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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 2008

 

A Model for the Thermoelastic Behavior of a Joint-Leg-Beam System for Space Applications

E.M. Cliff, Z. Liu and R. D. Spies*

Abstract. Rigidizable-Inflatable (RI) materials offer the possibility of deployable large space structures (C.H.M. Jenkins (ed.), Gossamer Spacecraft: Membrane and Inflatable Structures Technology for Space Applications, Pro-gress in Aeronautics and Astronautics, 191, AIAA Pubs., 2001) and so are of interest in applications where large optical or RF apertures are needed. In particular, in recent years there has been renewed interest in inflatable-rigidizable truss-structures because of the efficiency they offer in packaging during boost-to-orbit. However, much research is still needed to better understand dynamic response characteristics, including inherent damping, of truss structures fabricated with these advanced material systems. One of the most important characteristics of such space systems is their response to changing thermal loads, as they move in and out of the Earth's shadow. We study a model for the thermoelastic behavior of a basic truss componentconsisting of two RI beams connected through a joint subject to solar heating. Axial and transverse motions as well as thermal response of the beams with thermoelastic damping are taking into account. The model results in a couple PDE-ODE system. Well-posedness and stability results are shown and analyzed.

Key words and phrases. Truss structures, Euler-Bernoulli beams, thermoelastic system.
* Corresponding author

1. Introduction

In recent years there has been renewed interest in Rigidizable-Inflatable (RI) space structures because of the efficiency they offer in packaging during boost-to-orbit. RI materials offer the possibility of deploying large space structures ([7]) and so are of interest in applications where large optical or RF apertures are needed. Several proposed space antenna systems will require ultra-light trusses to provide the "backbone" of the structure (see Figure 1(a)). It has been widely recognized that practical precision requirements can only be achieved through the development of new high-fidelity mathematical models and corresponding numerical tools.

PIC (a) Rigidizable-Inflatable truss structure
PIC (b) Basic structure of the joint-legs-beams system
Figure 1.1: Truss (a) and basic structure of the joint-legs-beams system (b).

In this paper we study the dynamics of a basic truss component consisting of two RI beams connected through a joint (see Figure 1(b)). One of the more important characteristics of such space systems is their response to changing thermal loads, as they move in and out of the Earth's shadow. In this paper we study the thermoelastic behavior of a two-beam truss element subject to solar heating. The beams are fabricated as thin-walled circular cylinders.

2. Thermoelastic Model

The equations of motion for the Joint-Leg-Beam system depicted in Figure 1(b) are the following (see [1] for details):

 ∂2ui(t,si) ∂2ui(t,si) ∂2wi (t,si) ∂4wi (t,si) ρiAi------2---= EiAi -----2---, ρiAi------2--- = - EiIi------4---, ∂t ∂si ∂t ∂s i (2.1)
 2 M d-[ x(t)y(t)θ1(t)θ2(t) ]T = C [ M1 (t)N1 (t)M2 (t)N2 (t)F1(t)F2 (t) ]T dt (2.2)

for time t > 0 and spatial variable si ∈ [0,Li] , where M and C are 4 × 4 and 4 × 6 matrices give by

 ⌊ m 0 - m d cos φ m d cosφ ⌋ | 1 1 1 2 2 2 | M = |⌈ 0 m m1d1 sin φ12 m2d2 sin φ2 |⌉ , - m1d1 cos φ1 m1d1 sin φ1 I1ℓ + m1d 1 0 2 m2d2 cosφ2 m2d2 sin φ2 0 I2ℓ + m2d 2 (2.3)
 ⌊ ⌋ | 0 - cosφ1 0 cosφ2 sin φ1 sin φ2 | C = |⌈ 0 sinφ1 0 sinφ2 cosφ1 - cosφ2 |⌉ , 1 ℓ1 0 0 0 0 0 0 1 ℓ2 0 0 (2.4)

and the other functions and parameters are as follows (here the supra or sub-index i , i = 1,2 will always refer to beam or leg i ):  i i u (t,si),w (t,si) longitudinal and transversal displacement of the beam; x (t),y(t) horizontal and vertical displacement of the joint's tip; θi(t) rotation angle of the leg; ρi,Ai,Li,Ei, Ii mass density, cross section area, length, Young's modulus, moment of inertia of the beam; mi, di,ℓi,Iiℓ mass, center of mass, length, moment of inertia of the leg; m p mass of the joint, m = m1 + m2 + mp ; φ1 initial angle of leg 1 with positive y axis; φ2 initial angle of leg 2 with negative y axis; Fi(t) extensional force of beam at the end si = Li ; Ni (t) shear force of beam at the end si = Li ; Mi (t) bending moment of beam at the end si = Li .

Each beam is clamped at the end s = 0 i . Thus the boundary conditions at si = 0 are

 i i ∂wi- u (t,0) = w (t,0) = ∂si (t,0) = 0, i = 1,2. (2.5)

At the other end of each beam several obvious geometric compatibility conditions must be imposed. These conditions can be written in the form:

 ⌊ ⌋ ⌊ -∂- 1 ⌋ θ1(t) - ∂s11w (t,L1 ) | - x (t) cosφ1 + y(t)sinφ1 + ℓ1θ1(t) | ⌊ ⌋ || w (t,L1) || || θ (t) || x (t) || - ∂∂s2w2 (t,L2 ) || | 2 | T || y(t) || | w2(t,L2) | = || x (t) cosφ2 + y(t)sinφ2 + ℓ2θ2(t) || = C ⌈ θ1(t) ⌉. |⌈ - u1(t,L ) |⌉ |⌈ x(t)sinφ1 + y (t) cosφ1 |⌉ θ2(t) 2 1 x(t)sinφ2 - y (t) cosφ2 - u (t,L2) (2.6)

In [1], system (2.1)-(2.6) was re-cast as an abstract second-order ODE in an appropriate Hilbert space. Semigroup theory was then used to prove that the system is well-posed. Moreover, it was shown that if Kelvin-Voigt damping to both transverse and longitudinal motions is added, then the corresponding semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator of this semigroup was also characterized. The case of local damping was analyzed in [4] where it was shown that if only one of the beams is damped, then only polynomial stability is obtained even if additional rotational damping is assumed in the joint. Numerical approximations and several numerical results are shown in [2].

3. Thermal Dynamics

The external heat flux in the space normal to the beam's surface is given by (see [10])

 ( ∂wi) Si=. S0 cos ξi - ---- , ∂si (3.1)

where S 0 denotes the solar flux and ξ i the angle of orientation of the solar vector with respect to the beam. In this equation we shall neglect the contribution of ∂wi ∂si since we are assuming it is small. We denote by Ti(t,si,φi) the deviation of the temperature of the thin-walled circular beam i with respect to a reference temperature T i 0 at time t at the point on the beam corresponding to axial coordinate si and circumferential coordinate φi (here φi = 0 corresponds to the top of the beam while φi = π corresponds to the bottom). Conservation of energy for a small segment of circular cylinder including longitudinal and circumferential conduction in the cylinder wall and radiation from the cylinder's surface yields the following equation for  i T :

 ∂T-i kic∂2T-i i∂2T-i σεi i i 4 αis ρici ∂t - R2 ∂φ2 - ka ∂s2 + hi (T0 + T ) = hi Si cos(φi)δ(φi) i i i (3.2)

where kia and kic are the axial and circumferential thermal conductivity coefficients, respectively, ci is the specific heat, Ri the radius of the cylinder, hi is the thickness of the wall, εi is the surface emissivity and αi s is the surface absorptivity, σ is the Stefan-Boltzmann constant, δ is a function defined on [- π, π] by δ(φi) = 1 for  π π φi ∈ (- 2, 2) , and δ(φi) = 0 for  π π φi ∈ [- π,- 2 ] ∪ [2,π] . The heat flux distribution on the RHS of equation (3.2) can be written as

 ( ) 1- Si- Si cos(φi)δ(φi) = Si π + g(φi) = π + Sig(φi) (3.3)

where  . 1 g(φi)= cos(φi)χ π π (φi) - π [-2 ,2] (here χ denotes the characteristic function). Clearly g (φi) is continuous and it has zero average in [- π,π] .

For each beam, the temperature distribution is separated into two parts, namely:

 i i m,i T (t,si,φi) = T (t,si) + T (t,si)g(φi), (3.4)

where Ti(t,si) is independent of φi and corresponds to the uniform part of the flux, Siπ , in (3.3), and Tm,i(t,si) g(φi) amounts for the circumferential variation of the flux in (3.3). Note that for every si ∈ [0,Li ] and t ≥ 0 one has that T m,i(t,s ) = T i(t,s ,0) - T i(t,s ,π) = T i(t,s ,0) - T i(t,s ,φ) i i i i i for any  π π φ ∕∈ [- 2,2] . Hence,  m,i T (t,si) can be thought of as the thermal gradient between the top and the bottom of the beam at the axial location si .

Also, we approximate the thermal radiation term (T i0 + T i(t,si,φi))4 in (3.2) by linearizing T(t,si,φi) around T (t,si,φi) = T i s (where Ti s , to be determined later, is the steady-state constant temperature increment produced on the undeformed beam i by the solar flux Si ), i.e., we approximate  i i 4 (T 0 + T (t,si,φi)) by (T0i+ Tsi)4 + 4(T0i+ Tsi)3(T i(t,si) - Tis + Tm,i(t,si)g(φi)) . Hence equation (3.2) is replaced by pict

Since g has zero average, integration of equation (3.5) over the cylinder's cross sectional area yields pict

Since g ′(φi) is discontinuous at φi = ± π 2 the integration of g′′(φi) above must be performed in the distributional sense. The value of  i T s is now determined by setting the RHS, fi , equals to zero. By doing so we obtain

 ( i ) 1 T i= α-sSi 4 - Ti s πσεi 0 (3.7)

Note that with this value of  i Ts corresponds to the steady-state T i(t,si) = Tis for the case of homogeneous Neumann boundary conditions and, since usually T m,i(t,si) is small compared to Ti0 , the linearization of the thermal radiation term performed above, is justified near the steady state solution.

Now multiplying (3.5) by g(φi) and integrating over the cylinder's cross sectional area, we obtain for  m,i T the following equation:

 m,i 2 m,i ρ c ∂T----(t,si) - ki∂-T----(t,si) ii ∂t a ∂s2i ( i 2 i i 3) i + ---kcπ-----+ 4-σεi(T-0 +-T-s) T m,i(t,si) = αsSi. R2i(π2 - 4) hi hi (3.8)

Thermally induced vibrations in the system is taken into account by considering Hooke's law for the stress-strain relation in the form εi11 = E1σi11 + αiT i, i where αi is the thermal expansion coefficient, and T i is, as before, the deviation from the reference temperature T i 0 . Note that at  i T = 0 thermal strain vanishes, so that  i T0 is interpreted as the (uniform) temperature of beam i in the unstressed, rest-state. By the standard derivation of Euler-Bernoulli beam equation, we modify the Joint-Leg-Beam system (2.1) as follows:

 2 i ( i ) ρiAi ∂-u-(t,si) = EiAi -∂-- ∂u-(t,si)-- αiT i(t,si) , (3.9 ) ∂t2 ∂si ∂si ∂2wi(t,s ) ∂2 ( ∂2wi(t,s ) α ) ρiAi -----2--i- = - EiIi--2- ------2-i-+ --i-Tm,i(t,si) (3.10 ) ∂t ∂si ∂s i 2Ri

The above beam equations are coupled to the heat equations modified from equations (3.6) and (3.8) and with  i Ts chosen as in equation (3.7) (so that fi = 0 in (3.6) ), that is:

 i 2 i ρ c ∂T-(t,si)-= ki∂-T-(t,si) i i ∂t a ∂s2i i i 3( ) 2 - 4σεi(T-0 +-Ts)- T i(t,si) - Ti - αiEiT i-∂---ui(t,si) , hi s 0 ∂si∂t (3.11)

and

 m,i 2 m,i [ i 2 i i3 ] ρc ∂T----(t,si) = ki ∂-T---(t,si)-- ---kcπ-----+ 4-σεi(T-0 +-T-s) T m,i(t,s ) ii ∂t a ∂s2i R2i(π2 - 4) hi i i 3 i + αiEiIiT0---∂---wi(t,si) + α-sSi, (3.12) 2RiAi ∂s2i∂t hi

We impose Robin type boundary conditions for the temperature at both ends of each beam, i.e.

pict

∀t ≥ 0, φi ∈ [- π,π], i = 1,2 , where T * is the temperature of the surrounding medium and λi L , λi R , i = 1,2 , are nonnegative constants. By writing T i(t,s ,φ ) i i in terms of the decomposition given in (3.4) these equations take the form:

pict

Since these equations must hold for all φi ∈ [- π,π] it follows that

pict

and

pict

for all t ≥ 0 , i = 1,2 . So, in the same way that the dynamics for the temperature distribution (3.5) decouples into equations (3.11) and (3.12) for  i T and  m,i T , respectively, we observe that the boundary conditions also decouple. Note however in equation (3.13) that the boundary conditions for the axial component of the temperature, T i(t,si) , are non-homogeneous. By defining ˜Ti(t,s) =.T i(t,s ) - (T* - Ti) i i 0 , equation (3.11) can be written in the form

 ∂ ˜Ti(t,si) i∂2T˜i(t,si) ρici ∂t = ka ∂s2 i i i 3( ) 2 - 4σεi(T-0 +-Ts)- ˜Ti(t,s ) + T* - T i- T i - α E T i-∂---ui(t,s ), hi i 0 s i i 0∂si∂t i (3.15)

while the boundary conditions (3.13) now take the form

pict

Observe now that these boundary conditions are exactly the same as those in (3.14) for the circumferential component of the temperature. Finally, note also that in equation (3.9), T i(t,si) can be replaced by  ˜i T (t,si) without any changes.

System (3.9)-(3.12) (or equivalently (3.9), (3.10), (3.12), (3.15)), together with the joint-leg dynamics described by equation (2.2) constitute the thermoelastic Joint-Leg-Beam equations with the external solar heat source. The extensional forces, shear forces and bending moments of the beams at s = L i i are now given by:  ( i ) | F (t) = E A ∂u--(t,s ) - α T i(t,s ) || , (3.17 ) i i i ∂si i i i |s=L ( 2 i i i) | N (t) = E I ∂--- ∂-w--(t,s) + -αi-Tm,i(t,s) || , (3.18 ) i i i∂si ∂s2i i 2Ri i |s=L ( 2 i )| i i M (t) = E I ∂-w-(t,s ) + αi-T m,i(t,s ) || . (3.19 ) i i i ∂s2i i 2Ri i |si=Li

4. Well-posedness

In this section, we consider the well-posedness of the Joint-Leg-Beam system with solar heat flux, i.e., equations (3.9), (3.10), (3.12), (3.15) subject to the geometric beam-leg interface compatibility conditions (2.6), the dynamic boundary conditions (3.17), (3.18), (3.19) and the boundary conditions (2.5), (3.14), (3.16). We first rewrite the system as a first order evolution equation in an appropriate Hilbert space. Well-posedness is then obtained by using semigroup theory. Since the corresponding system without thermal effects has been studied in [1], we will follow the notation used there as much as possible for consistency. Numerical results for that case are reported in [2].

First, we define the following Hilbert spaces with their corresponding inner products:

( 2 2 2 2 |{ Hz = L (0,L1 ) ×2 L (0,L2) × L (0,L1 ) × L (0,L2), . ∑ [ i i i i ] |( ⟨z1,z2⟩Hz = ρiAi ⟨w1,w 2⟩ + ⟨u 1,u2⟩ ; i=1
{ Hb = [ker(C )]⊥ = range (CT ), ⟨b1,b2⟩H = ⟨b1,(CT M - 1C )†b2⟩lR6; b
( 2 2 2 2 |{ H ζ = L (0,L1 ) × L (0,L2) × L (0,L1 ) × L (0,L2), . ∑2 ρiciAi [ ] |( ⟨ζ1,ζ2⟩H ζ = ---i-- ⟨T m1,i,T m2,i⟩ + ⟨T˜1i,T˜i2⟩ ; i=1 T0

where  . ( )T . ( m,1 m,2 )T zj = w1j,w2j,u1j,u2j , ζj= Tj ,T j ,T˜1j ,T˜2j , and  T -1 † (C M C) denotes the Moore-Penrose generalized inverse of  T - 1 C M C . We also define the operators Az : Hz → Hz and Bz : H ζ → Hz by

 . 2 4 2 4 1 2 1 2 dom (Az ) = H(ℓ ∩ H (0, L1) × Hℓ ∩ H (0,L2) × H ℓ)∩ H (0, L1) × Hℓ ∩ H (0,L2), Eρ1AI1-D4 0 0 0 | 1 10 E2I2D4 0 0 | Az =. || ρ2A2 E1 2 || , ( 0 0 - ρ1 D 0 ) 0 0 0 - Eρ2 D2 2

dom (Bz ) .= H2 (0,L1 ) × H2 (0,L2 ) × H1 (0,L1 ) × H1 (0,L2 ), ( α1E1I1- 2 ) - 2R1ρ1A1 D 0 0 0 . || 0 - α2R22Eρ22IA22-D2 0 0 || Bz = |( 0 0 - α1E1 D 0 |) . ρ1 α2E2- 0 0 0 - ρ2 D

where  n . dn- D = dsni and for n ∈ IN ,  n H ℓ (0,L) denotes the space of functions in Hn (0,L) that vanish, together with all derivatives up to the order n - 1 , at the left boundary. With this notation, equations (3.9)-(3.10) can now be written as the following abstract second order ODE in Hz :

¨z(t) + Azz (t) - Bzζ(t) = 0. (4.1)

Next we define the operators Aζ : H ζ → H ζ and B ζ : Hz → H ζ by

 . 2 2 2 2 dom (A ζ) = H rb(0,L1 ) × H rb(0,L2) × H rb(0,L1 ) × H rb(0,L2),
 ( [ ] ) ( ) - -k1a-D2T m,1 + ---k1cπ2----+ 4σε1(T01+T-1s)3- Tm,1 T m,1 | ρ1c12 [ρ1c1R212(π22-4) ρ1c1h21 23] | | T m,2| . || - -ka-D2T m,2 + ---kc2π-2---+ 4σε2(T0+T-s)- Tm,2|| A ζ ζ = A ζ|( T˜1 |) = || ρ2c2 k1 ρ2c2R2(π4-σ4ε1)(T1+T1ρ)23c2h2 || , ˜2 ( - ρ1ac1-D2 ˜T1 + ---ρ1c01h1s--˜T1 ) T - -k2a-D2 ˜T2 + 4σε2(T20+T2s)3˜T2 ρ2c2 ρ2c2h2
dom (B )=. H2 (0,L ) × H2 (0,L ) × H1 (0,L ) × H1 (0,L ), ζ 1 2 1 2
 ( 1 ) α2R1Eρ1I1cT0A-D2 0 0 0 | 11 1 1 α2E2I2T20 2 | B z .= || 0 2R2ρ2c2A2 D 0 1 0 || . ζ |( 0 0 - α1E1T0 D 0 |) ρ1c1 α2E2T02 0 0 0 - ρ2c2 D

where H2rb(0,L ) denotes the space of functions in H2 (0,L ) satisfying the Robin boundary conditions (3.14) or equivalently (3.16). With this notation, equations (3.12), (3.15), can now be written as the following abstract first order ODE in H ζ :

ζ˙(t) - B z˙(t) + A ζ (t) = S ζ ζ (4.2)

where

 ( )T S =. ρ-αc1sh-S1,ρ-αc2sh-S2, 4σε1(ρTc10+Th1s)3(T1s+T10-T*), 4σε2ρ(T20c+hT2s)3(Ts2+T02-T*) . 1 11 22 2 1 11 2 22

We also define three boundary projection operators P B1 , P B2 from Hz into IR6 and P B 3 from H ζ into IR6 by

dom (P1B) .= H2 (0,L1 ) × H2 (0,L2 ) × H1 (0,L1 ) × H1 (0,L2 ), B . 4 4 2 2 dom (P2 ) = H (0,L1 ) × H (0,L2 ) × H (0,L1 ) × H (0,L2 ), dom (P3B) .= H2 (0,L1 ) × H2 (0,L2 ) × H1 (0,L1 ) × H1 (0,L2 ),

 ( 2 ) (- -∂w1(L1)) ∂∂s21w1 (L1) ( Tm,1(L1) ) (w1 ) | ∂ws11(L1) | ( w1) ||| ∂∂3s3w1 (L1)||| ( Tm,1) || ∂-Tm,1(L1)|| B |w2 | . ||- -∂w2(L2)|| B| w2| . || ∂12w2 (L2)|| B| Tm,2| . || ∂s1Tm,2(L2) || P1 |( u1|) = ||| ∂ws22(L2) ||| , P2 |( u1|) = || ∂∂s223- 2 || , P3 |( ˜T1 |) = || ∂∂sTm,2(L2)|| . u2 ( -u1(L1) ) u2 || ∂s32w 1(L2)|| ˜T2 |( 2˜T1(L1) |) -u2(L2) ( ∂∂s1u(L1)) ˜T2(L2) ∂∂s2u2(L2)
Now, by using the geometric compatibility conditions (2.6) and the dynamic boundary conditions (3.17)-(3.19), the equation for the leg-joint dynamics (2.2) can be written as the following abstract second order ODE in Hb :
 2 d--( B ) T -1 ( B B ) ˜ dt2 P 1 z(t) - C M CE P 2 z(t) + ΛP 3 ζ(t) = R (4.3)

where  . E = diag(E1I1, E1I1,E2I2,E2I2, E1A1, E2A2 ) ,  . α α α α Λ = diag (2R11-,21R1,2R22,2R22 , - α ,- α ) 1 2 and ˜R =.CT M -1C (0, 0, 0, 0, E A α (T * - T1), E A α (T * - T 2) )T 1 1 1 0 2 2 2 0 . Next we define the Hilbert space  . Hzb = Hz × Hb with the usual inner product inherited from those in Hz and Hb . In this Hilbert space we define the elastic operator Azb by

 { ( ) } . z B dom (Azb) = b ∈ dom (Az ) × Hb : P 1 z = b ( ) ( ) z . Azz and Azb b = - CT M - 1CE P B2 z .

Furthermore, we define Bzb : H ζ → Hzb by  . 2 2 1 1 dom (Bzb) = H (0, L1) × H (0, L2) × H (0, L1) × H (0, L2) and  ( ) Bzbζ =. T - B1zζ B . C M CE Λ P3 ζ Thus, equations (4.1) and (4.3) can be combined as

 ( ) ( ) d2- z(t) z(t) dt2 b(t) + Azb b(t) - Bzbζ(t) = R on Hzb, (4.4)

where  . R = (0,R˜)T . It has been proved in [1] that the operator Azb is self-adjoint and strictly positive. Thus, we can define the state space H =. dom (A1 ∕2) × H × H zb zb ζ with the inner product ⟨ (X ) (Y ) ⟩ 1 1 . 1∕2 1∕2 X2 , Y2 = ⟨A zb X1, A zb Y1⟩Hzb + ⟨X2, Y2⟩Hzb + ⟨X3, Y3⟩H ζ. X3 Y3 H Finally, we define operator A on H by  ( ( ) ) { X1 | } dom (A ) =. ( X2 ) ∈ H | X1 ∈ dom (Azb ),X2 ∈ dom (A1z∕b2), X3 ∈ dom (A ζ) ( X3 ) ,  (X ) ( 0 I 0 ) (X ) 1 . 1 A X2 = - Azb 0 Bzb X2 . X3 0 (B ζ,0) - Aζ X3 Then, equations (4.2) and (4.4) can be rewritten as a first order nonhomogeneous evolution equation

 ˙ X (t) = AX (t) + G on H (4.5)

where  ( ) ( ) . X1 . (z) . . . 0 X = ( X2 ) , X1 = , X2 = ˙X1, X3 = ζ and G = ( R) X3 b S .

Theorem 4.1. (Well-posedness): Let A : H → H be as defined above. Then A is the infinitesimal generator of a strongly continuous semigroup of contractions S (t) on H and hence, for any initial condition X0 = X (0) ∈ dom (A ) , system (4.5) has a unique global solution X (t) given by

 ∫ t X (t) = S(t) X + S (t - s)G ds. 0 0

Proof: It can be shown that A is dissipative and 0 ∈ ρ (A ) , the resolvent set of A ) . Since dom (A ) is dense in H , it then follows from Theorem 1.2.4 in [8] that A generates a strongly continuous semigroup of contractions S(t) on H . The existence and uniqueness of solutions for system (4.5) for any initial condition X0 = X (0 ) ∈ dom (A ) finally follows from Corollary 2.10 in [9]. For more details see [3]. _

5. Exponential Stability

We now turn our attention to the stability of system (4.5). It is well known that the semigroup associated with longitudinal and transversal motion of a thermoelastic Euler beam is exponentially stable ([5], [8]). System (4.5) consists of two thermoelastic beam equations plus the equations for the joint-leg dynamics. This type of system is often referred to as "hybrid system". It is certainly an interesting problem to determine whether the thermal damping is strong enough by itself to induce exponential stability of this kind of system. We shall prove this in the affirmative.

The following result by Huang [6] will be used:

Theorem 5.1. Let H be a Hilbert space, A : H → H a closed, densely defined linear operator. Assume that A generates a C0 -semigroup of contractions T(t) on H . Then T (t) is exponentially stable if and only if

 ilR ∩ σ(A ) = ∅, (5.1) -1 lβim→∞ ∥(iβ - A ) ∥ < ∞. (5.2)

Theorem 5.2. The C0 -semigroup of contractions S(t) generated by A (see Theorem 4.1) is exponentially stable.

Proof: If (5.2) is false then there exists a sequence {βn } ⊂ lR with βn → ∞ and a sequence {Xn } ⊂ D (A ) with ∥Xn ∥H = 1 ∀ n such that

nli→m∞ ∥(iβn - A )Xn ∥H = 0. (5.3)

Using the components related to the thermoelastic beam equations it can be show that (5.3) yields the contradiction ∥Xn ∥H → 0 as n → ∞ . Similarly, if the condition (5.1) is false, then there exist β ∈ lR and a sequence {Xn } ⊂ D(A ) with ∥Xn ∥H = 1 ∀n , such that

lim ∥ (iβ - A)X ∥ = 0. n→∞ n H (5.4)

By repeating the same arguments we get the contradiction ∥Xn ∥H → 0 . For complete details on these proofs, we refer the reader to [3]. Hence A satisfies conditions (5.1) and (5.2) and therefore, the C0 -semigroup of contractions S(t) generated by A is exponentially stable. __

6. Conclusions

In this article we considered a system of two thermoelastic Euler-Bernoulli beams coupled to a joint through two legs. By means of semigroup theory the well posedness of the system was proved and its exponential stability was derived. It is certainly of much interest to develop numerical approximations for our state-space model (4.5). Such numerical schemes will be useful in simulation and identification studies to predict and better understand the structural and thermal responses of space-borne observation systems. Efforts in this direction are already under way.

Acknowledgements

This work was supported by DARPA/SPO, NASA LaRC and the National Institute of Aerospace under grant VT-03-1, 2535, and in part by AFOSR Grants F49620-03-1-0243 and FA9550-07-1-0273. Acknowledgement is also given to CONICET and Universidad Nacional del Litoral of Argentina.

References

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[2]    J.A. Burns, E.M. Cliff, Z. Liu and R. D. Spies, Results on Transversal and Axial Motions of a System of Two Beams coupled to a Joint through Two Legs, submitted, 2008. ICAM Report 20070213-1.        [ Links ]

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[10]    E. A. Thornton and R. S. Foster, Dynamic Response of Rapidly Heated Space Structures. In: Computational Nonlinear Mechanics in Aerospace Engineering, edited by Atluri, S.N., Progress in Astronautics and Aeronautics, Vol. 146, AIAA, Washington D.C., 1992, p. 451-477.        [ Links ]

 

E. M. Cliff
Interdisciplinary Center for Applied Mathematics
Virginia Polytechnic Institute and State University
Blacksburg, VA, 24061-0531, USA.

Z. Liu
Department of Mathematics
University of Minnesota
Duluth, MN 55812-3000, USA.

R. D. Spies
Instituto de Matemática Aplicada del Litoral
IMAL, CONICET-UNL, Güemes 3450, and
Departamento de Matemática,
Facultad de Ingeniería Química, UNL,
Santa Fe, Argentina
rspies@imalpde.santafe-conicet.gov.ar

Recibido: 11 de mayo de 2008
Aceptado: 20 de mayo de 2008