Analysis of a viscoelastic spring-mass model

Analysis of a viscoelastic spring-mass model  M. Pellicer ∗,J.Sol`a-Morales Universitat Polit`ecnica de Catalunya, Departament de Matem`atica Aplicada 1, Avda. Diagonal 647, 08028 Barcelona, Spain Abstract In this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring-mass-damper ODE. Our purpose is to compare analytically these two approaches to the same physical system. We take a functional analysis point of view based on semigroup theory, spectral perturbation analysis and dominant eigenvalues. Key words: strongly damped wave equation, dynamical boundary conditions, asymptotic behavior, dominant eigenvalues. 1 Introduction Consider the motion of a system consisting of a spring of recovery constant kthat is fixed at one end and attached to a rigid mass mat the other one. Suppose also that the mass movement is linearly damped by a friction force of coefficient d. Typically, the dynamics of this system is modelled by the second order differential equation: mu (t)=−ku(t)−du (t)(1) where u(t) is the position of the mass at time t. This model considers the spring-mass-damper system as a problem with only two degrees of freedom. A more detailed point of view would lead us to treat the spring as a continuous medium where the deformation depends on the point, taking into account Partially supported by the projects PB98-0932-C02-01 of the MEC and BFM200204613-C03-01 of the MCT, Spain. ∗Corresponding author. Email addresses: [email protected] (M. Pellicer), [email protected] ( J. Sol`a-Morales). Preprint submitted to Elsevier Science 8 September 2003 possible internal deformation differences, and also to consider its internal viscosity, apart from the external damper dissipation. This gives us a partial differential equation model in which the action of the external damper onto the mass movement appears only in the boundary conditions. An example of such a system would be the car shock absorbers, where the damper acts onto the viscoelastic spring through the wheel of the car only. Our objective is to discuss this alternative partial differential equation model and analyze its solutions, comparing their asymptotic behavior with the ones of the ordinary differential equation model. The partial differential equation model in the appropriate variables system, justified in detail in section 2, is the following:              utt −uxx −αu txx =0,0<x<1,t>0 u(0,t)=0 utt(1,t)=−ε[ux(1,t)+ αu tx(1,t)+ ru t(1,t)] (2) where α>0 is a parameter related with the spring internal dissipation coeficient, r>0 comes from the damper viscosity coeficient and ε≥0isa parameter depending on the mass mand on the density and the length of the spring.Wedenotebyu(x, t) the displacement at time tof the xparticle of the spring. That means that x0+u(x0,t 0) is the position at time t0of the particle of position x0at equilibrium. Equation (2) is a wave equation with strong damping (also called Kelvin-Voigt damping) and dynamical boundary conditions. To our knowledge, this model with r=0andε=α= 1 was first proposed by Grobbelaar-van Dalsen in [6], who showed that it defines an analytic semigroup in an appropriate functional space. The functional framework for (2), discussed in section 3, is based on the spaces and norms she worked with and also on the pioneering work of P. Massat in [10] for the abstract equation utt +αAu t+ Au =f(t, u, ut), where Ais a sectorial operator and α>0. This kind of equation but with Neumann boundary conditions also has appeared in the work of N. C´onsul and J. Sol`a-Morales (see [2]). Models of wave equations but with weak (or Maxwell) damping instead of the strong one have been much more studied, even with the dynamical boundary conditions by authors like A. Freiria Neves and O. Lopes (see for example [9]). These type of boundary conditions have been considered also from a control theory point of view by authors like E. Zuazua or C.M. Castro in different works (see, for instance, [1]). To compare equation (2) with the classical ODE (1) our main tool will be the dominant eigenvalues. This is a very simple and well known idea: when 2 the solutions of a differential equation like (2) are of the form u(x, t)= aneλntun(x), their asymptotic behavior is dominated by the terms having the greatest Re(λn). This can simplify a model with infinitely many degrees of freedom to a finite dimensional one. Of course, this situation is not completely simple when one deals with nonselfadjoint linear operators that also have essential spectrum, apart from the eigenvalues, which is our case. But still then, the same ideas can be used. Let us be more precise: consider an abstract evolution equation d dtx(t)=Bx(t)(3) where Bis a linear operator in a Banach space Xwith spectrum σ(B). Supose that σ(B)haskisolated eigenvalues with finite algebraic multiplicities, λ1,...,λ k, and that there exist ω1,ω2∈Rsuch that Re λ < ω2<ω 1<Reλ i∀i=1,...,k , ∀λ∈σ(B)\{λ1,...,λ k} Then, we say that the operator Badmits {λ1,...,λ k}as a finite subset of dominant eigenvalues. In this situation we have a natural decomposition of the spectrum in σ1={λ1,···,λ k}and σ2=σ(B)\{λ1,···,λ k}, a decomposition of the total space X=X1⊕X 2,withdim(X1)<∞, and of the operator, B1=B|X1and B2=B|X2(for example see [7]). Then, the following result can be easily deduced from the general theory of analytic semigroups (see [7,11,4]). We point out that this result needs not to hold for general C0semigroups: this is why the analyticity proved in [6] is important for the application to (2). Theorem 1 Let X=X1⊕X2and B=(B1,B 2)as above, and supose that B is the infinitesimal generator of an analytic semigroup. Let x(t)=x1(t)+x2(t), x1(t)∈X 1and x2(t)∈X 2, be the solution of (3) with initial conditions x(0) = x1(0) + x2(0) and suppose also that x1(0) =0.Then lim t→∞ x(t)−x1(t) x(t)=0 Because of this result it is reasonable to say that the solutions of the finite dimensional ordinary differential equation x 1=B1x1are a good approximation for the solutions of the infinite dimensional evolution equation x=Bx for large time (see [12] for details). The main result of this paper is obtained in section 4 in which we prove that for small values of εthere are two complex conjugate dominant eigenvalues for (2), so we show that the PDE has an ODE of the type of (1) as a limit. The dependence of these dominant eigenvalues with respect to εis also calculated up to some reasonable approximation as well as the coefficients of the limit 3 ODE. The basic tools in the proof of this main result are the characteristic equation for the eigenvalues (equation (13)), a control of the essential spectrum of the infinitesimal generator of the semigroup and the notion of generalized convergence of closed operators. For these last two tools we need a precise functional formulation of the problem and, in particular, a characterization of the domains of the operators involved. This is done in section 3. Section 2 is devoted to modelling. Our personal reason to include it is to show how natural is to consider (2) as the first generalization of the classical spring-mass-damper ODE model, as well as to show that through this natural generalization one obtains a strong damping in the wave equation, and not a weak damping as one could perhaps suspect. We also want to say that our main result admits a nonlinear version, if one deals with a nonlinear perturbation of (2). In that case one could prove the existence of a globally attracting invariant manifold and a nonlinear limit ODE on it. This will be studied in a subsequent paper. It is also left for another occasion the study of the limit α→0andα= 0 (for fixed ε>0). In these cases the situation is very different: for instance, when small values of α>0 are considered the number of dominant eigenvalues can be up to four or more, all them with the same real part, and even there is no finite number of dominant eigenvalues when α= 0. Some results in these directions can be seen in [12] or in forthcoming publications. 2 Modelling. The mechanical behavior of a viscoelastic spring of length Lcan be modelled by the well known strongly damped wave equation, but then the action of the external damper onto the spring through the mass at the x=Lend is going to appear as a boundary condition. This boundary condition is slightly different from that considered in [6] in which the external damper does not appear. Let us derive it in detail from the rheological point of view. The rheological approach consists of discretising viscoelastic materials into different combinations of elementary units, which are springs and dashpots. As a spring models the material elastic behavior, its constitutive equation is given by Hooke’s law σe=Eε e,whereσeis the elastic stress, Ethe Young modulus and εethe elastic strain. And as a dashpot models the viscosity, its constitutive equation is σv=E1˙εv,whereσv,˙εvstand for viscous stress and strain rate and E1is the viscosity coefficient. These basic elements can be coupled either in series or in parallel (see [3] for more details). A parallelcoupled spring and dashpot system is known as the Kelvin-Voigt model, whose constitutive equation is σ=E1˙ε+Eε,whereσand εare now the total stress 4 m1··· E, E1 mim ··· ηd Fig. 1. Rheological model for the spring-mass-damper system. and strain. Let us think now our material as a sequence of increasingly many seriescoupled Kelvin-Voigt systems (that is, a continuous Kelvin-Voigt model). And the last system is parallel-coupled with a single external damper (see figure 1) across a rigid mass m. At the other end, the system is kept fixed. Following [3], the equation of motion is given by the balance of forces between the i-th and (i+ 1)-th components: mi d2ui(t) dt2=σi+1 −σi(4) where ui(t) is the displacement of the i-th mass. Replacing the single KelvinVoigt equation into (4), writing mi=ρih(being hthe length of each component and ρithe local density) and writing also the strain in terms of displacement, equation (4) becomes: ρi d2ui(t) dt2=E1 d dt ui+1 −2ui+ui−1 h2+Eui+1 −2ui+ui−1 h2(5) Taking the limit as h→0 in both sides of the equation, we obtain the continuous system, whose equation is: ρ(x)utt(x, t)=E1uxxt(x, t)+Eu xx(x, t).(6) Actually, in our model we will consider a constant density ρ. Concerning the boundary conditions, since the end x= 0 is fixed we have: u(0,t)=0 (7) For the boundary condition at x=L, only the action of the last Kelvin-Voigt component and the damper have to be considered (see again figure 1). As these two components are parallel-coupled, we have σ=(Eε n+E1˙εn)+ ηd˙εd, 5 where the n-subindex stands for the last Kelvin-Voigt component and the dsubindex means that of the damper. Following the same idea as before, this equation comes into: mu tt|x=L=−Eun−un−1 h+E1 d dt un−un−1 h+ηd Lutx=L (8) Taking limits as h→0 we obtain the dynamical boundary condition at x=L: mu tt(L, t)=−(Eux+qu t+E1uxt)(L, t)(9) where q=ηd/L. We now apply a change of variables to (6), (7) and (9) in order to obtain the non-dimensional model (2). The change of variables is: x←→ x L,t←→ tE ρ L So now the length of the system is 1. We also give a change of functions: u←→ u L And the non-dimensional parameter change is: α=E1 √EρL,ε=ρL m,r=q √Eρ Our model (2) now depends on the three nonnegative non-dimensional parameters α,rand ε. To get some intuition about these parameters we observe that for fixed E,ρand L,wehavethatαcomes from the internal spring viscosity E1,rcomes from the external damper coefficient qand 1/ε is proportional to m, the rigid mass at the end. 3 Functional setting. In this section, let us think in the model (2) as the following Cauchy problem: 6                                          utt −uxx −αu txx =0,0<x<1,t>0 u(0,t)=0,t>0 utt(1,t)+εu x(1,t)+εαu tx(1,t)+εru t(1,t)=0,t>0 u(x, 0) = u0(x),0<x<1 ut(x, 0) = v0(x),0<x<1 u(1,0) = η(= u0(1)) ut(1,0) = µ(= v0(1)) (10) with u=u(x, t), t∈[0,∞)andα>0, ε, r > 0. We now give a functional framework that will be appropriate for obtaining existence and uniqueness of solutions for (10), and also for discussing the convergence of the spectra of some of the involved operators. This convergence is obtained by using the notion of generalized convergence of operators (see section 4 below). This has been the motivation of our careful choice of some of the function spaces here. We want to write (10) as an evolution equation, in a similar way as it is done by Grobbelaar in [6] or by Massat in [10]. Let us consider the following spaces: X2={(u, γ)∈H2(0,1) ×C,u(1) = γ, u(0) = 0} as a subspace of H2(0,1) ×C; X1={(u, γ)∈H1(0,1) ×C,u(1) = γ, u(0) = 0} as a subspace of H1(0,1) ×C;and X0={(u, γ)∈L2(0,1) ×C}=L2(0,1) ×C In these subspaces, a natural inner product is defined: (u, u(1)),(v,v(1))X1=1 0 uxvxdx and (u, γ),(v,β)X0=1 0 uvdx +1 εγβ It can be easily proved that this products are equivalent to those defined in the Sobolev space in which are included (see [12]). This ε-dependence of the inner product on X0will be specially useful in some of the proofs below. 7 We define (Aα,D(Aα)) as follows. The domain D(Aα)is: D(Aα)=        (u, u(1)) (v,v(1))  ∈X1×X1,(u+αv)∈H2(0,1)    ⊂H where H=X1×X0is a Hilbert space with the inner product    (u1,u 1(1)) (u0,γ 0))   ,   (v1,v 1(1)) (v0,β 0))    H = =(u1,u 1(1)),(v1,v 1(1))X1 +(u0,γ 0),(v0,β 0)X0 If V=   (u, u(1)) (v,v(1))  ∈D(Aα) then we define the operator as: AαV=   (v,v(1)) ((u+αv)xx,−ε(u+αv)x(1) −εrv(1) )   Then, for V=   (u, u(1)) (ut,u t(1))  and V(0) = F0=   (u0(x),η) (v0(x),µ)  , the equation (10) can be written as the evolution equation:      d dt V=AαV, t ∈(0,∞) V(0) = F0 (11) The existence and uniqueness of the solutions for (11), in terms of the generated semigroup, follows the proof given by Grobbelaar in [6], who actually is based on Massat’s proof (see [10]). This result is summarized in the following theorem: Theorem 2 The operator (Aα,D(Aα)) with α>0is the infinitesimal generator of an analytic semigroup in H. Idea of the proof: The idea is simply to decompose −Aαinto −AαV=BV +KV =   (−v,−v(1)) ((−u−αv)xx,ε(u+αv)x(1) )  +   (0,0) (0,εrv(1))   8 Applying a result of [10], we see that Bis the infinitesimal generator of an analytic semigroup. As Aαis a bounded perturbation of B,itisalsothe infinitesimal generator of an analytic semigroup in the same space as B,which turns to be H(the details of this proof are given in [12]). 2 4 The case of εnear 0. The case of a small positive εand a fixed α>0 is a case of physical interest as it models a spring-mass system when the mass at the end is taken large. The linear operator is denoted now as Aα(ε), and for ε∼0 is going to be thought as a perturbation of the operator for the limit case ε= 0 (or an infinitely large mass), which is denoted as Aα(0). Our main result is the following. Theorem 3 For fixed α, r > 0there exists a certain ε0>0(depending on αand r)forwhich(Aα(ε),D(Aα)) when ε<ε 0admits λ+ 0(ε),λ − 0(ε)as a subset of two simple dominant eigenvalues, where: λ+ 0(ε)=i√ε−α+r 2ε−i4+3(α+r)2 24 (√ε)3+α+r 6ε2+ +i176 + 360(α+r)2−45(α+r)4 5760 (√ε)5−2α 45 +r 30ε3+O((√ε)7) and λ− 0(ε)=λ+ 0(ε). These two eigenvalues are perturbations of the double (not semi-simple) eigenvalue λ0(0) = 0 of Aα(0). Coming back to the dimensional variables, we obtain from theorem 3 the following result: Corollary 4 The solutions of the partial differential equation problem (2) when mis large can be approximated when t→∞by the solutions of the limit ODE: mw(t)+k1w(t)+k0w(t) = 0 (12) where k1=E1 L+q−1 3E1 L+qρL m+4E1 45 L+q 15ρL m2 +... k0=E L1−1 3ρL m+4 45 ρL m2 +q2 45 Eρ −16 945ρL m3 +... The solution w(t)can be interpreted as an approximation of u(L, t). Proof of corollary Theorem 3 gives an approximation for the dominant eigenvalues λ+ 0(ε)andλ− 0(ε). Then the corresponding second order ODE can 9