Search

Two variations of a basic model for a cochlea are described which consist of a basilar membrane coupled with a linear potential fluid. The basilar membrane is modeled as an array of oscillators which may or may not include longitudinal elasticity. The fluid is assumed to be a linear potential fluid described by Laplace's equation in a domain that surrounds the basilar membrane. The problem of controllability of the system is considered with control active on a portion of the basilar membrane. Approximate controllability is proved for both models and moreover lack of exact controllability is shown to hold when longitudinal stiffness is not included.

We consider the problem of boundary feedback stabilization of a multilayer Rao-Nakra sandwich beam. We show that the eigenfunctions of the decoupled system form a Riesz basis. This allows us to deduce that the decoupled system is exponentially stable. Since the coupling terms are compact, the exponential stability of the coupled system follows from the strong stability of the coupled system, which is proved using a unique continuation result for the overdetermined homogenous system in the case of zero feedback., Comment: 16 pages (Final version)

Exact controllability of a multilayer plate system with free boundary conditions are obtained by the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a three-layer "sandwich plate'' system due to Rao and Nakra. In the multilayer version, $m$ shear deformable layers alternate with $m+1$ layers modeled under Kirchoff plate assumptions. The resulting system involves $m+1$ Lame systems coupled with a scalar Kirchhoff plate equation. The controls are taken to be distributed in a neighborhood of the boundary. This paper is the sequel to [2] in which only clamped and hinged boundary conditions are considered.

Exact controllability results for a multilayer plate system are obtained from the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a classical three-layer "sandwich plate" system due to Rao and Nakra. The multilayer version involves a number of Lame systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controls are assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for the coupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lames ystem.

The classical Mead-Markus sandwich beam consists of two stiff outer layers modeled under Euler-Bernoulli beam assumptions and a compliant "core layer" that is elastic in shear. In this article we consider a multilayer analog consisting of $n = 2m + 1$ layers of alternating stiff and compliant beam layers ($m+1$ stiff and $m$ compliant) with viscous damping proportional to the shear in the compliant layers. We prove that the associated semigroup is analytic and describe the sector of analyticity. We also consider the problem of how to choose the damping parameters to optimize the angle of analyticity. We obtain an analytical solution to the optimization problem.

The multilayer Mead–Markus plate consists of m + 1 Kirchhoff plate layers bound together by m shear-deformable layers. We consider the case in which the Poisson ratios are the same in each Kirchhoff plate layer and linear viscous shear damping is included in the shear deformable layers. We show, via a contradiction argument, that the associated semigroup is analytic.

We consider a linear hybrid system composed by two rods of equal length connected by a point mass. We show that the system is null controllable with Dirichlet and Neumann controls. The results are based on a careful spectral spectral analysis together with the moment method., Comment: 12 pages, typos corrected, added references, matches version to be submitted to Systems and Control Letters

We examine the stability properties of a sandwich beam consisting of two outer layers and a thin core. The outer layers are modeled as Euler Bernoulli beams and the inner core provides both elastic and viscous resistance to shearing. We show for both clamped and hinged boundary conditions that (i) if rotational inertia terms are neglected, the model is described by an analytic semigroup, and (ii) if rotational inertia is retained in the outer layers, the model is uniformly exponentially stable.

In this article a criterion known as the “operator Carleson measure criterion” is applied to obtain regularity results for a general class of second-order boundary control problems. As specific examples, optimal regularity results are obtained for a wave equation and a plate equation, both with fractional order damping and on a rectangular domain with one edge controlled.

A dynamic model for a multilayered laminated plate is developed. The laminated plate consists of 2n plate layers and 2n-1 adhesive layers. The layers (both plate and adhesive layers) are assumed to be homogeneous, transversely isotropic and perfectly bonded to one another. In the initial modeling, the Reissner-Mindlin theory of shear deformable plates is applied to each layer, resulting in a high-order plate theory in which the shear motions of the layers are completely independent. Simpler, lower-order models are obtained from the initial model from asymptotic limits based upon the assumptions that (i) the adhesive layers are very thin, (ii) the elastic modulii of the adhesive layers are small compared to those of the plate layers, (iii) the shear stiffnesses of the plate layers are very large.

For industrial marketers to retain customers it is important to understand the behavior that buyers undertake following an unsatisfactory purchase. The responses that individual consumers undertake...

Understanding how industrial buyers complain when dissatisfied with a supplier is an increasingly important topic. Complaints from buyers can be expressed in a variety of behaviors, such as contracting the vendor about the problem, switching suppliers, and warning other buyers about poor vendor performance. The development of appropriate policies and procedures is essential in reducing buyer complaints that damage the seller's reputation. This article develops a typology of organizational buyer complaint response styles and identifies a set of variables useful in predicting these styles. Managerial implications drawn from the empirical results can help marketers develop programs to promote more favorable forms of buyer complaint behavior.

Three closely related models for two-layered beams in which slip can occur at the interface are described. In the first, beam layers are modelled under the assumptions of Timoshenko beam theory. Along the interface, an adhesive layer of negligible thickness bonds the surfaces so that a small amount of slip is possible. Friction is assumed to be proportional to the rate of slip. The second is obtained from the first by letting the shear stiffness of each beam tend to infinity. The third is obtained from the second by assuming that the moment of inertia parameter is negligible. In the last case, an analog of the Euler–Bernoulli beam is obtained which exhibits frequency proportional damping characteristics. The three models are compared, both numerically and analytically, and found to be in close agreement for low frequency motions. Optimal damping rates are calculated for these low frequency motions.

We show in this paper that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam. The optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted, as in the Euler–Bernoulli beam theory.

It is well known that an Euler-Bernoulli beam may be exactly controlled with a single control acting on an end of the beam. In this article we show that for certain boundary conditions, the same result holds for a beam that is surrounded by an incompressible, inviscid fluid with a sufficiently small density. The proof involves reducing the control problem to a moment problem and using compactness properties of the Neumann to Dirichlet map for the Laplacian operator to obtain the needed estimates.

In this paper a dynamic model for an n n -layered plate is developed based upon the assumptions of Reissner-Mindlin plate theory. Each plate layer is assumed to be transversely isotropic, transversely homogeneous and of a uniform thickness; however, no symmetry in the material properties or thicknesses of each plate is assumed. The layers are assumed to be perfectly bonded so that no slip occurs along the interface. No additional a priori kinematic restrictions are imposed upon the motion of the plates. The equations of motion are derived by the principle of virtual work. Existence and uniqueness results are obtained. In the case where the layers are symmetric we show that all solutions decouple into a bending solution (with antisymmetric displacements about the mid-plane) and an in-plane solution (with symmetric displacements).

This study suggests actions the industrial marketer can take to reduce the possibility of customer dissatisfaction complaints that are harmful to the seller's long-term business. The study reviews the literature on dissatisfaction response styles and supplier relationships that result in different dissatisfaction response styles. Data were gathered using the survey method and in-depth interviews. The study finds support for a theoretical framework that shows how an industrial marketer can better manage vendor relations in order to become “complaint friendly.”

This paper introduces CAPS (Client Adaptive Problem Solving) as an experiential learning method for marketing students. Experiential learning is discussed in terms of five dimensions: (1) learning ...

It is important for the industrial marketer to understand and effectively manage the process of complaining behavior in which dissatisfied buyers typically engage. By better managing this process, marketers can improve customer loyalty and prevent buyers from undertaking further complaining behavior which may damage the seller’s reputation. Reports on the types of order problems that occur with suppliers, as well as buyers’ perceptions of the effectiveness of supplier responses to different types of complaints from the buyer.

In this article we examine the problems of boundary control and stabilization for a one-dimensional wave equation with interior point masses. We show that singularities in waves are "smoothed one order" as they cross a point mass. Thus in the case of one interior point mass, with, e.g., $L^2$-Dirichlet control at the left end, the most general reachable space (from 0) that one can expect is $L^2\times H^{-1}$ to the left of the mass and $H^1\times L^2$ to the right of the mass. We show that this is in fact the optimal result (modulo certain compatibility conditions). Several related results for both control and stabilization of such systems are also given.

Boundary control of a linear partial differential equation that describes the temperature distribution and displacement within a one-dimensional thermoelastic rod is examined. In particular, it is shown that temperature or heat flux control at an endpoint is sufficient to obtain exact null-controllability. This improves earlier results for similar systems in which only partial null-controllability is obtained. Sharp regularity results for the controlled system are also obtained.

We examine the stability of solutions to a pair of coupled linear partial differential equations which describe the temperature distribution and displacement within a one-dimensional thermoelastic rod. For particular sets of natural boundary conditions, the eigenfunctions are shown to form a Riesz basis on the Hilbert space of finite energy states. For these cases, asymptotic eigenvalue estimates show that the spectrum is to the left of some vertical line Re λ = −β contained in the open left half of the complex plane. An energy decay estimate of the form E(t) ⩽ Me−βt E(0) is then obtained.

In this article, we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials {exp(− λ k t )}, where the { λ k } belong to a sector of the positive real axis. The bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties. The dependence of the bounds upon the growth and separation parameters is carefully examined. The main theorem then leads to some results regarding the solvability of moment problems of the form ∝ 0 T f ( t )exp(− λ k t ) dt = c k , which arise in control problems associated with damped or dissipative systems. As an application, we establish an exact null controllability result for boundary control of a rectangular structurally damped plate.

We consider a multilayer Rao-Nakra sandwich beam with shear damping included in alternate layers. In the case that the wave speeds of the layers are distinct, we show that by "tuning" the damping in each layer appropriately, it is possible to construct a scalar boundary control that exactly controls the entire layered system in a control time related to the sum of the control times of each separate layer. In the case where some waves speeds coincide, one can reduce the number of controls needed, but not to a single scalar control.

We consider the problem of boundary control of a system of three coupled partial differential equations that describe a three layer (Rao-Nakra type) sandwich beam with damping proportional to shear included in the core layer. In the case where one control is applied to each equation, we obtain exact controllablity modulo a finite dimensional quotient in a time determined by the three wave speeds. We show that in a longer time, under some mild conditions on the parameters, we can recover a similar exact controllability result using only two, or possibly even one appropriately chosen control function.

We consider a three layer Rao-Nakra sandwich beam with damping proportional to shear included in the core layer. We prove that eigenvectors of the beam form a Riesz basis for the natural energy space. In the damped case, we are able to give precise conditions under which solutions decay at a uniform exponential rate. We also consider the problem of boundary control using bending moment and lateral force control at one end. We prove that the space of exact controllability has finite co-dimension and provide sufficient conditions (related to small damping) for exact controllability to a zero energy state.

We describe some possible models for a multilayer sandwich beam consisting of alternating stiff and compliant beam layers. The stiff layers are modeled under Euler-Bernoulli assumptions while the compliant layers essentially only carry the shear. We include viscous damping in the compliant layers and consider the optimization problem of choosing the damping parameteres for each layer so that the maximal asymptotic damping angle in the system eigenvalues is obtained. The solution is obtained analytically as a closed-form function of the various material parameters.

We first describe an approach for the modeling of piezoelectric beams and plates that leads to a well-posed system of partial differential equations that retain the coupling between the mechanical system and a potential equation for the electrical components. We then describe how the same approach can be used to analyze a plate with a localized piezoelectric patch. In particular, since the higher order electrical terms have been retained, it is possible to deduce the proper sets of boundary conditions that should apply to an arbitrarily shaped patch.

We consider a system of coupled partial differential equations that describe the vibrations of laminated beam in which the layers are bonded together by a medium that dissipates energy at a rate proportional to the shear. We show that for the simplest model, in which only transverse inertial energy is accounted for, the associated semigroup is analytic.

We show that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam. The optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted, as in the Euler-Bernoulli beam theory.

A dynamic model for a multilayered laminated plate is developed. The laminated plate consists of 2n plate layers and 2n - 1 adhesive layers. The layers (both plate and adhesive layers) are assumed to be homogeneous, transversely isotropic and perfectly bonded to one another. In the initial modeling, the Reissner-Mindlin theory of shear deformable plates is applied to each layer, resulting in a high-order plate theory in which the shear motions of the layers are completely independent. Simpler, lower-order models can then be obtained from this initial model from asymptotic limits based upon the assumptions that (1) the adhesive layers are very thin, (2) the elastic modulii of the adhesive layers are small compared to those of the plate layers, (3) the shear stiffnesses of the plate layers are very large, (4) the rotational moments of inertia of the individual plate layers are very small.

Three models for two-layered beams in which slip can occur at the interface are described. In the first, beam layers are modelled under the assumptions of Timoshenko beam theory. Along the interface a `glue layer' of negligible thickness bonds the surfaces so that a small amount of slip is possible. Dissipation is assumed to be proportional to the rate of slip. The second is obtained from the first by letting the shear stiffness of each beam tend to infinity. The third is obtained from the second by assuming that the moment of inertia parameter is negligible. In this case an analog of the Euler-Bernoulli beam is obtained which exhibits frequency- proportional damping characteristics.