A thermoviscoelastic beam with a tip body

Computational Mechanics 33 (2004) 225–234 Ó Springer-Verlag 2004 DOI 10.1007/s00466-003-0523-3 A thermoviscoelastic beam with a tip body K. T. Andrews, J. R. Fernández, M. Shillor 225 in [1, 10]. However, the present setting requires more complicated boundary conditions than those treated previously. Such conditions were previously encountered in [2], where we also considered a beam with tip body problem but only in an isothermal context. Here we present a model which includes thermal effects and prove the existence of a solution. Then we develop a numerical algorithm and obtain numerical simulations of the system’s vibrations. The rest of the paper is organized as follows. In Sect. 2 we describe the classical model. A weak or abstract formulation is derived in Sect. 3, where the assumptions on Keywords Thermoviscoelastic beam, Tip body, the problem data are given and the main result is given in Vibrations, Numerical simulations Theorem 3.1. The algorithm for numerical approximations of the problem is described in Sect. 4, and the numerical 1 simulations are presented in Sect. 5. The paper concludes Introduction In this paper we model and analyze the thermomechanical in Sect. 6, where a short summary and a description of behaviour of a cantilever beam with an attached hollow tip open related problems are provided. body that contains granular material. For structures containing tip bodies most of the existing literature [5–7, 9, 16, 2 20] has focused on numerical or experimental analysis of The model vibration modes and on optimal control of the vibrations. In this section we model the dynamic evolution of a Such work is motivated by the fact that successful manip- thermoviscoelastic beam which is clamped at its left end, while a tip body is rigidly attached to the right end. The tip ulation of objects by robotic arms depends critically on damping any possible modes of vibration. More recently, body is a sealed container containing a granular material granular materials have been used as agents for damping such as sand. This granular material provides damping to the motion of the system, due to internal friction, and the vibrations of such structures. These applications use the energy loss generated by internal frictional contact of the system loses energy as heat exchange with its environment. The physical setting of the problem is depicted in Fig. 1. grains to cause internal damping of the vibrations of We assume that the beam is long and slender, and has a the structure. Some recent experimental and parameter identification results on the behavior of a cantilever beam uniform cross section. We use dimensionless variables, wrapped in a sleeve full of granular material can be found in and set 0
x
1 as the reference configuration for the beam center rod. Let u ¼ uðx; tÞ r ¼ rðx; tÞ and [3, 4]. From the mathematical point of view, problems h ¼ hðx; tÞ denote the vertical displacement, the shear involving tip bodies lead to initial-boundary value problems that contain acceleration terms in the boundary con- stress and the thermal moment at location x and time t, all ditions as well as in the equation of motion. Existence and in dimensionless units. We note that if the beam has a uniqueness results for problems of this type may be found uniform square cross section with vertical thickness h, then the thermal moment at ðx; tÞ is given by Abstract We present and investigate a model for the thermomechanical behaviour of a viscoelastic beam with an attached tip body. The model consists of a coupled energy–elastic system of equations for the beam together with an equation for the temperature of the tip body. We establish existence and uniqueness of a weak solution to the system. We also propose a finite element algorithm for solutions to the system and present numerical simulations based on the algorithm. Received: 18 August 2003 / Accepted: 20 October 2003 Published online: 16 January 2004 K. T. Andrews &, M. Shillor Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA e-mail: andrews@oakland.edu J. R. Fernández Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Sur s/n, 15706 Santiago de Compostela, Spain 12 h ¼ hðx; tÞ ¼ 3 h Zh=2 yHðx; y; tÞdy ; h=2 where H ¼ Hðx; y; tÞ is the temperature (measured relative to that of the environment) at ðx; tÞ and vertical location y. Consequently, in the discussion of the model that follows we will use the terms temperature and thermal moment interchangeably. We will also subscripts to denote partial derivatives. We use a linearized constitutive relation for a thermoviscoelastic material component of the inertial term is mðu þ d sin bÞtt , where b is the angle between the OO0 segment and the x-axis. We use the linear approximation, for small b, sin b ’ tan b ’ ux . The third term represents the damping that the granular material provides, which is assumed to be proportional to the velocity, and so c is the coefficient of internal damping. Thus, the force balance at the end x ¼ 1 is Fig. 1. The beam with the tip body rð1; tÞ ¼ a2 uxxx ð1; tÞ þ kuxxxt ð1; tÞ þ ahx ¼ mutt ð1; tÞ þ mduttx ð1; tÞ þ cut ð1; tÞ þ fB ðtÞ ; 2 226 r ¼  rðx; tÞ ¼ a uxxx ðx; tÞ þ kutxxx ðx; tÞ þ ahx ðx; tÞ ; ð2:1Þ where a2 is the scaled elasticity coefficient, k ð 0Þ is the scaled viscosity coefficient, and a is the coefficient of thermal expansion. We will allow for the case when k ¼ 0, i.e., when the material is purely elastic; otherwise, we will assume that all the process and material coefficients which appear in the model are positive constants. Under these assumptions the applied bending moment M of the beam is given by Here fB ðtÞ is a prescribed external force that acts on the tip body, such as gravity, and in which case fB ðtÞ ¼ mg. We next consider the balance of moments about an axis through the center of mass of the tip body and perpendicular to the plane of motion. Here we have Juttx ¼ M þ dr  dc utx þ dfB ðtÞ ; where J is the moment of inertia of the combined mass of the container and the granular material with respect to its center of mass, and c is another damping coefM ¼ Mðx; tÞ ficient describing the damping related to the angular  ð2:2Þ motion of the granular material. One can set c ¼ c ¼ a2 uxx ðx; tÞ þ kutxx ðx; tÞ þ ahðx; tÞ ; assuming that the moment of the damping force of the see, e.g., [8, Ch. 10] or [15, Ch. 1]. The beam’s equation of granular material is concentrated at the center of commotion, in dimensionless variables, is bined mass, but it seems more realistic to treat the two
2 ð2:3Þ separately. Combining the above expressions and using qutt þ a uxx þ kutxx þ ah xx ¼ f ; (2.2), we have that the balance of moments at the end where f denotes the density (per unit length) of applied x ¼ 1 is vertical forces and q is the material (linear) density. The mdutt ð1; tÞ þ ðJ þ md2 Þuttx ð1; tÞ þ a2 uxx ð1; tÞ beam is assumed to exchange heat with the environment, þ kutxx ð1; tÞ þ dcut ð1; tÞ þ dc utx ð1; tÞ þ ahð1; tÞ ¼ 0 : which is assumed to be at the reference temperature, chosen as zero. The beam’s energy equation is Finally, we assume that the tip body has a uniform ht  jhxx  autxx þ fh ¼ 0 ; ð2:4Þ temperature hB ¼ hB ðtÞ that may be different from the temperature of the beam’s right end. The heat flux from where j is a scaled thermal diffusivity coefficient and f denotes the scaled coefficient of heat exchange. The third the beam to the tip body is given by term on the left-hand side represents the heat generated by rapid changes in the mechanical strain rate. For initial conditions we have uðx; 0Þ ¼ u0 ðxÞ ut ðx; 0Þ ¼ v0 ðxÞ and hðx; 0Þ ¼ h0 ðxÞ for 0
x
1, where u0 ; v0 and h0 represent the beam’s initial displacement, velocity and temperature. The beam is rigidly attached to the left wall and that wall is assumed to be at the ambient temperature, thus uð0; tÞ ¼ ux ð0; tÞ ¼ 0 and hð0; tÞ ¼ 0, for 0
t
T. We turn to model the motion of the right end with the attached tip body. We assume that the container is rigidly attached to the end x ¼ 1, and that the container and its contents have mass m and a center of mass O0 located at distance d from the end of the beam O. We assume that the damping effect of the internal granular material can be represented by damping coefficients c and c whose precise contributions are described below. After rescaling the constants, we omit the overline. Now the force that the tip body exerts on the end of the beam is given by mutt þ mduttx þ cut : jhx ð1; tÞ ¼ hB ðhð1; tÞ  hB ðtÞÞ ; where hB is the coefficient of heat exchange. When the right-hand side is positive, energy flows from the beam’s right end into the tip body, which raises the body’s temperature. On the other hand, we suppose that the tip body loses energy to the surroundings with heat exchange coefficient hB , and we assume that the internal friction of the granular material produces heat proportional to the velocity squared. Thus, the energy balance in the tip body is mcb dhB ¼ hB ðhð1; tÞ  hB ðtÞÞ  hB hB ðtÞ þ bðut ð1; tÞÞ2 : dt Here b is a positive coefficient that measures the fraction of the kinetic energy lost as heat. Now let XT ¼ ð0; 1Þ  ð0; TÞ, for T > 0. Then the classical problem of the vibrations of a thermoviscoelastic beam with a damping tip body can be formulated as follows: Find a triple of functions fu ¼ uðx; tÞ; h ¼ hðx; tÞ; hB ¼ hB ðtÞg such that 2 Here the first two terms represent the contribution of the qutt þ a uxxxx þ kutxxxx þ ahxx ¼ f ; in XT ; inertia of the container; indeed, the precise vertical ht  jhxx  autxx þ fh ¼ 0; in XT ; ð2:5Þ ð2:6Þ uðx; 0Þ ¼ u0 ðxÞ; ut ðx; 0Þ ¼ v0 ðxÞ; ¼ rð1Þwð1Þ þ ða2 uxx ð1Þ þ kvxx ð1Þ Z1
þ ahð1ÞÞwx ð1Þ  a2 uxx þ kvxx þ ah ð2:7Þ hðx; 0Þ ¼ h0 ðxÞ ; uð0; tÞ ¼ 0; ux ð0; tÞ ¼ 0; hð0; tÞ ¼ 0 ; dhB mcb ¼ hB ðhð1; tÞ  hB ðtÞÞ  hB hB ðtÞ dt þ bðut ð1; tÞÞ2 ; jhx ð1; tÞ ¼ hB ðhð1; tÞ  hB ðtÞÞ ; ð2:8Þ 0  wxx dx þ ð2:9Þ ð2:10Þ 2 a uxxx ð1; tÞ þ kuxxxt ð1; tÞ þ ahx ð1; tÞ ð2:11Þ mdutt ð1; tÞ þ ðJ þ md2 Þuttx ð1; tÞ þ a2 uxx ð1; tÞ 0 Here we have also used the constitutive relation (2.1) and the fact that w 2 V. Now using the boundary condition and (2.8) in the above equation, we obtain q þ kutxx ð1; tÞ þ dcut ð1; tÞ Z1 0 þ dc utx ð1; tÞ þ ahð1; tÞ ¼ 0 : fw dx : rð1Þ ¼ mv0 ð1Þ  mdv0x ð1Þ  cvð1Þ  fB ; ¼ mutt ð1; tÞ þ mduttx ð1; tÞ þ cut ð1; tÞ þ fB ðtÞ ;  Z1 ð2:12Þ The problem thus involves a coupled system of hyperbolic and parabolic equations, with unusual boundary conditions. We show in the next section that the problem has a unique weak solution.
v0 w dx þ mv0 ð1Þwð1Þ þ J þ md2 v0x ð1Þwx ð1Þ   þ cvð1Þwð1Þ þ mdv0x ð1Þwð1Þ þ mdv0 ð1Þwx ð1Þ þ dc vx ð1Þwx ð1Þ þ dcvð1Þwx ð1Þ þ Z1 0
a2 uxx þ kvxx wxx dx 3 Z1 Z1 Weak formulation and existence result ð3:1Þ þ a hwxx dx ¼ fw dx  fB wð1Þ : In this section we reformulate the problem in an abstract 0 0 setting and establish its unique solvability. First, we define the spaces of test functions V and E by We turn to the heat equation (2.6). We multiply it by g 2 E, integrate by parts and use the boundary conditions V ¼fw 2 H 2 ð0; 1Þ : wð0Þ ¼ wx ð0Þ ¼ 0g ; at x ¼ 0 and (2.10). Thus, E ¼fg 2 H 1 ð0; 1Þ : gð0Þ ¼ 0g : Z1 Z1 Z1 Z1 R 1 2 1=2 0 h g dx þ j hx gx dx þ a vx gx dx þ f hg dx We note that ð 0 wxx Þ is equivalent to the usual R 1 2 1=2 2 H -norm of V, and ð 0 gx Þ is equivalent to the usual 0 0 0 0 H 1 -norm of E. We also let H ¼ L2 ð0; 1Þ. We denote by ð; Þ þ hB ðhð1; tÞ  hB ðtÞÞgð1Þ  avx ð1Þgð1Þ ¼ 0 : ð3:2Þ the inner product on H, by h; iV the duality pairing between V and V 0 , and by h; iE the duality pairing between Finally, for technical reasons we alter the last term in (2.9), by introducing, for fixed r > 0, the truncation E and E0 . We have operator 8 V  E  H ¼ H 0  E0  V 0 ; jwj
r, < w; with continuous inclusions. In what follows we will use the sr ðwÞ ¼ r; w > r, : prime on a variable to denote differentiation with respect r; w < r to t and we suppress the dependence on t to shorten the notation. We let v  ut denote the velocity. We begin by and modifying (2.9) to read multiplying (2.5) by w 2 V and integrating by parts to mcb h0B ðtÞ ¼ hB ðhð1; tÞ  hB ðtÞÞ  hB hB ðtÞ obtain ð3:3Þ þ bsr ðvð1Þ2 Þ : Z1 Z1 Z1 v0 w dx ¼ rwj10  q 0 rwx dx þ 0 0 ¼ rð1Þwð1Þ þ Z1 a2 uxxx þ kvxxx þ ahx 0  wx dx þ The problem thus reduces to finding a solution satisfying (3.1)–(3.3) and the conditions fw dx Z1 0 fw dx
uðtÞ ¼ u0 þ Zt vðsÞds; vð0Þ ¼ v0 and hð0Þ ¼ h0 : 0 ð3:4Þ We now put the problem in abstract form by defining the following operators. First, let A1 : V ! V 0 be given by 227 Z1 hA1 v; wi  q hB4 hB ; gi  hB hB gð1Þ : vw dx þ ½mdvx ð1Þwð1Þ þ mdvð1Þwx ð1Þ 0 2
þ mvð1Þwð1Þ þ J þ md vx ð1Þwx ð1Þ ; ð3:5Þ 228 ð3:15Þ Then Eq. (3.2) may be written as hB1 h; gi0 þ hB2 h; gi þ hB3 v; gi þ hB4 hB ; gi ¼ 0 : Finally, we define the following operators, which are related to the temperature equation of the tip body: and let A2 : V ! V 0 be given by C1 : R ! R; by C1 r ¼ mcb r ; hA2 v; wi  cvð1Þwð1Þ þ dcvð1Þwx ð1Þ Z1  þ dc vx ð1Þwx ð1Þ þ kvxx wxx dx ; C2 : E ! R; by C2 h ¼ hð1Þ ; C3 : R ! R; ð3:6Þ 0 Z1 hA3 u; wi  a ð3:7Þ uxx wxx dx ; 0 hA4 h; wi  a hwxx dx þ hB hð1Þgð1Þ : ð3:8Þ Finally, we let F 2 L2 ð0; T; V 0 Þ be defined by hFðtÞ; wi  f ðtÞw dx  fB ðtÞwð1Þ ; ð3:9Þ 0 for w 2 V. Now we may write Eq. (3.1) as 0 hA1 v; wi þ hA2 v; wi þ hA3 u; wi þ hA4 h; wi ¼ hF; wi : Note also that hA1 v; wi ¼ hA1 w; vi; and hA1 v; vi > 0 for all v 6¼ 0 ; ð3:10Þ since hA1 v; vi  q Z1 2 2 v dx þ mðdvx ð1Þ þ vð1ÞÞ : ð3:11Þ 0 hB1 h; gi  ð3:12Þ hg dx ; let B2 : E ! E0 be given by Z1 hx gx dx þ f Z1 hg dx ; ð3:13Þ 0 0 hB3 v; gi  a 0 A1 0 0 0 BA B 3 þB @ 0 0 10 110 u 0 0 C C B 0 0 CB v C CC CB CC B1 0 A@ h AA hB 0 C1 10 1 0 1 0 u R 0 0 B C C B A2 A4 0 CB v C B F C C CB C ¼ B C ; ð3:20Þ B3 B2 B4 A@ h A @ 0 A 0 hB C4 C2 C3 together with the initial condition 10 1 0 u R 0 0 0 B 0 A1 0 0 CB v C CB C B CB Cð0Þ B @ 0 0 B1 0 A@ h A 0 0 C1 0 R 0 0 B 0 A1 0 B ¼B @ 0 0 B1 0 0 0 0 hB 10 u0 1 C B 0 C CB v0 C C : CB 0 A@ h0 A C1 ð3:21Þ hB0 ðByÞ0 þAy ¼ g; Byð0Þ ¼ By0 ; ð3:22Þ 2 where y 2 L ð0; T; V  V  E  RÞ and ðByÞ0 2 L2 ð0; T; V 0  V 0  E0  RÞ. Let Y be the Banach space  Y  y 2 L2 ð0; T; V  V  E  RÞ : ðByÞ0 2 L2 ð0; T; V 0  V 0  E0  RÞ ; equipped with the product norm let B3 : V ! E0 be given by Z1 ð3:19Þ The implicit system (3.20) and (3.21) is of the form 0 hB2 h; gi  j R BB 0 BB BB @@ 0 0 0 0 Proceeding as above, we define B1 : E ! E0 by Z1 C4 v ¼ bsr ðv ð1ÞÞ : by ð3:18Þ If we let R : V ! V 0 denote the Riesz map, then we can write the conditions (3.1)–(3.4) as the implicit first order system 00 0 Z1 ; 2 ðC1 hB Þ0 ¼ C2 h þ C3 hB þ C4 v : and let A4 : E ! V 0 be given by Z1 hB Þr Then (3.3) may be written as let A3 : V ! V 0 be given by 2 C4 : V ! R; ð3:17Þ C3 r ¼ ðhB þ by ð3:16Þ kykY  kykL2 ð0;T;VVERÞ vx gx dx  avx ð1Þgð1Þ ; 0 and let B4 : R ! E0 be given by ð3:14Þ þ ðByÞ0 L2 ð0;T;V 0 V 0 E0 RÞ : Then it is straightforward to verify that if k > 0 and b is sufficiently large, then the following conditions hold: 1. There exists a b1 > 0 such that kAykL2 ð0;T;V 0 V 0 E0 RÞ
b1 kykL2 ð0;T;VVERÞ : 2. There exists b2 > 0 such that if Bb y  Bebt y, then hbBb y þ AðyÞ; yi  b2 kyk2L2 ð0;T;VVERÞ :
3. y ! bBb ðtÞ þ A ðyÞ is a pseudomonotone map [17, 18] from Y to Y 0 . for every  > 0. Under the assumption that dc
4c , we can apply Cauchy’s inequality with  ¼ ðd=2Þ to conclude that dc Zt vð1; sÞvx ð1; sÞ ds
c 0 u; v 2 L2 ð0; T; V Þ; h 2 L2 ð0; T; EÞ; v0 2 L2 ð0; T; V 0 Þ ; h0 2 L2 ð0; T; E0 Þ ; v2 ð1; sÞ ds 0 Under these conditions the abstract existence theorems in [13] (see also [14]) give the following existence and uniqueness result. Theorem 3.1 Let k; r > 0, let f 2 L2 ð0; T; V 0 Þ, let fB 2 H 1 ð0; TÞ, u0 ; v0 2 V; h0 2 E and hB0 2 R. Then there exists a unique solution ðu; v; h; hB Þ of (3.1)–(3.4) satisfying Zt þ dc  Zt v2x ð1; sÞ ds : ð3:25Þ 0 229 Using this fact in (3.23) we obtain 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Z t Z1 Zt 1 2 2 2 hvxx dx dt þ k jjvjjV ds þ a jjuðtÞjjV þ a 2 0 0 hB ; h0B 2 L2 ð0; T; RÞ : Zt 0 Zt 1 1 When k ¼ 0, we can still obtain existence of a solution jvðsÞj2H ds þ v2 ð1; sÞ ds ; ð3:26Þ
Cþ 2 2 2 under the additional hypotheses that f 2 L ð0; T; H Þ and 0 0 dc
4c . The idea is to choose a sequence of positive where C depends only on the initial conditions and the viscosity coefficients kn ! 0 with associated solutions ðukn ; vkn ; hn ; hBn Þ and pass to the limit. For this purpose, we prescribed functions. Next, we put g ¼ h in (3.2), and, develop apriori estimates for the solution by putting w ¼ v after routine manipulations, we find that in (3.1) and integrating over ð0; tÞ. Using Cauchy’s Z t Z1 Zt inequality with , this yields 1 2 2 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Zt Zt þ c v2 ð1; sÞ ds þ dc vð1; sÞvx ð1; sÞ ds 0  þ dc Zt 0 v2x ð1; sÞ ds þk þa 0 khðtÞkH þ j 1 ds þ a2 jjuðtÞjj2V 2 þf khðsÞk2H ds þ hB Z1 jhð1; sÞj2 ds 0 1 ¼ kh0 j2H þ hB 2 Z1 hð1; sÞhB ðsÞ ds : 0 0 1 1 mcb jhB ðtÞj2 ¼ mcb jhB0 j2 þ hB 2 2 ð3:27Þ vð1; sÞvx ð1; sÞ ds
dcð1=2Þ Zt  ðhB þ hB Þ Zt hð1; sÞhB ðsÞ ds Zt h2B ðsÞ ds 0 ð3:23Þ þb Zt sr ðv2 ð1; sÞÞhB ðsÞ ds : ð3:28Þ 0 Adding (3.26)–(3.28) and using Cauchy’s inequality with  yields v2 ð1; sÞ ds 0 0 Zt 0 Now Cauchy’s inequality with  implies that þ dcð=2Þ 0 Finally, we let (3.3) act on hB to produce 0 dc 0 0 Zt hvxx dx dt 0 hvxx dx dt 1 1
qjvð0Þj2H þ mðvð1; 0Þ þ dvx ð1; 0ÞÞ2 2 2 Zt 1 1 1 2 2 jvðsÞj2H ds þ Jjvx ð1; 0Þj þ jjf jjL2 ð0;T;HÞ þ 2 2 2 0 Zt Zt 1 2 1 1 2 2 þ a jjuð0ÞjjV þ jfB ðsÞj ds þ v2 ð1; sÞ ds : 2 2 2 Zt khðsÞkE ds  a 0 jjvjj2V 0 0 Z t Z1 Zt 2 v2x ð1; sÞ ds ; 0 ð3:24Þ 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Zt 1 1 þ k jjvjj2V ds þ a2 jjuðtÞjj2V þ khðtÞk2H 2 2 0 þj Zt khðsÞk2E Zt ds þ f 0 0 þ Zt hB khðsÞk2H u0 ; v0 2 V; h0 2 E; hB0 2 R. Then there exists a unique 1 2 solution ðu; v; h; hB Þ of ð3:1Þ  ð3:3Þ satisfying ds þ mcb jhB ðtÞj 2 u 2 L1 ð0; T; V Þ; v 2 L1 ð0; T; H Þ; v0 2 L2 ð0; T; V 0 Þ ; h2B ðsÞ ds h 2 L1 ð0; T; EÞ; 0 1 2
Cþ 230 Zt jvðsÞj2H ds þ 1 2 0 þb Zt Zt hB 2 L1 ð0; T; RÞ; v2 ð1; sÞ ds 0 sr ðv2 ð1; sÞÞhB ðsÞ ds : ð3:29Þ 0 An application of Gronwall’s inequality now produces 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Zt 1 1 þ k jjvjj2V ds þ a2 jjuðtÞjj2V þ khðtÞk2H 2 2 þj khðsÞk2E ds þ f 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Zt 1 1 þ a2 jjuðtÞjj2V þ khðtÞk2H þ j khðsÞk2E ds 2 2 0 khðsÞk2H þf 1 ds þ mcb jhB ðtÞj2 2 Zt Zt khðsÞk2H
h2B ðsÞ ds 1 2 Zt 0 þb ð3:30Þ where C is a constant that depends only on the data and is independent of k. This estimate implies the inequality kjhv; wij
kjjvjjL2 ð0;T;VÞ jjwjjL2 ð0;T;VÞ Zt h2B ðsÞ ds 0 jvðsÞj2H ds þ 0
C ; 1 ds þ mcb jhB ðtÞj2 þ hB 2 0 0 0 þ hB Zt h0B 2 L2 ð0; T; RÞ : Proof. We need only show the uniqueness of the solution. Let ðu1 ; v1 ; h1 ; hB1 Þ and ðu2 ; v2 ; h2 ; hB2 Þ be two solutions corresponding to the same data. Letting each solution act on the differences v ¼ v1  v2 , u ¼ u1  u2 , h ¼ h1  h2 , hB ¼ hB1  h2 in (3.1)–(3.3), subtracting the corresponding pairs, adding the resulting three equations and using Cauchy’s inequality with , we obtain 0 Zt h0 2 L2 ð0; T; E0 Þ ; 1 2 Zt v2 ð1; sÞ ds 0 Zt ðsr ðv21 ð1; sÞÞ  sr ðv22 ð1; sÞÞÞhB ðsÞ ds : 0 ð3:38Þ Since sr is Lipschitz continuous with Lipschitz constant 1 and since v1 ð1; Þ; v2 ð1; Þ 2 L1 ð0; T Þ we have that jsr ðv21 ð1; sÞÞ  sr ðv22 ð1; sÞÞj
Cvð1; sÞ :
k1=2 CjjwjjL2 ð0;T;VÞ and so we have that, for sequence of positive numbers kn ! 0 with associated solutions ðun ; vn ; hn ; hBn Þ, Using this fact, Cauchy’s inequality with  and Gronwall’s inequality now produces 1 1 1 qjvðtÞj2H þ mðvð1; tÞ þ dvx ð1; tÞÞ2 þ Jjvx ð1; tÞj2 2 2 2 Now (3.30) implies that there exists ðu; v; h; hB Þ such that, Zt by passing to a subsequence, 1 1 þ a2 jjuðtÞjj2V þ khðtÞk2H þ j khðsÞk2E ds  1 2 2 un ! u weak in L ð0; T; V Þ ; ð3:31Þ kn vn ! 0 in L2 ð0; T; VÞ : vn ! v weak in L1 ð0; T; H Þ ;  vn ð1; Þ ! vð1; Þ weak vxn ð1; Þ ! vx ð1; Þ weak  1 in L ð0; T Þ ; in L1 ð0; T Þ : 1 0 ð3:32Þ ð3:33Þ þf ð3:34Þ Zt khðsÞk2H 1 ds þ mcb jhB ðtÞj2 þ hB 2 0
0 : hn ! h weak in L ð0; T; H Þ ; ð3:35Þ hn ! h weak in L2 ð0; T; EÞ ; ð3:36Þ From this uniqueness follows. 1 ð3:37Þ 4 hBn ! hB  weak in L ð0; T; RÞ : Zt h2B ðsÞ ds 0 ð3:39Þ These results allow us to pass to the limit in (3.1)–(3.3) and Numerical algorithm obtain the following existence and uniqueness result. In this section we propose a FEM algorithm for the original untruncated problem. We will present some Theorem 3.2 Let k ¼ 0, let r > 0, let dc
4c , let numerical simulations based on this algorithm in the next f 2 L2 ð0; T; H Þ, let fL 2 H 1 ð0; TÞ and let section. Let 0 ¼ x0 < x1 <    < xM ¼ 1 be a partition of ½0; 1 , Z1 with Ii ¼ ½xi1 ; xi and hi ¼ xi  xi1 for i ¼ 1; . . . ; M, and q vhk wh dx þ mvhk ð1Þwh ð1Þ n n let h ¼ max1
i
M hi denote the maximal discretization step 0 size. Following [11], we define the following finite element
hk hk h spaces approximating the variational spaces V and E: þ J þ md2 vhk nx ð1Þwx ð1Þ þ kcvn ð1Þw ð1Þ   h hk h þ mdvhk nx ð1Þw ð1Þ þ mdvn ð1Þwx ð1Þ V h ¼fvh 2 C1 ð½0; 1 Þ ; vhjI is cubic, 1
i
Mg ; i h h E ¼fn 2 Cð½0; 1 Þ ; nhjI h hk hk þ kdc vhk nx ð1Þwnx ð1Þ þ kdcvn ð1Þwx ð1Þ Z1
h hk a2 kvhk þk nxx þ kvnxx wxx dx is affine, 1
i
Mg : i Thus, V h and Eh are composed of C 1 piecewise cubic and continuous affine functions, respectively. To approximate the time derivatives, we introduce a uniform partition of the time interval ½0; T , denoted by 0 ¼ t0 < t1 <    < tN ¼ T, and let k be the uniform time step size. For a continuous function wðtÞ we use the notation wn ¼ wðtn Þ and, for a sequence fwn gNn¼0 , we let dwn ¼ ðwn  wn1 Þ=k denote divided differences. A fully discrete scheme for the problem is as follows: N Find the discrete velocity field vhk ¼ fvhk n gn¼0 , the hk hk N discrete temperature field h ¼ fhn gn¼0 and the discrete hk N temperature of the tip body hhk B ¼ fhBn gn¼0 such that: h hhk 0 ¼ h0 ; h vhk 0 ¼ v0 ; 0 ¼q q h hk hk þ dc vhk nx ð1Þwnx ð1Þ þ dcvn ð1Þwx ð1Þ þ þ 0 ¼ Z1
h wxx dx þ a fn wh dx  fB wh ð1Þ; Z1 h hhk n1 wxx dx 0 h  hk h þ ðJ þ md2 Þvhk n1x ð1Þwx ð1Þ þ dc vn1x ð1Þwx ð1Þ hk h h þ mdvhk n1x ð1Þw ð1Þ þ mdvn1 ð1Þwx ð1Þ Z1 þ k fn wh dx  kf B wh ð1Þ; 8wh 2 V h : Z1 h dhhk n g dx þ j Z1 0 0 þf Z1 h hhk n g 8wh 2 V h ; ð4:1Þ ¼ ¼ hB ðhhk n1 ð1Þ  þ bvhk n1 ð1Þ ; Z1 hhk Bn1 Þ h hhk nx gx  hB hhk Bn1 Z1 h vhk nx gx dx 0 hk hk h h hhk n g dx þ hB ðhn ð1Þ  hBn Þg ð1Þ dx þ kj Z1 h hhk nx gx 0 hk þ khB hn ð1Þgh ð1Þ Z1 h hhk n1 g 0 ð4:2Þ dx þ a h h hk uhk n1xx w dx þ mvn1 ð1Þw ð1Þ 0 0 0 mcb dhhk Bn Z1 The equations can be viewed as a linear system in the unknowns vhk n with the right hand side as given. The matrix of coefficients in this system is not symmetric and so in creating the numerical examples of the next section the LU factorization of this matrix was employed (see [19]). In the same way, Eq. (4.3) can be written in the following recursive form:
hk hk h þ J þ md2 dvhk nx ð1Þwx ð1Þ þ cvn ð1Þw ð1Þ   h h hk þ mddvhk nx ð1Þw ð1Þ þ mddvn ð1Þwx ð1Þ kvhk nxx h hhk n1 wxx dx 0 0 a2 uhk nxx dx  ak Z1 0  ka2 h hhk B0 ¼ hB0 ; h h hk dvhk n w dx þ mdvn ð1Þw ð1Þ Z1 h vhk n1 w 0 and, for n ¼ 1; . . . ; N, Z1 Z1 dx  ak dx þ kf Z1 h hhk n g dx 0 Z1 h vhk nx gx dx 0 h hk h þ khB hhk Bn g ð1Þ þ kavnx ð1Þg ð1Þ; 8gh 2 Eh : Again, this can be viewed as a linear system with unknowns hhk n and, now since the matrix of coefficients is symmetric, the Cholesky method can be used to generate the examples of the next section. 5 Numerical simulations In this section we present the results obtained for some ¼ 0; 8gh 2 Eh ; ð4:3Þ numerical simulations which show the perfomance of the  h numerical algorithm given in the previous section. In all where uhk n denotes the discrete displacement field, and u0 , h h h v0 , h0 and hB0 are appropriate approximations of the initial cases, we use the discretization parameters k ¼ h ¼ 0:001 and study the behavior during a two unit time interval conditions u0 , v0 , h0 and hB0 . hk hk (T ¼ 2). We assume that there are no external forces Taking into account that uhk , we remark þ kv ¼ u n n1 n that Eq. (4.1) can be rewritten as the following recursion present; that is, f ¼ fB ¼ 0. We also suppose that there is zero initial velocity and temperature, and the initial relation: 0 h avhk nx ð1Þg ð1Þ 231 temperature of the tip body is also zero (v0 ¼ h0 ¼ 0 ¼ hB0 ). Finally, we assume that the beam has an initial upward displacement u0 ðxÞ ¼ 0:25x2 . Throughout these simulations the following parameters have these fixed values: c ¼ c ¼ 0:0001; d ¼ 0:05; cb ¼ 1; j ¼ 1; b ¼ 0:01; J ¼ 1; f ¼ 0:01; hB ¼ 1; hB ¼ 1 : The remaining parameters vary from example to example. 5.1 First example: a rigid beam In this first example we consider a fairly rigid beam where viscosity plays a minor role; that is, we take a ¼ 0:01; 232 Fig. 2. The tip body and the end of the beam k ¼ 0:01; a2 ¼ 10000; q ¼ 30 : Here we will consider three different values for the mass m of the tip body: 0:1, 1 and 5. In Fig. 3 the evolution of the displacement field at the contact point for a tip body of mass 0:1 is plotted on the left side, while on the right side the corresponding Fig. 3. Evolution of the displacement field at the contact point x ¼ 1 for m ¼ 0:1; 5 in Example 1 Fig. 4. Evolution of the temperature at the contact point x ¼ 1 and the temperature of the tip body for m ¼ 0:1; 1; 5 in Example 1 Fig. 5. Evolution of the displacement field at the contact point x ¼ 1 for m ¼ 0:1; 5 in Example 2 temperature of the tip body itself is seen to be quite distinct in each case, and the temperature of the lighter mass oscillates considerably. 5.2 Second example: a more viscous beam In this second example we alter parameters to make the beam more flexible and more subject to viscosity; that is, we take a ¼ 0:01; k ¼ 10; a2 ¼ 100; q¼1 : In Fig. 5, the left side shows the evolution in time of the displacement of the contact node x ¼ 1 for m ¼ 0:1 while the right side shows this evolution for m ¼ 5. We see that Fig. 6. Evolution of the temperature at x ¼ 1 for m ¼ 0:1, 1, 5 in the displacement of the beam may be viewed as a superExample 2 position of a higher frequency oscillation attributable to the beam together with a lower frequency oscillation caused by the mass. By contrast, with a heavier mass the fast oscillations are completely damped. The evolution in time of the temperature at the contact node is shown in Fig. 6 for the values m ¼ 0:1, 1, 5. The oscillations seem to be as one would expect them, larger amplitude for the smaller mass. Fig. 7. Evolution of the temperature at x ¼ 1 for a ¼ 0:1, 0:05, 0:01 in Example 3 5.3 Third example: thermal expansion effects In this final example, we explore the dependence of the temperature field on the thermal expansion coefficient a for a body with mass m ¼ 1. We will consider three different values for a and leave the other parameters with the same values as in the second example. In Fig. 7 the evolution in time of the temperature field at the contact point x ¼ 1 is shown for the values a ¼ 0:1, 0:05 and 0:01. It is seen that the pattern for the larger value of a is very different from the other two, and may warrant an additional investigation. Finally, the evolution of the temperature of the tip body is plotted in Fig. 8 for these values of a. Here, too, the behaviour of the larger a indicates that additional investigation may be of interest. 6 Conclusions A model for the vibrations of a thermomechanical system consisting of a beam made of a viscoelastic material, to which a tip body is attached, was presented and shown to possess a unique weak solution. The tip body was assumed to contain a granular material that provides damping to slow down any unwanted vibrations of the system. The Fig. 8. Evolution of the temperature of the tip body for a ¼ 0:1, damping was assumed to arise from the internal friction of 0:05, 0:01 in Example 3 the granular material, resulting in temperature increase and heat lost to the environment. displacement for a mass of 5 is drawn. It is seen that the A finite element algorithm for the numerical approxrelaxation of the vibrations of the heavier mass takes imations of the problem was proposed and implemented, longer. and the results of some numerical simulations were preIn Fig. 4 the left side shows the evolution of the tem- sented. These show typical types of behaviour of the sysperature field at the contact point for the three different tem, and also illustrate the performance of the algorithm. masses. On the right side, the evolution in time of the The latter appears to be well behaved. temperature of the tip body for these values of m is In the simulations, the evolution of the displacements of plotted. the tip were investigated for three different masses of the While the temperature at the contact point follows a tip body. For one choice of mass, the dependence of the similar oscillatory pattern in the three cases, the temperature field on the coefficient of thermal expansion 233 234 was presented as well. It seems that the behaviour of the field changes as the coefficient increases, and consequently there is some interest in further investigating this issue. The convergence of the algorithm and the development of its error estimates remain open problems. The system parameters may be obtained by using the parameter identification procedure illustrated in [4, 6, 7]. Finally, the main concern of the design engineer and the operators of such a system is in the control of the system. It might be of interest to apply the ideas of [12] to establish the boundary control of the system. 8. Boley BA, Weiner JH (1960) Theory of Thermal Stresses, Wiley, New York 9. Chang TP (1993) Forced vibration of a mass-loaded beam with a heavy tip body. J. Sound Vibration 164(3): 471–484 10. Grobbelaar-Van Dalsen M (1994) On the solvability of the boundary-value problem for the elastic beam with attached load Math. Models Meth. Appl. Sci. 4(1): 89–105 11. 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