displacement function considered here is w = j$, or, in matrix form,
ijxj-',
(12)
w = r+lK>~
where [$J] = [l x x2 x3 x4 x5 x6 x71 and {i} = [i, i2 13 i, As in section 3, the nodal displacements are expressed as
is
16 c-7 kc,]‘.
(4) = [Wf
(13)
where
(4) = Cwi- ‘i- 14i1
1
1 tii-
‘i
wi
I(/ilT,
4i
a3w/ax3,
J/ =
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
6
0
0
0
0
[Cl = 1
L
L2
L3
L4
LS
L6
L7
0
1
2L
3L2
4L3
5L4
6L5
7L6
0
0
2
6L
12L2
20L3
30L4
42Ls
0
0
0
6
24L
60L2
120L3
21OL4
(14)
The inverse of [C] is found to be 1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 1 6 -35/L4 -20/L3 -5/L' -2/3L 0
[cl-l
=
84/L’
0
45/L4
1 T
1O/L3
35/L4 - 15/L3
r/L= - 84/L’
- IO/L6-36/L' - 15/2L4 - 2/3L3 20/L’
IO/L6 2/L5
1/6L4
39/L4 - 7/L3
IO/L6- 34/L5 - 2O/L7
5/2L2 - 1/6L
(15)
l/2L2
13/2L4 - l/2L3
lo/L6-.2/L5
ll6LQ
The shape functions for this beam element are [N] = [N,, N,,N,,
N,,N,,
N,, NI: NH],
(16)
where N,, =
N,, = 1 - 35t4 + 844’ - 70c6 + 205’,
N,, = $(i’L)2(1 - IO<= + 20c3 - 15r4 +4:5) N,, = 35t4 - 845’ + 70t6 - 20t7,
N,, = $(
N,, =
Nr, = $(5L)2(5<2 - 14i3 + 13c4 - 45’),
N,, = 35L)3(352 - i - 3i3 + i4).
TAPERED BEAM FINITE ELEMENTS
39
As the element mass and stiffness matrices are symmetric only half of these matrices are given here. The diagonal and super-diagonal elements of the element mass matrix are as follows : + 1314432Oy,],
=
n[178557120
ml2
=
qL[33 268 320 + 10 208 160 'r', + 3 805 920 y2],
ml3
=
‘1L2[3 353 760 + 1 168 920 y, + 468 648 yZ]r
“11
+ 402696007,
ml4 = qL3[156264 + 583447, m15 = q[41983200
+ 24468y2],
+ 20991 6OOy, + 11 360 t6oy,],
qL[ 13 990 320 + 6 695 280 ;‘! + 3 470 760 y2],
m16
=
-
ml:
=
qL2[l
m18
=
-nL3[106284
897 200 + 878 220 ;‘, + 440 784 y2], + 47940~~ + 23472~~1,
mz2 = qL2[8078400 + 286416Oy,
+ 1 161 216~~1,
mz3 = qL3[899 640 + 345 168 y, + 147420~~1, mz4 = r,rL4[44064 + 17 748 y1 + 7848 y2], qL[ 13 990 320 + 7 295 040 y1 + 40 70 520 y2],
m25
=
m26
=
“2:
=
qL3[608 940 + 294 984 y1 + 153072 y,],
m28
=
-nL4[33660
-nL2[4565520
+ 228276Oy, + 1221 912y,],
+ 1591211, + 8064~~1,
m33 = qL4[105264 +4284Oy,
+ 1908Oy,],
m34 = qL5[5304 + 2244y, + 1029y,] m35
=
qL2[l
897200 + 101898Oy, + 581 544~~1,
%6 = -nL3[608940 m37
m 38
=
=
+ 313956y,
nL4[80172 + 40086y, -qL5[4386
+ 172044y,],
+ 21 312;‘?],
+ 2142y, + 1113yJ,
m44 = qL6[272 + 119 y1 + 56 y2], m45 = qL3[106284 + 58 344~~ + 33 876~~1, -nL4[33 660 + 17 748 y1 + 9900 y2],
rn46
=
m4:
=
qL5[4386 + 2244 y1 + 1315 y2],
m48
=
-qL6[238
+ 119 y, + 63 y,],
m55 = t(l78557120 =
m5:
=
m58
=
-nL3[156264
=
qL2[8078400
)7166
+ 111
16224Oy,],
qL[33 268 320 + 23 060 160 ;‘, + 16 657 920 yz],
m56
-
+ 1382875207,
qL2[3 353 760 + 2 184 840 ;‘1 + 1484 568 y2], + 979207, + 64044y,], + 521424Oy, + 3511 296~~1,
C. W. S. TO
rn6: m68
=
-qL3[899640
+ 5544727,
=
~I,~[44064 + 263167,
+ 356724yJ,
+ 16416y,],
= qL4[105 264 + 62424;1, + 38664~~1,
%
rnT8 = -@[5304 11288
=
+ 3060;‘, + 1845yJ,
uL6[272 + 153 ;‘, + 9Oy,], q = pA,_ ,I,/441 080640.
The diagonal and sub-diagonal elements of the element stiffness matrix are k 11 = H[27518400
+ 137592006,
+ 84672006,
+ 58212006,
+ 42336006,-j,
k,, = HL[l3759200
+ 58968006,
+ 3250800&
+ 21168006,
+ 15120006,],
k,,
= HL*[8424000
+ 32292006,
+ 15552006,
+9072OOfi, + 6048006,-J,
k 31
=
HL2[l
k 32
=
HL3[886860 + 4212006,
k,,
= HL4[234000
k,,
= HL3[49140
+ 327606,
+ 214206,
+ 15 1206, + 113406,],
k,,
= HL4[37440
+ 210606,
f 118806,
+ 73806, + 50406,-j,
k,,
= HL5[10920
+ 39006, + 19866, + 11886, + 7926,],
k,,
= HZ?[624 + 2346, + 1208, + 726, + 486,],
k,,
= -H[27518400
k,,
= -HL[13759200
k,,
= -HL*[l
k,,
= -HL3[49
k,,
= H[27518400
k,,
310400 + 7043406,
+ 702006,
+ 4208406? + 2835006, + 2197806,
+ 344406,
+ 1314006, + 202506,
+ 2066406,], + 880206,],
+ 133926,],
+ 137592006,
+ 84672006,
+ 58212006,
+ 58968006,
+ 32508006,
+ 2 1168006, + 15120006,],
310400 + 7043406, 140 + 327606,
+ 420840&
+ 214206,
+ 2835006,
+ 4233600&j,
+ 2066406,],
+ 15 1206, + 113406,],
+ 137592006,
+ 84672006,
+ 5821200 + 42336006,],
= HL[l3759200
+ 78624006,
+ 52164006,
+ 37044006,
+ 27216006,],
k 62 = HL*[5335200
+ 26676006,
+ 16956006,
+ 12096006,
+ 9072006,],
k,,
= HL3[423 540 + 283 1406, + 2010606,
k 64 = HL4[ll
+ 152 1006, + 1186206,-j,
700 + 117006, + 95406, + 77406, + 63006,],
k,,
= -HL[13759200
k,,
= HL*[8424000
k,, = -HL2[1310400 k:, = -HL3[423540
+ 78624006,
+ 52164006,
+ 5 1948006,
+ 35208006,
+ 6060606,
+ 3225606,
+ 1404006, + 583206,
+ 37044006, + 24948006,
+ 1764006,
+ 252006,
k 73 =
HL4[-2340
k 74
ZfL5[1950 + 7806, + 4716, + 3486, + 2946,],
=
k7, = HL2[1310400
+ 27216006,], + 18144006,],
+ 907206,],
+ 75606,-j,
- 11706, + 6606, + 15756, + 21966,],
+ 6060606,
+ 3225606,
+ 1764006, t- 907206,-j,
41
TAPERED BEAM FINITE ELEMENTS
k7, = -HL3[886860+ k,: = HL4[234000+
4656606, + 2642406, + 1512006, +831606,-j, 1638006, + 128040d2 t1064706, +922326,-j,
k,, = HL3[49140 + 163806, + 50406, - 252OS,], k,, = HL4[11 700 - 21606, - 25206, - 25206,], k,, = -HL5[1950 + 11706, + 8616, + 675b, + SSSS,], k,, = -Z-fL6[234-I+1176, + 756, + 546, + 426,-j, k,, = HL3[-49140-
163806, - 50406, + 25206,],
k,, = HL4[37440 + 163806, + 72006, + 25206,], ks7 = -HL5[10920+
70206, + 51066, + 39906, + 3276b,],
k,, = HL6[624 + 3906, + 2766, + 2106, + 1686,], H = Eli_ #081080L”. For uniform beams, that is when a = /I = 1, the above results reference [9], as obtained by using Hermitian polynomials.
5. VIBRATION
CHARACTERISTICS OF LINEARLY ELEMENTS
On assuming simple harmonic motion with radian of motion for the assembled structure can be written ([K]
- m2 CM])
frequency as
agree with those given in
TAPERED
BEAM
w the governing
equation
(Q,)= {O),
TABLE 2
Comparison of eigenvalues, (Al)* = wl*J’~ of three linearly tapered beam elements (cantilever) (a) a, = fi, = 0, pyramid First mode 8.7192
Second
mode
Third mode
21.146
38.453
Fourth
mode
Fifth mode
-_____ I
2
tB3 TB5 TB7
8.73517 8.74506 8.75096
25.18142 23.18407 22.83025
68.85968 55.37000
78.22758 169.30409
78.22758 178.53337
TB3 TB5
21.71782 21.25307 21.27920
41.92455 40.39639 40.25180
9486600 72.27633 70.13591
103.73917 128.51998 I1 7.87558
TB7
8.72388 8.72021 8.72089
4
TB3 TB5 TB7
8.71961 8.71927 8,71936
21.20345 21.14939 21.15169
39 10222 38.54472 38.57220
62.76983 61.57421 6 1.60905
98.77 139 92.29457 91.80001
IO
TB3 TB5 TB7
8.7 1927 8.71927 8.7 1927
21.14733 21.14571 21.14571
38.47887 38.45479 38.45556
6085846 6069236 60.69902
88.60268 87.92289 87.95364
(b) a, = fi, = 02 First mode Second mode
Third mode
Fourth
mode
71.242
Fifth mode 112.828
6.1964
18.3855
39.834
TB3 TB5 TB7
6.19986 6.20819
18.95919 19.03870
56.85606 48.69267
63.82743 147.25952
63.82743 154 19323
2
TB3 TB5 TB7
6.20186 6.19648 6.19667
18.88848 18.38837 1840247
41.11316 39.96101 4009945
92.82321 73.73720 73.37800
105.03897 131.03760 123.68543
4
TB3 TB5 TB7
6.19676 6.19641 6.19640
18.42262 18.38641 18.38556
4041611 39.84572 39.83513
75.30089 71.30942 71.25365
117.85937 113+)4873 112.89501
TB3 TB5 TB7
6.19640 6.19639 6.19639
18.38641 18.38547 18.38547
39.84648 3983367 39.83359
7 I.32205 7 1.24245 71.24177
113.16316 112.83180 I 12.82753
1
10
(c) a, = 8, = O-4 First mode
Second mode
Third mode
Fourth
mode
85.344
Fifth mode 138.0352
5.009032
19.06488
45.73836
TB3 TB5 TB7
501026 50J9760
19.11975 1916211
53-98838 4917535
6@98298
6098298
138.67416
148.06058
TB3 TB5 TB7
5.01154 5.009145 5009033
19.52378 1907756 19.06526
5001809 45.90603 45.74914
112.90697 86.13375 85.54191
13025277 146.41576 149.8967 I 138.38380 138.03534 138.37866 138.03620 138.03534
--
1
2
4
TB3 TB5 TB7
5.00925 5+IO9034 5.009033
1908500 19*06509 19.06483
46.06530 4574411 45.73838
88.90501 85-41801 85.34387
10
TB3 TB5 TB7
5w904 5.009033 5009033
1906569 1906483 19.06483
45.751 11 45.73838 45.73838
85.42747 85.34387 85.34378
140.99705
(d) a, = /3, = 1.O, uniform beam First mode 3.516015
1
TB3 TB5 TB7
Second
mode
22.03449
_~_____~_______~-________ 3.53270 34.8066 1 3.516019 22.22655 3.516019 22.03545
Third mode 61.69717
Fourth
mode
120.90192
Fifth mode 199.85953 __
_._.
6476539 61.76748
7421237 136.28122
7421237 151.70892
2
TB3 TB5 TB7
3.51772 3.516019 3.516019
22.22151 22.03474 2203449
75.15707 61.87149 61.69745
218.13809 123.27721 12093808
239.24153 206.46548 200.02710
4
TB3 TB5 TB7
3.51613 3.516019 3.516019
22.06019 2203449 22-03449
62.17492 61.69753 61.69719
122.65805 120.90648 12090220
228313769 200.13470 199*85971
10
TB3
3.516019
22.03525
T35
3.516019
2203448
61.71290 61.69718
121.01749 120-90220
200.36356 199.85969
TB7
3.516019
22.03448
61.69718
12090220
199.85969
43
TAPERED BEAM FINITE ELEMENTS (e) a, =
First mode
1 and 8, = 0, wedge Second
mode
Third
mode Fourth mode
Fifth mode
5.3150
15.2012
30.0198
TB3 TB5 TB7
5.3 1872 5.33 166 5.33991
17.30040 16.59495 16.57530
52.12928 43.70635
58.5744 1 137.69459
58.57441 146.05962
2
TB3 TB5 TB7
5.3 1585 5.31629 5.31735
15.44434 15.32421 15.37628
32.70286 3 1.92000 32.02802
78.57037 6099690 59.47300
84.80926 I 13.58248 104.34379
4
TB3 TB5 TBI
5.31515 5.31518 5.3 1526
15.22990 15.21513 1522119
30.37696 30.17491 30245 19
5 1.48493 51+0310 51.20251
85.8075 1 79.84908 79.69246
10
TB3 TB5 TB7
5.31510 5.3 I 502 5.31502
15.20778 15.20736 15.20761
30.03408 30.024 17 30.02785
49.88682 49.80433 49.83131
75.01 171 74.67446 74.78104
1
49.161
where ([K] - o2 [M]) is the dynamic stiffness matrix of the structure in question and {Q,} the column matrix of amplitude of displacement. The solution to the above algebraic equation is known as the eigenvalue solution [lo]. The computed results presented here are confined to vibration characteristics of clampedfree tapered beam structures. Consistent with the notation employed in Figure 1, the boundary conditions are as follows: at
x = 0,
w= 0
and
at
x = I,
a2~l/ax2 = 0
awlax = 0: and
~/dx(El(d2tllld.u2)) = 0.
It is to be noted that for the class of tapered beams considered, I is a function of x and therefore the last boundary condition amounts to EI(d3w/dx3) + E(al/dx)(Pw/Bx?
= 0.
Table 2 provides a comparison of the squared eigenvalues, US = of/*,!pA,/EI,, for five different cases. The values obtained by using the finite elements due to Lindberg, Handa, and the author are referred to as TB3, TB5 and TB7, respectively. The analytical solution is termed “exact”. In the case of a pyramid the exact solution is deduced from results in reference [4]. Exact solutions for the two cases a, = j?, = 02 and CI,= fi, = 04 are deduced from results presented in reference [I I], where a, = b,,/b, and j?, = d,,/d,, in which b, and d,,, and b, and d,, are the values of the cross-sectional dimensions at the tip and base respectively. The exact solution for a uniform beam is derived from reference [ 121 whereas that for a wedge is deduced from reference [6]. The mode shapes for the three special cases-namely, a pyramid, wedge and uniform beam-are presented in Table 3. The exact solutions for the first two cases are drawn from reference [4] while those of a uniform beam are obtained by using the computer program included in reference [ 121. Results in Table 2 lead to the following conclusions. 1. In general, a more correct representation of the displacement function within tapered beam elements can make up for the loss of accuracy resulting from coarser subdivision. In other words, the convergence of higher order tapered beam finite elements is more
0 0.0062 @0259 00614 01149 0.1893 0.2877 04138 0.5715 07656 i-0000
0 OGO62 @0259 O-0614 01149 @1892 0.2876 @4136 0.5713 0.7653 1GooO 0 00168 O-0639 Ck1365 0.2299 0.3395 04611 0.5909 0.1255 08624 1Woo
(cl a, = B, = 1, uniform beam 0.0 0 0 0.1 00168 0.0168 02 0.0639 0.0639 0.3 01365 @1365 D4 0.2299 0.2299 o-3395 05 Q3395 @6 0.4611 O-461 1 07 0.5909 0.5909 0.8 0.7255 07255 0.9 08624 0.8624 10000 I .o 1Woo 0.8168 0.0639 01365 0.2299 a3395 04611 0.5909 Cl.7255 08624 1NtOO
o-0501
00964 01635 02557 0.3787 G-5388 0.7435 l*ooOO
o-0206
- 0.0926 - 0.3011 - 05261 - 06834 - 07137 - 05895 -03171 00700 0.5238 1NWo
0
0 - 00926 - 0.3011 -0.5261 -0-6835 -0.7137 - 05895 -03171 0.0700 @5238 1SQOO
0
- 0.0078 - 0.0302 - oa640 - 01023 -0.1330 -0.1360 - 00803 PO801 0.4112 1@000
0
1moo
-- 0.0926 - 0.3011 - @5261 -0.6835 - 07137 - 0.5895 -03171 0.0700 0.5238
0
- 00037 -00147 -@0315 - 0.0498 -00612 - 00500 0.0097 0.1589 0.4604 1OOOo
- OGO78 - 0.0301 - 0.0637 -0.1018 - 0.1323 -0.!352 - 0.0198 0.0796 0.4090 1-0000
0
TB5
0 - 00078 - 00300 - @0636 -01017 -@I322 -0.1352 - @0798 00796 0.4087 1GOOil
TB3 _--^_----__-__--
w of second mode h
0 - 00037 - (fO146 -00313 - O-0495 - 00608 - 0.0496 0.0097 01580 0.4576 1MIO
I Exact
0 0.0037 00146 0.0313 CM495 QO607 bO496 0.0096 0.1578 04571 lN)OO
-
2048
TB7
0 00048 O-0206 00501 O-0964 0.1634 o-2556 03785 05386 0.7432 1WMO
TB5
0 0 00 @1 OGO48 QOO48 00206 00206 0‘2 00500 0.3 00500 00963 00963 04 01632 01632 @5 02553 0,6 02553 03781 @7 @3781 05379 &8 05379 0.7423 07423 0.9 1WOO 1*OoOO 1.0 (b) a, = 1 and & = 0, wedge 0 0.0 0 0.1 o&62 cGO62 02 00259 00259 @3 0.0613 00613 o-4 O-1148 01148 0.5 O-1891 01891 66 0.2874 0.2874 0.7 0.4133 04133 0.8 05709 0.5709 @9 0.7648 07648 1.0 l%IOtKI IWO
(a) a, = 8, = 0, pyramid (cone)
TB3
w of first mode
19ooo
- 00926 -@3011 - W261 - 0.6835 -0.7137 - @5895 - 0.3171 00700 05238
0
-t30078 -DO303 - @0642 --@lo26 -0.1333 -0.1363 - 00805 00803 0.4122 1@000
0
0 -00037 -00148 -0.0316 - QOJOO -0.0613 -@0501 MO97 01593 04615 1+KKlO
TB7
\
0.2281 0.6045 0.7562 0.5259 c-0197 - 0.4738 - 0.6574 - 0.3949 0.2285 1GOOO
0
0.0086 00304 QO552 00681 0*0515 -00063 -00915 -0.1282 00878 1WOtl
0
0 00030 O-0107 vo191 0.0216 00100 -@0196 -0-0541 - 00340 O-1991 l.OOOo
Exact
0
0.228 I 06045 07562 0.5259 00197 - 0.4738 - 06574 - 0.3949 0.2285 1GOOO
0
00088 0.0311 00566 OG698 @0527 - 00065 -00937 -0.1313 oWOO 1QGilo
0
0 00031 00109 @0196 00221 @0103 -002OI -0.0554 - 0.0348 0.2040 1MOO
TBS
o-228 I 0.6045 @7562 0.5259 0.0197 - 0.4738 - 06574 - 0.3949 0.2285 1QOOo
@0087 00306 0.0556 0%X85 00517 - 00064 - 0.0920 -@1286 0.0890 1%@00
0
00030 00108 00193 @0218 @OlOl -00198 --@OS44 -00339 0.201 I 1WOO
0
TB3
w of third mode
Comparison of mode shapes of cantilever beams (deflection w)
TABLE 3
0228 1 0.6045 0.7562 O-5259 0.0197 - Q4738 - 0.6574 - 0.3949 02285 l%IOOO
0
00089 0.0314 0.057 1 0.0704 0.0531 - 0.0066 - 0.0946 - a1324 0.0909 1@000
0
00031 0~0110 00197 0.0223 O-0104 - 0.0203 - 00558 -&0351 02055 1WOO
0
TB?
45
TAPERED BEAM FINITE ELEMENTS
rapid. This is particularly so for higher mode eigenvalues. This point is further confirmed by the results obtained for a uniform beam. In particular, results involving the approximation of a wedge by two elements show that increase in accuracy due to more correct representation of the displacement function within elements cannot make up for the loss of accuracy resulting from coarser subdivision. This is precisely the case considered and the conclusion reached by Thomas and Dokumaci [6]. In the cases of a pyramid and wedge anomalous results occur in TB7. The anomality indicates that the convergence of TB7 is less rapid than that of TB5. One possible explanation for this phenomenon is that it is due to the zero value of the crosssectional area (and also of the second moment of area) at the free end of the structure. In Table 3 two points are worth noting. First, for the pyramid and wedge the values of the mode shapes obtained by using elements TB3, TB5 and TB7 show excellent agreement with the exact values though the results by TB3 give the best values. Second, for a uniform beam results indicate that TB3, TB5 and TB7 all converge equally rapidly. Nevertheless. when a comparison to the exact solution obtained by the computer program given in reference [ 121 was made for the beam approximated by four elements, it was observed that the values of the mode shapes obtained by TB7 are superior, particularly for higher modes. 6. VIBRATION
OF TAPERED CANTILEVER BEAM INCORPORATING END MASS AND ITS ROTARY INERTIA
Having examined the three linearly tapered beam elements, one can now consider, the effects of taper ratio, end mass, and rotary inertia of the end mass on the eigenvalues and mode shapes of a double-tapered cantilever beam structure as predicted by using elements TB5 and TB7. To enable this investigation to be made, two discrete mass element matrices, one compatible with TB5 and the other with TB7, are required. By considering the kinetic energy, it can easily be shown that the discrete mass matrix associated with element TB5 is
[m] = IO0
I
,
md
0
0
0 0
0 Jd
0 0
I 1
111111,
,_ _ _.-
-.-
--------________
I
,
I
I
,-
A./
- /.
-,-.-
I,,
--_-______-_---
----
IO-
3
0.01
I
I
t,,,,,,
0.1
I
,
11,111
I.0
Figure 2. Eigenvalues u! of a double tapered cantilever beam with a free end. a, = 8,. -, -, third mode.
second mode: -
First mode:
-
-.
Mass ratio,+
3. Eigenvalues u? or a double second ~-, first mode:---,
Figure K/I
=
tapered cantilever beam with an end mass and its mode: ~.-. -. third mode.
rotary
inertia.
zr = B,
/ _._.&&::: 1 _.-.i
0;
p,-08 5
‘---_
.._
_
----‘_._
-.-
t
-
----._,
-.-
_.
-.--ilIII____,
Mass
Figure K/I
=
ratq
+
ui of a double tapered cantilever beam with an end mass and its rotary first mode; ---, second mode: -. .-. third mode.
4. Eigenvalues
02: e,
.
inertia.
2, = b,
TAPERED
Figure 5. Eigenvalues uf of a double tapered
K/I = 0.4: - --, first mode:
cantilever beam with an end mass and its rotary --, second mode: -.-.-. third mode.
and that associated with TB7 is
[ml=
47
BEAM FINITE ELEMENTS
I 1 mn 0 0 Jd
0 0
0 0
0
0
0
0’
0
0
00
inertia.
5~~= /I,:
where md and Jd are the discrete mass and its attached rotary inertia, respectively. Also, Jd = m$, where K is the radius of gyration of the attached mass. The computed results are presented in Figures 2 through 5. In all these computations, the beam structures were approximated by ten elements. It may be appropriate to point out that there is no exact solution available in the literature concerning a tapered cantilever beam with an end mass and rotary inertia of the end mass. TWO features appear in Figures 3 through 5. They can be best studied with Figure 4. First, the curve of the third mode with /?V= O-2 intersects those of the second mode with j?, = O-8 and 1.O at about 4 = 0.15 and 0.3, respectively. The other interesting feature in
48
C. W. S. TO
Figure 4 is to be found at about 4 = 0.35 and 0.9 for the first mode plots. At about C$= 0.35 all three cases with taper ratios /I, = 0.5, 08 and 1.0 have the same eigenvalue. Also, at about C$= 0.9 the two cases with /I, = 0.8 and 1.0 have a common eigenvalue. This feature is different from the first one in that the common eigenvalue occurs in the first mode. It implies that in the range of 0.5 d B, < 1.0 with the given mass and length ratios the lowest eigenvalue of the beam structure is independent of taper ratio.
7. CONCLUSIONS
To recapitulate, the investigation presented in this paper encompasses two integral parts: namely, (1) the examination and development of higher order tapered beam finite elements, and (2) the application of the higher order tapered beam elements to the transverse vibration of tapered cantilever beam structures with end mass and rotary inertia of the end mass representing a class of tapered mast antenna structures. The conclusions drawn from the first part of the investigation are as follows. (1) In general, the eigenvalues obtained by using higher order tapered beam elements converge more rapidly to the exact solution, but, for a pyramid and wedge, in particular, the adverse trend indicates that a more correct representation of the displacement function within elements cannot make up for the loss of accuracy resulting from coarser subdivision. This leads to a further conclusion that the contradictory general conclusions drawn by Handa and by Thomas and Dokumaci are both correct only with respect to the cases considered by them. Neither one of these two contradictory conclusions drawn by them should be regarded as a general conclusion for tapered beam elements. (2) The values of mode shapes obtained are in excellent agreement with the exact solution. (3) Additional information concerning curvature, a2w/ax2, and its gradient, a”w/ax’, is available. Thus, application of higher order tapered beam elements, such as the septimal order element presented here, permits problems of stress analysis of beam structures, including tapered ones, to be dealt with, circumventing the difficulties caused by discontinuous bending moments between elements when using a lower order tapered beam element. (4) The explicit element mass and stiffness matrices eliminate the loss of computer time and round-off errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. (5) Now that explicit expressions of mass and stiffness matrices of higher order tapered beam elements are available comparison between results obtained by using these expressions and those obtained by employing numerical integration and matrix operations for the element matrices can be made. (6) Greater accuracy can be achieved by using higher order uniform beam elements, which are special cases of the corresponding tapered beam elements. This finding is consistent with that of Fried [3]. In respect to the second part of the investigation, two points are to be noted. First, graphs such as those presented in Figures 2 through 5 are most useful in the design of similar structures. Second, the vibration characteristics of composite beam structures, with or without lumped masses and their rotary inertias, can be investigated by using the higher order tapered beam elements presented here.
TAPERED BEAMFINITE ELEMENTS
ACKNOWLEDGMENTS
The author gratefully acknowledges support by the Admiralty Surface Weapons Establishment, Ministry of Defence. He also wishes to express his gratitude to his supervisor, Professor B. L. Clarkson, for guidance; and to Dr M. Petyt for advice during the course of this investigation.
REFERENCES 1. T. Y. YANG and C. T. SUN 1973 Journal of Sound and Vibration 27,297-311. Finite elements for the vibration of framed shear walls. 2. C. T. SUN and S. N. HUANG 1975 Computers and Structures 5, 297-303. Transverse impact problems by higher order beam finite element. 3. I. FRIED 1971 American Institute of Aeronautics and Astronautics Journal 9, 2071-2073. Discretization and computational errors in high-order finite elements. 4. G. M. LINDBERG1963 Aeronautical Quarterfy 14,387-395. Vibration of non-uniform beams. 5. K. N. HANDA 1970 Ph.D. Thesis, University ofSouthampton. The response of tall structures to atmospheric turbulence. 6. J. THOMASand E. DOKUMACI 1973 Aeronautical Quarterly 24,39-46. Improved finite elements for vibration analysis of tapered beams. 7. W. J. WORLEYand F. D. BREUER1957 Product Engineering 28, 141-144. Areas, centroids and inertias for a family of elliptic-type closed curves. 8. E. BODEWIG1959 Matrix Calculus. Amsterdam: North-Holland Publishing Company, second edition. 9. E. C. PESTKL1965 Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, U.S.A., October 1965. Dynamic stiffness matrix formulation by means of hermitian polynomials. 10. 0. C. ZIENKIEWICZ1971 The Finite Element Method in Engineering Science. London: McGrawHill, second edition. 11. H. H. MABIEand C. B. ROGERS1974 Journal of the Acoustical Society of America 55, 986991. Transverse vibrations of double-tapered cantilever beams with end support and with end mass. 12. C. W. S. TO 1976 Institute of Sound and Vibration Research Contract Report No. 7717. A second mathematical model of ship mast antenna structures subjected to transient base disturbances.
APPENDIX:
NOTATION
In addition to the notation defined in what follows, symbols have been defined throughout text whenever it was felt necessary for clarity. Matrices, vectors, Roman upper and lower cases cross-sectional area A(x), A
[Bl b(x), b d(x), d E I(x), I
Jd
1;’ L
CM1
differential of matrix of element shape functions, [N] breadth of cross-section depth of cross-section Young’s modulus of elasticity second moment of area of cross-section node number rotary inertia of discrete mass, md integer assembled stiffness matrix of the structure element stiffness matrix length of element length of the structure assembled mass matrix of the structure
the
C. W. S. TO
50 element mass matrix discrete mass matrix of element shape functions nodal displacement vector surface area Greek lower case a breadth
; BS K P
4 w
taper ratio of the beam element breadth taper ratio of the structure depth taper ratio of the beam element depth taper ratio of the structure radius of gyration of the discrete mass, m,, density of material mass ratio: that is, the ratio of discrete mass md to mass of the structure radian frequency