A transmission problem for the Timoshenko system

A transmission problem for the Timoshenko system

C.A. Raposo^I; W.D. Bastos^II; M.L. Santos^III

^IDepartment of Mathematics, UFSJ, Praça Frei Orlando, 170 – 36307-352 São João del-Rei, MG

^IIDepartment of Mathematics, UNESP, Rua Cristovão Colombo, 2265 – 15054-000 São José do Rio Preto, SP

^IIIDepartment of Mathematics, UFPA, Rua do Una, 156 - 66113-200 Pará, PA E-mails: [e-mail] / [e-mail] / [e-mail]

ABSTRACT

In this work we study a transmission problem for the model of beams developed by S.P. Timoshenko [10]. We consider the case of mixed material, that is, a part of the beam has friction and the other is purely elastic. We show that for this type of material, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution, no matter how small is its size. We use the method of energy to prove exponential decay for the solution.

Mathematical subject classification: 35J55, 35J77, 93C20.

Key words: transmission, Timoshenko, beams, exponential decay, frictional damping.

1 Introduction

The transverse vibration of a beam is mathematically described by a system of two coupled differential equations given by

Here, L is the length of the beam in its equilibrium position, t is the time variable and x is the space coordinate along the beam. The function u = u(x,t) is the transverse displacement of the beam and y = y(x,t) is the rotation angle of a filament of the beam. The coefficients r, I _r, E, I and K are the mass per unit length, the polar moment of inertia of a cross section, Young's modulus of elasticity, the moment of inertia of a cross section and the shear modulus respectively. We denote r₁ = r, r₂ = I _r, b = EI, k = K and we obtain directly from (1.1) the following system

The mathematical model describing the vibrations of beam with fixed extremities is formed by the system (1.2), boundary conditions

u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0,

and initial data

u(·, 0) = f₀, u_t (·, 0) = f₁, y(·, 0) = y0, y_t (·, 0) = y₁, in (0, L).

If friction is taken into account, the system (1.2) becomes

where a and b are positive constants (we assume a = b = 1). The terms a u_t and b y _t represent the attrition acting in the vertical vibrations and in the angle of rotation of the filaments of the beam, respectively.

Dissipative properties associated to the system (1.3) have been studied by several authors by considering dissipative mechanism of frictional or viscoelastic type. The frictional dissipation, obtained by introduction of a frictional mechanism acting on the entire domain or on the boundary, was studied in [7, 8, 9]. The viscoelastic dissipation, given by a memory effect, was considered in [2] and [6].

An interesting problem comes out when the dissipation acts only on a part of the domain. In the present paper we consider a frictional mechanism acting only on the part of the domain given by 0 < x < L₀ with 0 < L₀ < L. We prove that for every L₀ the energy of the system decays exponentially to zero as time goes to infinity. In other words, our result states that dissipative properties of the system are transferred to the whole beam and stabilizes the system. The main result of this paper is Theorem 2 and its corollary, both in section 5.

The mathematical model which deals with this situation is called a transmission problem. From the mathematical point of view a transmission problem consist of an initial and boundary value problem for a hyperbolic equation for which the corresponding elliptic operator has discontinuous coefficients. Hence, we can not expect to have regular solutions in the role domain. In the next section we establish the transmission problem and define appropriately the notion of solution considered. We use H^m and L^p to denote the usual Sobolev and Lebesgue spaces [1].

2 The transmission problem

In this section we describe precisely the transmission problem treated in the paper and establish existence and regularity of solution. We begin by introducing the notation

Using the notation above, model (1.3) can be written in the following form:

with boundary conditions,

u(0, t) = v(L, t) = y(0, t) = f(L, t) = 0, t > 0,

transmission conditions,

and initial data

We define the notion of weak solution to the system (2.1)-(2.6) as follows:

Definition 1.Let, ^m and²be the spaces defined by

We say that (u,v,y,f) is a weak solution to the problem (2.1)-(2.6) if for every (,w) Î (0,T;²Ç) we have:

and

The transmission problem for a single hyperbolic equation was studied by Dautray and Lions [3], who proved the existence and regularity of solutions for the linear problem. The existence and regularity of solutions to the transmission problem for the Timoshenko system is given in the following theorem:

Theorem 1. If (u₀,v₀),(y₀,f₀) Î and (u₁,v₁),(y₁,f₁) Î ², then there exists a unique weak solution (u,v,y,f) to the system (2.1)-(2.6) satisfying:

(u, v), (y, f) Î C (0,¥;) Ç C¹(0,¥;²).

Moreover, if (u₀,v₀),(y₀,f₀) Î ²Ç and (u₁,v₁),(y₁,f₁) Î , then the weak solution is a strong solution and satisfies

(u, v), (y, f) Î C(0,¥;²Ç ) Ç C¹(0,¥;) Ç C²(0,¥;²).

Proof. For the proof we proceed in a quite similar manner as in [3].

The total energy associated to the system is defined by

Next we prove that the total energy associated to the system is decreasing for every t > 0.

Lemma 1. Let (u,v,y,f) be the strong solution to the system (2.1)-(2.6), then

Proof. Multiplying (2.1) by u_t and integrating by parts over the interval (0,L₀), we get

Now, multiplying (2.2) by y_t and integrating by parts over (0,L₀) we obtain

Multiplying (2.3) by v_t and integrating by parts on (L₀,L), we get

Multiplying (2.4) by f_t and integrating by parts on (L₀,L) leads to

Now observe that

and

Summing up (2.7), (2.8), (2.9) and (2.10), and using (2.11) and (2.12) together with the hypothesis of transmission we obtain

3 Technical lemmas

Now we develop a series of technical results in order to facilitate the proof of the main result of the paper. We begin by constructing a functional (t), equivalent to the energy functional, which satisfies (t) < C (0), C < 1. In order to do so, we use some multiplier techniques (usually associated to control problems) and the following restrictions on the boundary conditions for the elastic part of the beam:

Lemma 2. Let us define

Then

Proof. Multiply (2.1) by x u_x and integrate by parts over (0,L₀) to get

Multiply (2.2) by x y_x and integrate by parts over (0,L₀) to obtain

Multiplying (2.3) by x v_x and integrating by parts over (L₀,L) leads to

Now multiply (2.4) by x f_x and integrate over (0,L₀) to obtain

Summing up (3.3), (3.4), (3.5), (3.6), and making use of the hypothesis on the transmission, the punctual terms are canceled and we get

Now using (3.1), (3.2) and the Young's inequality [4] we obtain

Now, if we define

and choose N₁ > we conclude

It is worth noticing that the estimate above is important in two aspects. First, it recovers a part of energy with minus sign. Second, it will play a fundamental role in the next two lemmas controlling punctual terms which will come up in the search for other negative terms of the energy.

Lemma 3. Define

Then

Proof. Multiply (2.1) by x (u_x+y) and integrate by parts over (0,L₀) to obtain

Using Young's inequality, we get

Now, defining

we obtain

If we choose N₂ > we conclude that there exists > 0 such that

and then

Lemma 4. Let ₃ be defined as

Then

Proof. Multiply (2.3) by (x-L)(v_x+f) and integrate by parts over (L₀,L) to get

Using Young's inequality we obtain

Now we set

and verify that

It follows then

Observe that in the attempt to recover the total energy of the system with negative sign we introduced the lemmas 3 and 4. Now we need to control them. It will be achieved with the aid of the next section.

4 Compactness

This section is dedicated to discuss the argument of compactness employed in the proof of the main result of the paper. First we introduce a notation; the symbol is used to denote convergence in the norm of the Sobolev space L^¥ as in [5]. For sake of completeness, we state the following result due to J.U. Kim.

Lemma 5. Let (u^k) be a sequence of functions satisfying

u^ku in L^¥(0, T, H^b(0, L)),

u_t in L²(0, T, H^a(0, L)),

as k ® ¥, with a < b. Then

u^k ® u in C([0, T ], H ^r (0, L)),

for some r < b.

Proof. See [5].

Lemma 6 (Lemma of compactness). If we define

then, for every h > 0 there exists a constant C _h > 0 independent of the initial data, such that

for every strong solution (u,v,y,f) to the system (2.1)-(2.6) and sufficiently large T.

Proof. We use a contradiction argument. Define

Suppose that there exists a sequence of initial data

and a positive constant h₀ > 0 such that the corresponding solution (uⁿ,y ⁿ), (v ⁿ,f ⁿ) of the problem

with boundary conditions,

u ⁿ(0, t) = v ⁿ(L, t) = y ⁿ(0, t) = f ⁿ(L, t) = 0, t > 0,

transmission conditions,

and initial data

satisfies

and the following inequality

Then the integral

Eⁿ(t)dt is bounded for every n Î N,

and also,

and

Now we observe that Eⁿ(t) > 0 and that Eⁿ(t)dt is bounded. Hence Eⁿ(t) is bounded and we can take a subsequence of (uⁿ,y ⁿ),(vⁿ,f ⁿ) (for which we use the same notations) such that

Applying the lemma 5 we conclude that for r < 1

It follows from (4.1) that

We observe that the convergences (4.2) and (4.3) result in

Now, applying Poincare's inequality we obtain

This estimates implies

B(L₀, t) = 0,

which is a contradiction to (4.4). This completes the proof of the lemma.

We are now ready to prove the main result of this paper, that is, the exponential decay of the energy associated to the transmission problem for the Timoshenko System with frictional dissipation. This is the content of the next section.

5 Exponential decay

Theorem 2. Let (u,y,v,f) be a strong solution to the transmission problem for the Timoshenko System defined by (2.1)-(2.6). Then there exist positive constants C and w such that

E(t) <C E(0)e-^{w t} .

Proof. We start defining

(t) = N₃

₁(t) + ₂(t) + ₃(t).

It follows from lemmas 2, 3, and 4

Now, integrating this inequality over (0,T) and using the Lemma of Compactness we obtain

If we choose and fix h < C₀ and N₃ such that N₃C₁ > C _h, we get

Since E(t) decreases, we have

Using (5.2) in (5.1) we obtain

(T) - (0) < - T C₂(T).

Now observe that for sufficiently large N we have

from what follows that

(T) - (0) < - T (T),

or else

Note that a does not depend on the initial data, and hence, by using the semigroup property we have

For t > 0, there exists a natural n and a real r, 0 < r < T such that t = nT+r. This is equivalent to

Now, using the inequalities (5.3) and (5.4) n times we obtain

Observing once more that E(t) decreases we have

where w = -ln().

Finally, using (5.3) we obtain

E(t) < 4a^-1E(0) e^{-w t} ,

and conclude the proof.

We can extend the previous theorem to the weak solutions by using simple density argument and the laws of semi-continuity for the energy functional. In this direction we have the following corollary.

Corollary 1. Under the hypothesis of the previous theorem, there exist positive constants C and w, such that

E(t) <C E(0)e^{-w t} ,

for every weak solution (u,y,v,f) of the system (2.1)-(2.6).

6 Concluding remarks

During the past several decades, many authors have studied the same physical phenomenon for the Timoshenko system formulated into different mathematical models. Our approach to this problem is important not only from mathematical but mainly from the physical point of view with applications in Mechanics, amongst other branches of science. The system studied here is a model for vibrating beams subjected to two frictional mechanisms. More precisely, we proved that the presence of two frictional damping acting in a natural way on a small part of the beam, is enough to stabilize the whole beam. Moreover, it stabilizes quickly (at exponential rate). To the best of our knowledge, our result is the first in this direction. In this sense, this work generalizes the results previously obtained for Timoshenko's system where attrition acting in the whole beam was considered.

Received: 23/I/07. Accepted: 09/II/07.

#668/06.

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Publication Dates

• Publication in this collection
  24 July 2007
• Date of issue
  2007

History

• Accepted
  09 Feb 2007
• Received
  23 Jan 2007