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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating
CJA 1091 11 July 2018 Chinese Journal of Aeronautics, (2018), xxx(xx): xxx–xxx
No. of Pages 9
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating
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Xiaode YUAN, Zhihai XIANG *
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Department of Engineering Mechanics, Tsinghua University of Aerospace Engineering, Beijing 100084, China
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Received 1 March 2018; revised 16 April 2018; accepted 15 May 2018
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KEYWORDS
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Fourier finite element; Stability criteria; Thermal flutter; Thermally Induced Vibration (TIV); Thin-walled structures
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Abstract The flexible attachments of spacecraft may undergo Thermally Induced Vibration (TIV) on orbit due to the suddenly changed solar heating. The unstable TIV, called thermal-flutter, can cause serious damage to the spacecraft. In this paper, the coupled bending-torsion thermal vibration equations for an open thin-walled circular cantilever beam are established. By analyzing the stability of these equations based on the first Lyapunov method, the thermal-flutter criterion can be obtained. The criterion is very different form that of closed thin-walled beams because the torsion has great impact on the stability of the TIV for open thin-walled beams. Several numerical simulations are conducted to demonstrate that the theoretical predictions agree very well with the finite element results, which mean that the criterion are reliable. Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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The flexible attachments of spacecraft generally have the characteristics of large size, light weight, low stiffness and small heat capacity. Therefore, these structures are prone to experiencing the Thermally Induced Vibration (TIV) due to the suddenly applied solar heat flux when the spacecraft enter or leave eclipse.1–4 These vibrations could reduce the pointing accuracy of spacecraft and even introduce damage into the structure,
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* Corresponding author. E-mail address: [email protected] (Z. XIANG). Peer review under responsibility of Editorial Committee of CJA.
especially when the vibration is unstable, i.e., the thermalflutter. TIV was firstly predicted theoretically by Boley as early as 1956.5 Boley and Barber6 showed that when a very thin beam or plate is subjected to rapid surface heating, the vibration can be induced by a kind of time-dependent thermal moment due to the rapid temperature gradient in the structure. Later on, the Boley parameter B ¼ sT x1 was defined to characterize the severity of TIV for cantilever beams,7 where sT is the thermal characteristic time and x1 is the minimum angular frequency of the beam. The ratio of the maximum dynamic deflection over the quasi-static deflection of a cantilever beam pffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be expressed as 1 þ 1= 1 þ B2 , which means that the smaller B is, the more severe the TIV is. Although the Boley parameter B is a nice index for pure bending TIV of a cantilever beam, practical structures may have more complex TIV modes. For example, the structure composed of open
Production and hosting by Elsevier https://doi.org/10.1016/j.cja.2018.07.002 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: YUAN X, XIANG Z A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating, Chin J Aeronaut (2018), https://doi.org/10.1016/j.cja.2018.07.002
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thin-walled beams is apt to undergo torsional vibration due to its ultra-low torsional stiffness.8 Compared to stable TIV, the thermal flutter is more harmful to space structures. This phenomenon was first observed on orbit in 19689 and then it was realized in a laboratory environment.10 After that, more and more coupled thermal-structure analyses were conducted to investigate the condition of thermal flutter.11,12 Yu first established the stability criterion on the TIV of a closed thin-walled cantilever beam subject to solar heating13 and then that criterion was updated by Graham.14 In Graham’s criterion, the thermal flutter will only happen. when the beam axis points away from the sun, where the beam axis is defined as the vector pointing from the fixed end of the beam to the free end of the beam. An important conclusion of this criterion is that the normal-incident heat flux will not induce thermal flutter, which is contradictory to both experiment results15 and numerical simulations.16 Realizing that the stability analysis should be established on the deformed steady state instead of the original configuration of the beam, Zhang and Xiang proposed a new criterion, which conforms with the experimental and numerical results.17 All existing criteria of thermal flutter are only applicable to closed thin-walled beams. In contrast, the criterion for open thin-walled beams must consider the bending and torsion coupling deformations. Consequently, the circumferential incident angle of the heat flux should have great impact on the stability of the TIV. With a full consideration of these two points, this paper establishes a thermal-flutter criterion suitable for open thin-walled circular cantilever beams based on the first Lyapunov method.18
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2. Coupled thermal-structural dynamic analysis
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
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2.1. Analysis model and basic assumptions
76
As Fig. 1 shows, two sets of coordinate systems are defined to describe the deformation of the cantilever beam. OXYZ is a fixed spatial coordinate system, in which X axis is the initial centroid axis and Y axis points to the initial opening direction. Oxyz is a local coordinate system attached at a point on the beam, in which x axis is the deformed centroid axis and y axis always points to the opening direction of the rotated beam. The dimensions of the interested beam are defined as follows: l is the beam length; R and h are the midline radius and thickness of the beam cross-section, respectively. For a thin-walled slender beam, h=R << 1 and R=l << 1, so that Euler-Bernoulli beam theory is applicable. The solar heat flux vector S0 is uniformly distributed along the beam length. h0 is the angle between S0 and vector n, which is the normal of the beam and opposite to the projection of S0
77 78 79 80 81 82 83 84 85 86 87 88 89 90
Fig. 1 An open thin-walled circular cantilever beam subject to solar heat flux.
X. YUAN, Z. XIANG in plane OYZ. a is the angle between S0 and the Y axis in the YOZ plane. The following assumptions are adopted in the analysis: (1) Emission and radiation of heat to space is considered but convection and radiation between the different surfaces of the beam are neglected. (2) Heat transfer along beam length is neglected. (3) At positions of X = 0, X = l and the longitudinal opening sides of the beam are adiabatic. (4) The amplitude of the perturbation temperature is much smaller than the average temperature in the cross-section. (5) Damping is not considered. (6) Deflections and rotations are small before fluttering.
91 92 93
94 95 96 97 98 99 100 101 102 103 104 105
2.2. Basic equations
106
Bending and torsion of an open thin-walled beam are initiated mainly by the temperature gradients due to the absorbed heat flux. At the same time, the deformation also affects the incident angle of the heat flux. When the beam deforms, the absorbed solar heat flux is calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðx; u; tÞ ¼ as S0 dcosðu a0 Þ 1 w2 ð1Þ
107
where u is the circumferential angle along the midline of beam cross-section; as is the absorptivity of beam surface; S0 is the magnitude of solar heat flux S0; a0 2 ð0; 2pÞ denotes the equivalent circumferential incident angle; w is the angle between S0 and the deformed axis of the beam, and
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0
108 109 110 111 112 114
116 117 118 119 120
a ¼ a hx w ¼ cosh0 cosasinhzi coshyi cosh0 sinasinhyi þ sinh0 coshzi coshyi ð2Þ
122
hx is the torsion angle; hyi and hzi are the bending angles of the centroid around y and z axis, respectively; d is defined as 1 2np p2 6 u a0 6 2np þ p2 d¼ n ¼ 1; 2; . . . ð3Þ 0 2np þ p2 6 u a0 6 2np þ 3p 2
123
Based on Assumption (2), the beam temperature Tðx; u; tÞ is determined by cq
@T k @ 2 T er 4 qðx; u; tÞ þ T ¼ @t R2 @u2 h h
ð4Þ
124 125
127 128 129 130
132
where c is specific heat; q is mass density; k is thermal conductivity; e is the emissivity of beam surface; r is the Stefan–Boltzmann constant. Eq. (4) is a strong nonlinear equation, which is difficult to solve. However, it can be decomposed into two very simple equations by using the Fourier finite element method,16 which approximates the temperature Tðx; u; tÞ as the sum of an average temperature Ta ðx; tÞ and three perturbation temperatures: u Tðx; u; tÞ Ta ðx; tÞ þ Tp1 ðx; tÞcos þ Tp2 ðx; tÞcosu 2 3u ð5Þ þ Tp3 ðx; tÞcos 2
133
Substituting Eq. (5) into Eq. (4) and integrating it over the cross-section with respect to u, one can obtain two decoupled equations:
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating 3 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Ta er 4 as S0 wð0; tÞ ¼ 0 > > T ¼ þ 1 w2 ð6Þ > > > qch a pqch @t > > < w0 ð0; tÞ ¼ 0 " # ð18Þ 2 @Tpi i k 4er 3 > 00 > EI y w ðl; tÞ ¼ MTy ðl; tÞ > þ T þ T pi > > @t 2 qcR2 qch a > > : qffiffiffiffiffiffiffiffiffiffiffiffiffiffi EIy w000 ðl; tÞ ¼ 0 as S0 2 0 fi ða Þ 1 w i ¼ 1; 2; 3 ¼ ð7Þ 8 pqch hx ð0; tÞ ¼ 0 > > > 0 < where fi ða0 Þ are functions of a0 : h x ð0; tÞ ¼ 0 8 pffiffi ð19Þ 00 0 8 > > sina0 4 3 2 sin a2 0 6 a0 6 p2 > > > EIx hx ðl; tÞx ¼ BT ðl; tÞ < 3pffiffi : 0 hx 000 ðl; tÞ k2 hx 0 ðl; tÞ ¼ 0 p f1 ða0 Þ ¼ 4 3 2 cos a2 ð8Þ 6 a0 6 3p 2 2 > pffiffi > 0 :8 where Dp is the torsional rigidity;Iy and Iz are the moment of sina0 þ 4 3 2 sin a2 3p 6 a0 6 2p 3 2 inertia around y and z axis, respectively;Ix is the sectorial p quadratic moment; q is the density of the beam; v and w are f2 ða0 Þ ¼ cosa0 ð9Þ the deflections of the shear center (yc , zc) in y and z direction, 2 respectively; IA is the polar moment of inertia of the shear cen8 pffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 85 sina0 þ 4 5 2 sin 32 a0 0 6 a0 6 p2 ter; k is defined as Dp =EIx . For the open circular beam > < pffiffi p 0 3p shown in Fig. 1, f3 ða0 Þ ¼ 4 5 2 cos 32 a0 ð10Þ 6a 6 2 2 > pffiffi > : 8 yc ¼ 2R 5 sina0 4 5 2 sin 32 a0 3p 6 a0 6 2p 2 ð20Þ zc ¼ 0 Eq. (6) is easy to solve because it is much simpler than Eq. Let vi and wi denote the deflections of the centroid in y and z (4). Upon obtaining the average temperature Ta ðx; tÞ from Eq. direction, respectively. They can be calculated by (6), all perturbation temperatures can be solved by linear equa tion (7). vi ¼ v þ zc hx ð21Þ The nonuniform temperature distribution will result in wi ¼ w yc hx thermal loads, which include two thermal bending moments MTy and MTz and a thermal bimoment BT as follows: According to Euler-Bernoulli beam theory, it yields R 2p @vi @v MTy ðx; tÞ ¼ 0 EaT ðT T0 Þ ðRsinuÞhRdu þ zc v ¼ hzi ¼ ð22Þ ð11Þ 8 @x @x 2 8 ¼ 3 Tp1 5 Tp3 EaT R h
173
MTz ðx; tÞ ¼
178 179 180 181 182 183 184 185 187
R 2p 0
EaT ðT T0 Þ ðRcosuÞhRdu
¼ pEaT R hTp2 2
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R 2p BT ðx; tÞ ¼ 0 EaT ðT T0 Þ xhRdu 8 Tp3 EaT R3 h ¼ 3 Tp1 þ 184 45
ð12Þ
ð13Þ
where E is the elastic modulus of beam; T0 is the initial temperature of beam; aT is the thermal expansion coefficient; x is the sectorial area. According to Assumption (6), the change of thermal loads along beam length can be neglected. Thus, the vibration equations subject to these thermal loads are EIz
d4 v @2v @ 2 hx þ qA þ qAz ¼0 c @t2 dx4 @t2
ð14Þ
188 190
d4 w @2w @ 2 hx EIy 4 þ qA 2 qAyc 2 ¼ 0 @t dx @t
ð15Þ
191
d4 hx d2 hx @2w @2v @ 2 hx EIx Dp qAyc 2 þ qAzc 2 þ qIA 2 ¼ 0 4 2 @t @t dx dx @t 193 194 195
197
with the boundary conditions: 8 vð0; tÞ ¼ 0 > > > < v0 ð0; tÞ ¼ 0 > EIz v00 ðl; tÞ ¼ MTz ðl; tÞ > > : EIz v000 ðl; tÞ ¼ 0
ð16Þ
ð17Þ
hyi ¼
@wi @w þ yc v ¼ @x @x
ð23Þ
where v ¼ @hx =@x. Eq. (6), Eq. (7) and Eqs. (14)–(19) compose a set of coupled thermal-structural dynamic equations, which are difficult to solve due to these coupled terms and the nonlinearity of Eq. (6). However, the first Lyapunov method only investigates the linear approximations of these equations in their steady state for the stability analysis, so that it is not necessary to solve these equations directly.
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200 201
203 204 205 206 207 208 209 210 211
213 214 215 216
218 219 220 222 223 225 226 227 228 229 230 231 232 233
2.3. Coupled steady state
234
As aforementioned, the solution of coupled steady state is generally required for the stability analysis and it cannot be obtained analytically. However, this coupled steady solution is slightly different from the uncoupled steady solution for a closed thin-walled beam,17 in which the uncoupled solution is enough for stability analysis. Frustratingly, this is not true for an open thin-walled beam as illustrated by the numerical results shown in Section 3. Therefore, a simple iterative method has to be used to find an approximate solution. Let t ! 1 and x = l in Eqs. (6)–(19), and one can obtain the average temperature at iteration n (n = 0, 1, 2, . . ., nmax): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2ffi11=4 ðn1Þ 1 w a S ðnÞ C B s 0 C Ta ¼ B ð24Þ A @ pre
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Because Ta changes much more slowly than Tpi ,19 it can be regarded as a constant when Tpi is calculated. Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðnÞ as S0 ðnÞ 0ðn1Þ tTi fi ða Tpi ¼ Þ 1 wðn1Þ i ¼ 1; 2; 3 ð25Þ pqch
254
1 ðnÞ
¼
2 i k 4er ðnÞ þ T 2 2 qcR qch a
i ¼ 1; 2; 3
ð26Þ
256
tTi
257
where tTi are characteristic thermal time for different perturbation temperatures. Consequently, 8 ðnÞ ðnÞ ðnÞ > > MTy ¼ 83 Tp1 85 Tp3 EaT R2 h > > > > < ðnÞ ðnÞ ð27Þ MTz ¼ pEaT R2 hTp2 > > > > ðnÞ ðnÞ ðnÞ > > : BT ¼ 83 Tp1 þ 184 Tp3 EaT R3 h 45
258 259 260
262 263
265
ðnÞ
8 ðnÞ > > > hyi > > > > > ðnÞ > > > > hx > > > > > : vðnÞ
¼
ðnÞ MTy l
EIy
þ yc vðnÞ
¼
ðnÞ MTz l EIz
¼
T EI k2Bcosh ½coshðklÞ ðklÞ x
ð28Þ
ðnÞ
1
¼ EIBxT k tanhðklÞ
Then, one can update the following angles:
266 267
¼ cosh0 cosasinhzi coshyi cosh0 sinasinhyi þ sinh0 coshzi coshyi
ð0Þ
ð0Þ
hx ¼ hyi ¼ hzi ¼ 0
Considering the first-order vibration mode of the beam, the deformations can be represented as 8 > < vðx; tÞ ¼ VðtÞNðxÞ ð31Þ wðx; tÞ ¼ WðtÞNðxÞ > : hx ðx; tÞ ¼ HðtÞUðxÞ
289 290 291 292 294
309 310
ð37Þ
304 305 306 308
312
321
ðnÞ
278
288
where 0 a l ¼ a0 ðl; tÞ wl ¼ wðl; tÞ
303
320
ðnÞ
2.4. Approximate solutions
285 286
302
To analyze the stability of the deflection v, one can rewrite Eqs. (33) and (36) in state space as
ðnÞ
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284
The above equations are related to the perturbation temperatures at x = l. As mentioned in Section 2.3, the average temperature can be regarded as a constant. Therefore, the perturbation temperatures can be solved from Eq. (7): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Tpi ðl; tÞ Tpi ðl; tÞ as S0 fi ða0l Þ 1 wl 2 i ¼ 1; 2; 3 ð36Þ ¼ þ pqch @t tTi
2.5.1. Sub-criterion A
ðnÞ
To start the iteration, the initial values can be set as ð0Þ
283
301
Through Eqs. (24)–(30), the approximate solutions of coupled steady state at the beam free end can be obtained within a small number of iterations.
ðnÞ
ð29Þ
270 271
282
ð35Þ
319
ðnÞ
ðnÞ
¼ a hx
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298 299
2.5. Stability analysis
w
276
297
ð30Þ
275
where 8 Rl M11 ¼ qA 0 N2 ðxÞdx > > > Rl > > > M12 ¼ 2qAR 0 NðxÞUðxÞdx > > > Rl > > > M22 ¼ qIA 0 U2 ðxÞdx > > > R l 00 > 2 > > < Kv ¼ EIz 0 ðN ðxÞÞ dx R l 00 2 Kw ¼ EIy 0 ðN ðxÞÞ dx > > > R Rl 0 > l 2 2 00 > > > K2 ¼ EIx 0 ðU ðxÞÞ dx þ Dp 0 ðU ðxÞÞ dx > > > 0 > > Pv ðtÞ ¼ N ðlÞMTz ðl; tÞ > > > > Pw ðtÞ ¼ N0 ðlÞMTy ðl; tÞ > > : P2 ðtÞ ¼ U0 ðlÞBT ðl; tÞ
ð34Þ
313
a
274
295
€ þ Kw WðtÞ ¼ Pw ðtÞ € þ M12 HðtÞ M11 WðtÞ € € þ K2 HðtÞ ¼ P2 ðtÞ M22 HðtÞ þ M12 WðtÞ
According to Eqs. (33), (35) and (12), the deflection v is only related to the unknown variable Tp2 ðl; tÞ, regardless of Tp1 ðl; tÞ and Tp3 ðl; tÞ. For the same reason, Tp2 ðl; tÞ is useless when we analyze the stabilities of deflection w and torsion. Therefore, the stability of the deformations in different directions is discussed separately in the following.
0 ðnÞ
273
(
where NðxÞ and UðxÞ are the shape functions that satisfy the boundary conditions;VðtÞ, WðtÞ and HðtÞ are the functions of t by using the method of separation of variables. 8 0 000 > < Nð0Þ ¼ N ð0Þ ¼ N ðlÞ ¼ 0 ð32Þ Uð0Þ ¼ U0 ð0Þ ¼ 0 > : 000 U ðlÞ k2 U0 ðlÞ ¼ 0 Substituting Eq. (31) into Eqs. (14)–(16) and noticing Eq. (20), one can obtain the following equations by using the Galerkin weighted residual method: € þ Kv VðtÞ ¼ Pv ðtÞ M11 VðtÞ
ð33Þ
X_ ¼ fðXÞ ¼ AX þ B ð38Þ
T where X ¼ VðtÞ V_ ðtÞ Tp2 ðl; tÞ is the state variable vector, and 3 2 0 1 0 6 Kv pEaT R2 h 0 7 6 0 N ðl Þ 7 ð39Þ A ¼ 6 7 M11 5 4 M11 1 0 0 tT2 2 6 B¼4
3 0 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 5
as S0 cosa0 l 2qch
Y_ ¼ A Y
315 316 317 318
322 323 325 326 327 328
330 331
ð40Þ
1 w2l
333
Eq. (38) is nonlinear because of the coupled term in matrix B. According to the first Lyapunov method, its asymptotical stability is determined by its linear approximation in steady state. Therefore, Eq. (38) is approximated about the steady state by using the first-order Taylor expansion as
314
ð41Þ
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating 342 343 345 346
where
wl
Y¼XX 3 2 0 1 0 @f @B 6 Kv 0 pEaT R2 h N0 ðlÞ 7 A¼ ¼Aþ ¼4 5 M11 M11 @X X¼X @X X¼X 1 tT2 F N 0 ðl Þ 0 ð42Þ
348 349 351
355
357 358
as S0 wl @w 0 ffi l cosa qffiffiffiffiffiffiffiffiffiffiffiffiffiffi F¼ 2 2qch 1 wl @hz
356
ð43Þ
According to Eqs. (2), (22) and (37), F can be calculated by
ð44Þ
where wl ¼ w ðlÞ is the value of coupled steady state at x = l and
@wl
¼ sinh0 sinhzi coshyi þcosh0 cosa coshzi coshyi
ð45Þ
360
@hz
361 362
The characteristic polynomial of the matrix in Eq. (42) is det sI A ¼ s3 þ a2 s2 þ a1 s þ a0 ð46Þ
364 365 366 367 368 370 371 372
x¼l
where s is the characteristic root; a0, a1 and a2 are corresponding coefficients. A feasible shape function that satisfies Eq. (32) is NðxÞ ¼ x2
ð47Þ
Substituting Eq. (47) into Eqs. (42) and (46), one obtains 3
374
4R2 erTa þ kh a2 ¼ qchR2
@wl
@hz
0
cosal > 0 0
ð53Þ 397
where al ¼ a hx ðlÞ is the variable for the coupled steady state. Particularly, under the pure bending state in Y direction (around Z axis, a ¼ 0 or a ¼ 180 ), hx ¼ hy ¼ 0, and then this criterion becomes
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ as S0 1 2 0 cosa l 1 wl F¼ @VðtÞ 2qch N0 ðlÞ X¼X
352 353
5
ð48Þ
h0 > hzi
ð54Þ
398 399 400 401 402 403 405
2.5.2. Sub-criterion B
406
Compared to deflection v, the stability analysis of deflection in z direction and torsion is more difficult because these two deformations are coupled with each other, which leads to more unknown state variables. The shape functions that satisfy Eq. (32) are NðxÞ ¼ x2 ð55Þ UðxÞ ¼ ðchkx 1Þ
407
With these shape functions, Eq. (34) can be rewritten as ( € þ K11 W þ K12 H ¼ Pa ðtÞ MW ð56Þ € MH þ K21 H þ K22 W ¼ Pb ðtÞ
415 416
where 8 M ¼ M11 M22 M212 > > > > > K11 ¼ Kw M22 > > > > > > K ¼ K2 M12 > < 12 K21 ¼ K2 M11 > > > K22 ¼ Kw M12 > > > > > > Pa ¼ Ca1 Tp1 þ Ca3 Tp3 > > > : Pb ¼ Cb1 Tp1 þ Cb3 Tp3
419 420
408 409 410 411 412
414
418
ð57Þ
422
375 377
a1 ¼
381 382 383
385 386 387 388 389 390 391 393 394 395
ð49Þ
3 20E 4Iz R2 erTa þ Iz hk þ FR4 paT qch2 l
378
380
20EIz qAl4
a0 ¼
AR2 q2 chl4
ð50Þ
According to the Routh–Hurwitz criterion, the stability conditions for a third-order linear system are 8 a0 > 0 > > > 0 1 ð51Þ > a > 2>0 > : a1 a2 a0 > 0 Obviously, a1 > 0 and a2 > 0 are always true. According to Eq. (48), a0 > 0 is also satisfied because the term Iz hk is much greater than FR4 paT qch2 l for most space beams. Simplifying the last condition a1 a2 a0 > 0 in Eq. (51), one obtains F<0
ð52Þ
According to Eqs. (37) and (44), one can eventually get
And according to Eqs. (11), (13) and (35), one obtains 8 Ca1 ¼ 83 EaT R2 hð2lM22 ksinhðklÞRM12 Þ > > > > < Ca3 ¼ 8 EaT R2 h2lM22 23 ksinhðklÞRM12 5 9 ð58Þ 2 > > Cb1 ¼ 83 EaT R hð2lM12 þ ksinhðklÞRM11 Þ > > : Cb3 ¼ 85 EaT R2 h 2lM12 þ 239 ksinhðklÞRM11
423 424
Similar to Section 2.5.1, one can rewrite Eqs. (34) and (36) in the state space:
427
S_ ¼ fðSÞ ¼ CS þ D where
the state variable vector T _ _ ½ WðtÞ WðtÞ HðtÞ HðtÞ Tp1 ðtÞ Tp3 ðtÞ ; and 3 2 0 1 0 0 0 0 7 6 K Ca3 7 6 11 0 K12 0 Ca1 6 M M M M 7 7 6 6 0 0 0 1 0 0 7 7 6 7 C¼6 Cb3 7 6 K22 0 K21 0 Cb1 6 M M M M 7 7 6 6 0 1 0 0 0 tT1 0 7 7 6 5 4 1 0 0 0 0 0 tT3
is
426
428 429
ð59Þ
431
S¼
432 433 434 435
ð60Þ
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2
ð62Þ
T¼SS
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455 456 457
ðCa3 K22 Cb3 K11 ÞH3 Ul x1
448
2
3 0 0 0 6 0 CMa1 CMa3 7 6 7 6 7 6 1 0 07 @f @D 7 C¼ ¼Cþ ¼6 @S S¼S @S S¼S 6 0 CMb1 CMb3 7 6 7 6 7 4 G1 N0 ðlÞ 0 H1 UðlÞ 0 x1 0 5 G3 N0 ðlÞ 0 H3 UðlÞ 0 0 x3 ð63Þ 0 KM11 0 KM22
1 0 0 0
0 KM12 0 KM21
The parameters in Eq. (63) are 8 x1 ¼ 1=tT1 > > > > > x3 ¼ 1=tT3 > > 0 > > wl > as S0 > ffi @w l G1 ¼ pqch f1 a l pffiffiffiffiffiffiffiffi > 2 > @h y > 1wl > > > 0 < wl as S0 ffi @w l G3 ¼ pqch f3 a l pffiffiffiffiffiffiffiffi 2 @h y 1wl > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 0 > 2 df1 ða l Þ as S0 > H ¼ 1 w > 1 0 l da pqch > 0 > > a0 ¼a l > > q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > H3 ¼ as S0 1 w2 df3 ða0 0 l Þ : l pqch da 0 0 a ¼a
ð64Þ
l
The characteristic polynomial of the matrix in Eq. (63) is det sI C ¼ s6 þ b5 s5 þ b4 s4 þ b3 s3 þ b2 s2 þ b1 s þ b0 ¼ 0 ð65Þ
459
where
460 461 463
b5 ¼ x1 x3
ð66Þ
464 466
Mb4 ¼ Mx1 x3 þ K11 þ K21
ð67Þ
467
Mb3 ¼ 2ðCa1 G1 þ Ca3 G3 Þl þ K11 ðx1 þ x3 Þ þ ðCb1 H1 þ Cb3 H3 ÞUl þ K21 ðx1 þ x3 Þ
469
ð68Þ
470
M b2 ¼ 2Ca1 G1 Mlx3 þ 2Ca3 G3 Mlx1 þ Cb1 H1 MUl x3 2
þ Cb3 H3 MUl x1 þ ðK11 þ K21 ÞMx1 x3 472
þ K11 K21 K12 K22 M2 b1 ¼ ðCa1 H1 þ Ca3 H3 ÞK22 Ul 2 ðCa1 G1 þ Ca3 G3 ÞK21 l þ 2 ðCb1 G1 þ Cb3 G3 ÞK12 l ðCb1 H1 þ Cb3 H3 ÞK11 Ul ðK11 K21 K12 K22 Þ ðx1 þ x3 Þ
ðCa1 K22 Cb1 K11 ÞH1 Ul x3 þ ðK11 K21 K12 K22 Þx1 x3
ð71Þ
478
According to the Routh–Hurwitz criterion, the Stability conditions for a sixth-order linear system are 8 bi > 0 i ¼ 0; 1; :::; 5 > > > > > c1 > 0 > > > 0 1 ð72Þ > e1 > 0 > > > > > f1 > 0 > > : g1 > 0
479
where c1, d1, e1, 8 1 > 1 > c ¼ > 1 > b5 > b5 > > > > > b5 > > > d1 ¼ c11 < c1 > c > > e1 ¼ 1 1 > d1 > > d1 > > > > > d 1 > > : f1 ¼ e11 e1
f1 and g1 are the parameters of Routh table: 1 b2 b4 1 ; c3 ¼ b0 ¼ ; c 2 b5 b3 b5 b1 b b1 b3 1 5 ; d2 ¼ c1 c2 c1 c3 ð73Þ c2 ¼ b ; e 2 0 d2 d2 ; g1 ¼ b0 e2
484 485
Eq. (72) is the stability condition for the z-direction bending and torsion coupling vibration.
488
2.5.3. Thermal-flutter criterion
490
The thermal-flutter criterion for this open thin-walled circular cantilever beam composes of the sub-criterion A given in Eq. (53) and the sub-criterion B given in Eq. (72). Unstable TIV will happen when any one of these two sub-criteria is violated. The sub-criterion A establishes the relationship between the incident angle of the solar heat flux and the stability of deflection v. Under the pure bending state in Y direction, this new criterion (Eq. (54)) can degenerate into the existing criterion for a closed thin-walled beam17 free of torsion and warping. The sub-criterion B establishes the relationship between the incident angle of the solar heat flux and the stability of deflection in Z direction and torsion. It is too complex to find a clear physical meaning as that of the sub-criterion A. However, it can be easily verified numerically.
491
3. Numerical results
505
In this section, numerical simulations based on the Fourier finite element method20 will be conducted to obtain the dynamic responses of an open thin-walled circular cantilever beam subject to suddenly applied solar heat fluxes. The geometry dimensions and material properties are listed in Table 1. As shown in Fig. 1, two incident angles are interested in: the normal angle h0 and the circumferential angle a. Fig. 2 depicts a typical vibration curve, from which one can easily identify that the vibration period is about 5 s. Therefore,
506
480 481
483
487
489
492 493 494 495 496 497 498 499 500 501 502 503 504
ð69Þ
473
475
2 ðCa1 Cb3 Ca3 Cb1 ÞG3 H1 Ul l þ 2 ðCa3 K21 Cb3 K12 ÞG3 lx1 þ ðCa1 K21 Cb1 K12 ÞG1 lx3
T_ ¼ C T where
451
ð61Þ
The linear approximation of Eq. (58) is
445 446
449
M2 b0 ¼ 2 ðCa1 Cb3 Ca3 Cb1 ÞG1 H3 Ul l
0 6 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 7 6a S 7 D ¼ 6 s 0 f ða0 l Þ 1 w2 7 1 l 6 pqch 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 as S0 0 f ða l Þ 1 wl pqch 3
441 442 444
476
3
ð70Þ
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536
the time step is set to 0.25 s in the following numerical simulations to ensure the numerical accuracy. Fig. 3(a) and (b) compare the uncoupled and coupled thermal-structural response of the bending angles hy , hz and the torsion angle hx at beam free end under solar fluxes of different incident angles a and h0 . It is clear that the uncoupled displacements are quite different from the coupled displacements, because the torsion angle hx has great impact on the incident solar flux during the deformation of the beam. In order to get the coupled steady state values, the iterative method proposed in Section 2.3 with the maximum iterative number nmax ¼ 5 is utilized. The obtained results are hx ¼ 0:17 rad,hy ¼ 0:07 rad, hz ¼ 0:03 rad for Fig. 3(a) and hx ¼ 0:19 rad, hy ¼ 0:08 rad, hz ¼ 0:01 rad for Fig. 3(b), which are exactly the same as those from dynamic analysis. As Section 2 emphasizes, when either sub-criterion A or sub-criterion B is violated, the thermal-flutter will happen. Based on this rule, one can plot the stable and unstable zones of the TIV of this beam in Fig. 4. Since this open thin-walled beam has ultra-low torsional stiffness, it is not strange that most cases are unstable in Fig. 4. For example, when a ¼ 135 and h0 ¼ 0 , the steady
542 543
In this case, the steady angles are hx ¼ 0:21 rad, hyi ¼ 0:086 rad
538 539 540 541
Period of TIV when a ¼ 135 and h0 ¼ 0 .
angles are hx ¼ 0:10 rad, hyi ¼ 0:061 rad and hzi ¼ 0:0023 rad. In this case, both sub-criterion A and sub-criterion B are violated. Accordingly, in the numerical simulation results depicted in Fig. 5(a), all displacements are unstable. Thus, it verifies the prediction by the criterion. Fig. 4 also implies a stable TIV when a ¼ 75 and h0 ¼ 30 .
537
Fig. 2
7
544 545 546 547 548 549 550
and hzi ¼ 0:0016 rad, which satisfy both sub-criterion A and sub-criterion B. This conforms with the numerical simulation results depicted in Fig. 5(b). An interesting case is pure bending state in Y direction. For example, when a ¼ 180 and h0 ¼ 20 , it is obvious that deflection w and torsion angle hx are equal to zero due to the symmetry of this problem. In this case, the sub-criterion A
552
degenerates to h0 > hzi , which is the same as the criterion for a closed thin-walled beam.17 Since the steady angles
553
(hx ¼ 0 rad, hyi ¼ 0 rad and hzi ¼ 0:0035 rad) satisfy
551
554 555 556 557 558 559
h0 > hzi , v could be stable according to the sub-criterion A. However, the sub-criterion B is violated, so that w and hx must be unstable. These predictions are verified by the numerical results shown in Fig. 5(c), in which all deformations are stable before 3000 s, then w and hx gradually diverge, and finally v is also unstable due to the influence of w and hx . This example
Table 1
Fig. 3 Uncoupled and coupled thermal-structural responses at different incident angles.
Geometry dimensions and material parameters.
Parameter
L (m)
R (mm)
h (mm)
E (GPa)
Value Parameter
5.91 aT (1/K) 1.692 105 c (J/(kgK)) 502
10.92 q (kg/m3) 7010 as
0.6 S0 (W/m2) 1350 e
0.5
0.13
19.3 k (W/(mK)) 16.61 r (W/(m2K4)) 5.67 108
Value Parameter Value
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X. YUAN, Z. XIANG
Fig. 4
Fig. 5
Stable and unstable zones.
TIV at different incident angles.
demonstrates that the torsion has great impact on the stability of the TIV for open thin-walled beams.
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4. Conclusions
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This paper established a thermal-flutter criterion for an open thin-walled circular cantilever beam, which can be decomposed into two sub-criteria for the deflection v (sub-criterion A) and the coupled deformation of the bending state in z direction and torsion (sub-criterion B), respectively. In practice, thermalflutter happens when either the sub-criterion A or the subcriterion B is violated. The sub-criterion A can degenerate to the existing criterion for closed thin-walled beam free of torsion. However, the sub-criterion B shows that the torsion has great impact on the stability of the TIV for open thinwalled beams that have ultra-low torsional stiffness. However, this criterion does not consider the structural damping, so that it gives a conservative prediction for practical structures.
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References
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1. Frisch HP. Thermally induced vibrations of long thin-walled cylinders of open section. J Spacecraft Rockets 1970;7(8):897–905. 2. Thornton EA, Kim YA. Thermal induced bending vibrations of a flexible rolled-up solar array. J Spacecraft Rockets 1993;30 (4):438–48. 3. Ilsoung Y, Ohseop S, Librescu L. Vibration control of composite spacecraft booms subjected to solar heating. J Therm Stresses 2009;32(1–2):95–111. 4. Salehian A, Inman DJ. Thermally induced oscillations of an inflatable space structure with a repeated element pattern. Int J Acoust Vibr 2011;16(1):3–11. 5. Boley BA. Thermally induced vibrations of beams. J Aeronaut Sci 1956;23(2):179–81. 6. Boley BA, Barber AD. Dynamic response of beams and plates to rapid heating. J Appl Mech 1957;24(5):413–25. 7. Boley BA. Approximate analyses of thermally induced vibrations of beams and plates. J Appl Mech 1972;39(1):212–6. 8. Su XM, Zhang JH, Wang J, et al. Experimental investigation of the thermally induced vibration of a space boom section. Sci China Phys, Mech Astron 2015;58(4):1–9. 9. Thornton EA, Foster RS. Dynamic response of rapidly heated space structures. Prog Astronaut Aeronaut 1993;146:451–77. 10. Beam RM. On the phenomenon of thermoelastic instability/ thermal flutter/of booms with open cross section. Washington, D. C.: NASA; 1969. Report No.: NASA TN-D-5222. 11. Donohue JH., Frisch HP. Thermoelastic instability of open section booms. Washington, D.C.: NASA; 1969. Report No.: NASA TN-D-5310. 12. Connell GM, Chobotov V. Possible effects of boom flutter on attitude dynamics of OV1-10 satellite. J Spacecraft Rockets 1969;6 (1):90–2. 13. Yu YY. Thermally induced vibration and flutter of a flexible boom. J Spacecraft Rockets 1969;6(8):902–10. 14. Graham JD. Solar induced bending vibrations of a flexible member. AIAA J 1970;8(11):2031–6. 15. Rimrott FPJ, Abdel-Sayed R. Flexural thermal flutter under laboratory conditions. Trans Can Soc Mech Eng 1976;4(4):189–96. 16. Duan J, Xiang ZH, Xue MD. Thermal–dynamic coupling analysis of large space structures considering geometric nonlinearity. Int J Struct Stab Dyn 2008;8(4):569–96. 17. Zhang JH, Xiang ZH, Liu YH. Stability analysis on thermallyinduced vibration of a beam subjected to solar heating. AIAA J 2014;52(3):660–5.
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18. Lyapunov AM. The general problem of the stability of motion. Int J Control 1992;55(3):531–4. 19. Xue M, Ding Y. Two kinds of tube elements for transient thermal–structural analysis of large space structures. Int J Numer Meth Eng 2004;59(10):1335–53.
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20. Xue MD, Duan J, Xiang ZH. Thermally-induced bending-torsion coupling vibration of large scale space structures. Comput Mech 2007;40(4):707–23.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating
5
Xiaode YUAN, Zhihai XIANG *
6
Department of Engineering Mechanics, Tsinghua University of Aerospace Engineering, Beijing 100084, China
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Received 1 March 2018; revised 16 April 2018; accepted 15 May 2018
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KEYWORDS
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Fourier finite element; Stability criteria; Thermal flutter; Thermally Induced Vibration (TIV); Thin-walled structures
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Abstract The flexible attachments of spacecraft may undergo Thermally Induced Vibration (TIV) on orbit due to the suddenly changed solar heating. The unstable TIV, called thermal-flutter, can cause serious damage to the spacecraft. In this paper, the coupled bending-torsion thermal vibration equations for an open thin-walled circular cantilever beam are established. By analyzing the stability of these equations based on the first Lyapunov method, the thermal-flutter criterion can be obtained. The criterion is very different form that of closed thin-walled beams because the torsion has great impact on the stability of the TIV for open thin-walled beams. Several numerical simulations are conducted to demonstrate that the theoretical predictions agree very well with the finite element results, which mean that the criterion are reliable. Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
20
The flexible attachments of spacecraft generally have the characteristics of large size, light weight, low stiffness and small heat capacity. Therefore, these structures are prone to experiencing the Thermally Induced Vibration (TIV) due to the suddenly applied solar heat flux when the spacecraft enter or leave eclipse.1–4 These vibrations could reduce the pointing accuracy of spacecraft and even introduce damage into the structure,
21 22 23 24 25 26
* Corresponding author. E-mail address: [email protected] (Z. XIANG). Peer review under responsibility of Editorial Committee of CJA.
especially when the vibration is unstable, i.e., the thermalflutter. TIV was firstly predicted theoretically by Boley as early as 1956.5 Boley and Barber6 showed that when a very thin beam or plate is subjected to rapid surface heating, the vibration can be induced by a kind of time-dependent thermal moment due to the rapid temperature gradient in the structure. Later on, the Boley parameter B ¼ sT x1 was defined to characterize the severity of TIV for cantilever beams,7 where sT is the thermal characteristic time and x1 is the minimum angular frequency of the beam. The ratio of the maximum dynamic deflection over the quasi-static deflection of a cantilever beam pffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be expressed as 1 þ 1= 1 þ B2 , which means that the smaller B is, the more severe the TIV is. Although the Boley parameter B is a nice index for pure bending TIV of a cantilever beam, practical structures may have more complex TIV modes. For example, the structure composed of open
Production and hosting by Elsevier https://doi.org/10.1016/j.cja.2018.07.002 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: YUAN X, XIANG Z A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating, Chin J Aeronaut (2018), https://doi.org/10.1016/j.cja.2018.07.002
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thin-walled beams is apt to undergo torsional vibration due to its ultra-low torsional stiffness.8 Compared to stable TIV, the thermal flutter is more harmful to space structures. This phenomenon was first observed on orbit in 19689 and then it was realized in a laboratory environment.10 After that, more and more coupled thermal-structure analyses were conducted to investigate the condition of thermal flutter.11,12 Yu first established the stability criterion on the TIV of a closed thin-walled cantilever beam subject to solar heating13 and then that criterion was updated by Graham.14 In Graham’s criterion, the thermal flutter will only happen. when the beam axis points away from the sun, where the beam axis is defined as the vector pointing from the fixed end of the beam to the free end of the beam. An important conclusion of this criterion is that the normal-incident heat flux will not induce thermal flutter, which is contradictory to both experiment results15 and numerical simulations.16 Realizing that the stability analysis should be established on the deformed steady state instead of the original configuration of the beam, Zhang and Xiang proposed a new criterion, which conforms with the experimental and numerical results.17 All existing criteria of thermal flutter are only applicable to closed thin-walled beams. In contrast, the criterion for open thin-walled beams must consider the bending and torsion coupling deformations. Consequently, the circumferential incident angle of the heat flux should have great impact on the stability of the TIV. With a full consideration of these two points, this paper establishes a thermal-flutter criterion suitable for open thin-walled circular cantilever beams based on the first Lyapunov method.18
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2. Coupled thermal-structural dynamic analysis
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
75
2.1. Analysis model and basic assumptions
76
As Fig. 1 shows, two sets of coordinate systems are defined to describe the deformation of the cantilever beam. OXYZ is a fixed spatial coordinate system, in which X axis is the initial centroid axis and Y axis points to the initial opening direction. Oxyz is a local coordinate system attached at a point on the beam, in which x axis is the deformed centroid axis and y axis always points to the opening direction of the rotated beam. The dimensions of the interested beam are defined as follows: l is the beam length; R and h are the midline radius and thickness of the beam cross-section, respectively. For a thin-walled slender beam, h=R << 1 and R=l << 1, so that Euler-Bernoulli beam theory is applicable. The solar heat flux vector S0 is uniformly distributed along the beam length. h0 is the angle between S0 and vector n, which is the normal of the beam and opposite to the projection of S0
77 78 79 80 81 82 83 84 85 86 87 88 89 90
Fig. 1 An open thin-walled circular cantilever beam subject to solar heat flux.
X. YUAN, Z. XIANG in plane OYZ. a is the angle between S0 and the Y axis in the YOZ plane. The following assumptions are adopted in the analysis: (1) Emission and radiation of heat to space is considered but convection and radiation between the different surfaces of the beam are neglected. (2) Heat transfer along beam length is neglected. (3) At positions of X = 0, X = l and the longitudinal opening sides of the beam are adiabatic. (4) The amplitude of the perturbation temperature is much smaller than the average temperature in the cross-section. (5) Damping is not considered. (6) Deflections and rotations are small before fluttering.
91 92 93
94 95 96 97 98 99 100 101 102 103 104 105
2.2. Basic equations
106
Bending and torsion of an open thin-walled beam are initiated mainly by the temperature gradients due to the absorbed heat flux. At the same time, the deformation also affects the incident angle of the heat flux. When the beam deforms, the absorbed solar heat flux is calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðx; u; tÞ ¼ as S0 dcosðu a0 Þ 1 w2 ð1Þ
107
where u is the circumferential angle along the midline of beam cross-section; as is the absorptivity of beam surface; S0 is the magnitude of solar heat flux S0; a0 2 ð0; 2pÞ denotes the equivalent circumferential incident angle; w is the angle between S0 and the deformed axis of the beam, and
115
0
108 109 110 111 112 114
116 117 118 119 120
a ¼ a hx w ¼ cosh0 cosasinhzi coshyi cosh0 sinasinhyi þ sinh0 coshzi coshyi ð2Þ
122
hx is the torsion angle; hyi and hzi are the bending angles of the centroid around y and z axis, respectively; d is defined as 1 2np p2 6 u a0 6 2np þ p2 d¼ n ¼ 1; 2; . . . ð3Þ 0 2np þ p2 6 u a0 6 2np þ 3p 2
123
Based on Assumption (2), the beam temperature Tðx; u; tÞ is determined by cq
@T k @ 2 T er 4 qðx; u; tÞ þ T ¼ @t R2 @u2 h h
ð4Þ
124 125
127 128 129 130
132
where c is specific heat; q is mass density; k is thermal conductivity; e is the emissivity of beam surface; r is the Stefan–Boltzmann constant. Eq. (4) is a strong nonlinear equation, which is difficult to solve. However, it can be decomposed into two very simple equations by using the Fourier finite element method,16 which approximates the temperature Tðx; u; tÞ as the sum of an average temperature Ta ðx; tÞ and three perturbation temperatures: u Tðx; u; tÞ Ta ðx; tÞ þ Tp1 ðx; tÞcos þ Tp2 ðx; tÞcosu 2 3u ð5Þ þ Tp3 ðx; tÞcos 2
133
Substituting Eq. (5) into Eq. (4) and integrating it over the cross-section with respect to u, one can obtain two decoupled equations:
144
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating 3 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Ta er 4 as S0 wð0; tÞ ¼ 0 > > T ¼ þ 1 w2 ð6Þ > > > qch a pqch @t > > < w0 ð0; tÞ ¼ 0 " # ð18Þ 2 @Tpi i k 4er 3 > 00 > EI y w ðl; tÞ ¼ MTy ðl; tÞ > þ T þ T pi > > @t 2 qcR2 qch a > > : qffiffiffiffiffiffiffiffiffiffiffiffiffiffi EIy w000 ðl; tÞ ¼ 0 as S0 2 0 fi ða Þ 1 w i ¼ 1; 2; 3 ¼ ð7Þ 8 pqch hx ð0; tÞ ¼ 0 > > > 0 < where fi ða0 Þ are functions of a0 : h x ð0; tÞ ¼ 0 8 pffiffi ð19Þ 00 0 8 > > sina0 4 3 2 sin a2 0 6 a0 6 p2 > > > EIx hx ðl; tÞx ¼ BT ðl; tÞ < 3pffiffi : 0 hx 000 ðl; tÞ k2 hx 0 ðl; tÞ ¼ 0 p f1 ða0 Þ ¼ 4 3 2 cos a2 ð8Þ 6 a0 6 3p 2 2 > pffiffi > 0 :8 where Dp is the torsional rigidity;Iy and Iz are the moment of sina0 þ 4 3 2 sin a2 3p 6 a0 6 2p 3 2 inertia around y and z axis, respectively;Ix is the sectorial p quadratic moment; q is the density of the beam; v and w are f2 ða0 Þ ¼ cosa0 ð9Þ the deflections of the shear center (yc , zc) in y and z direction, 2 respectively; IA is the polar moment of inertia of the shear cen8 pffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 85 sina0 þ 4 5 2 sin 32 a0 0 6 a0 6 p2 ter; k is defined as Dp =EIx . For the open circular beam > < pffiffi p 0 3p shown in Fig. 1, f3 ða0 Þ ¼ 4 5 2 cos 32 a0 ð10Þ 6a 6 2 2 > pffiffi > : 8 yc ¼ 2R 5 sina0 4 5 2 sin 32 a0 3p 6 a0 6 2p 2 ð20Þ zc ¼ 0 Eq. (6) is easy to solve because it is much simpler than Eq. Let vi and wi denote the deflections of the centroid in y and z (4). Upon obtaining the average temperature Ta ðx; tÞ from Eq. direction, respectively. They can be calculated by (6), all perturbation temperatures can be solved by linear equa tion (7). vi ¼ v þ zc hx ð21Þ The nonuniform temperature distribution will result in wi ¼ w yc hx thermal loads, which include two thermal bending moments MTy and MTz and a thermal bimoment BT as follows: According to Euler-Bernoulli beam theory, it yields R 2p @vi @v MTy ðx; tÞ ¼ 0 EaT ðT T0 Þ ðRsinuÞhRdu þ zc v ¼ hzi ¼ ð22Þ ð11Þ 8 @x @x 2 8 ¼ 3 Tp1 5 Tp3 EaT R h
173
MTz ðx; tÞ ¼
178 179 180 181 182 183 184 185 187
R 2p 0
EaT ðT T0 Þ ðRcosuÞhRdu
¼ pEaT R hTp2 2
175 176
No. of Pages 9
R 2p BT ðx; tÞ ¼ 0 EaT ðT T0 Þ xhRdu 8 Tp3 EaT R3 h ¼ 3 Tp1 þ 184 45
ð12Þ
ð13Þ
where E is the elastic modulus of beam; T0 is the initial temperature of beam; aT is the thermal expansion coefficient; x is the sectorial area. According to Assumption (6), the change of thermal loads along beam length can be neglected. Thus, the vibration equations subject to these thermal loads are EIz
d4 v @2v @ 2 hx þ qA þ qAz ¼0 c @t2 dx4 @t2
ð14Þ
188 190
d4 w @2w @ 2 hx EIy 4 þ qA 2 qAyc 2 ¼ 0 @t dx @t
ð15Þ
191
d4 hx d2 hx @2w @2v @ 2 hx EIx Dp qAyc 2 þ qAzc 2 þ qIA 2 ¼ 0 4 2 @t @t dx dx @t 193 194 195
197
with the boundary conditions: 8 vð0; tÞ ¼ 0 > > > < v0 ð0; tÞ ¼ 0 > EIz v00 ðl; tÞ ¼ MTz ðl; tÞ > > : EIz v000 ðl; tÞ ¼ 0
ð16Þ
ð17Þ
hyi ¼
@wi @w þ yc v ¼ @x @x
ð23Þ
where v ¼ @hx =@x. Eq. (6), Eq. (7) and Eqs. (14)–(19) compose a set of coupled thermal-structural dynamic equations, which are difficult to solve due to these coupled terms and the nonlinearity of Eq. (6). However, the first Lyapunov method only investigates the linear approximations of these equations in their steady state for the stability analysis, so that it is not necessary to solve these equations directly.
198
200 201
203 204 205 206 207 208 209 210 211
213 214 215 216
218 219 220 222 223 225 226 227 228 229 230 231 232 233
2.3. Coupled steady state
234
As aforementioned, the solution of coupled steady state is generally required for the stability analysis and it cannot be obtained analytically. However, this coupled steady solution is slightly different from the uncoupled steady solution for a closed thin-walled beam,17 in which the uncoupled solution is enough for stability analysis. Frustratingly, this is not true for an open thin-walled beam as illustrated by the numerical results shown in Section 3. Therefore, a simple iterative method has to be used to find an approximate solution. Let t ! 1 and x = l in Eqs. (6)–(19), and one can obtain the average temperature at iteration n (n = 0, 1, 2, . . ., nmax): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2ffi11=4 ðn1Þ 1 w a S ðnÞ C B s 0 C Ta ¼ B ð24Þ A @ pre
235
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X. YUAN, Z. XIANG
Because Ta changes much more slowly than Tpi ,19 it can be regarded as a constant when Tpi is calculated. Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðnÞ as S0 ðnÞ 0ðn1Þ tTi fi ða Tpi ¼ Þ 1 wðn1Þ i ¼ 1; 2; 3 ð25Þ pqch
254
1 ðnÞ
¼
2 i k 4er ðnÞ þ T 2 2 qcR qch a
i ¼ 1; 2; 3
ð26Þ
256
tTi
257
where tTi are characteristic thermal time for different perturbation temperatures. Consequently, 8 ðnÞ ðnÞ ðnÞ > > MTy ¼ 83 Tp1 85 Tp3 EaT R2 h > > > > < ðnÞ ðnÞ ð27Þ MTz ¼ pEaT R2 hTp2 > > > > ðnÞ ðnÞ ðnÞ > > : BT ¼ 83 Tp1 þ 184 Tp3 EaT R3 h 45
258 259 260
262 263
265
ðnÞ
8 ðnÞ > > > hyi > > > > > ðnÞ >
¼
ðnÞ MTy l
EIy
þ yc vðnÞ
¼
ðnÞ MTz l EIz
¼
T EI k2Bcosh ½coshðklÞ ðklÞ x
ð28Þ
ðnÞ
1
¼ EIBxT k tanhðklÞ
Then, one can update the following angles:
266 267
¼ cosh0 cosasinhzi coshyi cosh0 sinasinhyi þ sinh0 coshzi coshyi
ð0Þ
ð0Þ
hx ¼ hyi ¼ hzi ¼ 0
Considering the first-order vibration mode of the beam, the deformations can be represented as 8 > < vðx; tÞ ¼ VðtÞNðxÞ ð31Þ wðx; tÞ ¼ WðtÞNðxÞ > : hx ðx; tÞ ¼ HðtÞUðxÞ
289 290 291 292 294
309 310
ð37Þ
304 305 306 308
312
321
ðnÞ
278
288
where 0 a l ¼ a0 ðl; tÞ wl ¼ wðl; tÞ
303
320
ðnÞ
2.4. Approximate solutions
285 286
302
To analyze the stability of the deflection v, one can rewrite Eqs. (33) and (36) in state space as
ðnÞ
277
284
The above equations are related to the perturbation temperatures at x = l. As mentioned in Section 2.3, the average temperature can be regarded as a constant. Therefore, the perturbation temperatures can be solved from Eq. (7): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Tpi ðl; tÞ Tpi ðl; tÞ as S0 fi ða0l Þ 1 wl 2 i ¼ 1; 2; 3 ð36Þ ¼ þ pqch @t tTi
2.5.1. Sub-criterion A
ðnÞ
To start the iteration, the initial values can be set as ð0Þ
283
301
Through Eqs. (24)–(30), the approximate solutions of coupled steady state at the beam free end can be obtained within a small number of iterations.
ðnÞ
ð29Þ
270 271
282
ð35Þ
319
ðnÞ
ðnÞ
¼ a hx
269
279 280
298 299
2.5. Stability analysis
w
276
297
ð30Þ
275
where 8 Rl M11 ¼ qA 0 N2 ðxÞdx > > > Rl > > > M12 ¼ 2qAR 0 NðxÞUðxÞdx > > > Rl > > > M22 ¼ qIA 0 U2 ðxÞdx > > > R l 00 > 2 > > < Kv ¼ EIz 0 ðN ðxÞÞ dx R l 00 2 Kw ¼ EIy 0 ðN ðxÞÞ dx > > > R Rl 0 > l 2 2 00 > > > K2 ¼ EIx 0 ðU ðxÞÞ dx þ Dp 0 ðU ðxÞÞ dx > > > 0 > > Pv ðtÞ ¼ N ðlÞMTz ðl; tÞ > > > > Pw ðtÞ ¼ N0 ðlÞMTy ðl; tÞ > > : P2 ðtÞ ¼ U0 ðlÞBT ðl; tÞ
ð34Þ
313
a
274
295
€ þ Kw WðtÞ ¼ Pw ðtÞ € þ M12 HðtÞ M11 WðtÞ € € þ K2 HðtÞ ¼ P2 ðtÞ M22 HðtÞ þ M12 WðtÞ
According to Eqs. (33), (35) and (12), the deflection v is only related to the unknown variable Tp2 ðl; tÞ, regardless of Tp1 ðl; tÞ and Tp3 ðl; tÞ. For the same reason, Tp2 ðl; tÞ is useless when we analyze the stabilities of deflection w and torsion. Therefore, the stability of the deformations in different directions is discussed separately in the following.
0 ðnÞ
273
(
where NðxÞ and UðxÞ are the shape functions that satisfy the boundary conditions;VðtÞ, WðtÞ and HðtÞ are the functions of t by using the method of separation of variables. 8 0 000 > < Nð0Þ ¼ N ð0Þ ¼ N ðlÞ ¼ 0 ð32Þ Uð0Þ ¼ U0 ð0Þ ¼ 0 > : 000 U ðlÞ k2 U0 ðlÞ ¼ 0 Substituting Eq. (31) into Eqs. (14)–(16) and noticing Eq. (20), one can obtain the following equations by using the Galerkin weighted residual method: € þ Kv VðtÞ ¼ Pv ðtÞ M11 VðtÞ
ð33Þ
X_ ¼ fðXÞ ¼ AX þ B ð38Þ
T where X ¼ VðtÞ V_ ðtÞ Tp2 ðl; tÞ is the state variable vector, and 3 2 0 1 0 6 Kv pEaT R2 h 0 7 6 0 N ðl Þ 7 ð39Þ A ¼ 6 7 M11 5 4 M11 1 0 0 tT2 2 6 B¼4
3 0 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 5
as S0 cosa0 l 2qch
Y_ ¼ A Y
315 316 317 318
322 323 325 326 327 328
330 331
ð40Þ
1 w2l
333
Eq. (38) is nonlinear because of the coupled term in matrix B. According to the first Lyapunov method, its asymptotical stability is determined by its linear approximation in steady state. Therefore, Eq. (38) is approximated about the steady state by using the first-order Taylor expansion as
314
ð41Þ
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where
wl
Y¼XX 3 2 0 1 0 @f @B 6 Kv 0 pEaT R2 h N0 ðlÞ 7 A¼ ¼Aþ ¼4 5 M11 M11 @X X¼X @X X¼X 1 tT2 F N 0 ðl Þ 0 ð42Þ
348 349 351
355
357 358
as S0 wl @w 0 ffi l cosa qffiffiffiffiffiffiffiffiffiffiffiffiffiffi F¼ 2 2qch 1 wl @hz
356
ð43Þ
According to Eqs. (2), (22) and (37), F can be calculated by
ð44Þ
where wl ¼ w ðlÞ is the value of coupled steady state at x = l and
@wl
¼ sinh0 sinhzi coshyi þcosh0 cosa coshzi coshyi
ð45Þ
360
@hz
361 362
The characteristic polynomial of the matrix in Eq. (42) is det sI A ¼ s3 þ a2 s2 þ a1 s þ a0 ð46Þ
364 365 366 367 368 370 371 372
x¼l
where s is the characteristic root; a0, a1 and a2 are corresponding coefficients. A feasible shape function that satisfies Eq. (32) is NðxÞ ¼ x2
ð47Þ
Substituting Eq. (47) into Eqs. (42) and (46), one obtains 3
374
4R2 erTa þ kh a2 ¼ qchR2
@wl
@hz
0
cosal > 0 0
ð53Þ 397
where al ¼ a hx ðlÞ is the variable for the coupled steady state. Particularly, under the pure bending state in Y direction (around Z axis, a ¼ 0 or a ¼ 180 ), hx ¼ hy ¼ 0, and then this criterion becomes
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ as S0 1 2 0 cosa l 1 wl F¼ @VðtÞ 2qch N0 ðlÞ X¼X
352 353
5
ð48Þ
h0 > hzi
ð54Þ
398 399 400 401 402 403 405
2.5.2. Sub-criterion B
406
Compared to deflection v, the stability analysis of deflection in z direction and torsion is more difficult because these two deformations are coupled with each other, which leads to more unknown state variables. The shape functions that satisfy Eq. (32) are NðxÞ ¼ x2 ð55Þ UðxÞ ¼ ðchkx 1Þ
407
With these shape functions, Eq. (34) can be rewritten as ( € þ K11 W þ K12 H ¼ Pa ðtÞ MW ð56Þ € MH þ K21 H þ K22 W ¼ Pb ðtÞ
415 416
where 8 M ¼ M11 M22 M212 > > > > > K11 ¼ Kw M22 > > > > > > K ¼ K2 M12 > < 12 K21 ¼ K2 M11 > > > K22 ¼ Kw M12 > > > > > > Pa ¼ Ca1 Tp1 þ Ca3 Tp3 > > > : Pb ¼ Cb1 Tp1 þ Cb3 Tp3
419 420
408 409 410 411 412
414
418
ð57Þ
422
375 377
a1 ¼
381 382 383
385 386 387 388 389 390 391 393 394 395
ð49Þ
3 20E 4Iz R2 erTa þ Iz hk þ FR4 paT qch2 l
378
380
20EIz qAl4
a0 ¼
AR2 q2 chl4
ð50Þ
According to the Routh–Hurwitz criterion, the stability conditions for a third-order linear system are 8 a0 > 0 > > > 0 1 ð51Þ > a > 2>0 > : a1 a2 a0 > 0 Obviously, a1 > 0 and a2 > 0 are always true. According to Eq. (48), a0 > 0 is also satisfied because the term Iz hk is much greater than FR4 paT qch2 l for most space beams. Simplifying the last condition a1 a2 a0 > 0 in Eq. (51), one obtains F<0
ð52Þ
According to Eqs. (37) and (44), one can eventually get
And according to Eqs. (11), (13) and (35), one obtains 8 Ca1 ¼ 83 EaT R2 hð2lM22 ksinhðklÞRM12 Þ > > > > < Ca3 ¼ 8 EaT R2 h2lM22 23 ksinhðklÞRM12 5 9 ð58Þ 2 > > Cb1 ¼ 83 EaT R hð2lM12 þ ksinhðklÞRM11 Þ > > : Cb3 ¼ 85 EaT R2 h 2lM12 þ 239 ksinhðklÞRM11
423 424
Similar to Section 2.5.1, one can rewrite Eqs. (34) and (36) in the state space:
427
S_ ¼ fðSÞ ¼ CS þ D where
the state variable vector T _ _ ½ WðtÞ WðtÞ HðtÞ HðtÞ Tp1 ðtÞ Tp3 ðtÞ ; and 3 2 0 1 0 0 0 0 7 6 K Ca3 7 6 11 0 K12 0 Ca1 6 M M M M 7 7 6 6 0 0 0 1 0 0 7 7 6 7 C¼6 Cb3 7 6 K22 0 K21 0 Cb1 6 M M M M 7 7 6 6 0 1 0 0 0 tT1 0 7 7 6 5 4 1 0 0 0 0 0 tT3
is
426
428 429
ð59Þ
431
S¼
432 433 434 435
ð60Þ
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438
440
2
ð62Þ
T¼SS
452 453
455 456 457
ðCa3 K22 Cb3 K11 ÞH3 Ul x1
448
2
3 0 0 0 6 0 CMa1 CMa3 7 6 7 6 7 6 1 0 07 @f @D 7 C¼ ¼Cþ ¼6 @S S¼S @S S¼S 6 0 CMb1 CMb3 7 6 7 6 7 4 G1 N0 ðlÞ 0 H1 UðlÞ 0 x1 0 5 G3 N0 ðlÞ 0 H3 UðlÞ 0 0 x3 ð63Þ 0 KM11 0 KM22
1 0 0 0
0 KM12 0 KM21
The parameters in Eq. (63) are 8 x1 ¼ 1=tT1 > > > > > x3 ¼ 1=tT3 > > 0 > > wl > as S0 > ffi @w l G1 ¼ pqch f1 a l pffiffiffiffiffiffiffiffi > 2 > @h y > 1wl > > > 0 < wl as S0 ffi @w l G3 ¼ pqch f3 a l pffiffiffiffiffiffiffiffi 2 @h y 1wl > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 0 > 2 df1 ða l Þ as S0 > H ¼ 1 w > 1 0 l da pqch > 0 > > a0 ¼a l > > q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > H3 ¼ as S0 1 w2 df3 ða0 0 l Þ : l pqch da 0 0 a ¼a
ð64Þ
l
The characteristic polynomial of the matrix in Eq. (63) is det sI C ¼ s6 þ b5 s5 þ b4 s4 þ b3 s3 þ b2 s2 þ b1 s þ b0 ¼ 0 ð65Þ
459
where
460 461 463
b5 ¼ x1 x3
ð66Þ
464 466
Mb4 ¼ Mx1 x3 þ K11 þ K21
ð67Þ
467
Mb3 ¼ 2ðCa1 G1 þ Ca3 G3 Þl þ K11 ðx1 þ x3 Þ þ ðCb1 H1 þ Cb3 H3 ÞUl þ K21 ðx1 þ x3 Þ
469
ð68Þ
470
M b2 ¼ 2Ca1 G1 Mlx3 þ 2Ca3 G3 Mlx1 þ Cb1 H1 MUl x3 2
þ Cb3 H3 MUl x1 þ ðK11 þ K21 ÞMx1 x3 472
þ K11 K21 K12 K22 M2 b1 ¼ ðCa1 H1 þ Ca3 H3 ÞK22 Ul 2 ðCa1 G1 þ Ca3 G3 ÞK21 l þ 2 ðCb1 G1 þ Cb3 G3 ÞK12 l ðCb1 H1 þ Cb3 H3 ÞK11 Ul ðK11 K21 K12 K22 Þ ðx1 þ x3 Þ
ðCa1 K22 Cb1 K11 ÞH1 Ul x3 þ ðK11 K21 K12 K22 Þx1 x3
ð71Þ
478
According to the Routh–Hurwitz criterion, the Stability conditions for a sixth-order linear system are 8 bi > 0 i ¼ 0; 1; :::; 5 > > > > > c1 > 0 > > >
479
where c1, d1, e1, 8 1 > 1 > c ¼ > 1 > b5 > b5 > > > > > b5 > > > d1 ¼ c11 < c1 > c > > e1 ¼ 1 1 > d1 > > d1 > > > > > d 1 > > : f1 ¼ e11 e1
f1 and g1 are the parameters of Routh table: 1 b2 b4 1 ; c3 ¼ b0 ¼ ; c 2 b5 b3 b5 b1 b b1 b3 1 5 ; d2 ¼ c1 c2 c1 c3 ð73Þ c2 ¼ b ; e 2 0 d2 d2 ; g1 ¼ b0 e2
484 485
Eq. (72) is the stability condition for the z-direction bending and torsion coupling vibration.
488
2.5.3. Thermal-flutter criterion
490
The thermal-flutter criterion for this open thin-walled circular cantilever beam composes of the sub-criterion A given in Eq. (53) and the sub-criterion B given in Eq. (72). Unstable TIV will happen when any one of these two sub-criteria is violated. The sub-criterion A establishes the relationship between the incident angle of the solar heat flux and the stability of deflection v. Under the pure bending state in Y direction, this new criterion (Eq. (54)) can degenerate into the existing criterion for a closed thin-walled beam17 free of torsion and warping. The sub-criterion B establishes the relationship between the incident angle of the solar heat flux and the stability of deflection in Z direction and torsion. It is too complex to find a clear physical meaning as that of the sub-criterion A. However, it can be easily verified numerically.
491
3. Numerical results
505
In this section, numerical simulations based on the Fourier finite element method20 will be conducted to obtain the dynamic responses of an open thin-walled circular cantilever beam subject to suddenly applied solar heat fluxes. The geometry dimensions and material properties are listed in Table 1. As shown in Fig. 1, two incident angles are interested in: the normal angle h0 and the circumferential angle a. Fig. 2 depicts a typical vibration curve, from which one can easily identify that the vibration period is about 5 s. Therefore,
506
480 481
483
487
489
492 493 494 495 496 497 498 499 500 501 502 503 504
ð69Þ
473
475
2 ðCa1 Cb3 Ca3 Cb1 ÞG3 H1 Ul l þ 2 ðCa3 K21 Cb3 K12 ÞG3 lx1 þ ðCa1 K21 Cb1 K12 ÞG1 lx3
T_ ¼ C T where
451
ð61Þ
The linear approximation of Eq. (58) is
445 446
449
M2 b0 ¼ 2 ðCa1 Cb3 Ca3 Cb1 ÞG1 H3 Ul l
0 6 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 7 6a S 7 D ¼ 6 s 0 f ða0 l Þ 1 w2 7 1 l 6 pqch 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 as S0 0 f ða l Þ 1 wl pqch 3
441 442 444
476
3
ð70Þ
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A thermal-flutter criterion for an open thin-walled circular cantilever beam subject to solar heating 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536
the time step is set to 0.25 s in the following numerical simulations to ensure the numerical accuracy. Fig. 3(a) and (b) compare the uncoupled and coupled thermal-structural response of the bending angles hy , hz and the torsion angle hx at beam free end under solar fluxes of different incident angles a and h0 . It is clear that the uncoupled displacements are quite different from the coupled displacements, because the torsion angle hx has great impact on the incident solar flux during the deformation of the beam. In order to get the coupled steady state values, the iterative method proposed in Section 2.3 with the maximum iterative number nmax ¼ 5 is utilized. The obtained results are hx ¼ 0:17 rad,hy ¼ 0:07 rad, hz ¼ 0:03 rad for Fig. 3(a) and hx ¼ 0:19 rad, hy ¼ 0:08 rad, hz ¼ 0:01 rad for Fig. 3(b), which are exactly the same as those from dynamic analysis. As Section 2 emphasizes, when either sub-criterion A or sub-criterion B is violated, the thermal-flutter will happen. Based on this rule, one can plot the stable and unstable zones of the TIV of this beam in Fig. 4. Since this open thin-walled beam has ultra-low torsional stiffness, it is not strange that most cases are unstable in Fig. 4. For example, when a ¼ 135 and h0 ¼ 0 , the steady
542 543
In this case, the steady angles are hx ¼ 0:21 rad, hyi ¼ 0:086 rad
538 539 540 541
Period of TIV when a ¼ 135 and h0 ¼ 0 .
angles are hx ¼ 0:10 rad, hyi ¼ 0:061 rad and hzi ¼ 0:0023 rad. In this case, both sub-criterion A and sub-criterion B are violated. Accordingly, in the numerical simulation results depicted in Fig. 5(a), all displacements are unstable. Thus, it verifies the prediction by the criterion. Fig. 4 also implies a stable TIV when a ¼ 75 and h0 ¼ 30 .
537
Fig. 2
7
544 545 546 547 548 549 550
and hzi ¼ 0:0016 rad, which satisfy both sub-criterion A and sub-criterion B. This conforms with the numerical simulation results depicted in Fig. 5(b). An interesting case is pure bending state in Y direction. For example, when a ¼ 180 and h0 ¼ 20 , it is obvious that deflection w and torsion angle hx are equal to zero due to the symmetry of this problem. In this case, the sub-criterion A
552
degenerates to h0 > hzi , which is the same as the criterion for a closed thin-walled beam.17 Since the steady angles
553
(hx ¼ 0 rad, hyi ¼ 0 rad and hzi ¼ 0:0035 rad) satisfy
551
554 555 556 557 558 559
h0 > hzi , v could be stable according to the sub-criterion A. However, the sub-criterion B is violated, so that w and hx must be unstable. These predictions are verified by the numerical results shown in Fig. 5(c), in which all deformations are stable before 3000 s, then w and hx gradually diverge, and finally v is also unstable due to the influence of w and hx . This example
Table 1
Fig. 3 Uncoupled and coupled thermal-structural responses at different incident angles.
Geometry dimensions and material parameters.
Parameter
L (m)
R (mm)
h (mm)
E (GPa)
Value Parameter
5.91 aT (1/K) 1.692 105 c (J/(kgK)) 502
10.92 q (kg/m3) 7010 as
0.6 S0 (W/m2) 1350 e
0.5
0.13
19.3 k (W/(mK)) 16.61 r (W/(m2K4)) 5.67 108
Value Parameter Value
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No. of Pages 9
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X. YUAN, Z. XIANG
Fig. 4
Fig. 5
Stable and unstable zones.
TIV at different incident angles.
demonstrates that the torsion has great impact on the stability of the TIV for open thin-walled beams.
560
4. Conclusions
562
This paper established a thermal-flutter criterion for an open thin-walled circular cantilever beam, which can be decomposed into two sub-criteria for the deflection v (sub-criterion A) and the coupled deformation of the bending state in z direction and torsion (sub-criterion B), respectively. In practice, thermalflutter happens when either the sub-criterion A or the subcriterion B is violated. The sub-criterion A can degenerate to the existing criterion for closed thin-walled beam free of torsion. However, the sub-criterion B shows that the torsion has great impact on the stability of the TIV for open thinwalled beams that have ultra-low torsional stiffness. However, this criterion does not consider the structural damping, so that it gives a conservative prediction for practical structures.
563
References
576
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